Download The Fermi-Hubbard model 11 The Hubbard model

Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
The Fermi-Hubbard model
Literature: Chapter 9. ”The BCS-BEC Crossover” by Meera M. Parish, and Chapter
3. ”Quantum Gases in Optical Lattices” by Peter Barmettler and Corinna Kollath
in ”Quantum Gas Experiments – Exploring Many-Body States”, P. Törmä and K.
Sengstock (Eds.), Ref. [1]. Corinna Kollath is thanked for providing a picture for
this lecture.
Learning goals
• To know the basics of the Fermi-Hubbard model.
• To understand the basic features of the BCS-BEC crossover in a lattice within
the attractive Hubbard model, and the di↵erences to the continuum case.
• To know the most important phases predicted for the repulsive Hubbard
model: Mott insulator, antiferromagnet and d-wave superfluid.
• To be aware of the mapping between the repulsive and attractive Hubbard
models.
11
The Hubbard model
In the ultracold gases context, the Bose-Hubbard model is often the one that is
taught first in courses since it was experimentally realized by ultracold gases well
before the Fermi-Hubbard model. This is the case also in these series of quantum gas
lectures at ETH: you have already learned about the Bose-Hubbard model and its
realization in optical lattices in the fall semester. However, it is important to keep in
mind that historically, it is the fermionic version that is the classic Hubbard model:
the Hubbard model was designed in 1963 describe electrons (which are fermions) in
solid state materials [31]. Only in eighties a bosonic version was considered[32, 33]
and in the nineties it was suggested to be realized by optical lattices [34]. Now we
will learn about the ”original” Hubbard model, that is, the Fermi-Hubbard model.
Since it indeed may describe electrons in materials and thereby various interesting
states of solid state materials, it is of great significance and has potential for instance
to shed light on the mechanism of high-temperature superconductivity which is still
not fully understood. In recent years, the realization of the Fermi-Hubbard model
has been achieved in optical lattices [35, 36]. This opens remarkable opportunities
for solving such questions with the help of quantum gas experiments.
The standard Fermi-Hubbard model considers two species of fermionic particles which are confined in a lattice and move by hopping to the nearest neighbour
sites. The two species for electrons are the spin-up and spin-down electrons. For
ultracold gases, they can be two di↵erent internal states of an atom or molecule
(e.g. hyperfine states), or even two di↵erent species of atoms; in any case, we label
them by a pseudospin =", #. The particles interact only when at the same site
within the standard Hubbard model. There exist various extended Hubbard models
with longer range interactions, but we do not discuss them here. In optical lattices,
the nearest neighbour hopping and on-site interaction are realized by having a sufficiently strong lattice potential and short range of the interparticle interaction.
The latter is the usual case for ultracold atoms where the interaction can be often
approximated even by contact interaction.
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
The Fermi-Hubbard Hamiltonian is
X
X
Ĥ =
J
ĉ†n, ĉn, + h.c. + U
n̂n," n̂n,# ,
n
hn,mi
where ĉ†n, is the creation operator for a fermion with (pseudo-) spin =", # and lattie site specified with discrete 3D (or 2D or 1D) set of integers marked with n. Here
hn, mi denoted summation over nearest neighbours. The operator n̂n, = ĉ†n, ĉn, is
the density operator. The hopping coefficient J and the interaction strength U are
given by optical lattice parameters by an expansion of the annihilation and creation
operators of each species with the help of Wannier functions:
✓
◆
Z
~2 2
Jn,m =
d3 r w⇤ (r n)
r + VL (r) w(r m)
2m
Z
1
=
d3 k eik·(n m) Ek and
(11.1)
V0 k21st BZ
Z
Un,m,n0 ,m0 = g d3 r w⇤ (r n)w⇤ (r m)w(r n0 )w(r m0 ).
The band energy Ek and the Wannier functions are obtained from the Bloch equations for a single particle
✓ 2
◆
~
(n)
(n) (n)
2
( ir + k) + VL (r) uk (r) = Ek uk (r) .
(11.2)
2m
For approximate analytical formulas for J and U see e.g. [37]. If you have forgotten
these issues, please have a look at the autumn semester course.
The Fermi-Hubbard model leads to rich physics both for repulsive (U > 0)
and attractive (U < 0) interactions. The fermionic statistics and the spin degree of
freedom are behind many of the interesting phenomena found in the model. Here we
choose to discuss the attractive interactions first because this allows the describe
the BCS-BEC crossover in a lattice and thus there is a direct connection to the
previous lecture. The repulsive interaction case is discussed after that. We will also
discuss a mapping between the attractive and repulsive Hubbard model.
