Download Recent progresses on diagrammatic determinant QMC

Recent progresses on
diagrammatic determinant QMC
Lei Wang, ETH Zürich
Trento 2015.10
better scaling
entanglement & fidelity
Iazzi and Troyer, PRB 2015
LW and Troyer, PRL 2014
LW, Iazzi, Corboz and Troyer, PRB 2015
LW, Liu, Imriška, Ma and Troyer, PRX 2015
sign problem
Huffman and Chandrasekharan, PRB 2014
Li, Jiang and Yao, PRB 2015
LW, Liu, Iazzi, Troyer and Harcos, 1506.05349
Recent progresses on
diagrammatic determinant QMC
Lei Wang, ETH Zürich
Trento 2015.10
better scaling
entanglement & fidelity
Iazzi and Troyer, PRB 2015
LW and Troyer, PRL 2014
LW, Iazzi, Corboz and Troyer, PRB 2015
LW, Liu, Imriška, Ma and Troyer, PRX 2015
sign problem
Huffman and Chandrasekharan, PRB 2014
Li, Jiang and Yao, PRB 2015
LW, Liu, Iazzi, Troyer and Harcos, 1506.05349
About me
Background: condensed matter physics
Interested in quantum many-body
systems, quantum phase transitions, etc
Temperature
Please forgive my ignorance!
Quantum
Critical
Phase 1
Phase 2
c
Hubbard model of fermions in this talk
Solid materials
Optical lattices
Quantum Monte Carlo
achieved. We shall continue our discussion of Markov-chain Monte Carlo
methods in Subsection 1.1.4, but want to first take a brief look at the
history of stochastic computing.
The first recorded
Monte Carlo simulation
Historical origins
The idea of direct sampling was introduced into modern science in the
late 1940s by the mathematician Ulam, not without pride, as one can
find out from his autobiography Adventures of a Mathematician (Ulam
(1991)). Much earlier, in 1777, the French naturalist Buffon (1707–1788)
imagined a legendary needle-throwing experiment, and analyzed it completely. All through thehits
eighteenth and nineteenth centuries, royal courts
and learned circles were intrigued by this game, and the theory was developed further. After a basic treatment of the Buffon needle problem,
we shall describe the particularly brilliant idea of Barbier (1860), which
foreshadows modern techniques of variance reduction.
The Count is shown in Fig. 1.6 randomly throwing needles of length
a onto a wooden floor with cracks a distance b apart. We introduce
hN
2`
i=
⇡d
rcenter
φ
0 xcenter
b
2b
3b
4b
Fig. 1.7 Variables xcenter and φ in Buffon’s needle experiment. The needles are of length a.
d{
` {
coordinates rcenter and φ as in Fig. 1.7, and assume that the needles’
centers rcenter are uniformly distributed on an infinite floor. The needles
do not roll into cracks, as they do in real life, nor do they interact with
each other. Furthermore, the angle φ is uniformly distributed between 0
and 2 . This is the mathematical model for Buffon’s experiment.
All the cracks in the floor are equivalent, and there are symmetries
xcenter ↔ b − xcenter and φ ↔ −φ. The variable y is irrelevant to the
Buffon 1777
Fig. 1.6 Georges Louis Leclerc, Co
of Buffon (1707–1788), performing
first recorded Monte Carlo simulati
Statistical
Mechanics:
in 1777.
(Published
with permission
Algorithms
and
Computations
Le Monde.)
Fig. 1.10 Buffon’s experiment with 2000 needles (a = b).
1.1.3
Werner Krauth
THE JOURNAL
OF
CHEMICAL
PHYSICS
VOLUME
21,
NUMBER
6
JUNE,
1953
Equation of State Calculations by Fast Computing Machines
NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER,
Los Alamos Scientific Laboratory, Los Alamos, New Mexico
AND
EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois
(Received March 6, 1953)
A general method, suitable for fast computing machines, for investigatiflg such properties as equations of
state for substances consisting of interacting individual molecules is described. The method consists of a
modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere
system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared
to the free volume equation of state and to a four-term virial coefficient expansion.
I. INTRODUCTION
T
HE purpose of this paper is to describe a general
method, suitable for fast electronic computing
disks
and
spheres
102 Hard disks and spheres
machines, of calculating the properties of any substance
which may be considered as composed of interacting
individual molecules. Classical statistics is assumed,
only two-body forces are considered, and the potential
field of a molecule is assumed spherically symmetric.
These are the usual assumptions made in theories of
liquids. Subject to the above assumptions, the method
is not restricted to any range of temperature or density.
This paper will also present results of a preliminary twodimensional calculation for the rigid-sphere system.
Work on the two-dimensional case with a Lennardbe reported
Jones potential is in progress and will
η = 0.48 in a
η = 0.48
II. THE GENERAL METHOD FOR AN ARBITRARY
POTENTIAL BETWEEN THE PARTICLES
In order to reduce the problem to a feasible size for
numerical work, we can, of course, consider only a finite
number of particles. This number N may be as high as
several hundred. Our system consists of a squaret containing N particles. In order to minimize the surface
effects we suppose the complete substance to be periodic,
consisting of many such squares, each square containing N particles in the same configuration. Thus we
define dAB, the minimum distance between particles A
and B, as the shortest distance between A and any of
the particles B, of which there is one in each of the
squares which comprise the complete substance. If we
have a potential which falls off rapidly with distance,
there ηwill
be at
η =most
0.72 one of the distances AB which
= 0.72
THE JOURNAL
OF
CHEMICAL
PHYSICS
VOLUME
21,
NUMBER
6
JUNE,
1953
Equation of State Calculations by Fast Computing Machines
NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER,
Los Alamos Scientific Laboratory, Los Alamos, New Mexico
AND
EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois
(Received March 6, 1953)
A general method, suitable for fast computing machines, for investigatiflg such properties as equations of
state for substances consisting of interacting individual molecules is described. The method consists of a
modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere
system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared
to the free volume equation of state and to a four-term virial coefficient expansion.
I. INTRODUCTION
T
HE purpose of this paper is to describe a general
method, suitable for fast electronic computing
disks
and
spheres
102 Hard disks and spheres
machines, of calculating the properties of any substance
which may be considered as composed of interacting
individual molecules. Classical statistics is assumed,
only two-body forces are considered, and the potential
field of a molecule is assumed spherically symmetric.
These are the usual assumptions made in theories of
liquids. Subject to the above assumptions, the method
is not restricted to any range of temperature or density.
This paper will also present results of a preliminary twodimensional calculation for the rigid-sphere system.
Work on the two-dimensional case with a Lennardbe reported
Jones potential is in progress and will
η = 0.48 in a
η = 0.48
II. THE GENERAL METHOD FOR AN ARBITRARY
POTENTIAL BETWEEN THE PARTICLES
In order to reduce the problem to a feasible size for
numerical work, we can, of course, consider only a finite
number of particles. This number N may be as high as
several hundred. Our system consists of a squaret containing N particles. In order to minimize the surface
effects we suppose the complete substance to be periodic,
consisting of many such squares, each square containing N particles in the same configuration. Thus we
define dAB, the minimum distance between particles A
and B, as the shortest distance between A and any of
the particles B, of which there is one in each of the
squares which comprise the complete substance. If we
have a potential which falls off rapidly with distance,
there ηwill
be at
η =most
0.72 one of the distances AB which
= 0.72
Quantum to classical mapping
⇣
Z = Tr e
Ĥ
⌘
Trotterization
⇣
imaginary-time
Z = Tr e
M
Ĥ
...e
Ĥ = Ĥ0 + Ĥ1
Diagrammatic approach
M
Ĥ
⌘
..
