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

Recent developments in mesoscopic physics

A calculation of the inelastic lifetime and the resistivity resulting from Coulomb coupling of conduction ( {ital s}'') electrons to a narrow ( {ital d}'') band is given for both the regular and the pair-transfer interactions in the ballistic regime. It is shown that in all cases for overlapping bands the scattering rate and the resistivity behave as {ital T}{sup 2} for a three-dimensional electron gas and as {ital T}{sup 2} ln{ital T} at two dimensions. This is the same temperature dependence as found for Coulomb scattering in a single band, but its contribution to the resistivity does not necessitate umklapp processes. The result contradicts recent arguments leading to a linear temperature dependence in a two-band picture at two dimensions. The latter can be obtained when a small amount of scattering by defects is added, as known for the single-band case. It is argued that such a linear contribution to the temperature-dependent resistivity may be more pronounced in the two-band case. Experimental manifestations of the results are discussed.

Inelastic Coulomb scattering in a two-band picture and the temperature dependence of the resistivity.

The Bloch (Boltzmann) theory of electronic transport in lightly disordered conductors has been quite successful in describing the impurity and temperature dependence of the conductivity in ordinary relatively pure conductors. Further transport properties such as magnetoconductivity, Hall effect, thermal conductivity and thermopower can also be handled with general success. However, when the amount of disorder (or impurity concentration) becomes very large, novel phenomena occur which are unexplainable within the weak-scattering theory. In particular, the temperature dependence of p becomes much weaker and eventually changes sign for high enough disorder (Wiesmann et al., 1979). The Mathiessen rule, according to which the disorder and temperature-contributions to p are additive, thus breaks down. An extremely interesting correlation was found by Mooij (1973), namely that in a large number of metallic conductors, dp/dT becomes negative when p becomes larger than a value ranging around 80-180 μΩ cm in something like a hundred disordered systems. This almost “universal” trend must have an explanation which is only weakly dependent on material properties, and we remember that dp/dT > 0 always in weak-scattering theory. More recently (and, in fact, following ideas from localization theory — for general references on localization see, e.g. articles in the books edited by Friedman and Tunstall, 1978; Balian et al., 1979, Stern, 1980; Castellani et al.., 1981; Nagaoka and Fukuyama, 1982; as well as the book by Mott and Davis, 1979 and lecture notes by Lee, 1981) it has been found that “restricted geometry” systems i.e. thin films and wires appear to always have dp/dT < 0 at low enough temperatures. This behavior of effectively one dimensional (1D) and 2D systems is also quite universal. A growing amount of evidence is indicating (Dolan and Osheroff, 1979; Giordano et al., 1979; Bishop et al., 1980; Pepper, 1980) that these latter systems are always (Thouless, 1977; Abrahams et al., 1979) insulators, i.e. p → ∞, as T → 0.

Localization and Superconductivity

We analyze some consequences of the Casimir-type zero-point radiation pressure. These include macroscopic ``vacuum'' forces on a metallic layer in between a dielectric medium and an inert [$ϵ(\ensuremath{\omega})=1$] one. Ways to control the sign of these forces, based on dielectric properties of the media, are thus suggested. Finally, the large positive Casimir pressure, due to surface plasmons on thin metallic layers, is evaluated and discussed.

https://arxiv.org/pdf/quant-ph/0501156.pdf

Yoseph Imry

Papers

Recent developments in mesoscopic physics

Inelastic Coulomb scattering in a two-band picture and the temperature dependence of the resistivity.

Localization and Superconductivity

Casimir Zero-Point Radiation Pressure

Is there Bragg — scattering off a two-dimensional crystal?