Quantum cluster theories

This article reviews quantum cluster theories, a set of approximations for infinite lattice models which treat correlations within the cluster explicitly, and correlations at longer length scales either perturbatively or within a mean-field approximation. These methods become exact when the cluster size diverges, and most recover the corresponding mean-field approximation when the cluster size becomes 1. Although quantum cluster theories were originally developed to treat disordered systems, they have more recently been applied to the study of ordered and disordered correlated systems, which will be the focus of this review. After a brief historical review, the authors provide detailed derivations of three cluster formalisms: the cluster perturbation theory, the dynamical cluster approximation, and the cellular dynamical mean-field theory. They compare their advantages and review their applications to common models of correlated electron systems.

Figure 1

The cluster and finite-size estimates of the uniform lattice susceptibility in the 1D Ising model vs cluster size when $J=1$ and $T=1.2$.

Figure 2

(Color in online edition) The inverse susceptibility vs temperature. The Weiss behavior at low $T$, where $1∕\chi \left(T\right)$ is linear in $T$, illustrates that the transition is always mean-field-like. The true critical behavior of the transition is reflected in the crossover region for $T\lesssim J$ and larger $N_c$. Inset: transition temperature for the 1D Ising model vs inverse cluster size when $J=1$, obtained with the cluster mean-field approach. For large clusters, $T_c\sim J{\pi }^2∕6N_c$, shown as a solid line.

Figure 3

(Color in online edition) Definition of the coordinates in real (left) and reciprocal (right) space illustrated for $N_c=4$ site clusters. The origin of a cluster is labeled by $\tilde{\mathbf{x}}$, the sites within a cluster by $\mathbf{X}$. The reciprocal space to $\mathbf{X}$ is labeled by $\mathbf{K}$, the wave vectors of the superlattice, i.e., within a cell, by $\tilde{\mathbf{k}}$.

Figure 4

A second-order term in the generating functional $\Phi$ of the Hubbard model. The dashed line represents the interaction $U$, and the solid line on the left-hand side (right-hand side) the lattice (coarse-grained) single-particle Green’s function $G\left(\overline{G}\right)$. With the dynamical cluster approximation (DCA) Laue function, the wave vectors collapse onto those of the cluster and each lattice Green’s function is replaced by its coarse-grained average.

Figure 5

The best bipartite periodic clusters with $N_c⩽22$ for the square lattice. All are used for scaling, with the exception of $16B$, which is highly imperfect. From 16.

Figure 6

Sketch of the cellular DMFT and DCA embedded-cluster algorithms. The iteration starts with computing the coarse-grained Green’s function $\overline{\mathbf{G}}$ using an initial guess for the cluster self-energy ${\mathbf{\Sigma }}_c$. The cluster excluded Green’s function $\mathcal{G}$ is then used to define the effective cluster problem which yields a new estimate of ${\mathbf{\Sigma }}_c$.

Figure 7

(Color in online edition) Transition temperature vs inverse cluster size: circles, calculated with the DCA Monte Carlo technique; squares, cellular DMFT Monte Carlo when $U=W=4t=1$. Inset: DCA order parameter $m\left(T\right)$ and inverse charge susceptibility $\chi {\left(Q\right)}^{-1}$ vs temperature. Taken from 112.

Figure 8

Noncompact (left) and compact (right) nonlocal corrections to the grand potential. Here the upper and lower circles are meant to represent a set of graphs that are closed except for the external lines shown, and restricted to site $n$ (the origin).

Figure 9

(Color in online edition) A few low-order diagrams in the irreducible self-energy of a quenched diagonally disordered system. Each circle represents the scattering of a state $\mathbf{k}$ from sites denoted by $\mathbf{X}$ and a dotted line. The correlations between the electrostatic potentials on different sites are denoted by the wavy line.

Figure 10

Schematic setup of the numerical renormalization group. The energy axis is partitioned into exponentially decreasing intervals.

Figure 11

The imaginary-time Green’s function $G(\mathbf{k},\tau )$ at $\mathbf{k}=(\pi ,0)$ in the half-filled 2D Hubbard model, calculated with finite-size quantum Monte Carlo (filled symbols) and DCA (open symbols) when $U=4.4t$ and $\beta =4∕t$. From 130. See also 9.

Figure 12

The imaginary-time Green’s function $G(\mathbf{k},\tau )$ at $\mathbf{k}=(\pi ,0)$, $\tau =\beta ∕2$ in the half-filled 2D Hubbard model, calculated with finite-size FLEX method (circles) and DCA/FLEX method (squares) when $U=1.57t$ for various cluster sizes $N_c=L\times L$. Both methods scale as $1∕L^2$ and coverge to a single value as $L\to \infty$. From 9.

Figure 13

Density of states, of the 2D Falicov-Kimball model for a $4\times 4$ cluster at various temperatures calculated with the DCA. The DOS develops a pseudogap as the temperature approaches $T_c\approx 0.189t(U=4t)$ due to the nonlocal CDW fluctuations present in the DCA $(N_c>1)$. In the DMFT $(N_c=1)$, there is no $T$ dependence of the DOS above $T_c$. From 69.