.
Z=
1
X
k=0
k
Z
h
d⌧1 . . .
0
Tr ( 1)k e
(
Z
⌧k
1
d⌧k ⇥
⌧k )Ĥ0
Ĥ1 . . . Ĥ1 e
/M
0
space
⌧1 Ĥ0
0
imaginary-time axis
Beard and Wiese, 1996
Prokof ’ev, Svistunov, Tupitsyn,1996
i
Time
11
10
9
8
7
6
5
4
3
2
1
0
p
World-line Approach
Stochastic Series Expansion
Prokof ’ev et al, JETP, 87, 310 (1998)
Sandvik et al, PRB, 43, 5950 (1991)
44 18
40 36 31
32 14
28 19
24 20 16 46
12 34
8 4 38
0 35
l=0
45 30
41 37 7
33 15
29 6
25 21 17 47
13 2
9 5 39
1 3
l=1
46 16
42 38 4
34 12
30 45
26 22 18 44
14 32
10 6 29
2 13
l=2
47 17
43 39 5
35 0
31 36
27 23 19 28
15 33
11 7 37
3 1
l=3
v X(v) v X(v) v X(v) v X(v)
bosons
quantum spins
1
0
0
2
1
3
Determinantal Methods
0
2
1
3
fermions
Gull et al, RMP, 83, 349 (2011)
0
2
1
3
RE 58. Allowed vertices for the isotropic S = 1/2 Heisenberg model. The numbering l = 0, 1, 2, 3
vertex legs corresponds to the position v = 4p + l, in linked-list storage illustrated in Fig. 57.
3
2
RE 57. Linked vertex storage of the configuration in Fig. 55. In the graphical representation to
ft, constant spin states between operators have been replaced by lines (links) connecting the spins
efore and after the operator acts. The links can be stored in a list X(v), where the four elements
p + l, l = 0, 1, 2, 3, correspond to the legs (with the numbering convention shown in Fig. 58) of the
x at position p in the sequence SL . For two linked legs v and v′ , X(v) = v′ and X(v′ ) = v.
Space
Diagrammatic approaches
Diagrammatic approaches
bosons
quantum spins
fermions
World-line Approach
Stochastic Series Expansion
Determinantal Methods
Prokof ’ev et al, JETP, 87, 310 (1998)
Sandvik et al, PRB, 43, 5950 (1991)
Gull et al, RMP, 83, 349 (2011)
0
Diagrammatic determinant QMC
Z=
1
X
k=0
=
1
X
k=0
k
Z
d⌧1 . . .
0
k
X
Ck
Z
⌧k
w(Ck )
1
h
d⌧k Tr ( 1)k e
(
⌧k )Ĥ0
Ĥ1 . . . Ĥ1 e
⌧1 Ĥ0
i
Diagrammatic determinant QMC
Z=
1
X
k=0
=
1
X
k=0
k
Z
d⌧1 . . .
0
k
X
Z
⌧k
w(Ck )
Ck
0
1
h
d⌧k Tr ( 1)k e
(
⌧k )Ĥ0
Ĥ1 . . . Ĥ1 e
⌧1 Ĥ0
i
Diagrammatic determinant QMC
Z=
1
X
k=0
=
1
X
k=0
k
Z
d⌧1 . . .
0
k
X
Z
⌧k
w(Ck )
Ck
0
1
h
d⌧k Tr ( 1)k e
(
⌧k )Ĥ0
Ĥ1 . . . Ĥ1 e
⌧1 Ĥ0
i
Diagrammatic determinant QMC
Z=
1
X
k=0
=
1
X
k=0
k
Z
d⌧1 . . .
0
k
X
Z
⌧k
1
h
d⌧k Tr ( 1)k e
(
⌧k )Ĥ0
⌧1 Ĥ0
Ĥ1 . . . Ĥ1 e
i
Rubtsov et al, PRB 2005 Gull et al, RMP 2011
w(Ck )
Ck
det
✓
hki ⇠
0
Noninteracting
Green0 s functions
N , scales as O(
◆
k⇥k
3 3
3
N )
Diagrammatic determinant QMC
Z=
1
X
k=0
=
1
X
k=0
k
Z
d⌧1 . . .
0
k
X
Z
⌧k
1
h
d⌧k Tr ( 1)k e
(
⌧k )Ĥ0
⌧1 Ĥ0
Ĥ1 . . . Ĥ1 e
i
Rubtsov et al, PRB 2005 Gull et al, RMP 2011
w(Ck )
Ck
det
✓
hki ⇠
Noninteracting
Green0 s functions
N , scales as O(
◆
k⇥k
3 3
3
N )
0
Rombouts, Heyde and Jachowicz, PRL 1999
Iazzi and Troyer, PRB 2015 LW, Iazzi, Corboz and Troyer, PRB 2015
LCT-QMC
Methods
⇣
det I + T e
R
0
thus achieving O(
d⌧ HCk (⌧ )
⌘
N ⇥N
N 3 ) scaling!
Fidelity susceptibility made simple!
LW, Liu, Imriška, Ma and Troyer, PRX 2015
F
=
hkL kR i
2
hkL i hkR i
2
kR = 4
0
kL = 2
Signifies quantum phase
transitions without need of
the local order parameter
kL
Time
=
kR
Worldline Algorithms Stochastic Series Expansion Determinantal Methods
(bosons)
44 18
40 36 31
32 14
28 19
24 20 16 46
12 34
8 4 38
0 35
l=0
45 30
41 37 7
33 15
29 6
25 21 17 47
13 2
9 5 39
1 3
l=1
46 16
42 38 4
34 12
30 45
26 22 18 44
14 32
10 6 29
2 13
l=2
47 17
43 39 5
35 0
31 36
27 23 19 28
15 33
11 7 37
3 1
l=3
v X(v) v X(v) v X(v) v X(v)
2
11
10
9
8
7
6
5
4
3
2
1
0
F
hkL kR i
p
LW, Liu, Imriška, Ma and Troyer, PRX 2015
hkL i hkR i
2
kL
Signifies quantum phase
transitions without need of
the local order parameter
(quantum spins)
kR
1
0
3
kL
0
2
kR
1
(fermions)
0
2
1
3
0
2
1
3
FIGURE 58. Allowed vertices for the isotropic S = 1/2 Heisenberg model. The numbering l = 0, 1, 2, 3
of the vertex legs corresponds to the position v = 4p + l, in linked-list storage illustrated in Fig. 57.
