generalized to describe also dynamical, namely quantum, fluctuations, which up to now have been ignored.
8. DYNAMICAL FLUCTUATIONS: THE ‘ALLOY ANALOGY’ AND THE
symmetry, disordered phase, and investigate its stability as the temperature is lowered. This is the strategy adopted in a phenomenological Landau Theory (Landau and Lifschitz, 1980). From the point of view of first-principles calcula- tions, the former seems easier as it involves ground state calculations. However, the ‘alloy analogy’ calculations highlighted in this review correspond to the latter.
Evidently, in these the role of the CPA is to describe thermal fluctuations of various local electronic configurations and hence the KKR-CPA procedure is an appropriate tool for the study of the high-temperature, high-symmetry equilib- rium states. In fact, when suitably generalized, it can be used for calculating the coefficients in the Landau expansion of the free energy (Gyorffy et al., 1989) and thereby turning the phenomenological theory into a material specific, quantitative first-principles theory.
Examining the accuracy and reliability of the calculations reviewed inSections 7.2 and 7.3in the light of the above remarks prompts the following observation:
the electronic structure relevant to these calculations is a smeared out version of that atT¼0. Namely, any coherent fluctuation lasting longer thanh/kBTcan be assumed to have been averaged to zero and hence such calculations are more forgiving than those nearT¼0 where timescales much longer thanh/kBTcneed to be accounted for. Thus, the first-principles Landau Theory alluded to above can be considered a robust and efficient theory of the phase diagrams. By contrast for T < Tc, asTtends to 0, longer and longer timescales make their presence felt and a more accurate description of the many-electron problem becomes necessary.
A particular general shortcoming of the ‘alloy analogy’ approximation is that it cannot describe quantum fluctuations such as the zero point fluctuations of spin waves. Evidently, these can be important at and near T ¼ 0. Moreover, an ensemble average of static fluctuations depicted by the ‘alloy’ configurations will, within the CPA, inevitably lead to quasi-particles with finite lifetime even atT¼0. Namely, the ground state is generically not that of a Fermi liquid as it mostly should be. In what follows we shall summarize briefly the current state of conceptual framework that needs to be invoked to deal with these issues.
In condensed matter at a site, an electron scatters from both the atomic nuclei and the other electrons in its vicinity. Such ‘target’ is not, in general, a static, spin- dependent electrostatic, c-number potential but a time-dependent quantum mechanical object which recoils during the scattering process. Clearly, the full complexity of scattering events produced by such ‘targets’ are not described by the static mean field theory language of the ‘alloy analogy’. However, it turns out that their essence is adequately captured by a dynamical generalization of this well tried methodology, namely the DMFT (Georges et al., 1996). Like the ‘alloy analogy’, this elegant procedure focuses on the single site nature of the many- electron problem in solids and it can be viewed as a time-dependent CPA in which the potential seen by an electron at a site changes with time during the scattering process and the averaging over configurations becomes average over all histories of such variations (Kakehashi, 2002).
A remarkable consequence of introducing time-dependent one-electron poten- tials into the calculation is that the well-known instability of a degenerate Fermi system to sudden local perturbations (Anderson, 1967) comes into play. This
effect, which is well understood in the context of the X-ray edge-singularity (Nozieres and DeDominicis, 1969) and the Kondo (Anderson et al., 1970) pro- blems, leads to qualitatively new features in the one-electron spectra predicted by the DMFT compared with the consequences of the static CPA-based theories. The most spectacular of these is the central peak, arising from the Kondo resonance at the Fermi energy, between the upper and lower Hubbard bands, which are already evident in the static alloy analogy calculations. As physical consequences of this peak, one might mention its role in the explanation of the metal-insulator Mott transition (Georges et al., 1996) and the spin-polarization Kondo cloud that screens magnetic impurities in dilute alloys (Anderson et al., 1970). Indeed, it may very well be relevant to the Cea–gtransition discussed inSection 7.3(Held et al., 2001).
Since their invention in the early 1990s, the DMFT technique (Georges et al., 1996) and its cluster generalization DCA (Hettler et al., 1998) have been studied intensively and have been successfully applied to many different problems in metal physics. But, while its virtues and limitations are well documented for simple tight binding model Hamiltonians, its implementation within the context of a fully first-principles theory remains an aspiration only. What has been done, repeatedly and with considerable success, is what may be called the LDAþUþ DMFT method (Georges et al., 1996). In these calculations, the LDAþUpart is a method for generating an effective, usually a multi-band Hubbard, Hamiltonian which serves as input into a DMFT procedure, but there is rarely an effort to recalculate the LDA bands (site energies and hopping integrals) or the electron–
electron interaction parameter U with the view of iterating to self-consistency.
Strangely, although technically the problem appears to be difficult conceptually, thanks to the numerous analytical and numerical results mentioned above, it is relatively simple. The fluctuations to be captured by the putative theoretical framework are tunnelling between atomic-like local electronic configurations, and of these, there are only few that are degenerate in energy and hence can be the source of a Kondo-like resonance. In other words, at an atomic centre, an electron scatters from a quantum mechanical two or few level systems instead of a classical effective electrostatic potential. Scaling arguments suggest that the occur- rences of such resonances are general consequences of degenerate Fermi systems being perturbed by sudden, local, quantum perturbations and their width is a new, emergent, low-energy scalekBTKwhereTKis usually referred as the ‘Kondo’
temperature (Cox and Zawadowski, 1999). From the point of view of our present concern, the importance of these results are twofold. First, this low-energy scale, which governs the behaviour of low-temperature fluctuations, has no counterpart in the static mean field theory. Second,kBTKvaries dramatically from material to material from 1 to 1000 K. Thus, there is need for a sophisticated first-principles theory which can make quantitative material specific predictions ofkBTK. Clearly, ifTKTcfor some particular order, the high-temperature phase is well described by the ensemble of static fluctuations depicted in the ‘alloy analogy’ calculation.
On the other hand, ifTK Tc, the static calculations miss out important aspects of the physics. To highlight the burden of this remark, we note that the Curie temperatures of ferromagnets and the critical temperatures of the conventional
superconductors can be calculated fairly reliably for the majority of materials.
Evidently, the challenge here is to do the same forkBTK.