arXiv:2205.04564 · Physical Review A

Three and four identical fermions near the unitary limit

Michael D. Higgins, Chris H. Greene

Published Aug 5, 2022• 11 citations

Abstract

This work analyzes the three and four equal-mass fermionic systems near and at the $s$- and $p$-wave unitary limits using hyperspherical methods. The unitary regime addressed here is where the two-body dimer energy is at zero energy. For fermionic systems near the $s$-wave unitary limit, the hyperradial potentials in the $N$-body continuum exhibit a universal long-range ${R}^{\ensuremath{-}3}$ behavior governed by the $s$-wave scattering length alone. The implications of this behavior on the low-energy phase shift are discussed. At the $p$-wave unitary limit, the four-body system is studied through a qualitative look at the structure of the hyperradial potentials at unitarity for the ${L}^{\ensuremath{\pi}}={0}^{+}$ symmetry. A quantitative analysis shows that there are tetramer states in the lowest hyperradial potentials for these systems. Correlations are made between these tetramers and the corresponding trimers in the two-body fragmentation channels. Universal properties related to the four-body recombination process $A+A+A+A\ensuremath{\leftrightarrow}{A}_{3}+A$ are discussed.

DOI: 10.1103/physreva.106.023304

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arXiv:2205.04564 · Physical Review A

Three and four identical fermions near the unitary limit

Michael D. Higgins, Chris H. Greene

Published Aug 5, 2022• 11 citations

arXiv HTML PDF

Abstract

This work analyzes the three and four equal-mass fermionic systems near and at the

s

- and

p

-wave unitary limits using hyperspherical methods. The unitary regime addressed here is where the two-body dimer energy is at zero energy. For fermionic systems near the

s

-wave unitary limit, the hyperradial potentials in the

N

-body continuum exhibit a universal long-range

{R}^{\ensuremath{-}3}

behavior governed by the

s

-wave scattering length alone. The implications of this behavior on the low-energy phase shift are discussed. At the

p

-wave unitary limit, the four-body system is studied through a qualitative look at the structure of the hyperradial potentials at unitarity for the

{L}^{\ensuremath{\pi}}={0}^{+}

symmetry. A quantitative analysis shows that there are tetramer states in the lowest hyperradial potentials for these systems. Correlations are made between these tetramers and the corresponding trimers in the two-body fragmentation channels. Universal properties related to the four-body recombination process

A+A+A+A\ensuremath{\leftrightarrow}{A}_{3}+A

are discussed.

DOI: 10.1103/physreva.106.023304

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Journal of Physics Conference SeriesMar 1, 20183 citations

This work analyzes the three and four equal-mass fermionic systems near and at the

s

- and

p

-wave unitary limits using hyperspherical methods. The unitary regime addressed here is where the two-body dimer energy is at zero energy. For fermionic systems near the

s

-wave unitary limit, the hyperradial potentials in the

N

-body continuum exhibit a universal long-range

{R}^{\ensuremath{-}3}

behavior governed by the

s

-wave scattering length alone. The implications of this behavior on the low-energy phase shift are discussed. At the

p

-wave unitary limit, the four-body system is studied through a qualitative look at the structure of the hyperradial potentials at unitarity for the

{L}^{\ensuremath{\pi}}={0}^{+}

A+A+A+A\ensuremath{\leftrightarrow}{A}_{3}+A

are discussed.

We study the behavior of the bound-state energy of a system consisting of two identical heavy fermions of mass

M

and a light particle of mass

m

. The heavy fermions interact with the light particle through a short-range two-body potential with positive

s

-wave scattering length

{a}_{s}

. We impose a short-range boundary condition on the logarithmic derivative of the hyperradial wave function and show that, in the regime where Efimov states are absent, a nonuniversal three-body state cuts through the universal three-body states previously described by Kartavtsev and Malykh [O. I. Kartavtsev and A. V. Malykh, J. Phys. B 40, 1429 (2007)]. The presence of the nonuniversal state alters the behavior of the universal states in certain regions of the parameter space. We show that the existence of the nonuniversal state is predicted accurately by a simple quantum defect theory model that utilizes hyperspherical coordinates. An empirical two-state model is employed to quantify the coupling of the nonuniversal state to the universal states.

Equal mass fermionic trimers with two different spin components near the unitary limit are shown to possess a universal van der Waals bound or resonance state near

s

-wave unitarity, when

p

-wave interactions are included between the particles with equal spin. Our treatment uses a single-channel Lennard-Jones interaction with long-range two-body van der Waals potentials. While it is well known that there is no true Efimov effect that would produce an infinite number of bound states in the unitary limit, we demonstrate that another type of universality emerges for the symmetry ${L}^{\mathrm{\ensuremath{\Pi}}}={1}^{\ensuremath{-}}$. The universality is a remnant of Efimov physics that exists in this system at

p

-wave unitarity, and it leads to modified threshold and scaling laws in that limit. Application of our model to the system of three lithium atoms studied experimentally by Du et al. [Phys. Rev. Lett. 102, 250402 (2009)] provides an interpretation of their measured three-body recombination loss rates.

a

. To this aim, we use the hyperspherical adiabatic expansion and we extract the

S

matrix through the integral relations recently derived. The results are compared to the universal form, $\ensuremath{\alpha}\ensuremath{\approx}67.1{\mathrm{sin}}^{2}[{s}_{0}\mathrm{ln}({\ensuremath{\kappa}}_{*}a)+\ensuremath{\gamma}]$, for different values of

a

and selected values of the three-body parameter ${\ensuremath{\kappa}}_{*}$. A good agreement with the universal formula is obtained after introducing a particular type of finite-range corrections, which have been recently proposed [A. Kievsky and M. Gattobigio, Phys. Rev. A 87, 052719 (2013)]. Furthermore, we analyze the validity of the above formula in the description of a very different system: neutron-neutron-proton recombination. Our analysis confirms the universal character of the process in systems of very different scales having a large two-body scattering length.