3
2
FIGURE 57. Linked vertex storage of the configuration in Fig. 55. In the graphical representation to
the left, constant spin states between operators have been replaced by lines (links) connecting the spins
just before and after the operator acts. The links can be stored in a list X(v), where the four elements
v = 4p + l, l = 0, 1, 2, 3, correspond to the legs (with the numbering convention shown in Fig. 58) of the
vertex at position p in the sequence SL . For two linked legs v and v′ , X(v) = v′ and X(v′ ) = v.
Space
Fidelity susceptibility made simple!
More advantages
1 X
hÔi =
Z
k
@hÔi
hÔki
=
@
hÔihki
w(Ck )O(Ck )
Ck
Histogram reweighing
Lee-Yang zeros
=
hÔi(k)
Observable derivatives
k
X
x
x
x x
x
x
x
x
x
x
k
Directly sample derivatives of
any observable
x
Can obtain observables in a
continuous range of coupling strengths
x
x
x
x x
<
Partition function zeros in the
complex coupling strength plane
Ferrenberg et al, 1988 Wang and Landau, 2001 Troyer et al, 2003
What about the sign problem ?
Sign problem free
time-reversal symmetry
M" =
w(Ck ) = det M" ⇥ det M#
= | det M" |
2
0
⇤
M#
Lang et al, Phys. Rev. C, 1993
Koonin et al, Phys. Rep, 1997
Hands et al, EPJC, 2000
Wu et al, PRB, 2005
uantum Monte Carlo simulations. The quantum spin liquid of the
k spin-orbit interaction, and is not adiabatically connected to the
ritical value of U > Uc both states are unstable toward magnetic
ate we study the spin, charge, and single-particle dynamics of the
the Hubbard interaction only on a ribbon edge. The Hubbard
currents along the edge and promotes edge magnetism but leaves
elical liquid intact.
What about the sign problem ?
PACS numbers: 75.10.!b, 03.65.Vf, 71.10.Pm, 71.30.+h
del on the
þ HU ,
cj ;
Sign problem free
time-reversal symmetry
Bulk phase diagram.—For bulk simulations we use the
projective auxiliary field QMC approach. The ground state
j!0 i is filtered out of a trial wave function j!T i with
h!T j!0 i ! 0; a very good choice is the ground state of
the KM model. For an arbitrary observable, h!0 jOj!0 i ¼
lim"!1 h!T je!"H=2 Oe!"H=2 j!T i=h!T je!"H j!T i. The
absence of the negative sign problem at half filling follows
from the fact that, after a discrete Hubbard-Stratonovich
transformation of HU and subsequent integration over the
fermionic degrees of freedom, the fermionic determinants
in the up and down spin sectors are linked via complex
conjugation such that their product is positive. We employ
an SUð2Þ invariant Hubbard-Stratonovich transformation
and an imaginary time discretization of #"t ¼ 0:1.
Projection parameters "t ¼ 40 prove sufficient for converged (within statistical errors) ground-state results. For
details of the algorithm, see [19].
The SO coupling reduces the SUð2Þ symmetry to a
Uð1Þ symmetry corresponding to spin rotations around
the z axis. The Hubbard interaction promotes transverse,
x-y magnetic ordering [15] which can be tracked by
M" =
w(Ck ) = det M" ⇥ det M#
= | det M" |
2
0
Lang et al, Phys. Rev. C, 1993
Koonin et al, Phys. Rep, 1997
Hands et al, EPJC, 2000
B
C
Wu et al, PRB, 2005
A
K
K
Attractive interaction at any filling on any lattice
Repulsive interaction at half-filling on bipartite lattices
And more …
PRL 111, 066401 (2013)
PHYSICAL REVIEW LETTERS
week ending
9 AUGUST 2013
exactly N=2 particles on each lattice site, H would reduce to
an exact SUðNÞ-symmetric Heisenberg(b)
model HH ¼
P λ
U
U
J=2N hi;ji
Si $ Sj , with Si being the vector of the SUðNÞ λ
P
y
a
a
t spin operators Si ¼ !;" ci! T!" ci" , expressed in terms of t
a
the generators T a ,2 a ¼ 1; . . . ; N 2 % 1; of SUðNÞ in the
δ
1
fundamental representation, with TrðT a T b Þ ¼ #ab =2 [e.g.,
T a ¼ $a =2 in terms of the Pauli matrices $a for SU(2)].
1 the large U limit of a model that in
This would result, e.g.,ain
δ3 to H also includes a local Hubbard-U interaction,
addition
δ2the local particle fluctuations around the
which reduces
mean value of N=2. In the large-N limit, and considering
only the paramagnetic saddle point, these fluctuations
become irrelevant and the Hubbard-U interaction merely
fixes the average particle number to N=2. The
FIG. 1 (color online).
Periodic
latticethe structure
the 1 (color online). Ground state phase diagram of fermions
Hamiltonian(a)
in Eq.
(1) indeed equals
U ¼ 0 limit ofof FIG.
the
Hubbard-Heisenberg
Hamiltonian
of
the
seminal
with SUðNÞ-symmetric flavor exchange on the honeycomb latKM-Hubbard model with nearest-neighbor hopping t, spin-orbit
works in Refs. [2–4], and represents an unrestricted
tice. Crosses denote parameters at which QMC simulations have
coupling !, and SUðNÞ-symmetric
Coulomb repulsion
U.It has
Arrows
indicate been
the carried out. For all considered (even) N the system undert % J model.
been considered
goes a quantum phase transition from a semimetal (SM) to an
using with
QMC simulations
on the square
lattice,
current direction previously
associated
the spin-orbit
term
for one
insulator. For N ( 6 the insulating state is a columnar valence
where an exotic gapless spin liquid was obtained for N ¼ 4
spin species and sublattice.
(b) Effective model on a semi-infinite
bond solid (cVBS), while at N ¼ 4 it is a valence bond solid with
flavors [27]. In the following, we consider the case of the
Fig. 1. (A) Fermi surface of the Fermi liquid phase of a single band model on the square
unit lattice spacing. The “hot spots” are denoted by the filled circles. (B) The reconstru
surface in the metal with SDW order. The dashed lines show the Fermi surface in the metal w
order, and its translation by K. Gaps have opened at the hot spots, leading to small “poc
surfaces. (C) A deformed Fermi surface of the metal without SDW order, in which the vicin
hot spots are unchanged from (A). The horizontal and vertical Fermi surfaces now belong t
electronic bands.
1
(a)
a Wannier
ree nearest
'"3 g, see
ext-nearest
=j"n ( "m j
article-hole
ted with a Hohenadler, Lang and Assaad, PRL 2011
ing the in-
Topological insulators
SU(2N) models
Lang, Meng, Muramatsu, Wessel
and Assaad, PRL 2013
resonating valence bond plaquettes (pVBS); both are depicted in
1
A
0
K
−0.5
−1
−1
B
0.5
0.5
ky /π
(1)
⇤
M#
ky /π
arises from
time reverdge states,
s spin and
ven by the
ces to two
gns of the
me reversal
ential scat[4,5], and
vious work
wo routes:
or (as here)
b repulsion
rlo (QMC)
transition
se of [18]
t magnetic
the helical
that the renormalization group and Feyn
expansions of the field theory flow to s
pling in two spatial dimensions, mak
analytical progress difficult.
