A concise, somewhat personal, review of the problem of superfluidity and quantum criticality in regular and disordered interacting Bose systems is given, concentrating on general features and important symmetries that are exhibited in different parts
of the phase diagram, and that govern the different possible types of critical behavior. A number of exact results for various insulating phase boundaries, which may be used to constrain the results of numerical simulations, can be derived using large rare region type arguments. The nature of the insulator-superfluid transition is explored through general scaling arguments, exact model calculations in one dimension, numerical results in two dimensions, and approximate renormalization group results in higher dimensions. Experiments on He-4 adsorbed in porous Vycor glass, on thin film superconductors, and magnetically trapped atomic vapors in a periodic optical potential, are used to illustrate many of the concepts.
We study the role of particle-hole symmetry on the universality class of various quantum phase transitions corresponding to the onset of superfluidity at zero temperature of bosons in a quenched random medium. The functional integral formulation of t
his problem in d spatial dimensions yields a (d+1)-dimensional classical XY-model with extended disorder--the so-called random rod problem. Particle-hole symmetry may then be broken by adding nonzero site energies. We may distinguish three cases: (i) exact particle-hole symmetry, in which the site energies all vanish, (ii) statistical particle-hole symmetry in which the site energy distribution is symmetric about zero, vanishing on average, and (iii) complete absence of particle-hole symmetry in which the distribution is generic. We explore in each case the nature of the excitations in the non-superfluid Mott insulating and Bose glass phases. We find that the Bose glass compressibility, which has the interpretation of a temporal spin stiffness or superfluid density, is positive in cases (ii) and (iii), but that it vanishes with an essential singularity as full particle-hole symmetry is restored. We then focus on the critical point and discuss the relevance of type (ii) particle-hole symmetry breaking perturbations to the random rod critical behavior. We argue that a perturbation of type (iii) is irrelevant to the resulting type (ii) critical behavior: the statistical symmetry is restored on large scales close to the critical point, and case (ii) therefore describes the dirty boson fixed point. To study higher dimensions we attempt, with partial success, to generalize the Dorogovtsev-Cardy-Boyanovsky double epsilon expansion technique to this problem. The qualitative renormalization group flow picture this technique provides is quite compelling.
The atomic Bose gas is studied across a Feshbach resonance, mapping out its phase diagram, and computing its thermodynamics and excitation spectra. It is shown that such a degenerate gas admits two distinct atomic and molecular superfluid phases, wit
h the latter distinguished by the absence of atomic off-diagonal long-range order, gapped atomic excitations, and deconfined atomic pi-vortices. The properties of the molecular superfluid are explored, and it is shown that across a Feshbach resonance it undergoes a quantum Ising transition to the atomic superfluid, where both atoms and molecules are condensed. In addition to its distinct thermodynamic signatures and deconfined half-vortices, in a trap a molecular superfluid should be identifiable by the absence of an atomic condensate peak and the presence of a molecular one.
