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Journal of the Physical Society of Japan

Spontaneous Magnetization in the Bounded XXZ Spin Chain

Yasuhiro Fujii, Miki Wadati

Published Apr 15, 1999• 2 citations
Publisher

Abstract

The bounded XXZ spin chain is studied. The operator appeared in the q -difference equation discovered by Cherednik is modified into the transfer matrix of the model. Its eigenstate is composed by the vertex operators of the quantum affine algebra. By use of a geometric symmetry called the “turning symmetry” that the spin chain without boundary magnetic fields satisfies, the asymptotic form of the eigenstate is found and the spontaneous magnetization at a boundary is calculated. It is roughly twice as large as that in the half-infinite XXZ spin chain.

DOI
10.1143/jpsj.68.1121

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The bounded XXZ spin chain is studied. The operator appeared in the q -difference equation discovered by Cherednik is modified into the transfer matrix of the model. Its eigenstate is composed by the vertex operators of the quantum affine algebra. By use of a geometric symmetry called the “turning symmetry” that the spin chain without boundary magnetic fields satisfies, the asymptotic form of the eigenstate is found and the spontaneous magnetization at a boundary is calculated. It is roughly twice as large as that in the half-infinite XXZ spin chain.
Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is constructed employing the representation theory of quantum groups. The spectrum of this Q-operator is discussed and novel functional relations which describe the finite temperature regime of the XXZ spin-chain are derived. For non-vanishing magnetic field the previously known Bethe ansatz equations can be replaced by a system of quadratic equations which is an important advantage for numerical studies. For vanishing magnetic field and rational coupling values it is argued that the quantum transfer matrix exhibits a loop algebra symmetry closely related to the one of the classical six-vertex transfer matrix at roots of unity.
The \\XXZ spin chain with a boundary magnetic field h is considered, using the vertex operator approach to diagonalize the Hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when h\\geq0, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuum states. We derive the boundary S-matrix and present an integral formula for the correlation functions. The boundary magnetization exhibits boundary hysteresis. We also discuss the rational limit, the \\XXX model.
We present a systematic way of constructing boundary quantum group generators associated to non-diagonal solutions of the reflection equation for the XXZ model. This is achieved by studying the asymptotics of the corresponding open spin chain. Linear intertwining relations for the K{\cal K}K matrix and Sklyanin's operator are also obtained, and the symmetry of the transfer matrix is investigated.
We consider correlation functions of the form , where |vac⟩ is the vacuum eigenstate of an infinite antiferromagnetic XXZ chain, |vac⟩′ is the vacuum eigenstate of an infinite XXZ chain which is split in two, and is a local operator. The Hamiltonian of the split chain has no coupling between sites 1 and 0 and has a staggered magnetic field at these two sites; it arises from a tensor product of left and right boundary transfer matrices. We find a simple, exact expression for ⟨vac|vac⟩′ and an exact integral expression for general using the vertex operator approach. We compute the integral when and find a conjectural expression that is analogous to the known formula for the XXZ spontaneous magnetization and reduces to it when the magnetic field is zero. We show that correlation functions obey a boundary qKZ equation of a different level to the infinite XXZ chain with one boundary.