Here, we show that the universa
field theory can be realized in lattice m
are free of the sign problem and so is
to large-scale QMC studies. Our claim
contradict the no-go theorem of (5), b
do not provide a general recipe for e
the sign problem. However, we will
it for the specific case of the onset
romagnetic order in a two-dimensio
provided the perturbative arguments
portance of the hot spots to the qua
theory (7, 8, 10, 11) apply. Our modi
model has at least two bands. Therefor
in which there is only a single active b
transition, such as in the electron-doped
our method requires modifying the
face far away from the hot spots; we
this can be done while preserving the
low-energy properties of the antiferr
critical point. On the other hand, o
applies to multiband situations (suc
sign problem. Such algorithms express the partition function as a sum over Feynman histories,
and the sign problem arises when the weights
assigned to the trajectories are not all positive
because of quantum interference effects. A general solution to the fermion sign problem has been
proved to be in the computational complexity class
of nondeterministic polynomial (NP) hard (5),
and so there has been little hope that the antiferromagnetic quantum critical point could be elucidated by computational studies.
Application of the methods of quantum field
theory and the renormalization group to the onset of antiferromagnetism in a metal (6) has
identified (7, 8) a universal quantum field theory
that captures all the singular low-energy quantum fluctuations that control the quantum critical point and deviations from the Fermi liquid
physics of traditional metals. The field theory
is expressed in terms of fermionic excitations in
the vicinity of a finite number of “hot spots” on
the Fermi surface, and is thus independent of the
details of the fermionic band structure, except for
the number of hot spots and Fermi velocities at
the hot spots (9). Recent work (10, 11) has shown
0
−0.5
0
kx /π
1
−1
−1
0
kx
Fig. 2. (A) Fermi surfaces (solid lines) of LF for free yx,y fermions with the parameters li
text. The dashed lines show the portion of the Fermi surface in Fig. 1C that was shifte
obtain the yy Fermi surface. The hot spots are now
at the intersections of the Fermi su
→
Mean-field yx,y Fermi surfaces with SDW order j〈f〉j = 0:25.
Two-orbital model
Berg, Metliski and Sachdev,
Science SCIENCE
2012
www.sciencemag.org
VOL 338
What about the sign problem ?
Sign problem free
time-reversal symmetry
M" =
w(Ck ) = det M" ⇥ det M#
= | det M" |
2
⇤
M#
Lang et al, Phys. Rev. C, 1993
Koonin et al, Phys. Rep, 1997
Hands et al, EPJC, 2000
Wu et al, PRB, 2005
0
But, how about this ?
spinless fermions Ĥ = t
X⇣
ĉ†i ĉj
hi,ji
+
ĉ†j ĉi
⌘
+V
X
n̂i n̂j
hi,ji
w(Ck ) = det M
Scalapino et al, PRB 1984 Gubernatis et al, PRB 1985
Meron cluster approach, Chandrasekharan and Wiese, PRL 1999
up to 8*8 square lattice and T≥0.3t
solves sign problem for V ≥ 2t
Solutions !
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 89, 111101(R) (2014)
Solution to sign problems in half-filled spin-polarized electronic systems
Emilie Fulton Huffman and Shailesh Chandrasekharan
Department of Physics, Duke University, Durham, North Carolina 27708, USA
(Received 19 December 2013; revised manuscript received 14 February 2014; published 12 March 2014)
We solve the sign problem in a particle-hole symmetric spin-polarized fermion model on bipartite lattices
using the idea of fermion bags. The solution can be extended to a class of models at half filling but without
particle-hole symmetry. Attractive Hubbard models with an odd number of fermion species can also be solved.
Our solutions should allow us to study quantum phase transitions that have remained unexplored so far due to
sign problems.
DOI: 10.1103/PhysRevB.89.111101
Quantum Monte Carlo methods for many-body fermionic
systems in thermal equilibrium usually require one to be
able to rewrite quantum partition functions as a sum over
classical configurations with positive Boltzmann weights that
are computable in polynomial time. Unfortunately, due to the
underlying quantum nature of the problem, the Boltzmann
weights can be negative or even complex in general. Such
expansions are said to suffer from a sign problem [1]. The
discovery of an expansion with positive Boltzmann weights
is referred to as a solution to the sign problem. Solutions to
sign problems in many quantum systems are considered to be
outstanding problems in computational complexity [2].
Traditionally, solutions are based on rewriting the interacting problem as a free fermion problem where fermions only interact with background auxiliary fields [3–6]. The Boltzmann
weight then depends on the determinant of the free fermion
matrix, which can still be negative or complex. However, in
electronic systems, a symmetric treatment of both spin components of the electron can sometimes make the Boltzmann
weight positive since it can be written as the product of two real
determinants that come with the same sign [7]. Sign problems
PACS number(s): 71.10.Fd, 71.27.+a, 71.30.+h
neighbor Hubbard-type interaction. The model is well known
as the tV model and was considered on square lattices a
long time ago [9,10]. Although the model has a particle-hole
symmetry, as far as we know its sign problem has not been
solved by traditional methods for any value of V . Thus, it
seems like the tV model at half filling in spin-polarized
systems is more difficult to solve than the traditional Hubbard
model with an on-site U interaction between the two spins.
Unlike the traditional Hubbard model, here the V < 0 model
cannot be mapped into the V > 0 model through a unitary
transformation. In the repulsive case for V ! 2t the sign
problem could indeed be solved using a nontraditional method
called the meron-cluster approach [11]. Unfortunately, that
solution could not be extended to smaller values of V .
The spin-polarized t-V model (1) is of interest from a fundamental quantum field theory perspective since it describes
a minimally doubled lattice fermion system [12]. A similar
minimally doubled fermion system can be obtained with
Hamiltonian staggered fermions on a square lattice [13,14].
These models contain an interesting quantum phase transition between a semimetal phase (containing massless Dirac
Solutions !
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 89, 111101(R) (2014)
Solution to sign problems in half-filled spin-polarized electronic systems
Emilie Fulton Huffman and Shailesh Chandrasekharan
Department of Physics, Duke University, Durham, North Carolina 27708, USA
(Received 19 December 2013; revised manuscript received 14 February 2014; published 12 March 2014)
RAPID COMMUNICATIONS
We solve the sign problem in a particle-hole symmetric spin-polarized fermion model on bipartite lattices
REVIEW
241117(R)
(2015)
using the idea of fermion bags. The solution can bePHYSICAL
extended to
a class Bof91,
models
at half
filling but without
particle-hole symmetry. Attractive Hubbard models with an odd number of fermion species can also be solved.
Solving
theallow
fermion
signquantum
problem
intransitions
quantum
Carlo unexplored
simulations
bydue
Majorana
Our solutions
should
us to study
phase
thatMonte
have remained
so far
to
sign problems.
Zi-Xiang Li,1 Yi-Fan Jiang,1,2 and Hong Yao1,3,*
DOI: 10.1103/PhysRevB.89.111101 1 Institute for Advanced Study,
PACSTsinghua
number(s):
71.10.Fd,
71.27.+a,
University,
Beijing
100084,71.30.+h
China
representation
2
Department of Physics, Stanford University, Stanford, California 94305, USA
Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
for (Received
many-body
fermionic
neighbor
Hubbard-type
interaction.
