We investigate the ground state of the one-dimensional fermionic system enclosed in a hard-wall trap with attractive contact
p-wave interactions. Based on the Bethe ansatz method, the explicit wave function is derived by numerically solving the Bethe ansatz equations for the full physical regimes $(\ensuremath{-}\ensuremath{\infty}\ensuremath{\le}{c}_{F}\ensuremath{\le}0)$. With the exact wave function some quantities which are important in many-body physics are obtained, including the one-body density matrix and the momentum distribution of the ground state for finite system. It is shown that the shell structure of the density profiles disappears with the increase of the interaction and in the fermionic Tonks-Girardeau limit the density distribution shows the same behavior as that of an ideal Bose gas. However, the one-body density matrix and the momentum distribution exhibit completely different structures compared with their bosonic counterparts.
We investigate theoretically strong-coupling and finite-temperature effects on the
p-wave contacts, as well as the asymptotic behavior of the momentum distribution in the large-momentum region in a one-component Fermi gas with a tunable
p-wave interaction. Including
p-wave pairing fluctuations within a strong-coupling theory, we calculate the
p-wave contacts above the superfluid transition temperature
Tc from the adiabatic energy relations. We show that, while the
p-wave contact related to the scattering volume monotonically increases with increasing interaction strength, the one related to the effective range depends nonmonotonically on interaction strength and its sign changes in the intermediate-coupling regime. The nonmonotonic interaction dependence of these quantities is shown to originate from the competition between the increase of the cutoff momentum and the decrease of the coupling constant of the
p-wave interaction with increasing effective range. We also analyze the asymptotic form of the momentum distribution in the large-momentum region. In contrast to the conventional
s-wave case, we show that the asymptotic behavior cannot be completely described by only the
p-wave contacts, and the extra terms, which are not related to the thermodynamic properties, appear. Furthermore, in the high-temperature region, we find that the extra terms dominate the subleading term of the large-momentum distribution. We also directly compare our results with the recent experimental measurement by including the effects of a harmonic trap potential within the local density approximation. We show that, while our model qualitatively explains the dependence on the interaction strength of the
p-wave contacts, there are some quantitative disagreements between our theory and recent experiment; for example, the peak position of the
p-wave contact related to the effective range. Since the
p-wave contacts connect the microscopic properties to the thermodynamic properties of the system, our results would be helpful for further understanding many-body properties of this anisotropic interacting Fermi gas in the normal phase.
We study odd-wave interacting identical fermions in one dimension with finite effective range. We show that to fully describe the high-momentum distribution $\ensuremath{\rho}(k)$ up to ${k}^{\ensuremath{-}4}$, one needs four parameters characterizing the properties when two particles contact each other. Two parameters are related to the variation of energy with respect to the odd-wave scattering length and the effective range, respectively, determining the ${k}^{\ensuremath{-}2}$ tail and part of the ${k}^{\ensuremath{-}4}$ tail in $\ensuremath{\rho}(k)$. The other two parameters are related to the center-of-mass motion of the system, respectively determining the ${k}^{\ensuremath{-}3}$ tail and the other part of the ${k}^{\ensuremath{-}4}$ tail. We point out that the unusual ${k}^{\ensuremath{-}3}$ tail, which has not been discovered before in atomic systems, is an intrinsic component to complete the general form of $\ensuremath{\rho}(k)$ and also is realistically detectable under certain experimental conditions. Various other universal relations are also derived in terms of those contact parameters, and finally the results are confirmed through the exact solution of a two-body problem.
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.
We investigate the lowest scattering state of a one-dimensional Bose gas with attractive interactions trapped in a hard-wall trap. By solving the Bethe ansatz equation numerically, we determine the full energy spectrum and the exact wave function for different attractive interaction parameters. The resultant density distribution, momentum distribution, reduced one-body density matrix, and two-body correlation show that a decreased attractive interaction induces rich density profiles and specific correlation properties in the weakly attractive Bose gas.
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.