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Acta Physica Polonica A

SU(2) Coherent State Path Integral Investigation of the Holstein Dimer

Tugrul Hakioglu, V.A. Ivanov, M. Ye. Zhuravlev

Published May 1, 2000• 0 citations
Publisher

Abstract

The SU(2) coherent state path integral is used to investigate the partition function of the Holstein dimer.This approach naturally takes into account the dynamical symmetry of the model.The ground-state energy and the number of the phonons are calculated as functions of the parameters of the Hamiltonian.The renormalizations of the phonon frequency and electron hopping for bonding and antibonding states are considered.The destruction of quasiclassical mean field solution is discussed.

DOI
10.12693/aphyspola.97.927

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The SU(2) coherent state path integral is used to investigate the partition function of the Holstein dimer.This approach naturally takes into account the dynamical symmetry of the model.The ground-state energy and the number of the phonons are calculated as functions of the parameters of the Hamiltonian.The renormalizations of the phonon frequency and electron hopping for bonding and antibonding states are considered.The destruction of quasiclassical mean field solution is discussed.
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A variational approach is developed to study the groundstate (GS) of the two-site Holstein model. By the extended coherent state, where the more phonon correlations are easily incorporated, we can get the very accurate ground state energy for all electron-phonon coupling range in typical values of hopping integral t = 0.5, 1.1, and 2.1 (in units of phonon frequency ωo ), which covers the crossover region from antiadiabatic limit to the adiabatic limit. Within a very wide t range [0, 2.7], the exact results for the GS energy are obtained with the twelfth (fourteenth) order corrections to the zeroth order wave function. Moreover, the present approach is more concise than any other analytical ones in this field, and hopefully can be easily generalized to many other Holstein models.
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 SU(2) coherent-state path integral is used to represent the matrix element of a propagator in the SU(2) coherent-state basis. It is argued that the continuum representation of this integral is correct provided the necessary boundary term is taken into account. In the case of the SU(2) dynamical symmetry the path integral is explicitly computed by means of a change of variables, the SU(2) motion of the underlying phase space. The correct stationary-phase expansion for the propagator in terms of the total action including boundary term and classical trajectories is obtained.
The thermodynamics of a two-level system coupled to a boson field is investigated by means of a functional-integral method. The model can be used to describe the motion of a small polaron between two sites on a molecule, or a spin-1/2 magnetic moment coupled to a phonon bath. The model is also relevant to the problem of dissipation in macroscopic quantum phenomena. We use a functional-integral method to investigate a two-level electronic system coupled linearly to a harmonic oscillator. The density matrix is cast as a functional integral over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kB{\mathit{k}}_{\mathit{B}}kB​T. The integration variables are c numbers for the oscillator and Grassmann numbers for the electrons. The oscillator variables are integrated out, leaving an effective density matrix for the electrons containing retarded interactions. The partition function is then calculated up to second order of an expansion about the exact fermionic classical paths. Then, the free energy, specific heat, and average hopping energy are obtained. We find that the most significant reduction in the hopping energy is manifested at intermediate temperatures.