Synthetic spin-orbit-coupled quantum gases have been widely studied both experimentally and theoretically in the past decade. As shown in previous studies, this modification of single-body dispersion will in general couple different partial waves of the two-body scattering and thus distort the wave function of few-body bound states which determines the short-distance behavior of many-body wave function. In this work, we focus on the two-component Fermi gas with one-dimensional or three-dimensional spin-orbit coupling (SOC) near an
s-wave resonance. Using the method of effective field theory and the operator product expansion, we derive universal relations for both systems, including the adiabatic theorem, viral theorem, and pressure relation, and obtain the momentum distribution matrix $\ensuremath{\langle}{\ensuremath{\psi}}_{a}^{\ifmmode\dagger\else\textdagger\fi{}}(\mathbf{q}){\ensuremath{\psi}}_{b}(\mathbf{q})\ensuremath{\rangle}$ at large
q (
a,b are spin indices). The momentum distribution matrix shows both spin-dependent and spatial anisotropic features. And the large momentum tail is modified at the subleading order thanks to the SOC. We also discuss the experimental implication of these results depending on the realization of the SOC.
In this work, we present an effective-field theory to describe a two-component Fermi gas near a
d-wave-interaction resonance. The effective-field theory is renormalizable by matching with the low-energy
d-wave scattering phase shift. Based on the effective-field theory, we derive universal properties of the Fermi gas by the operator-product-expansion method. We find that beyond the contacts defined by adiabatic theorems, the asymptotic expressions of the momentum distribution and the Raman spectroscopy involve two extra contacts which provide additional information on correlations of the system. Our formalism sets the stage for further explorations of many-body effects in a
d-wave resonant Fermi gas. Finally, we generalize our effective-field theory for interaction resonances of arbitrary higher partial waves.
We study the contacts, large-momentum tail, radio-frequency spectroscopy, and some other universal relations for an ultracold one-dimensional (1D) two-component Fermi gas with spin-orbit coupling (SOC). Different from previous studies, we find that the ${q}^{\ensuremath{-}8}$ tail in the spin-mixing (off-diagonal) terms of the momentum distribution matrix is dependent on the two SOC parameters in the laboratory frame for 1D systems, where
q is the relative momentum. This tail can be observed through time-of-flight measurement as a direct manifestation of the SOC effects on the many-body level. Besides the traditional 1D even-wave scattering length, we find that two new physical quantities must be introduced due to the SOC. Consequently, two new adiabatic energy relations with respect to the two SOC parameters are obtained. Furthermore, we derive the pressure relation and virial theorem at short distances for this system. To find how the SOC modifies the large-momentum behavior, we take the SOC parameters as perturbations since the strength of the SOC should be much smaller than the corresponding strength scale of the interatomic interactions. In addition, by using the operator product expansion method, we derive the asymptotic behavior of the large-momentum distribution matrix up to the ${q}^{\ensuremath{-}8}$ order and find that the diagonal terms of the distribution matrix include the contact of traditional 1D even-wave scattering length as the leading term and the SOC modified terms beyond the leading term, the off-diagonal term is beyond the subleading term and is corrected by the SOC parameters. We also find that the momentum distribution matrix shows spin-dependent and anisotropic features. Furthermore, we calculate the momentum distribution matrix in the laboratory frame for the experimental implication. In addition, we calculate the high-frequency tail of the radio-frequency spectroscopy and find that the presence of the contact related to the center-of-mass momentum in the radio-frequency spectral is due to the SOC effects. This paper paves the way for exploring the profound properties of many-body quantum systems with SOC in one dimension.
