Resummation of fermionic in-medium ladder diagrams to all orders

TL;DR

This paper develops a method to resum fermionic in-medium ladder diagrams to all orders, accurately capturing the energy per particle in fermionic systems and providing a new value for the Bertsch parameter at unitarity.

Contribution

The authors introduce a novel resummation technique for in-medium ladder diagrams that sums contributions to all orders in the scattering length, improving understanding of fermionic many-body systems.

Findings

Reproduces known low-density expansion results up to order a^4.

Derives an all-orders sum expressed as a double integral over an arctangent.

Calculates the Bertsch parameter as approximately 0.5067 at unitarity.

Abstract

A system of fermions with a short-range interaction proportional to the scattering length $a$ is studied at finite density. At any order $a^n$, we evaluate the complete contributions to the energy per particle $\bar E(k_f)$ arising from combined (multiple) particle-particle and hole-hole rescatterings in the medium. This novel result is achieved by simply decomposing the particle-hole propagator into the vacuum propagator plus a medium-insertion and correcting for certain symmetry factors in the $(n-1)$-th power of the in-medium loop. Known results for the low-density expansion up to and including order $a^4$ are accurately reproduced. The emerging series in $a k_f$ can be summed to all orders in the form of a double-integral over an arctangent function. In that representation the unitary limit $a\to \infty$ can be taken and one obtains the value $\xi= 0.5067$ for the universal Bertsch parameter. We discuss also applications to the equation of state of neutron matter at low densities and mention further extensions of the resummation method.