arXiv:cond-mat/0404055v1 [cond-mat.str-el] 2 Apr 2004
Quantum Cluster Theories
Thomas Maier
∗
Computational Science and Math Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6114
Mark Jarrell
Department of Physics, University of Cincinnati, Cincinnati, Ohio 45221-0011
Thomas Pruschke
Theoretical Physics, University of G¨ottingen, Tammannstr. 1, 37077 G¨ottingen, Germany
Matthias H. Hettler
Forschungszentrum Karlsruhe, Institut f¨ur Nanotechnologie, Postfach 3640, 76021 Karlsruhe, Germany
(Dated: 2nd February 2008)
Quantum cluster approaches offer new pers pectives to study the complexities of macroscopic cor-
related fermion systems. These approaches can be understood as generalized mean-field theories.
Quantum cluster approaches are non-perturbative and are always in the thermodynamic limit.
Their quality can be systematically improved, and they provide complementary information to
finite size simulations. They have been studied intensively in recent years and are now well es-
tablished. After a brief historical review, this article comparatively discusses the nature and
advantages of these cluster techniques. Applications to common models of correlated electron
systems are reviewed.
1
Contents
I. Introduction 1
A. Brief history 1
B. Corrections to Curie-Weiss theory 3
II. Quantum cluster theories 5
A. Cluster approximation to the locator expansion 6
B. Cluster approximation to the grand potential 9
1. Cluster perturbation theory 9
2. Cellular dynamical mean-field theory 10
3. Dynamical cluster approximation 11
4. Self-consistency scheme 12
C. Discussion 12
1. Conservation and thermodynamic consistency 13
2. Causality 13
3. Reducible and irreducible quantities 13
4. Comparison 14
D. Effective cluster model 17
E. Phases with broken symmetry 18
1. Uniform m agnetic field – Ferromagnetism 18
2. Superconductivity 19
3. Antiferromagnetic order 20
F. Calculation of susceptibilities 21
G. Disordered s ystems 22
H. Alternative cluster methods 24
1. Self-energy functional theory 24
2. Fictive impurity models 25
3. Non-local effects via spectral density approximation25
4. Non-local corrections via projection technique 25
5. Two-site correlations with composite operators 26
III. Quantum cluster solvers 26
∗
Electronic address: maierta@ornl.gov
1
This article has been submitted to Reviews of Modern Physics.
A. General remarks 26
B. Perturbative techniques 27
1. Second order perturbation theory 27
2. Fluctuation exchange approximation 28
3. Non-crossing approximation 28
C. Non-perturbative techniques 30
1. Quantum Monte Carlo 30
2. Exact diagonalization 33
3. Wilson’s numerical renormalization group 33
IV. Applications to strongly correlated models 34
A. Complementarity of finite size and quantum cluster simulations35
B. 2D Falicov-Kimball model 36
C. 1D Hubbard model 37
D. 2D Hubbard model 39
1. Metal-insulator transition 39
2. Antiferromagnetism and precursors 41
3. Pseudogap at finite doping 43
4. Superconductivity 45
5. Phase diagram 48
6. Studies of related models 48
V. Conclusions and perspectives 49
Acknowledgments 50
References 50
I. INTRODUCTION
A. Brief history
The theoretical description of interacting many-
particle systems remains one of the grand challenges
in condensed matter physics. Especially the field of
strongly co rrelated electron systems has regained theo-
retical and experimental interest with the discovery of
2
heavy Fermion compounds and high-temperature s uper-
conductors. In this class of systems the strength of the
interactions between particles is comparable to or larger
than their kinetic energy, i.e. any theory based on a per-
turbative expansion around the non-interacting limit is
at least questionable. Theoretical tools to des c ribe these
systems are therefore faced with extreme difficulties, due
to the non-perturba tive nature of the problem. A large
body of work has been devoted to a direct (numerically)
exact solution of finite size systems using exact diago-
nalization or Quantum Monte Carlo methods. Exact di-
agonalization however is severe ly limited by the expo-
nential growth of computational effort with system size,
while Quantum Monte Carlo methods suffer from the sign
problem at low temperatures. Another difficulty of these
methods arises from their strong finite size effects, of-
ten ruling out the reliable extraction of low energy s c ales
that are important to capture the competition between
different ground states often present in strongly corre-
lated systems.