11.1 BCS-BEC crossover in a lattice
In case of weak attractive interactions, the Hubbard model has a BCS-type superfluid ground state. When the strength of the attractive interaction is increased, one
can observe the BCS–BEC crossover, but with some characteristics that are specific
for lattices. In particular, while the critical temperature is low in the BCS limit and
higher at unitarity both for the lattice and continuum cases, in the BEC limit the
critical temperature of a continuum gas stays nearly constant while the one in the
lattice decreases dramatically. We will now learn what is the origin of this di↵erence.
Let us start from the Fermi-Hubbard Hamiltonian introduced above:
X
X
Ĥ =
J
ĉ†n, ĉn, + h.c. + U
n̂n," n̂n,# ,
(11.3)
n
hn,mi
where again J is the hopping energy and U now corresponds to an attractive onsite interaction (U < 0). Mean-field approximation introducing pairing fields can be
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
done for this Hamiltonian in a similar way as done in the lecture where we discussed
the BCS theory. The superfluid order parameter now becomes
= U hĉ†"n ĉ†#n i.
(11.4)
The Hamiltonian, including the chemical potential as should be the case when using
the grand canonical ensemble, is of the form (we again assume
real)
⌘
X⇣ †
Ĥ µN̂ =
µ" ĉ"n ĉ"n + µ# ĉ†#n ĉ#n
n
+
X✓
n
J
X
2
ĉ†"n ĉ†#n +
ĉ† m ĉ
ĉ#n ĉ"n
U
◆
(11.5)
n.
hn,mi,
To diagonalize the Hamiltonian and to understand the nature of pairing, it is convenient to write it in the (quasi-)momentum representation. This means doing a
Fourier transform by
1 X ik·n
ĉ" n = p
e
ĉ" k
M k
1 X ik·n †
ĉ†"n = p
e
ĉ"k
M k
1 X ik·n
ĉ#n = p
e
ĉ#k
M k
1 X ik·n †
ĉ†#n = p
e
ĉ#k ,
M k
(11.6)
where M is the (finite) number of lattice sites and k runs through the reciprocal
lattice. The Hamiltonian becomes
X✓
b =
H
⇠"k ĉ†"k ĉ"k + ⇠#k ĉ†#k ĉ#k
k
+
ĉ†"k ĉ†# k
+
ĉ#
k ĉ"k
◆
(11.7)
2
U
,
P
where ⇠ k = ✏ k µ = ↵ 2J(1 cos(k↵ )) µ . Here ↵ is x, y, z. We consider
here the case µ ⌘ µ. Note that in order to get the dispersion correspond to that
of a free particle in the limit of small k, the following terms have been added to the
Hamiltonian:
⌘ X
XX⇣ †
2
J ĉ k ĉ k =
2J N̂ .
(11.8)
↵
↵
k
The momenta are restricted to the first Brillouin zone |kx |, |ky |, |kz |  ⇡/a, where
a is the lattice spacing. Note, further, that U is finite in the lattice case, and a high
momentum cut-o↵ in the lattice is set by ⇡/a. Therefore renormalization procedures
that were applied in the usual BCS theory in the previous lectures are not needed
here. For the same reasons, the Hartree term cannot be formally neglected – in
practice, it leads to a constant shift U n of the chemical potential and is assumed
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
to be included in the chemical potentials above. This means that the interaction
energy of the normal Fermi liquid phase is always included in the BCS mean-field
theory, unlike in the continuum case without the lattice. The Hamiltonian (11.7)
can be diagonalized and gives solutions very similar to those in continuum BCS
theory, for certain value range of the chemical potential µ; we will go through the
details in the following lecture where the FFLO state in a lattice is discussed. The
extra length scale a given by the lattice means that the BCS-BEC crossover now
depends separately on the density, defined by the dimensionless parameter "F /J
and the dimensionless interaction |U |/J. In the lattice, "F is defined as the chemical
potential of the non-interacting Fermi gas with the same density n. Note that there
is a maximum density of n = 1 particle per site for each spin, corresponding to
"F = 12J. The system is then simply a band insulator. For low densities "F ⌧ 12J,
the system behaves similarly to the continuum case in the BCS limit (note that
one should be in general cautious with the statements that low densities in a lattice
correspond to the continuum limit; this is strictly speaking not the case in all
problems). In the regime of small attractive interactions U , the Hubbard model
therefore presents BCS-type pairing and superfluidity. Small interaction typically
means |U |/J < 1.
Increasing the interaction, the Hubbard model supports a two-body bound
state at |U |/J ' 7.9. The binding energy "B is given by
X
1
=
U
4J(3
k
cos(kx a)
1
cos(ky a)
cos(kz a)) + "B
.
