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The open XXZ chain at $Δ=-1/2$ and totally-symmetric alternating sign matrices

Jean Liénardy, Christian Walmsley Hagendorf

TL;DR

This work establishes a precise bridge between the open XXZ spin chain at $\Delta=-\frac{1}{2}$ with $x$-dependent boundary fields and the combinatorics of totally-symmetric alternating sign matrices (TSASMs). By solving the boundary quantum KZ equations via multivariate contour integrals, the authors construct a special eigenvector whose component sum $S_N$ is a polynomial in $x$ with integer coefficients and link it to a weighted TSASM generating function $A_{TS}(2N{+}1; t,\tau)$. The key technical device is a pair of parallel formalisms: a generalised sum of components $Z_N$ expressed through overlaps with six-vertex model partition functions on a triangular grid, and a suite of symmetry, degree, and reduction properties that guarantee a uniqueness result. The main theorem yields a concrete identity $S_N = (1+x(x-\tau))^{n/2} A_{TS}(2N{+}1; t,\tau)$ with explicit parameter relations, producing new contour-integral formulas for TSASM counts and clarifying the supersymmetric ($x=1$) case where $S_N$ recovers TSASM numbers of order $2N{+}3$. This work thus binds integrable model techniques to enumerative combinatorics in a novel and exact way, yielding both structural insight and practical counting formulas.

Abstract

The open XXZ spin chain with the anisotropy parameter $Δ=-\frac12$, diagonal boundary fields that depend on a parameter $x$, and finite length $N$ is studied. In a natural normalisation, the components of its ground-state vector are polynomials in $x$ with integer coefficients. It is shown that their sum is given by a generating function for the weighted enumeration of totally-symmetric alternating sign matrices with weights depending on $x$.

The open XXZ chain at $Δ=-1/2$ and totally-symmetric alternating sign matrices

TL;DR

This work establishes a precise bridge between the open XXZ spin chain at with -dependent boundary fields and the combinatorics of totally-symmetric alternating sign matrices (TSASMs). By solving the boundary quantum KZ equations via multivariate contour integrals, the authors construct a special eigenvector whose component sum is a polynomial in with integer coefficients and link it to a weighted TSASM generating function . The key technical device is a pair of parallel formalisms: a generalised sum of components expressed through overlaps with six-vertex model partition functions on a triangular grid, and a suite of symmetry, degree, and reduction properties that guarantee a uniqueness result. The main theorem yields a concrete identity with explicit parameter relations, producing new contour-integral formulas for TSASM counts and clarifying the supersymmetric () case where recovers TSASM numbers of order . This work thus binds integrable model techniques to enumerative combinatorics in a novel and exact way, yielding both structural insight and practical counting formulas.

Abstract

The open XXZ spin chain with the anisotropy parameter , diagonal boundary fields that depend on a parameter , and finite length is studied. In a natural normalisation, the components of its ground-state vector are polynomials in with integer coefficients. It is shown that their sum is given by a generating function for the weighted enumeration of totally-symmetric alternating sign matrices with weights depending on .

Paper Structure

This paper contains 25 sections, 32 theorems, 197 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The sum of components and its generalisation
  3. From multiple contour integrals to the special eigenvector
  4. Multiple contour integrals
  5. The special eigenvector
  6. Properties of the contour integrals
  7. Laurent polynomials.
  8. The exchange and reflection relations
  9. Reduction
  10. The generalised sum of components
  11. Properties
  12. Symmetries and degree
  13. Reduction
  14. Uniqueness
  15. The six-vertex model and TSASMs
  16. ...and 10 more sections

Key Result

Lemma 2.3

For $N\geqslant 2$ and each $i=1,\dots, N$, $(\Psi_N)_{a_1,\dots,a_n}$ is a centred Laurent polynomial in $z_i$ of degree width at most $4(n'-1)$ if $i \notin \{a_1,\dots,a_n\}$ and at most $2(2n-1)$ if $i\in \{a_1,\dots,a_n\}$.

Figures (4)

  • Figure 1: The triangular arrays corresponding to the elements of $\mathrm{TSASM(9)}$.
  • Figure 2: The triangular arrays corresponding to the elements of $\mathrm{TSASM}(11)$.
  • Figure 3: Illustration of the elements of $\mathrm{6V}_n^-$ for $n=2$.
  • Figure 4: Illustration of the elements of $\mathrm{6V}_n^+$ for $n=2$.

Theorems & Definitions (63)

  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • Lemma 2.4: Exchange relations
  • Lemma 2.5: Left reflection relation
  • Lemma 2.6: Reduction relation
  • Example 2.7
  • Example 2.8
  • Proposition 2.9
  • proof
  • ...and 53 more