arXiv:2012.08367 · Journal of High Energy Physics

Factorization identities and algebraic Bethe ansatz for $$ {D}_2^{(2)} $$ models

Rafael I. Nepomechie, Ana L. Retore

Published Mar 1, 2021• 0 citations

Abstract

Abstract We express $$ {D}_2^{(2)} $$ D 2 2 transfer matrices as products of $$ {A}_1^{(1)} $$ A 1 1 transfer matrices, for both closed and open spin chains. We use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. We also formulate and solve a new integrable XXZ-like open spin chain with an even number of sites that depends on a continuous parameter, which we interpret as the rapidity of the boundary.

DOI: 10.1007/jhep03(2021)089

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arXiv:2012.08367 · Journal of High Energy Physics

Factorization identities and algebraic Bethe ansatz for ${D}_2^{(2)}$ modelsFactorization identities and algebraic Bethe ansatz for $$ {D}_2^{(2)} $$ models

Rafael I. Nepomechie, Ana L. Retore

Published Mar 1, 2021• 0 citations

arXiv HTML PDF

Abstract

Abstract We express

{D}_2^{(2)}

D 2 2 transfer matrices as products of

{A}_1^{(1)}

A 1 1 transfer matrices, for both closed and open spin chains. We use these relations, which we call factorization identities, to solve the models by algebraic Bethe ansatz. We also formulate and solve a new integrable XXZ-like open spin chain with an even number of sites that depends on a continuous parameter, which we interpret as the rapidity of the boundary.

DOI: 10.1007/jhep03(2021)089

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A bstract We study the quantum integrable spin chain model associated with the twisted ${D}_2^{(2)}$ D 2 2 algebra (or simply the ${D}_2^{(2)}$ D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the ${D}_2^{(2)}$ D 2 2 model in different regimes of boundary parameters.A bstract We study the quantum integrable spin chain model associated with the twisted $$ {D}_2^{(2)} $$ D 2 2 algebra (or simply the $$ {D}_2^{(2)} $$ D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the $$ {D}_2^{(2)} $$ D 2 2 model in different regimes of boundary parameters.

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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.

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Factorization identities and algebraic Bethe ansatz for

{D}_2^{(2)}

models

Abstract We express

{D}_2^{(2)}

D 2 2 transfer matrices as products of

{A}_1^{(1)}

Spectrum of the quantum integrable

{D}_2^{(2)}

spin chain with generic boundary fields

Abstract Exact solution of the quantum integrable

{D}_2^{(2)}

D 2 2 spin chain with generic integrable boundary fields is constructed. It is found that the transfer matrix of this model can be factorized as the product of those of two open staggered anisotropic XXZ spin chains. Based on this identity, the eigenvalues and Bethe ansatz equations of the

{D}_2^{(2)}

D 2 2 model are derived via off-diagonal Bethe ansatz.

Exact physical quantities of the

{D}_2^{(2)}

spin chain model with generic open boundary conditions

A bstract We study the quantum integrable spin chain model associated with the twisted

{D}_2^{(2)}

D 2 2 algebra (or simply the

{D}_2^{(2)}

D 2 2 model) under generic open boundary conditions. The Hamiltonian of this model can be factorized into the sum of two staggered XXZ spin chains. Applying the t - W method, we derive the homogeneous Bethe ansatz equations for the zeros of the transfer matrix eigenvalues and the patterns of the corresponding zeros of the staggered XXZ spin chain with generic integrable boundaries. Based on these results, we analytically compute the surface energies and excitation energies of the

{D}_2^{(2)}

D 2 2 model in different regimes of boundary parameters.

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.

Off-diagonal Bethe Ansatz for the

{D}_3^{(1)}

model

Abstract The exact solutions of the

{D}_3^{(1)}

D 3 1 model (or the so(6) quantum spin chain) with either periodic or general integrable open boundary conditions are obtained by using the off-diagonal Bethe Ansatz. From the fusion, the complete operator product identities are obtained, which are sufficient to enable us to determine spectrum of the system. Eigenvalues of the fused transfer matrices are constructed by the T - Q relations for the periodic case and by the inhomogeneous T- Q one for the non-diagonal boundary reflection case. The present method can be generalized to deal with the

{D}_n^{(1)}

D n 1 model directly.

\mathfrak {sl}_2

sl 2 and diagonal K-matrices, we derive such an identity using the recently formulated theory of universal K-matrices for quantum affine algebras.