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Low-temperature behavior of density-functional theory for metals based on density-functional perturbation theory and Sommerfeld expansion

The temperature dependence of most solid-state properties is dominated by lattice vibrations, but metals display notable purely electronic effects at low temperature, such as the linear specific heat and the linear entropy, that were derived by Sommerfeld for the non-interacting electron gas via the low-temperature expansion of Fermi-Dirac integrals. Here we treat temperature as a perturbation within density-functional perturbation theory (DFPT). For finite temperature, we show how self-consistency screens the bare, temperature-induced density change obtained in the non-interacting picture: the inverse transpose of the electronic dielectric operator, that includes Adler-Wiser and a term related to the shift in Fermi level, links the self-consistent density response to the bare thermal density change. This approach is implemented in DFTK, and demonstrated by the computation of the second-order derivative of the free energy, and the first-order derivative of entropy for aluminum. Then, we examine the $T\!\to\!0$ limit. The finite temperature formalism contains divergences, that we cure using the Sommerfeld expansion to analyze metallic systems at 0 K. The electronic free energy is quadratic in $T$ provided the Fermi level is not at a Van Hove singularity of the density of states. If the latter happens, another temperature behavior might appear, depending on the type of Van Hove singularity, that we analyze. Our formulation applies to systems periodic in one, two, or three dimensions, and provides a basis for studying temperature-dependent electronic instabilities (e.g., charge-density waves) within density-functional theory and DFPT.