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arXiv:hep-th/0601107 · Journal of Statistical Mechanics Theory and Experiment

Equivalences between spin models induced by defects

Z Bajnok

Published Jun 15, 2006• 49 citations
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Abstract

The spectra of integrable spin chains are shown to be independent of the ordering of their spins. As an application we introduce defects (local spin inhomogeneities in homogenous chains) in two-boundary spin systems and, by changing their locations, we show the spectral equivalence of different boundary conditions. In particular we relate certain nondiagonal boundary conditions to diagonal ones.

DOI
10.1088/1742-5468/2006/06/p06010

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The spectra of integrable spin chains are shown to be independent of the ordering of their spins. As an application we introduce defects (local spin inhomogeneities in homogenous chains) in two-boundary spin systems and, by changing their locations, we show the spectral equivalence of different boundary conditions. In particular we relate certain nondiagonal boundary conditions to diagonal ones.
We consider the SU q ( N ) invariant spin chain with diagonal and non-diagonal integrable boundary terms. The algebraic study of spin chains with different kinds of boundary terms is used to motivate a set of spectral equivalences between integrable chains with purely diagonal boundary terms and ones with an arbitrary non-diagonal term at one end. For each choice of diagonal boundary terms there is an iso-spectral one-boundary problem and vice versa. The cases N = 2, 3, 4 are used as examples throughout to illustrate the general structure. The quantum group SU q ( N ) symmetry is broken by the presence of a non-diagonal boundary term; however, one can use the spectral equivalence with the diagonal chain to easily understand the residual symmetries of the system. This is described in detail for the case of SU (3).
The Motzkin spin chain is a spin-1 model introduced in \cite{shor} as an example of a system exhibiting a high degree of quantum fluctuations whose ground state can be mapped to Motzkin paths that are generated with local equivalence moves. This model is difficult to solve in general but keeping just the height preserving local equivalence moves we show that the model becomes integrable which when projected to certain subspaces of the full Hilbert space is isomorphic to the spin-12\frac{1}{2}21​ XXX chain. In fact in the full Hilbert space the system is akin to two non-interacting spin-12\frac{1}{2}21​ XXX chains making the spectrum the same as the latter with the change coming in the degeneracy of the states. We then show that including the height-changing local-equivalence move is the same as introducing interactions in the above system.
We determine the spectra of a class of quantum spin chains of Temperley–Lieb type by utilizing the concept of Temperley–Lieb equivalence with the S = 1/2 XXZ chain as a reference system. We consider open boundary conditions and, in particular, periodic boundary conditions. For both types of boundaries the identification with XXZ spectra is performed within isomorphic representations of the underlying Temperley–Lieb algebra. For open boundaries the spectra of these models differ from the spectrum of the associated XXZ chain only in the multiplicities of the eigenvalues. The periodic case is rather different. Here we show how the spectrum is obtained sector-wise from the spectra of globally twisted XXZ chains. As a spin-off, we obtain a compact formula for the degeneracy of the momentum operator eigenvalues. Our representative theoretical results allow for the study of the thermodynamics by establishing a TL-equivalence at finite temperature and finite field .
We solve the spectrum of quantum spin chains based on representations of the Temperley–Lieb algebra associated with the quantum groups [Formula: see text] for Xn=A1, Bn, Cn and Dn. The tool is a modified version of the coordinate Bethe ansatz through a suitable choice of the Bethe states which give to all models the same status relative to their diagonalization. All these models have equivalent spectra up to degeneracies and the spectra of the lower-dimensional representations are contained in the higher-dimensional ones. Periodic boundary conditions, free boundary conditions and closed nonlocal boundary conditions are considered. Periodic boundary conditions, unlike free boundary conditions, break quantum group invariance. For closed nonlocal cases the models are quantum group invariant as well as periodic in a certain sense.
We solve for the spectrum of quantum spin chains based on representations of the Temperley - Lieb algebra associated with the quantum groups for and . We employ a generalization of the coordinate Bethe ansatz developed previously for the deformed biquadratic spin-one chain. As expected, all of these models have equivalent spectra, i.e. they differ only in the degeneracy of their eigenvalues. This is true for finite length and open boundary conditions. For periodic boundary conditions the spectra of the lower-dimensional representations are contained entirely in the higher-dimensional ones. The Bethe states are highest-weight states of the quantum group, except in certain cases where the states have energy zero.