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Journal of the Physical Society of Japan

Nonlinear Excitations and Coherent States in the Ising Model in a Transverse Field

Shôzô Takeno

Published Apr 1, 1980• 9 citations
Publisher

Abstract

Nonlinear excitations in the Ising model in a transverse field or an equivalent Frenkel-exciton model for a system of two-level atoms are studied by using a method of diagonal coherent-state representation. It is shown that if the ratio of the Ising interaction to a transverse-field strength exceeds a certain critical value, stationary coherent states giving the ground state and excited states of the model Hamiltonian are characterized by two degenerate structures resulting from spatially uniform coherent states and static domain-wall type excitations, respectively. Particular solutions for the time evolution of spin operators corresponding to atomic polarization are then studied. It is shown that moving domain-wall solutions or moving solitary-wave solutions can exist under certain conditions.

DOI
10.1143/jpsj.48.1075

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Journal of the Physical Society of Japan•Sep 15, 1994•11 citations
Nonlinear excitations in the Ising model in a transverse field or an equivalent Frenkel-exciton model for a system of two-level atoms are studied by using a method of diagonal coherent-state representation. It is shown that if the ratio of the Ising interaction to a transverse-field strength exceeds a certain critical value, stationary coherent states giving the ground state and excited states of the model Hamiltonian are characterized by two degenerate structures resulting from spatially uniform coherent states and static domain-wall type excitations, respectively. Particular solutions for the time evolution of spin operators corresponding to atomic polarization are then studied. It is shown that moving domain-wall solutions or moving solitary-wave solutions can exist under certain conditions.
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.
A quantum theory of nonlinear excitations in a matter-photon system is developed by generalizing the conventional two-level-atom model of a matter in several directions, on the assumption that the overlapping of atomic wave functions on different atomic sites is negligible.The model matter Hamiltonian is shown to be expressible in a generalized form' of the Heisenberg model Hamiltonian with anisotropic long-range exchange interactions with spin operators replaced by quasi-Pauli operators.From approxirpate formal solutions of a set of nonlinear equations satisfied by the quasi-Pauli operators ~nd photon operators, brief discussion is given on nonlinear Frenkel excitons and nonlinear exciton-photon coupled modes.An attempt is made to formulate a theory of higher-order nonlinear physical properties of the matter-photon system by including interactions of magnetic as •well as electric dipoles of atoms with photons.A brief discussion is also given on the coherence properties. of the matter-photon system which are characterized by the simultaneo~s existence of •a photon coherent state and an atomic-polarization coherent state.*> Throughout this paper, we will often use a simplified .notation l, rather than l 1, whenever appropriate, to avoid the use of complicated symbols.Also, the complex conjugate of a quantity• A is denoted by A*.
Interactions of nonlinear Frenkel excitons with an intense external radiation field (rf) are studied by taking the atomic polarization, the atomic population difference and the electric field as a set of variables to study the coherence and nonlinearity properties of the exciton-rf system. To equations for these quantities, terms are added phenomenologicaly to take care of pumping and dissipative effects. Two kinds of solutions are obtained under certain conditions, one representing the existence of the stationary coherent polarization and the macroscopic number of photons with wave number equal to that of the excitons and another envelope-kink-like, steady exciton-rf coupled modes around the stationary coherent states
A formal, regorous theory of Frenkel excitons interacting with an external electromagnetic field is doveloped within the framework of a two-level-atom model in which the dipole approximation is employed in describing the atom-field (taken as a c-number) interaction. The model exciton Hamiltonian is shown to be identical in form with the Heisenberg model Hamiltonian for S=1/2 spins with anisotropic long-range exchange interactions. Instead of treating excitons as a non-ideal Bose gas, atomic polarization operators and level population defference are taken up as two fundamental quantities to describe Frenkel excitons. A pair of equations satisfied by these two quantities are derived. It is shown that Frenkel excitons are generally expressed as quantum-mechanical nonlinear polarization waves, the nonlinearity being characterized by the level population. A discussion is given on the interrelationship between such polarization waves and those derivable from the classical Drude-Lorentz model. With the aid of several approximation procedures similar to those employed in nonlinear optics, a set of equations describing nonlinear exciton-photon coupled modes are derived. Brief discussion is given on the eigenfrequencies of nonlinear Frenkel excitons and nonlinear polaritons. The existence of nonlinear exciton-photon coupled modes whose properties are different from those of polariton modes is suggested.
A variational calculation yields an eigenstate representing a domain wall-type solution for various quantum-mechanical spin models. Heisenberg Hamiltonians with local and exchange anisotropy, and an Ising model in a transverse field (including its critical regime) are discussed in detail. The domain walls are viewed as localized solutions of non-linear equations. The extent to which the results for the transverse field Ising model may be mapped on to a currently interesting “double well” anhurmonic lattice problem are analysed.
We study effects of a transverse magnetic field on the properties of the one-dimensional spin-1/2 Ising-like antiferromagnet. In order to emphasize the relevant role of domain walls between opposite antiferromagnetic domains, the spin Hamiltonian is exactly mapped to the interacting fermion Hamiltonian, via the dual transformation as well as the Jordan-Wigner transformation. Within the generalized Hartree-Fock approximation, we find that the excitation spectrum exhibits characteristic features, which are explained as a result of the different roles of the transverse field and of the transverse components of the exchange interactions in the propagation of the domain wall. At a critical field where the excitation gap vanishes, a ground-state phase transition is found to take place in the same manner as that in the transverse Ising model. In addition, we discuss the relevance of our magnetization curve with regard to experimental data in CsCoCl 3 .