Basis-parametrized determinant representations for the six-vertex model with partial domain wall boundary conditions

The partition function of the six-vertex model with domain wall boundary conditions (DWBC) is given by the Izergin-Korepin determinant. Minin, Pronko and Tarasov (2023) reorganized this theory by replacing Korepin's recursion characterization with a system of algebraic equations in one set of spectral parameters: they proved unique solvability and expressed the answer as a determinant parametrized by an arbitrary basis of polynomials, unifying the Izergin-Korepin, Kostov, and Foda-Wheeler determinant representations. They explicitly left open the extension of this construction to partial domain wall boundary conditions (pDWBC), calling the rational case straightforward and the trigonometric case subtle. Indeed, for trigonometric weights the only compact representations of the partial domain wall partition function (pDWPF) known to date are the Foda-Wheeler determinants, restricted to asymmetric weights obeying a stochasticity condition, and a Pfaffian valid only in the semi-infinite-strip limit; no compact representation is known for generic symmetric trigonometric weights. The problem is to construct the algebraic-equation characterization and the basis-parametrized determinant for the pDWPF in both weight regimes. No resolution is known in the literature to date.

Background

The six-vertex model is a statistical-mechanical model of arrows on the edges of a square lattice. At each vertex the ice rule holds: exactly two of the four incident arrows point into the vertex and two point out, which leaves six allowed vertex configurations with Boltzmann weights $w_1,\dots,w_6$. In the inhomogeneous version each horizontal line of the lattice carries a parameter $x_i$ and each vertical line a parameter $y_j$ (called spectral parameters), and the weight of a vertex depends only on the pair $(x_i,y_j)$ at its position. Throughout what follows the weights are taken invariant under reversal of all arrows, $w_1=w_2=:a$, $w_3=w_4=:b$, $w_5=w_6=:c$, with the standard rational parametrization $a(x,y)=x-y+1$, $b(x,y)=x-y$, $c=1$ and trigonometric parametrization $a(x,y)=\sin\gamma(x-y+1)$, $b(x,y)=\sin\gamma(x-y)$, $c=\sin\gamma$, where $\gamma$ is the anisotropy parameter; the trigonometric case is equivalent to the XXZ spin-$1/2$ chain.

Domain wall boundary conditions (DWBC), introduced by Korepin, fix all boundary arrows of an $N\times N$ lattice so that the arrows on each pair of opposite sides point in opposite directions; the allowed configurations are then in one-to-one correspondence with alternating sign matrices. Korepin proved that the DWBC partition function $Z_N$ is uniquely characterized by a short list of conditions, the key one being a recursion relation in the lattice size, and Izergin solved these conditions by a determinant of an $N\times N$ matrix (up to prefactors built from Vandermonde-type products $\prod_{j<k}(x_k-x_j)$) — the Izergin-Korepin determinant, valid for both rational and trigonometric weights (detailed exposition in Izergin-Coker-Korepin 1992). In the algebraic Bethe ansatz (ABA) — the framework in which transfer matrices are diagonalized through creation operators acting on a reference state, and in which partition functions arise as matrix elements of products of transfer-matrix entries — the natural generalization is partial domain wall boundary conditions (pDWBC): following Foda and Wheeler, the partial domain wall partition function $Z_n(\{x\}_n|\{y\}_N)$, $1\le n\le N$, is defined on the lattice obtained from the $N\times N$ domain-wall lattice by deleting the top $N-n$ rows, so that the left, right, and bottom boundaries retain the fixed domain-wall arrow assignments while the arrows on the new top boundary are summed over. Such objects arise as building blocks of three-point functions (structure constants) in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory and are closely related to scalar products of Bethe states: an on-shell Bethe state is one whose rapidities satisfy the Bethe equations (the consistency conditions for a transfer-matrix eigenstate), an off-shell state has arbitrary rapidities, and Slavnov's determinant gives the scalar product of an on-shell with an off-shell state.

