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Physical review. B, Condensed matter

Functional integral for the molecular polaron with q2 coupling

J. Florencio, Bernard Albert Goodman

Published Sep 15, 1986• 6 citations
Publisher

Abstract

The density matrix of a two-state electronic system coupled quadratically to an oscillator is expressed as functional integrals over commuting and anticommuting (Grassmann) elements satisfying one-sided boundary conditions. The functional integral over the oscillator coordinates is evaluated in closed form, and the resultant electronic integral which contains effective retarded interactions is evaluated in the semiclassical approximation. In the case of physical interest, N=1, we find that the semiclassical approximation already incorporates the essential physical features of the exact solution, for all values of the interaction strength and of the temperature.

DOI
10.1103/physrevb.34.3639

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Physical Review•Aug 15, 1962•401 citations
The density matrix of a two-state electronic system coupled quadratically to an oscillator is expressed as functional integrals over commuting and anticommuting (Grassmann) elements satisfying one-sided boundary conditions. The functional integral over the oscillator coordinates is evaluated in closed form, and the resultant electronic integral which contains effective retarded interactions is evaluated in the semiclassical approximation. In the case of physical interest, N=1, we find that the semiclassical approximation already incorporates the essential physical features of the exact solution, for all values of the interaction strength and of the temperature.
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.
We use a functional-integral technique to evaluate all the terms of the semiclassical expansion series for the density matrix of a mean-field two-level system coupled to an oscillator with q2{\mathit{q}}^{2}q2 interaction. Our calculation reproduces the exact result found by direct diagonalization. We also show that our theory yields a lower free energy than that obtained by using the adiabatic approximation.
Numerical evaluation of functional integrals usually involves a finite ( L -slice) discretization of the imaginary-time axis. In the auxiliary-field method, the L -slice approximant to the density matrix can be evaluated as a function of inverse temperature at any finite L as L (β) = [ 1 (β/ L )] L , if the density matrix 1 (β) in the static approximation is known. We investigate the convergence of the partition function Z L (β) ≡ Tr L (β), the internal energy and the density of states g L ( E ) (the inverse Laplace transform of Z L ), as L → ∞ . For the simple harmonic oscillator, g L ( E ) is a normalized truncated Fourier series for the exact density of states. When the auxiliary-field approach is applied to spin systems, approximants to the density of states and heat capacity can be negative. Approximants to the density matrix for a spin-½ dimer are found in closed form for all L by appending a self-interaction to the divergent Gaussian integral and analytically continuing to zero self-interaction. Because of this continuation, the coefficient of the singlet projector in the approximate density matrix can be negative. For a spin dimer, Z L is an even function of the coupling constant for L < 3: ferromagnetic and antiferromagnetic coupling can be distinguished only for L ⩾3, where a Berry phase appears in the functional integral. At any non-zero temperature, the exact partition function is recovered as L → ∞ .
We use a one-dimensional tight binding model with an impurity site characterized by electron-vibration coupling, to describe electron transfer and localization at zero temperature, aiming to examine the process of polaron formation in this system. In particular we focus on comparing a semiclassical approach that describes nuclear motion in this many vibronic-states system on the Ehrenfest dynamics level to a numerically exact fully quantum calculation based on the Bonca-Trugman method [J. Bonča and S. A. Trugman, Phys. Rev. Lett. 75, 2566 (1995)]. In both approaches, thermal relaxation in the nuclear subspace is implemented in equivalent approximate ways: In the Ehrenfest calculation the uncoupled (to the electronic subsystem) motion of the classical (harmonic) oscillator is simply damped as would be implied by coupling to a Markovian zero temperature bath. In the quantum calculation, thermal relaxation is implemented by augmenting the Liouville equation for the oscillator density matrix with kinetic terms that account for the same relaxation. In both cases we calculate the probability to trap the electron by forming a polaron and the probability that it escapes to infinity. Comparing these calculations, we find that while both result in similar long time yields for these processes, the Ehrenfest-dynamics based calculation fails to account for the correct time scale for the polaron formation. This failure results, as usual, from the fact that at the early stage of polaron formation the classical nuclear dynamics takes place on an unphysical average potential surface that reflects the distributed electronic population in the system, while the quantum calculation accounts fully for correlations between the electronic and vibrational subsystems.
We have obtained an approximate expression for the impedance function at all frequencies, temperatures, and coupling strengths of an electron coupled to a polar lattice (a system commonly called a polaron). The starting point for the calculation is the quantum mechanical expression for the expected current. The phonon coordinates are eliminated from this expression by well-known field-theory techniques. The resulting exact "influence functional" is then approximated by a corresponding quadratic "influence functional" which, it is hoped, imitates the real polaron. Correction terms are computed to account for the difference between the approximate impedance and the exact polaron impedance in a manner closely analogous to Feynman's treatment of the polaron self-energy. In fact, the analytic evaluation of the expression for the impedance obtained here is carried out using the approximate "influence functional" that was successfully employed in minimizing the binding (and free) energy of the polaron in earlier calculations. However, the accuracy obtained using this approximation, for the impedance calculation, is less satisfactory and its limitations are discussed. Nevertheless, beginning at intermediate coupling strengths, the approximate impedance produces a level structure of increasing complexity and narrowing resonances as the coupling strengthens. This suggests that further refinements may be fruitful. Methods for finding a better quadratic influence functional for use in our impedance expression as well as ways of improving the expression further are suggested. A comparison of our results with those of the Boltzmann equation points up interesting differences which arise from reversing the order of taking limits of zero frequency and coupling.