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 ^[1]. These systems typically include elements with partially filled [math]\displaystyle{ d }[/math] and [math]\displaystyle{ f }[/math] electron orbitals which are localized. 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. Below, you find methods relevant in the context of strongly correlated electrons. Many rely on estimating the on-site Coulomb interaction U.

Estimating the on-site Coulomb interaction U

DFT+U and DFT+DMFT calculations rely on the on-site Coulomb interaction U as an input parameter. There are some strategies to obtain a value for U:

• Estimate U based on available experimental results. This approach is not fully ab initio, yet it can be the most pragmatic strategy. Here one treats the effective on-site interaction [math]\displaystyle{ U }[/math] as an adjustable parameter, tuning it to reproduce selected experimental observables such as the band gap, magnetic moments, or lattice parameters. For instance, one may perform volume relaxations at different U values to obtain volume as a function of U, $v(U)$. Knowing the experimental value for the volume, a linear fit of $v(U)$ can give an estimate for a suitable U value. This approach relies on the fact that the expansion or localization of the strongly correlated d or f electron orbitals drives the ions to relax at a certain distance. One then use the fixed U value to obtain other quantities like band gap, optical properties. Another approach can be to fit the optical gap to ensure the band-structure properties resemble the experiment, which is crucial for, e.g., binding energies or excited states calculations.

• Linear-response calculation of [math]\displaystyle{ U }[/math] (LDAUTYPE = 3). Within this approach the effective interaction [math]\displaystyle{ U }[/math] can be determined via the linear response approach ^[2]

[math]\displaystyle{ 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}. }[/math]

The shortcoming of this method is that the effective interaction accounts for the response due to all electrons including the target states (localized [math]\displaystyle{ d }[/math] or [math]\displaystyle{ f }[/math] orbitals), thus leading to double counting in the derived effective potential U.

• Workflow for NiO Calculate U for LSDA+U calculations.

• Constrained random phase approximation (cRPA) is a first-principles method to estimate U. cRPA allows to separate the screening originating from the target states from the rest of the system and to determine the effective interaction [math]\displaystyle{ U }[/math] that is free of double counting.

The response function without the contribution of the target states or constrained polarizability [math]\displaystyle{ \chi_c }[/math] is calculated by explicitly removing the response in the target space [math]\displaystyle{ \chi_d }[/math] from the total response function:

[math]\displaystyle{ \chi_c(\omega) = \chi(\omega) - \chi_d(\omega) }[/math].

VASP provides several approaches for calculating [math]\displaystyle{ \chi_d }[/math]

• Lecture on the constrained random-phase approximation (cRPA).
• Tutorial for cRPA of NiO.
• Workflow for CRPA of SrVO3.

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 [math]\displaystyle{ U }[/math] that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to [math]\displaystyle{ d }[/math] or [math]\displaystyle{ f }[/math] states.

The total energy within DFT+U can be written as

[math]\displaystyle{ 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], }[/math]

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 [math]\displaystyle{ U^I }[/math] and [math]\displaystyle{ n_m^{I\sigma} }[/math] are occupation numbers that are defined as projections of occupied Kohn-Sham orbitals on the states of a localized basis set.

• Tutorial for DFT+U on NiO.
• Tutorial for GW and DFT+U on NiO.
• Tutorial for NiO DFT+U calculations.
• Tutorial for antiferromagnetic NiO.
• Tutorial for LSDA+U structure relaxation of NiO.
• Tutorial for Heisenberg model for NiO using DFT+U.
• Tutorial for magnetic anisotropy in FeO.

Dynamical mean-field theory (DMFT)

In DFT+DMFT calculations, DMFT^[3] augments the DFT calculation with an additional local correlated subproblem — typically a specific [math]\displaystyle{ d }[/math]- or [math]\displaystyle{ f }[/math]-shell. The key idea is to map the full lattice problem onto a quantum impurity model: a single correlated site embedded in a self-consistently determined effective bath representing the rest of the lattice. DMFT can be discussed using the Green's function formalism also used in the context of many-body perturbation theory. The main quantity is the electronic self-energy. Within DMFT it is defined as the sum of all one-particle irreducible diagrams. The central approximation is that the self-energy is frequency-dependent but momentum-independent, i.e., purely local. DFT+DMFT calculations capture many-body effects beyond the reach of DFT+U, such as quasiparticle mass renormalization, Hubbard bands, and the Mott metal–insulator transition. The interaction parameter entering DMFT can be determined as discussed in estimating the on-site Coulomb interaction U.

• Lecture on DFT+U and dynamical mean-field theory.
• Tutorial for DMFT on NiO.
• Workflow for NiO DFT+DMFT calculations.

Other methods

There are other methods that, while not specifically designed for strongly correlated systems, have nonetheless been demonstrated to improve their description and electronic structure.

Hybrid functionals

By incorporating a fraction of exact exchange, hybrid functionals partially mitigate the self-interaction error inherent to standard DFT. This reduction of the self-interaction error has been shown to yield an improved description of strongly correlated systems ^[4][5].

Hybrid functionals + U

A well-known limitation of hybrid functionals is their uniform treatment of all electronic states, which can result in markedly different levels of accuracy for states with varying degrees of localization. The inclusion of the Hubbard on-site interaction term within the hybrid functional framework (hybrid + U) has been shown to address inaccuracies arising from the overscreening of localized states ^[6].

Quasi-particle GW (QPGW)

The GW approximation in its simplest, non-self-consistent form (i.e., the one-shot approach) exhibits a strong dependence on the choice of starting point, and thus inherits the limitations of the underlying DFT description of localized states. In contrast, self-consistent GW schemes such as QPGW, which are independent of the starting electronic structure, have been shown to provide an accurate description of correlated electrons ^[7][8].

Additional resources

Books

• Interacting Electrons - Theory and Computational Approaches by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation ^[1].
• Dynamical Mean-Field Theory for Strongly Correlated Materials by Volodymyr Turkowski - a book about DMFT ^[9].

References