(11.9)
When |U |/J
12J, the size of the bound state is of order the lattice spacing a, with
"B ' U . Then the size of the dimer is essentially constant, limited by a. The dimers
may then Bose condense. However, increasing U has another remarkable e↵ect. One
can do second-order perturbation theory on Eq. (11.3) for small J/|U | and obtain
an e↵ective model where there are only on-site dimers which hop from one site to
the neighbouring one with the hopping energy of approximately J 2 /|U |. Therefore,
the hopping goes to zero as |U | ! 1; in e↵ect, the dimers have a very large e↵ective
mass. One can then anticipate that the BEC transition temperature goes down, as
is predicted by large masses in the simplest order of magnitude estimates of BEC
(i.e. that the thermal deBroglie wavelength of the particles should be similar to the
average distance). This is indeed what happens in the BEC regime of the Hubbard
model. While one still expects the system to approach a non-interacting BEC at
zero temperature, the critical temperature Tc scales with the dimer hopping energy,
i.e., Tc ⇠ J 2 /|U |, and it will thus zero instead of saturating like in Fig. 4, owing to
the localization of bosonic dimers in the lattice.
Figure 5 shows how Tc tends to zero in both the BCS and BEC limits, with
a pronounced maximum in between. For further details, see e.g. [39]. Note that the
Hubbard model displays particle-hole symmetry at half-filling, "F = 6J. Therefore
above half-filling, one observes the BCS-BEC crossover of holes instead of particles.
Note also that on the BEC side we may again have two relevant temperatures: a
temperature where on-site pairs form (basically of the order of U ) and a much lower
temperature where these pairs may condense into a superfluid. In contrast, on the
BCS side the pair formation and condensation happen at the same temperature.
In the context of quantum gases in optical lattices one has to bear in mind
that the one-band Hubbard model is unable to describe Feshbach-resonant gases at
unitarity 1/aS = 0. There interactions scale with the lattice depth and thus can
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
Figure 5: BCS-BEC crossover in a lattice. Open symbols are Quantum Monte Carlo
calculations for the critical temperature. The dashed line is not relevant here. Temperature is given in the units of Fermi energy of a free fermion system of the same
density, V0 is the lattice depth and ER is recoil energy. The main figure is for 6 Li
parameters and the inset for 40 K. The scattering lengths are small (close to background scattering length) and the crossover is driven by the strength of the lattice
potential V0 which e↵ectively determines the Hubbard interaction U . Figure from
[38].
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
never be made small with respect to the band gap. Moreover, once a > aS > 0 i.e.
deep in the BEC regime of the Feshbach resonances where tightly bound molecules
can exist, the inclusion of higher bands yields dimers that are smaller than the
lattice spacing. In [40] the limits of validity of the single band approximation are
calculated when the scattering length is increased.
At the time of this teaching, although fermionic atoms have been loaded to
optical lattices, low enough temperatures for observing the BCS-BEC crossover
have not been achieved yet.
11.2 Mott insulator, antiferromagnet, and d-wave superfluid
In case of repulsive interactions, the most prominent phases that the Fermi-Hubbard
model presents are the Mott insulator and the emergence of the antiferromagnetic
(Néel) order. The Mott insulator has alredy been observed with ultracold gas setups
[41, 36] and there are also observations of short-range antiferromagnetic correlations
[42, 43]. For a more detailed description of the phases in the context of ultracold
gases see Ref. [44].
At half filling, when the interaction energy U is large, charge fluctuations are
suppressed and a Mott-insulating state occurs in which charge degrees of freedom
are localized on single lattice sites. It is intuitively understandable that the particles
of the two species try to avoid being at the same site since the large repulsive
interaction dominates over kinetic energy (note that the particles of the same spin
naturally avoid being on the same site due to Pauli blocking). The lowest charge
excitations are of particle hole-type and their energy is approximately c ⇠ U .
That is, a gap is opened in the charge section of the excitation spectrum. Note
that ”charge” is terminology from the Hubbard model for electrons, and in case of
ultracold atoms means just the density (in contrast to the spin sector).
There exists a crossover between a liquid at low interaction strength and a
Mott-insulating state at strong interactions. At finite temperatures that are much
smaller than the charge gap, characteristic suppression of charge fluctuations of the
Mott-insulating state still takes place.
The spin degrees of freedom also lead to interesting physics. The spin degrees
lead to a highly degenerate ground state when U/J ! 1. One can think that deep
in the Mott insulator state where the particles are almost perfectly localized, it does
not matter whether the neighbour of a particle has the same or the opposite spin.
Therefore, there are a large number of energetically equivalent ways of arranging the
particles in the lattice. This degeneracy is lifted for smaller U , where an e↵ective
magnetic coupling between the spins emerges. This is because the system tries
to lower energy by having at least some tunneling. At large interaction U
J
this is done by the so-called superexchange process: neighboring fermions tunnel
(hop) via an intermediate highly energetic doubly occupied state. This is a second
order tunneling process. Due to Pauli blocking the doubly occupied state is only
possible for fermions of di↵erent spin. Therefore the superexchange can occur only
for fermions of opposite spins, and the arising e↵ective coupling is antiferromagnetic.
