  Including finite-temperature effects from the electronic degrees of freedom
in electronic structure calculations of semiconductors and metals is desired;
however, in practice it remains exceedingly difficult when using
zero-temperature methods, since these methods require an explicit evaluation of
multiple excited states in order to account for any finite-temperature effects.
Using a Matsubara Green's function formalism remains a viable alternative,
since in this formalism it is easier to include thermal effects and to connect
the dynamic quantities such as the self-energy with static thermodynamic
quantities such as the Helmholtz energy, entropy, and internal energy. However,
despite the promising properties of this formalism, little is know about the
multiple solutions of the non-linear equations present in the self-consistent
Matsubara formalism and only a few cases involving a full Coulomb Hamiltonian
were investigated in the past. Here, to shed some light onto the iterative
nature of the Green's function solutions, we self-consistently evaluate the
thermodynamic quantities for a one-dimensional (1D) hydrogen solid at various
interatomic separations and temperatures using the self-energy approximated to
second-order (GF2). At many points in the phase diagram of this system,
multiple phases such as a metal and an insulator exist, and we are able to
determine the most stable phase from the analysis of Helmholtz energies.
Additionally, we show the evolution of the spectrum of 1D boron nitride (BN) to
demonstrate that GF2 is capable of qualitatively describing the temperature
effects influencing the size of the band gap.
