Gutzwiller state

Gutzwiller M C 1967 Phase-integral approximation in momentum space and the bound states of an atom J Math. Phys. 8 1979... 

Rather than looking at the spectrum obtained from the secular determinant (5), we will here consider the spectrum SG for fixed wavenumber k and than average over k. One can write the spectrum in terms of a periodic orbit trace formula reminiscent to the celebrate Gutzwiller trace formula being a semiclassical approximation of the trace of the Green function (Gutzwiller 1990). We write the density of states in terms of the traces of SG, that is,... 

In the absence of the interaction U, the ground state is that of uncorrelated electrons j o) and has the form of a Slater determinant. As U is turned on, the weight of doubly occupied sites must be reduced because they cost an additional energy U per site. Accordingly, the trial Gutzwiller wave function (GWF) I/g) is built from the Hartree-like uncorraleted wave function (HWF) l/o),... 

Here p iaa occ, L() (respectively p iaa unocc, L()) represents the probability of the atomic configuration of site i, where the orbital a with spin a is occupied (resp. unoccupied) and where L[ is a configuration of the remaining orbitals of this site. This result is similar to the expression obtained by Biinemann et al. [22], but it is obtained more directly by the density matrix renormalization (5). To obtain the expression of the qiaa factors, an additional approximation to the density matrix of the uncorrelated state was necessary. This approximation can be viewed as the multiband generalization of the Gutzwiller approximation, exact in infinite dimension [23]... 

This ab-initio Gutzwiller approach is able to handle correctly the correlation aspects without loosing the ab-initio adjustable parameters free aspect of the more familiar DFT-LDA, and that way, corrects the deficiency of this method. It gives similar results to the methods that account for many-body effects like the LDA+DMFT of Ref. [10] from the ab-initio levels or that can have an orbital dependent potential like in the LDA+ 7 calculation of Ref. [36], which is impossible to DFT-LDA approach. On another hand, we stress again that our approach is clearly variational, and is able to provide an approximate ground state in contrast with those of Refs. [10] and [36]. 

Figure 5. Quasiparticles density of states obtained from Gutzwiller method for Plutonium in S phase.

About 50 years after Einstein, Gutzwiller applied the path integral method with a semiclassical approximation and succeeded to derive an approximate quantization condition for the system that has fully chaotic classical counterpart. His formula expresses the density of states in terms of unstable periodic orbits. It is now called the Gutzwiller trace formula [9,10]. In the last two decades, several physicists tested the Gutzwiiler trace formula for various... 

Quantum calculations for a classically chaotic system are extremely hard to perform. If more than just the ground state and a few excited states are required, semiclassical methods may be employed. But it was not before the work of Gutzwiller about two decades ago that a semiclassical quantization scheme became available that is powerful enough to deal with chaos. Gutzwiller s central result is the trace formula which is derived in Section 4.1.3. 

The most recent advance in the theory of the helium atom was the discovery of its classically chaotic nature. In connection with modern semiclassical techniques, such as Gutzwiller s periodic orbit theory and cycle expansion techniques, it was possible to obtain substantial new insight into the structure of doubly excited states of two-electron atoms and ions. This new direction in the application of chaos in atomic physics was initiated by Ezra et al. (1991), Kim and Ezra (1991), Richter (1991), and Bliimel and Reinhardt (1992). The discussion of the manifestations of chaos in the helium atom is the focus of this chapter. 

According to Gutzwiller s result (4.1.72) the density of states can be computed as a generalized Fourier sum which contains only classical periodic orbit information. Therefore, using a suitable projection technique it should be possible to invert the transform and extract classical periodic orbit information from the level density. The inversion can be achieved using scaled energy spectroscopy, a technique first introduced by... 

While the Hubbard model can be derived from the Hamiltonian in eqn.(2), the model was invented in an entirely different context, independently by Kanamori [19], Gutzwiller [20] and Hubbard [21]. It was first introduced to explain ferromagnetism in metals. While the simple Hubbard model is now known not to have a ferromagnetic ground state (except in some pathological cases [22]), it has become one of the most widely studied models in the context of metal-insulator transitions [23, 24]. 

The discovery that classical periodic motions of the electron can account for the oscillations in photoabsorption spectra is an elegant experimental connection between classical and quantal theories, as well as being a striking demonstration of the Gutzwiller formalism [2], Over the last few years, experimentally created wave-packets have been used to identify the corresponding periodic motions. In these empirical studies, a Rydberg wave-packet is formed from a low-lying state by a short laser pulse and the time evolution of this wave-packet is probed by a second laser pulse. 

A variety of munerical methods have been used to elucidate the existence of phase separation in the groimd states of the Hubbard and models of the Cu-0 planes of cuprate perovskites. States with charge density waves (CDW), which can be interpreted as stripes, were obtained in Refs. [24-26,29-32] by using the mean-field approximation, the variational principle with the Gutzwiller-type variational fimctions, and the density matrix renormalization group calculations. However, the results of the Monte Carlo simulations [33-35] and cluster calculations [36,37] cast doubt on this finding. Thus, the issue of whether a purely electronic mechanism can explain the stripe formation is still an open question. 

In the calculation of electronic structures, the presence of correlations thus always represents a difficulty. Perturbation expansions can account for the two extreme cases the delocalized limit in which the effective repulsion U is low compared to the band width, and the quasi-atomic limit where the electron delocalization modifies only slightly the correlated ground state (Anderson, 1959). Some variational techniques (Hubbard, 1964 Gutzwiller, 1965) allow a treatment of systems with U of the order of jS, but they are difficult to use. New methods have recently been developed for adding a part of the Hubbard Hamiltonian to the LDA (local density approximation) ground state (Czyzyk and Sawatzky, 1994). 

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