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arXiv:cond-mat/0601271 · Physical Review B

Quantum Monte Carlo study of strongly correlated electrons: Cellular dynamical mean-field theory

Bumsoo Kyung, Gabriel Kotliar, A.–M. S. Tremblay

Published May 10, 2006• 42 citations
arXivHTMLPDF

Abstract

We study the Hubbard model using the cellular dynamical mean-field theory (CDMFT) with quantum Monte Carlo (QMC) simulations. We present the algorithmic details of CDMFT with the Hirsch-Fye QMC method for the solution of the self-consistently embedded quantum cluster problem. We use the one- and two-dimensional half filled Hubbard model to gauge the performance of $\mathrm{CDMFT}+\mathrm{QMC}$ particularly for small clusters by comparing with the exact results and also with other quantum cluster methods. We calculate single-particle Green's functions and self-energies on small clusters to study their size dependence in one and two dimensions. It is shown that in one dimension, CDMFT with two sites in the cluster is already able to describe with high accuracy the evolution of the density as a function of the chemical potential and the compressibility divergence at the Mott transition, in good agreement with the exact Bethe ansatz result. With increasing $U$ the result on small clusters rapidly approaches that of the infinite size cluster. Large scattering rate and a positive slope in the real part of the self-energy in one dimension suggest that the system is a non-Fermi liquid for all the parameters studied here. In two dimensions, at intermediate to strong coupling, even the smallest cluster $({N}_{c}=2\ifmmode\times\else\texttimes\fi{}2)$ accounts for more than 95% of the correlation effect of the infinite-size cluster in the single particle spectrum, suggesting that some of the important problems in strongly correlated electron systems may be studied highly accurately with a reasonable computational effort. Finally, as an application that is sensitive to details of correlations, we show that $\mathrm{CDMFT}+\mathrm{QMC}$ can describe spin-charge separated Luttinger liquid physics in one dimension. The spinon and holon branches appear only for sufficiently large system sizes.

