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Journal de Physique IV (Proceedings)

Vortex matter in two dimensions: An introduction from a solid state physics viewpoint

Benoît Douçot

Published Oct 1, 2004• 0 citations
Publisher

Abstract

Motivated by possible applications to cold atomic gases in a rotating trap, we give an overview of some important concepts in two-dimensional vortex physics. In the context of Josephson junction arrays, we first discuss the crucial role played by vortices in both the thermal superconducting to normal metal transition, and the zero temperature superconducting to Mott insulator transition. Important notions on vortex dynamics in these arrays are then presented, with emphasis on experimental results indicating the quantum nature of vortex motion at low temperature. We move then to the Fractional Quantum Hall effect. as observed in electronic systems, and introduce the notions of fractional excitations, generalized statistics, and composite fermions, which have played a key role in the theoretical description of this remarkable phenomenon. Very likely, cold atomic gases in a rotating trap will soon provide a way to explore the bosonic version of the Fractional Quantum Hall effect, and thus shedding new light on these subtle strongly correlated quantum fluids.

DOI
10.1051/jp4:2004116006

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Motivated by possible applications to cold atomic gases in a rotating trap, we give an overview of some important concepts in two-dimensional vortex physics. In the context of Josephson junction arrays, we first discuss the crucial role played by vortices in both the thermal superconducting to normal metal transition, and the zero temperature superconducting to Mott insulator transition. Important notions on vortex dynamics in these arrays are then presented, with emphasis on experimental results indicating the quantum nature of vortex motion at low temperature. We move then to the Fractional Quantum Hall effect. as observed in electronic systems, and introduce the notions of fractional excitations, generalized statistics, and composite fermions, which have played a key role in the theoretical description of this remarkable phenomenon. Very likely, cold atomic gases in a rotating trap will soon provide a way to explore the bosonic version of the Fractional Quantum Hall effect, and thus shedding new light on these subtle strongly correlated quantum fluids.
Motivated by possible applications to cold atomic gases in a rotating trap, we give an overview of some important concepts in two-dimensional vortex physics. In the context of Josephson junction arrays, we first discuss the crucial role payed by vortices in both the thermal superconducting to normal metal transition, and the zero temperature superconducting to Mott insulator transition. Important notions on vortex dynamics in these arrays are then presented, with emphasis on experimental results indicating the quantum nature of vortex motion at low temperature. We move then to the Fractional Quantum Hall effect, as observed in electronic systems, and introduce the notions of fractional excitations, generalized statistics, and composite Fermions, which have played a key role in the theoretical description of this remarkable phenomenon. Very likely, cold atomic gases in a rotating trap will soon provide a way to explore the bosonic version of the Fractional Quantum Hall effect, and thus shedding new light on these subtle strongly correlated quantum fluids.
Ultracold atomic gases in rapidly rotating traps display a wealth of interesting physics. One of the most striking achievements in this field is the observation of th e Abrikosov lattice of vortices. When the rotation frequency is equal to the harmonic trap frequency, the atomic gases is similar to charged particles in a magnetic f ield. It has been predicted that for weak interactions and strong confinement along the rotation axis, this system will exhibit quantum phases equivalent to the one o f the fractional quantum Hall effect. The talk will be focused on bosonic atoms with only one hyperfine species (i.e spinless bosons). We will give numerical evidences of the fractional quantum Hall states in these systems. We will describe how they can be understood using the Laughlin wave function and the Jain's composite fermion theory. We will also discuss about more exotic quantum phases that have also been predicted in such regime. These quantum phases are described by the Read-Rezayi wave functions whose one of the most remarkable properties is that their excitations obey non-abelian statistics.
The main theme of this review is the many-body physics of vortices in quantum droplets of bosons or fermions in the limit of small particle numbers. Systems of interest include cold atoms in traps as well as electrons confined in quantum dots. When set to rotate, these in principle different quantum systems show remarkable analogies. The topics reviewed include the structure of the finite rotating many-body state, universality of vortex formation and localization of vortices in both bosonic and fermionic systems, and the emergence of particle-vortex composites in the quantum Hall regime. An overview of the computational many-body techniques sets focus on the configuration-interaction and density-functional methods. Studies of quantum droplets with one or several particle components, where vortices as well as coreless vortices may occur, are reviewed, and theoretical as well as experimental challenges are discussed.
The study of the correlated motion of electrons in solids is of increasing importance in condensed matter physics. In the past few years, the discovery of high-temperature superconductors has stimulated an enormous theoretical effort in this area, building on earlier theories of heavy-fermion and organic superconductors, and magnetic insulators. In a separate development, the discovery of the fractional quantum Hall effect stimulated research into the behaviour of the two-dimensional electron gas in a strong transverse magnetic field. The lectures give a systematic presentation of the current status of the theory in these areas. They cover the fractional quantum Hall effect and the many-body physics of the Hubbard model and its extension, paying particular attention to the properties of doped insulators which are relevant for high-temperature superconductivity. They detail discussions of situations for which controlled calculations may be carried out - specifically infinite dimensions, one dimension, and generalized models in which the fermions have N components and N to infinity.
The properties of phase transitions and the types of order present in the low-temperature states of matter are fundamentally dependent on the dimensionality of physical systems. Generally, highly ordered states are more robust in higher dimensions, while thermal and quantum fluctuations, which favour disordered states, play a more important role in lower dimensions. The case of a two-dimensional (2d) Bose fluid is particularly fascinating because of its "marginal" behaviour. In an infinite uniform 2d fluid thermal fluctuations at any non-zero temperature are strong enough to destroy the fully ordered state associated with Bose--Einstein condensation, but are not strong enough to suppress superfluidity in an interacting system at low, but non-zero temperatures. Further, the presence of residual "quasi-long-range" order at low temperatures leads to an interesting interplay between superfluidity and condensation in all experimentally relevant finite-size systems. In these notes we give an introduction to the physics of 2d Bose fluids from an atomic physics perspective. Our goal is to summarize the recent progress in theoretical understanding and experimental investigation of ultra-cold atomic gases confined to 2d geometry, and we also hope to provide a useful introduction to these systems for researchers working on related topics in other fields of physics.
The close theoretical analogy between the physics of rapidly rotating atomic Bose condensates and the quantum Hall effect (i.e. a two-dimensional electron gas in a strong magnetic field) was first pointed out ten years ago. As a consequence of this analogy, a large number of strongly correlated quantum-Hall-type states have been predicted to occur in rotating Bose systems, and suggestions have been made for how to manipulate and observe their fractional quasiparticle excitations. Due to a very rapid development in experimental techniques over the past years, experiments on BEC now appear to be close to reaching the quantum Hall regime. This paper reviews the theoretical and experimental work done to date in exploring quantum Hall physics in cold bosonic gases. Future perspectives are discussed briefly, in particular the idea of exploiting some of these strongly correlated states in the context of topological quantum computing.