Limitations of RPA and strongly correlated systems

[john_low1]

During a discussion of Georg Kresse's recent plenary talk entitled, "The Random Phase Approximation: A Practical Method Beyond DFT". In his response to a question about the limitations of RPA, Professor Kresse mentioned that strongly correlated systems could cause problems for RPA. This has spawned some controversy in our group and I hoped to get a confirmation that strongly correlated systems cause issues for the Random Phase Approximation.

[ferenc_karsai]

I asked Georg and got the following answer:

This is a tricky question, since the expertise is not yet huge in this area.
However, there are a couple of reasons, why we expect that the RPA will not be great
for strongly-correlated system.

But first a word of caution: in this context, strongly correlated materials means
that a single Slater determinantal wavefunction does not suffice to capture the physics of the system.
What kind of systems are then strongly correlated?
Some people believe that, if one needs LDA+U to describe the material,
the system is strongly correlated. That's, however, a misconception.
LDA, HSE, LDA+U (Kohn-Sham methods in general) and all the other mean field methods (HF) assume that the system can be mapped onto a reference system described
by a single Slater determinant. If this mapping is straightforward, i.e. there exists a one-particle
Hamiltonian that captures the essential physics, the system is not strongly correlated.
There are very few systems that fall outside this regime, for instance,
Kondo metals are prime examples of strongly correlated materials, in my opinion.
The relevant physics can simply not be captured by any mean field method, i.e.
the Kohn-Sham potential will (if it can be found at all) provide a poor description of
say spectral properties. Furthermore, none of the existing generalized Kohn-Sham
methods (LDA, PBE, HF) will yield a suitable reference state, and we would need
to invent a completely new DFT functional to obtain a reasonable Kohn-Sham potential.
In summary, RPA should be able to describe any system that can be described
using one of the following methods: LDA, PBE, SCAN, LDA+U, HSE, HF etc.
But the need for adjusting parameters has been dealt with.

Back to the question, why RPA is not highly accurate for correlated systems:
- RPA is not self-interaction free. It performs better than say, LDA, PBE, SCAN or even
HSE, but there is some residual self-interaction error. This is related to the neglect
of second order screened exchange, and it can cause issues when one tries to describe
systems with strongly localized electrons
(see Grüneis, A., Kresse, G., Hinuma, Y., & Oba, F. (2014). Ionization potentials of solids: the importance of vertex corrections. Physical review letters, 112(9), 096401.)
This also means, in some cases you might be better off with a hybrid functional,
after laborious adjustment of the amount of exact exchange.

- The calculations are always performed on top of a mean field method, usually PBE.
Hence, some of the deficiencies might be inherited from the preceding PBE calculations.
For instance, if PBE predicts a metal, single shot RPA or GW might not completely repair this deficiency.
This problem is particularly severe for materials that would require LDA+U or HSE
to describe their electronic structure reasonably well.
My advise on this: start from LDA+U, and use the smallest possible U to
obtain a gapped system, i.e. a small energy separation between occupied
and unoccupied d states.

Hope this clarifies the issue somewhat.

Georg Kresse

[john_low1]

Thank you. This is very helpful!