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XXZ chain with a boundary

Michio Jimbo, Rinat Kedem, Takeo Kojima, Hitoshi Konno, Tetsuji Miwa

Published Nov 17, 1994• 93 citations
OpenAlex

Abstract

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.

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Boundary K-matrices for the XYZ, XXZ AND XXX spin chains

Hector J. de Vega, A. González-Ruiz

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•Jun 18, 1993•127 citations
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.
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.
Abstract In this paper we continue our derivation of the correlation functions of open quantum spin 1/2 chains with unparallel magnetic fields on the edges; this time for the more involved case of the XXZ spin 1/2 chains. We develop our study in the framework of the quantum separation of variables, which gives us both the complete spectrum characterization and simple scalar product formulae for separate states, including transfer matrix eigenstates. Here, we leave the boundary magnetic field in the first site of the chain completely arbitrary, and we fix the boundary field in the last site N of the chain to be a specific value along the z -direction. This is a natural first choice for the unparallel boundary magnetic fields. We prove that under these special boundary conditions, on the one side, we have a simple enough complete spectrum description in terms of homogeneous Baxter like TQ -equation. On the other side, we prove a simple enough description of the action of a basis of local operators on transfer matrix eigenstates as linear combinations of separate states. Thanks to these results, we achieve our main goal to derive correlation functions for a set of local operators both for the finite and half-infinite chains, with multiple integral formulae in this last case.
The general solutions for the factorization equations of the reflection matrices Kpm(theta)K^{\\pm}(\\theta)Kpm(theta) for the eight vertex and six vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are explicitly derived, finding families dependig on several continuous as well as discrete parameters.