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The European Physical Journal Special Topics

Quantum Monte Carlo impurity solvers for multi-orbital problems and frequency-dependent interactions

Hiroshi Shinaoka, Fakher F. Assaad, Nils Blümer, Philipp Werner

Published Jul 1, 2017• 11 citations
Publisher

Abstract

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.

DOI
10.1140/epjst/e2017-70050-x

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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 present a continuous-time Monte Carlo impurity solver for multiorbital impurity models which combines a strong-coupling hybridization expansion and a weak-coupling expansion in the Hund's coupling parameter JJJ. This double-expansion approach allows to treat the dominant density-density interactions UUU within the efficient segment representation. We test the approach for a two-orbital model with static interactions, and then explain how the double expansion allows to simulate models with frequency dependent $U(\ensuremath{\omega})$ and $J(\ensuremath{\omega})$. The method is used to investigate spin-state transitions in a toy model for fullerides, with repulsive bare JJJ but attractive screened JJJ.
Multiorbital quantum impurity models with general interaction and hybridization terms appear in a wide range of applications, including embedding, quantum transport, and nanoscience. However, most quantum impurity solvers are restricted to a few impurity orbitals, discretized baths, diagonal hybridizations, or density-density interactions. Here, we generalize the inchworm quantum Monte Carlo method to the interaction expansion, and we explore its application to typical single- and multiorbital problems encountered in investigations of impurity and lattice models. Our implementation generically outperforms bare and bold-line quantum Monte Carlo algorithms in the interaction expansion. For the systems studied here, our implementation remains inferior to the more specialized hybridization expansion and auxiliary field algorithms. The problem of convergence to unphysical fixed points, which hampers so-called bold-line methods, is not encountered in inchworm Monte Carlo.
Andrew J. Millis Department of Physics, Columbia University, 538 West, 120th Street, New York, NY 10027, U.S.A. An atom or molecule embedded in an infinite host material defines a quantum impurity model (QIM). Exchange with the host causes the impurity to make transitions between different quantum states, leading to a nontrivial dynamics which may be described as a quantum field theory in zero space and one time dimension. QIMs exhibit many phenomena of intrinsic theoretical interest including quantum phase transitions (QPTs). The development over the past 15 years of dynamical mean field theory (DMFT) shows that they also serve as an auxiliary problem from whose solution important information relating to the physics of interacting lattice models can be obtained. Numerical simulation of these models remains a challenging task but the development of diagrammatic Monte Carlo techniques over the last few years has led to significant improvements in efficiency and flexibility. Highly precise results on few-orbital models can be computed and large Hilbert-space multiorbital models are now within reach. This chapter provides an overview of recent developments in quantum Monte Carlo methods and their application to multi-orbital systems. It presents both the weak-coupling approach, which scales favorably with system size and allows the efficient simulation of large impurity clusters, and the strong-coupling approach, which can handle QIMs with strong interactions.
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.
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.
The diagrammatic Monte Carlo method has so far been used mainly for weak-coupling expansions. Here we show that the strong-coupling expansion offers a key advantage: it can be implemented efficiently on both the real and imaginary frequency axes at finite temperature. Using a quantum-impurity solver for dynamical mean-field theory (DMFT) as an example, we find rapid convergence with expansion order. We derive closed-form real-axis Feynman rules for diagrams of arbitrary order, and implement them in a bold hybridization-expansion quantum Monte Carlo (BHQMC) impurity solver. Benchmarking against state-of-the-art numerical renormalization group (NRG) results for the DMFT Mott transition of the Hubbard model, we obtain a highly accurate frequency-dependent scattering rate at finite temperature. This enables reliable spectroscopy and provides benchmark transport results within DMFT and cluster-DMFT.