Kardar–Parisi–Zhang Physics in Integrable Rotationally Symmetric Dynamics on Discrete Space–Time Lattice

Krajnik, Žiga; Prosen, Tomaž

doi:10.1007/s10955-020-02523-1

Cited by 54 publications

(68 citation statements)

References 33 publications

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“…This is in agreement with the expression found previously in [24]. In the τ → 0 limit, this yields (modulo a constant term) the Hamiltonian of the isotropic Landau-Lifshitz ferromagnet lim τ→0 H (1,2) τ…”

Section: B1 Symplectic Generatorsupporting

confidence: 90%

“…T is a vector of Pauli matrices. The symplectic map Φ τ for this case has been studied previously in [24]…”

Section: Semi-discrete and Continuum Limitsmentioning

confidence: 99%

“…This work has been largely inspired by the recent discovery of superdiffusive magnetization transport in the isotropic Heisenberg spin-1/2 chain [66][67][68], subsequently scrutinized in a number of papers [41,42,69], collectively accumulating a convincing numerical evidence for the Kardar-Parisi-Zhang (KPZ) type universality [70] (see also [71,72] and [73][74][75]). It is remarkable that the same phenomenon is already visible at the classical level, namely in the integrable classical spin chains symmetric under global SO (3) rotations [24,76,77]. In spite of a phenomenological picture based on an effective noisy Burger's equation proposed recently in [41] and further refined in [42], a complete and quantitative understanding of this curious phenomen is still lacking at the moment.…”

Section: Introductionmentioning

confidence: 95%

“…To circumnavigate some of these inherent limitations it is fruitful to attempt a slightly different approach. To better understand certain peculiar features of integrable dynamical systems subject to non-trivial global symmetry constraints, we confine ourselves in this paper to a certain class of classical models in a discrete space-time geometry by following the spirit of a preceding work [24]. Our aim is to explore the possibility of realizing simple exactly solvable symplectic circuits which possess conserved non-abelian currents.…”

Section: Introductionmentioning

confidence: 99%

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Integrable matrix models in discrete space-time

2020

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We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic \sigmaσ-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau–Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar–Parisi–Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.

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Section: B1 Symplectic Generatorsupporting

confidence: 90%

“…T is a vector of Pauli matrices. The symplectic map Φ τ for this case has been studied previously in [24]…”

Section: Semi-discrete and Continuum Limitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

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Integrable matrix models in discrete space-time

2020

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“…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. An example of dual symplectic classical lattice dynamics with continuous local degrees of freedom that exhibits super-diffusive transport in the Kardar-Parisi-Zhang universality class has been recently proposed [22,23].…”

mentioning

confidence: 99%

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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KPZ scaling from the Krylov space

Gorsky,

Nechaev,

Valov

2024

J. High Energ. Phys.

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Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling in late-time correlators and autocorrelators of certain interacting many-body systems, such as Heisenberg spin chains, has been reported. Inspired by these results, we explore the KPZ scaling in correlation functions using their realization in the Krylov operator basis. We focus on the Heisenberg time scale, which approximately corresponds to the ramp–plateau transition for the Krylov complexity in systems with a large but finite number degrees of freedom. Two frameworks are under consideration: i) the system with growing Lanczos coefficients and an artificial cut-off, and ii) the system with the finite Hilbert space. In both cases via numerical analysis, we observe the transition from Gaussian to KPZ-like scaling, or equivalently, the diffusion-superdiffusion transition at the critical Euclidean time $$ {t}_E^{\ast } $$ t E ∗ = ccrK, for the Krylov chain of finite length K, and ccr = O(1). In particular, we find a scaling ~ K1/3 for fluctuations in the one-point correlation function and a dynamical scaling ~ K−2/3 associated with the return probability (Loschmidt echo) corresponding to autocorrelators in physical space. In the first case, the transition is of the 3rd order and can be considered as an example of dynamical quantum phase transition (DQPT), while in the second, it is a crossover. For case ii), utilizing the relationship between the spectrum of tridiagonal matrices at the spectral edge and the spectrum of the stochastic Airy operator, we demonstrate analytically the origin of the KPZ scaling for the particular Krylov chain using the results of the probability theory. We argue that there is interesting outcome of our study for the double scaling limit of matrix models. For the case of topological gravity, the white noise term of order O (1/N) is identified, which should be taken into account in the controversial issue of ensemble averaging in 2D/1D holography.

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Kardar–Parisi–Zhang Physics in Integrable Rotationally Symmetric Dynamics on Discrete Space–Time Lattice

Cited by 54 publications

References 33 publications

Integrable matrix models in discrete space-time

Integrable matrix models in discrete space-time

Space-like dynamics in a reversible cellular automaton

KPZ scaling from the Krylov space

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