Mean-field theories are defined in the thermodynamic
limit and therefore do not face the finite size problems.
With applications to a wide variety of extended systems
from spin models to models of correlated elec trons and/or
bosons, mean-field theorie s are extremely popular and
ubiquitous throughout science. The first mean-field the-
ory which gained wide acceptance was developed by P.
Weiss for spin systems (Weiss, 1907). The Curie-Weiss
mean-field theory reduces the complexity of the thermo-
dynamic lattice spin problem by mapping it onto that
of a magnetic impurity embedded in a self-consistently
determined mean magnetic field.
Generally, mean-field theories divide the infinite num-
ber o f degrees of freedom into two sets. A small set of
degrees of freedom is tre ated explicitly, while the effects
of the r e maining degrees of freedom are summarized as
a mean-field acting on the first set. Here, by mean-field
theory, we refer to the class of approximatio ns which ac-
count for the c orrelations between spatially localized de-
grees of freedom explicitly, while treating those at longer
length scales with an effective medium. Such local ap-
proximations become e xact in the limit of infinite coor-
dination number or equivalently infinite dimensions D
(Itzykson and Drouffe, 1989); however non-local correc-
tions become important in finite dimensions. The pur-
pose of this review is to discuss methods for incorporating
non-local corrections to local approximations.
Many different local approximations have been de-
veloped for systems with itinerant degrees of freedom.
Early attempts focused on disordered systems, and in-
cluded the virtual crystal approximation (Nordheim,
1931a,b; Parmenter, 1955; Schoen, 1969) and the
average-T matrix approximation (Beeby and E dwards,
1962; Schwartz et al., 1971). However, the most suc-
cessful local approximations for disordered systems is
the Coherent Potential Approximation (CPA) developed
by Soven (1967) and others (Shiba, 1971; Taylor, 1 967).
This method is distinguished from the others in that it
becomes ex act in both the limit of dilute and concen-
trated disordered impurity systems, as well as the limit
of infinite dimensions.
There have been many attempts to extend the CPA
formalism to correlated systems, starting with the Dy-
namical CPA (DCPA) of Kakehashi (2002); Sumi (1974),
the XNCA of Kim et al. (1990); K uramoto (1985) and
the LNCA of Grewe (1987); Grewe et al. (198 8). A great
breakthrough was achieved with the formulation of the
Dynamical Mean-Field Theory (DMFT) (for a review see
Georges et al., 1996; Pruschke et al., 1995) in the limit of
infinite dimensio ns by Metzner and Vollhardt (1989) and
M¨uller-Hartmann (1989b). The DCPA and the DMFT
have been the most successful approaches and employ the
same mapping between the cluster and the lattice prob-
lems. They differ mostly in their starting philosophy.
The DCPA employs the CPA equations to relate the im-
purity so lution to the lattice whereas in the DMFT the
irreducible quantities calcula ted on the impurity are used
to construct the lattice quantities.
Despite the success of these mean-field approaches,
they share the critica l flaw of neglecting the effects of
non-local fluctuations. Thus they are unable to capture
the physics of, e.g. spin waves in spin systems, local-
ization in disordered systems, or spin-liquid physics in
correla ted electronic systems. Non-lo c al corrections are
required to treat even the initial effects of these phenom-
ena and to describe phase transitions to states described
by a non-local order parameter.
The first attempt to add no n-local correc tions to mean-
field theories was due to Bethe (1935) by adding correc-
tions to the Curie-Weiss mean-field theory. This was
achieved by mapping the lattice problem onto a self-
consistently embedded finite-size spin cluster composed
of a central site and z nearest neighbors embedded in a
mean-field. For small z, the resulting theory provides a
remarkably large and accurate correction to the transi-
tion temperature (Kikuchi, 1951; Suzuki, 1986).
Many attempts have been made to apply similar idea s
to disordered electronic systems (Gonis, 1992). Most ap-
proaches were hampered by the difficulty of constructing
a fully causal theory, with positive spectral functions.
