NATO Advanced Study Institute

ESF Network

16 - 26 Février/February 1999

Défauts Topologiques et Dynamique des Transitions de Phase avec Brisure de Symétrie

Topological Defects and the Non-Equilibrium Dynamics of Symmetry-Breaking Phase transitions

Lectures:

Introduction

Tom Kibble (London),

Classification of Topological Defects and their Relevance to Cosmology and elsewhere.

(3 lectures)

These lectures will review symmetry-breaking phase transitions and the formation of topological defects, primarily in the context of cosmology but also with reference to condensed matter. Following a discussion of the basic ideas of spontaneous symmetry breaking, the classification of defects in terms of homotopy groups of the vacuum manifold will be reviewed, covering domain walls, cosmic strings or vortices, monopoles and textures and also composite objects of various kinds. The idea that early in its history the Universe went through a series of phase transitions will be discussed. The importance of the central problem of estimating the density of defects formed at a phase transition will be emphasized, with reference both to cosmology and to recent low-temperature experiments.

Wojciech Zurek (Los Alamos),

Dynamics of Symmetry Breaking (3 Lectures)

Special part: Cosmology:

Daniel Boyanovsky (Pittsburgh)

Lecture 1: i) Symmetry breaking from microphysics: spinodal decomposition in quantum systems, the main idea. ii) Implementing a scheme to understand quantum phase transitions: density matrices and probability description. iii) Connection with condensed matter physics: Large N and the Time Dependent Landau Ginzburg model without conservation. iv) Similarities and differences between quantum and classical descriptions: comparison of predictions.

Lecture 2: i) More on spinodal instabilities: particle production and phase coexistence. ii) Topological defects or coherent states of particles. iii) From quantum to classical physics: semiclassical description and probability interpretation. iv)Scaling, causality and condensates.

Lecture 3: i) Expanding universe: Big Bang 101 (the basics), ii) quantum mechanics and phase transitions in an expanding universe. iii) Phase separation and spinodal instabilities in FRW cosmologies. iv) ``Defect formation'' and redshift of the distribution. v) Scaling solutions and correlation functions, comparison with a TDGL description borrowed from condensed matter.

Tanmay Vachaspati (Cleveland)

Lecture 1. Defect Formation in First Order Phase Transitions

I will describe defect formation in strongly first order phase transitions in which bubbles nucleate and grow to form domains of the new phase. The structure of the domain lattice is vital to the network of defects formed at the phase transition. I will also describe how the analysis is also applicable to the distribution of galaxies.

Lectures 2&3. Observational Signatures of Cosmic Topological Defects

The observable signatures of topological defects in cosmology will be discussed.

Brandon Carter (Paris),

Mechanical elements and cosmological applications of vorton theory.

Ana Achucarro (Bilbao),

Lecture 1 An introduction to magnetic monopoles.

An elementary introduction to the subject of magnetic monopoles and a review of the status of experimental searches for their existence.

Lectures 2 Magnetic monopoles and vortices in particle accelerators.

The structure and properties of magnetic monopoles and vortices in the Glashow-Salam-Weinberg model of electroweak interactions will be discussed, and their observability in future accelerator experiments. These defects fall outside the usual topological classification of defects. Current numerical simulations of the formation of a network of such defects during a phase transition will be reviewed.

Ruth Durer (Geneve)

Defects and the large scale structure of the universe.

Special part: 3He as a test system:

Grigoriy Volovik (Moscow and Helsinki)

3He and Universe parallelism (3 lectures)

There are three main classes of homogeneous Fermi-systems, which are characterized by the topology of the fermionic spectrum: (1) Systems with the gap in the spectrum; (2) Gapless systems with Fermi-surface; (3) Gapless systems with topologically stable point nodes. Superfluid 3He-A and electroweak vacuum belong to the universality class (3). The fermionic quasparticles (particles) in this class are chiral. The collective bosonic modes of the system of class (3) are the effective gauge and gravitational fields. This allowed us to model in 3He-A experiments such phenomena as axial anomaly, baryoproduction and magnetogenesis. The 3He-A textures induce the nontrivial effective metrics of the space, in which the free quasiparticles move along geodesics. One can simulate the conical singularities, torsion strings, event horizons, Hawking radiation, etc.

Anthony Leggett (Urbana):

3He and its relation with modern physics.

Lecture 1: Basic description of liquid 3-He, Landau Fermi-liquid picture of the normal state, experimental parameters. Idea of Cooper pairing, effective interaction in 3-He, BCS wave function.

Lecture 2: General properties of an anisotropic Cooper-paired state,with special attention to the passage to a Ginzburg-Landau level of description.

Lecture 3: Specific properties of 3-He A and B, with emphasis on the spin and orbital dynamics and on the various kinds of defects.

Douglas Osheroff (Stanford):

The superfluid phase transitions in liquid 3He.

