Spectral functions of the strongly interacting 3D Fermi gas

Abstract

Computing dynamical properties of strongly interacting quantum many-body systems poses a major challenge to theoretical approaches. Usually, one has to resort to numerical analytic continuation of results on imaginary frequencies, which is a mathematically ill-defined procedure. Here, we present an efficient method to compute the spectral functions of the two-component Fermi gas near the strongly interacting unitary limit directly in real frequencies. To this end, we combine the Keldysh path integral that is defined in real time with the self-consistent T-matrix approximation. The latter is known to predict thermodynamic and transport properties in good agreement with experimental observations in ultracold atoms. We validate our method by comparison with thermodynamic quantities obtained from imaginary time calculations and by transforming our real-time propagators to imaginary time. By comparison with state-of-the-art numerical analytic continuation of the imaginary time results, we show that our real-time results give qualitative improvements for dynamical quantities. Moreover, we show that no significant pseudogap regime exists in the self-consistent T-matrix approximation above the critical temperature $T_c$, an issue that has been under significant debate. We close by pointing out the versatile nature of our method as it can be extended to other systems, like the spin- or mass-imbalanced Fermi gas, other Bose-Fermi models, 2D systems as well as systems out of equilibrium.

Figures (11)

• Figure 1: Time contour $\mathfrak{C}$ used to construct a real-time action.
• Figure 2: The $\Phi$-functional, the resulting self-energies, and the Dyson equations for both the pairing field and spin-$\uparrow$ fermion. The full fermion propagators are black bold lines, with an arrow indicating the species and the pair propagator is the grey box labeled with $\Gamma$. The arrows indicate the direction of propagation and dashed line is the bare contact interaction for the electrons, while the thin fermion lines represent the bare fermion propagators. The self-energy and Dyson equation for $\downarrow$ is obtained by flipping the spins of the spin-$\uparrow$ counterparts.
• Figure 3: Typical spectral function of an isotropic strongly interacting system with both broadening and a non-trivial structure at small momenta. The bare quadratic dispersion leads to a large change of the energy at large momentum in the original coordinate system shown in a). Performing the coordinate transformation makes the energy constant for large momentum as illustrated in b).
• Figure 4: The fermionic spectral function $\mathcal{A}(0,\omega)$ is known to decay as $\omega^{-5/2}$ and $(-\omega)^{-9/2}$ at very high and low frequencies respectively. Both limits are well reproduced by our method as is shown in the inset. Furthermore, the momentum distribution at high momenta behaves as $n(k \gg k_F)\approx \mathcal{C}/k^{4}$ with $\mathcal{C}$ the Tan contact density obtained from the pair propagator.
• Figure 5: Absolute value of fermionic imaginary time propagator at vanishing momentum $k=0$ (black) compared with the absolute value of the difference between its evaluation directly in imaginary times and via the generalized Laplace transform of the spectral function (blue).