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Journal of Physics Condensed Matter

Improved Hubbard-I approximation impurity solver for quantum impurity models

Huanhuan Qiu, Jianing Zhuang, Li Huang, Jianzhou Zhao, Liang Du

Published Nov 6, 2018• 0 citations
Publisher

Abstract

We develop a fast impurity solver which is based on the combination of Hubbard-I approximation and hybridization expansion continuous-time quantum Monte Carlo algorithm. This solver inherits the advantages of both algorithms. In order to demonstrate the power and usefulness of this solver, we use it to study the magnetic phase transitions of single-band and two-band Hubbard models in the framework of single-site dynamical mean-field theory. The calculated results agree well with those obtained by hybridization expansion quantum impurity solver. It is suggested that this solver is very suitable to solve the multi-orbital quantum impurity models efficiently and can be used to study more realistic systems with magnetic long-range order in the future.

DOI
10.1088/1361-648x/aaee95

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We develop a fast impurity solver which is based on the combination of Hubbard-I approximation and hybridization expansion continuous-time quantum Monte Carlo algorithm. This solver inherits the advantages of both algorithms. In order to demonstrate the power and usefulness of this solver, we use it to study the magnetic phase transitions of single-band and two-band Hubbard models in the framework of single-site dynamical mean-field theory. The calculated results agree well with those obtained by hybridization expansion quantum impurity solver. It is suggested that this solver is very suitable to solve the multi-orbital quantum impurity models efficiently and can be used to study more realistic systems with magnetic long-range order in the future.
An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to correlated-electron models. The method is based on performing an exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descent treatment of the charge part. We test this method by applying it to one- or two-site approximations to the single-band Hubbard model with different band structures, and comparing the results to quantum Monte Carlo and simplified exact diagonalization calculations. The resulting electron self-energies, densities of states, and magnetic transition temperatures show reasonable agreement with the quantum Monte Carlo simulation over wide parameter ranges, suggesting that the semiclassical method is useful for obtaining a reasonable picture of the physics in situations where other techniques are too expensive.
We propose a fast impurity solver for the general quantum impurity model based on the perturbation theory around the atomic limit, which can be used in combination with the local density approximation (LDA) and the dynamical mean-field theory (DMFT). We benchmark the solver in the two-band Hubbard model within DMFT against quantum Monte Carlo (QMC) and numerical renormalization-group (NRG) results. We find that the solver works very well in the paramagnetic Mott insulator phase. We also apply this impurity solver to the DMFT study of the antiferromagnetic phase transition in the unfrustrated Bethe lattice. The Neel temperature obtained by the fast impurity solver agrees very well with the QMC results in the large Hubbard U limit. The method is a promising tool to be used in combination with the LDA+DMFT\mathrm{LDA}+\mathrm{DMFT}LDA+DMFT to study Mott insulators starting from first principles.
The solution of an auxiliary quantum impurity system is the computationally expensive step in dynamical mean field theory simulations of lattice models and materials. In this review, we discuss Monte Carlo based impurity solvers, which are suitable for a wide range of applications. In particular, we present an efficient implementation of the hybridization expansion approach, which enables the simulation of multiorbital impurity problems with off-diagonal and complex hybridizations, and dynamically screened (retarded) density-density interactions. As a complementary approach, we discuss an impurity solver based on the determinant Monte Carlo method, which scales favorably with inverse temperature and hence provides access to the very low temperature regime. The usefulness of these state-of-the-art impurity solvers is demonstrated with applications to the downfolding problem, i.e., the systematic derivation of dynamically screened interactions for low-energy effective models, and to pyrochlore iridates, where the spin-orbit coupling leads to complex hybridization functions in a multi-orbital system. As a benchmark for cluster extensions of dynamical mean field theory, we also present results from lattice Monte Carlo simulations for the momentum dependence of the pseudo-gap in the half-filled two-dimensional Hubbard model.
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.
This paper proposes an impurity solver for the dynamical mean field theory (DMFT) study of the Mott insulators, which is based on the second order perturbation of the hybridization function. After careful benchmarking with quantum Monte Carlo results on the anti-ferromagnetic phase of the Hubbard model, it concludes that this impurity solver can capture the main physical features in the strong coupling regime and can be a very useful tool for the LDA (local density approximation) + DMFT studies of the Mott insulators with long range order.
We present two related customized software packages, Hubb_DMFT and Wan2mb_DMFT, designed to solve the Dynamical Mean-Field Theory (DMFT) equations for strongly correlated electron systems. Hubb_DMFT is adjusted for the single-band Hubbard model, providing a fast way to calculate the local self-energy, as well as the two-particle fermion and triangular fermion-boson charge and spin vertices, while Wan2mb_DMFT extends this capability to realistic multi-orbital systems, directly interfacing Wannier tight-binding Hamiltonians with many-body solvers. Both codes are based on iQIST v.0.7 impurity solver and utilize a modified and internally integrated Continuous-Time Quantum Monte Carlo (CT-QMC) core with the hybridization expansion (CT-HYB) framework and improved self-energy and vertex estimators for density-density interaction, ensuring numerically exact solutions for quantum impurity problems.