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International Journal of Modern Physics B

THE TRANSVERSE ISING MODEL BY CBF

J. W. Clark, MANFRED L. RISTIG, J. W. Kim

Published Mar 10, 1999• 1 citation
Publisher

Abstract

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.

DOI
10.1142/s0217979299000631

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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.
The method of correlated basis functions is applied at the variational level to the Ising model consisting of Pauli spins arranged on a simple cubic lattice, experiencing nearest-neighbor interactions through their xxx components and subject to a transverse field in the zzz direction of strength $\ensuremath{\lambda}$. Full optimization of a Hartree-Jastrow trial wave function is performed by solving two Euler-Lagrange equations: 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 ${\ensuremath{\lambda}}_{c}=5.10$. Explicit results are presented for (i) the magnetization order parameter, (ii) the energy per spin and its potential component, (iii) the static structure function at zero wave number, (iv) the spin-exchange strength, and (v) the magnon energy gap corresponding to a Feynman description of the elementary excitations.
Correlated-basis-function (CBF) theory, which has provided a firm foundation for ab initio calculations of the properties of quantum fluids such as liquid 4He^{4}\mathrm{He}4He, 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.
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.
The thermodynamic properties of the spin S=3/2 ferromagnetic Ising model in the presence of transverse and longitudinal crystal fields (equivalent to the Blume-Capel model with a transverse crystal field) have been studied by using two different approaches: (i) a zero-temperature mapping of the system onto a spin-1/2 quantum Ising model in longitudinal and transverse fields, together with time-independent quantum perturbation theory; and (ii) a standard mean-field approximation within the framework of the Bogoliubov inequality for the free energy. A very rich phase diagram, with different kinds of multicritical behavior, has been obtained. The results show first- and second-order transition lines, tricritical and tetracritical points, critical end points with a two-phase coexistence, double critical end points, and also double noncritical end points. Additionally, the behavior of the magnetization as a function of temperature, over a wide range of values of both longitudinal and transverse crystal fields, has also been analyzed in detail. While large magnitudes of the longitudinal crystal field select the z-spin components either in their states ±3/2 or ±1/2, it is surprising that a large transverse crystal field induces the spin component in the z direction to values ±1, which are completely different from any expected natural component. This comes indeed as a result of the zero-temperature mapping of the ground state with the superposition of the states 3/2 and -1/2, in one sector of the Hilbert space, and the states -3/2 and 1/2 on the other disjoint sector of the Hilbert space. This superposition for a large transverse crystal field prevails even for finite temperatures, implying that the exact critical points are obtained for the model on the one-dimensional lattice and the two-dimensional square lattice, and quite accurate estimates can be achieved for the three-dimensional simple cubic lattice.
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
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.