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Journal of the Physical Society of Japan

Spin Coherent State Representation in Non-Equilibrium Statistical Mechanics

Yoshinori Takahashi, Fumiaki Shibata

Published Mar 1, 1975• 74 citations
Publisher

Abstract

A new method of treating problems of spin systems is developed, in which spin operators are transformed into two kinds of boson operators according to Schwinger's method. Usual treatment of coherent state representation is generalized to spin systems with a fixed magnitude S . Some basic properties and the relations to the atomic coherent state are investigated rather in details. The method is applied to spin relaxation process and a corresponding Fokker-Planck type equation is obtained and solved to give a multiple relaxation process due to nonlinearities. It is shown that our formulation has several advantages over other ones in some respects.

DOI
10.1143/jpsj.38.656

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A new method of treating problems of spin systems is developed, in which spin operators are transformed into two kinds of boson operators according to Schwinger's method. Usual treatment of coherent state representation is generalized to spin systems with a fixed magnitude S . Some basic properties and the relations to the atomic coherent state are investigated rather in details. The method is applied to spin relaxation process and a corresponding Fokker-Planck type equation is obtained and solved to give a multiple relaxation process due to nonlinearities. It is shown that our formulation has several advantages over other ones in some respects.
Quantum statistical mechanics of spin systems is formulated by the use of the representation of atomic coherent state of spin, which is specified by a complex number. After introducing a newly defined distribution function of spin, the Bloch equation for the equilibrium density matrix, the von Neumann equation for the nonequilibrium density matrix and the Heisenberg equation of motion are converted into partial differential equations of the type of separating variables respectively. Formal solutions of them are presented. Any operator which is composed of spins can be converted uniquely into a differential operator. Thus we succeed in formulating equilibrium and nonequilibrium quantum statistical mechanics in the complex variable space, which differs from the previous one in the diagonal representation of density matrix. We further introduce the interaction representation for the perturbational expansion. The linear response formula is written down in the present representation.
We summarize some crucial results from mathematical models that help us understand the relevance of gener-alized coherent states (GCS), a fundamental method used for the description of non-linear problems in physics. We mainly concentrate our review in the following models: ferromagnetism, nonlinear Schr¨odinger equation with external potential, nonlinear quantum oscillators, Bose - Einstein condensation (BEC) and the DNA quasi-spin model. Such models and some applications of the wide variety these involve are outlined. As variational trial states, the coherent states (CS) allow the estimation of the ground state energies and properties, yielding results which become exact in a number of nonlinear differential equations for dynamical variable of each model. Mainly we concentrate our review on two types of coherent states, The first one are based on the Heisenberg - Weyl group where it is applied the bosonization effect. The second one is based on the SU(2)/U(1) Lie group. For this case when the Hamiltonian and all physical operators are constructed by the elements of a Lie algebra, the generalized coherent states approach is directly applied without bozonization.
A new method for describing the two-level pairing model in a many-fermion system is presented. The basic idea comes from the Schwinger boson representation for the su(2) ⊗ su(2)-algebra, in which four kinds of boson operators govern the dynamics. It is well known that in the original fermion space, it is almost impossible to introduce any trial state for the variation except the BCS state in the framework of the mean field approximation. However, in the Schwinger boson representation, various trial states can be defined on the condition that the mean field approximation is useful. In these states, the su(2) ⊗ su(1,1)-coherent state is mainly discussed in this paper.
We consider a spin coherent states description of a general quantum spin system. It is shown that it is possible to use the spin-1/2 representation to study the general spin-J case. We identify the 1/2 spinor components as the homogeneous coordinates of the projective space associated to the complex variable that labels the coherent states and establish a relation between the two-component spinor and the bosonic Schwinger representation of a spin operator. We rewrite the equations of motion, obtained from the path integral for the evolution operator or partition function, in terms of the 1/2 spinor and define the effective Hamiltonian of its evolution.
By introducing the holomorphic representation we obtain, in closed form, relations between the c-number functions, which represent an operator in spin-J space, without recurring to expansions in spherical harmonics. In particular, we obtain the diagonal representative from the diagonal matrix element through an integral relation. New analogies between these and the spin (Bloch) coherent states are established which avoid the introduction of Schwinger bosons, and the consequent projection to the physical subspace, by introducing a generalized Fourier transformation in spin-J space in a way similar to the treatment by Cahill and Glauber (1969) for boson coherent states. Also, we study the generalization to the spin case of the s-ordering of Cahill and Glauber.
Multipartite generalizations of spin coherent states are introduced and analyzed. These are the spin analogs of multimode optical coherent states as used in continuous variable quantum information, but generalized to possess full spin symmetry. Two possible generalizations are given, one of which is a simple tensor product of a given multipartite quantum state. The second generalization uses the bosonic formulation in the Jordan-Schwinger map, which we call spinor states. In the unipartite case, spinor states are equivalent to spin coherent states; however, in the multipartite case, they are no longer equivalent. Some fundamental properties of these states are discussed, such as their observables and covariances with respect to symmetric operators, form-preserving transformations, and entanglement. We discuss the utility of such multipartite spin coherent and spinor states as a way of storing quantum information.