Little attention was paid to Anderson's challenging paper on localization for the first ten years, but from 1968 onwards it generated a lot of interest. Around that time a number of important questions were raised by the community, on matters such as the existence of a sharp distinction between localized and extended states, or between conductors and insulators. For some of these questions the answers are unambiguous. There certainly are energy ranges in which states are exponentially localized, in the presence of a static disordered potential. In a weakly disordered one-dimensional potential, all states are localized. There is clear evidence, in three dimensions, for energy ranges in which states are extended, and ranges in which they are diffusive. Magnetic and spin-dependent interactions play an important part in reducing localization effects. For massive particles like electrons and atoms the lowest energy states are localized, but for massless particles like photons and acoustic phonons the lowest energy states are extended.Uncertainties remain. Scaling theory suggests that in two-dimensional systems all states are weakly localized, and that there is no minimum metallic conductivity. The interplay between disorder and mutual interactions is still an area of uncertainty, which is very important for electronic systems. Optical and dilute atomic systems provide experimental tests which allow interaction to be much less important. The quantum Hall effect provided a system where states on the Fermi surface are localized, but non-dissipative currents flow in response to an electric field.

An inertial mass of a vortex can be calculated by driving it around in a circle with a steadily revolving pinning potential. We show that in the low-frequency limit this gives precisely the same formula that was used by Baym and Chandler, but find that the result is not unique and depends on the force field used to cause the acceleration. We apply this method to the Gross-Pitaevskii model, and derive a simple formula for the vortex mass. We study both the long-range and short-range properties of the solution. We agree with earlier results that the nonzero compressibility leads to a divergent mass. From the short-range behavior of the solution we find that the mass is sensitive to the form of the pinning potential, and diverges logarithmically when the radius of this potential tends to zero.

We give a general review of recent developments in the theory of vortices in superfluids and superconductors, discussing why the dynamics of vortices is important, and why some key results are still controversial. We discuss work that we have done on the dynamics of quantized vortices in a superfluid. Despite the fact that this problem has been recognized as important for forty years, there is still a lot of controversy about the forces on and masses of quantized vortices. We think that one can get unambiguous answers by considering a broken symmetry state that consists of one vortex in an infinite ideal system. We argue for a Magnus force that is proportional to the superfluid density, and we find that the effective mass density of a vortex in a neutral superfluid is divergent at low frequencies. We have generalized some of the results for a neutral superfluid to a charged system.

We study the width-amplitude relation for three-dimensional Bernstein-Greene-Kruskal (BGK) electrostatic solitary waves in magnetized plasmas, taking into account the dynamics of both electrons and ions. We obtain two coupled inequalities that constrain the amplitude and the widths parallel and perpendicular to the magnetic field for a Gaussian potential, and demonstrate how the solution space is further constrained by the finite temperature ratio between electrons and ions. The description is valid for both the electron and ion mode solitary waves. Our results provide a quantitative basis for understanding the ubiquity of BGK waves in widely different classes of collisionless plasmas.

This book describes the key role played by thermally excited defects such as vortices, disclinations, dislocations, vacancies and interstitials in the physics of crystals, superfluids, superconductors, liquid crystals and polymer arrays. Geometrical aspects of statistical mechanics become particularly important when thermal fluctuations entangle or crumple extended line-like or surface-like objects in three dimensions. In the case of entangled vortices above the first-order flux lattice melting transition in hightemperature superconductors, the lines themselves are defects. A variety of theories combined with renormalization-group ideas are used to describe the delicate interplay among defects, statistical mechanics and geometry characteristic of these problems in condensed matter physics. This indispensible guide has its origins in Professor Nelson’s contributions to summer schools, conference proceedings and workshops over the past twenty years. It provides a coherent and pedagogic graduate-level introduction to the field of defects and geometry.

We study the pairing instability and mechanical collapse of a dilute homogeneous Bose gas with an attractive interaction. The pairing phase is found to be a saddle point and is unstable against pairing fluctuations. This pairing saddle point exists above a critical temperature. Below this critical temperature, the system is totally unstable in the pairing channel. Thus the system could collapse in the pairing channel in addition to mechanical collapse. The critical temperatures of pairing instability and mechanical collapse are higher than the Bose-Einstein condensation (BEC) temperature of an ideal Bose gas with the same density. When fluctuations are taken into account, we find that the critical temperature of mechanical collapse is even higher. The difference between the collapse temperature and the BEC temperature is proportional to (n\a(s)\(3))(2/9), where n is the density and a(s) is the scattering length.

