Nonlinear Sigma Model for Disordered Media: Replica Trick for Non-Perturbative Results and Interactions

In these lectures, given at the NATO ASI at Windsor (2001), applications of the replicas nonlinear sigma model to disordered systems are reviewed. A particular attention is given to two sets of issues. First, obtaining non-perturbative results in the replica limit is discussed, using as examples (i) an oscillatory behaviour of the two-level correlation function and (ii) long-tail asymptotes of different mesoscopic distributions. Second, a new variant of the sigma model for interacting electrons in disordered normal and superconducting systems is presented, with demonstrating how to reduce it, under certain controlled approximations, to known ``phase-only'' actions, including that of the ``dirty bosons'' model.Comment: 25 pages, Proceedings of the NATO ASI "Field Theory of Strongly Correlated Fermions and Bosons in Low - Dimensional Disordered Systems", Windsor, August, 2001; to be published by Kluwe

The Magic Angle "Mystery" in Electron Energy Loss Spectroscopy: Relativistic and Dielectric Corrections

Recently it has been demonstrated that a careful treatment of both longitudinal and transverse matrix elements in electron energy loss spectra can explain the mystery of relativistic effects on the {\it magic angle}. Here we show that there is an additional correction of order $(Z\alpha)^2$ where $Z$ is the atomic number and $\alpha$ the fine structure constant, which is not necessarily small for heavy elements. Moreover, we suggest that macroscopic electrodynamic effects can give further corrections which can break the sample-independence of the magic angle.Comment: 10 pages (double column), 6 figure

Random Matrix Theory and Chiral Symmetry in QCD

Random matrix theory is a powerful way to describe universal correlations of eigenvalues of complex systems. It also may serve as a schematic model for disorder in quantum systems. In this review, we discuss both types of applications of chiral random matrix theory to the QCD partition function. We show that constraints imposed by chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. We thus derive exact results for the low-lying eigenvalues of the QCD Dirac operator. We argue that the statistical properties of these eigenvalues are universal and can be described by a random matrix theory with the global symmetries of the QCD partition function. The total number of such eigenvalues increases with the square root of the Euclidean four-volume. The spectral density for larger eigenvalues (but still well below a typical hadronic mass scale) also follows from the same low-energy effective partition function. The validity of the random matrix approach has been confirmed by many lattice QCD simulations in a wide parameter range. Stimulated by the success of the chiral random matrix theory in the description of universal properties of the Dirac eigenvalues, the random matrix model is extended to nonzero temperature and chemical potential. In this way we obtain qualitative results for the QCD phase diagram and the spectrum of the QCD Dirac operator. We discuss the nature of the quenched approximation and analyze quenched Dirac spectra at nonzero baryon density in terms of an effective partition function. Relations with other fields are also discussed.Comment: invited review article for Ann. Rev. Nucl. Part. Sci., 61 pages, 11 figures, uses ar.sty (included); references added and typos correcte

Non-chiral current algebras for deformed supergroup WZW models

We study deformed WZW models on supergroups with vanishing Killing form. The deformation is generated by the isotropic current-current perturbation which is exactly marginal under these assumptions. It breaks half of the global isometries of the original supergroup. The current corresponding to the remaining symmetry is conserved but its components are neither holomorphic nor anti-holomorphic. We obtain the exact two- and three-point functions of this current and a four-point function in the first two leading orders of a 1/k expansion but to all orders in the deformation parameter. We further study the operator product algebra of the currents, the equal time commutators and the quantum equations of motion. The form of the equations of motion suggests the existence of non-local charges which generate a Yangian. Possible applications to string theory on Anti-de Sitter spaces and to condensed matter problems are briefly discussed.Comment: 43 pages, Latex, one eps figure; v.2: minor corrections, a reference adde

Keldysh technique and non-linear sigma-model: basic principles and applications

The purpose of this review is to provide a comprehensive pedagogical introduction into Keldysh technique for interacting out-of-equilibrium fermionic and bosonic systems. The emphasis is placed on a functional integral representation of underlying microscopic models. A large part of the review is devoted to derivation and applications of the non-linear sigma-model for disordered metals and superconductors. We discuss such topics as transport properties, mesoscopic effects, counting statistics, interaction corrections, kinetic equation, etc. The sections devoted to disordered superconductors include Usadel equation, fluctuation corrections, time-dependent Ginzburg-Landau theory, proximity and Josephson effects, etc. (This review is a substantial extension of arXiv:cond-mat/0412296.)Comment: Review: 103 pages, 19 figure

Gilbert Damping in Conducting Ferromagnets II: Model Tests of the Torque-Correlation Formula

We report on a study of Gilbert damping due to particle-hole pair excitations in conducting ferromagnets. We focus on a toy two-band model and on a four-band spherical model which provides an approximate description of ferromagnetic (Ga,Mn)As. These models are sufficiently simple that disorder-ladder-sum vertex corrections to the long-wavelength spin-spin response function can be summed to all orders. An important objective of this study is to assess the reliability of practical approximate expressions which can be combined with electronic structure calculations to estimate Gilbert damping in more complex systems.Comment: 10 pages, 10 figures. Submitted to Phys. Rev.

Universal quantum oscillations in the underdoped cuprate superconductors

The metallic state of the underdoped high-Tc cuprates has remained an enigma: How may seemingly disconnected Fermi surface segments, observed in zero magnetic field as a result of the opening of a partial gap (the pseudogap), possess conventional quasiparticle properties? How do the small Fermi-surface pockets evidenced by the observation of quantum oscillations (QO) emerge as superconductivity is suppressed in high magnetic fields? Such QO, discovered in underdoped YBa2Cu3O6.5 (Y123) and YBa2Cu4O8 (Y124), signify the existence of a conventional Fermi surface (FS). However, due to the complexity of the crystal structures of Y123 and Y124 (CuO2 double-layers, CuO chains, low structural symmetry), it has remained unclear if the QO are specific to this particular family of cuprates. Numerous theoretical proposals have been put forward to explain the route toward QO, including materials-specific scenarios involving CuO chains and scenarios involving the quintessential CuO2 planes. Here we report the observation of QO in underdoped HgBa2CuO4+{\delta} (Hg1201), a model cuprate superconductor with individual CuO2 layers, high tetragonal symmetry, and no CuO chains. This observation proves that QO are a universal property of the underdoped CuO2 planes, and it opens the door to quantitative future studies of the metallic state and of the Fermi-surface reconstruction phenomenon in this structurally simplest cuprate.Comment: 17 pages, 5 figure

Localization of preformed Cooper pairs in disordered superconductors

International audienceThe most profound effect of disorder on electronic systems is the localization of the electrons transforming an otherwise metallic system into an insulator. If the metal is also a superconductor then, at low temperatures, disorder can induce a pronounced transition from a superconducting into an insulating state. An outstanding question is whether the route to insulating behaviour proceeds through the direct localization of Cooper pairs or, alternatively, by a two-step process in which the Cooper pairing is first destroyed followed by the standard localization of single electrons. Here we address this question by studying the local superconducting gap of a highly disordered amorphous superconductor by means of scanning tunnelling spectroscopy. Our measurements reveal that, in the vicinity of the superconductor-insulator transition, the coherence peaks in the one-particle density of states disappear whereas the superconducting gap remains intact, indicating the presence of localized Cooper pairs. Our results provide the first direct evidence that the superconductor-insulator transition in some homogeneously disordered materials is driven by Cooper-pair localization