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Annalen der Physik

Quantum Theory of Localized Crystal

Vyacheslav I. Yukalov

Published Jan 1, 1979• 24 citations
Publisher

Abstract

Abstract A quantum crystal each particle of which is located to a proximity of one lattice point is considered. The causal Green functions at finite temperatures are used. The localized approximation for the iteration of propagator equations is formulated. New type conditions defining parameters of model propagator are proposed. The general method is illustrated with the example of a linear chain.

DOI
10.1002/andp.19794910106

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Abstract A quantum crystal each particle of which is located to a proximity of one lattice point is considered. The causal Green functions at finite temperatures are used. The localized approximation for the iteration of propagator equations is formulated. New type conditions defining parameters of model propagator are proposed. The general method is illustrated with the example of a linear chain.
The existence of Anderson localization, characterized by vanishing diffusion due to strong disorder, has been demonstrated in numerous ways. A systematic approach based on the Anderson quantum model of the Fermi gas in random lattices that can describe both diffusive and localized regimes has not yet been fully established. We build on a recent publication [V. Janiš, ] and present a microscopic theory of disordered electrons that covers both the metallic phase with extended Bloch waves and the localized phase, where a propagating particle forms a quantum bound state with the hole left behind at the origin. The general theory provides a framework for constructing controlled approximations to one- and two-particle Green functions that satisfy the necessary conservation laws and causality requirements across the full range of disorder strength. It is used explicitly to derive a local, mean-field-like approximation for the two-particle irreducible vertices, enabling quantitative analysis of the solution's dynamic properties in both metallic and localized phases, including critical behavior at the mobility edge. A new instability line for the dynamical electron-hole correlation function of the metallic phase is introduced.
The theory of Green's functions is briefly reviewed. It is discussed which properties of a quantum system and which physical processes can be described by Green's functions. A general approximation scheme to evaluate Green's functions and propagators is presented. Particular attention is paid to the one-particle Green's function and to the particle–particle propagator which is a relevant component of the two-particle Green's function. Several illustrative applications are presented and related to experiment.
A purely algebraic approach to Green's functions is presented. The new formulation allows the determination of a closed-form equation for Green's functions which depends only on the ground state of the system. By a straightforward perturbation expansion of this state an interesting approximation scheme is obtained which is nonperturbative in the final states (e.g., excited or ionic states). This method, which can be used for various propagators, is explicitly applied to the particle-particle propagator carrying out the scheme up to third order. The approach is discussed in detail and briefly compared with the well-established algebraic diagrammatic construction approximation scheme.
This is one of the two papers where the optimized perturbation theory was first formulated. The other paper is published in Theor. Math. Phys. 28, 652--660 (1976). The main idea of the theory is to reorganize the perturbative sequence by introducing control functions, defined by optimization conditions, so that the reorganized approximation sequence be convergent. In the present paper, the theory of perturbations is suggested for statistical systems in the absence of small interaction parameters. A new form is advanced for self-consistent conditions defining the optimal parameters for trial Green functions in iterating nonlinear propagator equations. Superharmonic, semiharmonic, and pseudoharmonic approximations for a molecular crystal are considered as examples.
The possibility of a perturbation treatment for quantum crystals is investigated. It is found that the usual form of the unperturbed Hamiltonian, being a single-particle operator, is not suitable for this situation, since it does not produce correlations necessary in a theory of quantum crystals. These correlations prevent the unperturbed ground state or low excited states from containing multiply occupied lattice sites. An unperturbed Hamiltonian is proposed which generates these correlations but is no longer a single-particle operator. The rules for a diagrammatic representation of a perturbation expansion starting from this unperturbed Hamiltonian are given. They differ from the usual rules for Goldstone diagrams. In particular, the linked-cluster theorem is not valid. Adding and subtracting pairs of mutually cancelling diagrams, in a manner similar to that in which Pauli-principle-violating diagrams are treated in the standard perturbation theory for fermions, we find new rules which, at least at low temperatures, are identical to those for spinless fermions. This result is essentially independent of the statistics of the actual lattice particles. In the case where the lattice particles are actually fermions with spin \textonehalf{}, e.g., 3He^{3}\mathrm{He}3He, we find an expansion formally identical to the problem of spinless fermions interacting among themselves and with a set of spins \textonehalf{} localized at lattice sites.