Category:Strongly correlated electrons: Difference between revisions

Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with $d-$ and $f-$electrons which are localized, and correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.

To model such systems, several extensions of DFT have been developed.

DFT+U

DFT+U is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term $U$ that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to $d$ or $f$ states.

\begin{equation} E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right], \end{equation} where the first term is the standard DFT energy, the second term is the Hubbard on-site interaction and the third term accounts for the double counting. The on-site interaction is described by $U^I$ and $n_m^{I\sigma}$ are occupation numbers that are defined as projections of occupied Kohn-Sham orbitals on the states of a localized basis set.

VASP provides the following approaches to include the Hubbard corrections:

- LDAUTYPE=1: a rotationally invariant formultion of the Hubbrd correction that eliminates the dependence on the specific choice of the localized basis set.

- LDAUTYPE=4: The same approach as LDAUTYPE=1, but uses spin-averaged expression that's simpler and assumes an average spin configuration.

- LDAUTYPE=2: a simplified approach which is also rotationally invariant but uses isotropic effective interaction $U^I_{eff}=U^I-J^I$ ^[1]. This approach neglects the anisotropy of the orbitals and thus the on-site interaction only depends on occupation but not the occupied orbitals.

The effective on-site interaction $U$ can be either found by a fitting procedure that yeilds the best description of the descired property, i.e., band gap, magnetic moments, lattice parameters, etc. Alternatively, there are ab initio approaches for determining the Hubbard on-site interaction.

- LDAUTYPE=3: The effective interaction $U$ can be determined via the linear response approach ^[2] \begin{equation} U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}. \end{equation} The shortcoming of this approach is that the effective interaction accounts for the response due to all electrons including the target states (localized $d-$ or $f-$ orbitals), thus leading to double counting in the derived effective potential U.

Constrained Random Phase Approximation (cRPA)

cRPA allows to separate the screening originating from the target states from the rest of the system. Thus, we find the on-site effective interaction that is free of double counting.

The response function without the contribution of the target states or constrained polarizability $\chi_c$ is calculated by explicitly removing the response in the target space $\chi_d$ from the total response function: $\chi_c(\omega) = \chi(\omega) - \chi_d(\omega)$.

VASP provides several approaches to calculating $\chi_d$

Dynamical Mean-Field Theory (DMFT)

DMFT is an advanced extension of DFT (DFT+DMFT) that provides an accurate treatment of strongly correlated materials including dynamical effects, which are fully neglected in DFT+U ^[3]. Within DMFT a lattice problem is mapped onto a self-consistent quantum impurity model by embedding a single correlated site in an effective bath that represents the rest of the system, and the key approximation is that the self-energy is local (frequency-dependent but momentum-independent). DMFT captures essential many-body effects such as quasiparticle renormalization, Hubbard bands, and Mott metal-insulator transitions. cRPA can be used as a preliminary step to determine the effective screening $U(\omega)$ without the contribution from the target space to avoid double-counting in the subsequent calculation of the self-energy within DMFT.

Other methods

There are other methods that are not specialized for strongly correlated system but nevertheless have been shown to improve the description of the electronic structure of the strongly correlated systems.

Hybrid functionals

By including a fraction of exact exchange, the hybrid functional approach can reduce the self-interaction error in DFT, which is required to improve the description of the physics of strong correlations ^[4],.

Hybrid functionals + U

A shortcoming of hybrid functionals is their uniform description of all states, which can show very different accuracy for states with different degrees of localization. The introduction of the Hubbard on-site interaction within the hybrid functional approach was shown to resolve issues caused by overscreening of localized states ^[5].

QPGW

The GW approximation in its simplest form (one-shot approach) is strongly dependent on the starting point and thus suffers from the shortcomings of the DFT for describing localized states. However, a self-consistent GW approach such as QPGW which does not depend on the starting electronic structure can yield an accurate description of the correlated electrons ^[6].

Tutorials: Tutorial for NiO LSDA+U calculations Tutorial for NiO DFT+U calculations Tutorial for NiO Calculate U for LSDA+U calculations Tutorial for CRPA of SrVO3 calculations Tutorial for Bandstructure and CRPA of SrVO3 calculations Tutorial for NiO DFT+DMFT calculations Tutorial for NiO GGA+U calculations

References