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Spin stiffness calculation in anisotropic XY model with Ring exchange interaction

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 $η+ λ>0 $, where $η= J^{\prime}/J$ and $λ= K/J$. We further show that anisotropy ($η=2$) decreases the spin stiffness by 5.7% of its isotropic ($η=1$) maximum value for some values of $λ$ and $δ= J_D/J$. A similar reduction shows up in the magnetization. The plot of the stiffness against $η(λ)$ reaches a maximum at $η=0(λ=0)$ for specific values of $δ$ and decreases rapidly as it approaches $η+ λ+ 1=0$. In general, the supersolid phase transition suggested earlier\cite{H} will occur in the regime $η+ λ+1 <0$.