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Physica Scripta

Coherent state path integral for the interaction of spin with external magnetic fields

Tahar Boudjedaa, Abdelhafid Bounames, Khireddine Nouicer, L. Chétouani, Théophile F. Hammann

Published Dec 1, 1997• 9 citations
Publisher

Abstract

Linear canonical transformation and time transformation are used within the formalism of path integrals in coherent state representation to solve the problem of an interacting spin with arbitrary time-dependent external magnetic fields. The Factorization technique is used to solve the equation of motion. Examples of the calculation of the transition probability in terms of elementary functions and special functions for solvable systems are presented. For completeness Berry's phase is also derived under adiabatic expansion.

DOI
10.1088/0031-8949/56/6/001

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Linear canonical transformation and time transformation are used within the formalism of path integrals in coherent state representation to solve the problem of an interacting spin with arbitrary time-dependent external magnetic fields. The Factorization technique is used to solve the equation of motion. Examples of the calculation of the transition probability in terms of elementary functions and special functions for solvable systems are presented. For completeness Berry's phase is also derived under adiabatic expansion.
Using the coherent states representation and the Schwinger boson model, the propagator of a spinning particle in interaction with an arbitrary magnetic field is derived in the framework of the path integral formalism. To show the relevance of the formalism to some applications, the Green function of spinning particle s=1/2 in special configuration of magnetic field is calculated. The transition amplitude and Berry’s phase for an arbitrary spin and time dependent magnetic field are deduced. The case of half integer spin is also considered.
Definition and formulas for harmonic oscillator coherent states and spin coherent states are reviewed in detail. The path integral formalism and its relation with the partition function of a system are also reviewed. The harmonic oscillator coherent state path integral is evaluated exactly at the discrete level and then used to find its continuum limit using various regularizations. The computation of the path integral for a particle of spin s put in a constant magnetic field is carried out using harmonic oscillator coherent states and spin coherent states, with a careful analysis of infinitesimal terms (in 1/N where N is the number of time slices) appearing in the Lagrangian. A mapping of the spin system into a CP1 model is shown explicitly. The theory of a spinless particle in the field of a magnetic monopole and its relation with the spin system are explained. The equivalence of these two models is established up to infinitesimal order by the introduction of an external field correction. This gives a new representation of a coherent state path integral in terms of a more familiar Feynman path integral.
The SU(2) coherent-state path-integral representation is used to derive a fairly explicit expression for the partition function Z0{\mathit{Z}}_{0}Z0​[\ensuremath{\psi}\ensuremath{\rightarrow}] of noninteracting quantum spins in the presence of an arbitrary time-dependent external field \ensuremath{\psi}\ensuremath{\rightarrow}. By performing then the standard Gaussian average of Z0{\mathit{Z}}_{0}Z0​[\ensuremath{\psi}\ensuremath{\rightarrow}] over \ensuremath{\psi}\ensuremath{\rightarrow} the partition function of the ferromagnetically coupled Heisenberg model is represented in a way which allows one to simplify the derivation of results such as obtained by Leibler and Orland [Ann. Phys. (N.Y.) 132, 277 (1981)] and to generalize them to arbitrary spin quantum number.
The spin depolarization of a quantum particle in random magnetic field is treated in terms of a path integral in the spin states as well as the translational motion. The path integral of the spin operator, represented in terms of Fermion operators, is performed in the Fermion coherent state representation. An expression for the spin depolarization is obtained for an arbitrary magnetic field distribution. A particular case, in which the magnetic field is oriented in the z -direction and the magnitude is Gaussianly distributed, is considered and the spin decays exponentially in time.