Dual symplectic classical circuits: An exactly solvable model of many-body chaos

TL;DR

This paper introduces an exact analytical method for calculating dynamical correlations in dual symplectic classical circuits, revealing light-cone constrained correlations and validating results with simulations.

Contribution

It develops a novel exact approach for dynamical correlations in classical many-body systems with symplectic structure, extending concepts from quantum dual-unitary circuits.

Findings

Correlations only along light cone edges

Transfer operator simplifies in spherical harmonics basis

Analytical predictions match Monte Carlo simulations

Abstract

We propose a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, we prove that two-point dynamical correlation functions are non-vanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a one-site Markov transfer operator, which is generally of infinite dimensionality. We test our theory in a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, expressing these models in the form of a composition of rotations leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows us to obtain analytical predictions for simple local observables. We demonstrate the validity of our theory by comparison with Monte Carlo simulations, displaying excellent agreement with the latter for different choices of observables.