This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet.

https://iopscience.iop.org/article/10.1088/0022-3719/6/7/010/pdf

Ordering, metastability and phase transitions in two-dimensional systems

The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground-state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.

https://arxiv.org/pdf/cond-mat/9806038.pdf

Theory of Bose-Einstein condensation in trapped gases

This review summarizes recent developments in the theory of spin glasses, as well as pertinent experimental data. The most characteristic properties of spin glass systems are described, and related phenomena in other glassy systems (dielectric and orientational glasses) are mentioned. The Edwards-Anderson model of spin glasses and its treatment within the replica method and mean-field theory are outlined, and concepts such as "frustration," "broken replica symmetry," "broken ergodicity," etc., are discussed. The dynamic approach to describing the spin glass transition is emphasized. Monte Carlo simulations of spin glasses and the insight gained by them are described. Other topics discussed include site-disorder models, phenomenological theories for the frozen phase and its excitations, phase diagrams in which spin glass order and ferromagnetism or antiferromagnetism compete, the Ne\'el model of superparamagnetism and related approaches, and possible connections between spin glasses and other topics in the theory of disordered condensed-matter systems.

Spin glasses: Experimental facts, theoretical concepts, and open questions

Statistical physics has proven to be a fruitful framework to describe phenomena outside the realm of traditional physics. Recent years have witnessed an attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. A wide list of topics are reviewed ranging from opinion and cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, and social spreading. The connections between these problems and other, more traditional, topics of statistical physics are highlighted. Comparison of model results with empirical data from social systems are also emphasized.

Statistical physics of social dynamics

The critical properties of the xy model with nearest-neighbour interactions on a two-dimensional square lattice are studied by a renormalization group technique. The mean magnetization is zero for all temperatures, and the transition is from a state of finite to one of infinite susceptibility. The correlation length is found to diverge faster than any power of the deviation from the critical temperature. Analogues of the strong scaling laws are derived and the critical exponents, eta , and delta , are the same as for the two-dimensional Ising model.

https://iopscience.iop.org/article/10.1088/0022-3719/7/6/005/pdf

The critical properties of the two-dimensional xy model

We observe that recent theories of phase transitions in the two-dimensional $\mathrm{XY}$ model predict a universal jump in the superfluid density of $^{4}\mathrm{He}$ films as ${T}_{c}$ is approached from below. Specifically, we find that ${\mathrm{lim}}_{T\ensuremath{\rightarrow}{T}_{c}}\ensuremath{-}{\ensuremath{\rho}}_{s}\frac{(T)}{T}=3.52\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ g/${\mathrm{cm}}^{2}$\ifmmode^\circ\else\textdegree\fi{}K. Analogous results should hold for two-dimensional planar magnets and liquid crystals.

https://link.aps.org/pdf/10.1103/PhysRevLett.39.1201

Universal Jump in the Superfluid Density of Two-Dimensional Superfluids

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.

https://iopscience.iop.org/article/10.1088/0022-3719/5/11/002/pdf

Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory)

Extensive simulations of growth in a stochastic ballistic deposition model on a (d-1)-dimensional substrate with a constraint on neighboring interface heights are described. The interface width obeys scaling even for small systems and grows as ${t}^{\ensuremath{\beta}}$ with \ensuremath{\beta}=1/(d+1). Generalizations to include irrevelant effects such as noise reduction are discussed as are possible reasons for the discrepancies in earlier results.

J. M. Kosterlitz

Papers

Ordering, metastability and phase transitions in two-dimensional systems

The critical properties of the two-dimensional xy model

Universal Jump in the Superfluid Density of Two-Dimensional Superfluids

Long range order and metastability in two dimensional solids and superfluids. (Application of dislocation theory)

Growth in a restricted solid-on-solid model.