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arXiv:2012.02543 · Journal of Statistical Mechanics Theory and Experiment

Solutions of the Yang–Baxter equation for ( n + 1) (2 n + 1)-vertex models using a differential approach

R S Vieira, A Lima-Santos

Published May 1, 2021• 7 citations
arXivHTMLPDF

Abstract

Abstract The formal derivatives of the Yang–Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which, two systems of polynomial equations are obtained in their place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nevertheless, the remaining unknowns can be found by solving a few simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work, we consider the Yang–Baxter equation for ( n + 1) (2 n + 1)-vertex models with a generalization based on the A n −1 symmetry. This differential approach allows us to solve the Yang–Baxter equation in a systematic way.

DOI
10.1088/1742-5468/abf7be

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Abstract The formal derivatives of the Yang–Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which, two systems of polynomial equations are obtained in their place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nevertheless, the remaining unknowns can be found by solving a few simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work, we consider the Yang–Baxter equation for ( n + 1) (2 n + 1)-vertex models with a generalization based on the A n −1 symmetry. This differential approach allows us to solve the Yang–Baxter equation in a systematic way.
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In this work, we employ the algebraic-differential method recently developed by the author to solve the Yang-Baxter equation for arbitrary fifteen-vertex models satisfying the ice-rule. We show that there are four different families of such regular RRR matrices containing several free-parameters. The corresponding reflection KKK matrices, solutions of the boundary Yang-Baxter equation, were also found and classified. We found that there are three different families of regular KKK matrices, regardless of what RRR matrix we choose.
On the multiparametric Uq[Dn+1(2)]{\mathcal{U}}_{q}[{D}_{n+1}^{(2)}]Uq​[Dn+1(2)​] vertex model
We study the solutions of the Yang–Baxter equation associated with 19-vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the respective Boltzmann weights and find that they possess a universal structure. This allows us to classify the integrable manifolds into four different families reproducing three known models, besides uncovering a novel 19-vertex model in a unified way. The introduction of the spectral parameter on the weights is made via the parameterization of the fundamental algebraic curve which is a conic. The diagonalization of the transfer matrix of the new vertex model and its thermodynamic limit properties are discussed. We point out a connection between the form of the main curve and the nature of the excitations of the corresponding spin-1 chains.