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

Abstract

We introduce a deterministic SO(3) invariant dynamics of classical spins on a discrete space-time lattice and prove its complete integrability by explicitly finding a related non-constant (baxterized) solution of the set-theoretic quantum Yang-Baxter equation over the 2-sphere. Equipping the algebraic structure with the corresponding Lax operator we derive an infinite sequence of conserved quantities with local densities. The dynamics depend on a single continuous spectral parameter and reduce to a (lattice) Landau-Lifshitz model in the limit of a small parameter which corresponds to the continuous time limit. Using quasiexact numerical simulations of deterministic dynamics and Monte Carlo sampling of initial conditions corresponding to a maximum entropy equilibrium state we determine spin-spin spatio-temporal (dynamical) correlation functions with relative accuracy of three orders of magnitude. We demonstrate that in the equilibrium state with a vanishing total magnetization the correlation function precisely follow Kardar-Parisi-Zhang scaling hence the spin transport belongs to the universality class with dynamical exponent z = 3/2, in accordance to recent related simulations in discrete and continuous time quantum Heisenberg spin 1/2 chains.

Integrable matrix models in discrete space-time

“…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…”

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

“…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.…”

“…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.…”

Space-like dynamics in a reversible cellular automaton

“…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].…”