Fermionic partition functions for a periodic soliton cellular automaton

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 $\Delta=\infty$.

💡 Research Summary

The paper investigates the combinatorial structure of the periodic box‑ball system (BBS), a cellular automaton that exhibits soliton‑like excitations, and derives a refined fermionic formula for its partition function. In the conventional combinatorial Bethe ansatz, fermionic expressions appear as sums of products of q‑binomial coefficients, each term corresponding to a particular set of quantum numbers (e.g., occupation numbers, string lengths). While these summed formulas correctly count states, they are cumbersome for explicit enumeration and obscure the direct relationship between a given soliton content and the weight of the corresponding configuration.

Recent work on non‑periodic BBS introduced a “refined” fermionic formula in which the summation disappears: the partition function for a fixed soliton content μ = (μ₁, μ₂, …, μ_g) is expressed as a simple product of q‑binomials,
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