diff --git a/doc/basicnotions/dft_dmft.rst b/doc/basicnotions/dft_dmft.rst index 602e992a..48aaadb4 100644 --- a/doc/basicnotions/dft_dmft.rst +++ b/doc/basicnotions/dft_dmft.rst @@ -1,5 +1,189 @@ Introduction to DFT+DMFT ======================== -.. warning:: - TO BE WRITTEN! +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 aluminium. + +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 reffered 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: + +.. math:: + 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 :math:`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 +:math:`\varepsilon_{\nu\mathbf{k}}` and the Kohn-Sham orbitals :math:`\psi_{\nu\mathbf{k}}(\mathbf{r})`. +This set of equations is exact, however, the exchange correlation +potential :math:`V_{xc}(\mathbf{r})` is not known explicitely. 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 +(localised) orbitals: Wannier functions. + +In principle, Wannier functions :math:`\chi_{\mu\sigma}(\mathbf{r})` +are nothing else than a Fourier transform of the Bloch basis set from +momentum space into real space, + +.. math:: + \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 :math:`\sigma`. The +unitary matrix :math:`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 localised 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 +:math:`P_{m\nu}(\mathbf{k})`, where :math:`m` is an orbital index and +:math:`\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 +:program:`dmftproj` program shipped with :program:`dft_tools`. + +Using projective Wannier functions for DMFT +------------------------------------------- + +In this scheme-that is used for the interface to Wien2k-the operators +:math:`P_{m\nu}(\mathbf{k})` are not unitary, since the two dimensions +:math:`m` and :math:`\nu` are not necessarily the same. They +allow, however, to project the local DFT Green function from Bloch band +space into Wannier space, + +.. math:: + 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 + +.. math:: + 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 :math:`G^0_{mn}(i\omega)` defines, +together with the interaction Hamiltonian, the Anderson impurity +model. The DMFT self-consitency cycle can now be formulated as +follows: + +#. Take :math:`G^0_{mn}(i\omega)` and the interaction Hamiltonian and + solve the impurity problem, to get the interacting Greens function + :math:`G_{mn}(i\omega)` and the self energy + :math:`\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 :ref:`CTHYB solver `. + +#. The self energy, written in orbital space, has to be corrected by + the double counting correction, and upfolded into Bloch band space: + + .. math:: + \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}) + +#. Use this :math:`\Sigma_{\nu\nu'}(\mathbf{k},i\omega)` as the DMFT + approximation to the true self energy in the lattice Dyson + equation: + + .. math:: + G^{latt}_{\nu\nu'}(\mathbf{k},i\omega) = \frac{1}{i\omega+\mu + -\varepsilon_{\nu\mathbf{k}}-\Sigma_{\nu\nu'}(\mathbf{k},i\omega)} + +#. Calculate from that the local downfolded Greens function in orbital space: + + .. math:: + 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}) + +#. Get a new :math:`G^0_{mn}(i\omega)` for the next DMFT iteration + from + + .. math:: + 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 :math:`\mu` +is set such that the electron density is correct. This can be achieved +by adjusting it for the lattice Greens 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 +:math:`\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 +:math:`\varepsilon_{\nu\mathbf{k}}`, +:math:`\psi_{\nu\mathbf{k}}(\mathbf{r})`, and projectors +:math:`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 +:program:`dft_tools` package is subject of the user guide in this documentation.