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Physical review. B, Condensed matter

Generating functional for electron-phonon systems from functional integrals with one-sided boundary conditions

Joo Florencio

Published Jan 15, 1987• 14 citations
Publisher

Abstract

We derive the generating functional of thermal Green's functions for an interacting electron-phonon system through the use of functional integrals in holomorphic representations (c numbers for phonons and Grassmann numbers for electrons). These integrals are over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kT, and do not require normalization factors unlike previous derivations. We recover the Matsubara perturbation formalism without using any results from the standard operator method.

DOI
10.1103/physrevb.35.452

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We derive the generating functional of thermal Green's functions for an interacting electron-phonon system through the use of functional integrals in holomorphic representations (c numbers for phonons and Grassmann numbers for electrons). These integrals are over paths obeying one-sided boundary conditions at either 0 or \ensuremath{\beta}=1/kT, and do not require normalization factors unlike previous derivations. We recover the Matsubara perturbation formalism without using any results from the standard operator method.
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We formulate the density matrix of many-body systems as functional integrals in the holomorphic representation, similar to the S-matrix formulation in field theory. Symmetry and antisymmetry requirements are built into the integration variables: c-numbers for bosons and Grassmann numbers for fermions. The integrals have well-defined normalization. They yield first-order differential equations for semiclassical extremal configurations and corrections to it, and do not require periodicity conditions on the individual integration paths, unlike the formulations found in the literature. We present some applications to simple boson and fermion systems.
Coherent-state functional-integral methods are used to develop approximation schemes for the description of coupled electron-phonon systems far from equilibrium. We consider an interacting-electron system which is coupled to a system of phonons and subjected to an external---not necessarily weak---perturbation. For systems where the electron-phonon interaction is linear in the phonon variables, the phonon degrees of freedom can be accounted for exactly. From a formally exact representation of a suitable generating function for the expectation value of electronic observables, approximation schemes based on mean-field solutions can be derived. Within the lowest order of approximation, the coupled electron-phonon system may be treated as a self-consistent system of indepenent electrons. Higher-order corrections are shown to give a random-phase approximation based on self-consistent mean fields. This approach is applied to electronic two-level systems coupled to phonons under a large external bias. We find that the coupling to acoustical-phonon modes leads to a damping of the quantum beats and drives the system into a new equilibrium state. The coupling to LO phonons is shown to account correctly for resonance effects which arise when the level splitting is close to the LO-phonon energy.
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 consider forward-scattering electron-electron and electron-phonon interactions in an extended Tomonaga-Luttinger model. The method of functional integration over anticommuting fields gives us a bosonized effective action. This enables us to obtain the spectrum and the correlation functions of the system exactly. We consider couplings to both acoustic and optical phonons. In the case of acoustic phonons, we obtain $T\ensuremath{\rightarrow}0$ phase diagram in the plane of coupling constants which indicates the existence of a normal phase between the regions of charge-density-wave and superconducting instabilities. This is qualitatively different from previous results.