There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.

Theory of Dynamic Critical Phenomena

Magnetoencephalography (MEG) is a noninvasive technique for investigating neuronal activity in the living human brain. The time resolution of the method is better than 1 ms and the spatial discrimination is, under favorable circumstances, 2-3 mm for sources in the cerebral cortex. In MEG studies, the weak 10 fT-1 pT magnetic fields produced by electric currents flowing in neurons are measured with multichannel SQUID (superconducting quantum interference device) gradiometers. The sites in the cerebral cortex that are activated by a stimulus can be found from the detected magnetic-field distribution, provided that appropriate assumptions about the source render the solution of the inverse problem unique. Many interesting properties of the working human brain can be studied, including spontaneous activity and signal processing following external stimuli. For clinical purposes, determination of the locations of epileptic foci is of interest. The authors begin with a general introduction and a short discussion of the neural basis of MEG. The mathematical theory of the method is then explained in detail, followed by a thorough description of MEG instrumentation, data analysis, and practical construction of multi-SQUID devices. Finally, several MEG experiments performed in the authors' laboratory are described, covering studies of evoked responses and of spontaneous activity in both healthy and diseased brains. Many MEG studies by other groups are discussed briefly as well.

/pdf/magnetoencephalography-theory-instrumentation-and-1oezfk882m.pdf

Magnetoencephalography—theory, instrumentation, and applications to noninvasive studies of the working human brain

Anyons in an exactly solved model and beyond

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

The orbital magnetic moment of localized electrons is considered, both in the absence and in the presence of the electron-phonon interaction. This magnetic moment, proportional to the persistent current circulating around groups of at least three localized states, arises from interference of the localized electronic wave functions. It is shown that for the isolated electronic system the magnetic moment has a definite sign which alternates as the location of the Fermi level is varied. When the electrons are coupled to the thermal bath, the orbital magnetic moment decreases due to two effects: (i) the appearance of a current that flows in the opposite direction to that of the isolated system\char22{}this countercurrent results from resonant electron-phonon processes, suggested by Holstein; (ii) an overall Debye-Waller factor that diminishes the magnitude of the orbital response to the external magnetic field. A possible experimental realization to observe the countercurrent is discussed.

Orbital magnetization in the hopping regime.

Harmonic and subharmonic microwave induced steps in the I-V characteristics of current fed Josephson junction were found and investigated by numerical and approximate analytical methods, for a purely sinusoidal Josephson current. The dependence of the steps on the various parameters is given and their local stability is discussed. Various types of I-V characteristics are shown. We have introduced thermal noise effects via a Langevin term in the equation of motion. The smaller steps are easily washed out by the thermal noise. The width of the fundamental step was found to be proportional to Txfor the range of parameters which we used, where x \gsim 1/2 .

Current noise effects on the microwave induced steps in current fed Josephson junctions

The Kondo picture for the electron-electron attraction due to transfer of pairs of charge carriers between two bands is investigated. This interaction may reduce considerably the Coulomb repulsion, leading to enhancement of the superconducting transition temperature and a substantial weakening of the isotope effect. The effect is large when the empty band is narrow and the energy gap between the two bands is small. The screening of the Coulomb interaction in a two-band model is considered in detail using the random-phase approximation, and the dynamic dielectric functions are derived. It is found that the intraband vertex of the empty band is almost unscreened and that of the occupied one has metallic screening. The screening of the pair-transfer vertex is independent of the intraband Coulomb interactions and may even be magnified. The possible relevance to the superconducting oxides is suggested.

Enhanced pairing interactions in a two-band model

The conductivity of indium oxide films was studied as a function of kFl (including kFl<or approximately=1), film thickness, temperature and magnetic field. For sufficiently thin films the authors observe a change from logarithmic to faster temperature dependence below a temperature determined by RSquare Operator . The analogous crossover for thicker samples is shown to depend on kFl. Whether a given system is 'thin' enough for RSquare Operator to be the relevant parameter in this respect is determined by the value of the correlation length xi .

On the role of the correlation length near the onset of non-metallic conduction

Considering N spinless Fermions in a random potential, we study how a short range pairwise interaction delocalizes the N-body states in the basis of the one-particle Slater determinants, and the spectral rigidity of the N-body spectrum. The maximum number g N of consecutive levels exhibiting the universal Wigner-Dyson rigidity (the Thouless number) is given as a function of the strength U of the interaction for the bulk of the spectrum. In the dilute limit, one finds two thresholds U c1 and U c2 . When U < U c1 , there is a perturbative mixing between a few Slater determinants (Rabi oscillations) and g N |U| P < 1, where P = N/2 (even N) or (N + 1)/2 (odd N). When U = U c1 , the matrix element of a Slater determinant to the first generation of determinants directly coupled to it by the interaction is of the order of the level spacing of the latter determinants, g N 1 and the level spacing distribution exhibits a crossover from Poisson to Wigner, related to the crossover between weak perturbative mixing and effective golden-rule decay. Moreover, we show that the same U c1 signifies also the breakdown of the perturbation theory in U. For U c1 < U < U c2 , the states are extended over the energetically nearby Slater determinants with a non-ergodic hierarchical structure related to the sparse form of the Hamiltonian. Above a second threshold U c2 , the sparsity becomes irrelevant, and the states are extended more or less ergodically over 9N consecutive Slater determinants. A self-consistent argument gives g N U N/(N-1) . We compare our predictions to a numerical study of three spinless Fermions in a disordered cubic lattice. Implications for the interaction-induced N-particle delocalization in real space are discussed. The applicability of Fermi's golden rule for decay in this dilute gas of real particles is compared with the one characterizing a finite-density Fermi gas. The latter is related to the recently suggested Anderson transition in Fock space.

https://hal.archives-ouvertes.fr/docs/00/24/74/72/PDF/ajp-jp1v7p1559.pdf

Yoseph Imry

Papers

Orbital magnetization in the hopping regime.

Current noise effects on the microwave induced steps in current fed Josephson junctions

Enhanced pairing interactions in a two-band model

On the role of the correlation length near the onset of non-metallic conduction

Thouless Numbers for Few-Particle Systems with Disorder and Interactions