The model
is well
known
27 August
2014; revised
manuscript
received 13 October
2014; published
30 June
2015)
3
Quantum Monte Carlo methods
systems in thermal equilibrium usually require one to be
as the tV model and was considered on square lattices a
discover as
a quantum
(QMC)
method
to
solve the
fermion sign problem
in interacting
fermion
able to rewrite quantum partition We
functions
a sumMonte
over Carlolong
time
agoPHYSICAL
[9,10].
Although
model
has
a particle-hole
REVIEWthe
B 91,
235151
(2015)
models
by employing
a Majorana
of complex
We call
the “Majorana
(MQMC).
classical configurations with positive
Boltzmann
weights
that representation
symmetry,
as far fermions.
as we know
itsit sign
problemQMC”
has not
been
MQMC
simulations
can
be
performed
efficiently
both
at
finite
and
zero
temperatures.
Especially,
MQMC
are computable in polynomial time. Unfortunately, due to the
solved by traditional methods for any value of V . Thus, isit
Efficient
continuous-time
Carlo
ground
fermion
sign free
in Boltzmann
simulating a class seems
of quantum
spinless
fermion
models
on bipartite
lattices
atfor
half
filling
and with anstate of correlated fermions
underlying quantum nature of the
problem,
the
like
theMonte
tV
model
at method
half
filling
inthe
spin-polarized
range of (unfrustrated) interactions. Moreover, we find a class of SU (N ) fermionic models with odd
weights can be negative or evenarbitrary
complex
in general. Such
systems is1more difficult1to solve than the traditional
Hubbard 1
2
N , which are sign free in MQMC but whose
sign problem
cannot
be Philippe
in solved in
other QMC
methods,
such
as
Lei
Wang,
Mauro
Iazzi,
Corboz,
and
Matthias
Troyer
expansions are said to suffer from a sign problem [1]. The
model1 with an on-site U interaction between the two spins.
continuous-time QMC. To the best of our knowledge,
MQMC
is theETH
first Zurich,
auxiliary8093
fieldZurich,
QMC method
to solve
Theoretische
Physik,
Switzerland
discovery of an expansion with positive
Boltzmann
weights
Unlike
the
traditional
Hubbard
model,
here
the
V
<
0
model
2
the fermion
sign problem
in spinless
(more generally,
number ofScience
species)Park
fermion
conjecture
Institute
for Theoretical
Physics,
Universityanofodd
Amsterdam,
904models.
PostbusWe
94485,
1090 GL Amsterdam, The Netherlands
is referred to as a solution to the that
signMQMC
problem.
tosolve cannot
be sign
mapped
into
the V
> 0 fermionic
model through
a unitary
couldSolutions
be applied
to
the
fermion
problem
in
more
generic
models.
January 2015; revised manuscript received 13 March 2015; published 30 June 2015)
sign problems in many quantum systems are considered(Received
to be 12transformation.
In the repulsive case for V ! 2t the sign
DOI: 10.1103/PhysRevB.91.241117
PACS
number(s):
05.30.Fk,
02.70.Ss,
71.27.+a
outstanding problems in computational
complexity [2].
problem
could
indeedof
be
using
a nontraditional
method
We present the ground
state
extension
thesolved
efficient
continuous-time
quantum
Monte Carlo algorithm for
Traditionally, solutions are based on rewriting lattice
the interactcalled
meron-cluster
[11]. Unfortunately,
fermions of M.
Iazzithe
and M.
Troyer, Phys. approach
Rev. B 91, 241118
(2015). Based on that
continuous-time expansion
ing problem as a free fermion
problem
where
fermions
only
insolution
could
not
be
extended
to
smaller
values
of
V
.
of an imaginary-time
projection
the algorithm
is free of fermions.
systematicConsequently,
error and scales
with
Introduction. Interacting fermionic
quantum systems
with operator,
represented
as two Majorana
welinearly
can
teract with background
auxiliary
fields and/or
[3–6].topological
Theprojection
Boltzmann
The
spin-polarized
t-V
model
(1)
is of
interest
from
funtime andhave
interaction
strength.
Compared
to the
conventional
quantum
Carlo methods
strong
correlations
properties
attracted
express
spinless
fermion
Hamiltonians
inMonte
aaMajorana
represen-for lattice
weight then depends increasing
on the determinant
of theNonetheless,
free
fermion
quantum
field
theory
perspective
sincepowerful
it describes
fermions,
thisinapproach
hashigher
greater
flexibility
and
is
easier
to combine
with
machinery
such as histogram
attention [1,2].
twodamental
and
tation
and
then
perform
Hubbard-Stratonovich
(HS) transformatrix, which can still
be negative
or complex.
However,
inand extended
spatial
dimensions,
strongly interacting
are doubled
mations
to decouple
by introducing
a systems
minimally
lattice
fermion
systemthe
[12].
A similar ofbackground
reweightingquantum
ensemble
simulation
techniques.
Weinteractions
discuss
implementation
the continuous-time
the reach
ofprojection
analytical
in thethe
sense
auxiliary
fields.
Under
certain
conditions,
as particle-hole
electronic systems, a generically
symmetricbeyond
treatment
of both
spin cominmethods
detailminimally
using
spinless
t-V fermion
model
as an
example
compare
thesuch
numerical
results with exact
doubled
system
canandbe
obtained
with
of solving
those quantum
in an unbiased
way.matrix
As anrenormalization
symmetry,
we can
find
symmetric
treatment
species
ponents of the electron
can sometimes
make models
thediagonalization,
Boltzmann
density
group,
and
infinite
projected
entangled-pair
states
calculations.
Hamiltonian
staggered
fermions
on
a asquare
lattice
[13,14].of two
unbiased
the the
quantum
of fermionic
Majorana
fermions,
namely,
freetransiMajorana
weight positive since itintrinsically
can be written
as thenumerical
productFinally
ofmethod,
twowereal
use
methodMonte
to study contain
the
quantum
critical
point the
of
spinless
fermions fermion
on a honeycomb
These
models
an interesting
quantum
phase
Carlo
(QMC)
simulation
plays
a
key
role
in
understanding
Hamiltonian
obtained
after
HS
transformations
is
a
sum
of two
determinants that come with the same sign [7]. Sign
problems
lattice
and confirm tion
previous
resultsaconcerning
its phase
critical (containing
exponents. massless Dirac
between
semimetal
Solutions !
RAPID COMMUNICATIONS
PHYSICAL REVIEW B 89, 111101(R) (2014)
Solution to sign problems in half-filled spin-polarized electronic systems
Emilie Fulton Huffman and Shailesh Chandrasekharan
Department of Physics, Duke University, Durham, North Carolina 27708, USA
(Received 19 December 2013; revised manuscript received 14 February 2014; published 12 March 2014)
RAPID COMMUNICATIONS
We solve the sign problem in a particle-hole symmetric spin-polarized fermion model on bipartite lattices
REVIEW
241117(R)
(2015)
using the idea of fermion bags. The solution can bePHYSICAL
extended to
a class Bof91,
models
at half
filling but without
particle-hole symmetry. Attractive Hubbard models with an odd number of fermion species can also be solved.
