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 Aharonov-Bohm effect in single normal-metal rings and show that averaging the conductance over many energies is equivalent to ensemble averaging. Thus raising the temperature T above a crossover ${\mathrm{T}}_{\mathrm{c}}$ changes the flux periodicity of magnetoresistance oscillations from h/e to h/2e. ${\mathrm{T}}_{\mathrm{c}}$ is determined by an energy correlation range, hD/${\mathrm{L}}^{2}$. The persistence of the h/e oscillations to high fields is explained.

Periodicity of the Aharonov-Bohm effect in normal-metal rings.

We review the adiabatic (slowly varying confining walls) approximation to the quantized conductance introduced by Glazman et al. We show that all the corrections to this approximation are exponentially small in the smoothness parameter of the constriction. A condition for accurate quantization is given, based on the result that the reflections due to the sudden widening of the constriction (existing in real devices) are highly suppressed if a small adiabatic widening and/or a potential barrier precede the sudden widening. An interesting collimation effect associated with the adiabatic picture is mentioned.

Quantization of the conductance of ballistic point contacts beyond the adiabatic approximation

For a quantum dot (QD) in the intermediate regime between integrable and fully chaotic, the widths of single-particle levels naturally differ by orders of magnitude. In particular, the width of one strongly coupled level may be larger than the spacing between other, very narrow, levels. In this case many consecutive Coulomb blockade peaks are due to occupation of the same broad level. Between the peaks the electron jumps from this level to one of the narrow levels, and the transmission through the dot at the next resonance essentially repeats that at the previous one. This offers a natural explanation to the recently observed behavior of the transmission phase in an interferometer with a QD.

Towards an explanation of the mesoscopic double-slit experiment: A new model for charging of a quantum Dot

Results of single-particle tunneling experiments performed on films of indium oxide are presented. The zero-bias anomalies change character with decreasing thickness from a three-dimensional behavior to a logarithmic energy dependence. Theoretical considerations suggest a crossover from a lnV to a V/sup 1/2/ correction to the density of states at a thickness-dependent energy. In the V/sup 1/2/ range, where the sample exhibits a three-dimensional density-of-states anomaly, its transport properties may still be two dimensional. These expectations are experimentally confirmed.

Density-of-states anomalies in a disordered conductor: A tunneling study

It is shown by a calculation analogous to that carried out by Wallace and Zia for the pure Ising model that the lower critical dimension of the Ising model in a random field is 3 and not 2 as suggested by domain energy arguments. Further, the critical dimension for the roughening transition is shown to be 5 as compared to 3 for the pure Ising model.

Yoseph Imry

Papers

Periodicity of the Aharonov-Bohm effect in normal-metal rings.

Quantization of the conductance of ballistic point contacts beyond the adiabatic approximation

Towards an explanation of the mesoscopic double-slit experiment: A new model for charging of a quantum Dot

Density-of-states anomalies in a disordered conductor: A tunneling study

Lower Critical Dimension and the Roughening Transition of the Random-Field Ising Model