# P1: Building effective models for correlated electron systems

## Project Description

Weakly correlated materials can usually be successfully described using density-functional approaches^{(1)}. These methods fail, however, even qualitatively, for materials with localized electrons, like transition-metal compounds, rare-earths, or organic crystals. For such strongly correlated materials we have to treat the electron-electron repulsion more accurately than by just using a static mean-field. A proper many-body treatment requires, however, enormous computational efforts which render it, in practice, impossible for the *ab-initio* Hamiltonian.

To simplify the problem we, therefore, focus on the electrons in the partially occupied, localized orbitals that dominate the low-energy physics of these systems. For these, we can set up a model Hamiltonian that contains the effects of the electrons not explicitly included by a mere renormalization of the model parameters ^{(2)}.

Approaches commonly used to obtain such realistic model parameters are density-functional calculations with constrained orbital-occupations^{(3)}, calculations of the screening within the random-phase approximation, where the contribution of the electrons in the correlated subspace are subtracted^{(4)}, or self-consistent calculations of the multipole-screening in organic crystals^{(5)}. An alternative approach is to use the functional renormalization-group^{(6),(7)}.

In the proposed project we plan to analyze the partitioning of electron states into correlated and uncorrelated subspaces, which forms the basis of the realistic modeling of strongly correlated materials, using numerically exact calculations and path-integral methods. In a first step we consider electrons strictly distinguished into correlated and uncorrelated, without allowing electron-fluctuations between the subspaces. The system can then be described, in close analogy to the Born-Oppenheimer approximation, in terms of a product wave function (8). For small models that are accessible to numerically exact solution using the Lanczos method, we can then assess the quality of this approximation as a function of the interaction between the two electron sub-systems and the characteristic

relaxation time of the uncorrelated electron system. In a path-integral description of the full system we can implement the factorization of the electrons by a saddle-point integration over the uncorrelated electrons for static configurations of the slow correlated electrons.

In a second step we will consider non-adiabatic effects by including higher-order corrections to the factorized Born-Oppenheimer form. The main non-adiabatic effects can also be approximately modeled by a renormalization of the hopping parameters^{(8)}, accounting for the screening cloud of uncorrelated electrons that is following a correlated electron. We finally plan to release the strict separation of the electrons into either correlated or uncorrelated, by allowing the transfer of electrons between the two subspaces.

(1) W. Kohn, Rev. Mod. Phys. **71**, 1253 (1999)

(2) O. Gunnarsson, *Strongly Correlated Electrons: Estimates of Model Parameters*
* in Correlated Electrons: From Models to Materials*, E. Pavarini, E. Koch, F. Anders and M. Jarrell (eds.) (Forschungszentrum Jülich, 2012)

(3) O. Gunnarsson, O. K. Anderssen, O. Jepsen and J. Zaanen, Phys. Rev. B **39**, 1708 (1989)

(4) F. Aryasetiawan, K. Karlsson, O. Jepsen and U. Schönberger, Phys. Rev. B **74**, 125106 (2006)

(5) L. Cano-Cortés, A. Dolfen, J. Merino, J. Behler, B. Delley, K. Reuter and E. Koch,

Europ. J. Phys. B **56**, 173 (2007)

(6) S. A. Maier and C. Honerkamp, Phys. Rev. B **85**, 064520 (2012)

(7) C. Honerkamp, Phys. Rev. B **85**, 195129 (2012)

(8) C. Adolphs, *Renormalization of the Coulomb Interaction in the Hubbard Model and in Dynamical Mean Field Theory* (Diploma thesis with E. Koch, RWTH Aachen University, 2010)