# Universal relations for spin-orbit-coupled Fermi gases in two and three dimensions

## TL;DR

This paper derives universal relations for spin-orbit-coupled Fermi gases in 2D and 3D, revealing new scattering parameters and extending known relations to include spin-orbit effects, aiding understanding of many-body quantum systems.

## Contribution

The work provides a comprehensive derivation of universal relations for spin-orbit-coupled Fermi gases, including new two-body parameters and extension to two dimensions.

## Key findings

- Universal relations derived for 2D and 3D systems.
- Identification of new scattering parameters due to spin-orbit coupling.
- Extension of high-frequency tail and short-range behavior analysis.

## Abstract

We present a comprehensive derivation of a set of universal relations for spin-orbit-coupled Fermi gases in three or two dimension, 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 of 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 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 do not depend on the short-range details of interatomic potentials. We find that 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. We show how these two-body parameters characterize the universal relations in the presence of spin-orbit coupling. This work probably shed light for understanding the profound properties of the many-body quantum systems in the presence of the spin-orbit coupling.