Recent advances in description of few two-component fermions

Abstract

Overview of the recent advances in description of the few two-component fermions is presented.The model of zero-range interaction is generally considered to discuss the principal aspects of the few-body dynamics. Particular attention is paid to detailed description of two identical fermions of mass m and a distinct particle of mass m 1 : it turns out that two L P = 1 − three-body bound states emerge if mass ratio m/m 1 increases up to the critical value µ c ≈ 13.607, above which the Efimov effect takes place. The topics considered include rigorous treatment of the few-fermion problem in the zero-range interaction limit, low-dimensional results, the four-body energy spectrum, crossover of the energy spectra for m/m 1 near µ c , and properties of potential-dependent states. At last, enlisted are the problems, whose solution is in due course.PACS numbers:

Few-body physics in resonantly interacting ultracold quantum gases

“…In particular, for F F F ′ systems the values for atom-molecule [152,161] and molecule-molecule [87,193] scattering lengths are important for describing the BEC-BCS crossover physics [194,195]. For the cases in which universal KM states [102][103][104][105] can occur (see Section III E), the derivation of the atomdimer scattering length closely follows the derivation of Eq. (114), but replaces s 0 ln(a a * ) → Ψ δ , where Ψ δ is a universal mass-ratio dependent phase [103], and e −4η → e −4η0 = 1 − (1 − e −4η )(r 0 a) 2p0 .…”

“…For a given value of a BX > 0, they found that up to two three-body states can exist for δ c ≤ δ XF ≤ 0.12236. Recent studies [103][104][105] have shown that such Kartartsev-Malykh (KM) states can also occur for BBX (J π ≠ 0-even) systems and other F F X (J π -odd) systems, showing that KM states represent a novel class of three-body states with interesting universal properties. We illustrate the typical energy spectrum of KM states (green dashed lines) in a F F X system in Fig.…”

“…In our model, the effects of universal KM three-body states [102][103][104][105] (see Section III E) can also be incorporated. For BBX (J π > 0-even) and F F X (J π -odd) systems with δ > δ c , the mass ratio dependence of such states can lead to resonant effects in atom-molecule collisions [103].…”

Energy lower bound for the unitary N + 1 fermionic model

“…The correct definition of the model, the occurrence of the Efimov effect and the analysis of the stability problem, i.e., the existence of a lower bound for the Hamiltonian, have been widely studied both in the physical ( [3], [4], [12], [13], [15], [16], [17]) and in the mathematical (see, e.g., [6], [7], [10], [14]) literature. It is well known that in the two-body case the entire class of Hamiltonians with zerorange interaction can be constructed and the spectral properties are completely characterized ( [1]) while, on the opposite, for more than two particles an explicit characterization is still lacking.…”

Ultracold Few-Body Systems

“…(49) show that for BBX systems, the Efimov effect occurs for J even Fig. Such states have since been intensively investigated, and recent studies have shown that they exist for every J -even BBX and J -odd FFX system with δ > δ (J ) c (Nishida et al, 2008;Endo et al, 2011Endo et al, , 2012Kartavtsev and Malykh, 2012). (49)] for both BBX and FFX systems at |a| = ∞.…”