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

After reviewing the ideal Bose-Einstein gas in a box and in a harmonic trap, the effect of interactions on the formation of a Bose-Einstein condensate are discussed, along with the dynamics of small-amplitude perturbations (the Bogoliubov equations). When the condensate rotates with angular velocity $\ensuremath{\Omega}$, one or several vortices nucleate, leading to many observable consequences. With more rapid rotation, the vortices form a dense triangular array, and the collective behavior of these vortices has additional experimental implications. For $\ensuremath{\Omega}$ near the radial trap frequency ${\ensuremath{\omega}}_{\ensuremath{\perp}}$, the lowest-Landau-level approximation becomes applicable, providing a simple picture of such rapidly rotating condensates. Eventually, as $\ensuremath{\Omega}\ensuremath{\rightarrow}{\ensuremath{\omega}}_{\ensuremath{\perp}}$, the rotating dilute gas is expected to undergo a quantum phase transition from a superfluid to various highly correlated (nonsuperfluid) states analogous to those familiar from the fractional quantum Hall effect for electrons in a strong perpendicular magnetic field.

Rotating trapped Bose-Einstein condensates

The physics of neutron star crusts is vast, involving many different research fields, from nuclear and condensed matter physics to general relativity. This review summarizes the progress, which has been achieved over the last few years, in modeling neutron star crusts, both at the microscopic and macroscopic levels. The confrontation of these theoretical models with observations is also briefly discussed.

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Physics of Neutron Star Crusts

Rotating relativistic stars have been studied extensively in recent years, both theoretically and observationally, because of the information they might yield about the equation of state of matter at extremely high densities and because they are considered to be promising sources of gravitational waves. The latest theoretical understanding of rotating stars in relativity is reviewed in this updated article. The sections on equilibrium properties and on nonaxisymmetric oscillations and instabilities in f-modes and r-modes have been updated. Several new sections have been added on equilibria in modified theories of gravity, approximate universal relationships, the one-arm spiral instability, on analytic solutions for the exterior spacetime, rotating stars in LMXBs, rotating strange stars, and on rotating stars in numerical relativity including both hydrodynamic and magnetohydrodynamic studies of these objects.

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Rotating Stars in Relativity

The first measurements on vortices in rotating superfluid $^{3}\mathrm{He}$ have been conducted in the Low Temperature Laboratory at Helsinki University of Technology during the past five years. These experiments have revealed unique vortex phenomena that are not observed in any other known superfluids. In this review, the concept of broken symmetry is applied to investigate the quantized vortex lines in superfluid $^{3}\mathrm{He}$. In the superfluid $A$ phase, vorticity can be supported by a continuous winding of the order parameter; this gives rise to continuous "coreless" vortices with two flow quanta. Novel vortices with a half-integer number of circulation quanta may also exist in $^{3}\mathrm{He}$-$A$ due to a combined symmetry of the superfluid state. In the superfluid $B$ phase, the vortices have a complicated core structure. The vortex-core matter is ferromagnetic and superfluid, and it displays broken parity. The ferromagnetism of the core is observed in NMR experiments due to a gyromagnetic effect. The calculated core structures exhibit an experimentally observed first-order phase transition. This vortex-core transition in rotating $^{3}\mathrm{He}$-$B$ may be understood in terms of a change in the topology for flaring-out of the vortex singularity into higher dimensions; the topological identification further suggests that the phase transition manifests a spontaneous bifurcation of vorticity---involving half-quantum vortices in $^{3}\mathrm{He}$-$B$. These recent advances of interest in quantum liquids are also of general relevance to a wide range of fields beyond low-temperature physics.

Quantized vortices in superfluid He-3

This review covers the progress in the study of vortex oscillations in rotating superfluids. The paper deals with the theory as its principal concern, but the experiments that one can compare with the theory considered are also discussed. Attention is focused mainly on the effects of crystalline order in the vortex lattice (the Tkachenko waves especially) and on the boundary problems arising in studies of vortex oscillations in finite containers. The approach is based mostly on the continuum hydrodynamic theory dealing with dense vortex arrays, and considerable attention is devoted to discussion of this theory in order to understand better the principles upon which the obtained results rest. The theory is traced from the simple description of a rotating classical fluid with continuous vorticity, through that of a perfect fluid with quantized vorticity in the form of an array of vortex lines, then the two-fluid theory of an isotropic superfluid, and finally the theory of rotating anisotropic superfluids such as $^{3}\mathrm{He}$-$A$. Applications of the theory to He II, the superfluid phases of $^{3}\mathrm{He}$, and the superfluid neutron matter in pulsars are discussed.

E.B. Sonin

Papers

Vortex oscillations and hydrodynamics of rotating superfluids