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arXiv:1611.09499 · Physical Review A

Strongly interacting p -wave Fermi gas in two dimensions: Universal relations and breathing mode

Yi‐Cai Zhang, Shizhong Zhang

Published Feb 6, 2017• 37 citations
arXivHTMLPDF

Abstract

The contact is an important concept that characterizes the universal properties of a strongly interacting quantum gas. It appears in both thermodynamic (energy, pressure, etc.) and dynamic quantities (radio-frequency and Bragg spectroscopies, etc.) of the system. Very recently, the concept of contact was extended to higher partial waves; in particular, the $p$-wave contacts have been experimentally probed in recent experiments. So far, discussions on $p$-wave contacts have been limited to three dimensions. In this paper, we generalize the $p$-wave contacts to two dimensions and derive a series of universal relations, including the adiabatic relations, high-momentum distribution, virial theorem, and pressure relation. At the high-temperature and low-density limit, we calculate the $p$-wave contacts explicitly using virial expansion. A formula which directly connects the shift of the breathing-mode frequency and the $p$-wave contacts is given in a harmonically trapped system. Finally, we also derive the relationships between interaction parameters in three- and two-dimensional Fermi gases and discuss possible experimental realization of a two-dimensional Fermi gas with $p$-wave interactions.

DOI
10.1103/physreva.95.023603

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The contact is an important concept that characterizes the universal properties of a strongly interacting quantum gas. It appears in both thermodynamic (energy, pressure, etc.) and dynamic quantities (radio-frequency and Bragg spectroscopies, etc.) of the system. Very recently, the concept of contact was extended to higher partial waves; in particular, the ppp-wave contacts have been experimentally probed in recent experiments. So far, discussions on ppp-wave contacts have been limited to three dimensions. In this paper, we generalize the ppp-wave contacts to two dimensions and derive a series of universal relations, including the adiabatic relations, high-momentum distribution, virial theorem, and pressure relation. At the high-temperature and low-density limit, we calculate the ppp-wave contacts explicitly using virial expansion. A formula which directly connects the shift of the breathing-mode frequency and the ppp-wave contacts is given in a harmonically trapped system. Finally, we also derive the relationships between interaction parameters in three- and two-dimensional Fermi gases and discuss possible experimental realization of a two-dimensional Fermi gas with ppp-wave interactions.
We investigate the properties of a spinless Fermi gas close to a p-wave interaction resonance. We show that the effects of interaction near a p-wave resonance are captured by two contacts, which are related to the variation of energy with the p-wave scattering volume v and with the effective range R in two adiabatic theorems. Exact pressure and virial relations are derived. We show how the two contacts determine the leading and subleading asymptotic behavior of the momentum distribution (∼1/k^{2} and ∼1/k^{4}) and how they can be measured experimentally by radio-frequency and photoassociation spectroscopies. Finally, we evaluate the two contacts at high temperature with a virial expansion.
Abstract We derive the full set of universal relations for spin-polarized Fermi gases with p -wave interaction in two dimensions, simply using the short-range asymptotic behavior of fermion-pair wave functions. For p -wave interactions, an additional contact related to the effective range needs to be introduced, besides the one related to the scattering volume. Since the subleading tail ( k −4 ) of the large-momentum distribution cannot fully be captured by the contacts defined by the adiabatic relations, an extra term resulted from the center-of-mass motions of the pairs gives rise to an additional divergence in the kinetic energy of the system, besides those related to the contacts defined. We show in Tan’s energy theorem that if only two-body correlations are taken into account, all these divergences are reasonably removed, leading to a finite internal energy of the system. In addition, we find that all the other universal relations, such as the high-frequency behavior of the radio-frequency response, short-range behavior of the pair correlation function, generalized virial theorem, and pressure relation, remain unaffected by the center-of-mass motions of the pairs, and are fully governed by the contacts defined by the adiabatic relations. Our results confirm the feasibility of generalizing the contact theory for higher-partial-wave scatterings, and could readily be confirmed in current experiments with ultracold 40 K and 6 Li atoms.
