Download  From Ultracold Atoms to Condensed Matter Physics

From ultracold atoms to condensed
matter physics
Charles Jean-Marc Mathy
A Dissertation
Presented to the Faculty
of Princeton University
in Candidacy for the Degree
of Doctor of Philosophy
Recommended for Acceptance
by the Department of
Physics
Adviser: Professor David A. Huse
September 2010
c Copyright by Charles Jean-Marc Mathy, 2010.
All rights reserved.
Abstract
We study the possibility of realizing strong coupled many-body quantum phases in
ultracold atomic systems. Motivated by recent experiments, we first analyze the
phase diagram of a Bose-Fermi mixture across a Feshbach resonance, and offer an
explanation for the collapse of the system observed close to the Feshbach resonance:
we find that phase separation leads to a high density phase which causes the collapse.
We then focus on the recent attempts to realize the three-dimensional fermionic Hubbard model in an optical lattice. One milestone on the experimentalists’ agenda is
to access the antiferromagnetic ordered Neél phase, which has so far been hindered
by the low ordering temperature. We ask which experimental parameters maximize
the antiferromagnetic interactions, which set the scale for the ordering temperature.
We find that the maximum is obtained in a regime where the effective Hamiltonian
describing the system no longer corresponds to a simple one-band Hubbard model,
and we characterize the physics of the system in this regime. The final system we
consider is mass imbalanced polarized two-component Fermi gases interacting via a
Feshbach resonance. By going to the strongly polarized limit, we use a recently developed method to obtain results which have been shown to be accurate in the mass
balanced case, and we find an intriguing set of competing phases in this limit. We
discuss what these results imply for the full phase diagram.
iii
Acknowledgements
First off, I would like to thank my advisor, David Huse. David is one of the sharpest
and most creative physicists I have ever met. He is a fantastic advisor : always
available, and constantly coming up with interesting problems to work on. He also
put me in touch with collaborators, and got me involved in the DARPA program,
which was tremendously beneficial to my career, and partly funded my Ph.D. Thank
you, David, for everything.
I would also like to thank Shivaji Sondhi and Duncan Haldane for the physics
discussions and for helping me with securing the next step. I owe a tremendous debt
of gratitude to Sander Bais for getting me on the condensed matter track, and working
with me in my first year at Princeton.
In a Ph.D. program, friends come and go, and it would be impossible to thank
everyone. But there was a friendship bedrock I would like to acknowledge. I’ll miss
the road trips with Abhi, listening to Will Smith, Dave Grusin or whatnot, riding
into the sunset. I’ll miss Fabio’s stories on the Peloponnesian war, pigeons used as
missile guides, and of course his arroz con tomato. Princeton would have been a lot
less exciting without Chris around : no ski trips, no camping, no themed parties.
Thanks to Said (sorry, Chris was taken) for all the adventures.
I would also like to thank everyone who made jadwin hall a second home: Meera,
Aakash, Katerina, Sid, Arijeet, Fiona, Xinxin, Tibi, Pablo, Diego, Arvind, Richard,
Anand, soccer ”capitan” John and the rest of the team, I’m sure I’ve missed out a lot
of people. There is also life outside of jadwin : Civo, Alex, Masha, Vanya, Catherine,
Ana, thank you all. There is also life outside of Princeton : Samantha, grazie per
tutto. Auntie Rebecca, thanks for providing an oasis where I could leave my woes
behind. Tio Ruben, gracias por mantener contacto todos estos años.
My family deserves an acknowledgment longer than this thesis, for their unwavering love and support throughout my life and my career. My mother and brother
iv
were instrumental in putting the US in my field of vision. It’s hard to live away from
family for so long, so I really appreciate that my mom, dad and brother came to
visit. Special thanks to mom, for taking the time to arrange that we saw each other
on a regular basis: from New York to Montreal, Milan, Buenos Aires, it was very
important for me to see you and know that you were with me the whole way. Thanks
to the all of my family, in the Netherlands, England, Belgium, Geneva, Argentina,
Australia, I dedicate this thesis to you all.
v
Relation to previously published work
Parts of this dissertation can be found in publications in APS journals [49, 51]. APS
permits the reproduction of material in its publications for the purpose of a Ph.D.
dissertation, provided that one includes the appropriate copyright notices in the bibliography.
The results of chapter 2 were published in [49]. Chapter 3 was based on [51] and
[50]. Most of the results of chapter 4 can be found in [52].
The work in chapters 3 and 4 was supported under ARO Award W911NF-07-10464 with funds from the DARPA OLE Program.
vi
To my family.
vii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
1 Introduction
1
1.1
Tunability and universality . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Strong coupling and phase transitions . . . . . . . . . . . . . . . . . .
4
1.3
Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2 Bose-Fermi mixtures across a Feshbach resonance
8
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
The phase diagram at T=0 . . . . . . . . . . . . . . . . . . . . . . . .
13
2.4
Experimental consequences . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5
Comparison to previous work . . . . . . . . . . . . . . . . . . . . . .
23
2.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3 Accessing the Néel phase of ultracold fermions in a simple-cubic
optical lattice
28
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.2
The incarnation of the Hubbard model in cold atoms . . . . . . . . .
31
viii
3.3
Strong lattice expansion and Néel temperature . . . . . . . . . . . . .
40
3.4
Hartree approximation . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.5
Experimental consequences . . . . . . . . . . . . . . . . . . . . . . . .
54
3.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4 Polarons, molecules and trimers in strongly polarized Fermi gases
59
4.1
Mean field theory of the imbalanced fermi gas . . . . . . . . . . . . .
61
4.2
FF, LO and FFLO . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.3
Bare polaron, molecule and trimer . . . . . . . . . . . . . . . . . . .
71
4.4
Dressed polaron and molecule, and bare trimer . . . . . . . . . . . . .
80
4.5
Unbinding transitions vs phase separation . . . . . . . . . . . . . . .
84
4.6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
A Single channel model of Feshbach resonances
A.1 Single channel model of the contact interaction . . . . . . . . . . . . .
94
95
B Two-channel model of Feshbach resonances
104
Bibliography
107
ix
List of Tables
3.1
The values of the various energies at the two TN maxima. . . . . . . .
x
46
List of Figures
3/2
2.1
Phase diagrams for different values of ν/γ 2 at λmb γ = 0.0063 . . . .
2.2
Bose Fermi hase diagram in the parameter space {ν (r) , µb , λ(r) } ≡
18
(r)
3/2
{(ν − µb )/|µf |, µb /γ|µf |1/2 , mb λ|µf |3/2 /γ 2 } . . . . . . . . . . . . . .
20
2.3
Density profiles in a harmonic trap at ν = 0. . . . . . . . . . . . . . .
22
3.1
Sketch of the generic phase diagram of High Temperature Superconductors, as a function of doping x and temperature T. . . . . . . . . .
3.2
29
The three lowest maximally localized Wannier functions in a one dimensional sinusoidal potential . . . . . . . . . . . . . . . . . . . . . .
34
3.3
Wannier states in a 2D square lattice. . . . . . . . . . . . . . . . . . .
34
3.4
Contour plots of Wannier states in a 3D simple cubic lattice . . . . .
35
3.5
Hopping in the lowest band . . . . . . . . . . . . . . . . . . . . . . .
37
3.6
Interaction terms in a three-dimensional optical lattice with atoms
scattering in the s-wave channel . . . . . . . . . . . . . . . . . . . . .
39
3.7
Approximate phase diagram for filling one fermion per lattice site . .
42
3.8
Our estimates of the optimal Néel temperature, TN , as a function of
as /d. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9
45
The strongest higher-order process contributing to the energy of the
antiferromagnetic Mott insulator at the maxima of TN
. . . . . . . .
46
3.10 Ground-state phase diagram for filling one fermion per lattice site . .
51
3.11 Plot of the Hartree estimate of the antiferromagnetic exchange coupling 53
xi
4.1
Mean field zero temperature phase diagram of mass imbalanced spin
polarized two component fermions, with mass ratio m↑ /m↓ = 10 . . .
65
4.2
M2 − P1 phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.3
Momentum Q of the bare FFLO molecule in units of kF ↑ , as a function
of r = m↑ /m↓ , along the M2 − P1 boundary . . . . . . . . . . . . . .
76
4.4
T3 − M2 − P1 phase diagram . . . . . . . . . . . . . . . . . . . . . . .
79
4.5
P3 − M4 − F F LO − T3 phase diagram . . . . . . . . . . . . . . . . .
85
4.6
Momentum Q of the dressed FFLO molecule in units of kF ↑ , as a
function of r = m↑ /m↓ , along the M4 − P3 boundary . . . . . . . . .
4.7
86
Schematics of two different scenarios for a molecule unbinding into a
polaron + particle
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
4.8
The molecular residue ZM4 , for 1/(kf as ) = 1.5, as a function of r. . .
87
4.9
The different approximations to the lines that mark the onset of the
phase separating region to their right. . . . . . . . . . . . . . . . . . .
xii
93
Chapter 1
Introduction
In the last two decades, the field of ultracold atomic physics has become interwoven
with condensed matter physics[6, 30]. What made this mariage possible was a series of
breakthroughs in cooling methods in the nineties, which allowed experimentalists to
bring a system of many atoms into the quantum regime. Some of the early milestones
were the observation of Bose-Einstein Condensation of
degenerate Fermi gas of
40
87
Rb [2] and
23
N a [19], and a
K [21].
The next important step was the realization that by using a clever combination of
electromagnetic fields, one could tune the background potential that the atoms felt,
and vary the interactions between them. This opened up a seemingly endless world of
possibilities in the use of ultracold atoms as quantum simulators, with the prospect
of resolving long standing issues in condensed matter physics. As cold atom systems
are in fact different from conventional condensed matter systems in several respects,
there are also new issues and questions which can be addressed.
1.1
Tunability and universality
Cold atom systems consist of neutral atoms at densities of about 1013 cm−3 , cooled
down to nanokelvin, or sometimes picokelvin temperatures, using a combination of
1
magnetic, laser and evaporative cooling. The atoms are kept in place thanks to a
harmonic confinement. On top of this harmonic trap, in analogy to the periodic
potential felt by electrons in a solid, one can introduce a periodic potential using a
combination of lasers. The lasers that create the lattice are detuned from an optical
absorption line and generate an electric dipole in the atoms that become trapped in
either the maxima (if it is red detuned) or the minima (if it is blue detuned) of the
laser light intensity [4]. The atoms feel an optical potential by way of the Stark effect:
~ given by d~ = αE,
~ where α is
a dipole moment d~ is formed due to the electric field E,
~ in an electric field,
the atom’s polarizability. An electric dipole has Hamiltonian d~ · E
~ 2 proportional to the square of the electric field.
thus the atom sees a potential αE
By superimposing a laser with its own retroreflection, one generates a standing
wave. If there is only one retroreflected laser, the atoms feel a one dimensional
potential. If the potential is made to be very deep, the system effectively becomes a
set of uncoupled ”pancakes”. Thus one can reduce the effective dimensionality of the
system. Two orthogonal retroreflected lasers lead to a set of one-dimensional tubes,
thus allowing one to study (quasi) one-dimensional physics. In general, it is possible
to generate a lattice of one’s choice, thus one can for example simulate atoms feeling
the potential of the Hubbard model, of frustrated systems, etc.
The atoms in cold atom experiments are typically alkali atoms, because they have
one electron in their outer shell, which makes their behavior relatively simple to
describe and predict. Such atoms have a set of hyperfine states, due to the nuclear
spin I and the electron spin S. The total spin is called F , and for a given value of
F there will be a manifold of 2F + 1 states. These states would be degenerate if
there were no magnetic field, and can be populated using RF spectroscopy. Thus
one can simulate systems with internal degrees of freedom, such as fermions with two
internal spin states, where in the cold atom context the internal degree of freedom is
the hyperfine states that one has chosen to occupy.
2
To vary the interaction between two distinguishable atoms, a static magnetic field
is tuned to a Feshbach resonance, which is defined as a value of the magnetic field
where the s-wave scattering length between the two states diverges[24]. Across a
Feshbach resonance, the scattering length as as a function of magnetic field behaves
as
as (B) = abg (1 −
∆B
),
B − B0
(1.1)
where abg is a background scattering length. The physics of Feshbach resonances is
briefly described in Appendix A. On one side of the resonance, where as < 0, the
low-energy scattering is attractive, while on the other side it is repulsive. Note that
the bare interaction is always attractive.
Thus, moving around a Feshbach resonance, one can realize interactions which
are repulsive or attractive, and one can vary the interaction strength by orders of
magnitude, depending on how precisely one can set the magnetic field close to the
Feshbach resonance.
One of the beautiful properties of cold atom systems is the universality of the
results: indeed one typically works in the dilute limit, at low temperatures, such
that the s-wave scattering length as is the only parameter needed to describe the
interactions. This means that the results obtained with one set of atoms will not
depend on the details of the short-range atomic physics, and can be described by a
restricted set of parameters, such as as , the mass of the species, and their densities.
The calculations are carried out using a s-wave pseudopotential, which is chosen to be
as simple as possible while capturing the essential features of the interactions at low
energies. The first applications of this pseudopotential was in the context of nuclear
physics [27, 8], as the interactions between neutrons at relatively low densities (such
as in the outer layers of a neutron star) can be modelled by a s-wave pseudopotential
[73].
3
Thus the physics of two-component fermions interacting via a Feshbach resonance
has direct consequences for other systems with the same effective description. This
universality has been shown time and time again in theory and experiments. One
must however work with dilute systems: indeed there is a set of bound states that
the atoms typically can fall into. If two atoms come together, they cannot fall into
the bound state because of kinematic rescritions. However if three atoms collide, two
can form a bound state and the third atom can carry off the excess energy[64]. Thus
three-body losses should be carefully avoided. Note that in some cases the three-body
losses can be used as a probe: if they suddenly increase, it suggests that the density
of the system is increasing, signalling an instability (see Chapter 2 for an example).
Thus in cold atoms one can vary parameters that are inaccessible in other experimental contexts, such as the number of species, their internal spin structure, the
dimensionality of the system, the masses of the particles, the interaction strength, the
potential seen by each species, etc. One can also introduce a rotating lattice, which
leads to Quantum Hall physics in the rotating frame [17]. Recently a combination
of Raman beams has allowed for the generation of artificial gauge fields [47]. An
active area of investigation is the trapping of dipolar molecules, which would allow
one to study systems with long range interactions [5]. In short, cold atoms are rapidly
coming into contact with many subfields of condensed matter physics.
1.2
Strong coupling and phase transitions
One of the main unresolved issues of modern condensed matter physics is the description and analysis of strongly coupled theories. A system is defined as strongly coupled
if the average kinetic and interaction energies of the atoms are of the same order. If
the kinetic term dominated, one could do perturbation theory in the interactions,
such as is done in Fermi liquids, for example. If the interaction term is dominant,
4
then the kinetic energy may be treated perturbatively, as in done in lattice models
around the atomic limit. Between these two regimes, there is no small parameter,
and it is not clear how to proceed. It is precisely in this regime that the most interesting physics, from High temperature superconductivity to Fractional Quantum
Hall physics to spin liquids, occurs.
Several approaches have been explored to realize strongly coupled physics in cold
atoms [6, 30, 36]. If one tunes a system to lie exactly at a Feshbach resonance, the
system is called a unitary gas. In such a gas, there is no small parameter to expand
around, and in that sense the system is strongly coupled. Perturbation theory will fail
around unitarity, and more accurate methods such as Quantum Monte Carlo (QMC)
must be employed [3]. However, the ubiquitous minus sign problem for fermions
makes the regime of applicability of QMC limited.
The canonical example of atoms interacting via a Feshbach resonance is twocomponent fermions (without an optical lattice). When there is an equal number of
fermions of both species, as one crosses from the repulsive to the attractive side of
the resonance, one realizes a BEC-BCS crossover [74]. The system’s ground state is
superfluid all the way, and crosses over from being a system of strongly bound fermion
pairs behaving like bosons and forming a BEC, to a system of weakly bound Cooper
pairs. At unitarity, we have what is called a crossover superfluid, where the size of
the Cooper pairs is of the order of the atomic spacing.
Instead of increasing the interactions, one can also reach strong coupling by reducing the kinetic energy. For example if one introduces an optical lattice in which
the atoms are confined into deep potential wells, with weak tunneling between the
wells, the kinetic energy is lowered, while simultaneously increasing the interactions
within one well, since the atoms in a well are closer together. Thus one can reach
strong coupling this way. If one starts with a BEC and introduces a three-dimensional
simple cubic optical lattice, as one deepens the well, the system goes through a quan5
tum phase transition from Superfluid to Mott Insulator [32]. This has been observed
experimentally. In the deep lattice limit, the bosons behave according to a BoseHubbard model.
The same approach can be applied to two-component fermions, leading to fermions
interacting via a Fermi Hubbard model[30, 36]. This model is one of the holy grails
of condensed matter physics, and may hold the key to high temperature superconductivity.
1.3
Thesis outline
In this thesis, we study the realization of strongly coupled many-body quantum phases
in cold atoms, in three specific contexts. In chapter 2 we look at Bose-Fermi mixtures across a Feshbach resonance. In analogy to the BEC-BCS phase diagram of
two-component fermions, we look at the phase diagram around unitarity. We were
motivated by experiments on Bose-Fermi mixtures showing a collapse as one approached the resonance. We offer an explanation for the collapse, namely that the
system is phase separating to a phase with high density, where three-body losses kick
in. We propose ways to test our predictions.
In chapter 3 we look at the attempts to realize the antiferromagnetically insulating
Néel phase of the three-dimensional Fermi Hubbard model in cold atom systems. We
find that to increase the robustness of the Néel phase, one must leave the region of
the phase diagram where the Hubbard model is a good approximation. We use an
expansion valid at relatively strong lattice potential, and a Hartree calculation at
weak to intermediate lattice poential to completely map out the phase diagram and
find the sweet spot to measure the Néel phase. Our two calculations agree well in the
intermediate regime.
6
Finally, chapter 4 deals with the strongly polarized limit of two-component
fermions interacting via a Feshbach resonance. Introducing mass imbalance, we find
an intriguing competition between polaron, molecule and trimer phases. The trimer
phase is competing directly with an FFLO phase, a phase which has so far eluded
experimental observation. We discuss the experimental consequences of these results.
7
Chapter 2
Bose-Fermi mixtures across a
Feshbach resonance
In this chapter, we analyze the zero-temperature phase diagram of a gas of bosonic and
fermionic atoms interacting through a Feshbach resonance, in a two-channel model
which explicitly includes the closed channel molecule as a separate species. We find a
rich phase diagram, comprising a mixture of Bose-condensed and non Bose-condensed
phases separated by both second order and first order phase transitions, and Fermi
Surface changing phase transitions. We show that close to unitarity there is a regime
in which the system phase separates. Finally we study the density profile in a trap
using LDA, and discuss in which experimentally available systems one is most likely
to see the predicted behaviour.
2.1
Introduction
The discovery of Feshbach resonances between bosonic and fermionic species has led
to a flurry of activity, both theoretical and experimental, in the study of Bose-Fermi
mixtures. Theoretical investigations have led to a prediction of a rich variety of
phases: phase separation of bosons and fermions [55, 78], BCS type Cooper pairing
8
mediated by the bosons [34], density waves in optical lattices [46], and polar molecules
with long range dipolar interactions.
We will be considering a single species of boson and a single species of fermion
interacting through a Feshbach resonance, in the low temperature limit where the
only interaction is in the s channel. In this limit, the fermions do not interact because of Pauli exclusion. The bosons interact repulsively amongst themselves, with a
background scattering length abb .
For details on the two-channel model of a Feshbach resonance, see Appendix . The
physical picture is that if there exist bound states between the boson and the fermion,
and a static magnetic field is applied to the system, the energy of the bound state
will change, as it carries a certain angular momentum. If the energy of the bound
state is made to cross the bottom of the continuum (i.e. the energy that the boson
and fermion have when they are far apart and at rest), then the s-wave scattering
length of the boson and fermion will diverge. We will define a parameter ν, called
the detuning of the bound state, which when varied will take us across the Feshbach
resonance. We include the bound state explicitly in the Hamiltonian, considering a
so-called two-channel model.
2.2
The model
Around the Feshbach resonance a bound state of a fermion and boson appears around
zero energy. Thus the fermions and bosons in the system can interact by forming a
molecule. The two-channel Hamiltonian is[67]
Z
d3 k d3 k 0
d3 k f †
ψ †
b †
ξ
f
f
+
ξ
b
b
+
ξ
ψ
ψ
+
g
ψ † 0 fk bk0 + h.c.
Ĥ =
3
3
3
k
k
k
k
k
k
k
k
k
k+k
(2π)
(2π) (2π)
Z
d3 k d3 k 0 d3 q † †
+λ
b b 0b 0 b
(2.1)
(2π)3 (2π)3 (2π)3 k k k +q k−q
Z
9
b, f , and ψ are respectively the destruction operators for the bosonic atom, the
fermionic atom, and the closed channel fermionic molecule. The molecule has a
binding energy which is called the detuning ν. ν will vary when a magnetic field
is applied, as the magnetic field couples to the total spin of the molecule, which we
assume to be nonzero.