We have used two-fluid dynamics to study the discrepancy between the work of Thouless, Ao and Niu (TAN) and that of Iordanskii. In TAN no transverse force on a vortex due to normal fluid flow was found, whereas the earlier work found a transverse force proportional to normal fluid velocity u and normal fluid density. We have linearized the time-independent two-fluid equations about the exact solution for a vortex, and find three solutions which are important in the region far from the vortex. Uniform superfluid flow gives rise to the usual superfluid Magnus force. Uniform normal fluid flow gives rise to no forces in the linear region, but does not satisfy reasonable boundary conditions at short distances. A logarithmically increasing normal fluid flow gives a viscous force. As in classical hydrodynamics, and as in the early work of Hall and Vinen, this logarithmic increase must be cut off by nonlinear effects at large distances; this gives a viscous force proportional to u/ln(u), and a transverse contribution which goes like u/(ln u)^2, even in the absence of an explicit Iordanskii force. In the limit u goes to zero the TAN result is obtained, but at nonzero u there are important corrections that were not found in TAN. We argue that the Magnus force in a superfluid at nonzero temperature is an example of a topological relation for which finite-size corrections may be large.

Dislocation theory is used to define long range order for two dimensional solids. An ordered state exists at low temperatures, and the rigidity modulus is nonzero at the transition temperature. Similar arguments show that the superfluid density is nonzero at the transition temperature of a two dimensional superfluid. Peierls (1934, 1935) has argued that no long range order exists in two dimensional solids because thermal motion of low energy phonons results in a mean square deviation of atoms from their equilibrium positions which increases logarithmically with the size of the system. The absence of long range order of this simple form has been shown rigorously by Mermin (1968). Similar arguments can be used to show that there is no spontaneous magnetization in a two dimensional Heisenberg magnet (Mermin and Wagner 1966) and that the expectation value of the superfluid order parameter in a two dimensional Bose liquid is zero (Hohenberg 1967). Numerical work on a two dimensional system of hard discs by Alder and Wainwright (1962) indicated a phase transition between a gaseous and a solid state. Stanley and Kaplan (1966) found that high temperature series expansions for two dimensional spin models indicated a phase transition at which the magnetic susceptibility becomes infinite. The evidence for such a transition is much stronger for the xy model (spins confined to a plane) than for the Heisenberg model, as can be seen in the papers of Stanley (1968) and Moore (1969). Low temperature expansions obtained by Wegner (1967) and Berezinskii (1970) give a magnetization proportional to some power of the field between zero and unity, and there may be a sharp transition between such behaviour, with infinite magnetic susceptibility, and the high temperature regime. In this paper we argue in favour of a different definition of long range order based on the overall properties of the system rather than on the behaviour of a two-point correlation function. This type of long range order, which we refer to as topological long range order, may exist for the two dimensional solid, neutral superfluid, and for the xy model, but not for a superconductor nor for the isotropic Heisenberg model. In the case of a solid the disappearance of topological long range order is associated with a transition from a rigid to a fluid response to a small external stress, while for a neutral superfluid it is associated with the instability of persistent currents. We have recently learnt that Berezinskii (1 971) has put forward similar arguments, but there are some important differences in our results. The definition of topological long range order which we adopt arises naturally in the

A topological argument is constructed and applied to explain subharmonic mode locking in a system of coupled oscillators with inertia. Via a series of transformations, the system is shown to be described by a classical XY model with periodic bond angles, which is in turn mapped onto a tight-binding particle in a periodic gauge field. It is then revealed that subharmonic quantization of the average phase velocity follows as a manifestation of topological invariance. Ubiquity of multistability and associated hysteresis are also pointed out.

These lecture notes were prepared rather soon after I completed my book on Topological quantum numbers in nonrelativistic physics, which was published by World Scientific Publishing Co. Pte. Ltd., Singapore, in early 1998. I have not attempted to make a completely fresh presentation, but have cannibalized the text of my book to produce something shorter, with a different ordering of topics. I wish to thank the publishers for allowing me to do this self-plagiarization.

D. J. Thouless, M. R. Geller, W. F. Vinen, J.-Y. Fortin, and S. W. Rhee Department of Physics, Box 351560, University of Washington, Seattle, Washington 98195 Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602 School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, England CNRS Laboratoire de Physique Theorique, Universite Louis Pasteur, 67084 Strasbourg Cedex, France (February 1, 2008)

It has been known since the pioneering work of Onsager and Feynman that the statistical mechanics and dynamics of vortices play an essential role in the behavior of superfluids and superconductors. However, the theory of vortices in quantum fluids remains in a most unsatisfactory state, with many conflicting results in the literature. In this paper we review the theory of Thouless, Ao and Niu, which gives an expression for the total transverse force acting on a quantized vortex that is in apparent disagreement with the word of lordanskii and of Lifshitz and Pitaevskii. In particular, no transverse force proportional to the asymptotic normal fluid velocity was found. We use two-fluid hydrodynamics to study this discrepancy.