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arXiv:cond-mat/0011043 · International Journal of Modern Physics B

AB INITIO TREATMENTS OF THE ISING MODEL IN A TRANSVERSE FIELD

R. F. Bishop, DAMIAN J. J. FARNELL, MANFRED L. RISTIG

Published Jun 20, 2000• 10 citations
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Abstract

In this article new results are presented for the zero-temperature ground-state properties of the spin-half transverse Ising model on various lattices using three different approximate techniques. These are, respectively, the coupled cluster method, the correlated basis function method, and the variational quantum Monte Carlo method. The methods, at different levels of approximation, are used to study the ground-state properties of these systems, and the results are found to be in excellent agreement both with each other and with results of exact calculations for the linear chain and results of exact cumulant series expansions for lattices of higher spatial dimension. The different techniques used are compared and contrasted in the light of these results, and the constructions of the approximate ground- state wave functions are especially discussed.

DOI
10.1142/s0217979200001618

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In this article new results are presented for the zero-temperature ground-state properties of the spin-half transverse Ising model on various lattices using three different approximate techniques. These are, respectively, the coupled cluster method, the correlated basis function method, and the variational quantum Monte Carlo method. The methods, at different levels of approximation, are used to study the ground-state properties of these systems, and the results are found to be in excellent agreement both with each other and with results of exact calculations for the linear chain and results of exact cumulant series expansions for lattices of higher spatial dimension. The different techniques used are compared and contrasted in the light of these results, and the constructions of the approximate ground- state wave functions are especially discussed.
In this article, the correlated basis-function (CBF) method is applied for the first time to the quantum spin-half XY model on the linear chain, the square lattice, and the simple cubic lattice. In this treatment of the quantum spin-half XY model a Jastrow ansatz is utilised to approximate the ground-state wave function. Results for the ground-state energy and the sublattice magnetisation are presented, and evidence that the CBF detects the quantum phase transition point in this model is also presented. The CBF results are compared to previous coupled cluster method (CCM) results for the spin-half XY model, and the two formalisms are then compared and contrasted. In this paper we consider the T = 0 properties of the quantum spin system known as the spin-half XY model, described by the Hamiltonian
A newly developed zero-temperature real-space renormalization-group method is applied to the quantum Ising model with a transverse field on different two-dimensional lattices: hexagonal, triangular, and quadratic. The order parameter, spin-spin correlation functions, the critical fields, and the critical exponents describing the low-field-large-field transition are calculated. The influence of size and shape of the spin blocks used in the iterative process is discussed. The results for triangular and hexagonal lattices are new; for the quadratic lattice the calculations show an encouraging agreement with previous series-expansion results. The expected equivalence to the critical behavior of the three-dimensional Ising model is discussed: A reasonable agreement is obtained for the hexagonal lattice.
In this article, the correlated basis-function (CBF) method is applied for the first time to the quantum spin-half {\it XY} model on the linear chain, the square lattice, and the simple cubic lattice. In this treatment of the quantum spin-half {\it XY} model a Jastrow ansatz is utilised to approximate the ground-state wave function. Results for the ground-state energy and the sublattice magnetisation are presented, and evidence that the CBF detects the quantum phase transition point in this model is also presented. The CBF results are compared to previous coupled cluster method (CCM) results for the spin-half {\it XY} model, and the two formalisms are then compared and contrasted.
The method of correlated basis functions (CBF) is applied at the variational level to the transverse Ising model in two and three spatial dimensions (D=2,3). The model consists of Pauli spins arranged on a simple square or cubic lattice, experiencing nearest-neighbor interactions through their x components and subject to a transverse field in the z direction of strength λ. Working at zero temperature, full optimization of a Hartree-Jastrow trial wave function is performed by solving two Euler-Lagrange equations, namely a renormalized Hartree equation for the order parameter characterizing the ferromagnetic phase and a paired-magnon equation for the optimal two-spin spatial distribution function. The optimized trial wave function yields a second-order transition with a numerically determined critical coupling of λc=3.14(D=2) or λc= 5.10(D=3). Numerical results have been obtained for the magnetization order parameter, the energy per spin and its potential component, the static structure function at zero wave number, and the magnon energy gap corresponding to a Feynman description of the elementary excitations. Correlated density matrix theory provides for a natural extension of this approach to finite temperature.
We have studied an Ising spin system in a transverse field, at zero temperature, under a time oscillating longitudinal field by means of a mean-field approximation and a Monte Carlo algorithm, appropriate to study the ground-state properties of quantum spin chains. For large values of the transverse field Γ or large amplitude h0 of the oscillating field, the magnetization oscillates around a zero value with the same frequency but delayed. Decreasing Γ, at small a h0, the system undergoes a phase transition to a state in which the magnetization oscillates around a nonzero value. We have also obtained the behavior of the hysteresis loop area near the Curie point.
We derive the moments, at $T=\ensuremath{\infty}$, of the longitudinal spin-spin correlation function of the Ising model in a transverse field. The first 20 moments are obtained for the linear chain, the first 12 moments for the fcc lattice. We use a method of Nickel [J. Phys. C 7, 1719 (1914)] to construct the relaxation function and find excellent agreement with the exact results of Capel and Perk [Physica (Utrecht) 87A, 211 (1977)] in one dimension and extremely good convergence in three dimensions. Our method of analysis provides much better convergence than methods based on truncation of the continued-fraction representation.