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Physical review. B, Condensed matter

Transverse dynamic susceptibility for an Ising spin system: An exact eigenvalue equation for the resonance frequencies

G. G. Cabrera, Alzira Cristina de Mello Stein-Barana

Published Jan 1, 1984• 7 citations

Abstract

The dynamics of an Ising system is studied assuming that a microwave radiation field is coupled to the magnetic system through the transverse spin components. The transverse dynamic susceptibility is calculated with the use of the Green-function formalism in an exact manner, without the use of any decoupling scheme. The method yields an exact eigenvalue equation for the resonance frequencies, valid for any dimension and lattice, and a physical meaning of these resonances can be given in terms of the low-lying excitations for an Ising system. Those excitations correspond to spin-flip transitions in elementary clusters which represent all the possible spin arrangements of nearest neighbors. The present method does not allow for a calculation of line intensities without the help of statistical mechanics to evaluate correlation functions.

DOI: 10.1103/physrevb.29.433

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The dynamics of an Ising system is studied assuming that a microwave radiation field is coupled to the magnetic system through the transverse spin components. The transverse dynamic susceptibility is calculated with the use of the Green-function formalism in an exact manner, without the use of any decoupling scheme. The method yields an exact eigenvalue equation for the resonance frequencies, valid for any dimension and lattice, and a physical meaning of these resonances can be given in terms of the low-lying excitations for an Ising system. Those excitations correspond to spin-flip transitions in elementary clusters which represent all the possible spin arrangements of nearest neighbors. The present method does not allow for a calculation of line intensities without the help of statistical mechanics to evaluate correlation functions.

Fisher has found exactly for the square and honeycomb lattice the static initial perpendicular susceptibility of the quantum-mechanical version of the Ising model defined by the Hamiltonian H=−J ∑ nnσizσjz−mHx(t) ∑ iσix,where all symbols have their usual meanings. As the magnetization, Mx = mΣiσix, does not commute with the Hamiltonian, the system can relax and Kubo's linear response theory can be applied. We have been able to evaluate the response function exactly and hence obtain simply the exact frequency-dependent initial perpendicular susceptibility for the honeycomb lattice Jχ⊥(ω,T)/Nm2=¾[〈σ0zσ1z〉−〈σ0zσ1zσ2zσ3z〉]ω02/(ω02−ω2)+3[¾〈σ0zσ1z〉+¼〈σ0zσ1zσ2zσ3z〉]ω02/(9ω02−ω2),where ω0 = 2J/ℏ and 〈σ0zσ1z〉 is the nearest-neighbor correlation evaluated by Onsager. A similar expression is obtained for the square lattice. For the four-particle correlation 〈σ0zσ1zσ2zσ3z〉 the other three spins are nearest neighbors to spin zero, and we have obtained its value from Fisher's transformation theory. The behavior of χ⊥ as a function of T and ω is described and its physical significance is discussed.

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 the two-time temperature-dependent Green's function has been used to analyze the Ising model of a ferromagnet in an external magnetic field. The selection of a particular Green's function enables us to write an exact expression for the equation of motion. We are then led to a differential difference equation for the correlation function corresponding to the Green's function. No decoupling assumptions have been made, so the equation is exact for both arbitrary spin and range of interaction. It is shown how various approximate theories may be extracted from our formalism. The exact differential difference equation may be reduced to a partial differential equation. The latter form allows us to generate relations among the magnetization and spin-spin correlation functions. These relations are given in detail for the case of spin \textonehalf{} and

z

nearest neighbors.