DOI
10.1103/physrevb.73.205106

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We study the Hubbard model using the cellular dynamical mean-field theory (CDMFT) with quantum Monte Carlo (QMC) simulations. We present the algorithmic details of CDMFT with the Hirsch-Fye QMC method for the solution of the self-consistently embedded quantum cluster problem. We use the one- and two-dimensional half filled Hubbard model to gauge the performance of CDMFT+QMC\mathrm{CDMFT}+\mathrm{QMC}CDMFT+QMC particularly for small clusters by comparing with the exact results and also with other quantum cluster methods. We calculate single-particle Green's functions and self-energies on small clusters to study their size dependence in one and two dimensions. It is shown that in one dimension, CDMFT with two sites in the cluster is already able to describe with high accuracy the evolution of the density as a function of the chemical potential and the compressibility divergence at the Mott transition, in good agreement with the exact Bethe ansatz result. With increasing UUU the result on small clusters rapidly approaches that of the infinite size cluster. Large scattering rate and a positive slope in the real part of the self-energy in one dimension suggest that the system is a non-Fermi liquid for all the parameters studied here. In two dimensions, at intermediate to strong coupling, even the smallest cluster (Nc=2\ifmmode×\else\texttimes\fi2)({N}_{c}=2\ifmmode\times\else\texttimes\fi{}2)(Nc​=2\ifmmode×\else\texttimes\fi2) accounts for more than 95% of the correlation effect of the infinite-size cluster in the single particle spectrum, suggesting that some of the important problems in strongly correlated electron systems may be studied highly accurately with a reasonable computational effort. Finally, as an application that is sensitive to details of correlations, we show that CDMFT+QMC\mathrm{CDMFT}+\mathrm{QMC}CDMFT+QMC can describe spin-charge separated Luttinger liquid physics in one dimension. The spinon and holon branches appear only for sufficiently large system sizes.
Even after decades of intense research, the single band Hubbard model representing the fundamental model for interacting quantum systems and electronic correlations remains an unsolved cornerstone paradigm in theoretical solid state physics. Within the non-perturbative dynamical mean-field theory (DMFT), the lattice problem is mapped to a self-consistent auxiliary quantum impurity model leading to a local approximation of the self-energy. Nonlocal correlations can be included by the real-space cluster extension of the DMFT (CDMFT). Enlarging the unit cell makes the calculation of a numerically exact solution in CDMFT very challenging. We here use a quantum Monte-Carlo approach in the imaginary-time space in order to solve the self-consistency equations for different cluster sizes. In order to restore the broken translational symmetry re-periodization schemes for the Green function, the self-energy, or its cumulants have been introduced. However, these suffer from ambiguity and may even lead to convergence problems when attempted inside the self-consistency loop. The comparison to numerically exact diagrammatic quantum Monte Carlo calculations shows that introducing a so-called centerfocused extrapolation (CFE) to approximate the lattice self-energy yields very accurate results. Moreover, the CFE converges faster with the cluster size than previous periodization schemes. We here perform a detailed CDMFT analysis for the single-band Hubbard model on the 2D square lattice, reaching real-space cluster sizes of up to 9x9 sites. In addition to spectral properties, we also compute two-particle correlation functions which are not accessible in DMFT. Using benchmarks against diagrammatic Monte Carlo at high temperature, we show that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size. In particular, the exponential decay of spin-spin correlations is very well captured by CDMFT calculations, even when correlations extend beyond the size of the cluster. We further present results at lower temperature T and larger UUU than the range currently accessible with diagrammatic Monte Carlo methods techniques. The CDMFT+CFE represents therefore a powerful computational tool to access the physics of non-local correlations beyond dynamical mean-field theory.
The one-dimensional half-filled Hubbard model is considered at zero temperature within the cellular dynamical mean-field theory (CDMFT). By the computation of the spectral gap and the energy density with various cluster and bath sizes we examine the accuracy of the CDMFT in a systematic way, which proves the accurate description of the one-dimensional systems by the CDMFT with small clusters. We also calculate the spectral weights in a full range of the momentum for various interaction strengths. The results do not only account for the spin-charge separation, but they also reproduce all the features of the Bethe ansatz dispersions, implying that the CDMFT provides an excellent description of the spectral properties of low-dimensional interacting systems.
We examine the cluster-size dependence of the cellular dynamical mean-field theory (CDMFT) applied to the two-dimensional Hubbard model. Employing the continuous-time quantum Monte Carlo method as the solver for the effective cluster model, we obtain CDMFT solutions for 4-, 8-, 12-, and 16-site clusters at a low temperature. Comparing various periodization schemes, which are used to construct the infinite-lattice quantities from the cluster results, we find that the cumulant periodization yields the fastest convergence for the hole-doped Mott insulator where the most severe size dependence is expected. We also find that the convergence is much faster around (0,0)(0,0)(0,0) and $(\frac{\ensuremath{\pi}}{2},\frac{\ensuremath{\pi}}{2})$ than around $(\ensuremath{\pi},0)$ and $(\ensuremath{\pi},\ensuremath{\pi})$. The cumulant-periodized self-energy seems to be close to its thermodynamic limit already for a 16-site cluster in the range of parameters studied. The 4-site results remarkably agree well with the 16-site results, indicating that the previous studies based on the 4-site cluster capture the essence of the physics of doped Mott insulators.
We revisit the cellular dynamical mean-field theory (CDMFT) for the single-band Hubbard model on the square lattice at half filling, reaching real-space cluster sizes of up to 9 × 9 sites.Using benchmarks against direct lattice diagrammatic Monte Carlo at high temperature, we show that the self-energy obtained from a cluster center-focused extrapolation converges faster with the cluster size than the periodization schemes previously introduced in the literature.The same benchmark also shows that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size, even though its spatial extension is larger than the cluster size.
We explore the use of exact diagonalization methods for solving the self-consistent equations of the cellular dynamical mean field theory for the one-dimensional regular and extended Hubbard models. We investigate the nature of the Mott transition and convergence of the method as a function of cluster size as well as the optimal allocation of computational resources between bath and ``cluster-impurity'' sites, with a view to develop a renormalization group method in higher dimensions. We assess the performance of the method by comparing results for the Green's functions in both the spin density wave and charge density wave phases with accurate density matrix renormalization group (DMRG) calculations.
We present the algorithmic details of the dynamical cluster approximation (DCA), with a quantum Monte Carlo (QMC) method used to solve the effective cluster problem. The DCA is a fully causal approach which systematically restores nonlocal correlations to the dynamical mean field approximation (DMFA) while preserving the lattice symmetries. The DCA becomes exact for an infinite cluster size, while reducing to the DMFA for a cluster size of unity. We present a generalization of the Hirsch-Fye QMC algorithm for the solution of the embedded cluster problem. We use the two-dimensional Hubbard model to illustrate the performance of the DCA technique. At half filling, we show that the DCA drives the spurious finite-temperature antiferromagnetic transition found in the DMFA slowly towards zero temperature as the cluster size increases, in conformity with the Mermin-Wagner theorem. Moreover, we find that there is a finite-temperature metal to insulator transition which persists into the weak-coupling regime. This suggests that the magnetism of the model is Heisenberg-like for all nonzero interactions. Away from half filling, we find that the sign problem that arises in QMC simulations is significantly less severe in the context of DCA. Hence, we were able to obtain good statistics for small clusters. For these clusters, the DCA results show evidence of non-Fermi-liquid behavior and superconductivity near half filling.