Grassmann time-evolving matrix product operators for equilibrium quantum impurity problems

Abstract

Tensor-network-based methods are promising candidates to solve quantum impurity problems (QIP). They are free of sampling noises and the sign problem compared to state-of-the-art continuous-time quantum Monte Carlo methods. Recent progress made in tensor-network-based impurity solvers is to use the Feynman–Vernon influence functional to integrate out the bath analytically, retaining only the impurity dynamics and representing it compactly as a matrix product state. The recently proposed Grassmann time-evolving matrix product operator (GTEMPO) method is one of the representative methods in this direction. In this work, we systematically study the performance of GTEMPO in solving equilibrium QIPs at a finite temperature with a semicircular spectrum density of the bath. Our results show that its computational cost would generally increase as the temperature goes down and scale exponentially with the number of orbitals. In particular, the single-orbital Anderson impurity model can be efficiently solved with this method, for two orbitals we estimate that one could possibly reach inverse temperature β ≈ 20 if high-performance computing techniques are utilized, while beyond that only very high-temperature regimes can be reached in the current formalism. Our work paves the way to apply GTEMPO as an imaginary-time impurity solver.

Keywords

• Grassmann time-evolving matrix product operators
• path integral
• Grassmann matrix product state
• Anderson impurity model
• imaginary time Green’s function