The dispersion relations are given by
f
ξk
= h̄2 k2 /2mf − µf
b
= h̄2 k2 /2mb − µb
ξk
ψ
ξk
= h̄2 k2 /2(mb + mf ) − µψ .
mb and µb are respectively the masses and chemical potentials for the bosons, and
similarly for the fermions and molecules. The chemical potential for the molecules is
given by µψ = µb + µf − ν, where ν is the detuning. It is given to lowest order in g
by [24]
ν = ∆µ(B − B0 )
(2.2)
where ∆µ is the difference in magnetic moments between open and closed channels,
and B0 is the value of B at which the Feshbach resonance occurs (see Eq.(1.1)).
For alkali atoms, to a good approximation one can think of the scattering problem
as being between the triplet and singlet state, thus ∆µ = µB , the Bohr magneton.
Positive ν corresponds to negative as , and negative ν to negative as vice versa. ν = 0
thus corresponds to unitarity 1 .
We define a mass ratio r =
mf
,
mb
m m
and a mass parameter m = 2 mff+mbb . We will work
in units where h̄ = m = 1. The relationship between the microscopic parameters and
the s-wave scattering length is derived in Appendix B.
ν = 0 only corresponds to unitarity to lowest order in g, in fact unitarity occurs at ν 0 = 0, where
ν is defined in Appendix B.
1
0
10
To study the mean field theory of this model, we equate the boson operator to a
scalar: bk → δk,0 φ. Defining ρ = φ2 , the mean field Hamiltonian becomes
ĤM F =
Z
Z
d3 k f †
d3 k †
ψ †
†
ξ
f
f
+
ξ
ψ
ψ
ψ
f
+
f
ψ
− µb ρ + λρ2 .
+
gφ
k
k
3
3
k
k
k
k
k
k
k
k
(2π)
(2π)
This Hamiltonian is quadratic, and can be diagonalized by defining mixed fermionic
operators:
ĤM F =
d3 k F †
Ψ †
2
Ψ
Ψ
F
F
+
ξ
ξ
k k k − µb ρ + λρ .
(2π)3 k k k
Z
(2.3)
The new operators are defined by
Fk = cos θk fk + sin θk ψk
(2.4)
Ψk = − sin θk fk + cos θk ψk ,
(2.5)
where θk is the mixing angle between the bands:
f
ψ
ξk
− ξk
1 1
cos θk = + q f
.
2 2 (ξ − ξ ψ )2 + 4g 2 ρ
k
k
2
(2.6)
The dispersion relations for the F and Ψ bands are
1
1q f
F,Ψ
(ξk − ξkψ )2 + 4g 2 ρ
ξk
= (ξkf + ξkψ ) ±
2
2
(2.7)
At zero temperatures, the F and Ψ bands are occupied up to their respective Fermi
momenta, k F and kΨ . The (free) energy becomes
E =
Z kF
0
dk 2 F Z kΨ dk 2 Ψ
k ξk +
k ξk − µb ρ + λρ2 ,
2π 2
2π 2
0
11
(2.8)
where ρ = φ2 is chosen so as to minimize E. We call φmin the value of φ that
minimizes the mean field energy, and define ρmin = φ2min .
The chemical potentials are fixed by setting the total number of fermionic and
bosonic atoms:
1 3
3
kF + kΨ
)
2
6π
Z kF
Z kΨ
k2
k2
2
nb = ρ +
dk 2 sin θk +
dk 2 cos2 θk
2π
2π
0
0
nf =
(2.9)
(2.10)
Since the molecular and fermionic bands are mixed, one finds bosons in both bands,
and in the BEC determined by ρ.
This model has a rather rich mean field phase diagram, as we will now see. There
are seven different phases. The phases are firstly characterized by whether the condensate φmin = 0 or not. One then has to state the number of Fermi surfaces in the
phase: there can be no Fermi Surface (FS), in which case one has either vacuum, if
φmin = 0, or a pure BEC with no fermions, φmin 6= 0 . If there is one FS, once again
there can be a BEC or no BEC. If there is a BEC, then there is no clear distinction
between a FS of fermions or molecules, since the bands are hybridized. However, if
there is no BEC, then one has to distinguish between having a FS of fermions or a
FS of molecules. Finally, one van have two FS and either a BEC or no BEC. All told,
we have seven different phases.
The way these phases are connected is rather intricate, and involves a phase
diagram with both second order and first phase transitions from a phase without to
a phase with a BEC, as well as a series of phase transitions where the number of FS
changes. We will elucidate the phase diagram in the next section.
12
2.3
The phase diagram at T=0
Our task is to determine the phase diagram, as a function of the parameters
r, µf , µψ , µb , g, λ. We will fix r to be the mass ratio relevant for
87
Rb −40 K, though
we have checked that the physics is qualitatively the same for different r. It turns
out that for fixed r, we can rescale our problem (by rescaling ρ and the energy) so
that we are left with three parameters. We have a certain freedom with regards to
which parameters we pick, which we will exploit later on. In fact, when studying the
phase diagram relevant to experiments it turns out to be favorable to work with four
parameters.
If there were only second order phase transitions present, we would find the line
of phase transitions by solving dE(ρ)/dρ|ρ=0 = 0. This would be the whole story if
higher order derivatives were always positive, but this is not the case. In fact, one can
simultaneously solve dE/dρρ=0 = 0 and dE/dρ2 |ρ=0 = 0 and obtain a line tricritical
points. One can finally solve dE/dρ|ρ=0 = dE/dρ2 |ρ=0 = dE/dρ3 |ρ=0 = 0 and obtain
tetracritical points. The existence of tetracritical points signals the richness of the
phase diagram to come.
Although it is possible, as we have just discussed, to plot the full phase diagram in
terms of three parameters, to relate the results to experiments it is more convenient
to work with four parameters: we choose the dimensionless parameters
3/2
{λ̃, ν̃, µ̃b , µ˜f } = {λmb γ, ν/γ 2 , µb /γ 2 , µf /γ 2 },
(2.11)
where we define (remember that h̄ = 1)
γ=
g 2 3/2
m .
8π
13
(2.12)
γ is related to the width ∆B of the Feshbach resonance [67]. Namely, within a mean
field approximation [24], g is given by
2
s
g = h̄
4πabg ∆µ∆B
.
m
(2.13)
Thus, large γ corresponds to a wide Feshbach resonance.
This choice of parameters is physically sensible, because for a given system and
fixed magnetic field, λ̃ and ν̃ are set, and µ̃b and µ˜f are fixed by the total number of
bosons and fermions one loads into the trap. To image the phase diagram, we fix λ̃
and ν̃, and look for phase transitions as one varies µ̃b and µ˜f .
We set the mass ratio to be the one for the
87
RB −40 K system: r = 0.46. We
also set λ̃ = 0.0063, which is a typical value [67]. Note that for a given system, one
can alter λ̃ by choosing a Feshbach resonance with a different width.
The resulting phase diagram, for different values of the detuning ν̃, is shown in
Fig.2.1. We show the phase diagram in chemical potential space for four values of ν̃:
ν̃ = −80, 0, 100 and 140, from left to right. Below each of these diagrams we show the
corresponding phase diagram in number space. The experiments so far have focused
on the ν > 0 (attractive) side, where they see collapse as they approach ν = 0.
Let us look at the diagram on the top right, with ν̃ = 140. At any µ˜f , for µ̃b
negative enough we are in the Normal (N) phase, where there is no BEC. As one
increases µ̃b , at some point the BEC appears, characterized by a nonzero value of ρ,
in the gray region. This phase transition is second order for µ˜f away from the red
line, and first order along the red line. The second order lines join the first order
line at conventional tricritical points, indicated by the circles. Now for µ̃b < 0 and
µ˜f < 0, we are in vacuum, because all states in the bands have positive energy. This
persists as we increase µ̃b until µ̃b > 0, where a BEC appears, due to the −µb ρ term
2
The discrepancy between the equation given here and the one cited in the reference is due to
the fact that we are dealing with scattering of distinguishable particles.
14
in the Hamiltonian. As µ̃b increase, ρ increases, which pushes the Ψ band down, until
it crosses the zero energy line, which happens at the grey dotted line in the figure.
Above this line there is one Fermi Surface (1 FS) for the Ψ band. This is an example
of a Fermi Surface changing phase transition. Since an increase in ρ pushes the F
band up and the Ψ band down, there are two possible Fermi Surface changing phase
transitions, induced by the appearance of a BEC: either the Ψ band starts filling
up, or the F band becomes empty. In this particular diagram, we haven’t indicated
the line where the second possibility takes place, it appears at positive values of µ˜f
beyond the values shown here. We do show this line in the number space diagram.
Another way of changing the number of FS is by varying the chemical potentials in
the normal region, where there is no BEC: the lines µf = 0 and µf + µb − ν = 0 are
Fermi Surface phase transition lines.
Let us now move closer to the resonance, to ν̃ = 100 (the second diagram from
the right). for µ˜f > 0 we once again have a conventional tricritical point. For µ˜f < 0,
however, the tricritical point gets preempted by a critical point, indicated by the
little square. The second order line joins the first order line at what is referred to
as a critical endpoint. Around this point, the energy has two minima as a function
of ρ. To the left of the critical endpoint, as one increase µ̃b one first encounters a
conventional second order phase transition, as the first minimum (i.e. the minimum
at a smaller value of ρ) shifts from ρ = 0 to nonzero ρ. Increasing µ̃b further, the value
of the energy at the second minimum decreases, until it becomes the global minimum
of the energy, at which point we have a first order phase transition. To the right of
the critical endpoint, the second minimum is always the global minimum, and the
conventional second order phase transition is preempted by the first order transition.
The first order line is surrounded by spinodal lines, which are the dotted blue lines
in the diagram. Along each line, one of the minima discussed above either appears or
disappears. The lower spinodal line indicates the appearance of the second minimum,
15
i.e. at the lower spinodal line there is a nonzero ρ for which dE/dρ = 0. Above the
lower spinodal line, this point becomes a local minimum, and as one crosses the first
order line this local minimum becomes the global minimum. The upper spinodal line
corresponds to the disappearance of the first minimum, which is determined by the
same criterion as the lower spinodal line. Numerically the first order lines take a long
time to calculate, but the calculation of the spinodal lines is much faster. This comes
in very handy, since one can first calculate the spinodal lines, after which one can look
for the first order line between them. The discussion of the Fermi Surface transitions
is the same here as it was for ν̃ = 140, except that in that case the 0F S → 1F S
transition at µf < 0 joined up wit the tricritical point, while here this transition
line does not join up with the critical point (this is clearer in the graphs for lower
value of ν̃). It joins the first order line at some point between the critical point and
the critical endpoint. Thus if we sit very slightly to the right of the critical point,
and vary µb from negative to positive values, we encounter three phase transitions: a
second transition from vacuum to a BEC, then a FS changing phase transition where
the Ψ band gets occupied, and finally a first order BEC → BEC transition in which
the value of ρ jumps. Closer but still to the left of the critical endpoint, the FS
transition disappears, and the rest is the same. To the right of the critical endpoint,
one encounters one first order phase transition.
Now for the resonance ν̃ = 0. At this point, we have two critical points, acoompanied by two critical endpoints. In this case, both BEC induced FS transitions
connect to the first order line between a critical point and a critical endpoint. Furthermore, here we actually see the 1F S → 2F S transition line in the normal phase.
Finally, the leftmost diagrams are at ν̃ = −80. Once again we have one critical
point, and one tricritical point.
16
If we decrease ν̃ far beyond −80 or increase it far beyond 140, the first order
line will shrink until it disappears completely, leaving us with second order phase
transitions, and FS transitions.
By using the equations for number densities given above, we can translate the
phase diagram to number space. The first order line becomes a region of phase
separation, which one obtains by calculating the numbers densities for the chemical
potentials just above and below the first order line. The darkened lines within the
Phase Separating (PS) region connect the two phases on the first order line that the
system will separate into, if one starts with (ñb , n˜f ) on that line (n˜b,f = nb,f /(m3/2 γ 3 )).
The spinodal lines delineate an unstable region. Inside the unstable region, there is
no local minimum of the energy, and it will immediately phase separate. Outside
of the unstable region, but still within the first order region, there is a metastable
minimum of the energy. In the metastable region, phase separation occurs through
nucleation, as there is an activation energy required to roll out of the metastable
state. The remaining lines in number space denote the FS transition lines.
Let us now address the full phase diagram. To this end, it is convenient to revert
to different parameters, this time three instead of four:
(r)
3/2
{ν (r) , µb , λ(r) } ≡ {(ν − µb )/|µf |, µb /γ|µf |1/2 , mb λ|µf |3/2 /γ 2 }.
(2.14)
As discussed earlier, the tritrical points will form a line in parameter space, because
they are set by fixing two derivatives of the energy at ρ = 0. Similarly the critical
endpoints, critical points and points where the FS transition lines join the first order
line, which we will call the FS endpoint, form lines in parameter space. However,
these points cannot be determined by studying derivatives of the energy at ρ = 0, so
instead one has to find the lines numerically. To obtain a planar phase diagram, one
can project down to the (ν (r) , λ(r) ) plane. We have scaled away µf , but there are still
17
3/2
Figure 2.1: Phase diagrams for different values of ν/γ 2 at λmb γ = 0.0063, where
the top and bottom rows correspond to chemical potential and density space, respectively, while the columns represent different detunings: ν/γ 2 = −80 (first), ν/γ 2 = 0
(second), ν/γ 2 = 100 (third) and ν/γ 2 = 140 (fourth). In chemical potential space,
the phase transition from the normal phase (in white) to the BEC phase (in gray)
can either be second order, along the thin (gray) lines, or first order, along the thick
(red) line. In (rescaled) number density space, this thick (red) line encircles a region
of phase separation (PS). Dark dotted lines within this region connect points on the
first order lines, such that a system whose total number densities lie on this line will
phase separate into the phases where the line intersects the red curve. The dotted
(blue) lines around the first order line in chemical potential space are the spinodal
lines, and in number space they encompass an unstable region within the PS region,
outside of which the system is metastable. The other dotted lines represent lines
where the number of Fermi Surfaces changes. The (blue) circles tricritical points,
while the (red) squares are critical points.
18
two cases one has to consider: µf > 0 and µf < 0. Thus we obtain two planar phase
diagrams, shown in Fig. 2.2. The line of tricritical points joins all the other lines,
namely the lines of critical endpoints, critical points, and FS endpoint, at tetracritical
points, which are found by solving
dE
|
dρ ρ=0
=
dE
|
dρ2 ρ=0
=
dE
|
dρ3 ρ=0
= 0. And that is where
the fun stops : we only have three parameters to vary (after rescaling), therefore we
can only set three derivatives to zero.
The lines in the phase diagram demarcate areas where there is a definite series of
phase transitions one encounters, as one varies µb from −∞ to +∞. Let us neglect
the FS transitions within the normal phase, and focus on the phase transitions to the
BEC phase and within it. The different sequences of phase transitions are as follows:
Below the critical endpoint line:
1st order N → BEC
Between the critical endpoint and FS endpoint lines:
2nd order N → BEC, 1st order BEC → BEC
Between the FS endpoint and critical point lines:
2nd order N → BEC, 2nd order 0F S → 1F S or 2F S → 1F S, 1st order BEC → BEC
Away from all lines:
2nd order N → BEC, 2nd order 0F S → 1F S or 2F S → 1F S
(2.15)
Now that we have fully discussed the phase diagram, we want to address phase
separation in an actual trap. In an experiment, the atoms are confined by an optical trap, which creates a harmonic potential, with a variable trap frequency. The
potential is given by
1
V (r) = mω 2 r2 .
2
19
(2.16)
(a) µf > 0
ΛHrL
Critical endpoints
FS endpoints
8
Tricritical points
Critical points
6
Tetracritical points
4
2
2
-2
4
6
8
ΝHrL
(b) µf < 0
ΛHrL
Critical endpoints
FS endpoints
25
Tricritical points
Critical points
20
Tetracritical points
15
10
5
Printed by Mathematica for Students
-4
2
-2
4
ΝHrL
(r)
Figure 2.2: Phase diagram in the parameter space {ν (r) , µb , λ(r) } ≡ {(ν −
3/2
µb )/|µf |, µb /γ|µf |1/2 , mb λ|µf |3/2 /γ 2 } projected onto the (ν (r) , λ(r) ) plane, for
(r)
(r)
(a)µf > 0 and (b)µf < 0. For λ(r) above a critical λcrit (λcrit = 8.44 for µf > 0,
and 25.5 for µf < 0), one only has second order phase transitions, which are inde(r)
pendent of λ(r) . As λ(r) is lowered below λcrit , one first encounters tricritical points.
(r)
When λ(r) < λcrit , one first encounters tricritical points, until one reaches a tetracritical point. Beyond such a point one has a line of critical points, a line of critical
endpoints, and a line of FS endpoints (see text), although the lines may overlap too
strongly to be visible.
20
Printed by Mathematica for Students
As one goes out from the center of the trap to the outside regions, the potential energy
increases, and the densities will decrease. Within the Local Density Approximation
(LDA), one assumes that the chemical potential follows the behaviour opposite to the
potential. In other words, within LDA we have µ(r) = µ0 − 12 mω 2 r2 . Experiments
have shown that the bosons and fermions feel the same potential in the trap[26] (in
other words, the trapping frequency depends on the mass precisely so as to cancel
the mass dependence of V (r)). Thus the LDA approximation leads to
µf,b (r)/γ 2 = µf,b (0)/γ 2 − (r/r0 )2 ,
(2.17)
where r0 is some arbitrary length scale. To obtain the density profile in the trap, one
chooses a value µf,b (0) of the chemical potentials in the center of the trap. As r varies,
so do µf,b (r), and therefore at each r we can calculate the number densities of the
fermions and bosons, and the density of condensed bosons. We will consider values
of µf,b (0) such that one crosses the first order line as r, so as to see the behaviour
we are interested in. We choose r0 to sit at the distance from the center of the trap
where one crosses the first order line.
We choose to sit at the unitary limit ν = 0, because all possible types of sequence
of phase transitions are represented there. The result of three different choices of
µf,b (0) are shown in Fig. 2.3.
2.4
Experimental consequences
After having laid out the physics of our model in detail, we address the question of
whether the rich behaviour we are predicting is accessible experimentally. To obtain
an estimate for g, we use Eq.(2.13). To estimate λ we use the mean field value[24]
λ=
2πabb
m
21
(2.18)
Figure 2.3: Density profiles in a harmonic trap at ν = 0 for the rescaled boson number
ñb = nb /(m3/2 γ 3 ), fermion number ñf = nf /(m3/2 γ 3 ) and condensed boson number
3/2
ncond = ρ, for λmb γ ' 0.0063. The profiles (a), (b) and (c) correspond to three
different sequences of phase transitions. The top left diagram shows which trajectory
is traced out in chemical potential space, with the arrow pointing along increasing
distance r from the center of the trap. r0 is chosen to coincide with the crossing of
the first order line.
22
where abb is the boson-boson scattering length.
From Fig. 2.1, we see that the phase separated region occurs at densities of order
100m3/2 γ 3 . In experiments, after cooling the atoms the trap has typically about 105
atoms, and the linear size of the cloud is about 0.1 mm, giving a density of about
1011 cm−3 . The density varies strongly across the trap, particularly if there is a BEC,
and the densities at the center of the trap are typically on the order of 1013 −1015 cm−3 .
We will now estimate the value of 100m3/2 γ 3 /h̄6 for the three pairs of bosonic and
fermionic atoms that have received the most experimental attention:
7
Li −6 Li[72], and
87
23
N a −6 Li[29],
Rb −40 K[26]. Using values quoted in the literature for all the
parameters that go into estimating g and λ, we can obtain estimates for 100m3/2 γ 3 /h̄6 .
The value depends on which Feshbach resonance one considers:
6
Li −7 Li : 100m3/2 γ 3 ∼ 7 × 1013 cm−3 for ∆B = 1G
87
Rb −40 K : 100m3/2 γ 3 ∼ 6.8 × 1014 cm−3 for ∆B = 51mG
87
Rb −40 K : 100m3/2 γ 3 ∼ 5.4 × 1018 cm−3 for ∆B = 1G
23
N a −6 Li : 100m3/2 γ 3 ∼ 3.1 × 1014 cm−3 for ∆B = 0.6G.
We see that these are reasonable densities, except for
87
Rb −40 K at a resonance of
width 1G. Therefore, to see the interesting physics in this pair, one would need to
tune to a rather narrow Feshbach resonance (the resonance with width ∆B = 51mG
has only been predicted theoretically [26], it hasn’t been measured yet).