2
The coupling strength is given by Jex = 4J
U [44] since it is a second order process
in tunneling coupling 2J, with the energy o↵set U related to the intermediate state
(c.f. the general formula for second order perturbation theory).
In three-dimensional cubic lattices, the superexchange coupling induces a
phase transition to an antiferromagnet with long-range order at low temperature
(Fig. 6a). The structure of the phase-boundary can be understood intuitively: at
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
large interaction the energy scale for the antiferromagnetic coupling diminishes as
J 2 /U , leading to a decreasing transition temperature with increasing interactions.
In contrast, at low interaction, the charge gap becomes small and charge fluctuations
can destroy the magnetic ordering. In this regime of weak repulsive interaction, antiferromagnetic order is due to a spin-density-wave transition, in which the opening
of the insulating gap and antiferromagnetic order occur simultaneously. See Figure
6.
Due to the Mermin-Wagner theorem, Néel order at finite temperature is restricted to the 3D case. In 2D lattices, long-range antiferromagnetic order exists
only in the limit T = 0. In a 1D system, the ground state at half-filling in the
presence of repulsive interaction is a Mott insulator with algebraically decaying
antiferromagnetic ordering. Ground state and finite temperature properties of the
one-dimensional case can be obtained analytically from the Bethe ansatz solution
[45].
The first experiments with non-interacting fermionic atoms loaded into a cubic optical lattice observed the change of the Fermi surface with increasing the
number of atoms [46, 47]. The characteristic suppression of particle fluctuations in
the Mott-insulating phase has also been detected [41, 36]. Only recently have the
first signs of short-range antiferromagnetic correlations been discovered by modulation spectroscopy [48, 49], and by using a superlattice [42] or Bragg spectroscopy
[43]. However, the spontaneous formation of long-range antiferromagnetic order has
not been realized so far. This is due to the relatively high temperatures, of the order
of the hopping amplitude [50]. An important challenge is therefore the design of efficient cooling schemes [51]. Even lower temperatures than for the antiferromagnet
are be required to address the long-standing question of unconventional superconductivity in the doped 2D Hubbard model. There are predictions that a d-wave
superfluid would exist in the doped case, but conclusive experimental as well as
numerical evidence is missing.
11.3 Mapping between the repulsive and attractive Hubbard models
The Hubbard model possesses certain symmetries with respect to the repulsive and
attractive interactions. There exist a unitary transformation U1 that connects the
two at half filling, namely
U1† ci" U = ✏(i)c†i"
(11.10)
U1† ci# U = c†i# ,
(11.11)
where ✏(i) = 1 for one sublattice of the bipartite lattice and ✏(i) =
With this transformation
U1† Ĥhalf
f illed (U )U1
= Ĥhalf
f illed (
U)
1 for the other.
(11.12)
which means that the Hamiltonians for both signs of U must have the same energy
spectrum and the same form of the ground state. Therefore, at half filling, the BCS
superfluid maps to the Néel-ordered anti-ferromagnet and the BEC of on-site pairs
to the Mott antiferromagnet at strong interactions. The existence of on-site pairs
before one reaches the superfluid corresponds to having a Mott state before reaching
a Mott antiferromagnet. This mapping explains the apparent similarity of Figures
5 and 6. The mapping when being away from half filling will be discussed in the
next lecture. The identical behaviour at negative and positive U has been seen for
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
(a)
kB TN /J or Energy/J
4.0
TN
3.5
Jex = 4J 2/U
3.0
c
2.5
2.0
L
⇠U
MI
1.5
1.0
0.5
0.0
0
1
SDW-Neel
2
3
MI-Neel
4
5
U/J
6
7
Figure 6: Qualitative phase diagram of the Fermi-Hubbard model in three dimensions at half-filling as a function of interaction U and temperature [44]. The Néelordered antiferromagnet at low temperatures is understood as a spin-density wave
(SDW-Néel) at weak interactions and a Mott antiferromagnet (MI-Néel) at strong
interactions. The latter is characterized by the superexchange constant Jex . The
charge gap c can be used to distinguish between the liquid (L) and Mott-insulating
(MI) regimes in the normal phase at temperatures beyond the Néel temperature
TN . Figure from [37].
instance in expansion experiments of ultracold Fermi gases [52, 53]. Note that in an
experimental situation where the symmetry of the system may be broken by some
unwanted or desired features, the mapping may not be exact.
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
Exotic superfluids: the Fulde-Ferrel-Larkin-Ovchinnikov
(FFLO) state
Literature: T. Koponen, Fermionic superfluidity in optical lattices, Ph.D. Thesis,
University of Jyväskylä, Finland (2008),
http://www.jyu.fi/static/fysiikka/vaitoskirjat/2008/timo-koponen-2008.pdf
and the journal references mentioned in the lecture notes below. Timo Koponen
is thanked for providing me the LaTex files of his Ph.D. thesis and, naturally, for
doing his Ph.D. in my group.
