数理解析研究所講究録

Integrable Vertex Models with General Twists(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

Marcio Jose Martins

Published Apr 1, 2006• 0 citations

OpenAlex

Abstract

Related papers

Integrable Vertex Models with General Twists

Marcio Jose Martins

arXiv:nlin/0512063Dec 21, 20050 citations

Exact solution of the quantum integrable model associated with the twisted $$ {\mathrm{D}}_3^{(2)} $$ algebra

Guang-Liang Li, Xiaotian Xu, Kun Hao +5 more

Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted $$ {D}_3^{(2)} $$ D 3 2 algebra (or the $$ {D}_3^{(2)} $$ D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the $$ {D}_{n+1}^{(2)} $$ D n + 1 2 model and other high rank integrable models.

arXiv:2110.02699Journal of High Energy PhysicsMar 1, 20223 citations

Exact solution of the B 1 model

Panpan Xue, Guang-Liang Li, Junpeng Cao +3 more

Journal of Physics A Mathematical and TheoreticalFeb 22, 20191 citation

The eight-vertex model with quasi-periodic boundary conditions

Giuliano Niccoli, V. Terras

arXiv:1508.03230Journal of Physics A Mathematical and TheoreticalDec 21, 201548 citations

Unified algebraic Bethe ansatz for two-dimensional lattice models

M. Martins

We develop a unified formulation of the quantum inverse scattering method for lattice vertex models associated to the nonexceptional A(2)(2r), A(2)(2r-1), B(1)(r), C(1)(r), D(1)(r+1), and D(2)(r+1) Lie algebras. We recast the Yang-Baxter algebra in terms of different commutation relations between creation, annihilation, and diagonal fields. The solution of the D(2)(r+1) model is based on an interesting 16-vertex model, which is solvable without recourse to a Bethe ansatz.

arXiv:solv-int/9901002Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 1, 199919 citations

Exact solution of the $SU_{q}(n)$ invariant quantum spin chains

Hector J. de Vega, González-Ruiz, A

Sep 3, 199380 citations

数理解析研究所講究録

Integrable Vertex Models with General Twists(Solvable Lattice Models 2004 : Recent Progress on Solvable Lattice Models )

Marcio Jose Martins

Published Apr 1, 2006• 0 citations

OpenAlex

Abstract

B_{n},

D_{n}

and

C_{n}

Lie algebras are presented.The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.

Related papers

Integrable Vertex Models with General Twists

Marcio Jose Martins

We review recent progress towards the solution of exactly solved isotropic vertex models with arbitrary toroidal boundary conditions. Quantum space transformations make it possible the diagonalization of the corresponding transfer matrices by means of the quantum inverse scattering method. Explicit expressions for the eigenvalues and Bethe ansatz equations of the twisted isotropic spin chains based on the $B_n$ , $D_n$ and $C_n$ Lie algebras are presented. The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.We review recent progress towards the solution of exactly solved isotropic vertex models with arbitrary toroidal boundary conditions. Quantum space transformations make it possible the diagonalization of the corresponding transfer matrices by means of the quantum inverse scattering method. Explicit expressions for the eigenvalues and Bethe ansatz equations of the twisted isotropic spin chains based on the $B_n$, $D_n$ and $C_n$ Lie algebras are presented. The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.

arXiv:nlin/0512063Dec 21, 20050 citations

Exact solution of the quantum integrable model associated with the twisted ${\mathrm{D}}_3^{(2)}$ algebraExact solution of the quantum integrable model associated with the twisted $$ {\mathrm{D}}_3^{(2)} $$ algebra

Guang-Liang Li, Xiaotian Xu, Kun Hao +5 more

Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted ${D}_3^{(2)}$ D 3 2 algebra (or the ${D}_3^{(2)}$ D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the ${D}_{n+1}^{(2)}$ D n + 1 2 model and other high rank integrable models.Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted $$ {D}_3^{(2)} $$ D 3 2 algebra (or the $$ {D}_3^{(2)} $$ D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the $$ {D}_{n+1}^{(2)} $$ D n + 1 2 model and other high rank integrable models.

