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arXiv:solv-int/9905009 · Journal of Mathematical Physics

The Uq(sln∧) analogue of the XXZ chain with a boundary

H. Furutsu, T. Kojima

Published Jul 1, 2000• 24 citations
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Abstract

We study the Uq(sln∧) analogue of the XXZ spin chain with a boundary magnetic field h. We construct explicit bosonic formulas of the vacuum vector and the dual vacuum vector with a boundary magnetic field. For an application of the explicit formula of the vacuum vector, we derive the integral representation of the boundary spontaneous magnetization.

DOI
10.1063/1.533351

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We study the Uq(sln∧) analogue of the XXZ spin chain with a boundary magnetic field h. We construct explicit bosonic formulas of the vacuum vector and the dual vacuum vector with a boundary magnetic field. For an application of the explicit formula of the vacuum vector, we derive the integral representation of the boundary spontaneous magnetization.
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 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.
We derive compact multiple integral formulae for several physical spin correlation functions in the semi-infinite XXZ chain with a longitudinal boundary magnetic field. Our formulae follow from several effective resummations of the multiple integral representation for the elementary blocks obtained in our previous paper (I). In the free fermion point we compute the local magnetization as well as the density of energy profiles. These quantities, in addition to their bulk behavior, exhibit Friedel-type oscillations induced by the boundary; their amplitudes depend on the boundary magnetic field and decay algebraically in terms of the distance to the boundary.
Abstract We derive an exact formula for the boundary free energy of the open Heisenberg XXZ spin chain. We allow for arbitrary boundary magnetic fields, but assume zero bulk magnetization. The result is completely analogous to earlier formulas for the so-called g -function: it is expressed as a combination of single integrals and two simple Fredholm determinants. Our expressions can be evaluated easily using numerical algorithms with arbitrary precision. We demonstrate that the boundary free energy can show a wide variety of behaviour as a function of the temperature, depending on the anisotropy and the boundary fields. We also compute the low temperature limit of the boundary free energy, and reproduce the known results for the ground state boundary energy, including the case of non-diagonal fields.