Mean-field theory andεexpansion for Anderson localization

The paper presents a general theory of coupling of eigenvalues of complex matrices of arbitrary dimension smoothly depending on real parameters. The cases of weak and strong coupling are distinguished and their geometric interpretation in two and three-dimensional spaces is given. General asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios. Two numerical exam- ples from crystal optics illustrate effectiveness and accuracy of the presented theory. I. INTRODUCTION Behavior of eigenvalues of matrices dependent on param- eters is a problem of general interest having many important applications in natural and engineering sciences. In modern physics, e.g. quantum mechanics, crystal optics, physical chemistry, acoustics and mechanics, multiple eigenvalues in matrix spectra associated with specific effects attract great interest of researchers since the papers (1), (2). In recent papers, see e.g. (3)-(6), two important cases are distinguished: the diabolic points (DPs) and the exceptional points (EPs). From mathematical point of view DP is a point where the eigenvalues coalesce, while corresponding eigenvectors remain different; and EP is a point where both eigenvalues and eigenvectors merge forming a Jordan block. Both the DP and EP cases are interesting in applications and were observed in experiments (6), (7). In this paper we present a general theory of coupling of eigenvalues of complex matrices of arbitrary dimen- sion smoothly depending on multiple real parameters. Two essential cases of weak and strong coupling based on a Jordan form of the system matrix are distinguished. These two cases correspond to diabolic and exceptional points, respectively. We derive general formulae describing coupling and decoupling of eigenvalues, crossing and avoided crossing of eigenvalue surfaces. It is emphasized that the presented theory of coupling of eigenvalues of complex matrices gives not only qualitative, but also quantitative results on behavior of eigenvalues based only on the information taken at the singular points. The paper is based on the author's previous research on interaction of eigenvalues for matrices and differential operators depending on multiple parameters (8)- (11); for more references see the recent book (12).

Two-particle Green's functions and the vertex functions play a critical role in theoretical frameworks for describing strongly correlated electron systems. However, numerical calculations at the two-particle level often suffer from large computation time and massive memory consumption. We derive a general expansion formula for the two-particle Green's functions in terms of an overcomplete representation based on the recently proposed ``intermediate representation'' basis. The expansion formula is obtained by decomposing the spectral representation of the two-particle Green's function. We demonstrate that the expansion coefficients decay exponentially, while all high-frequency and long-tail structures in the Matsubara-frequency domain are retained. This representation therefore enables efficient treatment of two-particle quantities and opens a route to the application of modern many-body theories to realistic strongly correlated electron systems.

The method of the derivation of two-particle Green's functions in non-equilibrium matter is developed. The closed set of equations for the vertex functions and also for the two-particle Green's functions is obtained by means of the summation of the series of irreducible diagrams. The solution of such equations completely defines the two-particle Green's functions in matter.

Ab initio green's function calculations on highly conducting polymers: Effects of electron correlation and aperiodicity

Averaged Green's functions for a disordered electronic system withn orbitals per site are expanded in powers of 1/n. These expansions should be valid in the region of extended states. The expansion coefficients for the d.c. conductivity are finite for dimensionalityd>2 and diverge asd approaches 2. Similarities of two types of two-particle Green's functions with the transverse and longitudinal susceptibilities of a ferromagnet with broken continuous symmetry are pointed out. Arguments for two being the lower critical dimensionality for the hydrodynamics and the mobility edge are given. Provided our series can be exponentiated we find that no metallic conductivity exists for finiten andd=2 in one of our models. Critical exponents ford infinitesimal above two are given. In this limitv diverges like 1/(d−2) and the conductivity vanishes linearly at the mobility edge. The diagrams of the Green's functions are given in terms of vertices of short-range order and of the two-particle propagators of then=∞ limit. Diagrams withs loops contribute in ordern −s . The diagrams can be rearranged so that a number of vertices vanishes like the square of the wavevector. This feature prevents infrared divergencies for the d.c. conductivity ford>2.

A new approach to localization in condensed matter: Electrons, phonons, and magnons

Quantum coherence in noise power spectrum in two quantum dots

Exact conformal-field-theory results on the multi-channel Kondo effect: Asymptotic three-dimensional space- and time-dependent multi-point and many-particle Green's functions

The tight-binding model for a chain, where the hopping constants follow a Fibonacci sequence, predicts multifractality in the spectrum and wave functions. Experimentally, we realize this model by chains of small dielectric resonators with a high refractive index (${\ensuremath{\epsilon}}_{r}\ensuremath{\approx}45$) of cylindrical form that exhibit evanescent coupling. We show that the fractality of the measured local density of state (LDOS) is best understood when the sites are rearranged according to the similarities in their local surrounding, i.e., their conumbers. This allows us to deduce simple recursive construction schemes for the LDOS for the two cases of dominant strong and weak coupling, despite our limited resolution due to nonzero resonance width and size constraints. We measure the singularity spectrum and the fractal dimensions of the wave functions, and we find good agreement with theoretical predictions for the multifractality based on a perturbative description in the quasiperiodic limit.

Using the path integral formalism and a model based on statistical arguments, we derive the correlation function which enters the description of a one-dimensional collective system coupled to an intrinsic system in terms of transport concepts. The derivation shows a finite time dependence of the correlation function in contradistinction to a former calculation. We discuss some points related to the derivation and estimate the relaxation times which characterise the memory effects in both cases of weak and strong coupling.

Polariton–polariton kinematic interaction modulation and kinematic bipolaritons in organic microcavities

We consider a non-interacting system of electrons on a clean one-channel Aharonov-Bohm ring which is threaded by a fluctuating magnetic flux. The flux derives from a Caldeira-Leggett bath of harmonic oscillators. We address the influence of the bath on the following properties: one- and two-particle Green's functions, dephasing, persistent current and visibility of the Aharonov-Bohm effect in cotunneling transport through the ring. For the bath spectra considered here (including Nyquist noise of an external coil), we find no dephasing in the linear transport regime at zero temperature. PACS numbers: 73.23.-b, 73.23.Hk, 73.23.Ra, 03.65.Yz

Two-particle Green's functions in the theory of an electron gas

Doubly ionized states of carbon tetrafluoride

The calculation of molecular Auger spectra