Correlated-basis-function analysis of the transverse Ising model

Abstract

Correlated-basis-function (CBF) theory, which has provided a firm foundation for ab initio calculations of the properties of quantum fluids such as liquid $^{4}\mathrm{He}$, is adapted and applied at the Jastrow-Feenberg variational level to give an optimized description of the structure and of the elementary excitations of the Ising-spin model in a transverse magnetic field. A set of trial wave functions of Hartree-Jastrow form is assumed to describe the spatial correlations present in the ordered as well as the disordered ground states, and the Feynman Ansatz is consistently adopted to represent the elementary magnon states. The CBF analysis of the spin system employs the hypernetted-chain (HNC) formalism for a substitutional binary mixture of bosons and derives HNC equations for the spatial distribution functions which determine the ground-state energy expectation value. Functional variation of this quantity with respect to the magnetic order parameter and the trial states leads to two Euler-Lagrange equations, which may be interpreted as a renormalized Hartree equation for the optimal magnetization and as a paired-magnon equation for the magnetic correlation function that is analogous to the familiar paired-phonon equation for conventional quantum fluids.Numerical calculations are based on simple cubic lattices and an optimized nearest-neighbor Ansatz for the generating pseudopotential. Results are reported on the order parameter, the energy per lattice site, the transverse magnetization, and the magnon excitation energies as functions of the coupling parameter 0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}\ensuremath{\infty} measuring the strength of the transverse magnetic field. We also present numerical results on the magnetic correlation function, the static structure function, and the correlation length. The system exhibits a second-order phase transition at a critical value ${\ensuremath{\lambda}}_{\mathit{c}}$ of the coupling strength. Our numerical calculations of the optimal order parameter yields ${\ensuremath{\lambda}}_{\mathit{c}}$\ensuremath{\simeq}5.17 for a simple cubic lattice and ${\ensuremath{\lambda}}_{\mathit{c}}$\ensuremath{\simeq}3.12 for a square planar lattice. The calculated data are in very good agreement with results derived from perturbation expansions in conjunction with Pad\'e techniques. \textcopyright{} 1996 The American Physical Society.