Figure 14

$T_c$ in the 2D Falicov-Kimball model as a function of the inverse linear cluster size for $U∕t=8,12$,16 calculated with the DCA. The Ising limit, and finite-size estimates of $T_c$ are shown for comparison. The inset shows that the influence of the cluster boundary conditions on $T_c$ disappears rapidly with increasing cluster size. From 69.

Figure 15

Specific heat vs temperature for single-site and $2\times 2$ clusters calculated with the DCA and exact enumeration when $U=8t$. For $N_c=1$, there is a single peak with integrated weight ln(2) associated with the suppression of local charge fluctuations. For $N_c=4$, there is an additional peak at lower temperatures associated with critical fluctuations near the charge-ordering transition temperature. $T_c$ for $N_c=4$ is indicated by an arrow. In the inset the entropy $S\left(T\right)={\int }_0^{T}d{T}^{\prime }[C\left(T^{\prime }\right)∕T^{\prime }]$ is shown in units of ln(2). From 69.

Figure 16

Spectral gap in the half-filled 1D Hubbard model as a function of $U∕t$ calculated with DMFT local-impurity self-consistent approximation (LISA) and cellular DMFT for a two-site cluster, $N_c=2$, compared to the exact result. From 26.

Figure 17

The spectral function $A(k,\omega )$ at $k=\pi ∕2$ in the half-filled 1D Hubbard model when $U=16t$ (top) and $U=8t$ (bottom), calculated with ordinary exact diagonalization and with cluster perturbation theory for cluster sizes $N=4$,8,12. From 165.

Figure 18

The spectral function $A(k,\omega )$ in the half-filled 1D Hubbard model when $U=4t$, calculated with CPT for $N_c=12$. From 165.

Figure 19

Double occupancy as a function of temperature for different values of $U$ in the half-filled 2D Hubbard model calculated with the DCA/QMC method for various cluster sizes $N_c$. Energies are in units of $4t$. Here, $t=0.25$, so the lines for $U=0.5$,1,2 correspond to $U∕t=2$,4,12. From 130.

Figure 20

Density of states when $U∕t=4$ at $T=0.125t$ in the half-filled 2D Hubbard model calculated with the DCA/QMC for various cluster sizes $N_c$. From 130.

Figure 21

(Color in online edition) Tiling of the first Brillouin zone for $N_c=2$.

Figure 22

DCA coarse-grained spectral functions for different values of $U$ obtained from $N_c=2$ DCA/numerical renormalization group calculations at $T=0$. For all $U>0$ a gap exists at the Fermi energy.

Figure 23

Density of states as a function of temperature at the chemical potential in the 1D and 2D half-filled Hubbard model when $U=2t$ calculated with a two-site composite-operator approximation. From 173.

Figure 24

Double occupancy as a function of $U$ in the frustrated 2D Hubbard model with $t^{\prime }=t$ for different temperatures (from top to bottom) $T∕t=1∕5$,$4∕30$,$1∕10$,$1∕11$, calculated with the cellular DMFT/QMC method. From 140.

Figure 25

Sublattice magnetization as a function of temperature in the half-filled 2D Hubbard model calculated with the DCA/NCA method for cluster sizes $N_c=1$,4 when $U=4t$. Inset: Neél temperature vs doping. From 111.

Figure 26

Inverse antiferromagnetic susceptibility vs temperature in the half-filled 2D Hubbard model calculated with the DCA/QMC method for various cluster sizes $N_c$ when $U=6t$. The lines are fits to the function ${(T-T_{\mathrm{N}})}^{\gamma }$. Inset: Corresponding Neél temperatures as a function of the cluster size. Energies are in units of $4t$. From 86.

Figure 27

Spectral function $A(\mathbf{k},\omega )$ and the real $\mathrm{Re}\Sigma (\mathbf{k},\omega )$ and imaginary $\mathrm{Im}\Sigma (\mathbf{k},\omega )$ parts of the self-energy for various temperatures at $\mathbf{k}=(\pi ,0)$ in the 2D half-filled Hubbard model for $U=5.2t$ calculated with the DCA/QMC method for a 64-site cluster $(N_c=64)$. From 77.

Figure 28

Illustration of the triangular lattice with (a) hopping amplitudes $t$ and $t^{\prime }$ and (b) DCA coarse-graining cells in the first Brillouin zone (dashed line) of the triangular lattice when $N_c=4$. Cluster $\mathbf{K}$ points are indicated by the dots. From 78.

Figure 29

The 2D half-filled frustrated Hubbard model: (a) density of states; coarse-grained single-particle spectral functions $\overline{A}(\mathbf{K},\omega )$ for (b) $\mathbf{K}=(0,0)$, (c) $\mathbf{K}=(\pi ,\pi ∕\sqrt{3})$, and (d) $\mathbf{K}=(0,2\pi ∕\sqrt{3})$ for various values of the frustration $t^{\prime }$ when $U=6t$ and $T=0.6t$. From 78.

