arXiv:0711.0425v1 [cond-mat.supr-con] 3 Nov 2007 Superfluidity and phase transitions in a resonant Bose gas Leo Radzihovsky 1 , Peter B. Weichman 2 , and Jae I. Park 1,3 1 Department of Physics, University of Colorado, Boulder, CO 80309 2 BAE Systems, Advanced Information Technologies, 6 New England Executive Park, Burlington, MA 01803 and 3 National Institute of Standards and Technology, 325 Broadway, Boulder, Colorado 80305-3328 (Dated: October 22, 2018) 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, with the latter distinguished by the absence of atomic off-diagonal long-range order, gapped atomic excitations, and deconfined atomic π-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. PACS numbers: Valid PACS appear here I. INTRODUCTION A. Background Remarkable experimental advances in manipulating degenerate atomic gases have opened a new era in stud- ies of highly coherent, interacting quantum many-body systems. One of the most striking advances is the ability to finely control atomic two-body interactions by tuning with a magnetic field the energy (detuning) of the molecular Feshbach resonance (FR) through the atomic continuum. 1,2 This technique has led to a realiza- tion of a long-sought-after s-wave paired superfluidity in bosonic 3,4 and fermionic atomic gases. 5,6,7 For fermionic atoms, it also allowed the system to be tuned between the BCS 8 regime of weakly-paired, strongly overlapping Cooper pairs (familiar from solid-state superconductors), and the BEC regime of tightly bound, weakly-interacting Bose-condensed diatomic molecules. Although this crossover has received considerable attention, 9,10,11,12,13,14,15 because of the absence of qual- itative differences between the BCS and BEC s-wave paired fermionic superfluids, their equilibrium proper- ties are already qualitatively well described by early sem- inal works. 16,17,18 In fact for a narrow FR (unfortunately not realized by most current experimental systems), the crossover can even be computed quantitatively, as a per- turbation series in the ratio of the FR width to the Fermi energy. 13,15 In such narrow FR systems the crossover to BEC takes place when the FR detuning ν (quasi- molecule’s rest energy) ranges from twice the Fermi en- ergy 2ǫ F (when it first becomes favorable to convert a finite fraction of the Fermi-sea into molecules stabi- lized by Pauli-blocking) down to zero energy, where all the fermions have become bound into Bose-condensed diatomic molecules. The complementary broad reso- nance regime of most experiments, 19 particularly near a universal unitary point 20 has been successfully stud- ied using quantum Monte Carlo 21,22,23 and field theo- retic ǫ-expansion 24,25 and 1/N -expansion 25,26 methods borrowed from critical phenomena. As was recently pointed out 27,28 and is the subject of this paper, the phenomenology of resonantly inter- acting degenerate bosonic atoms contrasts strongly and qualitatively with this picture. 29 For a large positive de- tuning, molecules are strongly energetically suppressed and unpaired atoms (as in any bosonic system at zero temperature) form an atomic superfluid (ASF), exhibit- ing atomic off-diagonal long-range order (ODLRO). 30 In the opposite extreme of a large negative detuning, free atoms are strongly disfavored (gapped), pairing up into stable bosonic molecules, that then, at T = 0, form a diatomic molecular superfluid characterized by a molec- ular ODLRO. The MSF does not exhibit atomic ODLRO, nor the associated atomic superfluidity. Together with a gapped atomic excitation spectrum and correlation func- tions (characteristics that extend to finite temperature), these features qualitatively distinguish it from the ASF. In a trapped, dilute atomic gas the existence of these two qualitatively distinct superfluid phases should be most directly detectable through independent images of atomic and molecular density profiles. As illustrated in Fig. 1(a), the atomic component should exhibit a BEC peak in the ASF phase, that is absent in the MSF phase, shown in Fig. 1(b). Both superfluid phases are distin- guished from the normal state by the BEC peak in the molecular density profile, as illustrated in the insets to these figures. Because of its paired nature, a complementary dis- tinguishing characteristic of a MSF are deconfined π− (half-) vortices, topological defects that, in contrast, are linearly confined in the ASF state. Consequently, as illustrated in Fig. 2, a thermodynamically sharp quan- tum phase transition, at an intermediate critical Fesh- bach resonance detuning ν c , must separate the MSF and ASF phases. Each in turn is also separated by a finite-