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 conductance of a sample scattering elastically and coupled to leads with many channels is derived. We assume that all the incident channels on one side of the sample are fed from the same chemical potential. The transmitted and reflected streams are determined by the incident streams through the multichannel scattering properties of the sample. We do not assume that the channels equilibrate with each other. Our result differs from those given earlier by other authors, except for that of Azbel [J. Phys. C 14, L225 (1981)], which is confirmed. We point out that a similar result is obtained for the conductance in a single channel at a temperature above zero. As an application, we obtain the dependence on channel number N of the contributions to the conductance of a small ring, periodic in the Aharonov-Bohm flux through it. Terms whose period is h/e as well as those with period h/2e vary with N as 1/N.

Generalized many-channel conductance formula with application to small rings.

Phase transitions are considered in systems where the field conjugate to the order parameter is static and random. It is demonstrated that when the order parameter has a continuous symmetry, the ordered state is unstable against an arbitrarily weak random field in less than four dimensions. The borderline dimensionality above which mean-field-theory results hold is six. (auth)

Random-Field Instability of the Ordered State of Continuous Symmetry

Preface Preface to the second edition List of symbols 1. Introduction and a brief review of experimental systems 2. Quantum transport, Anderson Localization 3. Dephasing by coupling with the environment, application to Coulomb electron-electron interactions in metals 4. Mesoscopic effects in equilibrium and static properties 5. Quantum interference effects in transport properties, the Landauer formulation and applications 6. The Quantum Hall Effect 7. Mesoscopics with superconductivity 8. Noise in mesoscopic systems 9. Concluding remarks A. The Kubo, linear response, formulation B. The Kubo-Greenwood Conductivity and the Edwards-Thouless Relationships C. The Aharonov-Bohm Effect and the Byers-Yang and Bloch Theorem D. Derivation of matrix elements in the diffusion regime E. Careful treatment of dephasing in 2D conductors at low temperatures F. Anomalies in the density of states (DOS) G. Quasiclassical theory of spectral correlations H. Details of the four-terminal formulation I. Universality of the conductance fluctuations in terms of the universal correlation of transmission eigenvalues J. The conductance of ballistic 'point contacts'

Introduction to mesoscopic physics

Josephson behavior in small normal one-dimensional rings

Early quantum theories of electrical conduction were semiclassical. Electrons were accelerated according to Bloch’s theorem; this was balanced by back scattering due to phonons and lattice defects. Cross sections for scattering, and band structures, were calculated quantum-mechanically, but the balancing process allowed only for occupation probabilities, not permitting a totally coherent process. Also, in most instances, scatterers at separate locations were presumed to act incoherently. Totally quantum-mechanical theories stem from the 1950s, and have diverse sources. Particularly intense concern with the need for more quantum mechanical approaches was manifested in Japan, and Kubo’s formulation became the most widely accepted version. Quantum theory, as described by the Schrodinger equation, is a theory of conservative systems, and does not allow for dissipation. The Schrodinger equation readily allows us to calculate polarizability for atoms, molecules, or other isolated systems that do not permit electrons to enter or leave. Kubo’s linear-response theory is essentially an extended theory of polarizability. Some supplementary handwaving is needed to calculate a dissipative effect such as conductance, for a sample with boundaries where electrons enter and leave (Anderson, 1997). After all, no theory that ignores the interfaces of a sample to the rest of its circuit can possibly calculate the resistance of such a sample of limited extent. Modern microelectronics has provided the techniques for fabricating very small samples. These permit us to study conductance in cases where the carriers have a totally quantum mechanically coherent history within the sample, making it essential to take the interfaces into account. Mesoscopic physics, concerned with samples that are intermediate in size between the atomic scale and the macroscopic one, can now demonstrate in manufactured structures much of the quantum mechanics we associate with atoms and molecules.

Yoseph Imry

Papers

Generalized many-channel conductance formula with application to small rings.

Random-Field Instability of the Ordered State of Continuous Symmetry

Introduction to mesoscopic physics

Josephson behavior in small normal one-dimensional rings

Conductance viewed as transmission