Solving
theallow
fermion
signquantum
problem
intransitions
quantum
Carlo unexplored
simulations
bydue
Majorana
Our solutions
should
us to study
phase
thatMonte
have remained
so far
to
sign problems.
Zi-Xiang Li,1 Yi-Fan Jiang,1,2 and Hong Yao1,3,*
DOI: 10.1103/PhysRevB.89.111101 1 Institute for Advanced Study,
PACSTsinghua
number(s):
71.10.Fd,
71.27.+a,
University,
Beijing
100084,71.30.+h
China
representation
2
Department of Physics, Stanford University, Stanford, California 94305, USA
Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
for (Received
many-body
fermionic
neighbor
Hubbard-type
interaction.
The model
is well
known
27 August
2014; revised
manuscript
received 13 October
2014; published
30 June
2015)
3
Quantum Monte Carlo methods
systems in thermal equilibrium usually require one to be
as the tV model and was considered on square lattices a
discover as
a quantum
(QMC)
method
to
solve the
fermion sign problem
in interacting
fermion
able to rewrite quantum partition We
functions
a sumMonte
over Carlolong
time
agoPHYSICAL
[9,10].
Although
model
has
a particle-hole
REVIEWthe
B 91,
235151
(2015)
models
by employing
a Majorana
of complex
We call
the “Majorana
(MQMC).
classical configurations with positive
Boltzmann
weights
that representation
symmetry,
as far fermions.
as we know
itsit sign
problemQMC”
has not
been
MQMC
simulations
can
be
performed
efficiently
both
at
finite
and
zero
temperatures.
Especially,
MQMC
are computable in polynomial time. Unfortunately, due to the
solved by traditional methods for any value of V . Thus, isit
Efficient
continuous-time
Carlo
ground
fermion
sign free
in Boltzmann
simulating a class seems
of quantum
spinless
fermion
models
on bipartite
lattices
atfor
half
filling
and with anstate of correlated fermions
underlying quantum nature of the
problem,
the
like
theMonte
tV
model
at method
half
filling
inthe
spin-polarized
range of (unfrustrated) interactions. Moreover, we find a class of SU (N ) fermionic models with odd
weights can be negative or evenarbitrary
complex
in general. Such
systems is1more difficult1to solve than the traditional
Hubbard 1
2
N , which are sign free in MQMC but whose
sign problem
cannot
be Philippe
in solved in
other QMC
methods,
such
as
Lei
Wang,
Mauro
Iazzi,
Corboz,
and
Matthias
Troyer
expansions are said to suffer from a sign problem [1]. The
model1 with an on-site U interaction between the two spins.
continuous-time QMC. To the best of our knowledge,
MQMC
is theETH
first Zurich,
auxiliary8093
fieldZurich,
QMC method
to solve
Theoretische
Physik,
Switzerland
discovery of an expansion with positive
Boltzmann
weights
Unlike
the
traditional
Hubbard
model,
here
the
V
<
0
model
2
the fermion
sign problem
in spinless
(more generally,
number ofScience
species)Park
fermion
conjecture
Institute
for Theoretical
Physics,
Universityanofodd
Amsterdam,
904models.
PostbusWe
94485,
1090 GL Amsterdam, The Netherlands
is referred to as a solution to the that
signMQMC
problem.
tosolve cannot
be sign
mapped
into
the V
> 0 fermionic
model through
a unitary
couldSolutions
be applied
to
the
fermion
problem
in
more
generic
models.
January 2015; revised manuscript received 13 March 2015; published 30 June 2015)
sign problems in many quantum systems are considered(Received
to be 12transformation.
In the repulsive case for V ! 2t the sign
DOI: 10.1103/PhysRevB.91.241117
PACS
number(s):
05.30.Fk,
02.70.Ss,
71.27.+a
outstanding problems in computational
complexity [2].
problem
could
indeedof
be
using
a nontraditional
method
We present the ground
state
extension
thesolved
efficient
continuous-time
quantum
Monte Carlo algorithm for
Traditionally, solutions are based on rewriting lattice
the interactcalled
meron-cluster
[11]. Unfortunately,
fermions of M.
Iazzithe
and M.
Troyer, Phys. approach
Rev. B 91, 241118
(2015). Based on that
continuous-time expansion
1506.05349
ing problem as a free
fermion
problem
where
fermions
only
insolution
could
not
be
extended
to
smaller
values
of
V
.
of an imaginary-time
projection
the algorithm
is free of fermions.
systematicConsequently,
error and scales
with
Introduction. Interacting fermionic
quantum
with operator,
represented
as two Majorana
welinearly
can
Splitsystems
orthogonal
group:
teract with background
auxiliary
[3–6].topological
The
Boltzmann
The
spin-polarized
t-V
model
(1)
is of
interest
from
funprojection
time
andhave
interaction
strength.
Compared
to the
conventional
quantum
Carlo methods
strong
correlations
and/or
properties
attracted
express
spinless
fermion
Hamiltonians
inMonte
aaMajorana
represen-for lattice
Afields
guiding
principle
for
sign-problem-free
fermionic
simulations
weight then depends increasing
on the determinant
of theNonetheless,
free
fermion
quantum
field
theory
perspective
sincepowerful
it describes
fermions,
thisinapproach
hashigher
greater
flexibility
and
is
easier
to combine
with
machinery
such as histogram
attention [1,2].
twodamental
and
tation
and
then
perform
Hubbard-Stratonovich
(HS) transformatrix, which can still
be negative
or complex.
However,
in
spatial
dimensions,
strongly
interacting
mations
to decouple
by introducing
a systems
minimally
lattice
fermion
systemthe
[12].
A2 similar ofbackground
reweightingquantum
ensemble
simulation
techniques.
Weinteractions
discuss
implementation
the continuous-time
1
1and extended
1are doubled
1
Lei
Wang
,
Ye-Hua
Liu
,
Mauro
Iazzi
,
Matthias
Troyer
and
Gergely
Harcos
the reach
ofprojection
analytical
in thethe
sense
auxiliary
fields.
Under
certain
conditions,
as particle-hole
electronic systems, a generically
symmetricbeyond
treatment
of
spin cominmethods
detail using
spinless
t-V fermion
model
as an
example
andbe
compare
thesuch
numerical
results with exact
doubled
system
canand
obtained
with
1 both
Theoretische Physik, minimally
ETH Zurich,
8093 Zurich,
Switzerland
of solving
models
in an unbiased
way.matrix
As anrenormalization
symmetry,
we can
find
symmetric
treatment
species
ponents of the electron
can sometimes
makeInstitute
thediagonalization,
Boltzmann
2those quantum
density
group,
and
infinite
projected
entangled-pair
states
calculations.
Hamiltonian
staggered
fermions
on
a asquare
lattice
[13,14].of two
Alfréd Rényi
of Mathematics,
Reáltanoda
utca 13-15.,
Budapest
H-1053,
Hungary
unbiased
the the
quantum
of fermionic
Majorana
fermions,
namely,
freetransiMajorana
weight positive since itintrinsically
can be written
as thenumerical
productFinally
ofmethod,
twowereal
use
methodMonte
to study contain
the
quantum
critical
point the
of
spinless
fermions fermion
on a honeycomb
These
models
an interesting
quantum
phase
Carlo
(QMC)
simulation
plays
a
key
role
in
understanding
Hamiltonian
obtained
after
HS
transformations
is
a
sum
of two
determinants that come with theWe
same
sign a[7].
