Pyqcm is a Python/C++ library that implements a few quantum cluster methods with an exact diagonalization impurity solver. Quantum cluster methods are used in the study of strongly correlated electrons to provide an approximate solution to Hubbard-like models. The methods covered by this library are Cluster Perturbation Theory (CPT), the Variational Cluster Approach (VCA) and Cellular (or Cluster) Dynamical Mean Field Theory (CDMFT). The impurity solver (the technique used to compute the cluster’s interacting Green function) is exact diagonalization from sparse matrices, using the Lanczos algorithm and variants thereof. The core library is written in C++ for performance, but the interface is in Python, for ease of use and inter-operability with the numerical Python ecosystem. The library is distributed under the GPL license.Pyqcm is a Python/C++ library that implements a few quantum cluster methods with an exact diagonalization impurity solver. Quantum cluster methods are used in the study of strongly correlated electrons to provide an approximate solution to Hubbard-like models. The methods covered by this library are Cluster Perturbation Theory (CPT), the Variational Cluster Approach (VCA) and Cellular (or Cluster) Dynamical Mean Field Theory (CDMFT). The impurity solver (the technique used to compute the cluster’s interacting Green function) is exact diagonalization from sparse matrices, using the Lanczos algorithm and variants thereof. The core library is written in C++ for performance, but the interface is in Python, for ease of use and inter-operability with the numerical Python ecosystem. The library is distributed under the GPL license.

arXiv:2305.18643SciPost Physics CodebasesDec 20, 202318 citations

An introduction to quantum cluster methods

David Sénéchal

These lecture notes provide an introduction to quantum cluster methods for strongly correlated systems. Cluster Perturbation Theory (CPT), the Variational Cluster Approximation (VCA) and Cellular Dynamical Mean Field Theory (CDMFT) are described, as well as the exact diagonalization solver for the cluster. Potthoff's self-energy functional formalism is reviewed. Some numerical procedures, in particular regarding the exact diagonalization method and the frequency-momentum integrals needed in VCA, are discussed in detail.These lecture notes provide an introduction to quantum cluster methods for strongly correlated systems. Cluster Perturbation Theory (CPT), the Variational Cluster Approximation (VCA) and Cellular Dynamical Mean Field Theory (CDMFT) are described, as well as the exact diagonalization solver for the cluster. Potthoff's self-energy functional formalism is reviewed. Some numerical procedures, in particular regarding the exact diagonalization method and the frequency-momentum integrals needed in VCA, are discussed in detail.

arXiv:0806.2690Jun 17, 200841 citations

Dynamical variational Monte Carlo as a quantum impurity solver: Application to cluster dynamical mean field theory

Peter Rosenberg, David Sénéchal, A.–M. S. Tremblay +1 more

Two of the primary sources of error in the Cluster dynamical mean field theory (CDMFT) technique arise from the use of finite size clusters and finite size baths, which makes the development of impurity solvers that can treat larger systems an essential goal. In this work we introduce an impurity solver based on the recently developed dynamical variational Monte Carlo (dVMC) method. Variational Monte Carlo possesses a favorable scaling as a function of system size, which enables the treatment of systems beyond the reach of current exact diagonalization solvers. To benchmark the technique, we perform a systematic set of CDMFT calculations on the single-band one-dimensional Hubbard model. We compare to results obtained with an exact diagonalization solver for small clusters, and against the exact solution in the thermodynamic limit obtained by Lieb and Wu for larger clusters. The development of improved impurity solvers will help extend the reach of quantum cluster methods, which can be applied to a wide range of strongly correlated electron systems, promising new insights into their emergent behavior.Two of the primary sources of error in the Cluster dynamical mean field theory (CDMFT) technique arise from the use of finite size clusters and finite size baths, which makes the development of impurity solvers that can treat larger systems an essential goal. In this work we introduce an impurity solver based on the recently developed dynamical variational Monte Carlo (dVMC) method. Variational Monte Carlo possesses a favorable scaling as a function of system size, which enables the treatment of systems beyond the reach of current exact diagonalization solvers. To benchmark the technique, we perform a systematic set of CDMFT calculations on the single-band one-dimensional Hubbard model. We compare to results obtained with an exact diagonalization solver for small clusters, and against the exact solution in the thermodynamic limit obtained by Lieb and Wu for larger clusters. The development of improved impurity solvers will help extend the reach of quantum cluster methods, which can be applied to a wide range of strongly correlated electron systems, promising new insights into their emergent behavior.

