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Progress of Theoretical Physics

Quantum Statistical Mechanics of Electron-Phonon System

Hiroshi Ichimura

Published Feb 1, 1956• 3 citations
Publisher

Abstract

The problem of the interaction between conduction electrons and the thermal vibration of the lattice ions in the metal is treated by a statistical mechanical method. We adopt the free electron and the phonon models as usual, using the Bloch-Bethe type interaction Hamiltonian. By the method of expanding the grand partition function in powers of the interaction parameter, we get a formula for the free energy of the system of conduction electrons in the presence of the interaction. A technique analogous to the renormalization method in the quantum mechanics is utilized in the calculation.

DOI
10.1143/ptp.15.151

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The problem of the interaction between conduction electrons and the thermal vibration of the lattice ions in the metal is treated by a statistical mechanical method. We adopt the free electron and the phonon models as usual, using the Bloch-Bethe type interaction Hamiltonian. By the method of expanding the grand partition function in powers of the interaction parameter, we get a formula for the free energy of the system of conduction electrons in the presence of the interaction. A technique analogous to the renormalization method in the quantum mechanics is utilized in the calculation.
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A new method of calculating the grand partition function of many-body system is developed, adopting extensively the techniques of calculus in quantum field theory. It is shown that the grand partition function, which is a trace of the density matrix expressed in terms of field operators, can be evaluated in a way almost parallel with the evaluation of the vacuum expectation value of the S-matrix in quantum field theory, provided that appropriate modifications in notation and definitions are made. As an example, the method is applied to electron-phonon system. Further, basing on this new formalism, various non-perturbational methods are discussed.
A general method of calculation is described for quantum statistical mechanics. It is based on a simplification of the Laplace transform of the density matrix which follows from a theorem due to Hugenholtz. The basic result is that an element of the density matrix can be written as a sum over graphs, with the contribution of each graph factored into contributions from connected or linked graphs. Applied to the grand partition function, the exponential formula of Bloch and DeDominicis is obtained in a simple way. A similar formula is then derived for the canonical ensemble for the case of a nondegenerate gas. In this way the familiar result of Uhlenbeck and Beth is obtained for the second virial coefficient. Techniques are also introduced for evaluating ensemble averages of operators. In this connection, some care must be exercised in the case of diagonal operators. Finally, these methods are used to calculate the pair-correlation function for a system of fermions interacting through short-range forces.
We perform investigation of the Hubbard model with interaction between strongly correlated conducting electrons on a lattice with Debye phonons. To solve the problem generalized dynamical mean-field $\text{DMFT}+\ensuremath{\Sigma}$ method is employed with ``external'' self-energy ${\ensuremath{\Sigma}}_{ph}$ corresponding to electron-phonon interaction. We present $\text{DMFT}+{\ensuremath{\Sigma}}_{ph}$ results for densities of states and kinks in energy dispersions for a variety of model parameters, analyzing the interplay of recently discovered kinks of purely electronic nature and usual phonon kinks in the electronic spectrum.