Sign
problems
lattice
and confirm
previous
results
its phase
criticaland
exponents.
present
guiding
principle
for designing
fermionic
Hamiltonians
quantummassless
Monte Carlo
tion
between
aconcerning
semimetal
(containing
Dirac
A tale of open science
⇣
w(Ck ) ⇠ det I + T e
R
0
d⌧ HCk (⌧ )
⌘ Free fermions with an
effective imaginary-time
dependent
A tale of open science
⇣
w(Ck ) ⇠ det I + T e
R
0
Let real matrices Ai =
then det I + e
A1 A2
e
d⌧ HCk (⌧ )
✓
0
BiT
AN
...e
⌘ Free fermions with an
Bi
0
effective imaginary-time
dependent
◆
0
http://mathoverflow.net/questions/204460/
how-to-prove-this-determinant-is-positive
A tale of open science
⇣
w(Ck ) ⇠ det I + T e
R
0
Let real matrices Ai =
then det I + e
A1 A2
e
d⌧ HCk (⌧ )
✓
0
BiT
⌘ Free fermions with an
Bi
0
AN
...e
The conjecture was
proved by Gergely
Harcos and Terence
Tao, with inputs from
others
https://terrytao.wordpress.com/2015/05/03/
the-standard-branch-of-the-matrix-logarithm/
Tao and Paul Erdős in 1985
effective imaginary-time
dependent
◆
0
http://mathoverflow.net/questions/204460/
how-to-prove-this-determinant-is-positive
A tale of open science
⇣
w(Ck ) ⇠ det I + T e
R
0
Let real matrices Ai =
then det I + e
A1 A2
e
d⌧ HCk (⌧ )
✓
0
BiT
⌘ Free fermions with an
Bi
0
AN
...e
The conjecture was
proved by Gergely
Harcos and Terence
Tao, with inputs from
others
https://terrytao.wordpress.com/2015/05/03/
the-standard-branch-of-the-matrix-logarithm/
Tao and Paul Erdős in 1985
effective imaginary-time
dependent
◆
0
http://mathoverflow.net/questions/204460/
how-to-prove-this-determinant-is-positive
A tale of open science
⇣
w(Ck ) ⇠ det I + T e
R
0
Let real matrices Ai =
then det I + e
A1 A2
e
d⌧ HCk (⌧ )
✓
0
BiT
⌘ Free fermions with an
Bi
0
AN
...e
The conjecture was
proved by Gergely
Harcos and Terence
Tao, with inputs from
others
https://terrytao.wordpress.com/2015/05/03/
the-standard-branch-of-the-matrix-logarithm/
Tao and Paul Erdős in 1985
effective imaginary-time
dependent
◆
0
http://mathoverflow.net/questions/204460/
how-to-prove-this-determinant-is-positive
A tale of open science
⇣
w(Ck ) ⇠ det I + T e
R
0
Let real matrices Ai =
then det I + e
A1 A2
e
d⌧ HCk (⌧ )
✓
0
BiT
⌘ Free fermions with an
Bi
0
AN
...e
The conjecture was
proved by Gergely
Harcos and Terence
Tao, with inputs from
others
https://terrytao.wordpress.com/2015/05/03/
the-standard-branch-of-the-matrix-logarithm/
Tao and Paul Erdős in 1985
effective imaginary-time
dependent
◆
0
http://mathoverflow.net/questions/204460/
how-to-prove-this-determinant-is-positive
A new de-sign principle
LW, Liu, Iazzi, Troyer and Harcos 1506.05349
M T ⌘M = ⌘
⌘ = diag(I, I)
A new de-sign principle
LW, Liu, Iazzi, Troyer and Harcos 1506.05349
M T ⌘M = ⌘
⌘ = diag(I, I)
O+ (n, n)
O++ (n, n)
O
O
M 2 O(n, n)
split orthogonal group
(n, n)
+
(n, n)
A new de-sign principle
LW, Liu, Iazzi, Troyer and Harcos 1506.05349
M T ⌘M = ⌘
⌘ = diag(I, I)
O+ (n, n)
0
⌘0
det (I + M )
has a definite sign
for each component !
O++ (n, n)
O
(n, n)
O
0
+
(n, n)
⌘0
A new de-sign principle
LW, Liu, Iazzi, Troyer and Harcos 1506.05349
M T ⌘M = ⌘
Te
R
0
d⌧ HCk (⌧ )
⌘ = diag(I, I)
O+ (n, n)
0
⌘0
det (I + M )
has a definite sign
for each component !
O++ (n, n)
O
(n, n)
O
0
+
(n, n)
⌘0
A new de-sign principle
LW, Liu, Iazzi, Troyer and Harcos 1506.05349
M T ⌘M = ⌘
Te
R
0
d⌧ HCk (⌧ )
⌘ = diag(I, I)
O+ (n, n)
0
⌘0
det (I + M )
has a definite sign
for each component !
O++ (n, n)
O
(n, n)
O
+
(n, n)
0
⌘0
ky
π
spinless fermions split Dirac cone
LW, Troyer, PRL 2014
LW, Corboz, Troyer, NJP 2014
LW, Iazzi, Corboz, Troyer, PRB, 2015
Liu and LW, 1510.00715
0
-π
-π
0
π
spin nematicity
k
x
SU(3)
A new de-sign principle
LW, Liu, Iazzi, Troyer and Harcos 1506.05349
M T ⌘M = ⌘
Te
R
0
d⌧ HCk (⌧ )
⌘ = diag(I, I)
O+ (n, n)
0
⌘0
det (I + M )
has a definite sign
for each component !
O++ (n, n)
O
(n, n)
O
+
(n, n)
0
⌘0
ky
π
spinless fermions split Dirac cone
LW, Troyer, PRL 2014
LW, Corboz, Troyer, NJP 2014
LW, Iazzi, Corboz, Troyer, PRB, 2015
Liu and LW, 1510.00715
0
-π
-π
0
π
spin nematicity
k
x
SU(3)
Asymmetric Hubbard model
t"
6=
t#
U
Realization: mixture of ultracold fermions (e.g.
Jotzu et al, PRL 2015
Now, continuously tunable by
t# /t" 2 ( 1, 1)
PRL 99, 220403 (2007)
PHYSICAL REVIEW LETTERS
al, PRL 2007
and many
others
FIG. 2. Lignier
Dynamicaletsuppression
of tunneling
in an optical
lattice. Shown here is jJeff =Jj as a function of the shaking parameter K0 for V0 =Erec ! 6, !=2! ! 1 kHz (squares), V0 =Erec ! 6,
!=2! ! 0:5 kHz (circles), and V0 =Erec ! 4, !=2! ! 1 kHz
week ending
30 NOVEMBER 2007
"
#
Dirac fermions with unequal Fermi velocities
Asymmetric Hubbard model
t"
6=
t#
U
Realization: mixture of ultracold fermions (e.g.