arXiv:2307.15738Physical review. B./Physical review. BDec 8, 20234 citations

Determinantal Quantum Monte Carlo solver for Cluster Perturbation Theory

Edwin W. Huang, Yao Wang

Cluster Perturbation Theory (CPT) is a technique for computing the spectral function of fermionic models with local interactions. By combining the solution of the model on a finite cluster with perturbation theory on intra-cluster hoppings, CPT provides access to single-particle properties with arbitrary momentum resolution while incurring low computational cost. Here, we introduce Determinantal Quantum Monte Carlo (DQMC) as a solver for CPT. Compared to the standard solver, exact diagonalization (ED), the DQMC solver reduces finite size effects through utilizing larger clusters, allows study of temperature dependence, and enables large-scale simulations of a greater set of models. We discuss the implementation of the DQMC solver for CPT and benchmark the CPT+DQMC method for the attractive and repulsive Hubbard models, showcasing its advantages over standard DQMC and CPT+ED simulations.Cluster Perturbation Theory (CPT) is a technique for computing the spectral function of fermionic models with local interactions. By combining the solution of the model on a finite cluster with perturbation theory on intra-cluster hoppings, CPT provides access to single-particle properties with arbitrary momentum resolution while incurring low computational cost. Here, we introduce Determinantal Quantum Monte Carlo (DQMC) as a solver for CPT. Compared to the standard solver, exact diagonalization (ED), the DQMC solver reduces finite size effects through utilizing larger clusters, allows study of temperature dependence, and enables large-scale simulations of a greater set of models. We discuss the implementation of the DQMC solver for CPT and benchmark the CPT+DQMC method for the attractive and repulsive Hubbard models, showcasing its advantages over standard DQMC and CPT+ED simulations.

arXiv:2107.11284Jul 23, 20210 citations

Codebase release 5.3 for EDIpack

Lorenzo Crippa, Igor S. Krivenko, Samuele Giuli +10 more

We present a next-generation version of EDIpack, a flexible, high-performance numerical library using Lanczos-based exact diagonalization to solve generic quantum impurity problems, such as those introduced in Dynamical Mean-Field Theory to describe extended strongly correlated materials. This new release efficiently solves impurity problems allowing for different broken-symmetry solutions, including superconductivity, featuring local spin-orbit coupling and/or electron-phonon coupling. It provides quick access to dynamical correlation functions on the entire complex frequency plane at zero and low-temperatures. The modular architecture of the software not only provides Fortran APIs but also includes bindings to C/C++, interfaces with Python and Julia or with TRIQS and w2dynamics research platforms, thus ensuring unprecedented level of inter-operability. The outlook includes further extensions to study quantum materials and cold atoms quantum simulators, as well as quantum information applications.We present a next-generation version of EDIpack, a flexible, high-performance numerical library using Lanczos-based exact diagonalization to solve generic quantum impurity problems, such as those introduced in Dynamical Mean-Field Theory to describe extended strongly correlated materials. This new release efficiently solves impurity problems allowing for different broken-symmetry solutions, including superconductivity, featuring local spin-orbit coupling and/or electron-phonon coupling. It provides quick access to dynamical correlation functions on the entire complex frequency plane at zero and low-temperatures. The modular architecture of the software not only provides Fortran APIs but also includes bindings to C/C++, interfaces with Python and Julia or with TRIQS and w2dynamics research platforms, thus ensuring unprecedented level of inter-operability. The outlook includes further extensions to study quantum materials and cold atoms quantum simulators, as well as quantum information applications.

SciPost Physics CodebasesSep 24, 20251 citation

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

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

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.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.

arXiv:cond-mat/0601271Physical Review BMay 10, 200642 citations

\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.