The nineteen-vertex model and combinatorics

Christian Hagendorf (IRMP, Universite catholique de Louvain)

Tuesday 29th April, 2014 15:00-16:00 Maths 326


In 2000, Razumov and Stroganov observed a surprising relation between spin chains and enumerative combinatorics: the components of the ground state of the quantum spin-$1/2$ XXZ chain with the particular anisotropy $\Delta=-1/2$ are related to the enumeration of so-called alternating sign matrices (ASMs). These are generalisations of permutation matrices which arise in various contexts such as the calculus of determinants, the enumeration of plane partitions, the six-vertex model. Since then, many relations between the XXZ chain and enumerative combinatorics have been discovered. Even for higher spin analogous phenomena have been found for very special values of the anisotropy, the so-called ``combinatorial points’’.

The topic of this talk is to present a relation between the ground state of the spin$-1$ XXZ chain with arbitrary anisotropy and weighted enumeration of alternating sign matrices. I will show that this connection can be understood by studying the related so-called nineteen-vertex model. The analysis reveals that the ground state can be constructed by the (algebraic) Bethe ansatz with completely explicit and simple Bethe roots. As an application I will discuss how to use results from quantum integrability in order to derive a sum rule which relates the square norm of this state to a generating function for weighted enumeration of ASMs.

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