Abstract
In this paper we study the space evolution in the Rule 54 reversible cellular automaton, which is a paradigmatic example of a deterministic interacting lattice gas. We show that the spatial translation of time configurations of the automaton is given in terms of local deterministic maps with the support that is small but bigger than that of the time evolution. The model is thus an example of space-time dual reversible cellular automaton, i.e. its dual is also (in general different) reversible cellular automaton. We provide two equivalent interpretations of the result; the first one relies on the dynamics of quasi-particles and follows from an exhaustive check of all the relevant time configurations, while the second one relies on purely algebraic considerations based on the circuit representation of the dynamics. Additionally, we use the properties of the local space evolution maps to provide an alternative derivation of the matrix product representation of multi-time correlation functions of local observables positioned at the same spatial coordinate.
Highlights
In this paper we have studied the properties of space evolution in Rule 54 reversible cellular automaton
We have shown that space translation of time configurations can be formulated as a reversible cellular automaton
We have provided two different interpretations of local space evolution; as 7-site local deterministic maps, or equivalently, as a composition of non-deterministic 3-site gates and 3-site projectors onto the subspace of allowed configurations
Summary
Studying exactly solvable models has been traditionally a fruitful approach towards explaining the emergence of macroscopic phenomena from microscopics [1, 2]. This motivates us to study the effect of exchanging time and space evolution in other 1 + 1 dimensional models, where the strict dual unitarity condition does not hold in hope of finding a generalized space-time duality allowing for potentially richer macroscopic physical properties, such as diffusive or super-diffusive transport This question can be rephrased in the context of classical deterministic interacting lattice systems, where the property analogous to unitarity is the symplectic feature of the dynamical evolution law. A recent study revealed that probability distributions of time configurations exhibit an efficient matrix-product description [32], suggesting that translating a given time configuration in space might be given by a composition of deterministic maps with a finite support Another indication that a reversible space evolution formulation of RCA54 should be possible comes from the quasi-particle interpretation of the dynamics; RCA54 rules describe solitons (kinks) that move with fixed velocities and interact pairwise acquiring a delay for one site after each scattering.
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