Spectral decomposition and high-accuracy Greens functions: Overcoming the Nyquist-Shannon limit via complex-time Krylov expansion

TL;DR

This paper introduces a novel complex-time Krylov expansion method that significantly improves the accuracy of spectral decomposition in strongly correlated quantum systems, surpassing traditional Nyquist-Shannon limits.

Contribution

The authors develop a complex-time Krylov expansion technique to overcome frequency resolution limits in Green's function computations for quantum many-body systems.

Findings

Enhanced spectral resolution in tensor network methods

Successful application to Heisenberg and SSH models

Demonstrated accuracy improvements over traditional Fourier methods

Abstract

The accurate computation of low-energy spectra of strongly correlated quantum many-body systems, typically accessed via Green's-functions, is a long-standing problem posing enormous challenges to numerical methods. When the spectral decomposition is obtained from Fourier transforming a time series, the Nyquist-Shannon theorem limits the frequency resolution $\Delta \omega$ according to the numerically accessible time domain size $T$ via $\Delta \omega = 2\pi/T$. In tensor network methods, increasing the domain size is exponentially hard due to the ubiquitous spread of correlations, limiting the frequency resolution and thereby restricting this ansatz class mostly to one-dimensional systems with small quasi\hyp particle velocities. Here, we show how this limitation can be overcome by augmenting the time series with complex-time Krylov states. At the example of the critical $S-1/2$ Heisenberg model and light bipolarons in the two-dimensional Su-Schrieffer-Heeger model, we demonstrate the enormous improvements in accuracy, which can be achieved using this method.