The Universality and stability for a dilute Bose gas with a Feshbach resonance Yu-Li Lee 1, ∗ and Yu-Wen Lee 2, † $^{1}$Department of Physics, National Changhua University of Education, Changhua, Taiwan, R.O.C. $^{2}$Department of Physics, Tunghai University, Taichung, Taiwan, R.O.C. We study the bosonic atoms with a wide Feshbach resonance at zero temperature in terms of the renormalization group. We indicate that this system will always collapse in the dilute limit. On the side with a positive scattering length, the atomic superfluid is an unstable local minimum in the dilute limit and it determines the thermodynamics of this system within its lifetime. We calculate the equilibrium properties at zero temperature in the unitary regime. They exhibit universal scaling forms in the dilute limit due to the presence of a nontrivial zero temperature, zero density fixed point. Moreover, we find that the T = 0 thermodynamics of this system in the unitary limit is exactly identical to the one for an ideal Fermi gas. PACS numbers: 03.75.-b, 67.85.Jk, 67.10.Ba, The bosonic atoms with Feshbach resonances have been one of the exciting ultrocold systems studied so far. On the side with a negative scattering length, the controlled collapsing and exploding dynamics has been observed $^{1}$. On the other side where the scattering length becomes positive, molecules can be produced and this system becomes a mixture of atoms and molecules. De- spite the reduced lifetime of molecules in the unitary regime due to the enhanced three-body recombination $^{2}$, quite remarkable progress has been made and the prop- erties of the strongly repulsive ultracold bosonic atoms have also been studied experimentally $^{3}$. A mean-field phase diagram for the Bose gas with an s -wave Feshbach resonance has been proposed $^{4}$. At zero temperature, there are two phases: the atomic super- fluid (ASF) in which both atoms and molecules condense and the molecular superfluid (MSF) in which only the molecules condense. Hence, there is a quantum phase transition between the ASF and MSF phases $^{4,5}$. The stability of these phases at T = 0 was examined within the mean-field level $^{6}$. Recently, a Nozi`eres-Schmitt-Rink formalism was developed to study the finite tempera- ture properties of this system $^{7}$. The ground state of the cold Bose gas with a large positive scattering length has also been studied by the variational wavefunction approach $^{8,9}$. Near the Feshbach resonance, various phys- ical quantities were found to exhibit universal scaling forms $^{9}$. Although the idea of the Feshbach resonance has its root in the few-body physics, it was shown that, for a dilute quantum gas, it is possible to cast the low-energy physics near the Feshbach resonance and the associated universal properties into the framework of the renormal- ization group (RG) due to the presence of a nontrivial zero-density fixed point $^{10}$. Bearing this picture in mind, the low-energy properties of a dilute Fermi gas with a wide Feshbach resonance in the unitary regime can be computed either by an ϵ expansion 11 ( ϵ = 4 − d ) or a large N expansion $^{10,12}$. Here we will extend the idea in The rest of this paper is organized as follows: In Sec. II, we present the RG theory to fix the notation. We discuss the stability of the dilute Bose gas with a wide Feshbach resonance and its phase diagram at T = 0 in Sec. III. The equilibrium properties of the ASF state at T = 0 are calculated in Sec. IV. The last section is devoted to a conclusive discussion and comparison with previous works. Ref. 10 to the dilute Bose gas with a wide Feshbach resonance. In contrast to the case of fermionic atoms, the self-interactions between atoms and molecules must be included in order to guarantee the stability of the uniform phases although these operators are irrelevant in the sense of RG. This point was also emphasized in Ref. 6. The resulting stability conditions impose a lower bound on the value of the chemical potential for atoms. Hence, we conclude that a Feshbach-resonant (FR) Bose gas will always collapse in the dilute limit. Moreover, in the dilute limit, the ASF state can appear in the guise of an unstable local minimum for the free energy den- sity, while the MSF state cannot exist. The ASF state will eventually collapse. However, within its lifetime, the properties of this system may be captured by the ASF state when we approach the unitary limit from the side with a positive scattering length. Then, we use the ϵ ex- pansion to calculate various properties of the ASF state at T = 0, including the relation between the chemical potential and detuning, the equation of state, and the sound velocity. We show that the thermodynamical prop- erties at T = 0 exhibit universal scaling forms near the Feshbach resonance due to the presence of a nontrivial zero-temperature, zero-density fixed point. Surprisingly, we find that the T = 0 thermodynamics of the ASF state in the unitary limit is exactly identical to the one for an ideal Fermi gas. This correspondence was also noticed in Refs. 8 and 9 from a different approach. Our main results are summarized in Figs. 1, 2, 3, and 4. arXiv:1001.2592v2 [cond-mat.quant-gas] 11 Jun 2010 I. INTRODUCTION