Resummation of Diagrammatic Series with Zero Convergence Radius for Strongly Correlated Fermions

We demonstrate that a summing up series of Feynman diagrams can yield unbiased accurate results for strongly correlated fermions even when the convergence radius vanishes. We consider the unitary Fermi gas, a model of nonrelativistic fermions in three-dimensional continuous space. Diagrams are built from partially dressed or fully dressed propagators of single particles and pairs. The series is resummed by a conformal-Borel transformation that incorporates the large-order behavior and the analytic structure in the Borel plane, which are found by the instanton approach. We report highly accurate numerical results for the equation of state in the normal unpolarized regime, and reconcile experimental data with the theoretically conjectured fourth virial coefficient.

Figure 1

First-order diagram (left) and an example of 9th-order diagram (right) for the single-particle self-energy. Each line represents a single-particle propagator while each box is a bosonic pair propagator.

Figure 2

Conformal mapping: the singularities of the Borel transform (in color) are mapped onto the unit circle. The two points $z_{\pm }$ are mapped onto $w_{\pm }$ and the real positive axis is mapped onto the segment [0, 1].

Figure 3

Resummed density vs maximal diagram order at $\beta \mu =0$ ($T/T_F\approx 0.6$). The ladder and bold diagrammatic schemes agree with each other and with experiment.

Figure 4

Density vs maximal diagram order at $\beta \mu =2$ ($T/T_F\approx 0.2$). The bold diagrammatic series is resummed by three variants of the conformal-Borel transformation (see text).

Figure 5

Equation of state and 4th virial coefficient: The difference between the density $n$ and its 3rd order virial expansion $n_{\text{virial}}^{(3)}$, divided by the appropriate factor, must tend to the 4th virial coefficient $b_4$ in the high-temperature limit $\zeta \to 0$.