Chuck-Hou Yee edited correlations.tex  over 7 years ago

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The basic feature of correlated materials is their electrons cannot be described as non-interacting particles. Since the behavior of the constituent electrons are strongly coupled to one another, studying the behavior of individual particles generally provides little insight into the macroscopic properties of a correlated material. While this conceptual definition is valuable for understanding the fundamental physics, it is of less use to a DFT practitioner, who seeks to make predictions for comparison with experimental observations. To arrive at an operational definition of a correlated material, we examine DFT and how it relates to the observed electronic spectra.  %% DFT is a workhorse of the materials science community, providing efficient and accurate computations of the total energy and distribution of electrons of a compound, requiring only the coordinates of the atoms in its crystal lattice as input. From the total energy, one can obtain lattice constants, equations of state and the spectrum of lattice vibrations. Furthermore, one can obtain electronic properties such as band gaps, electric polarization and topological numbers, which are by no means trivial for these "simple" compounds.  The idea behind DFT is that for ground state properties, we can use the total electron density $\rho(\vec{r})$ as the fundamental variable in our equations, rather than the complicated quantum many-body wavefunction $\Psi$. Kohn and Sham~\cite{Kohn_1965} recast the theory in practical form: solve the problem of non-interacting electrons in the presence of a periodic potential $V_\text{KS}(\vec{r})$ and the sum of their wavefunctions (squared) will give the exact density. Breaking the Kohn-Sham potential apart, we find three contributions: $V_\text{KS} = V_\text{ion} + V_\text{H} + V_\text{xc}$. The first is the one-body attractive potential of the nuclear ions. The other two arise from the electron-electron interaction: the (classical) Hartree component $V_\text{H}$ captures most of the Coulomb interaction, and the remaining contribution is contained in the exchange-correlation term $V_\text{xc}$. In practice, the exchange-correlation is challenging to capture, and is generally modeled by approximations known as the local density approximation (LDA) or generalized gradient approximation (GGA).  %% In the Kohn-Sham formulation, the potential $V_\text{KS}$ functionally depends on the total density $\rho$, which is not a problem as there are standard algorithms for iterative solution. The problem lies in the lack of a constructive prescription for $V_\text{KS}$.  % Technically, Mott insulators do not exist at T = 0. Magnetism will release the entropy of the Mott state.  DFT guarantees the correct ground state density and energy, but makes no claims about the electronic spectrum, given by the Green's function