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191 lines
8.7 KiB
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.. _dftplusdmft:
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Introduction to DFT+DMFT
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========================
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When describing the physical and also chemical properties of
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crystalline materials, there is a standard model that is used with
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great success for a large variety of systems: band theory. In simple
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terms it states that electrons in a crystal form bands of allowed
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states in momentum space. These states are then filled by the
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electrons according to Pauli's principle up the Fermi level. With this
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simple picture one can explain the electronic band structure of simple
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materials such as elementary copper or aluminum.
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Following this principle one can easily classify all existing
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materials into metals and insulators, with semiconductors being
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special insulators with a small gap in the excitation
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spectrum. Following this band theory, a system is a metal if there is
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an odd number of electrons in the valence bands, since this leads to a
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partially filled band, cutting the Fermi energy and, thus, producing a
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Fermi surface, i.e metallic behavior. On the other hand, an even
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number of electrons leads to completely filled bands with a finite
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excitation gap to the conduction bands, i.e. insulating behavior.
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This classification works pretty well for a large class of
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materials, where the electronic band structures are reproduced by
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methods based on wave function theories. Certain details such as the
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precise value of Fermi velocities and electronic masses, or the actual
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value of the gap in semi conductors may show difference between theory
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and experiment, but theoretical results agree at least qualitatively
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with measured data.
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However, there are certain compounds where this
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classification into metals and insulators fails dramatically. This
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happens in particular in systems with open d- and f-shells. There,
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band theory predicts metallic behavior because of the open-shell
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setting, but in experiments many-not all-of these materials show
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actually insulating behavior. This cannot be explained by band theory
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and the Pauli principle alone, and a different mechanism has to be
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invoked. The bottom line is that these materials do not conduct
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current
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because of the strong Coulomb repulsion between the electrons. With
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reference to Sir Nevill Mott, who contributed substantially to the
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explanation of this effect in the 1930's, these materials are in
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general referred to as Mott insulators.
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Density-functional theory in a (very small) nutshell
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----------------------------------------------------
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Density-functional theory tells that the ground state density
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determines uniquely all physical properties of a system, independent
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of the degree of correlations. Moreover, the theorems of Hohenberg,
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Kohn, and Sham state that the full interacting many-body problem can
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be replaced by independent electrons moving in an effective
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single-particle potential. These leads to the famous Kohn-Sham
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equations to be solved in DFT:
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.. math::
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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})
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= \varepsilon_{\nu\mathbf{k}}\psi_{\nu\mathbf{k}}(\mathbf{r}).
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Without going into details of the Kohn-Sham potential :math:`V_{KS}=V(\mathbf{r})+V_H(\mathbf{r})+V_{xc}(\mathbf{r})`
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that is discussed in the literature on DFT, let us just note that the
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main result of DFT calculations are the Kohn-Sham energies
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:math:`\varepsilon_{\nu\mathbf{k}}` and the Kohn-Sham orbitals :math:`\psi_{\nu\mathbf{k}}(\mathbf{r})`.
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This set of equations is exact, however, the exchange correlation
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potential :math:`V_{xc}(\mathbf{r})` is not known explicitly. In
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order to do actual calculations, it needs to be approximated in some
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way. The local density approximation is one of the most famous
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approximations used in this context. This approximation works well for
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itinerant systems and semiconductors, but fails completely for
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strongly-correlated systems.
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From DFT to DMFT
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----------------
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In order to extend our calculations to strong correlations, we need to
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go from a description by bands to a description in terms of
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(localized) orbitals: Wannier functions.
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In principle, Wannier functions :math:`\chi_{\mu\sigma}(\mathbf{r})`
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are nothing else than a Fourier transform of the Bloch basis set from
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momentum space into real space,
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.. math::
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\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
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where we introduced also the spin degree of freedom :math:`\sigma`. The
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unitary matrix :math:`U_{\mu\nu}` is not uniquely defined, but allows for a
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certain amount of freedom in the calculation of Wannier function. A
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very popular choice is the constraint that the resulting Wannier
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functions should be maximally localized in space. Another route,
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computationally much lighter and more stable, are projective Wannier
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functions. This scheme is used for the Wien2k interface in this
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package.
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A central quantity in this scheme is the projection operator
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:math:`P_{m\nu}(\mathbf{k})`, where :math:`m` is an orbital index and
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:math:`\nu` a Bloch band index.