2.5
Comparison to previous work
The experiments on
87
Rb −40 K see a collapse of the system[53, 54], as the Feshbach
resonance is approached from the a < 0 side. A collapse indicates that the bosons all
condense into a BEC. The standard theory used in the literature to study collapse is
23
a single-channel model, which is a model without a distinct molecular state. We can
easily see within mean field theory why such a model leads to collapse. The single
channel Hamiltonian is given by[55, 78, 12]
Z
d3 k d3 k 0 d3 q † †
d3 k f †
b †
ĤSC =
ξ f f + ξk bk bk ) + UBF
f b b
f
(2π)3 k k k
(2π)3 (2π)3 (2π)3 k k’ k’+q k’−q
Z
d3 k d3 k 0 d3 q † †
+λ
b b b
b
(2.19)
(2π)3 (2π)3 (2π)3 k k’ k’+q k’−q
Z
√
where UBF = 4πh̄2 aBF /m. By making the replacement bk → δk,0 ρ, the zerotemperature energy becomes
E = −µb ρ +
Z kf
0
For large ρ one can show that kf ∼
d3 k f
ξk + UBF ρ) + λρ2
3
(2π)
(2.20)
√
ρ. Therefore E ∼ UBF ρ5/2 . For aBF < 0 (on
the attractive side of the resonance), this implies a collapse.
For comparison, let us analyze the asymptotic behaviour of the energy for large ρ.
As ρ → ∞, we have here that kF = 0 and kΨ ∼ ρ1/4 , which leads to E ' Cρ5/4 . The
constant C depends only on the mass ratio r, and a careful analysis reveals that C is
negative for all mass ratios. Therefore, for λ = 0 the system is unstable with respect
to collapse: the energy is unbounded as a function of ρ, and all bosons condense into a
BEC. For λ > 0, the λρ2 term stabilizes the system, i.e. the energy becomes bounded
from below.
As a final point on the single channel model, one can show that it is obtained from
the two-channel model by taking g → ∞, and ν 7→ ∞, while keeping g 2 /ν constant,
because this keeps the effective interaction in the single channel model constant (see
appendix B). To see how the single channel model is obtained, consider the mean
√
field energy of the single channel model, by replacing bk with δk,0 ρ in Eq. (2.19):
24
ESC =
Z
k2
d3 k k 2
−
µ
+
U
ρ)θ(µ
−
− UBF ρ) + λρ2 .
f
BF
f
3
(2π) 2mf
2mf
(2.21)
Now take the limit ν → ∞, g → ∞, with g 2 /ν held constant. What happens is
F
that ξk
→ ∞, which means it drops out the problem, since it only contributes to the
Ψ
two-channel mean field energy when it is negative. On the other hand, ξk
→
k2
2mf
−
g2
µf − ρ ν , which implies that here kΨ ∼ ρ1/2 . Plugging this into the expression for the
free energy of the two channel model, Eq. (2.8), we recover the same expression as the
mean field energy for the single channel model, ESC , provided we set −g 2 /ν = UBF ,
which is precisely the expression derived in the appendix, Eq. (B.7) (with Γ replaced
by g, and g by UBF ).
Thus our model encompasses the single channel model, and allows for a more
detailed explanation of the physics close to resonance. Note that in the two-channel
model we showed that the energy is unbounded, but in the limit ν → ∞, g → ∞,
with g 2 /ν held constant, the minimum of the energy in the two-channel model goes
off to −∞. Therefore in this limit the two-channel model collapses, as does the
single-channel model.
2.6
Conclusion
To conclude, we have shown that contrary to the predictions in a single channel model,
our two channel model does not see a collapse to arbitrarily high density, since the
energy is always bounded from below. Far away from unitarity, we get that there
are only second order phase transitions, while close to unitarity and at low enough
densities, the system will tend to phase separate.
So far the experiments see what they interpret as a collapse, i.e. a sudden unbounded increase in the density. They may actually be seeing phase separation into a
25
phase with such a high density that three body effects dominate. In the experiments,
once the density has increased too strongly, three-body recombination effects become
important, which we have neglected in our treatment. These effects play an important role in the trap at positive ν: as the density increases, three-body interaction
leads to a significant loss of atoms, where one atom gains the kinetic energy that is
released when the other atoms go into a deep bound state. The rate for three-body
losses is given by [25]
ṅ/n ∼
h̄
(na2 )2
m
(2.22)
where n is the number density, and a is the scattering length. It therefore increases
strongly as the density increases. On the a < 0 side of the resonance, close to
unitarity the experiments see a sudden increase in density, after which the condensate
is lost due to three-body recombination losses[82, 59]. To avoid these losses, we want
to sit at low densities. If we want the dimensionless parameters nf,b /(m3/2 γ 3 ) to
remain constant, so that we are sitting in the interesting region of phase separation
(nf,b /(m3/2 γ 3 ) ∼ 100) , we have to reduce γ as we lower the density, which can be
achieved by reducing ∆B. Thus, in general one should tune to narrow Feshbach
resonances to see this behaviour, as we saw above for
87
Rb −40 K.
The physics in the repulsive regime, corresponding to negative ν, is different and
subtle, because the physics on that side depends on how one tunes to ν. Namely, if
one starts at large negative ν, where a(B) is small, and moves towards unitarity, one
is accessing a phase of strongly repulsive bosons and fermions. The single channel
theory then predicts that the atomic bosons and fermions phase separate. Namely,
within mean field theory they feel each other’s presence through an effective chemical
potential, and become immiscible if a is large enough. Close to unitarity there is a
bound state with small negative energy, but the only way for a boson and a fermion
to go into the bound state is through a three-body process, because of energy conversation. To access the phase we are describing, where there is a macroscopic number
26
of molecules, one should start at positive ν, and tune adiabatically to negative ν. At
positive ν, the “bound” state is virtual, which means there are scattering states with
the same energy. As one tunes to negative ν, the virtual state turns adiabatically
into a true bound state, and will have nonzero occupation. Only very recently have
experiments used this approach to form fermionic molecules [82, 58, 83]. To experimentally be able to access the phase separated region, one would want a narrow
Feshbach resonance, as we saw above, and a large value of λ, since a higher λ makes
the phase separating region shrink in number space, thus reducing the densities, and
therefore the 3-body losses, in that region.
27
Chapter 3
Accessing the Néel phase of
ultracold fermions in a
simple-cubic optical lattice
In this chapter, we examine the phase diagram of a simple-cubic optical lattice occupied by two species of fermionic atoms with a repulsive contact interaction, at a
density of one particle per site. In the limit of a deep lattice potential and weak
interactions, the effective Hamiltonian of the system is a Hubbard model, one of the
perennial models in condensed matter physics. We study the issues related to the
experimental realization of the antiferromagnetically ordered Néel phase. To determine which parameters will make the Néel phase most robust, we use an expansion
in a Wannier basis in the deep lattice regime, and a Hartree calculation in the regime
of weak to intermediate lattice. The regime where the Néel phase of this system is
likely to be most accessible to experiments is at intermediate lattice strengths and
interactions, and our two approximations match onto each other in this regime. We
discuss the various issues that may determine where in this phase diagram the Néel
28
Figure 3.1: Sketch of the generic phase diagram of High Temperature Superconductors, as a function of doping x and temperature T [45]. The parent compound, at
x=0, is a half filled band, which is a Mott insulator at low temperatures due to onsite Coulomb repulsion. Virtual superexchange leads to antiferromagnetic ordering,
thus giving the AFI phase. Doping leads to the superconducting dome (SC), with an
intervening pseudogap where the system has a soft gap (i.e. the spin susceptibility
follows an Arrhenius law) but there is no long range phase coherence.
phase is first produced and detected experimentally, and analyze the rich magnetic
phase diagram.
3.1
Introduction
Cold atoms hold exceptional promise for the quantum simulation of many-body systems. In particular, experimentalists have been able to generate systems whose dynamics is set by the Hubbard model, which is believed to capture the physics of
High-Temperature (High-Tc) Superconductors.
The generic phase diagram of High-Tc superconductors in cuprates is given in
Fig. 3.1. The parent compound is an antiferromagnetic Mott insulator. As one dopes
away from this limit at low temperatures, the system first enters a pseudogap regime,
in which there is tendency towards a spin gap but no superconductivity. Further
doping leads to the superconducting dome. There is no consensus on the precise
29
physics governing the dome, in particular the reason for the critical temperature Tc
at which superconductivity sets in dropping as one leaves the value of x at which
T c is maximal, called optimal doping. On the underdoped side, for example, there
are several proposals that there may be some static order that is competing with
superconductivity, such as stripes or incommensurate spin ordering. This physics is
possibly not captured in the Hubbard model, as it is indeed still a controversy to what
extent long range interactions are important for stripes [81]. Thus the precise details
of the phase diagram may in fact be different for the Hubbard model, and therefore in
its incarnation in cold atoms we do not precisely know what phase diagram to expect.
Another aspect in which cold atoms differ from cuprates is that the experiments so far
are done in three dimensions, for reasons that we will discuss, while cuprate physics
is believed to be captured by a two-dimensional Hubbard model. However this can be
remedied by going to a two-dimensional optical lattice. Thus a quantum simulation
of the Hubbard model allows us to address the crucial question of whether the model
indeed does capture the salient features of the physics of High-Tc superconductors.
Within the cold atom context, there is also the possibility of studying intriguing
physics beyond the Hubbard model, as will become clear later on.
The main challenge experimentally is to achieve temperatures low enough to see
the phases in Fig. 3.1, such as the Néel antiferromagnetic Mott insulating phase
(AFI), the pseudogap phase, and the superconducting phase. The first goal is to
measure the phase which is believed to be the easiest one to access, namely the AFI
phase. The main purpose of this chapter is to answer the following question: what are
the optimal parameters to obtain a robust and therefore experimentally most easily
accessible Néel phase?
First we will discuss the precise system used to simulate Hubbard physics in cold
atoms. For this system, we will derive an effective lattice model, and using this model
we will calculate our first estimate of where the Néel phase is most robust, using an
30
expansion around the deep lattice limit. This expansion allows us to explore part of
the magnetic phase diagram. Starting from the weak to intermediate lattice side, we
will use an unconstrained Hartree calculation to obtain a second estimate of which
parameters strengthen the Néel phase, and a more complete picture of the magnetic
phase diagram. These two methods match onto each other at intermediate lattice
strength, thus we obtain an estimate of the best parameters to see Néel order. We
then comment on the experimental consequences of our results.
3.2
The incarnation of the Hubbard model in cold
atoms
To realize Hubbard type physics in a cold atom context, one combines the use of lasers
to generate an optical lattice, and a Feshbach resonance to tune the interactions.
Three orthogonal retroreflected beams realize a simple-cubic optical lattice potential:
V (~x) = V0 (sin2 (kx) + sin2 (ky) + sin2 (kz)),
(3.1)
where k = π/d, d being the lattice period. One costumarily defines the lattice depth
in units of the recoil energy Er = h̄2 k 2 /(2m).
To realize a repulsive Hubbard model, we need fermions in two different hyperfine states, which one can populate using RF spectroscopy. To vary the interaction
between the two hyperfine states, a static magnetic field is then tuned to a Feshbach
resonance. The scattering length as as a function of magnetic field behaves as
as (B) = abg (1 −
∆B
),
B − B0
where abg can be positive or negative. We will assume that abg > 0.
31
(3.2)
We want the atoms to interact with a repulsive scattering length. This can be
achieved by setting B < B0 , thus sitting on the so-called repulsive side of the resonance, or going to large B0 , although the scattering length in that case cannot exceed
abg . This is only appropriate for atoms with large and positive abg , such as
40
K.
If one is on the repulsive side, there is a bound molecular state that the particles
can fall into. Kinematic restrictions forbid this in a two-body process, but one should
avoid three body interactions where two particles form a bound state and the remaining particle carries the leftover energy. It is so far unknown how the lattice affects
three-body losses : it may hamper them as it tends to reduce the overlaps of different
particles, but three particles in a single well of the lattice may see large three-body
losses due to the confinement of the wave functions.
When one is not too close to the unitary limit B = B0 , the interactions between the
two species of fermions, which we will call ↑ and ↓ for convenience, can be modelled as
an infinitely short-range interaction. The s-wave repulsive interaction is only between
atoms of opposite spin. We assume that the repulsion is weak, such that we can apply
the first order Born approximation to the scattering problem at zero momentum (see
appendix A for details).
To lowest order in the interactions, we approximate this 2-atom interaction as the
pseudopotential (see Eq. (A.20))
V2 (r↑ − r↓ ) =
4πh̄2 as
δ(r↑ − r↓ ) .
m
(3.3)
Only particles of opposite spin interact, since by Pauli exclusion fermions of like
spin do not feel a contact interaction.
Thus the Hamiltonian we are dealing with is
H=
Z
Z
h̄2 2
d~xΨ (x)( −
∇ + V (~x))Ψ(~x) + g d~xΨ†↑ (~x)Ψ↑ (~x)Ψ†↓ (~x)Ψ↓ (~x)
2m
†
32
where g =
4πh̄2 as
.
m
The canonical method to derive an effective lattice model for a given system in
a periodic potential is to expand the Hamiltonian in a complete basis of localized
atomic orbitals. In a simple-cubic optical lattice, it is common to employ maximally
localized Wannier orbitals[41, 35, 80].
The Wannier orbitals are defined in terms of the Bloch states of the non-interacting
problem (i.e. setting as = 0). To diagonalize the non-interacting Hamiltonian, one
can use separation of Cartesian variables, and thus solve a one-dimensional problem:
−
∂2
ψn,k (x) + V0 sin2 (kx)ψn,k (x) = En,k ψn,k (x).
2
∂x
(3.4)
The solutions are 1D Bloch states ψn,k (x), labelled by a band index n and a momentum k in the First Brillouin Zone (FBZ) : k ∈ [−π/d, π/d). To consider a system
with N lattice sites, we set the boundary condition Ψ(x + N d) = Ψ(x). This leads
to N states in each Bloch band: k = 2πn/N, n ∈ [−N, N − 1].
Using these Bloch states we define 1D Wannier functions
1
wn (x − xm ) = √
N
X
eiθ(k) e−ikxm ψn,k (x)
(3.5)
k∈F BZ
where xm = md is the coordinate of a lattice site. Using the defining relation of
Bloch states: ψn,k (x + xm ) = eikxm ψn,k (x), one can show that the Wannier functions
in one band are simple translations of each other (which is implied by our notation
wn (x − xm )). Each band of Bloch states leads to N Wannier functions, one for each
lattice site, labelled by m. From the orthonormality of the Bloch states, one can
prove that the Wannier functions are orthonormal.
The θ(k) are gauge degrees of freedom. One usually chooses them so as to maximize localization (i.e. minimize the variance) of the Wannier functions [41]. We can
also simultaneously choose the θ(k) so that the Wannier functions are real. Each
33
ΩnHxL
d
1.5
1.0
0.5
-2
-1
1
2
xd
-0.5
-1.0
Figure 3.2: The three lowest maximally localized Wannier functions in a one dimensional sinusoidal potential V (x) = V0 sin2 (πx/d) with V0 = 6Er : the lowest (n=0)
Wannier function is the full blue line, the n=1 function is the dashed red line, and
the n=2 function is the dot-dashed green line.
Bloch band leads to a well defined parity for its Wannier function centered around
the origin, and the parity alternates from one band to the next. The three lowest
Wannier functions are shown in Fig.3.2.
(a) A Wannier state in a 2D square lattice, in the lowest band, at V0 = 4Er . The
optical lattice is in green.
(b) A Wannier state in a 2D square lattice, in one of the two first excited bands,
at V0 = 4Er . The optical lattice is the
same as in ((a)).
Figure 3.3: Wannier states in a 2D square lattice.
34
Eq. (3.5) generalizes to any dimension, thus in three dimensions
wn (r − R) =
1
N 3/2
eiθ(k) e−ik·r ψn,k (r)
X
(3.6)
k∈F BZ
where N 3 is the number of lattice sites. Here the band label has become a set of
three band labels: n = (nx , ny , nz ). The lattice site is now a set of three coordinates:
R = (Rx , Ry , Rz ) where Ri = ni d, i.e. an integer times the lattice spacing. The
coordinate is r = (x, y, z). Since the non-interacting Hamiltonian is separable in x, y
and z, the Bloch states in three dimensions are simply products of one-dimensional
Bloch states, which immediately implies that Wannier functions in three dimensions
are products of one-dimensional Wannier functions:
wn (r − R) = wnx (x − Rx )wny (y − Ry )wnz (z − Rz )
(a) Contour plot of a Wannier state in
a 3D simple cubic lattice, in the lowest
band, at V0 = 4Er . The wave function is
constant positive on the light (blue), and
negative on the dark (red) surface.
(3.7)
(b) Contour plot of a Wannier state in
a 3D optical lattice, in one of the three
first excited bands, at V0 = 4Er . The
wave functions is constant and positive on
the light (blue), and negative on the dark
(red) surface.
Figure 3.4: Contour plots of Wannier states in a 3D simple cubic lattice
Wannier functions are also used in solid state systems to obtain an effective lattice
model. There is one important difference between Wannier functions in a solid state
systems and in cold atoms : the size of a Wannier function is set by the Coulomb
35
potential of the ionic lattice in solid state systems, which is deep, and forces the
Wannier function to be tightly confined on one lattice site, for a wide range of hopping
parameters. In cold atoms, on the other hand, the potential that leads to localized
orbitals is soft, and as one softens the lattice the Wannier functions will start to
strongly overlap with neighboring lattice sites, which leads to an effective lattice
model with a potentially large set of terms that cannot be neglected. We will now
show how to derive a effective lattice model from the Wannier functions.
In terms of Wannier functions wn (r − R), the fermion operator is given by
X
Ψσ (r) =
cnRσ wn (r − R.)
(3.8)
n,R
where cnRσ destroys a fermion in Wannier band n at site R. We will write a cnRσ as
cJσ , i.e. a capital letter labels both the band and lattice site indices: J = {nJ , RJ } =
{nJx , nJy , nJz , xJ , yJ , zJ }.
Ψσ (r) =
X
cJσ wnJ (r − RJ )
(3.9)
J
The non-interacting part of the Hamiltonian becomes
tIJ =
Z
dxwnIx (x − xI )(−
†
IJσ tIJ cIσ cJσ
P
where
∂2
+ V0 sin2 (kx))wnJx (x − xJ )δnIy ,nJy δyI ,yJ δnIz ,nJz δzI ,zJ
∂x2
+ x → y → z + x → z → y.
(3.10)
We see that there is no diagonal hopping, i.e. tIJ can be nonzero only if RI and
RJ differ by at most one coordinate. In other words, one can only hop from a point
to another by moving parallel to the x, y or z axis. The values of the hopping tIJ are
independent of dimension. We plot the nearest-neighbor and next-nearest-neighbor
hopping in the lowest band in Fig. (3.5). We denote the nearest neighbor hopping in
the lowest band by t0 .
36
Hopping in the lowest band
t
Er
0.200
0.100
0.050
0.020
0.010
0.005
Nearest
neighbor
Next-nearest
neighbor
5
10
15
V0
20 Er
Figure 3.5: Absolute value of the nearest-neighbor and next-nearest-neighbor hopping
in an optical lattice, as a function of lattice depth, in units of recoil Er (the nearestneighbor hopping is negative, the next-nearest-neighbor hopping is positive). The
dashed red line is obtained from an asymptotic expansion at large V0 of the analytic
expression for the bandwidth W of√Mathieu functions [33] (at large V0 , W = 4t in
√
1D): −t/Er = (4/ π)(V /Er )3/4 e−2 V /Er .
The interaction term becomes
X
UIJKL c†I,↑ cJ,↑ c†K,↓ cL,↓
(3.11)
IJKL
where UIJKL = g drwnI (r − RI )wnJ (r − RJ )wnK (r − RK )wnL (r − RL ).
R
Because the Wannier functions are localized on different sites, the largest interaction terms will be obtained when I, J, K and L are on the same site, or on neighboring
sites, as the overlap decays exponentially with distance. In the deep lattice limit, the
√
energy gap between the different bands grows like V0 , which means that the higher
bands grow increasingly irrelevant, for a fixed as . Indeed, in perturbation theory the
energy denominators will grow, such that the mixing of the higher bands decreases.
Thus the terms that are the most important are on site, on neighboring site, and
involving the lowest bands.
Let us group the terms of the interaction into different categories, classified according to how many of the {I, J, K, L} are different.
37
If I = J = K = L, we get UI n̂I,↑ n̂I,↓ , where we define UI = UIIII , and n̂Iσ = c†Iσ cIσ
is the density operator. These terms are on-site repulsions within a given band. We
call U0 the on-site repulsion in the lowest band.
If one of the indices is different, we group four terms together since UIJJJ =
UJIJJ = UJJIJ = UJJJI , and get If I 6= J = K = L, one gets
UIJJJ (t̂↑IJ n̂J↓ + t̂↓IJ n̂J↑ )
(3.12)
where t̂σIJ = c†Iσ cJσ +c†Jσ cIσ . These are density assisted tunneling operators: they only
allow a hop of particle of a certain spin to hop from one site to another if there is a
particle of the opposite spin sitting on either of those sites. As we will be focusing on
a half-filled state, with one particle per site, the density assisted tunneling operator
will renormalize the non-interacting hopping. We call tI the number in front of the
nearest-neighbor density assisted tunneling operator in the lowest band.
