Single-particle potential from resummed ladder diagrams

A recent work on the resummation of fermionic in-medium ladder diagrams to all orders is extended by calculating the complex single-particle potential $U(p,k_f)+ i\,W(p,k_f)$ for momenta $p<k_f$ as well as $p>k_f$. The on-shell single-particle potential is constructed by means of a complex-valued in-medium loop that includes corrections from a test-particle of momentum $\vec p$ added to the filled Fermi sea. The single-particle potential $U(k_f,k_f)$ at the Fermi surface as obtained from the resummation of the combined particle and hole ladder diagrams is shown to satisfy the Hugenholtz-Van-Hove theorem. The perturbative contributions at various orders $a^n$ in the scattering length are deduced and checked against the known analytical results at order $a^1$ and $a^2$. The limit $a\to\infty$ is studied as a special case and a strong momentum dependence of the real (and imaginary) single-particle potential is found. This indicates an instability against a phase transition to a state with an empty shell inside the Fermi sphere such that the density gets reduced by about 5%. For comparison, the same analysis is performed for the resummed particle-particle ladder diagrams alone. In this truncation an instability for hole-excitations near the Fermi surface is found at strong coupling. For the set of particle-hole ring diagrams the single-particle potential is calculated as well. Furthermore, the resummation of in-medium ladder diagrams to all orders is studied for a two-dimensional Fermi gas with a short-range two-body contact-interaction.