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arXiv:1308.1790 · Journal of Statistical Mechanics Theory and Experiment

A note on $\mathfrak {gl}_{{\cal N}}$ type-I integrable defects

Anastasia Doikou

Published Feb 14, 2014• 7 citations
arXivHTMLPDF

Abstract

Type-I quantum defects are considered in the context of the spin chain. The type-I defects are associated with the generalized harmonic oscillator algebra, and the chosen defect matrix is that of the vector nonlinear Schrödinger (NLS) model. The transmission matrices relevant to this particular type of defects are computed via the Bethe ansatz methodology.

DOI
10.1088/1742-5468/2014/02/p02002

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A note on glN\mathfrak {gl}_{{\cal N}}glN​ type-I integrable defects
Type-I quantum defects are considered in the context of the spin chain. The type-I defects are associated with the generalized harmonic oscillator algebra, and the chosen defect matrix is that of the vector nonlinear Schrödinger (NLS) model. The transmission matrices relevant to this particular type of defects are computed via the Bethe ansatz methodology.
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Spectrum and Bethe Ansatz Equations for the Uq(gl(N))U_{q} {\left( {gl(\mathcal{N})} \right)}Uq​(gl(N)) Closed and Open Spin Chains in any Representation
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.