In this Letter, we show that the classical SO(2,1) symmetry of a harmonically trapped Fermi gas in two dimensions is broken by quantum effects. The anomalous correction to the symmetry algebra is given by a two-body operator that is well known as the contact. Taking into account this modification, we are able to derive the virial theorem for the system and a universal relation for the pressure of a homogeneous gas. The existence of an undamped breathing mode is associated with the classical symmetry. We provide an estimate for the anomalous frequency shift of this oscillation at zero temperature and compare the result with a recent experiment by [E. Vogt et al., Phys. Rev. Lett. 108, 070404 (2012)]. Discrepancies are attributed to finite temperature effects.
An important tool for understanding the effects of interactions in harmonically trapped atomic gases is the examination of their collective modes. One such mode is the breathing or monopole mode, which is special as it is constrained to occur at twice the harmonic trapping frequency when the interactions are scale invariant. When the interactions are not scale invariant, the frequency of the breathing mode will deviate from twice the trap frequency. The deviation itself depends on the thermodynamic contacts, which describe how the energy changes with the interactions. In this work I examine how the thermodynamic contacts and the breathing-mode frequency of a spin-polarized one-dimensional (1D) ppp-wave Fermi gas depend on the 1D scattering volume $\ensuremath{\ell}$ and the effective range rrr in the high-temperature limit. Such dynamics can be studied in experiments and provide a tool for understanding how the dynamics depends on interactions with a finite effective range.
We elucidate universal many-body properties of a one-dimensional, two-component ultracold Fermi gas near the ppp-wave Feshbach resonance. The low-energy scattering in this system can be characterized by two parameters, that is, ppp-wave scattering length and effective range. At the unitarity limit where the ppp-wave scattering length diverges and the effective range is reduced to zero without conflicting with the causality bound, the system obeys universal thermodynamics as observed in a unitary Fermi gas with contact sss-wave interaction in three dimensions. It is in contrast to a Fermi gas with the ppp-wave resonance in three dimensions in which the effective range is inevitably finite. We present the universal equation of state in this unitary ppp-wave Fermi gas within the many-body TTT-matrix approach as well as the virial expansion method. Moreover, we examine the single-particle spectral function in the high-density regime where the virial expansion is no longer valid. On the basis of the Hartree-like self-energy shift at the divergent scattering length, we conjecture that the equivalence of the Bertsch parameter across spatial dimensions holds even for a one-dimensional unitary ppp-wave Fermi gas.
Inspired by the renewed experimental activities on ppp-wave resonantly interacting atomic Fermi gases, we investigate theoretically some experimental observables of such systems at zero temperature in two dimensions, using both mean-field theory and Gaussian pair fluctuation theory. These observables include the two ppp-wave contact parameters and the breathing mode frequency, which can be readily measured in current cold-atom setups with 40K^{40}\mathrm{K}40K and 6Li^{6}\mathrm{Li}6Li atoms. We find that the many-body component of the two contact parameters exhibits a pronounced peak slightly above the resonance and consequently leads to a dip in the breathing mode frequency. In the resonance limit, we discuss the dependence of the equation of state and the breathing mode frequency on the dimensionless effective range of the interaction ${k}_{F}{R}_{p}\ensuremath{\ll}1$, where kF{k}_{F}kF​ is the Fermi wave vector and Rp{R}_{p}Rp​ is the effective range. The breathing mode frequency ${\ensuremath{\omega}}_{B}$ deviates from the scale-invariant prediction of ${\ensuremath{\omega}}_{c}=2{\ensuremath{\omega}}_{0}$, where ${\ensuremath{\omega}}_{0}$ is the trapping frequency of the harmonic potential. This frequency shift is caused by the necessary existence of the effective range. In the small range limit, we predict that the mode frequency deviation at the leading order is given by $\ensuremath{\delta}{\ensuremath{\omega}}_{B}\ensuremath{\simeq}\ensuremath{-}({\ensuremath{\omega}}_{0}/4){ln}^{\ensuremath{-}1}({k}_{F}{R}_{p})$.