Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

AIP conference proceedings

The Dynamical Cluster Approximation with Quantum Monte Carlo Cluster Solvers

Mark Jarrell, Alexandru Macridin, Karlis Mikelsons, D. G. S. P. Doluweera, James E. Gubernatis, Adolfo Avella, Ferdinando Mancini

Published Jan 1, 2008• 5 citations
Publisher

Abstract

We present a pedagogical discussions of the dynamical mean field (DMFA) and dynamical cluster (DCA) approximations and associated Monte Carlo and entropy‐based methods of Bayesian data analysis. The DMFA and DCA methods are developed as coarse‐graining approximations and the relationship between the cluster and lattice problems are detailed. The Hirsch‐Fye and continuous time Quantum Monte Carlo (QMC) algorithms are used to solve the cluster problem. The algorithms are discussed, together with methods for efficient measurements and the modifications required by the self‐consistency of the DMFA/DCA. Then, several principles of Bayesian data analysis are presented. When coupled with information theory, this analysis produces a precise and systematic way to analytically continue Matsubara‐time QMC results to real frequencies. We show how to use Bayesian inference to qualify the solution of the continuation and optimize the inputs. Besides developing the Bayesian formalism, we also present a detailed description of the data qualification, sketch an efficient algorithm to solve for the optimal spectra, give cautionary notes where appropriate, and present two detailed case studies to demonstrate the method.

DOI
10.1063/1.2940445

Related papers

Quantum Monte Carlo algorithm for nonlocal corrections to the dynamical mean-field approximation

Mark Jarrell, Thomas Maier, Carey Huscroft +1 more

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.

arXiv:cond-mat/0108140•Physical review. B, Condensed matter•Oct 29, 2001•183 citations

Dynamical cluster approximation with continuous lattice self-energy

Peter Staar, Thomas Maier, T. C. Schulthess

The dynamical cluster approximation (DCA) is a systematic extension beyond the single-site approximation in dynamical mean field theory, to include spatially nonlocal correlations in quantum many-body simulations of strongly correlated systems. We extend the DCA with a continuous lattice self-energy in order to achieve better convergence with cluster size. This method, which we call DCA${}^{+}$, cures the cluster-shape dependence problems of the DCA, without suffering from causality violations of previous attempts to interpolate the cluster self-energy. A practical approach based on standard inference techniques is given to deduce the continuous lattice self-energy from an interpolated cluster self-energy. We study the pseudogap region of a hole-doped two-dimensional Hubbard model and find that, in the DCA${}^{+}\phantom{\rule{0.222222em}{0ex}}$algorithm, the self-energy and pseudogap temperature ${T}^{*}$ converge monotonously with cluster size. Introduction of a continuous lattice self-energy eliminates artificial long-range correlations and thus significantly reduces the sign problem of the quantum Monte Carlo cluster solver in the DCA${}^{+}\phantom{\rule{0.222222em}{0ex}}$algorithm compared to the normal DCA. Simulations with much larger cluster sizes thus become feasible, which, along with the improved convergence in cluster size, raises hope that precise extrapolations to the exact infinite cluster size limit can be reached for other physical quantities as well.

arXiv:1304.3624•Physical Review B•Sep 3, 2013•36 citations

Quantum Monte Carlo Methods and Extensions for the 2D Hubbard Model

Joseph Paki

This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.

•Jan 1, 2019•1 citation

Semiclassical approximation solved by Monte Carlo integration as an efficient impurity solver for dynamical mean field theory and its cluster extensions

Hunpyo Lee, Yu‐Zhong Zhang, Hoonkyung Lee +3 more

We propose that a combination of the semiclassical approximation with Monte Carlo simulations can be an efficient and reliable impurity solver for dynamical mean field theory equations and their cluster extensions with large cluster sizes. In order to show the reliability of the method, we consider two test cases: (i) the single-band Hubbard model within the dynamical cluster approximation with four- and eight-site clusters and (ii) the anisotropic two-orbital Hubbard model with orbitals of different bandwidth within the single-site dynamical mean field theory. We compare the critical interaction ${U}_{c}/t$ with those obtained from solving the dynamical mean field equations with the continuous-time and Hirsch-Fye quantum Monte Carlo approaches. In both test cases we observe reasonable values of the metal-insulator critical interaction strength ${U}_{c}/t$ and the nature of Mott physics in the self-energy behavior. While some details of the spectral functions cannot be captured by the semiclassical approximation due to the freezing of dynamical fluctuations, the main features are reproduced by the approach.

arXiv:1303.6813•Physical Review B•Oct 18, 2013•5 citations

Efficiency of quantum Monte Carlo impurity solvers for the dynamical mean-field theory

Nils Blümer

Since the inception of the dynamical mean-field theory, numerous numerical studies have relied on the Hirsch-Fye quantum Monte Carlo (HF-QMC) method for solving the associated impurity problem. Recently developed continuous-time algorithms (CT-QMC) avoid the Trotter discretization error and allow for faster configuration updates, which makes them candidates for replacing HF-QMC. We demonstrate, however, that a state-of-the-art implementation of HF-QMC (with extrapolation of discretization $\ensuremath{\Delta}\ensuremath{\tau}\ensuremath{\rightarrow}0$) is competitive with CT-QMC. A quantitative analysis of Trotter errors in HF-QMC estimates and of appropriate $\ensuremath{\Delta}\ensuremath{\tau}$ values is included.

arXiv:0708.1749•Physical Review B•Nov 28, 2007•31 citations

Dynamical cluster approximation: Nonlocal dynamics of correlated electron systems

