Zero-temperature renormalization-group method for quantum systems. IV. S=12XY model in a transverse field in two and three dimensions

Abstract

A zero-temperature real-space renormalization-group method for quantum systems is applied to the $S=\frac{1}{2}\mathrm{XY}$ model with a $Z$ field with the Hamiltonian $H=\ensuremath{-}\frac{1}{2}J\ensuremath{\Sigma}{〈i, j〉}^{}[(1+\ensuremath{\gamma}){S}_{i}^{x}{S}_{j}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{i}^{y}{S}_{j}^{y}]\ensuremath{-}h\ensuremath{\Sigma}{i}^{}{S}_{i}^{z}$ on quadratic, triangular, and hexagonal lattices in two dimensions (2D) and on a cubic lattice in 3D. In all the isotropic ($\ensuremath{\gamma}=0$) 2D cases the method gives no order in the ground state $|0〉$ but changes in the correlation functions at ${(\frac{h}{J})}_{c}=\frac{1}{2}{z}^{\ensuremath{'}}$ [${z}^{\ensuremath{'}}$ is the number of nearest neighbors (NN)] are observed. An argument is presented that at ${(\frac{h}{J})}_{c}$ the method becomes exact in the limit of infinite-size spin blocks. It is suggested that the correlation functions $〈0|{{S}_{0}}^{\ensuremath{\alpha}}{{S}_{R}}^{\ensuremath{\alpha}}|0〉$ ($\ensuremath{\alpha}=x, y$) for large $R$ have a power-law behavior $\ensuremath{\sim}{R}^{\ensuremath{-}\ensuremath{\eta}}$ for $0<~|\frac{h}{J}|<|{(\frac{h}{J})}_{c}|$ with $\ensuremath{\eta}\ensuremath{\approx}1.5$ for $|\frac{h}{J}|<{(\frac{h}{J})}_{c}|$ and $\ensuremath{\eta}=2$ at ${(\frac{h}{J})}_{c}$. The dynamical exponent $z$ is obtained to be \ensuremath{\approx} 0.3 for $|\frac{h}{J}|<\frac{1}{2}{z}^{\ensuremath{'}}$ and at ${(\frac{h}{J})}_{c}$ close to 2. An indirect method of calculating the ground-state energy is developed, and the results are compared with the known estimates. For $\ensuremath{\gamma}\ensuremath{\ne}0$ and $h=0$ the $\ensuremath{\gamma}$ dependences of the order in the ground state ${\ensuremath{\mu}}^{x}(\ensuremath{\gamma})$ is calculated to be ${\ensuremath{\mu}}^{x}\ensuremath{\sim}{\ensuremath{\gamma}}^{0.21}$ for $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ and $h=0$. In a 3D cubic lattice ($\ensuremath{\gamma}=0$) the analysis of recursion relations indicates the existence of order, in accord with recent exact theorems.