Algebraic Bethe ansatz construction of multiple-integral representations of multi-point local height probabilities for the unrestricted solid-on-solid model

The exactly solvable solid-on-solid (SOS) model describes a crystal interface by integer-valued heights whose local Boltzmann weights satisfy the dynamical Yang-Baxter equation. Levy-Bencheton and Terras (J. Stat. Mech. 2014 P04014) derived, within the algebraic Bethe ansatz (ABA), multiple-integral representations for the multi-point local height probabilities (LHPs) of the cyclic SOS model, where the crossing parameter $\eta=r/L$ is rational and heights are periodic modulo $L$, so that the relevant finite-size matrix elements are finite ($L$-term) sums of determinants of elliptic functions. They explicitly left open the extension to the unrestricted SOS model, in which $\eta$ is generic, the dynamical (height) parameter runs over the infinite lattice $s_0+\mathbb{Z}$, and the same construction produces infinite series whose convergence must be controlled. The open problem is to construct the rigorous ABA-derived multiple-integral representation of the multi-point LHPs of the unrestricted SOS model at the thermodynamic limit: establish convergence of the infinite height-series replacing the cyclic finite sums, justify the termwise manipulations (interchanges of height sums, thermodynamic limits, and contour integrals), and verify consistency with Baxter's corner-transfer-matrix one-point LHPs.

Background

The solid-on-solid (SOS) model is a two-dimensional lattice model of a crystal-vapor interface: an integer-valued 'height' variable is attached to every site of a square lattice, heights at neighbouring sites differ by exactly $\pm 1$, and each elementary face carries a Boltzmann weight depending on its four corner heights. Since the interaction runs around faces rather than vertices, it is the archetype of the class of interaction-round-a-face (IRF, or 'face') models, as opposed to vertex models such as the six-vertex model; historically it was the auxiliary tool in Baxter's solution of the eight-vertex model. The local weights can be encoded in an elliptic $R$-matrix acting on $\mathbb{C}^2\otimes\mathbb{C}^2$, depending on a spectral variable $u$ and on a 'dynamical' parameter $s$ (the local height), with respect to which the $R$-matrix is shift-covariant. Integrability means these weights satisfy the star-triangle relation, equivalently the dynamical (modified) Yang-Baxter equation of the elliptic quantum group $E_{\tau,\eta}(\mathrm{sl}_2)$, with nome $\tau$ and crossing parameter $\eta$.

Two regimes must be distinguished. In the cyclic SOS (CSOS) model, $\eta=r/L$ is rational (with $r,L$ relatively prime) and heights are periodic modulo $L$, so the dynamical parameter takes values in the finite set $s_0+\mathbb{Z}/L\mathbb{Z}$; the space of states is finite-dimensional and the infinite-volume model has $2(L-r)$ degenerate ground states. In the general (unrestricted) SOS model, $\eta$ is arbitrary and the dynamical parameter lives on the infinite lattice $s_0+\mathbb{Z}$; the space of states is infinite-dimensional even on a finite lattice, and there are infinitely many degenerate flat ground states (the alternating configurations $(s,s+1,s,s+1,\dots)$ labelled by $s\in s_0+\mathbb{Z}$).

Local height probabilities are the basic height-resolved observables: the multi-point LHP $P(s_1,\dots,s_m)$ is the probability that the heights at specified sites take the values $s_1,\dots,s_m$, evaluated in a given ground-state sector. One-point LHPs of the unrestricted SOS model were computed by Baxter's corner transfer matrix (CTM) method, which represents half-infinite lattice blocks through the spectrum of a one-dimensional 'corner Hamiltonian'; for the CSOS model the one-point LHPs are due to Pearce and Seaton. Multi-point LHPs have been obtained in two neighbouring settings: for the six-vertex model (equivalently the XXZ spin-1/2 chain) within the ABA correlation-function program of Kitanine, Maillet and Terras, which expresses thermodynamic-limit correlation functions as multiple contour integrals whose integrands factorize into an algebraic part and an analytic part built from known one-point functions; and for the restricted (RSOS) models, which have finite height sets, by Lukyanov and Pugai using bosonization (free-field) techniques.

The algebraic Bethe ansatz approach to correlation functions proceeds as follows: eigenstates of the transfer matrix ('Bethe states') are built by repeated action of the monodromy-matrix entry $B(u)$ on a pseudovacuum; local operators are re-expressed in terms of monodromy-matrix entries via the solution of the quantum inverse scattering problem; and their matrix elements between Bethe states are evaluated as scalar products of Bethe states, for which determinant representations are required. For face models the height dependence forces one to use 'partial' scalar products, which (unlike the six-vertex case) do not reduce to a single determinant: Levy-Bencheton and Terras showed that in the CSOS model they are finite sums, over $L$ terms, of determinants of elliptic functions, and used this to derive multiple-integral representations of the multi-point LHPs at adjacent sites at the thermodynamic limit, with a theta-function algebraic part and an analytic part involving a 'deformed' one-point LHP; the same finite-size representation was in fact shown to exist for the unrestricted SOS model as well, where however the sum over the height lattice is an infinite series already at finite lattice size.

For the unrestricted SOS model, the same quantities are available formally in the massive regime III through the free-field/CTM constructions of Lashkevich and Pugai developed for the eight-vertex model, which express SOS height correlations as bosonic traces admitting integral representations. Those constructions, however, rest on the CTM hypothesis and on unproven identifications of the free-field Fock spaces with the ground-state sectors (acknowledged by their authors as 'strong evidences' and 'expectations'), are tied to specific regimes, and do not derive the results from the finite-lattice Bethe ansatz. What is missing is the ABA route at the level of rigor that the cyclic case reached; this is the problem stated below.

Problem Statement

Establish, within the algebraic Bethe ansatz framework, the multiple-integral representation of the multi-point local height probabilities of the unrestricted (non-cyclic) solid-on-solid model at the thermodynamic limit, with control of convergence. The construction is required to satisfy the following conditions. The convergence of the finite-size building blocks — the multi-point matrix elements of monodromy-matrix entries between Bethe ground states of the unrestricted SOS model, which exist as sums of determinants of elliptic functions but are infinite series over the height lattice $s_0+\mathbb{Z}$ (the cyclic case truncates this to $L$ terms when $\eta=r/L$) — must be established on the domain of validity that the construction delimits, and the termwise manipulations that were automatic in the cyclic case, namely the interchanges of the infinite height sums with the thermodynamic limit and with the contour integrals, must be justified. The resulting thermodynamic-limit multiple-integral representation must be exhibited for the multi-point LHPs at adjacent sites (the building blocks of arbitrary correlation functions on the face lattice, general correlations being accessible through the solution of the inverse problem), identifying its algebraic (theta-function) part and its analytic part (built from deformed one-point LHPs), at generic crossing parameter $\eta$. As a consistency requirement, the one-point reduction must reproduce Baxter's corner-transfer-matrix one-point LHPs of the unrestricted SOS model. The problem also includes delimiting where the construction is valid. A resolution consists either in a rigorous construction meeting these conditions, or in a concrete obstruction showing that the algebraic Bethe ansatz method cannot be so adapted; divergence of the height series confined to particular parameter regimes, with the construction valid elsewhere, delimits the domain of validity rather than by itself establishing such an obstruction, and a rigorous construction on a delimited domain of validity, together with the delimitation itself, also constitutes a resolution.

References

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