Dynamical mean-field theory

Having discussed the static properties, we shall now consider the dynamics of the concentrated solution. A natural generalization of the static mean field theory to dynamics is to assume that each molecule does Brownian motion under the mean field potential ngid 

Introducing this into the kinetic equation (9.42), we have  

Equations (10.38) and (10.39) give a nonlinear integro-differential equation for W, and its mathematical handling is not easy. A guidance of how to proceed is obtained from the phenomenological theory in nematics. De Gennes showed that the dynamics of nematics is essentially described by the Landau theory of phase transition and proposed a phenomenological nonlinear equation fof the order parameter tensor 

Here L is a phenomenological kinetic coefficient and A is a free energy. Near the transition point, S p is small so that A can be expanded with respect to S p as in the series 

We thus aim at deriving a closed equation for from eqn (10.39). For simplicity, we first consider the case when there is no external field ( /,-0, Jf = 0). 

This constitutes the DMFT impurity problem. Subsequently, the Green s function for the solid is constructed from 

An intriguing aspect of DMFT is that the effective bath to be used in the impurity problem, Eq. (35), may be determined by averaging the solid state Green s function in Eq. (36) and subtracting the correlation term —E), namely 

we explicitly write the averaging as a sum over the BZ. Hence, the imposed self-consistency condition is that the Gbath put into Eq. (35) should equal the Gbath extracted from Eq. (38) with Eq. (36). This is usually obtained via an iteration process (the DMFT self-consistency cycle). 

Crucial to all DMFT calculations is how r(Gbath)is evaluated. There are many possibilities and the quality of the calculation depends on this. Two aspects need to be specified, namely the model adopted to describe the correlation effects of the impurity atom and the method/approximation used to solve the impurity problem (the impurity solver), once the model has been chosen. The choice of the model requires physical insight. The Hubbard model (Hubbard, 1963) is the 

Many DMFT calculations have been reported in recent years, the majority discussing 3d systems, but also several dealing with actinides. Less attention has been paid to the lanthanides, most probably because the atomic limit, as outlined Section 6.1, is sufficient for an accurate description in most cases. An important exception is the y-cc transition in cerium, which has been the subject of several studies (Held et al., 2001 Zolfl et al., 2001 McMahan et al., 2003 Haule et al., 2005 Amadon et al., 2006), all using the Hubbard model, but with various impurity solvers. Ce compoimds have also been studied based on the Hubbard model (Laegsgaard and Svane, 1998 Sakai et al., 2005 Sakai and Shimuzi, 2007). 

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The dynamic structure factor of block copolymer liquids (melts and solutions) has been accounted for using dynamical mean-field theory by Benmouna et al. (1987a,6). For a block copolymer melt, the dynamic structure factor can be written (Stepanek and Lodge 1996)... 

Freericks, J. K. Zlatic, V. 2003 Exact dynamical mean field theory of the Falicov-Kimball model, Rev. Mod. Phys. 75, 1333-1382. (doi 10.1103/RevModPhys.75.l333)... 

It is believed that electron correlation plays an important role with the anomalously high resistivity exhibited in marginal metals. Unfortunately, although the Mott-Hubbard model adequately explains behavior on the insulating side of the M-NM transition, on the metallic side, it does so only if the system is far from the transition. Electron dynamics of systems in which U is only slightly less than W (i.e. metallic systems close to the M-NM transition), are not well described by a simple itinerant or localized picture. The study of systems with almost localized electrons is still an area under intense investigation within the condensed matter physics community. A dynamical mean field theory (DMFT) has been developed for the Hubbard model, which enables one to describe both the insulating state and the metallic state, at least for weak correlation. 

The LDA-fU theory may be regarded as an approximate GW method [37]. The screened Coulomb and exchange parameters U and J are usually estimated in a supercell approximation [39]. However, there is some arbitrariness in the choice of the localized orbitals when performing the partitioning of the Hamiltonian. A further step in the improvement of LDA-I-U consists in adding dynamical effects — frequency dependence in H r,r u). This may be performed using a DMFT-type approach (DMFT= Dynamical Mean Field Theory) [40] as part of the so-called LDA-b-1- approaches [41]. 

Finally a method which shows promise for the future is d5mamic mean field theory. Dynamical mean field theory uses an approximation to the local spectral density functional (rather than energy density functional) and a set of correlated local orbitals. For solids this local description is combined with a periodic description such as DFT using EDA to provide a method of dealing with both localised and delocalised electrons." Anisimov et applied this method to the photo-... 

FLAPW GGA study of the structure and optical properties of AgFe02 and CuFe02 has been carried out by Blaha s group. " Ogata has reviewed band models of Na cCo02. This solid has also been the subject of a study by the recently-developed dynamic mean field theory (DMFT). This study used VASP with an LDA functional, a value of U = 3 eV for Co and an on-site potential for Na. It was found that the Na potential was the key to explaining the stronger correlation for X = 0.7 than for x = 0.3. 

G. Szamel and K. S. Schweizer, Reptation as a dynamic mean field theory -self and tracer diffusion in a simple model of rodlike polymers, J. Chem. Phys., 100 (1994) 3127-3141. 

Fratini, S., and Ciuchi, S., Dynamical mean-field theory of transport of small polarons, Phys. Rev. Lett., 91, 256403, 2003. 

A. Georges, G. Kotliar, W. Krauth, and M.J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Rev. Mod. Phys. 68, 13 (1996). 

Besides wave-function based methods (HF, Cl, CC) or DFT there is a third eategory that became important recently, namely many-body physics, which can handle correlation effects on a different level. Traditionally such schemes were often based on parameters but now they can be combined with DFT results. For example one can start with an EDA calculation and transform the basis set from a Bloch-picture to a Wannier description (see for example [8]). In the latter the correlated electrons can be described by the dynamical mean field theory (DMFT) which can... 

Coleman, 1999). The dynamical competition between the Kondo and RKKY interactions is analyzed by an extension (Smith and Si, 2000 Chitra and Kotliar, 2000) of the dynamical mean-field theory (see (Georges et al., 1996) and references therein). The resulting momentum-dependent susceptibility has the general form... 

Biermaim, S., Aryasetiawan, F., and Georges, A. (2003) First-principles approach to the electronic structure of strongly correlated systems combining the GW approximation and dynamical mean-field theory. Phys. Rev. Lett, 90 (8), 086402. 

Dynamical Mean-Field Theory of Strongly Correlated Fermion Systems and the Limit of Infinite Dimensions. 

Rev. Mod. Phys., 78, 865 (2006). Electronic Structure Calculations with Dynamical Mean-Field Theory. 

In the last 15 years, the Dynamical Mean Field Theory (DMFT) has been developed to investigate the properties for strongly correlated materials. Using this technique, the thermopower has been calculated, in particular in the case of a triangular lattice. Starting from the Hubbard model, the thermopower can be written at low T, as ... 

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