Strongly correlated electron models
Materials which contain atoms with highly compact electronic orbitals,
like U and Ce compounds or even pure transition metals, show a wide variety
of fascinating physical properties that still challenge physicists for
explanation. There are a few basic models that capture the essence of the
microscopic situation; the strong repulsion of electrons inside the localized
orbitals plus their itinerancy outside and, the Pauli exclusion principle.
For the past few years I have been studying these models (especially the
Hubbard model and the periodic Anderson model) to understand the mechanism
behind such novel phenomena as the formation of a spin-density-wave ground
state, superconductivity and the metal-insulator transition in the strongly
correlated electron materials.
Quantum many-body approximation schemes in these models often lead to
a very involved system of integral equations for spatial dimensions of
physical interest (2 and 3). The idea of performing calculations in the
limit of large spatial dimensions is productive here. Metzner and Vollhart
showed that the irreducible vertex functions of strongly correlated models
become purely local (momentum-independent) for large spatial dimensions
and hence much of the difficulty in calculating the multiple momentum integrals
in the perturbation techniques can be bypassed. Thereafter investigators
have shown that the infinite-dimensional results are indeed a good starting
point for approximating the physical quantities in the original two or
three dimensional models.
Cluster calculations: a first step towards a 1/D expansion
Metzner and Vollhardt showed that the irreducible vertex functions of strongly
correlated models become purely local (momentum-independent) in the limit
of large spatial dimensions. In this limit most lattice models can be mapped
onto a single site problem embedded in a self-consistently determined host.
The quantum Monte Carlo (QMC) algorithm of Hirsch and Fye can then take
advantage of this simplification and give an essentially exact solution
for the lattice model. But like in every mean-field type calculation, we
must justify the assumption of smallness of the spatial fluctuations if
we are to use the "local approximation" for small D. For this purpose we
are currently developing a "dynamical cluster algorithm" which includes
nonlocal dynamical correlations on a finite size cluster as corrections
to the local approximation (which corresponds to a single site cluster).
We believe this is the first step towards achieving a systematic 1/D expansion
for models of strong correlations. I am now working on computing the two-particle
properties of the Hubbard model in search for magnetic and/or nonlocal
superconducting instabilities.
Periodic Anderson model in infinite dimensions: dynamical properties
QMC simulations give the response function for the system as a function
of Matsubara frequencies. Hence one does not have a direct access to the
dynamical properties of the system unless one can analytically continue
this information to real frequencies. I extracted the dynamics of the periodic
Anderson model (PAM) by utilizing the maximum entropy method (MEM) for
analytical continuation of the QMC data. Concomitant with the differences
in screening behavior for the PAM and the single impurity Anderson model
(SIAM) I found that the temperature dependence of the Kondo peak in the
spectral function of the PAM is much weaker than what is expected from
the SIAM, consistent with the recent photoemission data in Ce-based materials.
I also found that the Kondo peak is dispersive and forms a narrow quasiparticle
band which crosses the Fermi energy as indicated in those experiments.
I could also calculate the resistivity for the PAM. Its temperature dependence
shows nicely the interplay between the two relevant energy scales for the
model. Around the larger "impurity screening scale" (the Kondo scale calculated
from a SIAM with the same model parameters) the resistivity shows the usual
Kondo type behavior (a log-linear dependence adjacent to a maximum). Around
the smaller "coherence scale" (the Kondo scale calculated directly from
the PAM) the resistivity flattens and then falls quickly toward zero as
the temperature decreases, indicating the emergence of a metallic state.
Periodic Anderson model in infinite dimensions: static properties
I used the QMC method along with the diagrammatic techniques to study the
static magnetic response function of the non-degenerate PAM in infinite
dimensions. Specifically, I studied this model in the limit where there
is on the average one electron in each local orbital on each site (Kondo
limit) and for various numbers of electrons in the delocalized band. I
found paramagnetic, antiferromagnetic as well as ferromagnetic regions
depending on the band filling. I also found that the Kondo scale for the
PAM is greatly reduced compared to that of the single impurity Anderson
model (SIAM) as the system is doped away from half-filling. This guided
me to another important observation; I found a quantitatively different,
non-universal behavior for the temperature dependence of screening in the
PAM as compared to the well-known universal behavior in the SIAM.
Three-dimensional Hubbard model in the local approximation
Quantum many-body approximation schemes in the models of strong correlations
often lead to a very involved system of integral equations for spatial
dimensions of physical interest ($D=2, 3$). The locality (momentum independence)
of the irreducible vertex functions in the limit of infinite dimensions
has proved to be very fruitful in many-body calculations for these models
since much of the difficulty in calculating the multiple momentum integrals
can be bypassed. Motivated by an existing result for the Hubbard model
in infinite dimensions, I applied the local approximation to the three-dimensional
Hubbard model. I applied a second-order perturbation expansion in the coupling
constant $U$ to calculate the magnetic susceptibility. I derived the instability
condition for the paramagnetic ground state towards the formation of a
spin-density-wave state for this approximation. This turned out to be the
original Stoner criterion modified by an additional term due to the second
order graph in the expansion of the self-energy which represents the lowest
order effect of quantum fluctuations. I formed a magnetic phase diagram
for the three-dimensional Hubbard model, which provided the transition
temperature from paramagnetic to commensurate or incommensurate antiferromagnetic
states depending the number of electrons or holes. Furthermore, I was able
to produce an excellent one-parameter fit between this transition curve
and that of the dilute Chromium alloys which showed the usefulness of the
approximation.
Statistical mechanical models of neural networks
For my masters degree dissertation I studied the neural networks from the
statistical mechanics point of view. Particularly, I worked on the Hopfield
model of associative memory with a simple Hebb learning rule, which resembles
very much to an Ising model for spin-glass systems in formalism. I developed
a Monte Carlo algorithm which could simulate the system in the presence
of an external source and hence calculate the corresponding response functions.
The resulting phase diagram showed that the correlation between the source
term with any memorized pattern can increase the basin of attraction of
that pattern.