Colour-independent partition functions in coloured vertex models

TL;DR

This paper demonstrates that certain partition functions in coloured vertex models are independent of the number of colours, extending previous results and explaining degenerations into simpler determinants.

Contribution

It proves n-independence of partition functions in coloured vertex models, generalizing to all n and including trigonometric models, linking to recent Bethe ansatz studies.

Findings

Partition functions are n-independent under specific boundary conditions.

Degeneration of S_2 into products of A_1 determinants in certain limits.

Results extend to trigonometric A_n models and all n.

Abstract

We study lattice configurations related to S_n, the scalar product of an off-shell state and an on-shell state in rational A_n integrable vertex models, n = {1, 2}. The lattice lines are colourless and oriented. The state variables are n conserved colours that flow along the line orientations, but do not necessarily cover every bond in the lattice. Choosing boundary conditions such that the positions where the colours flow into the lattice are fixed, and where they flow out are summed over, we show that the partition functions of these configurations, with these boundary conditions, are n-independent. Our results extend to trigonometric A_n models, and to all n. This n-independence explains, in vertex-model terms, results from recent studies of S_2 [1, 2]. Namely, 1. S_2 which depends on two sets of Bethe roots, b_1 and b_2, and cannot (as far as we know) be expressed in single determinant form, degenerates in the limit b_1 -> infinity, and/or b_2 -> infinity, into a product of determinants, 2. Each of the latter determinants is an A_1 vertex-model partition function.