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.

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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

We study the spectral properties of a class of random matrices where the matrix elements depend exponentially on the distance between uniformly and randomly distributed points. This model arises naturally in various physical contexts, such as the diffusion of particles, slow relaxations in glasses, and scalar phonon localization. Using a combination of a renormalization group procedure and a direct moment calculation, we find the eigenvalue distribution density (i.e., the spectrum), for low densities, and the localization properties of the eigenmodes, for arbitrary dimension. Finally, we discuss the physical implications of the results.

Localization, anomalous diffusion, and slow relaxations: a random distance matrix approach.

We discuss the effect of edge states on the de Haas-van Alphen oscillations in a small singly connected sample made of a 2D electron gas in the quantum-Hall-effect regime. We find that because of the "ring topology" of these states the magnetization oscillates with a period $\frac{\mathrm{hc}}{e}$ in the total flux through the system. This oscillation survives a substantial amount of disorder. Its detectability is discussed and a related oscillation in the longitudinal and transverse resistances suggested.

de Haas-Alphen and Aharonov-Bohm-type persistent current oscillations in singly connected quantum dots.

Coupling to anisotropic elastic media: Magnetic and liquid-crystal phase transitions

A comparison between infrared absorption and neutron inelastic‐scattering measurements indicates the existence of a broad low‐energy (around 400 cm−1) protonic level in KH2PO4 above the Curie point. This level is only weakly discerned in the ir but is seen quite clearly in neutron scattering. The mode is explained as resulting from splitting of the ground level of the proton in a slightly asymmetric double minimum potential well where tunneling of the proton takes place. The asymmetry is caused by the interaction between the different protons, and it changes slowly with time as the result of the collective motions of the protons. Prominent changes in ir spectra in the 400‐cm−1 range were found on cooling through the Curie point, indicating the disappearance of the tunneling mode.This picture allows a simplified treatment of the system, leading to the existence of two correlated phase transition points, and a negative thermal expansion coefficient of the H bond at temperatures between these points. The two...

Proton Dynamics in Hydrogen‐Bonded Ferroelectrics

The two-probe conductance, g, of a disordered quantum system with N transverse scattering channels is determined by N real parameters («levels») {λ i } characterizing the transfer matrix M. The appropriate measure for M combined with a «global» maximum entropy hypothesis, leads to a joint distribution for th En regime quantique coherent, la conductance g d'un systeme desordonne possedant N canaux de diffusion est determinee par N parametres reels («niveaux») {λ i } qui caracterisent sa matrice de transfert M. La mesure adequate pour M, combinee avec une hypothese d'entropie maximum «globale», conduit a une loi de distribution de ces niveaux identique a celle des ensembles de matrices aleatoires classiques, si l'on excepte la presence d'un comportement nouveau de la densite de niveaux ρ(λ), qui est loin d'etre uniforme, et depend de facon significative des parametres du systeme. Nous etudions en detail cette densite et ses implications pour les fluctuations de conductance, montrant que le comportement specifique de cet ensemble resulte de la loi de composition multiplicative de M. Pour N donne et quand la longueur L 2 du systeme augmente, ce sont les α i ≃(1/2L z )ln(λ i )(qui convergent vers les exposants de Lyapunov de M) qui ont une densite essentiellement uniforme. Utilisant cette densite, nous developpons une analogie coulombienne pour expliquer le changement des statistiques des niveaux et de la conductance qui accompagne la transition du regime metallique au regime isolant. Le regime metallique correspond a une «phase» de haute densite du gaz de coulomb, avec une statistique identique a celle des ensembles classiques avec interactions logarithmiques, si l'on excepte la presence d'une «interaction» avec des charges images pres de l'origine; ceci implique une loi normale pour p(g) avec une variance universelle. Des mecanismes possibles d'apparition de queues lognormales dans le regime metallique sont discutes

Yoseph Imry

Papers

Localization, anomalous diffusion, and slow relaxations: a random distance matrix approach.

de Haas-Alphen and Aharonov-Bohm-type persistent current oscillations in singly connected quantum dots.

Coupling to anisotropic elastic media: Magnetic and liquid-crystal phase transitions

Proton Dynamics in Hydrogen‐Bonded Ferroelectrics

Theory of random multiplicative transfer matrices and its implications for quantum transport