Now for (I = K) 6= (J = L). We bring together 6 terms, since UIIJJ = UIJIJ ,
etc. With a little manipulation of fermionic operators, one can write these terms in
a neat form:
1
UIIJJ (P HI,J − 2SI · SJ + nI nJ ),
2
(3.13)
where SI is the spin operator, and we defined a pair hopping term
P HI,J = c†I↑ c†I↓ cJ↓ cJ,↑ + c†J↑ c†J↓ cI↓ cI↑ .
(3.14)
We therefore obtain a direct ferromagnetic exchange term, which plays a crucial role
in limiting the Néel temperature, as we will show in the next section.
38
In the deep lattice, weakly interacting limit, the Hamiltonian becomes a Hubbard
model:
H=
X
<Ri ,Rj >,σ
t0 (c†R σ cRj σ + c†R σ cRi σ ) +
i
j
X
U0 nRi ↑ nRi ↓
(3.15)
Ri
where < Ri , Rj > denotes nearest neighbors.
In Fig. (3.2), we plot the interaction terms that are the most important, as
one starts to leave the deep lattice limit: nearest neighbor repulsion Unn , on-site
pair-hopping from the lowest to a first excited band which we denote by P H01 , and
density assisted tunneling in the lowest band tI . In the next section we will discuss
how these terms affect the Néel temperature as one lowers the lattice depth, and use
the effective Hamiltonian to obtain an estimate of the magnetic phase diagram.
Interaction terms in 3D
E d
E r as
Interaction terms in 3D
U0
50
20
10
5
2
1
E d
E r as
0.10
0.05
0.00
-0.05
-0.10
PH01
5
10
15
V0
20 Er
(a) Interaction terms in a three-dimensional
optical lattice with atoms scattering in the swave channel: On-site repulsion U0 in the lowest band, and on-site pair hopping P H from
the lowest Wannier states to one of the three
degenerate first excited Wannier states. The
dashed red line represents the deep lattice estimate of U0 , obtained by approximating the
deep p
lattice site as a harmonic square well :
U0 ' 8/πkas (V0 /Er )3/4 Er .
Unn
5
10
15
V0
20 Er
tI
(b) Interaction terms in a threedimensional optical lattice with atoms
scattering in the s-wave channel: Nearest
neighbor repulsion Unn in the lowest
band, and density-assisted nearestneighbor hopping tI .
Figure 3.6: Interaction terms in a three-dimensional optical lattice with atoms scattering in the s-wave channel
39
3.3
Strong lattice expansion and Néel temperature
When the optical lattice is sufficiently deep and the repulsive s-wave interaction
between the atoms is sufficiently weak, the Néel temperature TN for the case of one
atom per lattice site can be estimated by modeling the system as a one-band Hubbard
model, and one can analyze the possibility of reaching this phase by adiabatically
ramping up the interactions and the optical lattice [35, 80, 18, 40, 75]. The most
accessible conditions for first producing this ordered phase in an experiment will
most likely be some compromise between the highest TN and the highest entropy at
the transition S(TN ). If the parameters of the system, the intensity V0 of the optical
lattice and the s-wave scattering length as , can be tuned in a perfectly adiabatic
manner, then to access the Néel phase only requires achieving sufficiently low entropy
[80, 18, 40]. But in the more likely event that there is always some “background”
heating, so things are not perfectly adiabatic, the phase will be more accessible when
it occurs at higher temperature. Thus here we study how the Néel temperature TN
depends on the two tunable parameters V0 and as as one leaves the region where the
standard Hubbard model is a good approximation to this system. Away from the
Hubbard regime, theoretical studies have suggested that one may be able to access a
wealth of phases governed by quantum spin hamiltonians [35, 22, 23].
According to quantum Monte Carlo simulations [76] of the simple-cubic fermionic
Hubbard model, for a given nearest-neighbor hopping matrix element t the highest
kB TN ∼
= t/3 occurs at interaction U ∼
= 8 t, while for a given U the maximal kB TN ∼
=
U/20 occurs at t ∼
= 0.15 U . Thus to increase TN one wants to move to larger t, which
means a weaker optical lattice (smaller V0 ), and to larger U , which means larger as .
This necessarily moves the system away from the regime where it is well-approximated
by the usual Hubbard model. As we saw in the previous section, the mapping from
40
the real system to the Hubbard model uses the single-atom Wannier orbitals as the
basis states [41, 35, 80]. The standard one-band Hubbard model includes only the
lowest-energy Wannier orbital at each lattice site and only the on-site interaction
between two atoms of different hyperfine states occupying the same Wannier orbital.
We find that it is the corrections due to including the interactions between Wannier orbitals on nearest-neighbor lattice sites that are the leading effects that stop
and reverse the increase of TN as t and U are increased by decreasing V0 and increasing as . In particular, these interactions produce a “direct” ferromagnetic exchange
interaction favoring neighboring sites to be occupied by the same species. These
ferromagnetic interactions are of the opposite sign from the antiferromagnetic superexchange interactions that cause the Néel ordering, and thus they suppress TN .
Within the approximations that we make (discussed in detail below) the maximal
(max)
kB TN
∼
= 0.03Er occurs near V0 ∼
= 3Er and as ∼
= 0.15 d, where Er = (πh̄)2 /(2md2 )
is the recoil energy and d is the lattice spacing. For example, Er ∼
= 1.4 µK for 6 Li
with d = 532 nm, which puts the maximum Néel temperature near 40 nK, which
seems to be well within the reach of current experimental cooling techniques.
This regime of large repulsive as is attained by approaching a Feshbach resonance
from the repulsive side. But the atoms must scatter repulsively without “falling” in
to the weakly-bound molecular state. Ref. [38] studied the Mott insulator with 40 K at
as ∼
= 0.08 d and did not mention any problem with excessive molecule formation. It is
not clear whether this can be increased to the as ∼
= 0.15 d that maximizes TN [64]. It
is also not clear whether the optical lattice increases or decreases molecule formation.
The lattice breaks momentum conservation, thus possibly opening up channels for
molecule formation, while in the Mott insulator the atoms are kept apart in different
wells of the optical lattice, which, naively, reduces the opportunities for molecule
formation.
41
Phase diagram
asa
1.00 FM to FI
0.50
t0 = tI
0.20
0.10
0.05
É J f É = Js
U=14t
Optimal TN
0.02
0.01
2
3
4
5
6
7
8
V0HErL
Figure 3.7: Approximate phase diagram for filling one fermion per lattice site using
the deep lattice expansion. The line marked |Jf | = Js is our approximation to the
ground-state phase boundary separating the antiferromagnetic phase at smaller as
from the ferromagnetic phase at larger as . The ferromagnetic phase is mostly a fullypolarized band insulator, but there is a small sliver of polarized Fermi liquid at small
V0 between the lines marked |Jf | = Js and “FM to FI”. The line marked “Optimal
TN ” indicates where TN as a function of V0 is maximized for each given as . The dot
on that line is our estimate of the parameters that produce the overall maximum of
TN /Er , and at that point Jf ' −Js /4 (see text). The U = 14t line is near where the
entropy is maximized at TN [80] and TN on this line is maximized at the dot. The
t0 = tI line signals when the interaction correction to the hopping becomes strong.
There is presumably also a paramagnetic Fermi liquid ground state in the lower left
corner of this phase diagram, but the deep lattice approximations are not well-suited
to estimating where this phase is.
In the deep lattice limit, as in the Hubbard model, when adjacent sites i and j
are each singly-occupied by atoms with the same spin, then the hopping between
those two sites is Pauli-blocked. When these adjacent sites are each singly-occupied
by opposite spins, then virtual hopping between these sites, treated in second-order
perturbation theory, allows them to lower their energy and thus generates an antifer~i · S
~j − 1 ) with Js = 4t2 /U .
romagnetic superexchange interaction Js (S
4
42
The leading corrections to the Hubbard model approximation to this system in
the regime we are interested in are due to the interactions between atoms of opposite
spin occupying lowest Wannier orbitals on nearest-neighbor sites i and j. There are
2 contributions: First, and apparently most important in limiting how large TN can
be made, is the nearest neighbor repulsion [77]
Unn
Z
4πh̄2 as Z
2
4
dyw02 (y)w02 (y + d)
=
[ dxw0 (x)]
m
(3.16)
between atoms of opposite spin in adjacent orbitals. This term is due to the overlap of
the probability distributions of adjacent Wannier orbitals. It raises the energy of the
~i · S
~j − 1 )
Néel state. It thus produces a direct ferromagnetic exchange interaction Jf (S
4
with Jf = −2Unn < 0 that partially cancels the antiferromagnetic superexchange Js
that occurs in the Hubbard model. It is primarily this ferromagnetic interaction that
stops and reverses the increase in TN as one moves towards stronger interaction and a
weaker lattice while staying near the optimal values of U/t. At the global maximum
of TN , indicated in Fig. 3.7, we find Jf ' −Js /4.
For large enough as this direct ferromagnetic exchange is stronger than the superexchange and thus we have a ground-state phase transition to a ferromagnetic
phase, as indicated in Fig. 1, and discussed more below. [Very near this |Jf | = Js
line, effects due to weaker further-neighbor interaction might produce some other
magnetically-ordered phases.]
The direct nearest-neighbor interaction (4) also reduces the effective U that enters
in the superexchange interaction, so at this level of approximation our simple-cubic
Hubbard model has interaction U = U0 − 6Unn , since it is the change of the interaction energy due to moving the atom that enters in the energy denominator in the
superexchange process.
43
Also, the interaction generates the density assisted tunneling term, which is of the
same sign as t0 : [80]
Z
4πh̄2 as Z
4
2
tI = −
[ dxw0 (x)]
dyw03 (y)w0 (y + d)
m
(3.17)
that is operative when the two sites are each singly-occupied by opposite spins. The
resulting effective hopping that enters in the superexchange process at this level of
approximation is thus t = t0 + tI .
Thus once we include these leading effects due to the nearest-neighbor interaction,
the effective Hamiltonian in the vicinity of the ground state of the half-filled Mott
insulator has hopping t = t0 + tI , an effective on-site interaction U = U0 − 6Unn , and
an additional ferromagnetic nearest-neighbor exchange interaction Jf = −2Unn when
both sites are singly-occupied. To estimate the Néel temperature of our system we
propose the following approximation: For the Hubbard model without Jf , we have
(H)
estimates of its Néel temperature TN (t, U ) from quantum Monte Carlo simulations
[76]. This Néel ordering is due to the antiferromagnetic superexchange interaction
Js = 4t2 /U between neighboring singly-occupied sites. When we include Jf < 0 this
reduces this magnetic interaction, and we will approximate the resulting reduction
of TN as being simply in proportion to the reduction of the total nearest-neighbor
exchange interaction:
Jf (H)
TN (V0 , as ) ∼
= (1 + )TN (t, U ) .
Js
(3.18)
In Fig. 3.7 we plot the ”Optimal TN ” line, which shows the lattice strength V0 /Er
that maximizes this approximation to TN for each value of the interaction as /d. The
highest TN occurs near as /d = 0.15, but the system at this value of as is may be too
close to the Feshbach resonance and thus not stable against formation of molecules.
The highest achievable TN thus may be somewhere along this line at a lower value of
44
TN at the optimal V
TN HErL
0.04
0.03
Optimal TN
0.02
U=14t
0.01
0.00
0.01
0.025
0.05
0.1
0.2
0.4
asa
Figure 3.8: Our estimates of the optimal Néel temperature, TN , as a function of as /d.
For each value of as , TN is maximized by varying the lattice depth V0 . We also plot
TN at the line U = 14t, which is near where the critical entropy is maximized [80].
as and thus a stronger lattice V0 . We note that a recent experiment [38] has studied
as /d ' 0.08 for 40 K, albeit at a temperature well above TN , without noting any strong
instability towards molecule formation. We also show on Fig. 3.8 the line along which
U = 14t, since this is near where the critical entropy S(TN ) is maximal [80], so if the
system can be adjusted adiabatically this is where the Néel phase is most accessible.
In Fig. 3.8 we show kB TN /Er as a function of as /d at the value of V0 that
maximizes our estimate of TN , as well as at the value of V0 that gives U = 14t and
thus is near the maximum of S(TN ). Note that in Fig. 3.8 the horizontal scale for
as /d is logarithmic, so TN drops rather weakly as as is reduced, meaning that the
possible limitation in how large as can be made will not “cost” a lot in terms of the
resulting reduction of TN .
Our approximations are clearly beginning to break down in the vicinity of the
parameter values that maximize TN . Thus, although we expect these approximations
45
to give reasonably reliable rough estimates of the maximal values of TN , there are
many higher-order effects that we are ignoring that may alter these estimates by a
little (our calculations suggest on the ∼ 10% level). At the maximum of TN , |Jf | is
about 25% of Js . The correction to Js due to tI is also of roughly this size, but its
dependence on as is much weaker, which is why Jf is the important actor in causing
the maximum in TN .
Figure 3.9: The strongest higher-order process contributing to the energy of the
antiferromagnetic Mott insulator at the maxima of TN shown in Fig. 3.7. It consists
of (1) a nearest-neighbor hop in the lowest (S) band, (2) an on-site “pair hopping” of
both fermions up to the next (P) band, (3) on-site pair hopping back to the S band,
and (4) a nearest-neighbor hop back to the original configuration. At both maxima
of TN , this process corrects Js by about 10%.
Table 3.1: The values of the various energies at the two TN maxima. The “Js correction” corresponds to the process detailed in fig. 3.9.
Global maximum
Maximum
of TN
of TN at U = 14t
0 (Er )
4.2
5.5
U0 (Er )
0.9
1.3
t0 (Er )
0.11
0.07
tI (Er )
0.02
0.02
0.08
0.03
Js (Er )
Jf (Er )
-0.02
-0.008
Js correction (Er )
0.007
0.004
The approximations we have used are those appropriate for the Mott insulator, and
are based on the inequalities on energy scales 0 > U > t, where 0 is the expectation
value of the single-particle energy in a lowest Wannier orbital. We have analyzed in
46
perturbation theory many corrections beyond those included above. We find that at
the maximum of TN (both the global maximum and the maximum along the U = 14t
line) the strongest next correction is the fourth-order process illustrated in Fig. 3.9; it
alters Js by about 10%. Since our perturbatively-based approximations are breaking
down near this regime of interest where TN is maximized, it would be nice to have
a more systematic approach that can obtain more precise and reliable estimates of
the phase diagram in this regime. For example, quantum Monte Carlo simulations
might be possible for temperatures near TN , although of course the famous fermionic
“minus signs” may prevent this from being feasible in the near term.
The ferromagnetic phase of this model at strong repulsion is mostly a band insulator, with a band gap between the spin-polarized bands. But in the weaker lattice
regime there should also be a partially-polarized Fermi liquid ground state near the
phase boundary to the Néel state. The transition from the fully-polarized band insulator to the partially-polarized ferromagnet occurs when the spin-polarized bands
overlap, so the system can lower its energy by flipping spins. A single spin flip makes a
hole and a doubly-occupied site that are each moving freely within the fully-polarized
background state. At the level of approximation we have used in this paper, the hole
moves freely with hopping t0 , so its lowest energy is −6t0 . The doubly-occupied state
costs interaction energy U0 +6Unn and moves freely with effective hopping t2 = t0 +2tI
because its motion is the hopping of the flipped spin between sites that are both occupied by unflipped spins. The total energy of this particle-hole pair can be negative
when U0 + 6Unn < 12(t0 + tI ) = 6(t0 + t2 ); this occurs below the line indicated in Fig.
1 as “FM to FI” (ferromagnetic metal to ferromagnetic insulator). We show this for
completeness, although these ferromagnetic phases at high as are very likely to be
inaccessible in experiments with cold fermionic atoms in optical lattices. Also, the
present approximations are probably not very reliable in this regime of large as /d.
47
There is also a paramagnetic Fermi liquid phase at weak enough lattice and at
weak enough interaction, which the strong lattice expansion cannot address. In the
next section, we will use consider a Hartree calculation, which allows us to address
the weak and intermediate lattice regimes.
3.4
Hartree approximation
In the Hartree-Fock approximation, one considers a mean-field decoupling of the
R
interaction term, i.e. one turns g drρ↑ (r)ρ↓ (r) into
g
Z
dr < ρ↑ (r) > ρ↓ (r) + g
Z
drρ↑ (r) < ρ↓ (r) > −g
Z
dr < ρ↑ (r) >< ρ↓ (r) > (3.19)
where g = (4πh̄2 )/m.
When considering the regularized contact potential in mean-field theory, there is
no exchange term, implying that the Hartree and Hartree-Fock approximations are
identical here. The total effective potential “seen” by the atoms with Sz = −σ in the
Hartree approximation is thus
(ef f )
V−σ (r)
4πh̄2 as
=
nσ (r) + V (r) ,
m
(3.20)
where nσ (r) is the number density of atoms with Sz = σ at position r, and −σ is the
spin opposite to σ.
For each point in the ground-state phase diagram, specified by the lattice intensity
V0 and the interaction as , we solve the Hartree equations numerically by discretizing
them in momentum space, and iteratively achieving self-consistency.
We obtain up to 3 different low energy self-consistent Hartree many-body states
at density one atom per lattice site, and determine which state is of the lowest energy
and thus is the Hartree ground state. In the paramagnetic state we impose the
48
constraints n↑ (r) = n↓ (r) = n↑ (r + ~δ) at all r, where ~δ is any nearest neighbor vector
of the simple cubic lattice. This paramagnetic Hartree state exists for all values of as
and V0 , since this constraint is preserved by the iterations. In this state the lowest
Hartree bands are always partially occupied, so there are Fermi surfaces.
The approach to solving the Hartree equations is as follows: choose a filling, i.e.
a number of particles per site. One first picks a discretization of momentum space
k = (nx N2πd , ny N2πd , nz N2πd ) with ni ∈ Z, and N some even integer. One then starts with
an initial guess for n↑ (r). The Fourier transformed density will only have momenta in
the reciprocal lattice of the simple cubic lattice, which is simple cubic as well. This
implies that the eigenstates of the Hamiltonian
Hψknσ (r) = (−
∇2
4πh̄2 as
+ V (r) +
n−σ (r))ψknσ (r),
2m
m
(3.21)
can be labelled by a k ∈ F BZ, and a Hartree band index n. The FBZ is {k =
(nx N2πd , ny N2πd , nz N2πd ): ni ∈ [0, N − 1]}.
Symmetries speed up the calculation tremendously. There are 48 symmetry elements in the symmetry group of a cube, called the orthogonal group. All elements of
the FBZ that are related by a symmetry give the same bloch states. Thus we need only
consider elements of the reduced FBZ : {k = (nx 2π/(N d), ny 2π/(N d), nz 2π/(N d):
0 ≤ nx ≤ ny ≤ nz }. Each element of the reduced FBZ is assigned a weight, corresponding to how many times it appears in the FBZ when all the symmetries are
applied to it. For example, (0, 0, 0) has weight 1, while (2π/(N d), 0, 0) has weight 6,
etc. Once we have the Hartree states, we occupy the lowest energy states until we
have the right filling. The density is then calculated:
n↑ (r) =
(Weight(i))|ψi↑ (r)|2 .
X
i occupied
49
(3.22)
where i runs over all states in the reduced FBZ. Once again, this is all done in momentum space (using Fast Fourier transforms). The Hartree states are then recalculated,
the density obtained from occupying the states that lead to the right filling is fed
back into the calculation, and the process is repeated until convergence is attained.
For the Néel antiferromagnetic state we impose only the constraints n↑ (~r) = n↓ (~r +
~δ), thus permitting two-sublattice Néel order. As the unit cell is doubled, the FBZ is
halved, as the density can now have a Fourier component ( Nπd , Nπd , Nπd ). Apart from
that, the method is the same.
At weak enough lattice and interactions, the only self-consistent Hartree state is
in fact a paramagnetic state, i.e. it has no magnetic order. When an ordered Néel
state exists, it can be either Mott insulating, with the lowest Hartree bands full and a
Mott-Hubbard gap to the next bands, or the two lowest Hartree bands for each species
can overlap in energy and each be partially occupied, with Fermi surfaces. We also
look at ferromagnetic states that have different densities of ↑ and ↓ atoms but do not
break the discrete translational symmetries of the lattice. Again, there is a portion
of the phase diagram where there is no self-consistent ferromagnetic state. And when
there is such a state, it can be either a band insulator or have Fermi surface(s), and it
can be either fully or partially spin-polarized. In the latter case (PPF in Fig. 1) there
are 3 partially occupied Hartree bands. In these candidate Hartree ground states, the
fermions occupy only the lowest Hartree bands, but in terms of free noninteracting
fermions, we have included states extending out to many Brillouin zones and thus
the Hartree states are admixtures of multiple bands of the noninteracting system.
