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Acta Physica Sinica

SOLUTIONS OF YANG-BAXTER EQUATIONWITH SPECTRAL PARAMETER FORSIX-VERTEX MODEL

S Wang

Published Jan 1, 1995• 0 citations
OpenAlex

Abstract

In this paper, we obtain all solutions of Yang-Baxter equation with spectral parameter for six-vertex model in theory of exactly solved statistical models by four sets of non-degenarate basic solutions and seven sets of degenerate basic solutions up to five solution transformations. The unitary condition of the solutions is also discussed.

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In this note, we claim that we have obtained all nondegenerate solutions of Yang–Baxter equation with spectral parameter for square-lattice eight-vertex model without “zero-field” conditions by several solution transformations and six basic solutions with the help of Wu algorithm.

Communications in Theoretical Physics•Jan 30, 1995•5 citations

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

R S Vieira, A Lima-Santos

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.

arXiv:2012.02543•Journal of Statistical Mechanics Theory and Experiment•May 1, 2021•7 citations

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The Six-Vertex Yang-Baxter Groupoid

Daniel Bump, Slava Naprienko

A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid $\mathfrak{G}$ together with a map $π:\mathfrak{G}\to\operatorname{End}(V\otimes V)$ for some vector space $V$ such that the Yang-Baxter commutator $[[ π(u),π(w),π(v)]]=0$ if $u,v\in\mathfrak{G}$ are such that the groupoid composition $w=u\star v$ is defined. An important role is played by an object map $Δ:\mathfrak{G}\to M$ for some set $M$ such that $Δ(u)=Δ(v')$, $Δ(w)=Δ(v)$ and $Δ(w')=Δ(u')$, where $v\mapsto v'$ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group $\operatorname{GL}(2)\times\operatorname{GL}(1)$. For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.

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Six-Vertex Model

Hans-Peter Eckle

Abstract This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.

•Jul 25, 2019•12 citations

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

R. S. Vieira

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 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. Nonetheless, the remaining unknowns can be found by solving a few number of 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 considered the Yang-Baxter equation for two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.

arXiv:1712.02341•Journal of High Energy Physics•Oct 1, 2018•31 citations
In this paper, we obtain all solutions of Yang-Baxter equation with spectral parameter for six-vertex model in theory of exactly solved statistical models by four sets of non-degenarate basic solutions and seven sets of degenerate basic solutions up to five solution transformations. The unitary condition of the solutions is also discussed.
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.
A parametrized Yang-Baxter equation is usually defined to be a map from a group to a set of R-matrices, satisfying the Yang-Baxter commutation relation. These are a mainstay of solvable lattice models. We will show how the parameter space can sometimes be enlarged to a groupoid, and give two examples of such groupoid parametrized Yang-Baxter equations, within the six vertex model. A groupoid parametrized Yang-Baxter equation consists of a groupoid G\mathfrak{G}G together with a map π:G→End⁡(V⊗V)π:\mathfrak{G}\to\operatorname{End}(V\otimes V)π:G→End(V⊗V) for some vector space VVV such that the Yang-Baxter commutator [[π(u),π(w),π(v)]]=0[[ π(u),π(w),π(v)]]=0[[π(u),π(w),π(v)]]=0 if u,v∈Gu,v\in\mathfrak{G}u,v∈G are such that the groupoid composition w=u⋆vw=u\star vw=u⋆v is defined. An important role is played by an object map Δ:G→MΔ:\mathfrak{G}\to MΔ:G→M for some set MMM such that Δ(u)=Δ(v′)Δ(u)=Δ(v')Δ(u)=Δ(v′), Δ(w)=Δ(v)Δ(w)=Δ(v)Δ(w)=Δ(v) and Δ(w′)=Δ(u′)Δ(w')=Δ(u')Δ(w′)=Δ(u′), where v↦v′v\mapsto v'v↦v′ is the groupoid inverse map. There are two main regimes of the six-vertex model: the free-fermionic point, and everything else. For the free-fermionic point, there exists a parametrized Yang-Baxter equation with a large parameter group GL⁡(2)×GL⁡(1)\operatorname{GL}(2)\times\operatorname{GL}(1)GL(2)×GL(1). For non-free-fermionic six-vertex matrices, there are also well-known (group) parametrized Yang-Baxter equations, but these do not account for all possible interactions. Instead we will construct a groupoid parametrized Yang-Baxter equation that accounts for essentially all possible Yang-Baxter equations in the six-vertex model. We will also exhibit a separate groupoid for the five-vertex model. We will show how to construct solvable lattice models based on groupoid parametrized Yang-Baxter equations.
Abstract This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.
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 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. Nonetheless, the remaining unknowns can be found by solving a few number of 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 considered the Yang-Baxter equation for two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.