Spectral functions of the strongly interacting three-dimensional Fermi gas

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, two-dimensional systems, and systems out of equilibrium.

Figure 1

Time contour $\mathfrak{C}$ used to construct a real-time action.

Figure 2

The $\mathrm{\Phi }$ 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 gray box labeled with $\mathrm{\Gamma }$. The arrows indicate the direction of propagation and the 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 nontrivial structure at small momenta. (a) The bare quadratic dispersion leads to a large change of the energy at high momentum in the original coordinate system. (b) Performing the coordinate transformation makes the energy constant for high momentum.

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 the fermionic imaginary-time propagator of the balanced unitary Fermi gas at $T/T_F=0.2$ and 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). For comparison, also the result obtained with the alternative real-time approach developed in Ref. [69] is shown in red

Figure 6

As for the fermionic propagator in Fig. 5, the difference between the pairing propagator in imaginary times obtained from real frequencies and in Matsubara formalism (blue) is small compared to the absolute value of the paring propagator (black).

Figure 7

The integrated relative difference between single-particle imaginary-time propagators obtained in the Matsubara and Keldysh formalism as defined in Eq. (49) shows very good agreement between the two independent numerical methods. The parameters are equivalent to those in Figs. 5 and 6.

Figure 8

Comparison between single-particle spectral functions at $\mathit{k}=0$. Here ${\mathcal{A}}_{\text{Keldysh}}$ denotes the spectrum obtained by the method developed here and ${\mathcal{A}}_{\text{NAC}}$ the result from numerical analytic continuation explained in Sec. 6. The spectrum ${\mathcal{A}}_{\text{mixed}}$, which agrees almost perfectly with ${\mathcal{A}}_{\text{Keldysh}}$, is obtained by analytic continuation with ${\mathcal{A}}_{\text{Keldysh}}$ as the default model, indicating that the latter is fully consistent with the imaginary-time propagator used for ${\mathcal{A}}_{\text{NAC}}$.

Figure 9

Spectral functions at criticality for different values of the scattering length ${\left(k_{F}a\right)}^{-1}=\{-1,0,1\}$ where the effect of the interactions increases from left to right. The critical temperatures as determined by our methods from left to right are $T_c/T_F=\{0.068,0.156,0.205\}$ and the chemical potential $\mu /{\epsilon }_F=\{0.743,0.407,-0.803\}$ is indicated by the white dashed line. Both agree with the values obtained in imaginary time.

Figure 10

The single-particle density of states of the unitary Fermi gas exhibits a very weak pseudogap only for temperature $T\lesssim 0.19T_F$, which is close to the critical temperature $T_c=0.156T_F$. Here ${\rho }_0\left({\epsilon }_F\right)=m{k}_F/2{\pi }^2$ denotes the density of states of the noninteracting Fermi gas.

Figure 11

Spectral function of the pair susceptibility for the unitary balanced system at the critical temperature $T/T_F=0.156$. The logarithmic color scale highlights structures beyond the singularity at $(k=0,\omega =0)$.