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Journal of Physics A Mathematical and General

Critical properties of S= 1/2 antiferromagnetic XXZ chain with next-nearest-neighbour interactions

K. Nomura, Kazuki Okamoto

Published Sep 7, 1994• 207 citations
Publisher

Abstract

We investigate numerically the critical lines and the critical properties of the fluid-dimer and the Neel-dimer transitions of the S= 1/2 antiferromagnetic XXZ chain with next-nearest interactions, and we confirm that the universality class of this model belongs to the quantum sine-Gordon model, as is expected from the bosonization. The method which we use in this paper to calculate the critical lines is free from the logarithmic corrections on the Kosterlitz-Thouless (K-T)-type transition line, which have made the K-T critical point difficult to obtain. By the use of this method, it is possible to determine the K-T critical line with high precision from small size data, and to identify the universality class.

DOI
10.1088/0305-4470/27/17/012

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Positions of critical temperatures of antiferromagnetic Ising model with ferromagnetic next nearest neighbour interactions on the triangular lattice are investigated by making use of their properties of the Kosterlitz-Thouless type of phase transition. In particular, finite-size scaling method and Monte Carlo Renormalization Group methods (MCRG) with flow diagrams of the anomalous dimension η show the fixed line structure where the correlation length diverges. Another MCRG with flow diagram of the stiffness constant also shows a critical property clearly. The temperature dependence of the long range order is also discussed in order to locate the lower critical point.
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We calculate the quantum phase diagram of an XXZ chain with nearest-neighbor (NN) J1{J}_{1}J1​ and next-NN exchange J2{J}_{2}J2​ with anisotropies ${\ensuremath{\Delta}}_{1}$ and ${\ensuremath{\Delta}}_{2},$ respectively. In particular we consider the case ${\ensuremath{\Delta}}_{1}=\ensuremath{-}{\ensuremath{\Delta}}_{2},$ to interpolate between the XX chain $({\ensuremath{\Delta}}_{i}=0)$ and the isotropic model with ferromagnetic phase J2.{J}_{2}.J2​. For ${\ensuremath{\Delta}}_{1}<\ensuremath{-}1,$ a ferromagnetic phase and two antiferromagnetic phases exist. For $|{\ensuremath{\Delta}}_{i}|<1,$ the boundary between the dimer and spin fluid phases is determined by the method of crossing of excitation spectra. For large J2/J1,{J}_{2}{/J}_{1},J2​/J1​, this method seems to indicate the existence of a second spin fluid critical phase. However, an analysis of the spin stiffness and magnetic susceptibility for ${\ensuremath{\Delta}}_{1}={\ensuremath{\Delta}}_{2}=1$ suggests that a small gap is present.
The ground state phase diagram of an XXZ chain with asymmetric second-neighbour interactions is studied. We first prove that this model has the doubly degenerate exact dimer ground state under some conditions. Then, by using the level spectroscopy method, we calculate the ground state phase diagram and find that the dependence of the dimer–fluid critical line on the asymmetric interactions is much stronger than those of the dimer–Neel critical line and the ferro-fluid first-order transition line. In order to understand this result, we analyse eigenvectors of the ground state and excitations characterizing each of phases, and show that only the doublet excitation can be used to detect the asymmetric interactions.