A Career in Theoretical Physics

The following sections are included:

• INTRODUCTORY ESSAY

• CONTENTS

Please refer to full text.

Please refer to full text.

Please refer to full text.

The phase transition of BaTiO₃ is remarkably free of fluctuation effects relative to other similar cooperative transitions. This is related to the fact that the singularity which causes it involves only a very few of the modes of motion of the lattice, the remainder staying regular and harmonic. Various experimental consequences of these ideas are explored. The concept that the individual ions “rattle” in multiple-minimum potentials is shown to be misleading.

A B.C.S. type of theory (see BARDEEN, COOPER and SCHREIFFER, Phys. Rev. 108, 1175 (1957)) is sketched for very dirty superconductors, where elastic scattering from physical and chemical impurities is large compared with the energy gap. This theory is based on pairing each one-electron state with its exact time reverse, a generalization of the k up, –k down pairing of the B.C.S. theory which is independent of such scattering. Such a theory has many qualitative and a few quantitative points of agreement with experiment, in particular with specific-heat data, energy-gap measurements, and transition-temperature versus impurity curves. Other types of pairing which have been suggested are not compatible with the existence of dirty superconductors.

The energy gap and other parameters of the superconducting state are calculated from the Bardeen-Cooper-Schrieffer theory in Gor'kov-Eliashberg form, using a realistic retarded electron-electron interaction via phonons and including the Coulomb repulsion. The solution is facilitated by observing that only the local phonon interaction, mediated entirely by short-wavelength phonons, is important, and that a good approximation for the phonon spectrum is therefore an Einstein model rather than Debye model. The resulting equation is solved by an approximate iteration procedure. The results are similar to earlier gap equations but the derivation gives a precise meaning to the interaction and cutoff parameters of earlier theories. The numerical results are in good order-of-magnitude agreement with the observed transition temperatures but lead to an isotope effect at least 15% less than the accepted exponent (T_c proportional to ). Also, the present theory predicts that all metals should be superconductors, although those not observed to do so would have remarkably low transition temperatures.

Schwinger has pointed out that the Yang-Mills vector boson implied by associating a generalized gauge transformation with a conservation law (of baryonic charge, for instance) does not necessarily have zero mass, if a certain criterion on the vacuum fluctuations of the generalized current is satisfied. We show that the theory of plasma oscillations is a simple nonrelativistic example exhibiting all of the features of Schwinger's idea. It is also shown that Schwinger's criterion that the vector field m≠0 implies that the matter spectrum before including the Yang-Mills interaction contains m=0, but that the example of superconductivity illustrates that the physical spectrum need not. Some comments on the relationship between these ideas and the zero-mass difficulty in theories with broken symmetries are given.

These are notes on lectures presented at the Banff Summer School of Theoretical Physics. They were transcribed mostly by Dr. J. M. Vail, directly from tapes of the lectures as given, and my rough notes. After a background discussion of superconductivity, they review the microscopic derivation of the Ginzburg-Landau equations and the fundamental paper of Abrikosov on type II superconductors.

Please refer to full text.

Please refer to full text.

Please refer to full text.

Please refer to full text.

First, we show that the most important equations of the dynamics of the two types of superfluids, He II and superconductors, follow quite directly from the simple assumption that the quantum field of the particles has a mean value which may be treated as a macroscopic variable. The background of this ansatz is also discussed. Second, we apply these equations to various physical situations in He II, notably the orifice geometry and the superfluid film, and show how they, and particularly the idea of phase slippage accompanying all dissipative processes, can be applied and what kinds of macroscopic interference phenomena may be expected. The effect of synchronization in the ac interference experiment is discussed.

Please refer to full text.

Please refer to full text.

Please refer to full text.

MACROSCOPIC COHERENCE AND SUPERFLUIDITY. The problem of the coherence in a superfluid, namely a substance which possesses ODLRO in the Yang sense, is treated, with a particular account of the problem of dissipation in the superfluid. The relation between dissipation and quantized vortices is also explained, both for superfluids and superconductors.

Please refer to full text.

The essential difficulties of the problem of the ground state of the Kondo Hamiltonian have been cleared up by: (1) showing the equivalence of to the partition function of a classical one-dimensional statistical problem with long-range interactions. (In one version, an Ising “n = 2” problem). The two parameters of the anisotropic Kondo problem, J_±ρ and J_Zρ, go into two parameters of the classical problem, T and the ratio of short to long-range interaction. (2) Then a set of “scaling laws” are derived which connect different parameter sets to each other (this may also be done by “cutoff renormalization” or “renormalization group” techniques in the original Kondo problem). All ferromagnetic Kondo problems are equivalent, and are equivalent to the classical Ising problem at its critical point; the nature of the singularity which prevents solution of the Kondo problem by perturbation methods is thus revealed. The antiferromagnetic Kondo problem, on the other hand, scales into the Ising problem at high temperatures, which is nonsingular and rather trivial. The equivalent Kondo problem, however, is in its strong coupling limit, so that again perturbation theory in J is not relevant. The low temperature behavior of the isolated Kondo system is highly nonsingular and thus it appears that many of the experiments are dominated by interaction effects.

Using current superconductivity theory and some assumptions about the normal state properties of solids, estimates of the maximum superconducting transition temperature are made. The optimum resonant frequency for an attractive interaction, the role of umklapp scattering, and the appearance of lattice instabilities are discussed.

