Since the dynamical mean-field theory (DMFT) have appeared earlier than the SDFT itself, it could be represented as a separate approach without reviewing the SDFT basics. However, one can trace better the analogy with the DFT-LDA if it is presented as a sort of mean-field approximation to SDFT. DMFT is based on the work of Metzner and Vollhardt [50] who derived the non-trivial limit of the Hubbard model for the large coordination number and the work of Georges et. al [21] who made the connection between the localized and the delocalized descriptions. Detailed overviews of DMFT can be found in Refs. [86] and [87].
First of all, similar to the LDA+U, the DMFT considers the two- particle interactions within a certain electronic subsystem, namely in the d-, or f-shell which is well-localized within the atomic site. Thus, the localization domain is site-centered instead of travelling throughout the solid as illustrated in Fig. 5.3. This leads to a certain drawback,
BATH
V(r,r’,t) r’
r
Fig. 5.3. Localization domains in SDFT (upper left side) and in the DMFT (upper right side). Lower panel: illustration of the Anderson model. Localizedd-electrons coupled by the non-local time-dependent interaction V(r,r′, t) with the surrounding bath. The
5.5. The dynamical-mean field approximation 59 namely, it is not possible any more to reproduce the exact density of states from Gloc and as a result to recover the Hartree energy directly. On the other hand, this situation is rather close to the real systems containing 3d- and 5f-elements in which the correlated electrons are well-localized within the atomic sites. Thus, it is natural to consider the non-local correlation only for the d- or f-electron subsystem within its localization domain.
However, in real solids the situation is complicated by the addi- tional coupling of the d-electrons with the s and p delocalized shell. In the DMFT this problem is avoided by the mean-field description of the coupling. Namely, the d-electrons feel the time-dependent non- local interaction with the delocalized shell as a sort of time-dependent but local mean-field, which is created by the spatial integration over the delocalized shell. In the literature the latter is usually referred as “bath”. To perform such an integration we have to know the many- body wave function of the bath. On the other hand, by presenting the coupling as an interaction of the randomly situated bath particles coupled to the certain space point r (see Fig. 5.3), the integral influ- ence, namely the random variable Vbath(r, t) = Pr′ V(r,r′, t) will have normal (r, t)-parameterized distribution that does not depend on the particular distribution of V(r,r′, t) once the number of particles in the bath increases to infinity (Central Limit theorem). As was numerically shown by Metzner and Vollhardt this sum converges very fast. For ex- ample, in the model which is equivalent to 6 bath particles it is already close to the infinite limit [86].
Having incorporated all external influence with respect to the local- ized electron subsystem into the local scalar potential Vbath, the condi- tions of the Baym-Kadanoff approach are satisfied and the many-body problem of the localized shell can be solved exactly using the pertur- bational technique discussed in Chapter 3. Since the DMFT model is equivalent to the famous Anderson effective impurity model (AIM) [3, 24, 90] the corresponding many-body technique solving the Baym- Kadanoff equations is referred as “impurity solver”.
Of course, the function Vbath(r) is unknown. On the other hand, if the self-energy and the corresponding Green’s function of the interact- ing d-system are known, it is possible to find the effective one-particle Green’s function of the bath G0 from the following saddle-point equa-
60 Chapter 5. The energy functional description
tion:
G0−1(r,r′, ǫ) = G−loc1(r,r′, ǫ)−Σ(r,r′, ǫ). (5.48) Then, it can be used as an input quantity for the Baym-Kadanoff prob- lem of the interactingd-electrons which can be approximately solved by one of the impurity solvers which delivers the corresponding self-energy. Thus we have a closed relationship between all introduced quantities and the overall problem can be solved self-consistently as shown in Fig. 5.4. The particular feature of the DMFT is that the whole inter-
G−1 = G−01 + Σ Gloc = G ·θloc G0−1 = G−loc1 −Σ Impurity solver Σ = Σnew G0,Σ Σ Σnew G Gloc G0 Σ,G
Fig. 5.4.Illustration of the self-consistent solution of the DMFT problem. Starting from the non-interacting functionG0 and the initial guess for the localized self-energyΣwe obtain the auxiliary functionG within the whole space as G−1 = G−1
0 + Σ. Then the localized partGlocis extracted byGloc =Gθloc. In the next step the Green’s functionG0 accounting for the interaction only between the conduction and localized systems is created; for that reason the many-body interactionΣis extracted from Gloc. G0 is used as an input for the many-body problem within the localized shell. The latter is solved by applying a certain many-body technique (impurity solver) which derives the self-energyΣnew. The loop is repeated until the self-consistent self-energy is found. All quantities are evaluated for fixed time (or energy) and space arguments.
acting system is split into one being treated exactly and another one treated within the mean-field approximation.
5.6. LDA+DMFT 61