1. Theory of high-temperature superconductivity
2. Nanoscale superconductivity
3. Strong electron correlations in solids
4. The Electron-phonon Interaction in Solids
5. Spin current-induced Spin-flip interactions

All five of these topics are closely related; nonetheless a separate description is provided for each:

1. Theory of high-temperature superconductivity

There is no general consensus on the mechanism of superconductivity in the high temperature oxide materials. Our work has focused on understanding various anomalous superconducting and normal state properties of these materials, both in terms of the conventional electron-phonon mechanism of superconductivity, and in terms of other novel mechanisms. Primary amongst these has been the so-called "hole-mechanism" of superconductivity, first proposed by J.E. Hirsch. We have written a number of papers since the late 1980's, which work out many of the consequences of this mechanism. Ongoing work is directed towards discovering the "smoking gun" signature which will point towards the correct theory for high temperature superconductivity.

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2. Nanoscale Superconductivity

In 1959, Richard Feynman wrote, in a remarkably prophetic account, entitled "There's Plenty of Room at the Bottom" that in the future it would be possible to read and write information on a tiny scale. Much of what he foresaw has now been realized in the last two decades of the 20th century. Our group has begun to examine the process of miniaturization on superconductivity. What has to be modified? First, the Bardeen, Cooper, and Schrieffer (BCS) Theory of Superconductivity is based on the grand canonical ensemble. Since present day experiments can detect the difference between an even and an odd number of electrons in ultrasmall superconducting grains the grand canonical ensemble is clearly not adequate. We have investigated the predictions of a CANONICAL formulation of BCS theory (link to paper), where even/odd effects, for example, emerge naturally. A next obvious requirement for a proper description of superconductivity in small nanograins is a correct description of surfaces and impurities. There are several formulations one can use; we have adopted the Bogoliubov-de Gennes formalism. These calculations resemble BCS calculations, except that the order parameter (and other quantities of interest) are allowed to acquire a spatial dependence. Thus, even the occurence of surfaces results in a dramatic change in the order parameter (link to paper), particularly in the case where the order parameter has d-wave symmetry. We have also examined the effects of single impurities, where, locally, they give rise to a strong suppression of the order parameter. As our ability to fabricate small superconducting nanograins improves, and our probes for observing the effect of superconductivity in these grains (for example the scanning tunnelling microscope (STM)) improve in resolution, the theory will have to keep pace to critically examine the many fascinating properties of these grains.

Such a formalism is also required to properly describe the results of surface-sensitive probes of the high temperature superconductors, such as recent STM work and photoemission spectroscopy.

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3. Strong Electron Correlations in Solids

The motivation for this work comes from the high-temperature superconductors, though, in many ways this represents a long-standing unsolved problem in general. Perhaps because of the quasi-two-dimensional nature of the cuprates, or perhaps because of the high critical temperatures involved, fluctuations appear to play an important role in these systems above Tc. One possible source of these fluctuations is the tendency to pair itself, a contribution that can be summarized by the ladder diagrams in the particle-particle channel. In collaboration with Robert Gooding our group has been examining a systematic approach to this problem in the low density limit. This requires going beyond the simple ladder sum of diagrams, i.e. including renormalization or self-consistent effects. As a first step, long range order is suppressed properly in two dimensions by including self-consistent feedback on the single electron Green functions. We are currently tackling the dynamics, and have already utilized a variety of promising techniques (link to paper A) (link to paper B) (link to paper C). The long term goal will be to have a completely controlled diagrammatic theory in the low density limit.

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4. The Electron-phonon Interaction in Solids

(a) conventional superconductors

In collaboration with Jules Carbotte we have examined various possible signatures of the electron-phonon interaction in conventional electron-phonon superconductors. The primary source of evidence for this interaction comes from the tunneling inversion procedure (McMillan and Rowell, in Superconductivity (edited by Parks), 1969). We have recently extended early work by Allen, Phys. Rev. B3, 305 (1971) and Farnworth and Timusk Phys. Rev. B10, 2799 (1974), to show how the optical conductivity can also be inverted, in the normal state, for K₃C₆₀, to infer the electron-phonon interaction in the fullerene family.

(b) polarons

In more extreme cases of very strong electron-phonon interaction, the electrons and phonons form quasiparticles called polarons. Recent advances have been made using numerical diagonalization and strong coupling techniques. Examples of work in this area are The Spectral Function of a One-Dimensional Holstein Polaron, by F. Marsiglio, in Phys. Lett. A180, 280-284 (1993) and Pairing in the Holstein Model in the Dilute Limit, in Physica C244, 21-34 (1995). In this latter paper retardation effects are examined in the Holstein-Hubbard Hamiltonian.

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5. Spin current-induced Spin-flip interactions

Much of the computer hard drive memory business relies on the ability to flip magnetic spins quickly. One new approach, often discussed in the past decade, is to use spin currents to perform the flipping. This process can be described by the Landau-Lifshitz-Gilbert-based approach, a semiclassical formalism that works reasonably well in certain parameter regimes. However, to gain further insight and to probe regimes where the semi-classical approach is likely to break down, we have performed fully quantum mechanical calculations to describe the spin-flip process of Heisenberg-coupled spins subjected to spin current wave packets. A density-matrix-based approach is required to avoid entanglement difficulties. Recent work is in ( EPL paper) and in (PRB paper ), and references therein.

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