Equilibrium Quantum Impurity Problems via Matrix Product State Encoding of the Retarded Action

TL;DR

This paper introduces RAMPS, a tensor-network method that effectively models quantum impurity problems, especially in the challenging Kondo regime, by representing the retarded action as a matrix product state.

Contribution

The study demonstrates that RAMPS reliably captures the Kondo regime in the single-impurity Anderson model with polynomial error scaling, offering a promising alternative to existing methods.

Findings

RAMPS accurately reaches the Kondo regime for various interaction strengths

Error scales as a weak power law with inverse temperature

Polynomial convergence with bond dimension and time discretization

Abstract

In the $0+1$ dimensional imaginary-time path integral formulation of quantum impurity problems, the retarded action encodes the hybridization of the impurity with the bath. In this Article, we explore the computational power of representing the retarded action as matrix product state (RAMPS). We focus on the challenging Kondo regime of the single-impurity Anderson model, where non-perturbative strong-correlation effects arise at very low energy scales. We demonstrate that the RAMPS approach reliably reaches the Kondo regime for a range of interaction strengths $U$, with a numerical error scaling as a weak power law with inverse temperature. We investigate the convergence behavior of the method with respect to bond dimension and time discretization by analyzing the error of local observables in the full interacting problem and find polynomial scaling in both parameters. Our results show that the RAMPS approach offers promise as an alternative tool for studying quantum impurity problems in regimes that challenge established methods, such as multi-orbital systems. Overall, our study contributes to the development of efficient and accurate non-wavefunction-based tensor-network methods for quantum impurity problems.