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arXiv:2307.11646 · Physical review. B./Physical review. B

Surrogate model solver for impurity-induced superconducting subgap states

V. Baran, Emil J. P. Frost, Jens Paaske

Published Dec 20, 2023• 22 citations
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Abstract

A simple impurity solver is shown to capture impurity-induced superconducting subgap states in quantitative agreement with the numerical renormalization group and quantum Monte Carlo simulations. The solver is based on the exact diagonalization of a single-impurity Anderson model with discretized superconducting reservoirs including only a small number of effective levels. Their energies and couplings to the impurity $d$ level are chosen so as to best reproduce the Matsubara frequency dependence of the hybridization function. We provide a number of critical benchmarks and demonstrate the solver's efficiency in combination with the reduced basis method [V. V. Baran and D. R. Nichita, Phys. Rev. B 107, 144503 (2023)] by calculating the phase diagram for an interacting three-terminal junction.

DOI
10.1103/physrevb.108.l220506

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A simple impurity solver is shown to capture impurity-induced superconducting subgap states in quantitative agreement with the numerical renormalization group and quantum Monte Carlo simulations. The solver is based on the exact diagonalization of a single-impurity Anderson model with discretized superconducting reservoirs including only a small number of effective levels. Their energies and couplings to the impurity ddd level are chosen so as to best reproduce the Matsubara frequency dependence of the hybridization function. We provide a number of critical benchmarks and demonstrate the solver's efficiency in combination with the reduced basis method [V. V. Baran and D. R. Nichita, Phys. Rev. B 107, 144503 (2023)] by calculating the phase diagram for an interacting three-terminal junction.
We present a very efficient solver for the general Anderson impurity problem. It is based on the perturbation around a solution obtained from exact diagonalization, using a small number of bath sites. We formulate a perturbation theory which is valid for both weak and strong coupling and interpolates between these limits. Good agreement with numerically exact quantum Monte Carlo results is found for a single bath site over a wide range of parameters. In particular, the Kondo resonance in the intermediate-coupling regime is well reproduced for a single bath site and the lowest-order correction. The method is particularly suited for low temperatures, alleviates analytical continuation of imaginary time data due to the absence of statistical noise and can be generalized to obtain dynamical quantities directly on the real axis.
We present a very efficient solver for the general Anderson impurity problem. It is based on the perturbation around a solution obtained from exact diagonalization, using a small number of bath sites. We formulate a perturbation theory which is valid for both weak and strong coupling and interpolates between these limits. Good agreement with numerically exact quantum Monte Carlo results is found for a single bath site over a wide range of parameters. In particular, the Kondo resonance in the intermediate-coupling regime is well reproduced for a single bath site and the lowest-order correction. The method is particularly suited for low temperatures, alleviates analytical continuation of imaginary time data due to the absence of statistical noise and can be generalized to obtain dynamical quantities directly on the real axis.
The Anderson impurity model (AIM) has long served as a cornerstone in the study of correlated electron systems. While numerical renormalization group (RG) offers great flexibility for metallic reservoirs, it becomes impossible in an unbiased way when a spectral gap $\mathrm{\ensuremath{\Delta}}$ opens up in the tunneling density of states. The only known exception is provided by the superconducting bath. In this paper, we lift these limitations by developing a numerical RG procedure that employs a discretization of the gapped tunneling densities of states into patches that accumulate at the gap edges. This reveals an unusual double scaling, which is a shared behavior between the superconducting and the scalar gapped AIMs. Moreover, it requires a special iterative diagonalization procedure with an alternating scheme for discarding states only every second iteration. The discretization and the diagonalization scheme form together what we refer to as the log-gap numerical RG. It is successfully applied to the superconducting and to the scalar gapped AIM. Consequently, it reveals that both models belong to the same RG equivalence class, which manifests physically in common singlet-doublet quantum phase transitions accompanied by in-gap bound states of given parities. While superconducting AIM is mainly used for benchmarking the log-gap numerical RG, we also rigorously confirm the phenomenon of in-gap states escaping into the continuum. The gapped AIM is then tackled in an exact numerical RG approach and confirms quantitatively the assertions based on models with auxiliary metallic leads, but reveals that the other common approach used, for example, in the work of Moca and Roman [Phys. Rev. B 81, 235106 (2010)] is of strictly approximate nature.
We describe a variational approach to solving Anderson impurity models by means of exact diagonalization. Optimized parameters of a discretized auxiliary model are obtained on the basis of the Peierls-Feynman-Bogoliubov principle. Thereby, the variational approach resolves ambiguities related to the bath discretization, which is generally necessary to make Anderson impurity models tractable by exact diagonalization. The choice of variational degrees of freedom made here allows systematic improvements of total energies over mean-field decouplings like Hartree-Fock. Furthermore, our approach allows us to embed arbitrary bath discretization schemes in total-energy calculations and to systematically optimize and improve on traditional routes to the discretization problem such as fitting of hybridization functions on Matsubara frequencies. Benchmarks in terms of a single orbital Anderson model demonstrate that the variational exact diagonalization method accurately reproduces free energies as well as several single- and two-particle observables obtained from an exact solution. Finally, we demonstrate the applicability of the variational exact diagonalization approach to realistic five-orbital problems with the example system of Co impurities in bulk Cu and compare it to continuous-time Monte Carlo calculations. The accuracy of established bath discretization schemes is assessed in the framework of the variational approach introduced here.
We use the numerical renormalization group method (NRG) to investigate a single-impurity Anderson model with a coupling of the impurity to a superconducting host. Analysis of the energy flow shows that, contrary to previous belief, NRG iterations can be performed up to a large number of sites, corresponding to energy differences far below the superconducting gap Δ. This allows us to calculate the impurity spectral function A(ω) very accurately for frequencies |ω|∼Δ, and to resolve, in a certain parameter regime, sharp peaks in A(ω) close to the gap edge.
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.