Introduction to DFT+DMFT

When describing the physical and also chemical properties of crystalline materials, there is a standard model that is used with great success for a large variety of systems: band theory. In simple terms it states that electrons in a crystal form bands of allowed states in momentum space. These states are then filled by the electrons according to Pauli’s principle up the Fermi level. With this simple picture one can explain the electronic band structure of simple materials such as elementary copper or aluminum.

Following this principle one can easily classify all existing materials into metals and insulators, with semiconductors being special insulators with a small gap in the excitation spectrum. Following this band theory, a system is a metal if there is an odd number of electrons in the valence bands, since this leads to a partially filled band, cutting the Fermi energy and, thus, producing a Fermi surface, i.e metallic behavior. On the other hand, an even number of electrons leads to completely filled bands with a finite excitation gap to the conduction bands, i.e. insulating behavior.

This classification works pretty well for a large class of materials, where the electronic band structures are reproduced by methods based on wave function theories. Certain details such as the precise value of Fermi velocities and electronic masses, or the actual value of the gap in semi conductors may show difference between theory and experiment, but theoretical results agree at least qualitatively with measured data.

However, there are certain compounds where this classification into metals and insulators fails dramatically. This happens in particular in systems with open d- and f-shells. There, band theory predicts metallic behavior because of the open-shell setting, but in experiments many-not all-of these materials show actually insulating behavior. This cannot be explained by band theory and the Pauli principle alone, and a different mechanism has to be invoked. The bottom line is that these materials do not conduct current because of the strong Coulomb repulsion between the electrons. With reference to Sir Nevill Mott, who contributed substantially to the explanation of this effect in the 1930’s, these materials are in general referred to as Mott insulators.

Density-functional theory in a (very small) nutshell

Density-functional theory tells that the ground state density determines uniquely all physical properties of a system, independent of the degree of correlations. Moreover, the theorems of Hohenberg, Kohn, and Sham state that the full interacting many-body problem can be replaced by independent electrons moving in an effective single-particle potential. These leads to the famous Kohn-Sham equations to be solved in DFT:

\[H_{KS}\psi_{\nu\mathbf{k}}(\mathbf{r})=\left[-\frac{1}{2m_e}\nabla^2+V_{KS}[\rho](\mathbf{r})\right]\psi_{\nu\mathbf{k}}(\mathbf{r}) = \varepsilon_{\nu\mathbf{k}}\psi_{\nu\mathbf{k}}(\mathbf{r}).\]

Without going into details of the Kohn-Sham potential \(V_{KS}=V(\mathbf{r})+V_H(\mathbf{r})+V_{xc}(\mathbf{r})\) that is discussed in the literature on DFT, let us just note that the main result of DFT calculations are the Kohn-Sham energies \(\varepsilon_{\nu\mathbf{k}}\) and the Kohn-Sham orbitals \(\psi_{\nu\mathbf{k}}(\mathbf{r})\). This set of equations is exact, however, the exchange correlation potential \(V_{xc}(\mathbf{r})\) is not known explicitly. In order to do actual calculations, it needs to be approximated in some way. The local density approximation is one of the most famous approximations used in this context. This approximation works well for itinerant systems and semiconductors, but fails completely for strongly-correlated systems.

From DFT to DMFT

In order to extend our calculations to strong correlations, we need to go from a description by bands to a description in terms of (localized) orbitals: Wannier functions.

In principle, Wannier functions \(\chi_{\mu\sigma}(\mathbf{r})\) are nothing else than a Fourier transform of the Bloch basis set from momentum space into real space,

\[\chi_{\mu\sigma}(\mathbf{r})=\frac{1}{V}\sum_\mathbf{k} e^{-i\mathbf{k}\mathbf{r}}\sum_\nu U_{\mu\nu}\psi_{\mathbf{k}\nu}^\sigma\]

where we introduced also the spin degree of freedom \(\sigma\). The unitary matrix \(U_{\mu\nu}\) is not uniquely defined, but allows for a certain amount of freedom in the calculation of Wannier function. A very popular choice is the constraint that the resulting Wannier functions should be maximally localized in space. Another route, computationally much lighter and more stable, are projective Wannier functions. This scheme is used for the Wien2k interface in this package.