Matthias H. Hettler, Manash Mukherjee, Mark Jarrell +1 more

We recently introduced the dynamical cluster approximation (DCA), a technique that includes short-ranged dynamical correlations in addition to the local dynamics of the dynamical mean-field approximation while preserving causality. The technique is based on an iterative self-consistency scheme on a finite-size periodic cluster. The dynamical mean-field approximation (exact result) is obtained by taking the cluster to a single site (the thermodynamic limit). Here, we provide details of our method, explicitly show that it is causal, systematic, $\ensuremath{\Phi}$ derivable, and that it becomes conserving as the cluster size increases. We demonstrate the DCA by applying it to a quantum Monte Carlo and exact enumeration study of the two-dimensional Falicov-Kimball model. The resulting spectral functions preserve causality, and the spectra and the charge-density-wave transition temperature converge quickly and systematically to the thermodynamic limit as the cluster size increases.

arXiv:cond-mat/9903273•Physical review. B, Condensed matter•May 15, 2000•323 citations
We present a pedagogical discussions of the dynamical mean field (DMFA) and dynamical cluster (DCA) approximations and associated Monte Carlo and entropy‐based methods of Bayesian data analysis. The DMFA and DCA methods are developed as coarse‐graining approximations and the relationship between the cluster and lattice problems are detailed. The Hirsch‐Fye and continuous time Quantum Monte Carlo (QMC) algorithms are used to solve the cluster problem. The algorithms are discussed, together with methods for efficient measurements and the modifications required by the self‐consistency of the DMFA/DCA. Then, several principles of Bayesian data analysis are presented. When coupled with information theory, this analysis produces a precise and systematic way to analytically continue Matsubara‐time QMC results to real frequencies. We show how to use Bayesian inference to qualify the solution of the continuation and optimize the inputs. Besides developing the Bayesian formalism, we also present a detailed description of the data qualification, sketch an efficient algorithm to solve for the optimal spectra, give cautionary notes where appropriate, and present two detailed case studies to demonstrate the method.
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.
The dynamical cluster approximation (DCA) is a systematic extension beyond the single-site approximation in dynamical mean field theory, to include spatially nonlocal correlations in quantum many-body simulations of strongly correlated systems. We extend the DCA with a continuous lattice self-energy in order to achieve better convergence with cluster size. This method, which we call DCA+{}^{+}+, cures the cluster-shape dependence problems of the DCA, without suffering from causality violations of previous attempts to interpolate the cluster self-energy. A practical approach based on standard inference techniques is given to deduce the continuous lattice self-energy from an interpolated cluster self-energy. We study the pseudogap region of a hole-doped two-dimensional Hubbard model and find that, in the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm, the self-energy and pseudogap temperature T∗{T}^{*}T∗ converge monotonously with cluster size. Introduction of a continuous lattice self-energy eliminates artificial long-range correlations and thus significantly reduces the sign problem of the quantum Monte Carlo cluster solver in the DCA+{}^{+}\phantom{\rule{0.222222em}{0ex}}+algorithm compared to the normal DCA. Simulations with much larger cluster sizes thus become feasible, which, along with the improved convergence in cluster size, raises hope that precise extrapolations to the exact infinite cluster size limit can be reached for other physical quantities as well.
This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.
We propose that a combination of the semiclassical approximation with Monte Carlo simulations can be an efficient and reliable impurity solver for dynamical mean field theory equations and their cluster extensions with large cluster sizes. In order to show the reliability of the method, we consider two test cases: (i) the single-band Hubbard model within the dynamical cluster approximation with four- and eight-site clusters and (ii) the anisotropic two-orbital Hubbard model with orbitals of different bandwidth within the single-site dynamical mean field theory. We compare the critical interaction Uc/t{U}_{c}/tUc​/t with those obtained from solving the dynamical mean field equations with the continuous-time and Hirsch-Fye quantum Monte Carlo approaches. In both test cases we observe reasonable values of the metal-insulator critical interaction strength Uc/t{U}_{c}/tUc​/t and the nature of Mott physics in the self-energy behavior. While some details of the spectral functions cannot be captured by the semiclassical approximation due to the freezing of dynamical fluctuations, the main features are reproduced by the approach.
Since the inception of the dynamical mean-field theory, numerous numerical studies have relied on the Hirsch-Fye quantum Monte Carlo (HF-QMC) method for solving the associated impurity problem. Recently developed continuous-time algorithms (CT-QMC) avoid the Trotter discretization error and allow for faster configuration updates, which makes them candidates for replacing HF-QMC. We demonstrate, however, that a state-of-the-art implementation of HF-QMC (with extrapolation of discretization $\ensuremath{\Delta}\ensuremath{\tau}\ensuremath{\rightarrow}0$) is competitive with CT-QMC. A quantitative analysis of Trotter errors in HF-QMC estimates and of appropriate $\ensuremath{\Delta}\ensuremath{\tau}$ values is included.
We recently introduced the dynamical cluster approximation (DCA), a technique that includes short-ranged dynamical correlations in addition to the local dynamics of the dynamical mean-field approximation while preserving causality. The technique is based on an iterative self-consistency scheme on a finite-size periodic cluster. The dynamical mean-field approximation (exact result) is obtained by taking the cluster to a single site (the thermodynamic limit). Here, we provide details of our method, explicitly show that it is causal, systematic, $\ensuremath{\Phi}$ derivable, and that it becomes conserving as the cluster size increases. We demonstrate the DCA by applying it to a quantum Monte Carlo and exact enumeration study of the two-dimensional Falicov-Kimball model. The resulting spectral functions preserve causality, and the spectra and the charge-density-wave transition temperature converge quickly and systematically to the thermodynamic limit as the cluster size increases.