Jotzu et al, PRL 2015
Now, continuously tunable by
t# /t" 2 ( 1, 1)
PRL 99, 220403 (2007)
PHYSICAL REVIEW LETTERS
week ending
30 NOVEMBER 2007
"
#
U
+
al, PRL 2007
and many
others
FIG. 2. Lignier
Dynamicaletsuppression
of tunneling
in an optical
lattice. Shown here is jJeff =Jj as a function of the shaking parameter K0 for V0 =Erec ! 6, !=2! ! 1 kHz (squares), V0 =Erec ! 6,
!=2! ! 0:5 kHz (circles), and V0 =Erec ! 4, !=2! ! 1 kHz
?
Dirac fermions with unequal Fermi velocities
Two limiting cases
VOLUME 22, NUMBER
19
t"
✘
PHYSICAL REVIEW LETTERS
t#
U
12 MA+ 1969
t"
K. Gustafson, J. P. Taran, H. A. Haus, J. R.
Lifsitz, and P. L. Kelley, Phys. Rev. 177, 306 (1969).
Y. R. Shen and N. Bloembergen, Phys. Rev. 137,
1787 {1965). The calculation on the saturation effect
in this reference has recently been verified by M. Maier and W. Kaiser, to be published.
~P. L. Kelley, Phys. Rev. Letters 15, 1005 (1965).
~
We have been informed that T. K. Gustafson and
J. P. Taran have obtained similar results.
M. Naier, W. Kaiser, and J. A. Giordmaine, Phys.
Rev. Letters 17, 1275 (1966).
~2J. H. Marburger and E. L. Dawes, Phys. Rev. Letters 21, 556 (1968), and Phys. Rev. (to be published).
i Teor. Fiz. —Pis'ma Redakt. 7, 153 (1968) [translation: JETP Letters 7, 117 (1968)].
8M. T. Loy and Y. R. Shen, to be published. The essence of the technique is to extract the pulse signal
s (t) from the convolution integral R(t) = f ~ s (v)g(t-v)
x «, where R{t) and g(t) are response functions of the
detection system to the pulse signal aad to a pulse of
6 function, respectively.
Allowing the possibility of
different pulse shapes, we were able to measure the
pulse width with an accuracy of &0 psec for a 200psec pulse and of &0 psec for a 100-psec pulse. The
YT.
Falicov-Kamball Limit
time constant of our detection system was about 400
psec.
t# ⌧ U
Strong Coupling Limit
SIMPLE MODEL FOR SEMICONDUCTOR-METAL
SmB~ AND TRANSITION-METAL
TRANSITIONS:
OXIDES
L. M. Falicov*
Department
of Physics, University of California,
Berkeley, California 94720
and
J. C. Kimballg
Department
of Physics, and The James Franck Institute, University of Chicago, Chicago, Illinois 60637
(Received 12 March 1969)
We propose a simple model for a semiconductor-metal
transition, based on the existence of both localized (ionic) and band (Bloch) states. It differs from other theories in
that we assume the one-electron states to be essentially unchanged by the transition.
The electron-hole interaction is responsible for the anomalous temperature dependence
of the number of conduction electrons. For interactions larger than a critical value, a
first-order semiconductor-metal phase transition takes place.
SmBS' and a numtions and (b) a set of localized states centered at
ber of transition-metal oxides, ' exhibit semiconthe sites of the metallic ions in the crystal. As
ductor-metal transitions. ' The transitions are
T —0 the localized states are lower in energy
in many cases first-order phase transitions (e.g. ,
than the band states and are fully occupied by
in V, O, ); however, they can also result from a
electrons. Therefore the quasiparticle excitagradual but anomalously large increase in cantions are either localized holes or itinerant elecductivity over a range of temperatures (e.g. , in
trons. In the language of second quantization and
SmB, and Ti, O, ). In addition, measurements of
in the spirit of the Landau theory of Fermi liqlarge magnetic susceptibilities with anomalous
uids, we write the one-particle terms as
temperature dependences suggest that in many of
(k)a-~aH
+gab ig tb ig'
these materials localized magnetic moments ex0
v
vkg vkg .
vkg
Sg
ist and that they are intimately connected with
Kennedy
and
Lieb
the transition. As an example, it has been hywhere avko4 creates an electron in state k, band
pothesized' that in SmB, the conduction electrons
v, with spin o, and b, o~ creates a hole with spin
and the localized moments are produced simulo at site i. The energies ev(k) and F. are positive
taneously by the promotion of a single localized
definite and such that
electron from the spherically symmetric Sm++
ion (4= 0) into a conduction band The Sm+. ++ ion
n =— min[E + e (k) j & 0
') acts as a localized moment.
left behind (J= —,
We present here a simple theory of the semiis the energy
for the formation
of an elecFreericks
and
RMP, 2003
gapZlatić,
conductor-metal transition based on a model havtron-hole pair. We further assume that the quaing both localized and itinerant interacting quasisiparticle interaction is screened, and its range
particle states. The relevant single-electron
short enough so that only intra-atomic terms
states consist of (a) bands of extended Bloch funcneed be considered. In this case
Many substances,
including
Long-range spin order on bipartite
lattices with infinitesimal
repulsion
=Pe
“Fruit fly” of DMFT
Jxy
⇣
x x
Ŝi Ŝj
+
y y
Ŝi Ŝj
⌘
+
z z
Jz Ŝi Ŝj
2(t"2 + t#2 )
4t" t#

U
U
XXZ model with
t# /t"
Phase diagram
0
U/t"
t# /t"
Phase diagram
Dirac
Fermions
0
U/t"
t# /t"
Phase diagram
Dirac
Fermions
AF-Ising
0
Falicov-Kimball limit
U/t"
t# /t"
Phase diagram
Dirac
Fermions
XXZ limit
AF-Ising
0
Falicov-Kimball limit
U/t"
t# /t"
Phase diagram
1
AF-Heisenberg
Dirac
Fermions
XXZ limit
AF-Ising
0
Falicov-Kimball limit
U/t"
t# /t"
Phase diagram
1
Meng et al 2010
Sorella et al 2012
Assaad et al 2013
⇡ 3.8
AF-Heisenberg
Dirac
Fermions
XXZ limit
AF-Ising
0
Falicov-Kimball limit
U/t"
t# /t"
Phase diagram
1
Meng et al 2010
Sorella et al 2012
Assaad et al 2013
⇡ 3.8
AF-Heisenberg
Dirac
Fermions
XXZ limit
AF-Ising
0
Falicov-Kimball limit
U/t"
How to connect the phase boundary ?
What is the
Binder ratio
M2 =
*
1 X iQ·r n̂r" n̂r#
e
N r
2
!2 +
M4 =
*
t# /t" = 0.15
1 X iQ·r n̂r" n̂r#
e
N r
2
!4 +
Scaling analysis
⌫ = 0.84(4)
z + ⌘ = 1.395(7)
Summary
Exciting time for QMC simulation of lattice fermions
Thanks to my collaborators!
Mauro
Iazzi
Philippe
Corboz
Ye-Hua
Liu
Jakub
Imriška
Ping Nang
Ma
Gergely
Harcos
Matthias
Troyer