Single-particle potential from resummed ladder diagrams

Abstract

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 ) 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 → ∞ is studied as a special case and a strong momentum dependence of the real (and imaginary) single-particle potential is found. This feature 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%. The imaginary single-particle potential vanishes linearly at the Fermi surface. 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.