Two-species hardcore reversible cellular automaton: matrix ansatz for dynamics and nonequilibrium stationary state

In this paper we study the statistical properties of a reversible cellular automaton in two out-of-equilibrium settings. In the first part we consider two instances of the initial value problem, corresponding to the inhomogeneous quench and the local quench. Our main result is an exact matrix product expression of the time evolution of the probability distribution, which we use to determine the time evolution of the density profiles analytically. In the second part we study the model on a finite lattice coupled with stochastic boundaries. Once again we derive an exact matrix product expression of the stationary distribution, as well as the particle current and density profiles in the stationary state. The exact expressions reveal the existence of different phases with either ballistic or diffusive transport depending on the boundary parameters.

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Exact matrix product decay modes of a boundary driven cellular automaton

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8121/aa85a3 ◽

2017 ◽

Vol 50 (39) ◽

pp. 395002 ◽

Cited By ~ 14

Author(s):

Tomaž Prosen ◽

Berislav Buča

Keyword(s):

Cellular Automaton ◽

Matrix Product ◽

Decay Modes ◽

Exact Matrix

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Space-like dynamics in a reversible cellular automaton

SciPost Physics Core ◽

10.21468/scipostphyscore.2.2.010 ◽

2020 ◽

Vol 2 (2) ◽

Cited By ~ 1

Author(s):

Katja Klobas ◽

Tomaz Prosen

Keyword(s):

Cellular Automaton ◽

Lattice Gas ◽

Time Correlation ◽

Matrix Product ◽

Product Representation ◽

Circuit Representation ◽

Reversible Cellular Automaton ◽

The Matrix ◽

Local Space ◽

Local Observables

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.

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Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution

Entropy ◽

10.3390/e22090984 ◽

2020 ◽

Vol 22 (9) ◽

pp. 984

Author(s):

Regina Finsterhölzl ◽

Manuel Katzer ◽

Andreas Knorr ◽

Alexander Carmele

Keyword(s):

Time Evolution ◽

Nearest Neighbor ◽

Matrix Product ◽

Body System ◽

Many Body ◽

Initial State ◽

Matrix Product States ◽

Open Quantum ◽

Many Body Systems ◽

The Many

This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are circumvented. It is applicable to systems with a Markovian type of interaction, where only the present state of the reservoir needs to be taken into account. Its adaption to a non-Markovian type of interaction between the many-body system and the reservoir is demonstrated, where the information backflow from the reservoir needs to be included in the computation. Also, the derivation of the basis in the quantum stochastic Schrödinger picture is shown. As a paradigmatic model, the Heisenberg spin chain with nearest-neighbor interaction is used. It is demonstrated that the algorithm allows for the access of large systems sizes. As an example for a non-Markovian type of interaction, the generation of highly unusual steady states in the many-body system with coherent feedback control is demonstrated for a chain length of N=30.

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