Accuracy of the finite-temperature Lanczos method compared to simple typicality-based estimates

TL;DR

This paper evaluates the accuracy of the finite-temperature Lanczos method and similar trace estimators based on typicality for quantum spin systems, finding they are generally reliable except near the lowest energy gap.

Contribution

The study systematically compares the FTLM and typicality-based estimates across various quantum spin systems, highlighting their robustness and limitations.

Findings

Methods are reliable for most temperature ranges.

Convergence slows near the lowest energy gap.

No significant failures identified in studied cases.

Abstract

We study trace estimators for equilibrium thermodynamic observables that rely on the idea of typicality and derivatives thereof such as the finite-temperature Lanczos method (FTLM). As numerical examples quantum spin systems are studied. Our initial aim was to identify pathological examples or circumstances, such as strong frustration or unusual densities of states, where these methods could fail. Instead we failed with the attempt. All investigated systems allow such approximations, only at temperatures of the order of the lowest energy gap the convergence is somewhat slower in the number of random vectors over which observables are averaged.