Learning goals
• To understand the question of whether fermionic superfluidity may coexist
with magnetization, and to know about the Chandrasekhar-Clogston limit.
• To learn in detail the theoretical description of the FFLO state in lattices.
• To know that a similar treatment applies in continuum, and understand why
the FFLO state is more stable in a lattice than in continuum.
• To understand Luttinger’s theorem and (dis)appearance of Fermi surfaces.
• To know why the FFLO state fulfills the Bloch theorem i.e. does not carry
net current in the ground state.
• To be aware of modern developments of the FFLO theory, for instance beyond
mean-field calculations, 1D systems, inclusion of spin-orbit coupling.
12
Superconductivity and magnetization: the
Chandrasekhar-Clogston limit
Already in 1962 Chandrasekhar [54] and Clogston [55] asked the question whether
superconductivity would survive in presence of magnetization. I follow here Clogston’s
argument. What one usually expects for superconductors in presence of magnetic
field is the Meissner e↵ect. A metal in a magnetic field may go from the superconducting state to the normal due to free energy associated with the Meissner e↵ect
(repelling of the magnetic field from the metal). The critical field H0 for this is
given by
H2
FN = F S + 0 ,
(12.1)
8⇡
where the F ’s are the free energies of the normal and the superconducting states.
There is, however, a finite penetration depth for the magnetic field and therefore if
the conductor has the right geometry, e.g. it is a thin cylinder, the Meissner e↵ect
may be very small and the critical field associated with it very high. One may then
ask is there another limit for the field that penetrates into the superconductor? A
limit could come from the paramagnetic susceptibility P of a normal metal, that
is, that the spins would like to align with the magnetic field (also referred to as
Pauli paramagnetism). Notably in the BCS theory, there is no such susceptibility
since the spin-up and spin-down particles are correlated in pairs. From this, one
gets another constraint for the field, namely
FN = F S +
2
P H0
2
.
(12.2)
The free energy di↵erence between a superfluid and a normal state is given by
FN FS = N (0) 2 /2 where N (0) is the density of states at the Fermi level, while
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
P
= 2µ2B N (0) where µB is the Bohr magneton. This gives the critical field
µ B H0 = p .
2
(12.3)
Often also a scaled version of the magnetic field is used:
h= p .
2
(12.4)
Chandrasekhar’s argument is similar but produces only rough estimate of h ' .
The limit (12.3) is referred to as the Chandrasekhar-Clogston limit. It has been
observed in ultracold Fermi gases [56] in trapped geometries which favour phase
separation.
Now, the above argument is based on the simple BCS theory. People soon
asked whether it would be modified if one allows a di↵erent type of paired superfluid state. One of such proposals is the Fulde-Ferrel-Larkin-Ovchinnikov state
[57, 58, 58]. We will discuss it in detail in the following, and later mention also some
other proposals. The FFLO state was originally proposed for a continuum system.
However, is has been shown that it occurs in a rather small parameter window in
3D continuum systems [59], and also in 3D traps as a small edge e↵ect [60]. In
2007, we predicted [61] that the FFLO state appears in a much larger parameter
area in lattice geometries. Therefore we present the basic theory of the FFLO state
below using the lattice context. As you have seen already from the BCS theory,
the continuum and lattice descriptions are very similar, the di↵erence being just
the di↵erent dispersion in the energies, and issues related to the renormalization of
the interaction and the treatment of the Hartree field. Once you know the FFLO
formalism in the lattice, it will be easy to do the continuum calculation. A nice
classic reference about the continuum FFLO theory is [62].
13
FFLO state in a lattice
The simplest FFLO ansatz assumes that the Cooper pairs carry a finite momentum 2q and the order parameter is of the form (notation is otherwise the same as
introduced before in these lectures; now the Hubbard U is negative)
U hĉ†"n ĉ†#n i =
e2iq·n ,
(13.1)
where
0. This is equivalent with the standard BCS-theory in the limit of q = 0.
This plane wave form of order parameter was first considered by Fulde and Ferrel
[57]. Larkin and Ovchinnikov [58, 63] actually considered an ansatz which contains
both q and q, thus the order parameter would be of the form
U hĉ†"n ĉ†#n i = 2 cos(2q · n).
(13.2)
It seems to be the case in many known systems that the cosine wave LO ansatz
gives a lower energy than the FF plane wave. This is actually fortunate because
it is often simpler theoretically to use the FF ansatz. The argument then goes
that if FF minimizes the energy, LO would do it even better. So it is a reasonable
first approach to try the FF ansatz. In general, the FFLO order parameter could
be composed of any number of Fourier components with di↵erent q’s, leading to
basically arbitrarily complicated spatial dependence (n) (or (x) in general).