Specifically, for the parameters in Fig. 1, a 20 × 20 × 20 grid of momentum points in
each Brillouin zone and a 9 × 9 × 9 grid of zones was enough to achieve convergence
everywhere.
The resulting ground-state Hartree phase diagram of this system is presented in
Fig. 3.10. At any lattice strength, the paramagnetic Fermi liquid phase exists at
50
Figure 3.10: Ground-state phase diagram for filling one fermion per lattice site in the
Hartree approximation. The phases shown are the antiferromagnetic Mott insulator
(AFI); paramagnetic (P), antiferromagnetic (AFM) and partly- (PPF) and fullypolarized (FM) ferromagnetic Fermi liquids; and the ferromagnetic band insulator
(FI). The solid line marked Jsmax indicates where the Hartree estimate of the effective
exchange interaction J as a function of the lattice strength V0 is maximized for each
interaction as , and the dashed line near it shows where our estimate of the Néel
temperature TN gets maximized under the same prescription (see text). The dashdotted U = 14t line is near where the DMFT estimate of the entropy is maximized at
the Néel ordering temperature TN [80]. The square on each line denotes the overall
maximum of J or TN along that line. The lattice intensity V0 is given in units of the
recoil energy Er = (πh̄)2 /(2md2 ).
51
weak enough interaction. At strong interaction, the Hartree approximation always
produces a ferromagnetic ground state, due to the classic Stoner instability. The
ferromagnetic phase is a band insulator for V0 > 2.2Er , with a band gap above the
filled lowest fully spin-polarized band. For V0 < 2.2Er the lowest two bands for
the majority-spin atoms overlap and there is instead a ferromagnetic Fermi liquid.
We show this ferromagnetic part of the phase diagram for completeness, but it is
important to emphasize both that the Hartree approximation is not to be trusted
at such strong interactions, and that systems of ultracold fermionic atoms are likely
to be highly unstable to Feshbach molecule formation when brought this close to
the Feshbach resonance. Thus we do not expect such a ferromagnetic phase to be
experimentally accessible for these systems, even if it does exist in a model that
ignores the instability towards Feshbach molecules. Finally, for lattices above a certain
minimum strength there is a Néel-ordered ground state at intermediate values of the
repulsive interaction. This Néel state is a Mott insulator over most of the phase
diagram, and becomes an antiferromagnetic Fermi liquid over a small sliver of the
phase diagram at weak lattice and weak interaction between the paramagnetic and
Mott insulating phases. It is in this lower (smaller as ) portion of the phase diagram
where we believe the Hartree approximation is qualitatively correct, producing the
paramagnetic and antiferromagnetic Fermi liquid phases and the Mott insulating Néel
phase.
To quantify the energy scale associated with Néel ordering in the antiferromagnetic
phase, we (crudely) estimate the nearest-neighbor antiferromagnetic exchange J by
taking J/2 to be the energy difference per bond between the Hartree antiferromagnet
and ferromagnet (at low as and/or V0 , the ferromagnet has zero magnetization, so is
really the paramagnet). For each given interaction as , we locate the lattice intensity
V0 where this Hartree estimate of J has its maximum, and indicate these maxima
by the red (full) line marked Jsmax in Fig. 3.10. It is somewhere near this line
52
Figure 3.11: Plot of the Hartree estimate of the antiferromagnetic exchange coupling,
J, as a function of V0 for as /d = 0.08, compared with the estimate from the perturbative expansion at strong lattice. The red (full) line shows the Hartree estimate of
J, while the blue (dotted) line gives the perturbative estimate, discussed in the text.
They match very well at high lattice depth V0 .
that antiferromagnetic interactions are strongest, and the Néel phase survives to the
highest temperature. The overall global maximum in this estimate of J occurs at
as /d ∼
= 0.15 and V0 ∼
= 3Er , where Er = (πh̄)2 /(2md2 ) is the “recoil energy”. This
point is marked with a small square on the full red line in Fig. 3.10.
As an example, in Fig. 3.11 we show our Hartree estimate of J vs. V0 at as /d =
0.08, which is the largest interaction explored experimentally in the study of ref. [38],
although these experiments only reached temperatures far above those of the Néel
phase. In Fig. 3.11 we also plot the estimate of J from a strong lattice expansion
which we discussed in the previous section, and it coincides well with the Hartree
estimate for V0 > 5Er .
The line in Fig. 3.10 where the Hartree estimate of J is maximal not only indicates
roughly where TN is maximized, but it also occurs at the location of a crossover in the
nature of the antiferromagnetic state. In the regime of larger V0 and as , the system is a
“local moment” magnet: almost every well of the optical lattice is occupied by a single
53
atom with a “spin”, with very few wells either empty or multiply-occupied. Here we
can make a Hartree ferromagnet as well as an antiferromagnet and compare their
energies to obtain our estimate of J. In fact we could make many other metastable
magnetic Hartree states with ordering at essentially any momentum in the Brillouin
zone. Here J is primarily a superexchange interaction J ∼ t2 /U , and it increases as
the lattice depth V0 is reduced (see Fig. 2), since this increases the hopping t between
wells and reduces the interaction U between two unlike atoms in the same well.
In contrast, the regime of the antiferromagnetic phase that is at lower V0 and as
near the paramagnet is not a local-moment regime, but instead is a spin density wave
(SDW). Here one can make a magnetically-ordered Hartree state only at momenta
near those that cause substantial Fermi-surface nesting, which is initially only near
the corners of the first Brillouin zone. In this regime we do not have a ferromagnetic
Hartree state and our estimate of J is obtained by subtracting the energy of the Néel
state from the paramagnet, so this J should not be interpreted simply as a spin-spin
interaction. We now understand why this estimate of J increases with increasing V0
(see Fig. 3.11), since this increases the interaction that causes the magnetic ordering,
and decreases the hopping that favors the paramagnetic Fermi liquid. The effective
J is maximized at the crossover between the SDW and local moment regimes. This
crossover shows up in the metastable ferromagnetic Hartree state as two closelyspaced phase transitions between un-, partially-, and fully-polarized.
3.5
Experimental consequences
We are now in a position to discuss the experimental consequences of the calculations
above. The assumption usually found in the literature is that experiments will be
able to evolve the system adiabatically, and that the natural variable is therefore the
entropy, instead of the temperature. We would like to point out some caveats to this.
54
The system must be cooled to low temperature T and thus low entropy S, and
equilibrated at the point in the phase diagram that is being measured. This cooling
may be done under some other conditions (e.g., with the lattice turned off and other
parameters optimized for cooling), with the “pre-cooled” system then moved adiabatically to the conditions of interest for measurement [35, 80, 18, 40, 75]. For this
to work, the time scales of the system must be such that this can be done without
strongly violating adiabaticity. For the Néel phase of the Mott insulator, this will limit
how small the hopping energy t and the antiferromagnetic superexchange interaction
J can be, since the system must be able to remain near equilibrium, adiabatically
rearranging the atoms so that there is one atom per lattice site and these atoms are
antiferromagnetically correlated. Alternatively, the system might be actively cooled
under the conditions of measurement, but this again requires the system to be able
to equilibrate under those conditions. Thus, either way, this constraint limits equilibrium access to the strong-lattice portion of the phase diagram, where the exchange J
and/or hopping t are too slow to allow equilibration and adiabaticity. Thus the Néel
phase is going to be most accessible to experiment in some regime of intermediate
lattice strength V0 . These considerations may also limit how large the interaction can
be made, since strong repulsion as suppresses the superexchange rates and thus can
limit spin equilibration in the Mott insulating phase.
Another constraining issue is that as one approaches the Feshbach resonance from
the repulsive side, two atoms will interact repulsively only as long as they scatter in a
way that is orthogonal to the molecular bound states. Thus the atoms must remain
metastable against forming Feshbach molecules. In the absence of the lattice, the rate
of molecule formation grows as ∼ a6s as the Feshbach resonance is approached [64].
This will limit how large the interaction as can be made, in a way that is presumably
less of a constraint as the lattice is made stronger so that it keeps the atoms apart.
55
As we mentioned earlier, the DMFT estimate of the maximum of S(TN ) is near
U = 14t [80]. In fact, the maximum in S(TN ) is rather weak, being only a little
higher than that of the Heisenberg limit U t. Thus one might more usefully say
that the critical entropy is nearly maximized for any U > 12t. As we can infer from
Fig. 1, U > 12t occurs at relatively large as and V0 , and the issues raised above may
force experiments away from the maximal entropy line, and towards the line where
the exchange interactions and TN are maximized.
So far we have assumed a homogeneous system, but the trapping potential is
actually nonuniform, which leads to inhomogeneity of the local equilibrium state in
the trap. The spatial size of the region occupied by the antiferromagnetic Mott phase
will increase with increasing U/t, since U increases the Mott “charge” gap. This effect
means larger U/t should favor detection of the Néel phase; of course the optimal U/t
will be some compromise between this and the other issues discussed above. Using
our Hartree calculation to obtain the charge gap in the Mott phase, we find within
the local-density approximation (LDA) that at the point in the phase diagram where
J is maximized, the Mott phase should occupy about half of the linear size of the
trap. This is encouraging, but the Hartree approximation likely overestimates the
Mott gap, as the true gap should be renormalized downwards by spin fluctuations.
Finally, we come to novel experimental predictions that result from our calculations. The Hartree calculation predicts that there is interesting and as yet unexplored
physics at low to intermediate lattice strength, and weak coupling. We mentioned
already that the effective model at around optimal values of V0 and as is that of a
Hubbard model with ferromagnetic correlations, which is interesting in its own right.
The different phase boundaries that we uncovered may also be worth exploring. For
example, the quantum phase transitions between the Mott and metallic Néel phases
and the paramagnet occur in parameter regimes that are quite accessible to the exper-
56
iments, although it may not be possible to see their effects at accessible temperatures,
since TN decreases strongly as this weak-coupling regime is approached.
3.6
Conclusion
We have shown that to maximize antiferromagnetic interactions for fermionic atoms in
an optical lattice one must explore the regime of intermediate lattice depths, where
the system has significant deviations from the standard one-band Hubbard model.
We have found that the nearest-neighbor direct ferromagnetic exchange is the most
important correction to the Hubbard model that limits the maximal exchange J, and
therefore the maximal Néel temperature TN .
There are also higher-order corrections to the Hubbard model: virtual hopping
into higher bands and other higher-order processes. The relative contribution of the
higher-order corrections in the vicinity of the optimal J drops exponentially as one
goes to smaller interaction as and thus a larger V0 . Thus our perturbative calculation
should yield accurate results in the large V0 (strong lattice) regime.
We included a subset of all higher-order corrections via a Hartree calculation, and
used it to find an estimate of the line where the exchange J is maximized. This line
coincides well with the line where our estimate of TN is maximized, obtained by using
quantum Monte Carlo results and the strong lattice expansion.
For quantitatively more accurate results in the intermediate lattice depth regime,
one needs to resort to more systematic quantum calculations. Of course this is a
system of many fermions, so it is not clear whether this regime can be accurately
treated in some form of quantum Monte Carlo simulations.
Experiments on these systems still need to reduce the temperature by a substantial
factor before they are able to access magnetically ordered phases of fermions in optical
lattices. Once they bridge this gap, our results above suggest that the Néel phase
57
will be most accessible in the intermediate lattice strength regime. If that is the case,
then this “quantum simulator” should be able teach us about more than just the
standard one-band Hubbard model.
58
Chapter 4
Polarons, molecules and trimers in
strongly polarized Fermi gases
In this chapter, we will study mass imbalanced two-component Fermi gases interacting via a contact interaction, in the strongly polarized limit. Motivated by recent
experimental [71] and theoretical work [56, 14, 15, 13, 56, 70] in the mass balanced
case, we consider the limit where there is a single minority atom interacting with a
Fermi sea of majority spins. This limit is in itself very rich, and one can do calculations which have been shown in previous works to match onto Quantum Monte Carlo
(QMC) calculations [69]. The phase diagram in this limit is not only relevant for
two-component fermi gases: the single minority atom could also be a boson. Thus we
learn valuable information about a variety of systems, and we will discuss in depth
how to interpret the results.
The limit we are considering is directly related to the Cooper problem [16], which
was a crucial step towards understanding low-temperature superconductors. In the
Cooper problem, one considers an up and a down spin interacting with an attractive
potential, in the presence of a Fermi sea (FS) for the up spins, and a FS for the
down spins. The interaction only acts on momenta close to the FS. It was shown
59
that an arbitrarily weak attraction leads to the formation of a bound state, and the
formation of this bound state signals the instability towards superconductivity (this is
known as the Thouless criterion). In this chapter, we are considering generalizations
of and improvements on the Cooper problem: indeed, in the Cooper problem one
assumes that the interactions do no generate particle-hole pairs, which means that
one underestimates the binding energy. We will be including particle-hole pairs, thus
dressing our wave functions, to the extent that it is numerically feasible. Also we will
be considering other possible bound states above a Fermi sea: instead of having an
up and a down spin forming a bound state, we will be looking at the possiblity of a
single down spin binding to an up spin FS, which has been called a polaron in the
literature; a down spin binding to two up spins, also known as a trimer; a down spin
binding to three up spins to form a tetramer (although we haven’t found a region of
the phase diagram where the tetramer is the ground state). In the Cooper problem,
the actual interactions are Coulomb interactions, while in our case we are dealing
with ultracold fermions interacting via a contact interaction. The main difference
with Coulomb interactions is that particles in the same Fermi sea do not interact
because of Pauli exclusion.
To understand how our results fit into the existing paradigm on polarized Fermi
gases, we first briefly discuss the phase diagram of mass imbalanced polarized twocomponent Fermi gases interacting via a Feshbach resonance [61]. We use a mean
field theory, which more advanced methods have shown gives the right qualitative
picture. We then describe the variational wavefunctions used to analyze the strongly
polarized limit, and get the phase diagram to lowest order. One very attractive aspect
of the approach is that it can be systematically improved by broadening the class of
variational wave functions, and in doing so we obtain a refined phase diagram. Finally
we detail the physics behind the phase transitions that the wavefunctions predict, and
how they are related to the phase diagram at intermediate polarization.
60
4.1
Mean field theory of the imbalanced fermi gas
Let us first briefly discuss the phase diagram of spin polarized two-component fermions
interacting via a contact interaction. We use a mean field prescription [74, 62, 1, 60].
All calculations to date agree with the qualitative picture obtained from mean field
theory, though the lines may be strongly shifted from their mean field values, as QMC
calculations have shown [3, 11].
The Hamiltonian is
H=
X
g
k,σ c†k,σ ck,σ +
V
X
k,k’,Q
c†k+Q↑ c†k’−Q↓ ck’↓ ck↑ ,
(4.1)
k2 , m being the mass of each species. We define a mass ratio r =
where k,σ = 2m
σ
σ
m↑ /m↓ . We will work in units where m↓ = 1. Here we have assumed a single channel
model, which is valid for broad Feshbach resonances. The momenta have a cutoff
Λ, which is sent to zero as g is sent to zero, according to a standard regularization
prescription (see Appendix A for details). This prescription gives results that are
independent of the microscopic details of the two-body interactions, and only depend
on the scattering length. In that sense they are universal : one can use any two
species of fermions as long as the interaction is short range.
In BCS mean field theory, one works in the grand canonical ensemble and assumes
two types of operators have non trivial expectation value in the ground state: the
pairing operator and the density operator.
< c†k↑ ck’↓ >= δk,k’ nk + δn̂k
(4.2)
< ck↓ ck’↑ >= δk,−k’ bk + δ b̂k ,
(4.3)
61
where nk and bk are numbers, indicating expectations values in the ground state.
δn̂k and δ b̂k are operators which capture the fluctuations around the mean field
values. One then considers them to be small, in the sense that one drops any product
of two such as operators, such as δn̂k δn̂k0 . The resulting Hamiltonian no longer
conserves particle number, thus one works in the grand canonical ensemble, replacing
the Hamiltonian with HM F − µ↑ N↑ − µ↓ N↓ .
The gap parameter ∆ is defined as ∆ =
g
V
P
k bk . The mean field Hamiltonian
becomes
HM F =
X
k,σ
(k,σ −µσ )c†kσ ckσ +∆∗
where µσ = µσ −
g
V
X
c−k↓ ck↑ +∆
k
X †
k
ck↑ c†−k↓ −
V
g
N↑ N↓ − ∆2 , (4.4)
V
g
N−σ are the chemical potentials of the two species, renormalized
by the Hartree terms. Nσ =
†
k < ckσ ckσ > is the total number of particles with
P
spin σ. We call the thermodynamic potential ΩM F (∆) =< HM F (∆) >.
For this contact potential, the Hartree terms are zero. We should have expected
that : the Hartree terms look like g(Λ) < Nσ > N̂−σ , and since as Λ → ∞, g(Λ) → 0,
while < nσ > is finite, this term is zero. Therefore µσ = µσ , and gN↑ N↓ = 0. It turns
out that this doesn’t apply for the gap parameter : ∆ = g
P
~k
< c~k↑ c−~k↓ > remains
finite, as the k sum diverges in just the right way as Λ → 0 to get a finite result.
From now on we assume that ∆ is real.
We define µ = 12 (µ↑ + µ↓ ), h = 21 (µ↑ − µ↓ ). µ acts as an average chemical potential,
and h as a magnetic field.
The Hamiltonian is quadratic, and can therefore be diagonalized. The operators
that diagonalize HM F are called the quasiparticle operators, which are hybridizations
of particle of one species and hole of the other species. The operators are
†
γk
= uk c†k↑ + vk c−k↓
1
62
(4.5)
†
γk
= vk ck↑ − uk c†−k↓
2
Defining Ek =
q
(4.6)
2k + ∆2 , k = 12 (k↑ + k↓ ) − µ, hk = 12 (k↑ − k↓ ) − h, we find
that
uk = q
Ek + k
∆
, vk = q
.
2
2
2
∆ + (Ek + k )
∆ + (Ek + k )2
(4.7)
The energies of the two quasiparticle bands are respectively
2
1
= Ek − hk .
= Ek + hk , ξk
ξk
(4.8)
1
2
If for all k, ξk
> 0 and ξk
> 0, we have a balanced superfluid. More generally,
for ∆ > 0, at most one of the bands can have zero energy states. The region of
1
2
momentum space where ξk
and ξk
have opposite signs is fully polarized, we will call
it the Pauli blocked (PB) region. If the PB region exists and ∆ > 0, then we have
an imbalanced superfluid, also known as the Sarma phase (SFM ). The boundaries of
the PB region are Fermi surfaces, with gapless excitations. Elsewhere in momentum
space, the Sarma phase has gapped quasiparticles, analogous to BCS superconductors.
It was shown previously that mean field theory actually predicts that the Sarma phase
always has only one Fermi surface, i.e. that the so-called ”breached-pair” Sarma phase
with two Fermi Surfaces is never thermodynamically stable [61].
In terms of these operators, the mean-field Hamiltonian becomes
HM F =
†
(−Ek + hk )γk
γ +
1 k1
†
(Ek + hk )γk2 γk
+
2
V
(k↓ − µ↓ ) − ∆2 . (4.9)
g
X
X
X
k
k
k
The ground state becomes
|GS >=
Y
k’∈P B
c†k’↑
Y
k∈P
/ B
63
γk1 γk2 |vaci ,
(4.10)
where |vaci is the ground state with respect to the fermionic operators. The thermodynamic potential is
ΩM F
V
∆2
1 X
=
(−E~k + k +
)
V
k↑ + k↓
k∈P
/ B
1 X
∆2
mr ∆2
+
(hk sign(µ↑ − µ↓ /r) + k +
)−
.
V
k↑ + k↓
4πas
k∈P B
(4.11)
We call k1 and k2 the ”Fermi momenta”, i.e. the momenta bounding the PB
region (if there is no bounding region, they are zero):
k2/1 = θ((1 + 1/r)µ + (1 − 1/r)h(+/−)|µ(1/r − 1) − h(1 + 1/r)|)
q
× (1 + 1/r)µ + (1 − 1/r)h(+/−)|µ(1/r − 1) − h(1 + 1/r)| (4.12)
The number equations are
nk↑ = θ(k1 − k)u2k + θ(k2 − k)θ(k − k1 )θ(rµ↑ − µ↓ ) + θ(k − k2 )u2k
(4.13)
nk↓ = θ(k1 − k)u2k + θ(k2 − k)θ(k − k1 )θ(−rµ↑ + µ↓ ) + θ(k − k2 )u2k .(4.14)
One can rescale µ, i.e. use the fact that
ΩM F (as , h, µ, ∆) = |µ|5/2 ΩM F (as|µ|1/2 , h/|µ|, sign(µ), ∆|µ|).
(4.15)
Thus we only need to vary one chemical potential, h, and consider three cases: µ =
1, −1, 0 in order to get the phase diagram.
Now the optimized thermodynamic potential Ωmin
M F =< HM F > (∆min ) is found
by picking the value ∆min of the gap parameter that minimizes the mean-field energy.