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Its definition and how it is calculated can be found in the original
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literature or in the extensive documentation of the
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:program:`dmftproj` program shipped with :program:`DFTTools`.
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Using projective Wannier functions for DMFT
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-------------------------------------------
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In this scheme-that is used for the interface to Wien2k-the operators
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:math:`P_{m\nu}(\mathbf{k})` are not unitary, since the two dimensions
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:math:`m` and :math:`\nu` are not necessarily the same. They
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allow, however, to project the local DFT Green function from Bloch band
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space into Wannier space,
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.. math::
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G^0_{mn}(i\omega) =
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\sum_{\mathbf{k}}\sum_{\nu\nu'}P_{m\nu}(\mathbf{k})G^{DFT}_{\nu\nu'}(\mathbf{k},i\omega)P^*_{\nu'
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n}(\mathbf{k})
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with the DFT Green function
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.. math::
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G^{DFT}_{\nu\nu'}(\mathbf{k},i\omega) = \frac{1}{i\omega +\mu-\varepsilon_{\nu\mathbf{k}}}\delta_{\nu\nu'}
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This non-interacting Green function :math:`G^0_{mn}(i\omega)` defines,
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together with the interaction Hamiltonian, the Anderson impurity
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model. The DMFT self-consistency cycle can now be formulated as
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follows:
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#. Take :math:`G^0_{mn}(i\omega)` and the interaction Hamiltonian and
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solve the impurity problem, to get the interacting Green function
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:math:`G_{mn}(i\omega)` and the self energy
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:math:`\Sigma_{mn}(i\omega)`. For the details of how to do
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this in practice, we refer to the documentation of one of the
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Solver applications, for instance the `CTHYB solver <https://triqs.github.io/cthyb/>`_.
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#. The self energy, written in orbital space, has to be corrected by
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the double counting correction, and upfolded into Bloch band space:
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.. math::
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\Sigma_{\nu\nu'}(\mathbf{k},i\omega) = \sum_{mn}P^*_{\nu
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m}(\mathbf{k}) (\Sigma_{mn}(i\omega) -\Sigma^{DC})P_{n\nu'}(\mathbf{k})
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#. Use this :math:`\Sigma_{\nu\nu'}(\mathbf{k},i\omega)` as the DMFT
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approximation to the true self energy in the lattice Dyson
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equation:
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.. math::
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G^{latt}_{\nu\nu'}(\mathbf{k},i\omega) = \frac{1}{i\omega+\mu
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-\varepsilon_{\nu\mathbf{k}}-\Sigma_{\nu\nu'}(\mathbf{k},i\omega)}
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#. Calculate from that the local downfolded Green function in orbital space:
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.. math::
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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'
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n}(\mathbf{k})
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#. Get a new :math:`G^0_{mn}(i\omega)` for the next DMFT iteration
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from
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.. math::
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G^0_{mn}(i\omega) = \left[
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\left(G^{loc}_{mn}(i\omega)\right)^{-1} + \Sigma_{mn}(i\omega)
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\right]^{-1}
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Now go back to step 1 and iterate until convergence.
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This is the basic scheme for one-shot DFT+DMFT calculations. Of
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course, one has to make sure, that the chemical potential :math:`\mu`
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is set such that the electron density is correct. This can be achieved
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by adjusting it for the lattice Green function such that the electron
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count is fulfilled.
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Full charge self-consistency
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----------------------------
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The feedback of the electronic correlations to the Kohn-Sham orbitals
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is included by the interacting density matrix. With going into the
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details, it basically consists of calculating the Kohn-Sham density
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:math:`\rho(\mathbf{r})` in the presence of this interacting density
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matrix. This new density now defines a new Kohn-Sham
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exchange-correlation potential, which in turn leads to new
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:math:`\varepsilon_{\nu\mathbf{k}}`,
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:math:`\psi_{\nu\mathbf{k}}(\mathbf{r})`, and projectors
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:math:`P_{m\nu}(\mathbf{k})`. The update of these
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quantities can easily be included in the above
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self-consistency cycle, for instance after
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step 3, before the local lattice Green
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function is downfolded again into orbital space.
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How all these calculations can be done in practice with this
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:program:`DFTTools` package is subject of the user guide in this documentation.
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