Several causal theories were developed including the em-
bedded cluster method (Gonis, 1992) and the molecu-
lar C PA (MCPA) by Tsukada (19 69) (for a review see
Ducastelle, 1974). These methods generally are obtained
from the local approximation by replacing the impurity
by a finite size cluster in real space. As a re sult these ap-
proaches suffer from the lack of translational invariance,
since the clus ter has open b oundary conditions and only
the surface sites couple to the mean-field.
Similar effort has been e xpended to find cluster ex-
tensions to the DMFT, including most notably the Dy-
namical Cluster Approximation (DCA) (Hettler et al.,
2000, 1998) a nd the Cellular Dynamical Mean-Field The-
ory (CDMFT) (Kotliar et al., 2001). Both cluster ap-
proaches reduce the complexity of the lattice problem by
mapping it to a finite size cluster self-consistently em-
3
bedded in a mean-field. As in the classical case, the
self-consistency condition reflects the translationally in-
variant nature of the original lattice problem. The main
difference with their classical counterparts arises from the
presence of quantum fluctuations. Mean-field theories for
quantum systems with itinerant degrees of fre e dom cut
off spatial fluctuations but take full account of tempo-
ral fluctuations. As a result the mean-field is a time-
or respectively frequency depe ndent quantity. Even an
effective cluster problem consisting of only a single site
(DMFT) is hence a highly non-trivial many-body prob-
lem. CDMFT and DCA mainly differ in the nature of the
effective c luster problem. The CDMFT shares an identi-
cal mapping of the lattice to the cluster problem with the
MCPA, and hence also violates translational symmetries
on the cluster. The DCA maps the lattice to a periodic
and therefo re trans lationally invariant cluster.
A numerically more tractable cluster approxima-
tion to the thermodynamic limit was developed by
Gros and Valenti (1994). In this formalism the self-
consistent coupling to a mean-field is neglected. This
leads to a theory in which the self-energy of an isolated
finite size cluster is used to approximate the lattice prop-
agator . As shown by S´en´echal et al. (2000), this cluster
extension of the Hubbard-I approximation is obta ined as
the leading o rder approximation in a strong-coupling ex-
pansion in the hopping amplitude and hence this method
was named Cluster Perturbation Theory (CPT).
Generally, cluster formalisms share the basic idea to
approximate the effects of correlations in the infinite lat-
tice problem with those on a finite size quantum cluster.
We refer to this class of techniques as quantum cluster
theories. In contrast to Finite System Simulations (FSS),
these techniques are built for the thermodynamic limit.
In this review we focus on the three most established
quantum cluster approaches, the DCA, the CDMFT and
the CPT formalisms. The CDMFT approach was origi-
nally fo rmulated for general, possibly non-orthogonal ba-
sis sets. In this review we restrict the discussion to the
usual, completely localized orthogonal basis set and refer
the reader to Kotliar et al. (2001) for the generalization
to arbitrary basis sets.
The organization of this article is as follows: To famil-
iarize the reader with the concept of cluster approaches,
we develop in section I.B a cluster generalization of the
Curie-Weiss mean-field theory for spin systems. Section
II sets up the theoretica l framework of the CDMFT,
DCA and CPT formalisms by pr e senting two derivations
based on different starting philosophies. The derivation
based on the locator expansion in Sec. II.A is analogous
to the clus ter generalization of the Curie-Weiss mean-
field method and thus is physically very intuitive. The
derivation based on the cluster approximation to dia -
grams defining the grand potential in Sec. II.B is closely
related to the reciprocal space derivation of the DMFT
by M¨uller-Hartmann (19 89b). The nature of the different
quantum cluster approaches together with their a dvan-
tages and weaknesses are assessed in Sec. II.C. Discus-
sions of the effective cluster problem, generalizations to
symmetry broken states and the calculation of response
functions are pres ented in Sec s. II.D, II.E and II.F. The
remainder of this section is devoted to describe the ap-
plication of the DCA formalism to disordered sys tems in
Sec. II.G and to a brief discussion of alterna tive methods
proposed to introduce non-local cor rections to the DMFT
method in Sec. II.H. In Sec. III we review the various
perturbative and non-perturbative techniques available
to solve the effective self-consistent cluster problem of
quantum cluster approaches. We include a detailed as-
sessment of their advantages and limitations. Although
numerous applications of quantum cluster approaches to
models of many-particle systems are found in the litera-
ture, this field is still in its footsteps a nd currently very
active. A large body of work has been concentrated on
the Hubbard model. We r eview the progress made on
this model in Section IV together with applications to
several other strongly correlated models. Finally, Sec. V
concludes the review by stressing the limitations of quan-
tum cluster approaches and proposing possible directions
for future research in this field.