For pDWBC the state of the determinant art is asymmetric between the two weight regimes. In the rational case, Kostov derived a determinant for the pDWPF from Slavnov's scalar-product determinant, and Foda and Wheeler derived further rational pDWPF determinants as limits of the Izergin-Korepin determinant, proving the equivalence of the two formulas. In the trigonometric case, Foda and Wheeler obtained a generalization of Kostov's formula only for a restricted parametrization of the weights that breaks arrow-reversal symmetry and obeys a stochasticity condition (a normalization of the type that makes the row-to-row transfer matrix stochastic); they emphasized that with these weights the two lattice representations of the pDWPF are no longer equivalent and that the weights carry phases varying from configuration to configuration. For DWBC such restrictions are harmless, because in every configuration the vertex-type counts satisfy $n_1=n_2$, $n_3=n_4$, $n_5+N=n_6$, so the asymmetry factors out; for pDWBC this mechanism fails, which is one reason the trigonometric case is harder. Additionally, Pronko and Pronko proved that the trigonometric pDWPF admits a Pfaffian representation (an alternating signed sum over pairings whose square is a determinant) only in the limit where the lattice degenerates into a semi-infinite strip. Outside the stochastic class and that limit, no compact representation of the trigonometric pDWPF is known.

Minin, Pronko and Tarasov (2023) reorganized the DWBC theory itself. They replaced Korepin's recursion relation by a system of algebraic equations with respect to one of the two sets of spectral parameters, supplemented by symmetry and polynomiality requirements, proved that this system has a unique solution equal to the DWBC partition function (separately for rational and trigonometric weights), and observed that the solution is a determinant parametrized by an arbitrary basis of polynomials: the Lagrange interpolation basis in the second set of spectral parameters reproduces the Izergin-Korepin determinant, while the monomial basis reproduces the Kostov and Foda-Wheeler determinants. The authors state explicitly that they restrict themselves to DWBC, that they believe the construction can be extended to pDWBC — the rational case straightforwardly, the trigonometric case requiring more thorough analysis — and they leave this extension, as well as the related questions of more general fixed boundary conditions and correlation functions in these models, for future work.

Problem Statement

Let $Z_n(\{x\}_n|\{y\}_N)$, $1\le n\le N$, denote the partition function of the six-vertex model with partial domain wall boundary conditions in the sense of Foda and Wheeler: the lattice obtained from the $N\times N$ domain-wall lattice by deleting the top $N-n$ rows, so that the left, right, and bottom boundaries retain the fixed domain-wall arrow assignments while the arrows on the new top boundary are summed over. The problem is to extend the basis-parametrized determinant construction of Minin-Pronko-Tarasov from the domain-wall case $n=N$ to this object, in both weight regimes.

(a) Rational weights. Formulate the analogue of the algebraic-equation characterization — a system of equations with respect to one set of spectral parameters, supplemented by symmetry and polynomiality requirements, replacing Korepin's recursion — prove that this system has a unique solution equal to $Z_n$, and derive the determinant representation parametrized by an arbitrary basis of polynomials, showing that suitable choices of the basis reproduce the determinants of Kostov and of Foda-Wheeler.

(b) Trigonometric weights. Carry out the same program for the symmetric trigonometric weights $a(x,y)=\sin\gamma(x-y+1)$, $b(x,y)=\sin\gamma(x-y)$, $c=\sin\gamma$ with arbitrary spectral parameters — the case the source authors describe as subtle: construct, for these generic trigonometric weights, a single determinant representation of $Z_n$ parametrized by an arbitrary basis of polynomials as in part (a), with prefactors built from the same Vandermonde-type products as in the Izergin-Korepin determinant — going beyond the asymmetric, stochasticity-restricted class of Foda-Wheeler and beyond the semi-infinite-strip Pfaffian of Pronko-Pronko — together with the corresponding characterization and uniqueness theorem.

Parts (a) and (b) are two regimes of one scientific objective: an algebraic characterization with a unique solution, and a basis-parametrized determinant, for the six-vertex partial domain wall partition function, completing the extension of the source paper's construction from DWBC to pDWBC. The source paper additionally lists, as separate further directions rather than components of this problem, partition functions with more general fixed boundary conditions and the computation of correlation functions in DWBC/pDWBC models.

References

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