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Functional relations in solvable lattice models II: Applications

Atsuo Kuniba, Tomoki Nakanishi, Junji Suzuki

Published Jan 31, 2013• 72 citations
OpenAlex

Abstract

Abstract. We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious phenomenon that the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r + 1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent publication.

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Functional relations and the Yang-Baxter algebra

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Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are inti-mately connected with Baxter’s concept of commuting transfer matri-ces. This concept has culminated in the celebrated Yang-Baxter equa-tion which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spec-trum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.

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Bethe eigenvectors of higher transfer matrices

Evgeny Mukhin, V. Tarasov, A Varchenko

We consider the XXX -type and Gaudin quantum integrable models associated with the Lie algebra . The models are defined on a tensor product of irreducible -modules. For each model, there exist N one-parameter families of commuting operators on , called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.

arXiv:math/0605015•Journal of Statistical Mechanics Theory and Experiment•Aug 3, 2006•53 citations
Abstract. We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite dimensional quantum group modules is clarified. We find a curious phenomenon that the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r + 1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent publication.
We study a system of functional relations among a commuting family of row-to-row transfer matrices in solvable lattice models. The role of exact sequences of the finite-dimensional quantum group modules is clarified. We find a curious phenomenon where the solutions of those functional relations also solve the so-called thermodynamic Bethe ansatz equations in the high temperature limit for sl(r+1) models. Based on this observation, we propose possible functional relations for models associated with all the simple Lie algebras. We show that these functional relations certainly fulfill strong constraints coming from the fusion procedure analysis. The application to the calculations of physical quantities will be presented in the subsequent paper.
Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are intimately connected with Baxter's concept of commuting transfer matrices. This concept has culminated in the celebrated Yang-Baxter equation which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spectrum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.
Functional equations methods are a fundamental part of the theory of Exactly Solvable Models in Statistical Mechanics and they are inti-mately connected with Baxter’s concept of commuting transfer matri-ces. This concept has culminated in the celebrated Yang-Baxter equa-tion which plays a fundamental role for the construction of quantum integrable systems and also for obtaining their exact solution. Here I shall discuss a proposal that has been put forward in the past years, in which the Yang-Baxter algebra is viewed as a source of functional equations describing quantities of physical interest. For instance, this method has been successfully applied for the description of the spec-trum of open spin chains, partition functions of elliptic models with domain wall boundaries and scalar product of Bethe vectors. Further applications of this method are also discussed.