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Journal of the Physical Society of Japan

Dynamics and Statistical Mechanics of the Classical Ising Model in a Transverse Field

Shigeo Homma, Shôzô Takeno

Published Nov 15, 1980• 5 citations
Publisher

Abstract

Dynamics and the approximate thermodynamical properties of the classical Ising model in a transverse field (CIMTF) are studied. This is done by elucidating a close correspondence of the CIMTF to a nonlinear lattice field in which each atom, while coupled harmonically with its neighbours, sits on on-site anharmonic potential. It is shown that when the ratio of the Ising interaction to a transverse field-strength exceeds a certain critical value, the on-site potential is of double-well type with saturable nonlinearity. Domain wall modes and magnon modes in continuous one-dimensional CIMTF are studied for both systems to substantiate the correspondence. By exploiting this, a brief study is made on the approximate properties of the free energy and of the dynamical structure factor S ( k ,ω) of longitudinal spin fluctuations of the CIMTF. These quantities are shown to be composed of two contributions, one from domain-wall modes and the other from magnon modes, which manifest themselves as a central peak and side peaks in S ( k ,ω), respectively.

DOI
10.1143/jpsj.49.1671

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Dynamics and the approximate thermodynamical properties of the classical Ising model in a transverse field (CIMTF) are studied. This is done by elucidating a close correspondence of the CIMTF to a nonlinear lattice field in which each atom, while coupled harmonically with its neighbours, sits on on-site anharmonic potential. It is shown that when the ratio of the Ising interaction to a transverse field-strength exceeds a certain critical value, the on-site potential is of double-well type with saturable nonlinearity. Domain wall modes and magnon modes in continuous one-dimensional CIMTF are studied for both systems to substantiate the correspondence. By exploiting this, a brief study is made on the approximate properties of the free energy and of the dynamical structure factor S ( k ,ω) of longitudinal spin fluctuations of the CIMTF. These quantities are shown to be composed of two contributions, one from domain-wall modes and the other from magnon modes, which manifest themselves as a central peak and side peaks in S ( k ,ω), respectively.
The molecular-dynamics technique is applied to the study of the classical ($S=\ensuremath{\infty}$) Ising model in a transverse field on a simple-cubic lattice. The results for the longitudinal- and transverse-relaxation shape function are presented for a set of temperatures in the disordered phase at one value of the transverse field. The longitudinal spectra at $T=\ensuremath{\infty}$ and in the critical regime are analyzed via the Mori's continued-fraction expansion and compared with various approximate results within this approach. The relation of the observed central mode to a domain-wall motion is also discussed. However, due to heavy-damping phenomena no well-defined propagating modes are observed in the critical region. Finally, transverse correlation functions are tested against their diffusive character.
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Dynamics and statistical mechanics of kinks in one-dimensional classical Ising model in a transverse field is studied in a dilute kink-gas approximation along the line with the theory of Krumhansl and Schrieffer. The energy of a single kink is written in a simple analytic form by approximating an exact one-kink solution. Kinks in the system are then treated as gas of quasi-particles. It is shown that the partition function so obtained is coincident with that of the result obtained previously by using the transfermatrix method. The dynamical structure of the system is also calculated to show the existence of central peak due to kink excitations.
We investigate analytically and numerically the dynamics of domain walls in a spin chain with ferromagnetic Ising interaction and subject to an external magnetic field perpendicular to the easy magnetization axis (transverse field Ising model). The analytical results obtained within the continuum approximation and numerical simulations performed for the discrete classical model are used to analyze the quantum properties of domain walls using the semiclassical approximation. We show that the domain wall spectrum shows a band structure consisting of 2S2S2S nonintersecting zones.
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.