Accuracy of the typicality approach using Chebyshev polynomials

TL;DR

This paper compares Chebyshev polynomial-based trace estimators to the finite-temperature Lanczos method for quantum spin systems, highlighting their accuracy and systematic errors at low temperatures.

Contribution

It introduces and evaluates a Chebyshev polynomial approach as an alternative to the FTLM for approximating thermodynamic observables.

Findings

Chebyshev method is generally accurate for trace estimation.

Systematic inaccuracies occur at low temperatures due to density of states approximation.

Applications demonstrate the method's effectiveness on quantum spin systems.

Abstract

Trace estimators allow to approximate thermodynamic equilibrium observables with astonishing accuracy. A prominent representative is the finite-temperature Lanczos method (FTLM) which relies on a Krylov space expansion of the exponential describing the Boltzmann weights. Here we report investigations of an alternative approach which employs Chebyshev polynomials. This method turns out to be also very accurate in general, but shows systematic inaccuracies at low temperatures that can be traced back to an improper behavior of the approximated density of states with and without smoothing kernel. Applications to archetypical quantum spin systems are discussed as examples.