B. Corrections to Curie-Weiss theory
As an intuitive example of the formalism developed in
the next sections we consider a sy stematic c lus ter ex ten-
sion of the Curie-Weiss mean-field theory for a lattice
of classical interacting spins. This dis c ussion is es pe-
cially helpful to illustrate many new aspects of cluster
approaches as compared to finite size simulations. The
quality of this approach, and its convergence and critica l
properties will be demonstrated with a simple ex ample,
the one-dimensional Ising model
H = −J
X
i
σ
i
σ
i+1
− h
X
i
σ
i
(1)
where σ
i
= ±1 are classical spins, h is an external mag-
netic field and the exchange integral J > 0 acts b e tween
nearest neighbors only, favoring ferromagnetism. The
generalizatio n o f this approach to higher dimensions and
quantum spin systems is stra ightforward.
We start by dividing the infinite lattice into N/N
c
clus-
ters of size N
c
(see Fig. 3) with origin ˜x and the exchange
integral J
ij
into intra- (J
c
) and inter-cluster (δJ) parts
J(˜x
i
− ˜x
j
) = J
c
δ
˜x
i
,˜x
j
+ δJ(˜x
i
− ˜x
j
) (2)
where each of the terms is a matrix in the N
c
cluster sites.
The central approximation of cluster theories is to retain
correla tio n effects within the cluster and neglect them
between the clusters. A natural formalism to implement
this approximation is the locator expa ns ion. The spin-
susceptibility χ
ij
= β(hσ
i
σ
j
i− hσ
i
ihσ
j
i), where β = 1/T
is the inverse temperature, can be written as a locator
expansion in the inter-cluster part δJ of the exchange
4
interaction, around the cluster limit χ
χ
χ
o
= χ
χ
χ(δJ = 0) as
χ
χ
χ(˜x
i
− ˜x
j
) = χ
χ
χ
o
δ
˜x
i
,˜x
j
+ χ
χ
χ
o
X
l
δJ(˜x
i
− ˜x
l
)χ
χ
χ(˜x
l
− ˜x
j
) (3)
where we used again a matrix notation in the N
c
cluster
sites. By using the translational invariance of quantities
in the superlattice ˜x, this expression can be simplified in
the reciprocal spa c e ˜q of ˜x to
χ
χ
χ(˜q) = χ
χ
χ
o
+ χ
χ
χ
o
δJ(˜q)χ
χ
χ(˜q) . (4)
This locator expansion has two well-defined limits. For
an infinite size c lus ter it recovers the exa ct result since
the sur fa c e to volume ratio vanishes making δJ irrelevant,
and thus χ
χ
χ = χ
χ
χ
o
. For a single site cluster, N
c
= 1,
it recovers the Curie-Weiss mean-field theory. This is
intuitively clear since for N
c
= 1 fluctuations between all
sites ar e neglected. With the susce ptibility of a single
isolated site χ
o
= 1/T a nd δJ(˜q = 0) = J(q = 0) = J,
we obtain for the uniform susceptibility
χ(q = 0) =
1
1/χ
o
− J(q = 0)
=
1
T − T
c
(5)
the mean-field result with critical temperature T
c
= J.
For clus ter sizes larger than one, translational sym-
metry within the cluster is v iolated since the clusters
have open boundary conditions and δJ only couples sites
on the surface of the clusters. As detailed in the next
section, this shortcoming can be formally over come and
translational invariance restored by considering an anal-
ogous expres sion to the locator expansion (4) in the
Fourier space Q of the cluster
χ(Q, ˜q) = χ
o
(Q) + χ
o
(Q)δJ(Q, ˜q)χ(Q, ˜q)
=
1
1/χ
o
(Q) − δJ(Q, ˜q)
, (6)
with analogous relations for the intra- and inter-cluster
parts of J
δJ(Q, ˜q) = J(Q + ˜q) −
¯
J(Q) (7)
¯
J(Q) =
N
c
N
X
˜q
J(Q + ˜q) . (8)
Here, ˜q is a vector in the rec iproc al space of ˜x, and Q
is a vector in the reciproc al space of the cluster sites.
