# Dynamical Screening in Correlated Electron Materials 

$^{2}$Columbia University, 538 West, 120th Street, New York, NY 10027, USA 

$^{1}$Theoretische Physik, ETH Zurich, 8093 Z¨ urich, Switzerland 

Philipp Werner 1 and Andrew J. Millis 2 

(Dated: January 8, 2010) 

We present an efficient method for incorporating the dynamical effects of the screening of the Hubbard U by electronic degrees of freedom in the solid into the single site dynamical mean field approximation. The formalism is illustrated by model system calculations which capture the es- sential features of the frequency dependent interactions proposed for Gd, Ni, SrVO 3 and other compounds. Screening leads to shifts in the metal-insulator phase boundary, changes in the spectral function near the Mott-Hubbard gap edge and to a renormalization of the quasiparticle weight. Hubbard bands are generically neither separated by the screened nor the unscreened interaction en- ergy, implying that the common practice of extracting the Hubbard U from the energies of features in photoemission and inverse photoemission spectra requires reexamination. 

‘Strongly correlated electron systems’ are a central topic in electronic condensed matter physics [1]. The low energy physics of these systems is typically described by an effective Hamiltonian which models the behavior of relatively localized d or f orbitals and is obtained (at least notionally) via a “downfolding” procedure in which other degrees of freedom are integrated out. A crucial as- pect of the effective Hamiltonian is an interaction which acts to suppress local number fluctuations. This inter- action is typically parametrized by a number, the “Hub- bard U ”. However, U is generically dynamical: a density fluctuation in a correlated orbital produces electric fields, which other degrees of freedom will act to screen, result- ing in a frequency-dependent renormalization. Screen- ing has been observed as a shift in excitation energies in experiments comparing the surface of solid C$_{60}$ to C$_{60}$ films on silver [2] and has been computed using variations of the ‘random phase approximation’ (RPA) [3–5]. The renormalizations are found to be strong in many cases. In Gd, U ( ω ) rises from a static value of about 6.5 eV to about 17 eV as the frequency ω is raised from 0 to ∼ 3 eV, while in Ce, U ( ω ) changes from ∼ 3 . 5 eV to 7 eV as ω is increased from 0 to ∼ 4 eV [4]. 

While the study of the strong correlation effects in- duced by an instantaneous interaction is well advanced thanks to the development of dynamical mean field the- ory (DMFT) [6], our ability to treat frequency depen- dent interactions has been limited. The most widely used method for solving the DMFT equations has been the Hirsch-Fye algorithm [7], which is based on a time- discretization and decoupling of interaction terms by auxiliary fields. Frequency dependent interactions lead to a proliferation of decoupling fields which become pro- hibitively expensive to sample. A treatment of screening effects within exact diagonalization, numerical renormal- ization group and other Hamiltonian based methods re- quires the explicit introduction of many bosonic modes, leading to a Hilbert space which is too large to be handled 

PACS numbers: 71.27.+a,71.30.+h,71.10.Fd 

numerically. Considerations of this sort have led to the belief (see e.g. Ref. [3]) that DMFT simulations with fre- quency dependent interactions are exceedingly difficult. 

Here we show that this is not the case: the recently developed ‘hybridization expansion’ diagrammatic quan- tum Monte Carlo method [8, 9] can be used to treat models with an arbitrary frequency dependence of the on-site repulsion U ( ω ) at negligible additional computa- tional cost, opening the door to a systematic investiga- tion of screening effects in correlated electron materials. We begin our discussion by recalling that the downfolded models used to describe the correlated degrees of freedom in a transition metal or actinide involve a one electron part and an interaction part. The parameters describ- ing the one-electron physics are obtained by projecting a band theory calculation onto a set of distinguished (“ d ”) orbitals and are in principle energy dependent. If the d orbitals are correctly chosen the energy dependence is negligible [10, 11] so the one electron part may be mod- elled as a tight-binding-like Hamiltonian H$_{band}$ . 

The interaction part is obtained (see, e.g. Ref. 3) by screening the bare Coulomb interaction e $^{2}$/ | r − r $^{′}$| with real and virtual transitions involving the orbitals which are integrated out, projecting the result onto the d manifold and retaining only the fully site-diagonal terms. One finds two kinds of terms: an instantaneous interaction and a screening contribution. The instanta- neous interaction may be represented as a Hamiltonian term H$_{int}$ which takes the usual Slater-Kanamori form H$_{int}$ = ∑ i 1 $_{2}$U ˆ N 2 i + .... with ˆ N$_{i}$ the number operator for electrons in the d manifold on site i and the ellipsis denot- ing exchange, ‘pair hopping’ and other terms which in- volve operators such as the spin and angular momentum which commute with ˆ N . The screening contribution cou- ples the site densities at different times and is expressed as a contribution S$_{W}$ = 1 2 ∫ dτdτ ′ N ( τ ) W ( τ − τ ′ ) N ( τ ′ ) to the effective action. The screening function W ( τ ) is negative, depends only on one time (or frequency) argu- 

