Matrix product state of multi-time correlations

TL;DR

This paper introduces a matrix product state framework for multi-time correlations in an interacting lattice system, specifically applied to the Rule 54 automaton, enabling explicit calculations of multi-time probabilities and correlation functions.

Contribution

It develops a novel matrix product state representation for multi-time correlations in a lattice system, with explicit construction for the Rule 54 automaton, linking time state distributions to equilibrium properties.

Findings

Explicit matrix product form for the time state in Rule 54

Agreement of correlation functions with previous results at maximum-entropy parameters

Proof of the absence of decoupling of timescales in the model

Abstract

For an interacting spatio-temporal lattice system we introduce a formal way of expressing multi-time correlation functions of local observables located at the same spatial point with a time state, i.e. a statistical distribution of configurations observed along a time lattice. Such a time state is defined with respect to a particular equilibrium state that is invariant under space and time translations. The concept is developed within the Rule 54 reversible cellular automaton, for which we explicitly construct a matrix product form of the time state, with matrices that act on the 3-dimensional auxiliary space. We use the matrix-product state to express equal-space time-dependent density-density correlation function, which, for special maximum-entropy values of equilibrium parameters, agrees with the previous results. Additionally, we obtain an explicit expression for the probabilities of observing all multi-time configurations, which enables us to study distributions of times between consecutive excitations and prove the absence of decoupling of timescales in the Rule 54 model.