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Journal of Mathematical Physics

The exact correspondence between conserved quantities of a periodic box-ball system and string solutions of the Bethe ansatz equations

Jun Mada, Makoto Idzumi, Tetsuji Tokihiro

Published May 1, 2006• 7 citations
Publisher

Abstract

We investigate the link between a periodic box-ball system (PBBS) and a solvable lattice model. Introducing a PBBS with an integer parameter corresponding to the dimensionality of the auxiliary space for the lattice model, we prove an important relationship between the conserved quantities of states of the PBBS and eigenvectors constructed through the string hypothesis.

DOI
10.1063/1.2200142

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We investigate the link between a periodic box-ball system (PBBS) and a solvable lattice model. Introducing a PBBS with an integer parameter corresponding to the dimensionality of the auxiliary space for the lattice model, we prove an important relationship between the conserved quantities of states of the PBBS and eigenvectors constructed through the string hypothesis.
To analyze integrable multiparticle systems in the thermodynamic limit, one typically confines the system to a large periodic box, ensuring that the Bethe wave function resides in the asymptotic regime of free motion. The resulting periodic boundary conditions give rise to a system of coupled nonlinear equations for particle momenta, known as the Bethe equations. Solving these equations requires constructing common eigenvectors for a family of commuting operators acting on the Hilbert space of an auxiliary spin chain. This approach, known as the coordinate Bethe ansatz, is grounded in the generalized Bethe hypothesis, which posits the coordinate-space structure of eigenstates. This chapter introduces the formalism of the coordinate Bethe ansatz and its algebraic underpinnings, including the role of the Bethe–Yang equation, permutation modules, and Specht modules. The chapter concludes by applying this framework to the Lieb–Liniger model for spin-1/2 particles, setting the stage for more advanced developments.
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.
For the one-dimensional XXX model under the periodic boundary conditions, we discuss two types of eigenvectors, regular eigenvectors which have finite-valued rapidities satisfying the Bethe ansatz equations and non-regular eigenvectors which are descendants of some regular eigenvectors under the action of the SU (2) spin-lowering operator. It has been pointed out by many authors that the non-regular eigenvectors should correspond to the Bethe ansatz wavefunctions which have multiple infinite rapidities. However, it has not been explicitly shown whether such a delicate limiting procedure is possible. In this paper, we discuss it explicitly at the level of wavefunctions: we prove that any non-regular eigenvector of the XXX model is derived from the Bethe ansatz wavefunctions through some limit of infinite rapidities. We formulate the regularization also in terms of the algebraic Bethe ansatz method. As an application of infinite rapidity, we discuss the period of the spectral flow under the twisted periodic boundary conditions.