Lecture 1: Liquid 3He is perhaps the purest and most isotropic fluid known to man at temperatures a few mK above absolute zero. Yet starting at 2.49mK, depending upon sample pressure, the liquid undergoes a series of phase transitions to p-wave BCS states, with three known bulk states having been observed. These states are all anisotropic to some degree, and exhibit behavior which had never been predicted prior to their discovery.The speaker shall describe the unusual properties of these phases from an experimental prospective, in a talk which will introduce the subject prior to a series of more technical (and theoretical) discussions by Tony Leggett.

Lecture 2: There are three stable bulk phases of superfluid 3He. While the transition at Tc is a rather mean-field like second order transition, the transition from the superfluid A phase to the B phase is first order, and this transition can be strongly supercooled. Measurements of the interfacial surface energy between the two phases and the buld free energy difference between them suggests that nucleation of the low temperature B phase cannot occur due to homogeneous nucleation, and that the critical radius for growth of the B phase from a small droplet within the A phase is macroscopic (of order 1 micrometer). It has been found that ionizing radiation can bring about the transition, however the nature of the transition depends strongly upon the degree of surface roughness in contact with the superfluid. The speaker will discuss what is known experimentally about the behavior of this phase transition, and compare the observed behavior with existing theoretical models where possible

Yuriy Bunkov (Grenoble):

Spin supercurrents and non-linear NMR in superfluid 3He

Owing to its p-pairing 3He can be considered as a superposition of three superfluids (with different projections of the spin) and one normal liquid. The counterflow of superfluid components transports spin without transporting mass. This unique process can be relevant for cosmology.

Matti Krusius (Helsinki):

Topological defects of 3He superfluids

The two major superfluid 3He phases, 3He-A and 3He-B, display the most intricate broken-symmetry properties which by now are well understood and experimentally accessible. Temperature, pressure, magnetic field, and uniform rotation of the superfluid provide the driving fields as a function of which topologically stable defects of the order parameter field are formed. These have different dimensionality, starting from point-like defects up to 3-dimensional textures. Quantized vorticity in different forms, including formation, structure, and dynamics, has been the most important area of both experimental and theoretical work.

Henri Godfrin (Grenoble)

Grenoble cosmological experiment

The process of creation of cosmic strings shortly after the Big Bang, according to the Kibble-Zurek theory, is analogous to the creation of vortices in liquid Helium after a rapid transition into the superfluid phase. We describe measurements of neutron capture by superfluid $^3$He at ultra-low temperatures which have led to the observation of a deficit in the energy released after the exothermic neutron-3He nuclear reaction. The deficit can be ascribed to the formation of a vortex tangle during the rapid cooling of the liquid 3He. In particular, its magnitude agrees well with the quantitative predictions of Zurek's scenario of the Kibble mechanism.

George Pickett (Lancaster):

Superfluid 3He in the ballistic regime.

Special part: other systems:

Alexandr Andreev (Moscow):

Mesoscopic phase transitions and fundamental properties of space - time.

Adriaan Schakel (Berlin),

Lecture 1: Time-dependent Ginzburg-Landau Theory

The time-dependent Ginzburg-Landau theory is derived for (i) a weakly-interacting Bose gas and (ii) a superconductor both in the weak-coupling as well as in the strong-coupling limit. As starting point, the respective microscopic model is taken. Various aspects of the dynamics of the phase transitions which these systems undergo, such as the time scale of the nucleation of the condensate, are discussed.

Lecture 2: Dual Theory

The concept of duality is exemplified by deriving the dual theory of a superconducting film. The magnetic vortices, which in the Ginzburg-Landau formulation are described as topological defects, become the elementary excitations of the dual formulation.

Lecture 3: Bose-Einstein Condensation

The Bose-Einstein condensation of a free Bose gas is described as the proliferation of strings. The connection with Feynman's space-time approach to the lambda transition in 4He is discussed.

Giuseppe Vitiello (Salerno),

"Defect formation through the boson condensation mechanism in QFT"

(2 lectures)

Kinks, vortices, monopoles are extended objects, or defects, of quantum origin with topologically non-trivial properties and macroscopic behavior. They are described in Quantum Field Theory in terms of non-homogeneous boson condensation. I will review the related QFT formalism, the spontaneous breakdown of symmetry framework in which the defects appear and discuss finite temperature effects, also in connection with phase transition problematics.

James Sauls (Evanston),

Quantum fluids far from equilibrium

Eric Varoquaux (Saclay)

Nucleation and unpinning of vortices in superfluid 4He

Lecture 1: will give an overview of superfluid vortices, mainly in 4He. A great deal of information about these stable hydrodynamical entities has been inferred from a maze of experimental observations. Most of the experimental studies involving the behaviour of isolated vortices comes from phase slippage experiments. Some of the experimental evidence will be presented. In particular, it has led to the recognition: 1) that these defects can be nucleated both by thermal fluctuations or through quantum tunnelling; 2) or that they can go forth and multiply in self-sustaining tangles.

Lecture 2: will cover the mechanism for phase slips, its relation to the uperfluid Josephson effects, the half-ring nucleation model, the influence of 3He impurities in 4He and their use as microscopic probe of the nucleation process, the effect of wall contamination which brings about the phenomena of flow collapses (i.e. sudden bursts of vorticity) and vortex pinning.

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