Bethe ansatz atq= 0 and periodic box–ball systems

Abstract

Abstract. A class of periodic soliton cellular automata is introduced associated with crystals of non-exceptional quantum affine algebras. Based on the Bethe ansatz at q = 0, we propose explicit formulae for the dynamical period and the size of certain orbits under the time evolution in the A (1) n case.

Periodic Cellular Automata and Bethe Ansatz

“…Let us turn to another Bethe ansatz result, the character formula called combinatorial completeness of the string hypothesis at q = 0 [11]: (12) and (13). We introduce T (P (m)) = n a=1 j≥1 {T (a) j (p) | p ∈ P (m)}, which is the subset of B consisting of all kinds of one step time evolutions of P (m).…”

“…In [13], a class of periodic soliton cellular automata is introduced associated with non-exceptional quantum affine algebras. The dynamical period and a state counting formula are proposed by the Bethe ansatz at q = 0 [11].…”

“…Here we develop the approach to the periodic case in [13] further by combining the commuting transfer matrix method [1] and the Bethe ansatz [2] at q = 0. In section 2 we formulate the most general periodic automaton for g n = A (1) in terms of the crystal theory.…”

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for An(1)

“…The KKR bijection on the other hand is a canonical algorithm allowing generalizations not only to type A (1) n [18] but also beyond [32]. Conjecturally the approach based on the KKR type bijection will work universally for the periodic SCA [25] associated with Kirillov-Reshetikhin crystals as exemplified for A (1) 1 higher spin case [22] and A (1) n [25, 26, this paper]. In fact, the essential results like torus, dynamical period and phase space volume formula are all presented in a universal form by the data from Bethe ansatz and the root system 2 .…”

“…In [25], a class of soliton cellular automata (SCA) on a periodic one dimensional lattice was introduced associated with the non-exceptional quantum affine algebras U q (g n ) at q = 0. In this paper we focus on the type g n = A (1) n .…”