The excitonic mechanism of superconductivity in a metal -semiconductor system is studied from the point of view of the complete electron-electron interaction. It is shown that recent calculations of an enhancement of the superconducting transition temperature by way of virtual excitons involves a double counting of these processes. Once this is taken into account the enhancement disappears. The local-field effects in the semiconductor are discussed but it is shown that the off-diagonal elements of the interaction are no help in recovering the enhanced superconductivity.

Please refer to full text.

The following sections are included:

• Introduction

• The Kondo Problem: A Discrete Path-Integral Approach

• Eliminating the Fermi Gas

• The Method of Schotte and Schotte

• Summing Up over the Paths

• Finite Temperatures

• Numerical Results

• The Scaling Method

• The Kondo Temperature

• Physical Implications

• “Renormalization Group” Methods in the Kondo Problem

• References

Please refer to full text.

Please refer to full text.

I have proposed, and Mott and others elaborated, a model of amorphous semiconductors in which there is a fairly high density of localized centers near E_F which have effectively negative U and can hence accommodate zero or two electrons, in conjunction with a mobility gap which is of order | U |. Two new aspects of these centers will be mentioned here: (1) Varma and Pandey (private communication) have proposed that such centers form at metal-insulator contacts and there may be direct experimental evidence for them. They could be responsible for the wellknown Fermi level pinning effect ascribed by Bardeen to surface states (2). Several arguments suggest that the one-electron gap itself will be a function of the pair state occupation. If so, and the relaxation rates of these states are very slow and cover a broad range as expected, they may be seen to lead to : (1) 1/f resistance noise; (2) Long-period photo-electric phenomena ; (3) Switching.

Please refer to full text.

Please refer to full text.

Please refer to full text.

After a discussion of the general physics of neutron stars, we give a brief discussion of the ‘glitch’ phenomenon and its relation to superfluidity, and finally a rather detailed study of the physics of vortex-line pinning in the crust lattice.

I survey recent progress in the field of localization. Theory and experiment are in excellent agreement in two-dimensional systems such as inversion layers and thin metal films. The metal insulator transition in three-dimensional systems remains an interesting problem as exemplified by highly accurate data on doped semiconductors, but on the metallic side perturbational theory has had a number of successes.

The subject of so-called chemical or localized pseudopotentials will be reviewed. These pseudopotentials are based on the concept of Wannier functions and one can derive a self -consistent wave equation which such localized functions satisfy. As pointed out by Adams and by Gilbert, the optimum such functions are not orthonormalized, as was remarked by Wannier himself many years ago. These functions have been the subject of many successful chemical calculations by D.J. Bullett, which will be reviewed.

Please refer to full text.

Please refer to full text.

The following sections are included:

• 1. An approach to the Bohr-Einstein debate

• 2. Canonical Stern-Gerlach experiment

• 3. Phase variable of superfluid helium. Broken symmetry

• 4. Conclusions

• References

• Discussion

Please refer to full text.

Please refer to full text.

Please refer to full text.

Please refer to full text.

Please refer to full text.

The educated layman is used to thinking of science as having aesthetic values in two senses. Often he can recognize the grandeur and sweep of the scientific vision: the cosmological overview of the universe, the long climb of evolution towards complexity, the slow crunch of the tectonic plates, the delicately concentrated energy of the massive accelerator. Also, many visual images from science have aesthetic meaning: images of galaxies, of the complex structures of crystals or of the double helix, the fascinating diversity of organisms and their traces in Nature. What I want to discuss here, however, is the internal, intellectual aesthetic of science, which is often what the scientist himself alludes to when he calls a certain piece of science “sweet” or “beautiful”. This is very often a comprehension of internal intellectual connections among diverse phenomena or even fields of science — that the same intellectual structure, for instance, may govern the formation of elementary particles and the flow of electricity in a superconducting wire; anoiher may relate a complex magnetic alloy with the functioning of neuronal circuits. In summary, I will try to describe what the scientist (or, at least, one scientist) finds beautiful in science.

The following sections are included:

Please refer to full text.

Please refer to full text.

We give an historical discussion of the “infrared catastrophe” and the “x-ray edge anomalies” of Mahan associated with scatterers in a Fermi sea of electrons. The infrared catastrophe provides a perspicuous way into understanding the difficulties with manybody perturbation theory which have recently been discovered as a result of a study of high T_c superconductivity, and we show how this “catastrophe” is avoided in some cases, but cannot be avoided in the one and 2-dimensional electron gas systems. Finally, we indicate the new type of theory which is necessary in the event of such a breakdown.

Analysis of the many experiments on high-temperature superconductivity indicate several essential aspects of any theory. The conductivity and other transport properties as a function of disorder, temperature, and frequency point to a non–Fermi liquid–like behavior, whereas photoemission experiments and magnetic properties indicate the presence of a Fermi surface in momentum space. To reconcile this apparent contradiction, a new type of electron liquid, called a Luttinger liquid, has been postulated, and the present article aims to show the need for this postulate. Theory and experiment indicate that the suitable phenomenological electronic structure model of the CuO layers is that of the one-band Hubbard model. It is also argued that experiment clearly indicates that interlayer interactions strongly affect the superconducting transition temperature, T_c, consistent with the fact that no theoretical calculations on two-dimensional Hubbard models have resulted in the prediction of high transition temperatures, and that anyon models are not favored by experiment.