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arXiv:2206.14271 · International Journal of Modern Physics B

A comment on the solutions of the generalized Faddeev–Volkov model

Mehmet Dede

Published Aug 17, 2023• 0 citations
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Abstract

We consider two recent solutions of the generalized Faddeev–Volkov model, which is an exactly solvable Ising-type lattice spin model. The first solution is obtained by using the noncompact quantum dilogarithm, and the second one is constructed in a recent study via the gauge/YBE correspondence. We show that the weight functions of these models obtained by different techniques are the same upto a constant.

DOI
10.1142/s0217979224503119

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We consider two recent solutions of the generalized Faddeev–Volkov model, which is an exactly solvable Ising-type lattice spin model. The first solution is obtained by using the noncompact quantum dilogarithm, and the second one is constructed in a recent study via the gauge/YBE correspondence. We show that the weight functions of these models obtained by different techniques are the same upto a constant.
Abstract Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group A , we construct an operator solution of the Yang–Baxter equation generalizing the solution of the Faddeev–Volkov model. Based on a specific choice of a subgroup and by using the Weil transformation, we also give a new non-operator interpretation of the Yang–Baxter relation. That allows us to construct a lattice QFT-model of IRF-type with gauge invariance under independent B -translations of local ‘spin’ variables.
We have introduced a type of universal transformation between spin variables si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ to map the general spin-1 model onto the spin-1/2 Ising model and have employed the exact results of the latter to find the exactly solvable cases of the spin-1 model. Our transformations are not one-to-one correspondences between si{\mathit{s}}_{\mathit{i}}si​ and ${\mathrm{\ensuremath{\sigma}}}_{\mathit{i}}$ and they are applicable to all lattices, depending on the lattice structure only through the coordination number. On square, triangle, and honeycomb lattices the exact critical points are shown in certain subspaces of parameter space (K, J, L, H, and \ensuremath{\Delta}).
The goal of this thesis is to present some novel results for solvable lattice models. In chapter 2 a limit is taken of the master solution that gives an integrable model generalising the Faddeev-Volkov model, first described by Spiridonov. There are properties of this model that arise quite similarly to that of the Faddeev-Volkov model, that are given in this chapter. This includes an integrable "dual spin'' lattice model, with Boltzmann weights parameterised by gamma functions, that appears in the strong coupling regime, and the quasi-classical limit of the model and its relation to the discrete hyperbolic geometry. In chapter 3 hyperbolic and rational degenerations of the master solution are taken that give various solutions of the star-triangle relation. These limits can be viewed as unphysical lattice models in the sense that their Boltzmann weights are not always positive. The aim of this chapter is to show that most of the discrete equations in the Adler, Bobenko, and Suris (ABS) list arise in a quasi-classical expansion of these models. This extends previously known limits of the master solution and Faddeev-Volkov model to the rest of the ABS list, including asymmetric H equations recently classified by Boll and Suris. In chapter 4 a star-star relation conjectured by Bazhanov and Sergeev is proven by relating it to a multivariate elliptic hypergeometric integral identity of Rains. Also the hyperbolic degeneration of the star-star relation is shown to give a new multi-spin model parameterised by the non-compact quantum dilogarithm. A quasi-classical expansion of this model gives a multi-spin generalisation of the Q3 discrete Laplace type integrable system. The corner transfer matrix algorithm was successfully used by Mangazeev et al. to study the two dimensional Ising model around the critical point. Chapter 5 describes a modification of their algorithm to work with the chiral models whose Boltzmann weights don't possess the spin reflection symmetry property. Convergence is tested against the exact values of observables in the integrable Kashiwara-Miwa and chiral Potts models. The convergence decreases approaching the critical point as expected, while away from criticality the convergence is limited by the precision of the software.
A new type of Yang-Baxter equation (YBE) for RRR-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the ZNZ_NZN​ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for N>2N>2N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated RRR-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter RRR-matrix---realized as an intertwiner of cyclic representations of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​) at a root of unity---and Shastry's two-parameter RRR-operator obtained from the decorated YBE.