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Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
Remember here the lecture about the BCS-BEC crossover. There the initial
(non-mean-field) Hamiltonian in the momentum space (10.1) clearly indicates that
the pairs might have a finite momentum. Now in the FFLO state, the condensation,
i.e. macroscopic population of Cooper pairs gathers to the pair momentum state 2q
instead of q = 0.
The Hamiltonian is now in the form (we allow here for spin-dependent and
anisotropic hopping)
⌘
X⇣ †
H µ" N" µ# N# =
µ" ĉ"n ĉ"n + µ# ĉ†#n ĉ#n
n
+
X✓
n
X
e2iq·n ĉ†"n ĉ†#n +
X
J
↵
X
e
2iq·n
ĉ† m ĉ
2
ĉ#n ĉ"n
U
◆
(13.3)
n.
hn,mi↵
↵2{x,y,z}
It is convenient to write the Hamiltonian in the (quasi-)momentum representation,
i.e. represent the operators in the plane wave basis. This essentially means doing a
Fourier transform on (13.3) by
1 X ik·n
ĉ"n = p
e
ĉ"k
M k
1 X ik·n †
ĉ†"n = p
e
ĉ"k
M k
1 X ik·n
ĉ#n = p
e
ĉ#k
M k
1 X ik·n †
ĉ†#n = p
e
ĉ#k ,
M k
(13.4)
where M is the (finite) number of lattice sites and k runs through the reciprocal
lattice. The density terms transform as
!
!
X †
X
1 X ik·n †
1 X ik0 ·n
p
p
ĉ"n ĉ"n =
e
ĉ"k
e
ĉ"k0
M k
M k0
n
n
(13.5)
X †
1 X X i(k0 k)·n †
=
e
ĉ"k ĉ"k0 =
ĉ"k ĉ"k ,
M n
0
k,k
k
where the following identity is used:
1 X i(k0
e
M n
k)·n
=
k,k0 .
This equation holds for all reciprocal lattice vectors k, k0 .
42
(13.6)
Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
Similarly, the interaction term becomes
⌘ 1 X ⇣
X⇣
e2iq·n ĉ†"n ĉ†#n =
e2iq·n e
M
0
n
n,k,k
1 X ⇣ i(2q k
=
e
M
0
n,k,k
X
1 X i(2q
=
e
M n
k,k0
|
{z
=
X
k
=
0
ik·n †
ĉ"k e ik ·n ĉ†#k0
k0 )·n † †
ĉ"k ĉ#k0
k k0 )·n
!
ĉ†"k+q ĉ†#
k+q .
(13.7)
ĉ†"k ĉ†#k0
}
k+k0 ,2q
⌘
⌘
The nearest neighbour hopping term gives rise to a cosine dispersion, for
example the x-direction looks like
⌘
X ⇣ †
ĉ"m ĉ"n + ĉ†#m ĉ#n
hn,mix
=
X⇣
n
(ĉ†"n+(1,0,0) + ĉ†"n
1 X
=
e
M
0
i((k k0 )·n)
n,k,k
=
X
k
(1,0,0) )ĉ"n
eikx + e
|
{z
+ (ĉ†#n+(1,0,0) + ĉ†#n
ikx
=2 cos kx
⇣
⌘
2 cos kx ĉ†"k ĉ"k + ĉ†#k ĉ#k .
}
⇣
(1,0,0) )ĉ#n
⌘
ĉ†"k ĉ"k0 + ĉ†#k ĉ#k0
⌘
(13.8)
Finally, we have arrived at the mean field Hubbard Hamiltonian in momentum
space,
b =
H
X✓
k
⇠"k ĉ†"k ĉ"k + ⇠#k ĉ†#k ĉ#k
+
ĉ†"k+q ĉ†# k+q
+
ĉ#
k+q ĉ"k+q
◆
(13.9)
2
U
,
P
where ⇠ k = ✏ k µ = ↵ 2J ↵ (1 cos(k↵ )) µ . Note that in order to get the
dispersion correspond to that of a free particle in the limit of small k, the following
terms have been added to the Hamiltonian:
⌘ X
XX⇣
2
J ↵ ĉ† k ĉ k =
2J ↵ N .
(13.10)
↵
↵
k
13.1 Bogoliubov transformation in the FFLO case
The Hamiltonian in (13.9) can be written in an equivalent form as
X✓
b
H=
⇠"k+q ĉ†"k+q ĉ"k+q + ⇠# k+q (1 ĉ# k+q ĉ†# k+q )
k
+
ĉ†"k+q ĉ†# k+q
+
ĉ#
43
k+q ĉ"k+q
◆
2
U
.
