main focus of my research is to describe the electron correlation
effects in the condensed matters from the theoretical point of view. So
far, I have developed a new path-integral theory which is free from
Grassmann algebra and applicable for spectroscopy, such as
photoemission, light absorption, and x-ray scattering. The strong
coupling between the spin and charge excitations, inherent in the
strongly correlated electron systems, are clarified through the
numerical calculations of these spectra. I have also developed a
resonating Hartree-Fock method to describe the quantum fluctuations in
the strongly correlated electron systems. More information is given
Path-integral theory for the optical response functions
path-integral theory, the two-body Coulomb interaction is reduced into
the linear interaction between an electron and scaler boson field. As a
result, we can easily diagonalize such a one-body Hamiltonian once a
configuration of bonson fields is determined. However, the possible
number of the configurations is so huge that we cannot completely sum
them up. (Note that the summation over the configurations of boson
fields corresponds to the path-integration.) Therefore, we estimate
this summation by means of the Monte Carlo simulation.
The merit of
this mehod lies in the easy access to the spectroscopy,
such as photoemission, light absorption and emission. We can calculate
the spectral functions almost exactly even for the strongly correlated
electrons systems. So far, we have successfully described the
experimentally observed photoemission spectra of the one-dimensional
Cu-oxides and halogen-bridged Ni complex within the framework of the
Hubbard and extended Hubbard models, respectively.
One important result obtained by this path-integral approach is that
the photoemission spectrum of the strongly correlated electron system
is predominated by the incoherent component caused by the nonlinear
coupling between the photo-generated hole and magnetic fluctuations. We
reached this important result in the following. First, we calculated a
photoemission spectrum for the electron system having the gap in the
magnetic excitatiion. In such a sytem, we can clearly distinguish a
sharp magnon-free (coherent) peak and broad
magnon-coupled (incoherent) component because of the finite magnon gap.
Then, by tracing the magnon gap-dependence of the coherent peak, we can
clearly conclude that the the photoemission spectrum consists of
the multi-magnon incoherent component almost completely.
On the other hand, in the weakly correlated electron system, the
photoemission spectrum still has a coherent component near the Fermi
energy. We are now trying to establish this result and also to clarify
the mechanism of the metal-insulator transition in the weakly
correlated electron system.
Hartree-Fock theory for quantum fluctuations
all know that Hartree-Fock theory no longer works well on the strongly
interacting electron system. Theorists have tried to descirbe the
electron correlations beyond a Hartree-Fock state. The exact
diagonalization method is the most straightforward approach to explain
the electron correlation completely. However, this method is applicable
only for the samll systems up to about 24 electrons. The quantum Monte
Carlo method is another strong approach accessible to more than 100
electrons. However, the quantum Monte Carlo method has a drawback
called the negative sign problem. The probability to update the
configuration becomes negative when the electron-hole symmetry is
broken. As a result, we cannot make an
importance sampling and get accurate observavles. For example, we
cannot access to the doped systems by the quantum Monte Carlo method.
Density matrix renormalization group (DMRG) method is also a powerful
approach accessible to the almost inifinite systems. However, the DMRG
method is accessible only to the one-dimensional system.
The resonating Hartree-Fock method was originally developed by H.
Fukutome to descirbe both the electron correlations and quantum
fluctautions. These two concepts are of-course closely related, but the
previous theories cannot even suggest how they are. This method
constructs a many-body wave function by superposition of non-orthogonal
Slater deteminants (S-dets), such as
Here, not only the superpostion coefficients C's but also the molecular
orbitals in all the S-dets are variational determined. The orbital
optimization is very time-consuming, but makes the wave function much
better than the simple superposition without the orbital optimization.
It was shown for the one-dimensional Hubbard model that the resonating
Hartree-Fock wave function describe the more correlation energies than
the well-established variational Monte Carlo method.
The most important merit to employ the resonating Hartree-Fock method
is that we can direclty obtain information on quantum fluctuations
through the S-dets constituting the wave function. For example, in the
one-dimensional Hubbard system, the Hartree-Fock ground state is a spin
density wave, where the energy is degenerate for the spin density
alternations up down up down ..., and down up down up ... .In such a
broken symmetry state, we have a low energy excitation connecting the
two degenated alternating phases called a soliton. Then, we showed for
the one-dimensional Hubbard system, the dominant quantum fluctuations
are described as a translational, vibrational and breathing motions.
Furhtermore, we also clarified that in the spin density wave-charge
density wave phase transition, the domain wall connecting the two sates
workd as precursor of the quantum nucleation.
The resonating Hartree-Fock method is thus a very promissing approach
to describe the electron correlation effects quite efficiently. I am
now trying to extend this method to the spectroscopy.