Spin stiffness calculation in anisotropic XY model with Ring exchange interaction

TL;DR

This paper develops a spin wave theory for an anisotropic XY model with ring exchange on a square lattice, analyzing thermodynamic properties and phase transitions influenced by anisotropy, exchange interactions, and ring exchange.

Contribution

It introduces a comprehensive spin wave analysis incorporating anisotropic interactions and ring exchange, revealing their effects on spin stiffness, magnetization, and phase transitions.

Findings

Anisotropy reduces spin stiffness and magnetization.

No soft modes develop when η + λ > 0.

Phase transition occurs when η + λ + 1 < 0.

Abstract

We present the spin wave theory of XY model with anisotropic nearest neighbour (NN) interactions $J(J^{\prime})$ along the $x(y)$ directions, next nearest (NNN) neighbour interaction $J_D$ and the ring exchange interaction $K$ on the square lattice. We calculate the thermodynamic quantities: Zero temperature spin stiffness, internal energy, specific heat and the magnetization. Using the diagonalized Hamiltonian, we show that no soft modes develop when $\eta + \lambda >0 $, where $\eta = J^{\prime}/J$ and $\lambda = K/J$. We further show that anisotropy ($\eta =2$) decreases the spin stiffness by 5.7% of its isotropic ($\eta =1$) maximum value for some values of $\lambda$ and $\delta = J_D/J$. A similar reduction shows up in the magnetization. The plot of the stiffness against $\eta (\lambda)$ reaches a maximum at $\eta=0(\lambda =0)$ for specific values of $\delta$ and decreases rapidly as it approaches $\eta + \lambda + 1=0$. In general, the supersolid phase transition suggested earlier\cite{H} will occur in the regime $\eta + \lambda +1 <0$.