We present a comprehensive derivation of a set of universal relations for spin-orbit-coupled Fermi gases in three or two dimensions, which follow from the short-range behavior of the two-body physics. Besides the adiabatic energy relations, the large-momentum distribution, the grand canonical potential, and pressure relation derived in our previous work for three-dimensional systems [Phys. Rev. Lett. 120, 060408 (2018)], we further derive high-frequency tail of the radio-frequency spectroscopy and the short-range behavior of the pair correlation function. In addition, we also extend the derivation to two-dimensional systems with Rashba-type spin-orbit coupling. To simply demonstrate how the spin-orbit-coupling effect modifies the two-body short-range behavior, we solve the two-body problem in the sub-Hilbert space of zero center-of-mass momentum and zero total angular momentum, and we perturbatively take the spin-orbit-coupling effect into account at short distance, since the strength of the spin-orbit coupling should be much smaller than the corresponding scale of the finite range of interatomic interactions. The universal asymptotic forms of the two-body wave function at short distance are then derived, which are independent on the short-range details of interatomic potentials. We find that new scattering parameters emerge in the universal asymptotic forms, apart from the traditional
s- and
p-wave scattering length (volume) and effective ranges. This is a general and unique feature for spin-orbit-coupled systems. We show how these two-body parameters characterize the universal relations in the presence of spin-orbit coupling. This work probably sheds light on understanding the profound properties of the many-body quantum systems in the presence of the spin-orbit coupling.
We explore the two-body spectra of spin-
1/2 fermions in isotropic harmonic traps with external spin-orbit potentials and short-range two-body interactions. Using a truncated basis of total angular momentum eigenstates, nonperturbative results are presented for experimentally realistic forms of the spin-orbit coupling: a pure Rashba coupling, Rashba and Dresselhaus couplings in equal parts, and a Weyl-type coupling. The technique is easily adapted to bosonic systems and other forms of spin-orbit coupling.
We present exact solutions of a two-body problem for spin-
1/2 fermions with isotropic spin-orbit (SO) coupling and interacting with an arbitrary short-range potential. We find that in each partial-wave scattering channel, the parametrization of a two-body wave function at short interparticle distance depends on the scattering amplitudes of all channels. This reveals the mixed-partial-wave scattering induced by SO couplings. By comparing with results from a square-well potential, we investigate the validity of original pseudopotential models in the presence of SO coupling. We find the
s-wave pseudopotential provides a good approximation for low-energy solutions near
s-wave resonances, given the length scale of SO coupling much longer than the potential range. However, near
p-wave resonance the
p-wave pseudopotential gives low-energy solutions that are qualitatively different from exact ones, based on which we conclude that the
p-wave model can not be applied to the fermion system if the SO coupling strength is larger or comparable to the Fermi momentum.
We develop the contact theory for spin-orbit-coupled Fermi gases. By using a perturbation method, we derive analytically the universal two-body behavior at short distance, which does not depend on the short-range details of interatomic potentials. We find that two new scattering parameters need to be introduced because of spin-orbit coupling, besides the traditional s- and p-wave scattering length (volume) and effective ranges. This is a general and unique feature for spin-orbit-coupled systems. Consequently, two new adiabatic energy relations with respect to the new scattering parameters are obtained, in which a new contact is involved because of spin-orbit coupling. In addition, we derive the asymptotic behavior of the large-momentum distribution, and find that the subleading tail is corrected by the new contact. This work paves the way for exploring the profound properties of spin-orbit-coupled many-body systems, according to two-body solutions.
The coupling between the spin degrees of freedom and the orbital angular momentum has a profound effect on the properties of nuclei, atoms, and condensed-matter systems. Recently, synthetic gauge fields have been realized experimentally in neutral cold-atom systems, giving rise to a spin-orbit coupling term with ``strength''
kso. This paper investigates the interplay between the single-particle spin-orbit coupling term of Rashba type and the short-range two-body
s-wave interaction for cold atoms under external confinement. Specifically, we consider two different harmonically trapped two-atom systems. The first system consists of an atom with spin-orbit coupling that interacts with a structureless particle through a short-range two-body potential. The second system consists of two atoms that both feel the spin-orbit coupling term and that interact through a short-range two-body potential. Treating the spin-orbit term perturbatively, we determine the correction to the ground-state energy for various generic parameter combinations. Selected excited states are also treated. An important aspect of our study is that the perturbative treatment is not limited to small
s-wave scattering lengths but provides insights into the system behavior over a wide range of scattering lengths, including the strongly interacting unitary regime. We find that the interplay between the spin-orbit coupling term and the
s-wave interaction generically enters, depending on the exact parameter combinations of the
s-wave scattering lengths, at order
kso2 or
kso4 for the ground state and leads to a shift of the energy of either sign. While the absence of a term proportional to
kso follows straightforwardly from the functional form of the spin-orbit coupling term, the absence of a term proportional to
kso2 for certain parameter combinations is unexpected. The well-known fact that the spin-orbit coupling term couples the relative and center-of-mass degrees of freedom has interesting consequences for the trapped two-particle systems. For example, we find that, for certain parameter combinations, the spin-orbit coupling term turns sharp crossings into avoided crossings with an energy splitting proportional to
kso. Our perturbative results are confirmed by numerical calculations that expand the eigenfunctions of the two-particle Hamiltonian in terms of basis functions that contain explicitly correlated Gaussians.
We study the short-range correlation strength of three-dimensional spin-half dilute atomic Fermi gases with spin-orbit coupling. The interatomic interaction is modeled by the contact pseudopotential. In the high-temperature limit, we derive the expression for the second-order virial expansion of the thermodynamic potential via the ladder diagrams. We further evaluate the second-order virial expansion in the limit that the spin-orbit coupling constants are small and find that the correlation strength between the fermions increases as the fourth power of the spin-orbit coupling constants. At zero temperature, we consider the cases in which there are symmetric spin-orbit couplings in two or three directions. In such cases, there is always a two-body bound state of zero net momentum. In the limit that the average interparticle distance is much larger than the dimension of the two-body bound state, the system primarily consists of condensed bosonic molecules that fermions pair to form; we find that the correlation strength also becomes bigger compared to that in the absence of spin-orbit coupling. Our results indicate that generic spin-orbit coupling enhances the short-range correlations of the Fermi gases. Measurement of such enhancement by photoassociation experiment is also discussed.