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arXiv:2007.03955 · Physical review. B./Physical review. B

Sparse modeling of large-scale quantum impurity models with low symmetries

Hiroshi Shinaoka, Yuki Nagai

Published Jan 19, 2021• 12 citations
arXivHTMLPDF

Abstract

Quantum embedding theories can be used for obtaining quantitative descriptions of correlated materials. However, a critical challenge is solving an effective impurity model of correlated orbitals embedded in an electron bath. Many advanced impurity solvers require the approximation of a bath continuum using a finite number of bath levels, producing a highly nonconvex, ill-conditioned inverse problem. To address this drawback, this study proposes an efficient fitting algorithm for matrix-valued hybridization functions based on a data-science approach, sparse modeling, and a compact representation of Matsubara Green's functions. The efficiency of the proposed method is demonstrated by fitting random hybridization functions with large off-diagonal elements and those of a 20-orbital impurity model for a high-${T}_{\mathrm{c}}$ compound, LaAsFeO, at low temperatures $(T)$. The results set quantitative goals for the future development of impurity solvers toward quantum embedding simulations of complex correlated materials.

DOI
10.1103/physrevb.103.045120

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Quantitative descriptions of strongly correlated materials pose a considerable challenge in condensed matter physics and chemistry. A promising approach to address this problem is quantum embedding methods. In particular, the dynamical mean-field theory (DMFT) maps the original system to an effective quantum impurity model comprising correlated orbitals embedded in an electron bath. The biggest bottleneck in DMFT calculations is numerically solving the quantum impurity model, i.e., computing the Green's function. Past studies have proposed theoretical methods to compute the Green's function of a quantum impurity model in polynomial time using a quantum computer. So far, however, efficient methods for computing the imaginary-time Green's functions have not been established despite the advantages of the imaginary-time formulation. We propose a quantum-classical hybrid algorithm for computing imaginary-time Green's functions on quantum devices with limited hardware resources by applying the variational quantum simulation. Using a quantum circuit simulator, we verified this algorithm by computing Green's functions for a dimer model as well as a four-site impurity model obtained by DMFT calculations of the single-band Hubbard model, although our method can be applied to general imaginary-time correlation functions.
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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.
An impurity model describes an atom or molecule embedded in some host or bath, with which it can exchange electrons. This exchange of electrons allows the impurity to make transitions between different quantum states, and leads to a non-trivial dynamics. Therefore, despite the zero dimensional nature (which makes impurity problems computationally much more tractable than fermionic lattice models), their numerical simulation remains a challenging task. Methods such as exact diagonalization or numerical RG, which explicitly treat a finite number of bath states, work well for single orbital models. However, because the number of bath states must be increased proportional to the number of orbitals, the computational effort grows exponentially with system size, and requires severe truncations of the bath already for two orbitals. Monte Carlo methods have the advantage that the bath is integrated out and thus the (infinite) size of the bath Hilbert space does not affect the simulation. While restricted to finite temperature, Monte Carlo methods are thus the method of choice for the solution of large multi-orbital or cluster impurity problems. Over the last few years, significant progress has been made (both in terms of efficiency and flexibility) with the development of diagrammatic Monte Carlo techniques. These notes provide an overview over two recently developed, complementary methods: (i) the weakcoupling approach, which scales favorably with system size and allows the efficient simulation of large impurity clusters, and (ii) the strong-coupling approach, which can handle impurity