ΩM F (∆) has at most two minima, one of which is at zero. We therefore have three
64
r=10
P
1.0
F S
T
Unstable
0.8
0.6
0.4
N
0.2
0.0
SFM
Metastable
0
5
10
15
20
1Hk f asL
Figure 4.1: Mean field zero temperature phase diagram of mass imbalanced spin
polarized two component fermions, as a function of 1/(kF as ) and polarization P =
(N↑ −N↓ )/(N↑ +N↓ ), with mass ratio m↑ /m↓ = 10. SFM is the magnetized superfluid,
or Sarma, phase, with one Fermi Surface ; N is the normal phase. In the unstable
regions, there is no thermodynamically stable phases, and the dashdotted lines are
the tie curves, which connect the phases that the system separates into if it starts
somewhere along the curves. The metastable regions are bordered by the spinodal
lines. There are three important points along the P=1 line, from left to right : F is
the point where the first order line touches the P=1 line, S is the point where the
spinodal line touches the P=1 line, and T is the tricritical point.
thermodynamically stable phases :
Normal phase (N) : ∆min = 0
(4.16)
Balanced superfluid (SF ) : ∆min 6= 0, ξ~k± 6= 0∀~k
(4.17)
Sarma phase with 1FS (SFM ) : ∆min 6= 0, k2 > 0
(4.18)
In mean field, the thermodynamic potential density ΩM F /V is in fact the pressure,
since by standard thermodynamic arguments (E − T S − µN )/V = p.
65
nk
EΕkF
1.0
5
0
Ξk2
k2
0.5
1.0
Εk -
1.5
Μ­
0.8
2.0
-5
-Εk + Μ¯
-10
k
nk
PB
0.6
0.4
nk
0.2
-Ξk1
0.5
1.0 k 1.5
2
2.0
2.5
3.0
k
(a) An example of the dispersions in the (b) The momentum distribution in the
Sarma phase
Sarma phase
In the mean field diagram, the P = 0 line is a crossover line : ∆min 6= 0 all along
this line, where the system system goes from being a condensate of tightly bound
bosons, in the limit 1/(kF as ) → ∞ also known as the BEC limit, to a superfluid
with weakly bound Cooper pairs, in the limit 1/(kF as ) → −∞ also known as the BCS
limit. In passing, it goes through the unitary limit, where 1/(kF as ) = 0, and there
is no simple perturbative understanding of the physics. It is also called a crossover
superfluid, where the size of the Cooper pairs is of the order of the interparticle
spacing.
The key to understanding the phase diagram once we move away from P = 0 is
to start at the tricritical point T. T is defined as the point where dΩM F /d(∆2 )|∆=0 =
d2 ΩM F /d(∆2 )2 |∆=0 = 0. To its right, the horizontal line it connects to is in fact a
second order phase transition line. Thus to the right of T , the SFM phase survives
all the way up to P = 1, where the phase a fully polarized Free Fermi Gas. The
transition is second order because d2 ΩM F /d(∆2 )2 |∆=0 > 0.
T is the point where first order physics first appears in the SFM region at P = 1, as
one goes from the BEC limit towards unitarity. We then obtain the phase separating
regions by finding values of as , µ and h where ΩM F (∆) has two degenerate minima,
i.e. ΩM F (0) = ΩM F (∆min ) for some nonzero value of ∆. The two phases one thus
obtains, i.e. the one with ∆ = 0 and the one with ∆ = ∆min , have equal chemical
potential and pressures (since ΩM F = pV ). Therefore they can coexist. In the phase
66
diagram, we connect these points with so-called tie curves [42] : if one starts the
system somewhere along the tie curve, it will phase separate into the two phases.
The calculation of the tie curves is a standard calculation in thermodynamics[43] : if
the axes of the phase diagram were (n↑ a3s , n↓ a3s ), the tie curve would be a straight line
joining the two phases. Thus we simply translate that line to (1/(kF as ), P ) space .
There are two distinct regions in the phase separating region : the unstable and
the metastable regions. The lines that bound the unstable region are called the
spinodal lines. In the metastable region, there is a nonzero value of ∆ such that
dΩM F /d(∆2 )|∆ = 0. If one starts with no interactions outside the spinodal region, and
rapidly turns on the interactions, the system will find itself in a metastable minimum,
and therefore have to nucleate the phase corresponding to the real thermodynamic
minimum. In the unstable region, however, there is no metastable minimum and the
system will evolve via a linear instability [42].
In this chapter we will be doing calculations along the P = 1 line. From our
previous discussion, we see that there are three special points in the phase diagram
along the P = 1 line : T is the tricritical point ; F is the point where the phase
separation line hits the P = 1 line ; and S is the point where the spinodal line hits
the P = 1 line.
All calculations to date on mass balanced polarized Fermi gases agree with the
qualitative picture obtained from the mean field theory in the single channel model,
though the positions of the lines are strongly renormalized.
When the system is unpolarized, one can in fact improve on mean field theory by
including fluctuations in the Cooper pair condensate [63], i.e. one sums the diagrams
where a single up and down spin scatter off each other an arbitrary number of times.
This is known as the T matrix approximation. Mean field plus these corrections
gives results which are very close to values obtained from QMC, for all values of
1/(kF as ). However, once one allows for polarization and/or mass imbalance, the T
67
matrix approximation is expected break down. Indeed, polarization is relevant in an
RG sense, for dimensions larger than 1. The physical reason why the T matrix works
for the mass balanced case is that two particles close to the Fermi sea have several
restricted scattering possibilities, due to the Fermi seas blocking the phase space.
Thus one can ignore the possibility of a up spin scattering with several down spins,
and consider its scattering with a single down spin. However spin polarization opens
up the phase space and a new set of diagrams become relevant. Mass imbalance has
also been shown to make a new set of diagrams relevant : it is again a question of
phase space, because if the Fermi surfaces match in momentum, they do not match
in energy, once there is mass imbalance.
When the system is unpolarized, one can do QMC calculations without a sign
problem [9], for the same reason that the Hubbard model at half filling has no sign
problem: one can write the QMC claculation in a way that involves sampling the
product of a determinant coming from the up spins, and one from the down spins. In
the unpolarized case these two determinants are equal and one is sampling a positive
function. Once the system is polarized, the function can go negative, leading to a sign
problem. Thus people have resorted to variational QMC calculations, coupled with
fixed node diffusion monte carlo (DMC) [66]. These calculations are not as reliable,
because one strongly biases them by choosing a variational wave function.
Thus an important aspect in this field is to find methods that are quantitatively
reliable, especially if one wants to compare with experiments. In this chapter, we will
be doing exactly that, using a method that was developed and explored only recently,
and has led to a flurry of activity [15, 13, 56, 14, 70]. Since we find interesting results
bearing on the FFLO phase, we will first briefly discuss this phase.
68
4.2
FF, LO and FFLO
Since the FFLO phase plays an important role in this chapter, let us introduce it
now. The original FFLO phase was discussed in the context of superconductors
in a magnetic field, which corresponds to imbalanced fermi gases in our context.
Instead of phase separating the excess fermions and forming a balanced superfluid
(or superconductor), another possibility is to introduce a condensate with nonzero
momentum Q. Through interactions with this condensate, fermions |k ↑> and | −
k+Q ↓> can now pair. This allows the fermions on the Fermi seas of different sizes to
exploit the phase space, which is what leads to the Cooper problem and BCS theory
in the balanced case.
There are a couple of possibilities : the condensate can break time reversal symmetry (i.e. conjugation) , and maintain spatial symmetry. This is called the FF
phase, after the original proposal of Fulde and Ferrell [28]:
∆(r) = ∆Q eiQ·r .
(4.19)
Note that the gap varies in space, but it is not a gauge invariant quantity : |∆2 |
is homogeneous.
Another possibility is to maintain time reversal symmetry, but break spatial symmetry, thus obtaining the FFLO phase, after Larkin and Ovchinnikov [44] :
∆(r) = ∆Q sin(Q · r).
(4.20)
One can think of the FFLO phase as the formation of standing waves by superimposing two F F phases at momenta Q and −Q. One of the difficulties in studying the
F F LO phase is that there are an infinity of possibilities : ∆(r) =
P
i Qi · r
Qi ∆Qi e
, for
any set of momenta Qi . It is not even settled within mean field which set of momenta
give the lowest energy state. Adding momenta in a clever arrangement leads to small
69
improvements in the energy, thus one finds a manifold of metastable states. So far
it has been shown that in fermi gases the simplest LO phase given above is favored
[37]. In contrast, in QCD plasmas the face-centered cubic FFLO lattice was shown
to have the lowest energy [7]. Whether these are the best set of momenta, or how
fluctuations affect these calculations is unknown.
In ultracold atoms, the FFLO phase has been shown, for the mass balanced case,
to occupy a very thin shell around part of the line between the N phase and the
unstable region (see [74] for an extensive review), which has made the community
skeptical about being able to measure this phase in Fermi gases. The size of the
FFLO region, however, is unknown and its estimates depend on the optimism of the
author.
Several proposals have been put forward to increase the size of the FFLO region.
One idea was to make the system effectively one dimensional, by turning it into a
array of one-dimensional tubes using a two-dimensional optical lattice. To understand
why this should favor FFLO, an understanding of the physics behind FFLO is of the
order. The mean field picture of FFLO is exactly the same as the mean field theory
discussed above for imbalanced Fermi gases, except that now the hybridization is
between majority particles at momentum k and minority particles at momentum
−k + Q. If Q = kF ↑ − kF ↓ , then the condensate allows particles of the majority atom
Fermi sea to pair up with particles of the minority atom Fermi sea. Indeed, in the
BCS picture the wavefunction is a condensate of Cooper pairs with finite center-ofmass momentum Q. The reason the simplest, FF version of this scenario occupies
such a small sliver of the phase diagram at mean field level is that Q has to point
in a certain direction, only allowing a small part of the two Fermi surfaces to profit
from pairing. The rest is Pauli blocked. If, on the other hand, the Fermi surfaces
were nested, as they would approximately be in a quasi one-dimensional system, then
pairing would be enhanced.
70
In this chapter, we look for another scenario that might enhance the FFLO phase :
the introduction of mass imbalance. Little is known about the effect of mass imbalance
on the FFLO phase. We will study it in the strongly polarized limit, close to P = 1,
where there is a very clear picture of why mass imbalance might enhance FFLO.
However, as we shall see, there is a trimer phase which may supercede FFLO.
4.3
Bare polaron, molecule and trimer
We will be adopting a variational approach, in the limit of a single minority spin on top
of a Fermi sea of majority spins. In general consider an unnormalized wavefunction
|Ψi that depends on a set of variational parameters αi . We minimize the energy
E=
hΨ|H|Ψi
hΨ|Ψi
with respect to these parameters :
∂
∂
∂
E=0⇒
hΨ|H|Ψi = E
hΨ|Ψi.
∂αi
∂αi
∂αi
(4.21)
In this single impurity limit, we find a phase transition as one varies 1/(kf as ),
which has an interesting relation to the transition from N to SFM [70, 14]. On the
N side, the ground state with a single impurity is a down spin dressed by particlehole excitations on the Fermi sea, which has been called a polaron in the literature.
On the SFM side, the impurity will bind to an up spin out of the Fermi sea and
make a molecule (also called a dimer). As we cross from the SFM to the N side, the
molecule will unbind : it will emit an up spin at the Fermi sea, leaving a polaron
behind. However, there is an issue of momentum conservation : if the molecule
and the polaron are at momentum zero, then the molecule cannot simply unbind
into a polaron at zero momentum, plus an up spin at the Fermi surface which has
momentum kF . However, the Fermi sea takes care of this: one can create a particlehole pair at arbitrarily small energy, thus absorbing the momentum. This means that
the molecule unbinds into four particles: a polaron, two up particles and one up hole.
71
We will label our eigenstates with a subindex describing the maximum number
of creation and/or desctruction operators we let act on the Fermi sea to create the
eigenstate, and with the total momentum Q. The simplest wave function for the
polaron, the bare polaron, is
|P1 (Q)i = c†Q↓ |F SiN ,
(4.22)
where |F SiN is a Fermi sea of up spins with N particles. The energy of the bare
polaron (with the energy of the fermi sea substracted) is simply E =
Q2
.
2m↓
The wave function for the bare molecule is
|M2 (Q)i =
X (Q) †
k
γk cQ−k↓ c†k↑ |F SiN −1 .
(4.23)
Using the anticommutation relations of fermionic operators : {ckσ , c†k’σ0 } =
δk,k’ δσ,σ0 , {ckσ , ck’σ0 } = 0, we obtain, using the Hamiltonian given in Eq.(4.1),
< M2 (Q)|H|M2 (Q) > =
X
k
g
( Q)
|γk |2 (Q−k↓ + k↑ + (N − 1) + EF S )
V
g X (Q)∗ (Q)
γk γk’
V
k,k’
X ( Q)
< M2 (Q)|M2 (Q) > =
|γk |2
k
+
The variational equations Eq.(4.21) give :
∂
∂
< M2 (Q)|H|M2 (Q) >= E
< M2 (Q)|M2 (Q) >
(Q)∗
(Q)∗
∂γk
∂γk
g
g X (Q)
(Q)
⇒ (E − Q−k↓ − k↑ − (N − 1))γk =
γk’ ,
V
V
k’
where E = E − EF S .
72
(4.24)
Now consider the limit of g → 0. This means the Hartree term
g
V
(N − 1) becomes
zero. Indeed, as we saw in the mean field calculation, the Hartree term is zero for
P
(Q)
the regularized contact potential. However, the term Vg k’ γk’ is finite, because
(Q)
as we can see from the equation, for large k’, γk’ ∼ 1/(k 02 ). Combined with the
regularization prescription Eq. (A.12), this gives a finite result.
(Q)
The variational equations are easily solved : solve for γk , sum over k, then
P
( Q)
eliminate k’ γk’ from the equation. Using Eq. (A.12), this leads to
1 X
1
1 X
1
mr
=
+
.
4πas
V
k↑ + k↓ V k>kF E − Q−k↓ − k↑
k
(4.25)
For a given as and Q, finding the E that solves this equation gives us the variational energy of the bare molecule. In this equation, the cutoff can be sent to ∞. The
b
binding energy of the molecule is then EM
(Q) = kF2 /2 − E. Indeed one must remove
2
kF2 /2, or in other words measure the energy of the up spin with respect to the Fermi
energy. Now define Qmin is the momentum that minimizes the molecule energy. If
Qmin > 0 we have an FFLO molecule.
As one increases as , the binding of the molecule decreases, and there comes a
point where EM2 (Qmin ) = 0. At this point the molecule unbinds : it decays into
a bare polaron, i.e. a single down spin at momentum zero, plus an up spin that
joins the Fermi sea. Note that in order to do that, it must also emit a particle-hole
pair sitting at the Fermi sea to soak up the remaining momentum. Thus the decay
process is M2 → Polaron + Particle + Particle + Hole, where the particles and holes
are understood to be majority spins. If Qmin = kF , on the other hand the decay
process is M2 → Polaron + Particle.
Note that the equation for the M2 binding diverges at the unbinding, because
of the E − Q−k↓ − k↑ in the denominator. Indeed in the derivation we lose the
unbound states along the way, but since in this case the continuum is at a different
73
momentum it doesn’t actually diverge. One could study scattering states by going
back to the original variational equations.
The M2 − P1 transition is directly related to the mean field calculation of the
previous section: when Qmin = 0, it is precisely the spinodal point S obtained from
mean-field. To prove this, we simply take the P = 1 limit of the spinodal line we
are interested in. This line is found by solving dΩM F /d(∆2 )|∆=0 = 0. Indeed just to
the right of this line in the phase diagram, dΩM F /d(∆2 )|∆=0 > 0, which implies that
there is a mestastable ∆ = 0 phase. The spinodal line hits the P = 1 precisely when
µ = h, or µ↓ = 0. The first derivative of the thermodynamic potential is given by
d(ΩM F /V )/d(∆2 )| =
1 X
1
1 X 1
mr
.
−
+ −
V
k↑ + k↓ V
2Ek 4πas
k
k∈P
/ B
(4.26)
Setting ∆ = 0, µ↓ = 0, the Pauli Blocking region becomes {k : k < kF }, so
d(ΩM F /V )/d(∆2 )|∆=0 =
1
1 X
1
mr
1 X
−
−
. (4.27)
V
k↑ + k↓ V k>kF k↑ + k↓ − µ↑ 4πas
k
Setting this equal to zero reproduces Eq.(4.25) for the M2 − P1 transition with
E = µ↑ .
An interesting question arises : what if one improves the calculation of the polaronmolecule transition? At the M2 − P1 level, the transition will always be to the right
of the F point where first order physics kicks in. Indeed one can show that the
spinodal line always touches the P = 1 line. But as we will see once we improve
the calculation, it is possible for the polaron-molecule transition to sit outside of the
region of phase separation. In that case, the transition is really a separate entity.
This does not happen in the mass balanced case, which is why experiments on the
strongly polarized fermi gases see a transition at F [71]. However, we will provide a
calculation later on that suggests that these quantum phase transitions in the single
74
r
10
8
6
4
FFLO
P1
M2
2
0
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
1Hk f asL
Figure 4.2: M2 − P1 phase diagram, as a function of 1/(kF as ) and r = m↑ /m↓ . in the
F F LO region, the minimum of the M2 (Q) dispersion is at nonzero Q.
impurity limit can indeed survive when a finite density of impurities is introduced.
To motivate these we will need a calculation beyond mean field of the F point.
Close to the M2 − F F LO transition line, Q is small and one can consider Taylor
expanding the energy of the FFLO in terms of Q. By symmetry, the energy can only
depend on Q2 , so one can write E = aQ2 + bQ4 . As one crosses into the FFLO, a
goes negative, and one gets Q =
q
−a/(2b). Thus Q should grow like a square root in
a. In Fig.4.3 we plotted Q as a function of r along the M2 − P1 line, and see that at
the transition Q comes in as a square root. As one goes deep into the FFLO phase,
Q grows until it hits kF , then hovers around that value. Indeed once the down spin
has dipped to the bottom of the Fermi sea there is nowhere for it to go. Note that at
higher kF , Q can become bigger than kF , but not by much.
Thus far we have found an FFLO phase in this limit of a single down spin. But in
fact there are other possible states that can compete with what we have found : the
down spin could bind two up spins at the Fermi sea to form a trimer, or bind three
up spins to form a tetramer, etc.
75
Q
1.5
1.0
0.5
0.0
2
4
6
8
r
10
Figure 4.3: Momentum Q of the bare FFLO molecule in units of kF ↑ , as a function
r = m↑ /m↓ , along the M2 − P1 boundary.
To understand these wave functions we have to think more carefully about the
quantum numbers of our wave functions. If the wave function has momentum zero,
then it can be labelled by angular momentum L. If, on the other hand, it has
nonzero momentum, then it cannot be labelled with angular momentum, since angular
momentum and momentum do not commute.
The molecule has no reason to have nonzero angular momentum in its ground
state, but consider instead a trimer, i.e. a bound state of two majority fermions
and one minority atom. Without a Fermi sea, the problem has been solved exactly
[39], and indeed the trimer ground state has orbital angular momentum L = 1. It
was also shown that the trimer only exists at relatively large mass imbalance. The
physical picture is that the minority atom is then very light, therefore it has low
momentum, which translates in real space into a wavefunction with large extent.
This wave function then drapes itself over the two majority atoms, which are heavy
and can therefore be thought of, in the Born Oppenheimer picture, as semiclassical
small charged objects that feel an attraction with the light atom. The situation is
analogous to the H2+ molecule. The reason that L = 1 is as follows : consider the limit
76
where the two-body bound state, made up of one up and one down spin, is tightly
bound. A second spin up cannot have angular momentum zero relative to the other
up spin due to Pauli exclusion. Therefore it is likely to have angular momentum one
relative to the bound state, giving a total angular momentum L = 1. The centrifugal
barrier leads to the trimer only binding at large mass imbalance, because at a given
angular momentum, the barrier goes down with increasing mass.
Without further ado, we consider the bare trimer T3 , with total momentum Q = 0:
|T3 (0) >=
X
k1 k2
τk1 k2 c†−k −k ↓ c†k ↑ c†k ↑ |F S > .
1
2
1
2
(4.28)
The variational equations are
gN↑ τk1 k2 +
g X
(τk1 k − τk2 k ) = (E − −k1 −k2 ↓ − k1 ↑ − k2 ↑ )τk1 k2
V
k
(4.29)
As for M2 and all calculations in this approach, in the g → 0 limit the Hartree
term gN↑ τk1 k2 → 0. We define η(k) =
g
V
P
k’ τkk’ , and the variational equation can
be turned into an integral equation for η(k):
η(k)(
mr
1 X
1
1
1 X
−
)
−
4πas V
k’↑ + −k’↓ V
E − Q−k−k’↓ − k↑ − k’↑
k’
k’
1 X
1
η(k’)
=−
V
E − Q−k−k’↓ − k↑ − k’↑
k’
(4.30)
We can solve this integral equation using standard methods in the solution of
integral equations [20]. The basic idea is to discretize the momentum k. The integral
equation can be written in the form
X
k
Akk0 (E)η(k0 ) = 0
77
(4.31)
for some matrix A that depends on E. The ground state energy is the value of E
such that the highest eigenvalue of A is zero.
