Multigrid Hirsch-Fye quantum Monte Carlo method for dynamical mean-field theory

Performance analysis of continuous-time solvers for quantum impurity models

Impurity solvers play an essential role in the numerical investigation of strongly correlated electrons systems within the "dynamical mean field" approximation. Recently, a new class of continuous-time solvers has been developed, based on a diagrammatic expansion of the partition function in either the interactions or the impurity-bath hybridization. We investigate the performance of these two complementary approaches and compare them to the well-established Hirsch-Fye method. The results show that the continuous-time methods, and in particular the version which expands in the hybridization, p…

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

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 delta_tau -> 0) is competitive with CT-QMC. A quantitative analysis of Trotter errors in HF-QMC estimates …

Quantum Monte Carlo Study of Strongly Correlated Electrons: Cellular Dynamical Mean-Field Theory

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 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 depende…

Cluster solver for dynamical mean-field theory with linear scaling in inverse temperature

Dynamical mean field theory and its cluster extensions provide a very useful approach for examining phase transitions in model Hamiltonians, and, in combination with electronic structure theory, constitute powerful methods to treat strongly correlated materials. The key advantage to the technique is that, unlike competing real space methods, the sign problem is well controlled in the Hirsch-Fye (HF) quantum Monte Carlo used as an exact cluster solver. However, an important computational bottleneck remains; the HF method scales as the cube of the inverse temperature, $β$. This often makes simul…

Quasi continuous-time impurity solver for the dynamical mean-field theory with linear scaling in the inverse temperature

We present an algorithm for solving the self-consistency equations of the dynamical mean-field theory (DMFT) with high precision and efficiency at low temperatures. In each DMFT iteration, the impurity problem is mapped to an auxiliary Hamiltonian, for which the Green function is computed by combining determinantal quantum Monte Carlo (BSS-QMC) calculations with a multigrid extrapolation procedure. The method is numerically exact, i.e., yields results which are free of significant Trotter errors, but retains the BSS advantage, compared to direct QMC impurity solvers, of linear (instead of cubi…

Self-learning Monte Carlo with Deep Neural Networks

Self-learning Monte Carlo (SLMC) method is a general algorithm to speedup MC simulations. Its efficiency has been demonstrated in various systems by introducing an effective model to propose global moves in the configuration space. In this paper, we show that deep neural networks can be naturally incorporated into SLMC, and without any prior knowledge, can learn the original model accurately and efficiently. Demonstrated in quantum impurity models, we reduce the complexity for a local update from $ \mathcal{O}(β^2) $ in Hirsch-Fye algorithm to $ \mathcal{O}(β\ln β) $, which is a significant sp…

Imaginary-time formulation of steady-state nonequilibrium in quantum dot models

We examine the recently proposed imaginary-time formulation for strongly correlated steady-state nonequilibrium for its range of validity and discuss significant improvements in the analytic continuation of the Matsubara voltage as well as the fermionic Matsubara frequency. The discretization error in the conventional Hirsch-Fye algorithm has been compensated in the Fourier transformation with reliable small frequency behavior of self-energy. Here we give detailed discussions for generalized spectral representation ansatz by including high order vertex corrections and its numerical analytic co…

Combined approach of density functional theory and quantum Monte Carlo method to electron correlation in dilute magnetic semiconductors

We present a realistic study for electronic and magnetic properties in dilute magnetic semiconductor (Ga,Mn)As. A multi-orbital Haldane-Anderson model parameterized by density-functional calculations is presented and solved with the Hirsch-Fye quantum Monte Carlo algorithm. Results well reproduce experimental results in the dilute limit. When the chemical potential is located between the top of the valence band and an impurity bound state, a long-range ferromagnetic correlations between the impurities, mediated by antiferromagnetic impurity-host couplings, are drastically developed. We observe…

An Impurity Solver Using the Time-Dependent Variational Matrix Product State Approach

We use the time dependent variational matrix product state (tVMPS) approach to investigate the dynamical properties of the single impurity Anderson model (SIAM). Under the Jordan-Wigner transformation, the SIAM is reformulated into two spin-1/2 XY chains with local magnetic fields along the z-axis. The chains are connected by the longitudinal Ising coupling at the end points. The ground state of the model is searched variationally within the space spanned by the matrix product state (MPS). The temporal Green's functions are calculated both by the imaginary and real time evolutions, from which …

A Quantum Monte Carlo algorithm for non-local corrections to the Dynamical Mean-Field Approximation

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 non-local 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 mod…

利用子矩阵更新增强行列式量子蒙特卡洛：揭示三维 Hubbard 模型的相图

Boosting Determinant Quantum Monte Carlo with Submatrix Updates: Unveiling the Phase Diagram of the 3D Hubbard Model

本文首次将子矩阵更新全面引入 DQMC，支持扩展相互作用与零温度模拟，效率提升一个数量级，并据此确定半填充三维 Hubbard 模型的有限温度相图。