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

The Kubo fluctuation-dissipation theorem relates the current
fluctuations of a system in an equilibrium state with the linear AC-conductance. This theorem holds also out of equilibrium provided that the system is in a stationary state and that the linear conductance is replaced by the (dynamic) conductance with respect to the nonequilibrium state. We provide a simple proof for that statement and then apply it in two cases. We first show that in an excess noise measurement at zero temperature, in which the impedance matching is maintained while driving a mesoscopic sample out of equilibrium, it is the nonsymmetrized noise power spectrum which is
measured, even if the bare measurement, i.e. without extracting the excess part of the noise, obtains the symmetrized noise. As a second application we derive a commutation relation for the two components
of fermionic or bosonic currents which holds in every stationary state
and which is a generalization of the one valid only for bosonic currents. As is usually the case, such a commutation relation can be used e.g. to derive Heisenberg uncertainty relationships among these current components.

Applications of nonequilibrium Kubo formula to the detection of quantum noise

A light-scattering experiment in 3He-4He mixtures is analysed in the perspective of the 'non-ordering' parameters scaling-laws. The tricritical behaviour is found to be characterized by the specific-heat exponent alpha =-0.9+or-0.2 and by the correlation-length exponent nu =1.15+or-0.1. These tricritical exponents are renormalised in the manner of Fisher, the corresponding ideal critical exponents being alpha G=0.48+or-0.1, nu G=0.6+or-0.1. The critical behaviour near the lambda -line, at the concentration x3=0.6318 is found to be strongly influenced by its proximity to the tricritical point. The effective renormalized critical exponents are alpha RI=-0.36+or-0.04, nu RI=0.9+or-0.06 and the effective ideal critical exponents are alpha I=0.26+or-0.04, nu I=0.66+or-0.88.

Critical and tricritical exponents of 3He-4He solutions from light-scattering data

Phase diagrams and dimensionality shift in “Random site” spin glasses

The conduction electrons' dephasing rate, $\tau_{\phi}^{-1}$, is expected to vanish with the temperature. A very intriguing apparent saturation of this dephasing rate in several systems was recently reported at very low temperatures. The suggestion that this represents dephasing by zero-point fluctuations has generated both theoretical and experimental controversies. We start by proving that the dephasing rate must vanish at the $T\to 0$ limit, unless a large ground state degeneracy exists. This thermodynamic proof includes most systems of relevance and it is valid for any determination of $\tau_{\phi}$ from {\em linear} transport measurements. In fact, our experiments demonstrate unequivocally that indeed when strictly linear transport is used, the apparent low-temperature saturation of $\tau_{\phi}$ is eliminated. However, the conditions to be in the linear transport regime are more strict than hitherto expected. Another novel result of the experiments is that introducing heavy nonmagnetic impurities (gold) in our samples produces, even in linear transport, a shoulder in the dephasing rate at very low temperatures. We then show theoretically that low-lying local defects may produce a relatively large dephasing rate at low temperatures. However, as expected, this rate in fact vanishes when $T \to 0$, in agreement with our experimental observations.

https://arxiv.org/pdf/cond-mat/0312135.pdf

Decoherence in Disordered Conductors at Low Temperatures, the effect of Soft Local Excitations

We consider a quantum Hall effect system whose dimension in the Hall-voltage direction is not too large compared with the magnetic length. When a potential (a barrier, random impurities, walls) exists in the current direction, ac oscillations follow with a Josephson-type frequency ω= eV H h . These oscillations exist both in the dc current and voltage directions and their amplitudes are obtained. It is pointed out that in the latter case the amplitude can be larger than in the former one. Shapiro-step phenomena are found to exist with impressed rf radiation. Specific geometrical arrangements are briefly discussed.

Yoseph Imry

Papers

Applications of nonequilibrium Kubo formula to the detection of quantum noise

Critical and tricritical exponents of 3He-4He solutions from light-scattering data

Phase diagrams and dimensionality shift in “Random site” spin glasses

Decoherence in Disordered Conductors at Low Temperatures, the effect of Soft Local Excitations

How to observe the AC Josephson-type oscillation around the quantum hall effect