The calculation is considerably simplified by exploiting the symmetries of the wave
function. The first observation is that the angular momentum of the function η(k)
(i.e. its properties under rotations) is precisely the angular momentum of the wave
function. Indeed, τk1 k2 has the angular momentum of the wave function, and η(k)
is obtained by the rotationally symmetric process of summing out one momentum.
To study the trimer at L = 1, Lz = 0, we can therefore assume η(k) = η(k)k̂ · ẑ,
where kF < k < Λ and 0 < k̂ · ẑ < π, as all functions of a single three-dimensional
variable with L = 1 and Lz = 0 are of this form. The wave functions at Lz = ±1
will be degenerate in energy to the trimer at Lz = 0, by symmetry. The numerics are
aided by a clever discretization of the relevant variables, also known as quadrature1 .
In the limit kF ↑ → 0, this L = 1 trimer becomes the 3-particle bound state in a
vacuum, for which analytical solutions have been found [39]. Here, it has been shown
that the trimer is bound relative to a molecule and an extra particle for r > rC1 ∼
=
8.17. However, for r > rC2 ∼
= 13.6, the energy of the trimer is no longer finite in
the limit Λ → ∞. In calculations, this shows up as a wave function with weight at
increasingly high momenta as one approaches rC2 from below. This critical rC2 is
independent of kF ↑ , since it relies on high momenta. Thus, the results for the trimer
are always cutoff dependent once r > rC2 , and therefore no longer universal.
It is not impossible for our trimer to have angular momentum L = 0 (higher
angular momenta are unlikely due to the centrifugal barrier). We therefore repeated
the calculation at angular momentum L = 0, which is done by setting η(k) = η(k)
for kF < k < Λ. We found that there is no region of the phase diagram where |T3 (0)i
at L = 0 beats the |T3 (0)i at L = 1.
1
In short[20] : for periodic variables use rectangular quadrature, for variables of finite extent
use Gauss-Legendre quadrature, and for variables going up to infinity use either Gauss-Rational,
Gauss-Hermite or Gauss-Laguerre depending on the behavior at large values of the variables. For
the trimer, Gauss-Rational is optimal, as the function decays as a power as a function of momentum.
78
r
4.5
4.0
3.5
3.0
2.5
2.0
1.5
1.0
T3
P1
FFLO
M2
0.3
0.4
0.5
0.6
0.7
1Hk f asL
0.8
Figure 4.4: T3 − M2 − P1 phase diagram. The trimer T3 takes up most of the FFLO
phase.
It is understandable that the trimer competes with the FFLO phase in this limit :
indeed, as the impurity atom is made lighter, for a fixed momentum its kinetic energy
increases. Thus the molecule eventually favors sitting at nonzero momentum, as the
impurity dips down below the Fermi momentum. As it does so, it loses some of the
phase space available for pairing : before it dipped down, the molecule could form a
superposition of states involving the up and down spin with opposite momenta, and
explore This is no longer the case once the momentum of the molecule is nonzero.
The trimer state, on the other hand, restores the availability of phase space : the
down spin can sit at or close to the bottom of the Fermi sea, which lowers its kinetic
energy, and the two up spins can sit at opposite momenta, exploring all the phase
space above the Fermi sea. The surprise is that it competes so effectively, and only
leaves a tiny sliver of the FFLO region, when comparing the bare wave functions.
79
4.4
Dressed polaron and molecule,
and bare
trimer
The bare wave functions we looked at so far are exact in the limit of kF → 0. The
prescription to improve on the calculation above is straightforward : consider a more
general class of wave functions, including particle-hole pairs, which are generated by
the interactions. It was shown for the mass balanced case[15, 13] that the inclusion of
a single particle-hole pair gives a bound state energy that agrees with the best values
obtained through QMC calculations[68, 69, 48], to within a percent. The inclusion of
a second particle-hole pair gave results indistinguishable from the QMC calculations.
We will therefore dress the molecule and polaron to obtain more accurate results.
Therefore the size of the trimer phase is likely to be underestimated in the calculations
to come, since we will consider only the ‘bare’ trimer2 , while we will the polaron and
molecule with a particle-hole pair.
Our first dressed wavefunction is the polaron wave function with a particle-hole
pair :
|P3 (Q)i = α(Q) c†Q↓ |F Si +
( Q) †
†
k,q βkq cQ+q−k↓ ck↑ cq↑ |F Si
P
(4.32)
(Q)
Note that kF < k < Λ, and q < kF . Now optimize with respect to α(Q) and βkq ,
obtaining
g (Q) X (Q) X (Q)
(Q)
(Q)
(α
−
βkq’ +
βk’q + N↑ βkq ) = (E − Q+q−k↓ − k↑ + q↑ )βkq
V
q’
k’
g
g X (Q)
N↑ α +
βkq = (E − Q↓ )α,
(4.33)
V
V
q, k
2
For numerical reasons we have not yet dressed the trimer. As one approaches rC2 , the cutoff
needed for convergence of the results for T3 diverges, which means that the number of points needed
in the quadrature also diverges.
80
where E = E − EF S , EF S being the energy of the noninteracting up spin Fermi sea.
(Q)
Sending g → 0, again implies that the Hartree terms, Vg N↑ βkq and Vg N↑ α, are
zero, just as in the mean field. Now we can tell from the equations that for large k,
P
P
(Q)
(Q)
(Q)
βkq ∼ k12 . So this means that Vg q’ βkq’ = 0, while Vg k’ βk’q is finite.
(Q)
As we did for the trimer, we sum out one of the momenta in βkq , i.e. define
P
(Q)
γ(q) = Vg k’ βk’q . Now we can get an expression for γ(q) from both equations, set
those equal, and use the regularization prescription Eq. (A.12), to obtain
1 X
1
1 X
1 X mr
(
−
−
(E − Q+q−k↓ − k↑ + q↑ )−1 )−1
E = Q↑ +
V q 4πas V
+ k↓ V
k k↑
k
(4.34)
This is an integral equation for E, which can be solved numerically.
The wave function for the molecule, with at most one particle-hole pair, is given
by
|M4 (Q)i =
X ( Q) †
k
+
γk cQ−k↓ c†k↑ |F Si
X
k,k’,q
( Q)
δkk’q c†Q+q−k−k’↓ c†k↑ c†k’↑ cq↑ |F Si..
Once again, we can sum out a momentum, and define ζ(q, k) =
Also define ∆ =
g
V
(4.35)
(4.36)
g
V
( Q)
k’ δkk’q .
P
P
k γk , then the variational equations are
∆−2
X
q
(Q)
ζ(q, k) = (E − ρk )γk
ζ(q, k) − ζ(q, k’) −
q
(Q)
(α − αk’ ) = (E − χq,k,k’ )δq,k,k’ ,
2V k
where
(Q)
χq,k,k’ = Q+q−k−k’↓ + k↑ + k’↑ − q↑
81
(4.37)
(4.38)
(Q)
ρk
= Q−k↓ + k↑
and kF < k, k 0 < Λ, q < kF .
With a little work, and droppings term that are zero when g → 0, one finds a
integral equation for ζ(q, k) :
(
mr
1 X
(Q)
− Ωq,k )ζ(q, k) = −(−
(E − χq,k,k’ )−1 ζ(q, k’)
4πas
V
k’
1 X
∆
+ (E − ρk )−1 (
ζ(q’, k) − )),
V
2
q’
(4.39)
where
1 X
1
1 X
1
1
mr
1 X
1 X
−
−
η(q, k)
)−1
4πas V k>0 k↑ + k↓ V k>kF E − ρk
V
E − ρk V q
k
1
1
1 X
1 X
Ωq,k =
+
V k0 >0 k’↑ + k’↓ V k0 >kF E − χ(Q)
q,k,k’
∆ = −2(
We use the same method to solve this equation as we did for T3 , except that now
the function we are solving for, ζ(q, k), depends on several variables. To simpify the
calculation, the integrals in the equation should be done analytically when possible,
for example if there is an integral over an angle that ζ(q, k) does not depend on.
Let us first consider |M4 (0)i, at L = 0. The function ζ(q, k) is then invariant
under rotations, which means it can only depend on the three scalars one can build
out of q and k : ζ(k, q) = ζ(k, q, θ), where k = k > kF , q = q < kF , and cosθ = k̂ · q̂.
So we discretize all three variables, and write the variational equations in the form
X
Bkqθ,k0 q0 θ0 ζ(k 0 , q 0 , θ0 ) = 0
(4.40)
k0 q 0 θ0
where cosθ = k̂ · q̂, cosθ0 = q̂0 · kˆ0 , and B is a matrix. The integral equation
for ζ(q, k) can be solved using standard methods for solving integral equations with
82
multiple variables. To simply the calculation, the symmetries of the Hamiltonian help
tremendously. Also, the integrals in the equation should be done analytically when
possible, for example if there is an integral over an angle that η(q, k) does not depend
on.
Moving from the M4 to the T3 phase, one could envisage the M4 moving to angular
momentum L = 1. Indeed, the term with a particle-hole pair in M4 can be thought
of as a trimer plus an up spin, thus since the trimer is at L = 1, if it became bound
to an up spin such that the total angular momentum L = 1, we would have an M4
phase at L = 1.
To check this, we return to the variational equation Eq.(4.39) for M4 . We need to
determine the most general form of the function ζ(q, k) at L = 1. A bit of work with
the angular momentum algebra, and considering the Taylor expansion of ζ(q, k) in
terms of kx , ky , kz , qx , qy , qz , allows one to prove the most general form for a function
of two three-dimensional variables with L = 1 and Lz = 0:
ζ(q, k) = ζ1 (q, k, k · q)k̂ · ẑ + ζ2 (q, k, k · q)q̂ · ẑ + ζ3 (q, k, k · q)(k̂ × ẑ)z
(4.41)
In fact, there is another symmetry of the Hamiltonian we haven’t considered yet,
parity. The first two terms have odd parity, the last term even parity. We find that
the odd parity M4 at L = 1 never beats M4 at L = 0. The even parity one beat
M4 at L = 0 within the T3 part of the phase diagram, however it never beats the
trimer: it has an energy close to to the trimer, but we have not found a region where
it beats it. However, we were not able to explore the region where 1/(kF as ) is large,
because in that region the cutoff one needs diverges, and the size of the matrix grows
polynomially in the number of points in the quadrature. In other words, there may
83
be a region where the even parity M4 at L = 1 beats the trimer at large 1/(kF as ),
but we have not found it.
We finally explore one last possibility for M4 : that the minimum of its dispersion
move to nonzero momentum Q, thus leading to an FFLO phase. One a finite momentum is introduced, a lot of symmetry is lost. Write the momenta k and q in polar coordinates: k = k(sinθk cosφk , sinθk sinφk , cosθk ), q = q(sinθq cosφq , sinθq sinφq , cosθq ),
where Q is pointing along the ẑ axis. Exploiting the symmetry of rotation around Q,
we can set φk = 0, thus ζ(q, k) = ζ(q, k, θk , θq , φq ). Obviously solving for the energy
will be numerically quite costly : we have to discretize 5 variables. If each variable
requires about ten quadrature points to achieve converge, the matrix is 105 × 105 .
Luckily we only need the highest eigenvalue, which means that we can resort to the
lanczos algorithm3 .
Finally, we have all the energies we need to make an improved phase diagram,
Fig. 4.5: the energy of |M4 (Q) >, the energy of |P3 (Q) > and the energy of |T3 (0) >.
We haven’t yet dressed the trimer, this will be the object of future work.
4.5
Unbinding transitions vs phase separation
All the phase transitions we have discussed so far are unbinding transitions. In Fig.
4.7 we discuss the difference between a first and a second order unbinding transitions.
A phase transition is typically accompanied with an order parameter showing
nonanalytic behaviour at the transition. We now discuss a possible order parameter
for the M4 − T3 phase transition.
3
Note that the lanczos algorithm requires a real symmetric matrix. The matrix obtained is
real, but not symmetric. However, there
P is considerable freedom in the solution of the integral
equation. For example, if one is solving j Ai,j vj = 0, one can multiply for the left by a function
fP(i) that depends only on i, and one can write vj = g(j)wj . Thus and equivalent problem is
j f (j)Ai,j g(j)wj , and look for a zero eigenvalue of the new matrix Bi,j = f (i)Ai,j g(j). Exploiting
this freedom one can make the matrix symmetric.
84
14
T3H0L
r
12
10
8
6
4
0.5
P3H0L
1.0
FFLO
1.5
1HkF­ asL
M4H0L
2.0
2.5
Figure 4.5: The ground-state phase diagram as a function of mass ratio r and interaction strength 1/kF ↑ as for the polaron (P3 ), molecule (M4 ) and trimer (T3 ) wave
functions. The FFLO region corresponds to M4 with non-zero momentum; the momentum of the FFLO molecule goes continuously to zero at the FFLO-M4 (0) transition line. The T3 -M4 boundary approaches the 3-body transition rC1 ' 8.17 in the
limit 1/kF ↑ as → ∞, as expected. Above the black dashed-dotted (r = rC2 ) line, the
results for T3 become cutoff-dependent, and are therefore no longer universal. The
calculations become numerically more demanding as r approaches rC2 , so we have
not yet precisely determine how far the FFLO phase extends towards large r. The
shaded region marks where the system is unstable to phase separation. See text for
the approximations used.
85
Q
1.0
0.8
0.6
0.4
0.2
6.6
6.7
6.8
6.9
7.0
r
0
E
E
Figure 4.6: Momentum Q of the dressed FFLO molecule, as a function r, along the
M4 − P3 boundary.
kF-
Q
0
kF-
Q
Figure 4.7: Schematics of two different scenarios for a molecule unbinding into a
polaron + particle. The solid (red) lines represent the molecule dispersion E(Q)
and the shaded regions correspond to the polaron + particle (two-body) continuum.
When both the molecule and polaron have their minimum energies at Q = 0 (left),
the transition is first-order. However, we have a continuous unbinding transition
(where the bound state fully “mixes” with the continuum) when the molecule has
ground-state momentum Q = kF ↑ (right).
86
Z M4
0.8
0.6
0.4
0.2
r
0.05 0.10 0.15 0.20 0.25 0.30
Figure 4.8: The molecular residue ZM4 , defined in the text, for 1/(kf as ) = 1.5, as
a function of r. The sharp drop of ZM4 as one reduces r coincides happens at the
boundary between T3 and M4 . It signals that M4 is looking increasingly like a trimer
bound to a hole.
As one approaches the T3 from the molecule phase, one would expect that the
weight of the M4 wave function shifts from the ”bare” part, i.e. that the part with
two operators acting on the Fermi sea, to the dressed part. Namely, the dressed part
can be interpreted as a trimer binding to a hole. Thus we define a molecular ”residue”
ZM4 :
(Q) 2
k |γk |
P
.
ZM4 = P
P
(Q) 2
(Q)
2
0
|γ
|
+
|δ
|
k k
k, k , Q k, k0 , Q
(4.42)
We can evaluate by returning to the variational equations. We plot ZM4 for 1/(kf as ) =
1.5, as a function of r. We see that, as expected, ZM4 drops to zero as one enters the
trimer phase. Note, however, that here we calculated ZM4 for |M4 (0) >, while the
molecule goes into an FFLO phase before it enters T3 . For numerical reasons, we did
not calculate ZM4 in the FFLO phase, but we suspect similar behavior.
87
We have now characterized the phase transitions in our limit, but there is one
question which the single impurity limit cannot address : what happens when there
is a finite density of impurities? Naively one might think that the phase separations
in the finite density case match onto the single impurity limit when the density is
sent to zero, but that is in fact wrong. A case in point is the experiment of Zwierlein
et al [71], on mass balanced spin polarized fermi gases. In that case, phase separation
occurs at a lower value of 1/(kf as ) than the quantum phase transition calculated in
the single impurity limit.
As we introduce mass imbalance, we would like to obtain an estimate of the critical
1/(kf as ) at which phase separation sets in. To that end, we will first use a mean field
calculation, and then a more inspired calculation beyond mean field.
We are thus led to analyzing the value of 1/(kf as ) at P=1 where phase separation
first kicks in. The system will phase separate into a fully polarized normal phase, and
a weakly polarized superfluid (as QMC calculations have shown). The two phases the
system separates into will have equal pressure, and chemical potential of each species:
P N = P SF
(4.43)
SF
µN
σ = µσ
(4.44)
In the mean field calculation, this is easily achieved. Namely, the “energy” calculated
in mean field is an approximation to the thermodynamic potential Ω = E − T S − µN ,
where T = 0 in our case. Now by a standard argument from thermodynamics (using
Euler’s theorem on homogeneous functions), Ω = −pV . The points that are connected
by tie lines have the same chemical potentials, and the same Ω/V = −p, which means
that they satisfy all the conditions for coexistence. Thus in mean field, we simply
take the 1/(kf as ) at which phase separation sets in. In chemical potential space, this
is the point where µ = h.
88
To go beyond mean field, we can exploit the existing body of knowledge on BEC
BCS crossover. If we have the energy per particle E/N of a phase, then the pressure
)
is given by p = n2 d(E/N
, and the chemical potentials are µσ =
dn
N
E
N
)
,
+ n ∂(E/N
∂nσ
where nσ = Nσ /V and n = n↑ + n↓ . The average chemical potential is µ =
V
E
N
+ np .
For the normal phase, the energy per particle is simply that of a fully polarized free
Fermi gas, which is
3
E
= EF ↑
N
5
(4.45)
where EF ↑ = kF2 ↑ /(2m↑ ) and kF ↑ = (6π 2 n↑ )1/3 . Thus the pressure, and the chemical
potential for the majority spin in the normal phase are
pN =
(2m↑ )5/2 5/2
EF ↑
30π 2
µN
↑ = EF ↑ .
(4.46)
(4.47)
The chemical potential of the down spins is the binding energy of the impurities.
We have an estimate of this binding energy from our variational wave function calculations: indeed, the chemical potential of the minority in the fully polarized phase is
precisely what the variational wave functions are estimating. Thus we must take the
binding energy of either the polaron, the molecule or the trimer, depending on which
of these has lowest energy. However, when the ground state for the system with a
single impurity is no longer the polaron, the situation is more intricate. Indeed, in
the phase diagram we obtained from mean field theory, the normal phase is a polaron
phase at strong polarization, but once that is no longer the case the do not know what
the phase diagram looks like at intermediate polarizations. For example, the phase
separating region might grow or shrink. Thus we will first simply take the chemical
potential of the down spins to be the binding energy of P3 , and compare this to the
result obtained when one takes our best estimate of the ground state binding energy.
We will discuss how to interpret the different results.
89
We know in the mass balanced case that the system separates into a fully polarized
normal phase, and a very weakly polarized superfluid. The mean field calculation
suggests that as r increases, the superfluid one separates into becomes less and less
polarized. Thus we will assume that the superfluid is unpolarized. We can then use
the fact that the equation of states is known in and around the BEC and BCS limits,
and at unitarity, for a balanced superfluid. We simply extrapolate from these limits
to obtain an estimate of the equation of state for all 1/(kF as ).
In the BEC limit, it is known that the system behaves as a gas of boson interacting
with a repulsive dimer-dimer interaction add . Thus its equation of state depends solely
on add , and we exploit the fact that add /as has been calculated as a function of mass
ratio [79, 65].
The energy in the BEC limit has been calculated in pertubation theory, and is
b kF add
128
E
= +
(1 + √
(kF add )3/2 + . . .)kF2 /m↑ + m↓
3
N
2
6π
15 6π
(4.48)
This is obtained from the mass balanced case, by replacing the mass of the boson,
2m, by m↑ + m↓ .
At unitarity, the equation of state is
3
E
= ξs EF
N
5
where EF =
(4.49)
2
kF
.
2mr
In the BCS limit, the energy contains both Hartree shifts and a contribution from
the superfluidity [31]:
E
3
10
4(11 − 2log2)
40 k π a
2
= EF (1 +
kF a +
(k
a)
−
e F + . . .)
F
N
5
9π
21π 2
e4
Thus we obtain for the pressure
90
(4.50)



5
2

nEF (1 + 3π
kF as

5



pSF = 






+
8(11−2log2)
(kF as )2
21π 2
− (1 −
π
π
) 40 e kF as )
2kF as e4
2
nξs EF
5
k2
5
2
n F
k a (1
5 m↑ +m↓ 12π F dd
+
if 1/(kF as ) << 0,
if 1/(kF as ) = 0,
√64 (kF add )3/2 )
5 6π 3
if 1/(kF as ) >> 0.