(13.11)
Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
Now the operator part can be expressed as a matrix:
X✓
⇠"k+q ĉ†"k+q ĉ"k+q ⇠# k+q ĉ# k+q ĉ†#
k
+
ĉ†"k+q ĉ†#
X⇣
k
ĉ†"k+q
k+q
◆
ĉ# k+q ĉ"k+q =
◆✓
◆
⌘ ✓⇠
ĉ"k+q
"k+q
.
k+q
⇠# k+q
ĉ†# k+q
k+q +
ĉ#
(13.12)
The essential point in this representation is that it is a sum of independent 2 ⇥ 2
matrices that can be diagonalized separately. It is due to this feature that the
mean field approach under discussion is the most simple and straightforward way
to address superfluidity in a lattice theoretically.
Let us now derive a Bogoliubov transformation, B, for the terms of (13.12).
The requirements are that B diagonalizes the matrix and that B is canonical, i.e. it
preserves the fermionic anticommutation relations. Without any loss of generality
B can be assumed real in this case. Denoting the new basis operators as ˆ+ and
ˆ , we have
!
✓ ◆
✓
◆
ĉ"k+q
B11 ĉ" + B12 ĉ†#
ˆ+
=B †
=
(13.13)
ĉ# k+q
ˆ†
B21 ĉ" + B22 ĉ†#
and we require
B
✓
⇠"k+q
⇠#
k+q
◆
1
B
=
✓
E+,k,q
0
0
E
,k,q
◆
.
(13.14)
It should be pointed out that the sign in front of E is a choice of notation at this
point and does not a↵ect the results. From the anticommutation relations it follows
that
†
2
2
1 = {ˆ+ , ˆ+
} = {B11 ĉ" + B12 ĉ†# , B11 ĉ†" + B12 ĉ# } = B11
+ B12
2
2
1 = {ˆ , ˆ † } = {B21 ĉ†" + B22 ĉ# , B21 ĉ" + B22 ĉ†# } = B21
+ B22
0 = {ˆ+ , ˆ } = {B11 ĉ" +
B12 ĉ†# , B21 ĉ†"
(13.15)
+ B22 ĉ# } = B11 B21 + B12 B22 .
2
2
2
2
These conditions imply B11
= B22
, B12
= B21
, and B11 B21 + B12 B22 = 0, which
has several physically equivalent solutions that di↵er only by the locations of minus
signs. It is now possible to choose
✓
◆
u
v
B=
,
(13.16)
v u
where u, v 2 R and u2 + v 2 = 1. This form satisfies all the requirements given
above. Because B is a unitary matrix, it is enough to solve the eigenvalue problem
of the Hamiltonian and the columns of B are the eigenvectors. The eigenvalues are
s✓
◆2
⇠"k+q ⇠# k+q
⇠"k+q + ⇠# k+q
=
±
+ 2,
(13.17)
±
2
2
and therefore
E±,k
⇠k
= Ek ±
:=
2
s✓
⇠"k+q + ⇠#
2
k+q
44
◆2
+
2
±
⇠"k+q
2
⇠#
k+q
.
(13.18)
Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
For comparison with the standard BCS theory it is good to note that if both species
experience the same potential, i.e. the single particle dispersions are identical, ✏"k =
✏k = ✏#k , and in addition q = 0, this reduces to
s✓
◆2
µ# µ"
µ" + µ#
E±,k =
✏k
+ 2±
.
(13.19)
2
2
When µ" = µ# , the minimum value of both E+ and E is , which shows that
quasiparticle excitations have a minimum energy, i.e. energy gap, the magnitude of
which is .
For further applications, it is relevant to know the values of u2 , v 2 , and uv;
they are
✓
◆
1
⇠"k+q + ⇠# k+q
2
uk =
1+
2
2Ek
✓
◆
1
⇠"k+q + ⇠# k+q
2
(13.20)
vk =
1
2
2Ek
uk v k =
2Ek
.
The Hamiltonian can now be written in the form
◆✓
◆
⌘ ✓E
X ✓⇣ †
0
ˆ+,k
+,k
b =
H
+ ⇠#
ˆ+,k ˆ ,k
0
E ,k
ˆ ,k
k+q
k
◆
2
U
,
(13.21)
with E± defined in (13.28). The operators ˆ± are the quasiparticle operators. The
anticommutation relations are preserved by the canonical transformation. Thus the
system is now described as an ideal Fermi gas with dispersions given by E+ and
E :
b =
H
X⇣
†
E+,k ˆ+,k
ˆ+,k + E
†
,k ˆ ,k ˆ ,k
k
where the additional E
+ ⇠#
k+q
E
,k
⌘
2
U
,
(13.22)
appears because of the normal ordering.