arXiv:2110.02699Journal of High Energy PhysicsMar 1, 20223 citations

Exact solution of the B 1 model

Panpan Xue, Guang-Liang Li, Junpeng Cao +3 more

Abstract The B 1 vertex model is studied by the off-diagonal Bethe ansatz method. New closed operator product identities of the fused transfer matrices are obtained by using the fusion technique. Based on them and the asymptotic behaviors as well as the values of the fused transfer matrices at certain points, the exact solutions of the B 1 model with periodic and with integrable off-diagonal open boundary conditions are obtained. We find that the degree of the polynomial of the inhomogeneous T − Q relation is lower. The B 1 model is equivalent to the integrable spin-1 Heisenberg chain, thus the method and the results in this paper can be generalized to the high spin Heisenberg model and the integrable models associated with B n Lie algebra.Abstract The B 1 vertex model is studied by the off-diagonal Bethe ansatz method. New closed operator product identities of the fused transfer matrices are obtained by using the fusion technique. Based on them and the asymptotic behaviors as well as the values of the fused transfer matrices at certain points, the exact solutions of the B 1 model with periodic and with integrable off-diagonal open boundary conditions are obtained. We find that the degree of the polynomial of the inhomogeneous T − Q relation is lower. The B 1 model is equivalent to the integrable spin-1 Heisenberg chain, thus the method and the results in this paper can be generalized to the high spin Heisenberg model and the integrable models associated with B n Lie algebra.

Journal of Physics A Mathematical and TheoreticalFeb 22, 20191 citation

The eight-vertex model with quasi-periodic boundary conditions

Giuliano Niccoli, V. Terras

Abstract We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T – Q functional equations, where the Q -functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.Abstract We study the inhomogeneous eight-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, -twisted, -twisted or -twisted. We show that in all these cases but the periodic one with an even number of sites , the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical six-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin’s separation of variables approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical six-vertex model, the twisted eight-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the eight-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical six-vertex antiperiodic transfer matrix. In all these cases we can adapt to the eight-vertex case the reformulations of the dynamical six-vertex transfer matrix spectrum and eigenstates that had been obtained by T – Q functional equations, where the Q -functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enable one to characterize the eight-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.

arXiv:1508.03230Journal of Physics A Mathematical and TheoreticalDec 21, 201548 citations

Unified algebraic Bethe ansatz for two-dimensional lattice models

M. Martins

arXiv:solv-int/9901002Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topicsJun 1, 199919 citations

Exact solution of the $SU_{q}(n)$ invariant quantum spin chainsExact solution of the $SU_{q}(n)$ invariant quantum spin chains

Hector J. de Vega, González-Ruiz, A

The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex model with fixed open boundary conditions and the corresponding $SU_{q}(n)$ invariant hamiltonian. The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions.The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the $A_{n-1}$ vertex model with fixed open boundary conditions and the corresponding $SU_{q}(n)$ invariant hamiltonian. The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions.

Sep 3, 199380 citations

B_{n},

D_{n}

and

C_{n}

Lie algebras are presented.The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.

B_n

D_n

and

C_n

Lie algebras are presented. The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.

Exact solution of the quantum integrable model associated with the twisted

{\mathrm{D}}_3^{(2)}

algebra

Abstract We generalize the nested off-diagonal Bethe ansatz method to study the quantum chain associated with the twisted

{D}_3^{(2)}

D 3 2 algebra (or the

{D}_3^{(2)}

D 3 2 model) with either periodic or integrable open boundary conditions. We obtain the intrinsic operator product identities among the fused transfer matrices and find a way to close the recursive fusion relations, which makes it possible to determinate eigenvalues of transfer matrices with an arbitrary anisotropic parameter η. Based on them, and the asymptotic behaviors and values at certain points, we construct eigenvalues of transfer matrices in terms of homogeneous T − Q relations for the periodic case and inhomogeneous ones for the open case with some off-diagonal boundary reflections. The associated Bethe ansatz equations are also given. The method and results in this paper can be generalized to the

{D}_{n+1}^{(2)}

D n + 1 2 model and other high rank integrable models.

Exact solution of the

SU_{q}(n)

invariant quantum spin chains

The Nested Bethe Ansatz is generalized to open boundary conditions. This is used to find the exact eigenvectors and eigenvalues of the

A_{n-1}

vertex model with fixed open boundary conditions and the corresponding

SU_{q}(n)

invariant hamiltonian. The Bethe Ansatz equations obtained are solved in the thermodynamic limit giving the vertex model free energy and the hamiltonian ground state energy including the corresponding boundary contributions.