Perturbing finite temperature multicomponent DFT 1D Kohn-Sham systems: Peierls gap & Kohn anomaly

One of the greatest challenges when designing new technologies that make use of non-trivial quantum materials is the difficulty associated with predicting material-specific properties, such as critical temperature, gap parameter, etc. There is naturally a great amount of interest in these types of condensed matter systems because of their application to quantum sensing, quantum electronics, and quantum computation; however, they are exceedingly difficult to address from first principles because of the famous many-body problem. For this reason, a full electron-nuclear quantum calculation will likely remain completely out of reach for the foreseeable future. A practical alternative is provided by finite temperature, multi component density functional theory, which is a formally exact method of computing the equilibrium state energy of a many-body quantum system. In this work, we use this construction alongside a perturbative scheme to demonstrate that the phenomena Peierls effect and Kohn anomaly are both natural features of the Kohn-Sham (KS) equations without additional structure needed. We find the temperature dependent ionic density for a simple 1D lattice which is then used to derive the ionic densities temperature dependent affect on the electronic band structure. This is accomplished by Fourier transforming the ionic density term found within this KS electronic equation. Using the Peierls effect phonon distortion gap openings in relation to the Fermi level, we then perturb the KS ionic equation with a conduction electron density, deriving the Kohn anomaly. This provides a workable predictive strategy for interesting electro-phonon related material properties which could be extended to 2D and 3D real materials while retaining the otherwise complicated temperature dependence.