Correlations in Perturbed Dual-Unitary Circuits: Efficient Path-Integral Formula

TL;DR

This paper develops an efficient path-integral formula to compute correlation functions in perturbed dual-unitary circuits, extending the understanding of their dynamical features beyond fine-tuned models.

Contribution

It introduces a rigorous path-sum approach for correlation functions in perturbed dual-unitary circuits, valid in the dilute limit and supported by numerical evidence.

Findings

Correlation functions can be expressed as sums over paths in perturbed circuits.

The path-sum formula remains valid beyond fine-tuned dual-unitary models.

Four types of non-dual-unitary systems with exact correlation functions are identified.

Abstract

Interacting many-body systems with explicitly accessible spatio-temporal correlation functions are extremely rare, especially in the absence of integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brick-wall type local quantum circuits whose dynamics are unitary in both time and space. For these systems the spatio-temporal correlation functions are non-trivial only at the edge of the causal light cone and can be computed in terms of one-dimensional transfer matrices. Dual-unitarity, however, requires fine-tuning and the degree of generality of the observed dynamical features remained unclear. Here we address this question by introducing arbitrary perturbations of the local gates. Considering fixed perturbations, we prove that for a particular class of unperturbed elementary dual-unitary gates the correlation functions are still expressed in terms of one-dimensional transfer matrices. These matrices, however, are now contracted over generic paths connecting the origin to a fixed endpoint inside the causal light cone. The correlation function is given as a sum over all such paths. Our statement is rigorous in the "dilute limit", where only a small fraction of the gates is perturbed, and in the presence of random longitudinal fields, but we provide theoretical arguments and stringent numerical checks supporting its validity even in the clean case and when all gates are perturbed. As a byproduct, in the case of random longitudinal fields -- which turns out to be equivalent to certain classical Markov chains -- we find four types of non-dual-unitary(and non-integrable) interacting many-body systems where the correlation functions are exactly given by the path-sum formula.