Symbolic recursion method for strongly correlated fermions in two and three dimensions

TL;DR

This paper introduces a symbolic recursion method to analyze strongly correlated fermions in multiple dimensions, confirming universal operator growth and enabling precise computation of transport properties like charge diffusion.

Contribution

The work develops a symbolic implementation of the recursion method for fermionic systems, allowing direct confirmation of the universal operator growth hypothesis and efficient calculation of autocorrelation functions and transport coefficients.

Findings

Universal linear growth of Lanczos coefficients confirmed.

Charge diffusion constant follows a 1/V^2 dependence.

Method applicable in the thermodynamic limit.

Abstract

We present a symbolic implementation of recursion method for the dynamics of strongly correlated fermions on one-, two- and three-dimensional lattices. Focusing on two paradigmatic models, interacting spinless fermions and the Hubbard model, we first directly confirm that the universal operator growth hypothesis holds for interacting fermionic systems, manifested by the linear growth of Lanczos coefficients. Equipped with symbolically computed Lanczos coefficients and knowledge of their asymptotics, we are able to compute infinite-temperature autocorrelation functions up to times long enough for thermalization to occur. In turn, the knowledge of autocorrelation functions unlocks transport properties. We compute with high precision the charge diffusion constant over a broad range of interaction strengths, $V$. Surprisingly, we observe that these results are well described by a simple functional dependence $\sim 1/V^2$ universally valid both for small and large $V$. All results are obtained directly in the thermodynamic limit. Our results highlight the promise of symbolic computational paradigm where the most costly step is performed once and outputs symbolic results that can further be used multiple times to easily compute physical quantities for specific values of model parameters.