13.2 Self-consistent crossover equations
It is now possible to derive a set of equations from which , µ" , and µ# can be
solved. These equations are called crossover equations. To start with the number
equations, note that the following holdsDfor the total
number of particles in either
E
P
one of the spin components: N = k ĉ† ,k ĉ ,k . It is straightforward to use the
inverse of the Bogoliubov transformation U to write the particle operators as linear
combinations of the quasiparticle operators as
✓ ◆
✓ ◆
✓ ◆ ✓
◆✓ ◆ ✓
◆
ĉ"
ˆ+
ˆ+
ˆ+
u v
uˆ+ + vˆ †
=B 1
= B†
=
=
. (13.23)
†
†
†
†
v u
ĉ#
ˆ
ˆ
ˆ
uˆ † vˆ+
45
Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
Thus we arrive at
N" =
=
XD
k
E XD
E
ĉ†"k ĉ"k =
ĉ†"k+q ĉ"k+q
X D⇣
k
†
uˆ+
⌘⇣
⌘E
uˆ+ + vˆ †
+ vˆ
k
=
XD
k
=
X
k
=
X
†
† †
u2 ˆ+
ˆ+ + v 2 ˆ ˆ † + uvˆ ˆ+ + uvˆ+
ˆ
D
E
D
E
†
u2 ˆ+
ˆ+ + v 2 ˆ ˆ †
u2k nF (E+,k ) + vk2 nF ( E
E
(13.24)
,k ).
k
Note that expectation values of type hˆ+ ˆ i are automatically zero. A similar equation holds for the number of down particles:
E
XD †
N# =
ĉ#k ĉ#k
k
X D⇣
=
k
X
=
k
X
=
uˆ †
vˆ+
⌘⇣
uˆ
†
vˆ+
D
E
D
E
†
u2 ˆ † ˆ + v 2 ˆ+ ˆ+
u2k nF (E
⌘E
(13.25)
+ vk2 nF ( E+,k ).
,k )
k
Using the original definition of the order parameter, U hĉ†"n ĉ†#n i = e2iq·n , it
is possible to derive the so called gap equation, which is derived here in the q = 0
limit:
*
+
D
E
X
0
† †
†
†
= ĉ"n ĉ#n =
e i(k+k )·n ĉ"k ĉ#k0
U
0
k,k
*
⇣
⌘
X
0
†
=
e i(k+k )·n uk ˆ+,k
+ vk ˆ , k ·
k,k0
⇣
=
v
X
k0 ˆ+, k0
†
+u
k0 ˆ ,k0
i(k+k0 )·n
e
uk v
k0
k,k0
+u
=
k0 v k
X
D
|
†
ˆ
, k ˆ ,k0
/
uk vk (1
{z
k0 ,
k
E
}
!
nF (E+,k )
k
46
⌘
D
+
†
ˆ+,k
ˆ+,
|
{z
/
k0 ,
nF (E
k0
k
E
}
,k )) .
(13.26)
Päivi Törmä, Frontiers of Quantum Gas Research – Lectures 7-12, ETH Zürich
Finally, after substituting
1=
U
/2U for uv and dividing by
, the gap equation is
X1
.
k
nF (E+,k ) nF (E
2Ek
,k )
(13.27)
Since U was defined as negative, the prefactor on the right hand side of the equation
is positive.
The number equations (13.24) and (13.25), and the gap equation (13.27)
together are equivalent with the standard BCS-Leggett theory. In the balanced
case, where N" = N# , it is possible to choose µ" = µ# by hand and eliminate
one of the number equations. If the interactions are weak, i.e. |U | is small, the
chemical potential can be approximated with the Fermi energy. However, when the
interaction strength increases, the chemical potential has to be solved from the
number equation in order to get the correct results.
The calculation becomes more involved once non-zero values for q are allowed.
In this case it is more feasible to solve
and q by minimizing the relevant free
energy, as described in the next section.
13.3 The nature of the eigenenergies in the FFLO state
Now, let us have a closer look at the energy eigenvalues
s✓
◆2
⇠k
⇠"k+q + ⇠# k+q
⇠"k+q ⇠#
E±,k = Ek ±
:=
+ 2±
2
2
2
k+q
.
(13.28)
It turns out that when solving for the FFLO state by minimizing the energy,
the FFLO wave vector will be q ' kF " kF # . The bigger the di↵erence in chemical
potentials or particle numbers, the bigger is q. It is thus plausible that the ⇠k above
becomes large when the spin-density imbalance grows. This means that some of the
energy eigenvalues become negative! In the BCS state, all of them are positive and
thus there are no quasiparticles at zero temperature. In the FFLO state, however,
there may exist single quasiparticles even in the ground state at zero temperature.
These are particles that do not participate in pairing. In other words, a Fermi
surface may exist in the system at the same time as it is a superfluid. This is called
a gapless superfluid. This means that even when most particles (momentum states)
are associated with the pairing gap , there are some momentum values where
gapless excitations are possible.
FFLO TO BE CONTINUED IN THE NEXT LECTURE...
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