Finding local integrals of motion in quantum lattice models in the thermodynamic limit

TL;DR

This paper introduces an efficient numerical method to identify local integrals of motion in infinite quantum lattice models, aiding the understanding of long-term dynamics and slow modes in nearly integrable systems.

Contribution

The authors develop a simple algorithm that finds LIOMs directly in the thermodynamic limit, bypassing the need for exact diagonalization of large Hamiltonians.

Findings

Successfully calculates LIOMs and correlation bounds for infinite integrable spin chains.

Identifies slow modes and estimates relaxation times in nearly integrable models.

Abstract

Local integrals of motion (LIOMs) play a key role in understanding the long-time properties of closed macroscopic systems. They were found for selected integrable systems via complex analytical calculations. The existence of LIOMs and their structure can also be studied via numerical methods, which, however, involve exact diagonalization of Hamiltonians, posing a bottleneck for such studies. We show that finding LIOMs in translationally invariant lattice models or unitary quantum circuits can be reduced to a problem for which one may numerically find an exact solution in the thermodynamic limit. We develop a simple algorithm and demonstrate its efficiency by calculating LIOMs and bounds on correlations (the Mazur bounds) for infinite integrable spin chains and unitary circuits. Finally, we demonstrate that this approach identifies slow modes in nearly integrable spin models and estimates their relaxation times.