Abstract
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.
Highlights
The XXZ spin chain with the antiperiodic boundary condition is a very interesting quantum system [1, 2, 3, 4]
We find that the contribution of the inhomogeneous term in the associated T − Q relation to the ground state energy can be neglected when the system-size N tends to infinity
Due to the fact that b1 < 0, the value of Eignh tends to zero when the system-size N tends to infinity, which means that the contribution of the inhomogeneous term for the ground state can be neglected in the thermodynamic limit
Summary
The XXZ spin chain with the antiperiodic boundary condition (or the twisted boundary condition) is a very interesting quantum system [1, 2, 3, 4]. The most important observation in the paper is that the contribution of the inhomogeneous term for the ground state, in the gapless region, can be neglected when the system-size N tends to infinity. Such a fact has been confirmed recently by the studies of other integrable models [19, 20, 21, 22] whose eigenvalue of the transfer matrix is given in terms of the inhomogeneous T − Q relation. We propose a method to study the thermodynamic limit of the XXZ spin chain with the twisted boundary condition at the antiferromagnetic region (i.e., η being a real number). Some supporting detailed calculations are given in Appendices A&B
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