(4.51)
and for the average chemical potential








EF (1 +
4kF as
3π
+
4(11−2log2)
(kF as )2
15π 2
µSF = 






+
8
( π
e4 kF as
ξs EF
EF (− (kF 1as )2 +
4rkF add
(1
3π(r+1)2
π
− 5)e kF as ) if 1/(kF as ) << 0,
if 1/(kF as ) = 0,
+
√32 (kF add )3/2 )
5 6π 3
if 1/(kF as ) >> 0.
(4.52)
We then use numerical extrapolation to obtain an approximation to µSF and pSF
for all values of 1/(kF as ). We verified that our results we only weakly dependent on
how one carries out the extrapolation. The results in the BEC and BCS limits should
be used for values of 1/(kF as ) where the terms that are second or higher order in
1/(kF as ) are smaller than the lowest order term.
For a fixed mass imbalance m↑ /m↓ , we then have two equations: µSF = µN and
pSF = pN , and two unknowns: the 1/(kF as ) corresponding to the balanced superfluid,
and the 1/(kF ↑ as ) at which the normal phase first starts to phase separate, the point
we called F in Fig. 4.1. Note that we do not set the separate chemical potentials
equal because we do not know the chemical potentials in the superfluid phase. They
need not be equal: as long as their difference is less than the gap, the superfluid will
be unpolarized.
Finally we solve the two equations. We show the results for the phase separation
line in Fig. 4.9. The mean field result shows the right trend, but it is strongly
renormalized. Beyond mean field, we consider two approximations for the chemical
potential of the down spins in the normal phase : for the dotted red line, we take
91
the binding energy of P3 . Once the line crosses into the molecular phase, however,
the binding energy obtained fom P3 will be an underestimate of the binding energy.
This implies that this approximation overestimates the size of the phase separating
region : namely an increase in binding implies an increase in stability of the N phase.
Once the dotted red line enters the F F LO phase, we should really take the binding
energy of the F F LO molecule as the chemical potential of the down spin. Doing so
leads to the full green line. We find that the phase separation line then nevers enters
the T3 phase: it goes off to 1/(kF ↑ as ) = ∞ within the molecule phase. The true
phase separation line will lie somewhere between the dotted red and the full green
lines. Namely, as the phase separation line moves off to large 1/(kF ↑ as ), eventually
the superfluid phase that one separates into is no longer unpolarized, as we assumed.
4.6
Conclusion
We have found that, for sufficiently large r, the polaron-trimer transition extends
well outside of the regime of phase separation, and thus should be observable in
the polarized gas. This appears to be a robust result, since if anything we have
underestimated the stability of the trimer phase and overestimated the region of
phase separation. Part of the FFLO phase also extends into this stable regime, but,
as we note above, our present approximations may overestimate the stability of the
FFLO phase, so more precise calculations and/or experiments are needed to determine
whether or not the FFLO phase can be seen in this high polarization limit away from
phase separation.
Another question is what happens to the trimer phase when there is a finite density of spin-down atoms. One might expect a Fermi liquid of trimers for low densities
n↓ /n↑ 1/2. However, one may also have a mixture of trimers, polarons and/or
molecules as we approach the single-impurity binding transition. In one dimen92
10
r
8
6
4
2
0.0 0.5 1.0 1.5 2.0 2.5
1HkF ­ asL
Figure 4.9: The different approximations to the lines that mark the onset of the phase
separating region to their right. The dot-dashed blue line is obtained from mean field
theory; the dashed red line from the calculation beyond mean field theory discussed
in the text, assuming that the spin down chemical potential for the fully polarized
normal phase is the binding energy of P3 ; the full green line from taking for the
chemical potential of the down spin, our best approximation of the binding energy,
i.e. taking the binding energy of the molecule when the line enters the molecular
regime of the phase diagram. The true phase separating line will lie somewhere
between the dotted red line and the full green line.
sion, a trimer phase exists for n↓ /n↑ = 1/2, provided the interactions are sufficiently
large [57].
93
Appendix A
Single channel model of Feshbach
resonances
Throughout this dissertation we use a regularized contact potential to model a Feshbach resonance, both in a two channel and a single channel model. In this Appendix
we derive the relationship between the s-wave scattering length and the microscopic
parameters of the Hamiltonian with the contact potential, using the single channel
model.
A Feshbach resonance occurs when the scattering state of two particles at low
energies is coupled to a bound state with different internal spin than the scattering
state [24]. The spin state of the low energy scattering is fixed because the background
magnetic field will split the spin degeneracy. As one varies the magnetic field, the
relative energy, also known as the detuning, of the open channel and closed channel
will change. It is possible for the energy of the closed channel to cross the energy of
the scattering state : this is called a Feshbach resonance. Close to the point where
this happens, the scattering length will diverge. Calling the coupling between the
94
open and closed channels Γ, we define the width of the resonance to be
γ=
Γ2 3/2
m ,
8π r
(A.1)
where 2/mr = (1/m↑ + 1/m↓ ). γ is a measure of the strength of the coupling between
the closed and open channels. One distinguishes two limits: when γ >> 1, we have a
broad Feshbach resonance. This is the case that most experimentally used Feshbach
resonances find themselves in. In this case, one can neglect the occupation of the
closed channel state, and use a single channel model. Around unitarity, there is no
small parameter, and the single channel model is an uncontrolled approximation. It
has been shown to produce qualitatively accurate results, though. When γ << 1,
the Feshbach resonance is narrow, and one needs a two-channel model to capture
the occupation of the closed-channel molecule. There is another reason to use the
two-channel model : it has been shown that there is a small parameter in this limit,
and that mean field theory is in fact exact in the limit γ → 0. Therefore one can
consider a systematic expansion away from this limit. Indeed fluctuations around the
mean field solution scale like γ 2 /EF , where EF is the Fermi energy.
A.1
Single channel model of the contact interaction
In this section we discuss the single channel model, which is valid for broad Feshbach
resonances.
Since we are interesting in working in the dilute, low temperature limit, where
the only property of two-body scattering that matters is the s-wave scattering length,
we seek the simplest possible potential that reproduces this physics. Since the range
of the potential is much smaller than all other length scales, we seek to work with
95
an infinitely short range potential. We do require that it be able to reproduce the
physics of a Feshbach resonance.
This immediately implies that the interaction must be attractive. Indeed, a repulsive potential can only lead to a positive scattering length. If it has a finite range,
then the most its scattering length can be is its range, which it would have if it were
hard core. Thus a repulsive contact potential is trivial. A short-range attractive
potential, on the other hand, can have both positive and negative scattering lengths,
as it can have a bound state which strongly affects the scattering length, as discussed
in Chapter 1. For a short range potential to have a positive scattering length in the
limit of zero range, it must be attractive, and indeed attractive enough to form a
bound state, at which point level repulsion kicks in and particles incoming at low
energy will scatter repulsively. The point where the interaction potential first forms
a bound state is therefore our model of the Feshbach resonance.
However, there are some subtleties involved with taking the limit of zero range.
Naively one would write down a delta function, but in fact in three dimensions the
scattering length of the delta function is zero [10]. We will see that this can be
remedied by introducing a cutoff.
One way of understanding the contact potential is to start with a finite square
well[24], and consider the limit where its range is sent to zero. Calling r the distance
between two particles, the interaction potential is
V (r) = V0 θ(R − r),
(A.2)
where V0 < 0 is the strength, and R is the range of the square well. The general
theory of scattering[43] states that whenever a bound state first appears in a twobody problem as one varies a parameter of the Hamiltonian, the s-wave scattering
length as diverges. On one side of the resonance the scattering length goes to ∞, and
96
on the other side to −∞. Furthermore, close to the resonance and on the side where
as > 0, the binding energy goes like −1/(ma2s ). This can be checked explicitly for the
finite square well[24].
Now consider the square well for a value of V0 and R such that one is close to a
resonance. Our goal is to send R → 0, and vary V0 so that we stay close this resonance.
To achieve this goal, we solve the problem of a bound state in the finite square well.
Consider a bound state with angular momentum zero, and use the standard partial
wave analysis. There will be no centrifugal barrier, and the equation one must solve
for u(r) = ψ(r)/r is
−
∂2
u(r) + V (r)u(r) = Eu(r),
∂r2
(A.3)
with the boundary condition u(0) = 0, and V0 < E < 0. The wave function is
√
therefore u(r) = Asin(kr)θ(R − r) + Be−κr θ(r − R), where k = E − V0 and κ =
√
−E. Matching the wave function and its first derivative at r = R, one can derive
the following equation for the binding energy :
√
q
q
−E = − E − V0 cot( E − V0 R).
(A.4)
A graphical study of this equation shows that, as |V0 | increases, a new bound state
q
forms whenever R |V0 | = (2n + 1) π2 . Therefore, if we fix the number of bound states,
we see that V0 ∼ 1/R2 .
Say we replaced the finite square well with a delta function potential gδ(r), where
r = r↑ − r↓ , the distance between the two interacting particles which we call spin up
and spin down for convenience. Then the integral over all space of the interaction
potential would be g. But the integral of the finite square well potential gives 43 πR3 V0 ,
which goes to zero as R → 0, since V0 ∼ 1/R2 . In this limit, the finite square well
becomes a delta function times a parameter that goes to zero as the range of the
potential goes to zero.
97
This is similar to the prescription we adopt, except that instead of keeping the
range of the potential in our theory, we introduce a momentum cutoff Λ, which
can be thought of as 1/R. This is convenient because our calculations are done in
momentum space. It isn’t precisely that, because a finite square well can scatter into
momenta above 1/R, but the idea is the same : the finite square well regularizes
the problem. We therefore adopt the infinitely short range potential of the form
g(Λ)δreg (r), where g(Λ) is a running coupling constant that depends on the cutoff,
and δreg (r) is a regularized delta function that only scatters into states with absolute
value of the momentum below Λ. We will show that this regularized delta function
has precisely one bound state at finite energy on the positive as side, so it is analogous
to a finite square well with a single bound state, where the range of the well has been
sent to zero. The potential may have another set of bound states whose energies
are −∞ in this limit, which will not affect the physics as their wave functions are
delta functions. These infinitely deeply bound states correspond to deeply bound
states which typically exist in the real physical problem. One avoids falling into these
states by keeping the density low enough to avoid three-body interactions. Two-body
interactions will be kinematically forbidden to have a real transition into one of these
bound states.
Since the interaction is an infinitely short range contact interaction, it only scatters
in the s-channel : because of Pauli exclusion particles with the same internal state
won’t interact.
Let us now calculate the s-wave scattering length obtained from this regularized
interaction. Consider two particles scattering in the center of mass frame. The
wavefunction is written as
Ψ(~r) = α(eikz + f (θ)
98
eikr
),
r
(A.5)
where the energy of the scattering state is E =
k2
,
2mr
where 2/mr = 1/m↑ + 1/m↓
(in units where h̄ = 1), and α is an arbitrary overall factor, that sets the incoming
flux. This is normally how the scattering wavefunction behaves at large r, but for an
infinitely short range potential it is true everywhere. The s-wave scattering length is
obtained when scattering at zero energy, where the wavefunction becomes
Ψ(~r) = α(1 −
as
),
r
(A.6)
so that
as = − lim f (θ).
(A.7)
k→0
In momentum space, the scattering wave function Eq.(A.5) becomes (using the regularized Fourier transform) :
|Ψ >= α(|k ↑, −k ↓> −
1 X
4πas
|k 0 ↑, −k 0 ↓>),
02
V k0 6=0 k − k 2 − imr δ
(A.8)
where |k ↑, −k ↓> is a state where the up spin has momentum ~k, and the down spin
has momentum −~k. The infinitesimal imr δ is needed to ensure the Fourier transform
converges, and it selects the outgoing waves. The reason for mr is that are in fact
going to consider eigenstates of the Hamiltonian with slightly complex energies E +iδ.
For a scattering problem, the energy is the non-interacting energy E = k 2 /mr of the
incoming particles. In real space the wave function with the iδ including looks like
√
2
eik·r − ars ei k +imr δr , so that it indeed has energy E+iδ. This is standard in scattering
problems : one must slightly rotate the Hamiltonian into the complex plane, so that
the Green’s function (E − H0 + iδ)−1 is non-singular.
For this regularized delta function interaction, the problem is easily solved in
momentum space, so we write |Ψ >= α|0 ↑, 0 ↓> +
99
k βk |k ↑, −k ↓>.
P
The Hamiltonian is
H=
X
k,σ
g
kσ c†kσ ckσ +
V
X
0
k, k , Q
c†k+Q↑ c† 0
c 0 c .
k −Q↓ k ↓ k↑
0
(A.9)
Setting H|Ψ >= (E + iδ)|Ψ >, we obtain
X
g
βk00 ) = (E + iδ)βk0 for k0 6= k
(k0 ↑ + −k0 ↓ )βk0 + (α +
V
k00
X
g
(k↑ + −k↓ )(α + βk ) + (α +
βk00 ) = (E + iδ)(α + βk )
V
00
k
(A.10)
Now divide by α, and use
βk 0
α
s
= − (k02 −k4πa
2 −im δ)V = − m V (
r
r
4πas
k0 ↑ +k0 ↓ −E−iδ)
. Both equa-
tions give the same result :
Λ
1
mr
1
1 X
=
−
g(Λ)
4πas V
+ −k↓ − E − iδ
k k↑
(A.11)
Finally, setting E = 0 and δ = 0 gives the regularization prescription:
Λ
1
1
mr
1 X
=
−
g(Λ)
4πas V ~ k↑ + −k↓
k
(A.12)
Note that the momentum integral has an upper cutoff Λ, which is essential to
keep it finite. Thus this equation gives us the dependence of the running coupling
constant g(Λ) on the scattering length. In the calculations that follow, we will simply
write g instead of g(Λ), and we will assume Λ will have been sent to infinity and g
sent to 0 in such a way that Eq.(A.12) is satisfied for a given as .
Armed with the regularization prescription, we can work out the bound state of
this potential. Simply solve for H|Ψ >= E|Ψ > with |Ψ >=
k βk |k ↑, −k ↓>.
P
Note that we needn’t rotate the energy into the complex plane for bound states, since
100
the energy is no longer in the continuum. The resulting equations
g X
(k↑ + −k↓ )βk +
β 0 = Eβk
V 0 k
k
Solve for βk , and sum out k. One gets that
(A.13)
P
k βk drops out of the equation, and
we are left with, using Eq.(A.12)
mr
1 X
1
1 X
1
=
+
4πas
V
+ −k↓ V
E − k↑ − −k↓
k k↑
k
(A.14)
Thanks to the regularization prescription, the equation no longer depends on g, and
the cutoff can be sent to infinity, as the divergent parts of the two sums cancel.
Replacing
P
k with
1
(2π)3
R
dk, one can do the integral exactly, resulting in
E=−
1
mr a2s
(A.15)
Thus the relation between as which is guaranteed by scattering theory to be valid
close to the resonance is actually valid for all as > 0, for this regularized contact
potential.
The derivation thus far applies to distinguishable particles. When they are indistinguishable, there are two cases : for identical fermions, there is no s-wave scattering,
as indeed the wave function has to be antisymmetric, while an s-wave scattering function cannot be antisymmetric. For identical bosons, the Hamiltonian is
H=
X
k
g
k b†k bk +
V
X
0
k,k ,Q
b†k+Q b† 0 bk0 bk .
k −Q
0
(A.16)
The s-wave scattering length is defined the same way as above : Ψ(r) = α(1 − ars )
for large r. the only difference with the above derivation is that if two bosons scatter
into states of momentum k and k0 , one cannot distinguish which of the original bosons
101
went into which momentum state, so the two possibilities must be added coherently.
The wave function for scattering at zero energy is |Ψ >= α0 |0, 0 > +
k αk |k, −k >,
P
where |k, k0 >= b†k b† 0 |vac >, and setting H|Ψ >= E|Ψ >= 0 results in (using
k
k0 bk0 b−k0 |k, −k >= 2|k, −k > for bosons)
P
g
g X
2(k)αk + 2 α0 + 2
αk’ = Eαk = 0.
V
V
k’
Setting
αk
α0
(A.17)
s
= − m4πa
, we obtain
r V 2k
Λ
2
1
mr
1 X
=
−
.
g(Λ)
4πas V ~ 2k
k
(A.18)
Finally, we consider the situation where the interactions are small and the
cutoff is finite, so that first order perturbation theory is valid.
Thus we apply
first order perturbation theory to the state |0 ↑, 0 ↓> with the contact interaction
g
V
†
†
k,k0 ,Q ck-Q↑ ck’+Q↓ ck’↓ ck↑ , and obtain obtaining
P
|Ψ >= |0 ↑, 0 ↓> +
1
g X
|k ↑, −k ↓>
V
−k↑ − k↓
k
(A.19)
Now in terms of the relative coordinate r between the two particles, the scattering
wavefunction is Ψ(r) = 1 −
−
P
k
4πas
|k
k2
as
,
r
which in momentum space reads |Ψ >= |0 ↑, 0 ↓>
↑, −k ↓>, thus we obtain
g=
4πas
mr
(A.20)
For indistinguishable particles (in which case the masses must be equal), we obtain
|Ψ >= |0 ↑, 0 ↓> +2
g X
1
|k ↑, −k ↓>
V
−k↑ − k↓
k
102
(A.21)
and therefore (for a finite Λ)
g=
2πas
m
103
(A.22)
Appendix B
Two-channel model of Feshbach
resonances
In Appendix A we have discussed the single channel model in detail. Now let us consider a two channel model, and show when the single channel model is an appropriate
description.
Consider the case of two distinguishable particles, whose creation operators we
call b†k and fk† . These could be completely different particles, or the same particle in
†
different internal spin states. We call ψk
the creation operator for the closed-channel
molecular state. The closed-channel state is tightly bound, so we can set its internal
structure be fixed, thus we need only specify its center-of-mass momentum. The
two-channel Hamiltonian is1
H=
Λ
Γ X
†
(fk† b† 0 ψk+k0 + h.c.),
(kf fk† fk + kb b†k bk + kψ ψk
ψk ) + √
k
V
k
k,k0
X
(B.1)
√
R
To understand the V , start with the hybridization term in real space : g drf † (r)b† (r)ψ(r),
and use the relation between creation operators in real space and in momentum space in a finite box,
R
P † ik·r
, where f † is unitless, and dreik·r = V δk,0 where V is the volume.
e.g. f †(r) = √1V
k fk e
k
1
104
where Λ is the cutoff, and
h̄2 k2
h̄2 k2
h̄2 k2
kf =
, =
, =
+ ν.
2mf kb
2mb kψ
2mψ
(B.2)
ν is called the detuning of the closed channel. In the center-of-mass frame, the
wave function is
X
|Ψ >= (
k
αk fk† b†−k + βψ0† )|0 > .
(B.3)
Now solving forH|Ψ >= E|Ψ > gives
Γ
(kf + kb )αk + √ β = Eαk
V
Λ
Γ X
α = Eβ.
νβ + √
V k k
(B.4)
(B.5)
We can use these equations to eliminate β and find an integral equation for αk :
X
Γ2
(k + kb )αk +
αk = Eαk .
V (E − ν)
k
(B.6)
This is precisely the same equation that was obtained when solving the
Schrodinger equation of the single channel model, Eq.(A.13), with the interaction being g = Γ2 /(E − ν). Setting E = 0, we obtain
g=−
Γ2
.
ν
(B.7)
Thus we can use our efforts in the single channel case, Eq. (A.12), to immediately
write down the relation between Γ and the s-wave scattering length as :
Λ
mr
E−ν
1 X
1
.
=
+
2
4πas
Γ
V
kf + −kb
k
105
(B.8)
Solving for as , we obtain
4πas
=
mr
1
1
V
PΛ
k
1
kf +−kb
−
(B.9)
ν
Γ2
To obtain a broad Feshbach resonance, send Γ → ∞, and ν → ∞ in such a way
that as remains constant. Then we can see from Eq. (B.4) that β → 0. In other
words, in the 2 body ground state the occupation of the closed channel goes to zero.
This is why one can get away with a single channel model. However, we immediately
see the problem with a pertubation theory analysis : we are actually in the infinite
coupling limit. The molecule ψ can be integrated out of the Hamiltonian, but at the
price of introducing an infinite interaction, unless one is in the weak coupling limit
(as → 0− ). All one can hope for out of a mean field theory in the single channel limit,
then, is a qualitatively accurate picture.
In the other limit, Γ → 0, and ν →
Γ2
V
PΛ
k
1
kf +−kb
in such a way that as is
constant, we have a narrow Feshbach resonance. In this limit of weak coupling to the
closed channel molecular state, perturbation theory in Γ2 should apply.
2
Defining ν 0 = ν − ΓV
that
Γ2
V
PΛ
k
1
kf +−kb
PΛ
k
1
.
kf +−kb
Unitarity occurs when ν 0 = 0. If Γ is small such
is small, then we can set ν 0 = ν, and unitarity occurs at ν = 0.
This relation is used in Chapter 2 (note that the g in that chapter corresponds to Γ
in this chapter).
106
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