Fermionic partition functions for a periodic soliton cellular automaton

TL;DR

This paper extends refined fermionic formulas to the periodic box-ball system and provides a q-analogue of the Bethe root counting formula for the XXZ chain at infinite anisotropy, linking combinatorial and quantum integrable models.

Contribution

It introduces a refined fermionic formula for the periodic box-ball system and connects it to the Bethe root counting in the XXZ chain at .

Findings

Extended fermionic formulas to periodic systems.

Established a q-analogue of Bethe root counting.

Linked combinatorial models with quantum integrable systems.

Abstract

Fermionic formulas in combinatorial Bethe ansatz consist of sums of products of q-binomial coefficients. There exist refinements without a sum that are known to yield partition functions of box-ball systems with a prescribed soliton content. In this paper, such a refined fermionic formula is extended to the periodic box-ball system and a q-analogue of the Bethe root counting formula for XXZ chain at $\Delta=\infty$.