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arXiv:2011.08052 · SciPost Physics

Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

Atsuo Kuniba, Grégoire Misguich, Vincent Pasquier

Published Apr 30, 2021• 5 citations
arXivHTMLPDF

Abstract

We introduce the complete box-ball system (cBBS), which is an integrable cellular automaton on 1D lattice associated with the quantum group U_q(\widehat{sl}_n) U q ( s l ̂ n ) . Compared with the conventional (n-1) ( n − 1 ) -color BBS, it enjoys a remarkable simplification that scattering of solitons is totally diagonal. We also submit the cBBS to randomized initial conditions and study its non-equilibrium behavior by thermodynamic Bethe ansatz and generalized hydrodynamics. Excellent agreement is demonstrated between theoretical predictions and numerical simulation on the density plateaux generated from domain wall initial conditions including their diffusive broadening.

DOI
10.21468/scipostphys.10.4.095

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Generalized hydrodynamics in complete box-ball system for Uq(sl^n)U_q(\widehat{sl}_n)Uq​(sln​)
We introduce the complete box-ball system (cBBS), which is an integrable cellular automaton on 1D lattice associated with the quantum group U_q(\widehat{sl}_n) U q ( s l ̂ n ) . Compared with the conventional (n-1) ( n − 1 ) -color BBS, it enjoys a remarkable simplification that scattering of solitons is totally diagonal. We also submit the cBBS to randomized initial conditions and study its non-equilibrium behavior by thermodynamic Bethe ansatz and generalized hydrodynamics. Excellent agreement is demonstrated between theoretical predictions and numerical simulation on the density plateaux generated from domain wall initial conditions including their diffusive broadening.
Abstract Box-ball system (BBS) is a prominent example of integrable cellular automata in one dimension connected to quantum groups, Bethe ansatz, ultradiscretization, tropical geometry and so forth. In this paper we study the generalized Gibbs ensemble of BBS soliton gas by thermodynamic Bethe ansatz and generalized hydrodynamics. The results include the solution to the speed equation for solitons, an intriguing connection of the effective speed with the period matrix of the tropical Riemann theta function, an explicit description of the density plateaux that emerge from domain wall initial conditions including their diffusive corrections.
Abstract We explore several aspects of the current fluctuations and correlations in the box–ball system, an integrable cellular automaton in one space dimension. The state we consider is an ensemble of microscopic configurations where the box occupancies are independent random variables (i.i.d. state), with a given mean ball density. We compute several quantities exactly in such homogeneous stationary state: the mean value and the variance of the number of balls N t crossing the origin during time t , and the scaled cumulants generating function associated to N t . We also compute two spatially integrated current–current correlations. The first one, involving the long-time limit of the current–current correlations, is the so-called Drude weight and is obtained with thermodynamic Bethe ansatz (TBA). The second one, involving equal time current–current correlations is calculated using a transfer matrix approach. A family of generalized currents, associated to the conserved charges and to the different time evolutions of the models are constructed. The long-time limits of their correlations generalize the Drude weight and the second cumulant of N t and are found to obey nontrivial symmetry relations. They are computed using TBA and the results are found to be in good agreement with microscopic simulations of the model. TBA is also used to compute explicitly the whole family of flux Jacobian matrices. Finally, some of these results are extended to a (non-i.i.d.) two-temperatures generalized Gibbs state (with one parameter coupled to the total number of balls, and another one coupled to the total number of solitons).
The conventional theory of hydrodynamics describes the evolution in time of chaotic many-particle systems from local to global equilibrium. In a quantum integrable system, local equilibrium is characterized by a local generalized Gibbs ensemble or equivalently a local distribution of pseudomomenta. We study time evolution from local equilibria in such models by solving a certain kinetic equation, the "Bethe-Boltzmann" equation satisfied by the local pseudomomentum density. Explicit comparison with density matrix renormalization group time evolution of a thermal expansion in the XXZ model shows that hydrodynamical predictions from smooth initial conditions can be remarkably accurate, even for small system sizes. Solutions are also obtained in the Lieb-Liniger model for free expansion into vacuum and collisions between clouds of particles, which model experiments on ultracold one-dimensional Bose gases.
We study an integrable vertex model with a periodic boundary condition associated with U q A (1) n at the crystallizing point q = 0.It is an (n + 1)-state cellular automaton describing the factorized scattering of solitons.The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system.Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures.The action-angle variables are constructed representing the amplitudes and phases of solitons.By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety.We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus.The weight multiplicity formula derived from the q = 0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori.The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set.The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data.As an application, time average of some local variable is calculated.