Grassmann time-evolving matrix product operators for fermionic impurities coupled to a superconducting bath

TL;DR

The paper introduces an extension of the GTEMPO method to solve fermionic impurity problems in the Nambu formalism, enabling accurate real-time and imaginary-time calculations for superconducting baths.

Contribution

We adapt the GTEMPO method to handle superconducting baths using the Bogoliubov transformation, broadening its applicability to Nambu formalism impurity problems.

Findings

Benchmark against exact diagonalization shows high accuracy.

Comparison with quantum Monte Carlo confirms reliability.

Method performs well in both equilibrium and non-equilibrium scenarios.

Abstract

The Grassmann time-evolving matrix product operator (GTEMPO) method, which represents the Feynman-Vernon influence functional as a temporal matrix product state, has been shown to be a flexible and potentially scalable solution for fermionic quantum impurity problems. In this work, we extend GTEMPO to solve fermionic impurity problems in the Nambu formalism, in which the impurity is coupled to a superconducting bath. A key insight is that by employing the Bogoliubov transformation for the superconducting bath, one could obtain the analytic expression of the Feynman-Vernon influence functional in a similar form to the case of a normal bath, after which the core algorithms of GTEMPO can be straightforwardly adapted. We demonstrate the accuracy of our method by benchmarking it against exact diagonalization in several exactly solvable cases, and against the continuous-time quantum Monte Carlo method using converged dynamical mean field theory (DMFT) iterations on the imaginary contour in the non-integrable case. In all cases, we perform both imaginary- and real-time calculations to illustrate the flexibility of our method. These results illustrate that our method could be potentially useful as an impurity solver in DMFT as well as its non-equilibrium extension for fermionic impurity problems in the Nambu formalism.