The Fourier trans form of the exchange integral is given
by J(Q + ˜q) = J cos(Q + ˜q), the intra-cluster exchange
is
¯
J(Q), while the inter -cluster exchange is δJ(Q, ˜q). As
we will se e in the next section, the resulting formalism
is ana logous to the dynamical cluster approximation for
itinerant fermion s ystems.
In analogy to the Curie-Weiss theory, the lattice sys-
tem can now be mapped onto an effective cluster model
embedded in a mean-field since correlations between the
clusters are neglected. The susceptibility res tricted to
cluster sites is obtained by averaging or coarse-graining
over the superlattice wave-vectors ˜q
¯χ(Q) =
N
c
N
X
˜q
χ(Q, ˜q) =
1
1/χ
o
(Q) − Γ(Q)
. (9)
with the hybridization function
Γ(Q) =
N
c
N
P
˜q
δJ
2
(Q, ˜q)χ(Q, ˜q)
1 +
N
c
N
P
˜q
δJ(Q, ˜q)χ(Q, ˜q)
. (10)
This follows from the fact that the isolated cluster s us cep-
tibility χ
o
(Q) does not depend on the integration varia ble
˜q in Eq. (9).
This expression defines the effective cluster model
H
c
= −
X
Q
¯
J(Q)σ(Q)σ(−Q) − hσ(Q = 0)
−
X
Q,˜q
δJ(Q, ˜q)σ(Q)hσ(−Q − ˜q)i, (11)
where σ(Q) (σ(q)) denotes the cluster (lattice) Fourier
transform of σ
i
and h. . .i the expectation value calcu-
lated with respect to the cluster Hamiltonian H
c
. As in
the Curie-Weiss theory, the cluster model is used to self-
consistently determine the order parameter hσ(Q + ˜q)i =
hσ(Q)iδ(˜q) in the ferromagnetic state. In the paramag-
netic state, the susceptibility calculated in the cluster
model takes the same form as the coarse-grained result
Eq. (9) obtained from the locator expansion.
The uniform susceptibility χ(Q = 0, ˜q = 0) contains
information about the nature of this cluster appr oach, its
critical properties and its convergence with cluster size.
The sum in Eq . (8) may be solved analytically
¯
J(Q = 0) = J(N
c
/π) sin(π/N
c
) . (12)
The isolated cluster sus ceptibility χ
o
(Q) can also be ca l-
culated analytically by using the transfer matr ix method
to give (Goldenfeld, 1992)
χ
o
(Q = 0) = β exp(2K)
1 − (tanh(K))
N
c
1 + (tanh(K))
N
c
, (13)
where K = β
¯
J(Q = 0) = βJ(N
c
/π) sin(π/N
c
).
With these express ions the uniform la ttice susceptibility
Eq. (6) becomes
χ(T ) =
1
1/χ
o
(Q = 0) − δJ(Q = 0, ˜q = 0)
(14)
=
1
1/χ
o
(Q = 0) − J (1 − (N
c
/π) sin(π/N
c
))
.
The cluster estimate of the lattice susceptibility interpo-
lates between the Cur ie -Weiss result and the exact lattice
result as N
c
increases. It may be used to reveal some of
the properties of cluster appr oximations and to compare
the cluster results to both the finite-size calculation and
the exact result in the thermodynamic limit.
5
0 10 20 30 40
N
c
0
1
2
3
4
5
χ(T)
Cluster mean-field
Finite size
Figure 1 The cluster and finite-size estimates of the un iform
lattice susceptibility versus cluster size when J = 1 and T =
0.7.
First, both the cluster mean-field result Eq. (14) and
the finite-size result Eq. (13) with K = βJ may be re-
garded as an approximation to the thermodynamic re-
sult. However, as illustrated in Fig. 1, the cluster mean-
field result converges more quickly as a function of cluster
size N
c
than the finite size result. This reflects the supe-
rior starting point o f the cluster approximation compared
to the finite-size calculation. The cluster approximation
is an expansion about the mean-field result, whereas the
finite-size calculation is an expansion about the atomic
limit.
