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Nested algebraic Bethe ansaetze for orthogonal and symplectic spin chains

Allan John Gerrard

Published Sep 1, 2020• 0 citations
OpenAlex

Abstract

In this thesis the nested algebraic Bethe ansatz technique is applied to various orthogonal and symplectic closed and open spin chain models. Each spin chain considered is regarded as a representation of an underlying quantum group algebra, and expressions for eigenvectors of transfer matrices associated to these models are constructed using the algebra relations, reducing the problem to a set of Bethe equations. The specific models considered are the Ol'shanskii twisted Yangian spin chain, where gl_n bulk symmetry is broken to orthogonal or symplectic symmetry; the MacKay twisted Yangian spin chain, an open spin chain with bulk orthogonal or symplectic symmetry and various boundary types; and the q-deformed orthogonal or symplectic closed spin chain. For the first and third cases, a closed 'trace formula' expression for the eigenvector is also provided.

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In this thesis the nested algebraic Bethe ansatz technique is applied to various orthogonal and symplectic closed and open spin chain models. Each spin chain considered is regarded as a representation of an underlying quantum group algebra, and expressions for eigenvectors of transfer matrices associated to these models are constructed using the algebra relations, reducing the problem to a set of Bethe equations. The specific models considered are the Ol'shanskii twisted Yangian spin chain, where gl_n bulk symmetry is broken to orthogonal or symplectic symmetry; the MacKay twisted Yangian spin chain, an open spin chain with bulk orthogonal or symplectic symmetry and various boundary types; and the q-deformed orthogonal or symplectic closed spin chain. For the first and third cases, a closed 'trace formula' expression for the eigenvector is also provided.
We present a nested algebraic Bethe ansatz for one-dimensional so2n- and sp2n-symmetric open spin chains with diagonal boundary conditions. The monodromy matrix of these spin chains satisfies the defining relations on the extended twisted Yangians Xρ(so2n,so2nρ)tw and Xρ(sp2n,sp2nρ)tw, respectively. We use a generalisation of the De Vega and Karowski approach allowing us to relate the spectral problem of so2n- or sp2n-symmetric open spin chain to that of gln-symmetric open spin chain studied by Belliard and Ragoucy. We explicitly derive the structure of Bethe vectors, their eigenvalues and the nested Bethe equations. We also provide a proof of Belliard and Ragoucy's trace formula for Bethe vectors of gln-symmetric open spin chains.
We present in a unified and detailed way the nested Bethe ansatz for open spin chains based on(super)algebras, with arbitrary representations (i.e., 'spins') on each site of the chain and diagonal boundary matrices (K + (u), K -(u)).The nested Bethe ansatz applies for a general K -(u), but a particular form of the K + (u) matrix.The construction extends and unifies the results already obtained for open spin chains based on the fundamental representation and for some particular superspin chains.We give the eigenvalues, Bethe equations and the explicit form of the Bethe vectors for the corresponding models.The Bethe vectors are also expressed using a trace formula.
We present an 'algebraic treatment' of the analytical Bethe ansatz. For this purpose, we introduce abstract monodromy and transfer matrices which provide an algebraic framework for the analytical Bethe ansatz. It allows us to deal with a generic gl ( )-spin chain possessing on each site an arbitrary gl ( )-representation. For open spin chains, we use the classification of the reflection matrices to treat all the diagonal boundary cases. As a result, we obtain the Bethe equations in their full generality for closed and open spin chains. The classifications of finite-dimensional irreducible representations for the Yangian (closed spin chains) and for the reflection algebras (open spin chains) are directly linked to the calculation of the transfer matrix eigenvalues. The local Hamiltonian associated with a given integrable spin chain needs to be computed case by case. A general formula is still lacking at that point. As examples, we recover the usual Hamiltonian and Bethe equations for closed and open spin chains; we treat the alternating spin chains and the closed spin chain with an impurity. We also compute the Hamiltonian and Bethe equations for the open spin chain model used in the context of large- N QCD.
We study Olshanski twisted Yangian based models, known as one-dimensional “soliton non-preserving” open spin chains, by means of algebraic Bethe Ansatz. The even case, when the bulk symmetry is \mathfrak{gl}_{2n} 𝔤 𝔩 2 n and the boundary symmetry is \mathfrak{sp}_{2n} 𝔰 𝔭 2 n or \mathfrak{so}_{2n} 𝔰 𝔬 2 n , was studied in [Ann. Henri Poincaré 20, 339 (2018)]. In the present work, we focus on the odd case, when the bulk symmetry is \mathfrak{gl}_{2n+1} 𝔤 𝔩 2 n + 1 and the boundary symmetry is \mathfrak{so}_{2n+1} 𝔰 𝔬 2 n + 1 . We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and Y(\mathfrak{gl}_n) Y ( 𝔤 𝔩 n ) -type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.