Matrix product state of multi-time correlations

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2020-08, Vol.53 (33), p.335001
• Main Authors: Klobas, Katja, Vanicat, Matthieu, Garrahan, Juan P, Prosen, Tomaž
• Format: Article
• Language: English
• Subjects: cellular automata; exact results; matrix product ansatz; multi-time correlations; solvable lattice models
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/ab8c62

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 three-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.