It is instructive to explore the convergence of the clus-
ter result a nalytically. For large N
c
, the character of the
susceptibility Eq. (14) can be split into three regimes. At
very high temperatures
χ(T ) ≈
1
T − Θ
for T ≫ J (15)
where Θ ≈ 2J +
J
6
π
N
c
2
. At intermediate temperatures,
χ(T ) ≈ βe
2βJ
1 −
βJ
3
π
N
c
2
!
for J ≫ T ≫ T
c
.
(16)
The true critical behavior of the system can be resolved
by studying the properties of this intermediate temper-
ature regime. At both high and intermediate tempera-
tures, the s us c eptibility differs from the exact result by
corrections of order O(1/N
2
c
). In general, cluster meth-
ods with periodic boundary conditions have co rrections
of order O(1/L
2
c
), where L
c
= N
1/D
c
is the linear size of
the cluster.
At low temper atures, very close to the transition to
the ferromagnetic state, dev iations from the exact result
are far larger. Here, for large clusters
χ(T ) ∼
N
c
T − T
c
, (17)
with the c ritical temperature T
c
> 0, wher eas the exact
susceptibility in this regime χ(T ) ≈ β e xp (2βJ) do e s not
diverge until zero temperature. This discrepancy is ex-
pected in cluster approximations, since they treat long
length scales which drive the transition in a mean-field
way and therefore neglect long wave-length modes which
eventually suppress the transition. Hence, cluster ap-
proximations generally predict finite tr ansition tempera-
tures independent of dimensionality due to their r esidual
mean-field character. With increasing cluster size how-
ever, the tr ansitions are expected to be s ystematically
suppressed by the inclusion of longer-rang e d fluctuations.
For cluster sizes larger than one, all three regions are
evident in the plot of the cluster mean-field estimate of
the inver se susceptibility, shown in the inset to Fig. 2.
For N
c
= 8 and N
c
= 16, the high and low tempera-
ture parts are seen as straight lines on the plot in the
inset, with the cro ssover region in between. In numeri-
cal simulations with significant source s of numerical noise
especially close to the transition, it is extremely difficult
to resolve the true low-temperature mean-field behav-
ior. Exponents extracted from fits to the susceptibility
in these simulations will more likely reflect the properties
of the intermediate temperature regime.
Despite the large deviations of the cluster result from
the exact result low temperatures, we may still ex-
tract the correct physics through finite size extrapola-
tion. In general, for a system where the correlatio ns
build like ξ ∼
T −T
c
T
c
−ν
, we ex pect T
c
∼ T
∗
c
− aL
−1/ν
c
,
where T
∗
c
is the exact transition temperature, L
c
is
the linear cluster size, and a is a positive real con-
stant (Suzuki, 1986). However, for the 1D Ising system,
ξ ∼
1
2
exp (βJ), so more care must be ta ken. Fortu-
nately, an analytic expr e ssion for the transition temper-
ature may be extracted from Eq. (14). For large clus-
ters, T
c
≈ JN
c
1
6
π
N
c
2
−
1
120
π
N
c
4
. This behavior
is shown in the main frame of Fig. 2 with the circles de-
picting the numerical values for T
c
and the solid line their
asymptotic behavior.
II. QUANTUM CLUSTER THEORIES
In this section we provide two derivations of qua ntum
cluster appr oaches for systems with itinerant quantum
degrees of freedom. The locator expansion in Sec. II.A
is analo gous to the cluster extension of the Curie-Weiss
mean-field theory developed in the pr eceding section.
Sec. II.B provides a micro scopic derivation based on clus-
ter approximations to the thermodynamic grand poten-
tial. A detailed discus sion of the nature of quantum
cluster approaches and the effective cluster model is pr e-
sented in Secs. II.C and II.D. Generalizations for sym-
metry broken phases, the calculation of susceptibilities
and the application to disordered systems is e xplained in
Secs. II.E, II.F and II.G and a brief discussion of alter-
native c lus ter methods is presented in Sec. I I.H.
