Tensor cross interpolation approach for quantum impurity problems based on the weak-coupling expansion

TL;DR

This paper introduces a tensor cross interpolation (TCI) method for solving quantum impurity problems efficiently and accurately, avoiding the sign problem and enabling direct free energy calculations, with applications to the Hubbard model.

Contribution

The paper presents a novel TCI algorithm that factorizes high-dimensional integrals in quantum impurity problems, improving computational efficiency and accuracy over existing methods.

Findings

Accurately solves impurity models with high precision.

Successfully describes the metal-to-Mott insulator transition.

Avoids the sign problem common in quantum Monte Carlo methods.

Abstract

We apply the tensor cross interpolation (TCI) algorithm to solve equilibrium quantum impurity problems with high precision based on the weak-coupling expansion. The TCI algorithm, a kind of active learning method, factorizes high-dimensional integrals that appear in the perturbative expansion into a product of low-dimensional ones, enabling us to evaluate higher-order terms efficiently. This method is free from the sign problem which quantum Monte Carlo methods sometimes suffer from, and allows one to directly calculate the free energy. We benchmark the TCI impurity solver on an exactly solvable impurity model, and find good agreement with the exact solutions. We also incorporate the TCI impurity solver into the dynamical mean-field theory to solve the Hubbard model, and show that the metal-to-Mott insulator transition is correctly described with comparable accuracy to the Monte Carlo methods. Behind the effectiveness of the TCI approach for quantum impurity problems lies the fact that the integrands in the weak-coupling expansion naturally have a low-rank structure in the tensor-train representation.