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arXiv:1901.01514 · Journal of Physics A Mathematical and Theoretical

Surface energy and elementary excitations of the XXZ spin chain with arbitrary boundary fields

Pei Sun, Zhi-Rong Xin, Yi Qiao, Kun Hao, Like Cao, Junpeng Cao, Tao Yang, Wen‐Li Yang

Published May 17, 2019• 12 citations
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Abstract

Abstract The thermodynamic properties of the XXZ spin chain with integrable open boundary conditions at the gaped region (i.e. the anisotropic parameter being a real number) are investigated. It is shown that the contribution of the inhomogeneous term in the T − Q relation of the ground state and elementary excited state can be neglected when the size of the system N tends to infinity. The surface energy and elementary excitations induced by the unparallel boundary magnetic fields are obtained.

DOI
10.1088/1751-8121/ab2259

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Abstract The thermodynamic properties of the XXZ spin chain with integrable open boundary conditions at the gaped region (i.e. the anisotropic parameter being a real number) are investigated. It is shown that the contribution of the inhomogeneous term in the T − Q relation of the ground state and elementary excited state can be neglected when the size of the system N tends to infinity. The surface energy and elementary excitations induced by the unparallel boundary magnetic fields are obtained.
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An analytic method is proposed to compute the surface energy and elementary excitations of the XXZ spin chain with generic nondiagonal boundary fields. For the gapped case, in some boundary parameter regimes the contributions of the two boundary fields to the surface energy are nonadditive. Such a correlation effect between the two boundaries also depends on the parity of the site number NNN even in the thermodynamic limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. For the gapless case, contributions of the two boundary fields to the surface energy are additive due to the absence of long-range correlation in the bulk. Although the U(1)U(1)U(1) symmetry of the system is broken, exact spinonlike excitations, which obviously do not carry spin-12\frac{1}{2}21​, are observed. The present method provides a universal procedure to deal with quantum integrable systems either with or without U(1)U(1)U(1) symmetry.
By using a set of gauge transformations, the exact solutions of the XXZ spin chain with unparallel boundary magnetic fields are derived in the framework of the algebraic Bethe ansatz. In the easy-plane case, we show the elementary excitations are some kind of spinons without definite spin because the U(1) symmetry is broken, while in the easy-axis ferromagnetic case a spiral state is realized in the ground state. The correlation functions, the spin torque as well as the spin voltage for the later case are also derived.
We investigate the thermodynamic limit of the inhomogeneous T−Q relation of the antiferromagnetic XXZ spin chain with antiperiodic boundary condition. It is shown that the contribution of the inhomogeneous term for the ground state can be neglected when the system-size N tends to infinity, which enables us to reduce the inhomogeneous Bethe ansatz equations (BAEs) to the homogeneous ones. Then the quantum numbers at the ground states are obtained, by which the system with arbitrary size can be studied. We also calculate the twisted boundary energy of the system.
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.