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

Aharonov-Bohm mesoscopic solid-state interferometers yield a conductance which contains a term cos(phi+beta), where phi relates to the magnetic flux. Experiments with a quantum dot on one of the interfering paths aim to relate beta to the dot's intrinsic Friedel transmission phase alpha(1). For closed systems, which conserve the electron current (unitarity), the Onsager relation requires that beta = 0 or pi. For open systems, we show that in general beta depends on the details of the broken unitarity. Although it gives information on the resonances of the dot, beta is generally not equal to alpha(1). A direct relation between beta and alpha(1) requires specific ways of opening the system, which are discussed.

http://arxiv.org/pdf/cond-mat/0108064.pdf

Broken unitarity and phase measurements in Aharonov-Bohm interferometers.

Flux quantization without off-diagonal long range order in a thin hollow cylinder

Steady oscillatory states of a finite Josephson junction

Superconducting-quantum-interference-device magnetometry is used to study the long-time cooling-rate dependence of the susceptibility cusp of the spin-glasses, $\mathrm{Cu}\mathrm{Mn}$ 4.6 at.% and $Al\mathrm{Gd}$ 37 at.%. Over a range of time delay $\ensuremath{\Delta}T=6 \mathrm{to} 2400$ min per temperature change $\ensuremath{\Delta}T=0.2$ K, no change is observed in the position of the cusp. These long-time results do not fit an extrapolation from Tholence's earlier ac susceptibility data as a function of frequency ($\ensuremath{\nu}=10 \mathrm{to} 30000$ Hz). Thus, they appear to be inconsistent with current nonequilibrium models for spin-glass freezing and rather suggest a phase-transition phenomenon.

Yoseph Imry

Papers

Broken unitarity and phase measurements in Aharonov-Bohm interferometers.

Flux quantization without off-diagonal long range order in a thin hollow cylinder

Steady oscillatory states of a finite Josephson junction

Long-time cooling-rate dependence of spin-glass freezing

Tricritical Points in Compressible Magnetic Systems