The Perturbative Approach at Zero Temperature

Jaime Merino, A. Levy Yeyati

S. Ciccariello, A. De Col

Using the zero-temperature perturbative expansion, we show how to obtain the correct value of the magnetic susceptibility of a system of free fermions at zero temperature. To this aim, it is first observed that the Feynman rules for the perturbative expansion of the Green function substantially do not change when the ground state is substituted with a generic eigenstate of the unperturbed system. Then, the requirement that the resulting Green function has the correct Lehmann representation uniquely determines the unperturbed eigenstate which yields the interacting ground state by adiabatically switching on the interaction.

Corbinian Wellenhofer

The order-by-order renormalization of the self-consistent mean-field potential in many-body perturbation theory for normal Fermi systems is investigated in detail. Building on previous work mainly by Balian and de Dominicis, as a key result we derive a thermodynamic perturbation series that manifests the consistency of the adiabatic zero-temperature formalism with perturbative statistical mechanics---for both isotropic and anisotropic systems---and satisfies at each order and for all temperatures the thermodynamic relations associated with Fermi-liquid theory. These properties are proved to all orders.

Wolfgang Nolting

Yi Sun, Hugh G. A. Burton

We investigate the convergence properties of finite-temperature perturbation theory by considering the mathematical structure of thermodynamic potentials using complex analysis. We discover that zeros of the partition function lead to poles in the internal energy and logarithmic singularities in the Helmholtz free energy that create divergent expansions in the canonical ensemble. Analyzing these zeros reveals that the radius of convergence increases at higher temperatures. In contrast, when the reference state is degenerate, these poles in the internal energy create a zero radius of convergence in the zero-temperature limit. Finally, by showing that the poles in the internal energy reduce to exceptional points in the zero-temperature limit, we unify the two main mathematical representations of quantum phase transitions.