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 study the contacts, large-momentum tail, and universal relations for an ultracold one-dimensional (1D) two-component Fermi gas with spin-orbit coupling (SOC). Different from previous studies, we find that the
q−8 tail in the spin-mixing (off-diagonal) terms of the momentum distribution matrix is dependent on the two SOC parameters in the laboratory frame for 1D systems, where
q is the relative momentum. This tail can be observed through time-of-flight measurement as a direct manifestation of the SOC effects on the many-body level. Besides the traditional
s-wave scattering length, we find that two new contacts need to be introduced due to the SOC. Consequently, two new adiabatic energy relations with respect to the two SOC parameters are obtained. Further, we derive the pressure relation and virial theorem at short distance for this system. To find how the SOC modifies the large-momentun behavior, we take the SOC parameters as perturbations, since the strength of the SOC should be much smaller than the corresponding strength scale of the interatomic interactions. In addition, by using the operator product expansion method, we derive the asymptotic behavior of the large-momentum distribution matrix up to the
q−8 order, and find that the diagonal terms of the distribution matrix include the contact of traditional
s-wave scattering length as the leading term and the SOC modified terms beyond the leading term, the off-diagonal term is beyond subleading term and is corrected by the SOC parameters. We also find that the momentum distribution matrix shows spin-dependent and anisotropic features. Furthermore, we calculate the momentum distribution matrix in the laboratory frame for the experimental implication. This work paves the way for exploring the profound properties of many-body quantum systems with SOC in one dimension.
We derive the universal relations for an ultracold two-component Fermi gas with a spin-orbit coupling (SOC) ${\ensuremath{\sum}}_{\ensuremath{\alpha},\ensuremath{\beta}=x,y,z}{\ensuremath{\lambda}}_{\ensuremath{\alpha}\ensuremath{\beta}}{\ensuremath{\sigma}}_{\ensuremath{\alpha}}{p}_{\ensuremath{\beta}}$, where
px,y,z and ${\ensuremath{\sigma}}_{x,y,z}$ are the single-atom momentum and Pauli operators for pseudospin, respectively, and the SOC intensity ${\ensuremath{\lambda}}_{\ensuremath{\alpha}\ensuremath{\beta}}$ could take an arbitrary value. We consider the system with an
s-wave short-range interspecies interaction, and ignore the SOC-induced modification for the value of the scattering length. Using the first-quantized approach developed by Tan [S. Tan, Phys. Rev. Lett. 107, 145302 (2011)], we obtain the short-range and high-momentum expansions for the one-body real-space correlation function and momentum distribution function, respectively. For our system these functions are a $2\ifmmode\times\else\texttimes\fi{}2$ matrix in the pseudospin basis. We find that the leading-order (
1/k4) behavior of the diagonal elements of the momentum distribution function, i.e., ${n}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}(\mathbf{k})$ and ${n}_{\ensuremath{\downarrow}\ensuremath{\downarrow}}(\mathbf{k})$, are not modified by the SOC. However, the SOC can significantly modify the large-
k behaviors of the distribution difference $\ensuremath{\delta}n(\mathbf{k})\ensuremath{\equiv}{n}_{\ensuremath{\uparrow}\ensuremath{\uparrow}}(\mathbf{k})\ensuremath{-}{n}_{\ensuremath{\downarrow}\ensuremath{\downarrow}}(\mathbf{k})$ as well as the nondiagonal elements of the momentum distribution function, i.e., ${n}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}(\mathbf{k})$ and ${n}_{\ensuremath{\downarrow}\ensuremath{\uparrow}}(\mathbf{k})$. In the absence of the SOC, the leading order of $\ensuremath{\delta}n(\mathbf{k})$, ${n}_{\ensuremath{\uparrow}\ensuremath{\downarrow}}(\mathbf{k})$, and ${n}_{\ensuremath{\downarrow}\ensuremath{\uparrow}}(\mathbf{k})$ is
O(1/k6). When SOC appears, it can induce a term on the order of
1/k5 for these elements. We further derive the adiabatic relation and the energy functional. Our results show that the SOC can induce an additional term in the energy functional, which describes the contribution from the SOC to the total energy. In addition, the form of the adiabatic relation for our system is not modified by the SOC. Our results are applicable for the systems with any type of single-atom trapping potential, which could be either diagonal or nondiagonal in the pseudospin basis.