A central quantity in this scheme is the projection operator \(P_{m\nu}(\mathbf{k})\), where \(m\) is an orbital index and \(\nu\) a Bloch band index. Its definition and how it is calculated can be found in the original literature or in the extensive documentation of the dmftproj program shipped with DFTTools.

Using projective Wannier functions for DMFT

In this scheme-that is used for the interface to Wien2k-the operators \(P_{m\nu}(\mathbf{k})\) are not unitary, since the two dimensions \(m\) and \(\nu\) are not necessarily the same. They allow, however, to project the local DFT Green function from Bloch band space into Wannier space,

\[G^0_{mn}(i\omega) = \sum_{\mathbf{k}}\sum_{\nu\nu'}P_{m\nu}(\mathbf{k})G^{DFT}_{\nu\nu'}(\mathbf{k},i\omega)P^*_{\nu' n}(\mathbf{k})\]

with the DFT Green function

\[G^{DFT}_{\nu\nu'}(\mathbf{k},i\omega) = \frac{1}{i\omega +\mu-\varepsilon_{\nu\mathbf{k}}}\delta_{\nu\nu'}\]

This non-interacting Green function \(G^0_{mn}(i\omega)\) defines, together with the interaction Hamiltonian, the Anderson impurity model. The DMFT self-consistency cycle can now be formulated as follows:

  1. Take \(G^0_{mn}(i\omega)\) and the interaction Hamiltonian and solve the impurity problem, to get the interacting Green function \(G_{mn}(i\omega)\) and the self energy \(\Sigma_{mn}(i\omega)\). For the details of how to do this in practice, we refer to the documentation of one of the Solver applications, for instance the CTHYB solver.

  2. The self energy, written in orbital space, has to be corrected by the double counting correction, and upfolded into Bloch band space:

    \[\Sigma_{\nu\nu'}(\mathbf{k},i\omega) = \sum_{mn}P^*_{\nu m}(\mathbf{k}) (\Sigma_{mn}(i\omega) -\Sigma^{DC})P_{n\nu'}(\mathbf{k})\]
  3. Use this \(\Sigma_{\nu\nu'}(\mathbf{k},i\omega)\) as the DMFT approximation to the true self energy in the lattice Dyson equation:

    \[G^{latt}_{\nu\nu'}(\mathbf{k},i\omega) = \frac{1}{i\omega+\mu -\varepsilon_{\nu\mathbf{k}}-\Sigma_{\nu\nu'}(\mathbf{k},i\omega)}\]
  4. Calculate from that the local downfolded Green function in orbital space:

    \[G^{loc}_{mn}(i\omega) = \sum_{\mathbf{k}}\sum_{\nu\nu'}P_{m\nu}(\mathbf{k})G^{latt}_{\nu\nu'}(\mathbf{k},i\omega)P^*_{\nu' n}(\mathbf{k})\]
  5. Get a new \(G^0_{mn}(i\omega)\) for the next DMFT iteration from

    \[G^0_{mn}(i\omega) = \left[ \left(G^{loc}_{mn}(i\omega)\right)^{-1} + \Sigma_{mn}(i\omega) \right]^{-1}\]

    Now go back to step 1 and iterate until convergence.

This is the basic scheme for one-shot DFT+DMFT calculations. Of course, one has to make sure, that the chemical potential \(\mu\) is set such that the electron density is correct. This can be achieved by adjusting it for the lattice Green function such that the electron count is fulfilled.

Full charge self-consistency

The feedback of the electronic correlations to the Kohn-Sham orbitals is included by the interacting density matrix. With going into the details, it basically consists of calculating the Kohn-Sham density \(\rho(\mathbf{r})\) in the presence of this interacting density matrix. This new density now defines a new Kohn-Sham exchange-correlation potential, which in turn leads to new \(\varepsilon_{\nu\mathbf{k}}\), \(\psi_{\nu\mathbf{k}}(\mathbf{r})\), and projectors \(P_{m\nu}(\mathbf{k})\). The update of these quantities can easily be included in the above self-consistency cycle, for instance after step 3, before the local lattice Green function is downfolded again into orbital space.

How all these calculations can be done in practice with this DFTTools package is subject of the user guide in this documentation.