A dual construction of the isotropic Landau–Lifshitz model

Findlay, Iain

doi:10.1016/j.physd.2019.06.003

Cited by 3 publications

(4 citation statements)

References 23 publications

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“…These states correspond to probability distributions of time configurations observed under the assumption of the system being in the equilibrium state p as introduced by Eq. (13). Explicitly, the equilibrium time-states are uniquely determined by the expectation values of multi-time correlation functions of one-site observables at the same position, 4 C a 1 ,a 2 ,...,a 2m (p) = lim n→∞ C (2n) a 1 ,a 2 ,...,a 2m…”

Section: Time Statesmentioning

confidence: 99%

“…A particularly interesting class of local quantum circuits that are exactly solvable in the statistical sense, yet they are not Bethe-ansatz or Yang-Baxter integrable, are dual unitary quantum circuits [11]. These are local interacting models in discrete space and discrete time where the roles of space and time can be exchanged while keeping dynamics unitary (a similar space-time duality has been explored in integrable field theories [12][13][14]). This property implies a nontrivial structure that enables exact computation of numerous physical quantities, such as local correlation functions [11,15], entanglement spreading [16][17][18], operator entanglement [19,20], and OTOCs [21].…”

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confidence: 99%

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Space-like dynamics in a reversible cellular automaton

Klobas

Prosen

2020

SciPost Phys. Core

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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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Section: Time Statesmentioning

confidence: 99%

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confidence: 99%

Space-like dynamics in a reversible cellular automaton

Klobas

Prosen

2020

SciPost Phys. Core

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“…[27,53] and references therein), to mention a few. Our approach is mainly based on algebraic considerations and is greatly inspired by earlier works on spacetime dualities [7,8,11,15,22] and the existence of underlying spatial and temporal Poisson structures. To illustrate the algebraic approach we present two distinct fully discrete versions of the NLS-type hierarchy based on the existence of classical and quantum r-matrices and the underlying deformed algebras: (1) the fully discrete version of the system introduced in [40,41] (fully DNLS), which is the more natural discretization of the NLS-type systems (AKNS scheme generally) from the algebraic point of view, and is associated to a rational r-matrix.…”

Section: Introductionmentioning

confidence: 99%

An algebraic approach to discrete time integrability

Doikou¹,

Findlay²

2021

J. Phys. A: Math. Theor.

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We propose the systematic construction of classical and quantum two-dimensional space-time lattices primarily based on algebraic considerations, i.e. on the existence of associated r-matrices and underlying spatial and temporal classical and quantum algebras. This is a novel construction that leads to the derivation of fully discrete integrable systems governed by sets of consistent integrable non-linear space-time difference equations. To illustrate the proposed methodology, we derive two versions of the fully discrete non-linear Schrödinger type system. The first one is based on the existence of a rational r-matrix, whereas the second one is the fully discrete Ablowitz–Ladik model and is associated to a trigonometric r-matrix. The Darboux-dressing method is also applied for the first discretization scheme, mostly as a consistency check, and solitonic as well as general solutions, in terms of solutions of the fully discrete heat equation, are also derived. The quantization of the fully discrete systems is then quite natural in this context and the two-dimensional quantum lattice is thus also examined.

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mentioning

confidence: 99%

Space-like dynamics in a reversible cellular automaton

Klobas,

Prosen

2020

Preprint

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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, where the spatial evolution is configurationally constrained. 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. ContentsC Factorization of the equilibrium state and the local few-site relations 24 D Matrix-product form of multi-time correlation functions 25 References 26

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A dual construction of the isotropic Landau–Lifshitz model

Cited by 3 publications

References 23 publications

Space-like dynamics in a reversible cellular automaton

Space-like dynamics in a reversible cellular automaton

An algebraic approach to discrete time integrability

Space-like dynamics in a reversible cellular automaton

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