Conservation laws in disordered electron systems: Thermodynamic limit and configurational averaging

We discuss conservation of probability in noniteracting disordered electron systems. We argue that although the norm of the electron wave function is conserved in individual realizations of the random potential, we cannot extend this conservation law easily to configurationally averaged systems in the thermodynamic limit. A direct generalization of the norm conservation to averaged functions is hindered by the existence of localized states breaking translational invariance. Such states are elusive to the description with periodic Bloch waves. Mathematically this difficulty is manifested through the diffusion pole in the electron-hole irreducible vertex. The pole leads to a clash with analyticity of the self-energy, reflecting causality of the theory, when norm conservation is enforced by the Ward identity between one- and two-particle averaged Green functions.Comment: REVTeX4, 8 pages, no figure

Causality vs. Ward identity in disordered electron systems

We address the problem of fulfilling consistency conditions in solutions for disordered noninteracting electrons. We prove that if we assume the existence of the diffusion pole in an electron-hole symmetric theory we cannot achieve a solution with a causal self-energy that would fully fit the Ward identity. Since the self-energy must be causal, we conclude that the Ward identity is partly violated in the diffusive transport regime of disordered electrons. We explain this violation in physical terms and discuss its consequences.Comment: 4 pages, REVTeX, 6 EPS figure

Thermodynamic origin of order parameters in mean-field models of spin glasses

We analyze thermodynamic behavior of general $n$-component mean-field spin glass models in order to identify origin of the hierarchical structure of the order parameters from the replica-symmetry breaking solution. We derive a configurationally dependent free energy with local magnetizations and averaged local susceptibilities as order parameters. On an example of the replicated Ising spin glass we demonstrate that the hierarchy of order parameters in mean-field models results from the structure of inter-replica susceptibilities. These susceptibilities serve for lifting the degeneracy due to the existence of many metastable states and for recovering thermodynamic homogeneity of the free energy.Comment: REVTeX4, 13 pages, 8 EPS figure

Quantum diffusion in a random potential: A consistent perturbation theory

We scrutinize the diagrammatic perturbation theory of noninteracting electrons in a random potential with the aim to accomplish a consistent comprehensive theory of quantum diffusion. Ward identity between the one-electron self-energy and the two-particle irreducible vertex is generally not guaranteed in the perturbation theory with only elastic scatterings. We show how the Ward identity can be established in practical approximations and how the functions from the perturbation expansion should be used to obtain a fully consistent conserving theory. We derive the low-energy asymptotics of the conserving full two-particle vertex from which we find an exact representation of the diffusion pole and of the static diffusion constant in terms of Green functions of the perturbation expansion. We illustrate the construction on the leading vertex corrections to the mean-field diffusion due to maximally-crossed diagrams responsible for weak localization.Comment: 12 pages, 3 figure

Spin-symmetric solution of an interacting quantum dot attached to superconducting leads: Andreev states and the 0−π0-\pi transition

Behavior of Andreev gap states in a quantum dot with Coulomb repulsion symmetrically attached to superconducting leads is studied via the perturbation expansion in the interaction strength. We find the exact asymptotic form of the spin-symmetric solution for the Andreev states continuously approaching the Fermi level. We thereby derive a critical interaction at which the Andreev states at zero temperature merge at the Fermi energy, being the upper bound for the $0-\pi$ transition. We show that the spin-symmetric solution becomes degenerate beyond this interaction, in the $\pi$ phase, and the Andreev states do not split unless the degeneracy is lifted. We further demonstrate that the degeneracy of the spin-symmetric state extends also into the $0$ phase in which the solutions with zero and non-zero frequencies of the Andreev states may coexist.Comment: 12 pages, 4 figure

Analytic impurity solver with the Kondo strong-coupling asymptotics

We present an analytic universal impurity solver for strongly correlated electrons. We extend the many-body perturbation expansion via suitable two-particle renormalizations from the Fermi-liquid regime to the critical region of the metal-insulator transition. The reliability of the approximation in the strong-coupling limit is demonstrated by reproducing the Kondo scale in the single-impurity Anderson model. We disclose the origin of the Kondo resonance in terms of Feynman diagrams and find criteria for the existence of the proper Kondo asymptotic behavior in approximate theories.Comment: 7 pages REVTeX4, 5 EPS figures, extended versio

Linked Cluster Expansion Around Mean-Field Theories of Interacting Electrons

A general expansion scheme based on the concept of linked cluster expansion from the theory of classical spin systems is constructed for models of interacting electrons. It is shown that with a suitable variational formulation of mean-field theories at weak (Hartree-Fock) and strong (Hubbard-III) coupling the expansion represents a universal and comprehensive tool for systematic improvements of static mean-field theories. As an example of the general formalism we investigate in detail an analytically tractable series of ring diagrams that correctly capture dynamical fluctuations at weak coupling. We introduce renormalizations of the diagrammatic expansion at various levels and show how the resultant theories are related to other approximations of similar origin. We demonstrate that only fully self-consistent approximations produce global and thermodynamically consistent extensions of static mean field theories. A fully self-consistent theory for the ring diagrams is reached by summing the so-called noncrossing diagrams.Comment: 17 pages, REVTEX, 13 uuencoded postscript figures in 2 separate file