Finite Temperature Dynamical Correlations using the Microcanonical Ensemble and the Lanczos Algorithm

TL;DR

This paper extends the Lanczos method to finite temperatures using the microcanonical ensemble, enabling efficient calculation of dynamical correlations in large quantum systems with results consistent with traditional methods.

Contribution

It introduces a microcanonical ensemble approach to finite-temperature dynamical correlations, replacing the canonical ensemble with a single state for improved computational efficiency.

Findings

Spectra of large systems are smooth and comparable to canonical ensemble results.

The method accurately reproduces spin conductivity spectra of the Heisenberg model.

Microcanonical approach reduces statistical fluctuations in large system calculations.

Abstract

We show how to generalise the zero temperature Lanczos method for calculating dynamical correlation functions to finite temperatures. The key is the microcanonical ensemble, which allows us to replace the involved canonical ensemble with a single appropriately chosen state; in the thermodynamic limit it provides the same physics as the canonical ensemble but with the evaluation of a single expectation value. We can employ the same system sizes as for zero temperature, but whereas the statistical fluctuations present in small systems are prohibitive, the spectra of the largest system sizes are surprisingly smooth. We investigate, as a test case, the spin conductivity of the spin-1/2 anisotropic Heisenberg model and in particular we present a comparison of spectra obtained by the canonical and microcanonical ensemble methods.