Figure 30

Density of states for various temperatures in the 2D Hubbard model at 5% doping when $U=12t$ near the chemical potential $(\omega =0)$ calculated with the DCA/NCA for a four-site cluster, $N_c=4$. From 111.

Figure 31

Density of states for various dopings $\delta$ in the 2D Hubbard model at $T=0.092t$ and $U=8t$ calculated with DCA/quantum Monte Carlo for a four-site cluster, $N_c=4$. Inset: Uniform spin susceptibility as a function of temperature. Energies are in units of $4t$. From 85.

Figure 32

The spectral function $A(\mathbf{k},\omega )$ near the chemical potential in the 5% doped 2D Hubbard model at $T=0.88t$ and $U=8t$ along high-symmetry directions in the first Brillouin zone between $\Gamma =(0,0)$, $X=(\pi ,0)$, and $M=(\pi ,\pi )$ calculated with the DCA/QMC method for a four-site cluster $N_c=4$. The DCA self-energy was interpolated using a smooth spline (see Sec. 2D4).

Figure 33

The doping dependence of the chemical potential in the 2D Hubbard model calculated with the two-site composite-operator method; dashed line, $T=0.15t$; solid line, $T=0.07t$. Right inset: DOS for various values of $U$ at 5% doping $(n=0.95)$. Left inset: Imaginary part of the self-energy evaluated at a Fermi momentum (0.3,2.10) for $n=0.97$, (0.3,1.84) for $n=0.80$, and (0.3,1.06) for $n=0.3$. From 174.

Figure 34

(Color in online edition) Intensity plot of the spectral function $A(\mathbf{k},0)$ (top) and imaginary part of the self-energy $\mathrm{Im}\Sigma (\mathbf{k},0)$ (bottom) of the 2D Hubbard model calculated with cluster perturbation theory on a $3\times 4$-site cluster. The left side shows the results in the first quadrant of the Brillouin zone for the 17% electron-doped system at $U=4t$ and the right side for the 17% hole-doped system at $U=8t$. The dashed line represents the Fermi surface of the noninteracting system. From 167.

Figure 35

Comparison of different DCA cells: (a) density of states near the chemical potential; (b), (c), and (d) coarse-grained anomalous Green’s function ${\overline{G}}_{12}(\mathbf{K},\omega )\equiv \overline{F}(\mathbf{K},\omega )$ in the superconducting state of the 2D Hubbard model at 19% doping, $T=0.047t$, $U=12t$ for different cluster K points calculated with DCA/NCA for a four-site cluster, $N_c=4$. From 116.

Figure 36

Pair-field susceptibilities vs temperature in the even-frequency $s$-wave, extended $s$-wave $\left(xs\right)$, $d$-wave, and odd-frequency $s$-wave channels in the 2D Hubbard model at 5% doping, $U=8t$ calculated with the DCA/QMC method for a four-site cluster, $N_c=4$. Inset: Inverse $d$-wave pair-field susceptibility vs temperature for different dopings and cluster sizes. The solid line is a fit to $b{(T-T_c)}^{\gamma }$ with $T_c=0.084t$ and $\gamma =0.72$. Temperatures are in units of $4t$. From 85.

Figure 37

DOS (left), local dynamic spin susceptibility (center), and local dynamic charge susceptibility (right) in the 2D Hubbard model at 5% doping at different temperatures above and below the critical temperature $T_c=0.0218=0.087t$ calculated with the DCA/QMC method for a four-site cluster, $N_c=4$. Temperatures are in units of $4t$. From 114.

Figure 38

Kinetic (top) and potential (bottom) energies vs temperature in the normal (NS) and superconducting (SC) states in the 2D Hubbard model at 5% (left) and 20% (right) doping for $U=8t$ calculated with the DCA/QMC method on a four-site cluster $(N_c=4)$. $T_c$ is indicated by the vertical dotted lines. Temperatures are in units of $4t$. From 114.

Figure 39

(Color in online edition) Four-site cluster calculation for the 2D Hubbard model: (a) Schematic representation of an antiferromagnetic $d$-wave $2\times 2$ periodically repeated cluster. (b) Generic phase diagram of high-temperature superconductor. (c) Magnetic (M) and $d$-wave superconducting (F) order parameters versus hole doping in the 2D Hubbard model at $\beta t=15$, $t^{\prime }=-0.15t$, $U=4.8t$ calculated with a four-site cluster approach similar to the DCA/QMC method. From 107.

Figure 40

Temperature-doping phase diagram of the 2D Hubbard model when $U=8t$ calculated with DCA/QMC for a four-site cluster, $N_c=4$. The error bars on $T^{*}$ result from the difficulty in locating the maximum in the uniform spin susceptibility. Regions of antiferromagnetism, $d$-wave superconducting, and pseudogap behavior can be seen.