Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54

Klobas, Katja; Bertini, Bruno

doi:10.48550/arxiv.2104.04511

Cited by 15 publications

(29 citation statements)

References 91 publications

(125 reference statements)

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“…The correlation function decays exponentially for all values of parameters, and furthermore, one can show [35,36] that the correlations exponentially decay also when we move away from x = 0 as long as |x/t| < 1 3 . This is compatible with the hydrodynamic picture, according to which the correlations move with velocity larger than 1 3 (cf ( 102)), and therefore we cannot observe power-law decay on rays with |x/t| < 1 3 .…”

Section: Correlation Functions At One Sitementioning

confidence: 83%

“…If these equations are satisfied, the pair of states p, p written in terms of MPAs ( 38) and ( 39), solves the staggered eigenvalue equations (36). This can be straightforwardly verified by plugging the MPAs into e.g.…”

Section: Ness: Matrix Product Ansatzmentioning

confidence: 95%

“…Furthermore, analogous treatment can be straightforwardly recast for the quantum version of the model [35][36][37]. The main conceptual difference between the quantum and classical case is the size of the vector space, which is in the quantum case doubled (squared in dimensionality), as we are considering density matrices rather than probability distributions.…”

Section: J Stat Mech (2021) 074001mentioning

confidence: 99%

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Rule 54: exactly solvable model of nonequilibrium statistical mechanics

2021

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We review recent results on an exactly solvable model of nonequilibrium statistical mechanics, specifically the classical rule 54 reversible cellular automaton and some of its quantum extensions. We discuss the exact microscopic description of nonequilibrium dynamics as well as the equilibrium and nonequilibrium stationary states. This allows us to obtain a rigorous handle on the corresponding emergent hydrodynamic description, which is treated as well. Specifically, we focus on two different paradigms of rule 54 dynamics. Firstly, we consider a finite chain driven by stochastic boundaries, where we provide exact matrix product descriptions of the nonequilibrium steady state, most relevant decay modes, as well as the eigenvector of the tilted Markov chain yielding exact large deviations for a broad class of local and extensive observables. Secondly, we treat the explicit dynamics of macro-states on an infinite lattice and discuss exact closed form results for dynamical structure factor, multi-time-correlation functions and inhomogeneous quenches. Remarkably, these results prove that the model, despite its simplicity, behaves like a regular fluid with coexistence of ballistic (sound) and diffusive (heat) transport. Finally, we briefly discuss quantum interpretation of rule 54 dynamics and explicit results on dynamical spreading of local operators and operator entanglement.

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Section: Correlation Functions At One Sitementioning

confidence: 83%

Section: Ness: Matrix Product Ansatzmentioning

confidence: 95%

Section: J Stat Mech (2021) 074001mentioning

confidence: 99%

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Rule 54: exactly solvable model of nonequilibrium statistical mechanics

2021

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“…The most important example is the Rule54 model, which is often called the simplest interacting integrable model [70]. It is a very special model with soliton-like behaviour, which allows for exact solutions [71][72][73] and integrable quantum deformations [74] (see also [75]). However, the connection with the standard Yang-Baxter integrability remained unknown.…”

Section: B Integrable Quantum Circuitsmentioning

confidence: 99%

Integrable spin chains and cellular automata with medium range interaction

Gombor,

Pozsgay

2021

Preprint

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We study integrable spin chains and quantum and classical cellular automata with interaction range ≥ 3. This is a family of integrable models for which there was no general theory so far. We develop an algebraic framework for such models, generalizing known methods from nearest neighbor interacting chains. This leads to a new integrability condition for medium range Hamiltonians, which can be used to classify such models. A partial classification is performed in specific cases, including U (1)-symmetric three site interacting models, and Hamiltonians that are relevant for interactionround-a-face models. We find a number of models which appear to be new. As an application we consider quantum brickwork circuits of various types, including those that can accommodate the classical elementary cellular automata on light cone lattices. In this family we find that the so-called Rule150 and Rule105 models are Yang-Baxter integrable with three site interactions. We present integrable quantum deformations of these models, and derive a set of local conserved charges for them. For the famous Rule54 model we find that it does not belong to the family of integrable three site models, but we can not exclude Yang-Baxter integrability with longer interaction ranges.

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“…For instance, although in principle the quench action can provide the full post-quench dynamics [26], this task has to be performed numerically [56,57], with only few special cases (usually mappable to free systems) where some analytic solutions can be found in full glory [26,58,59]. Interestingly, however, in recent years a number of exact results concerning the finite time dynamics have been found in a special class of integrable systems, which can be thought of as strong coupling limits [60][61][62][63][64][65][66].…”

Section: Introductionmentioning

confidence: 99%

Real-Time Evolution in the Hubbard Model with Infinite Repulsion

Tartaglia,

Calabrese,

Bertini

2021

Preprint

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We consider the real-time evolution of the Hubbard model in the limit of infinite coupling. In this limit the Hamiltonian of the system is mapped into a number-conserving quadratic form of spinless fermions, i.e. the tight binding model. The relevant local observables, however, do not transform well under this mapping and take very complicated expressions in terms of the spinless fermions. Here we show that for two classes of interesting observables the quench dynamics from product states in the occupation basis can be determined exactly in terms of correlations in the tight-binding model. In particular, we show that the time evolution of any function of the total density of particles is mapped directly into that of the same function of the density of spinless fermions in the tight-binding model. Moreover, we express the two-point functions of the spin-full fermions at any time after the quench in terms of correlations of the tight binding model. This sum is generically very complicated but we show that it leads to simple explicit expressions for the time evolution of the densities of the two separate species and the correlations between a point at the boundary and one in the bulk when evolving from the so called generalised nested Néel states.

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Exact relaxation to Gibbs and non-equilibrium steady states in the quantum cellular automaton Rule 54

Cited by 15 publications

References 91 publications

Rule 54: exactly solvable model of nonequilibrium statistical mechanics

Rule 54: exactly solvable model of nonequilibrium statistical mechanics

Integrable spin chains and cellular automata with medium range interaction

Real-Time Evolution in the Hubbard Model with Infinite Repulsion

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