My interest lies in basic problems and phenomena
in statistical mechanics, superconductivity and superfluidity.
Below are some of the recent achievements.
Superconductivity | Superfluidity | Nonequilibrium statistical mechanics
- A new self-consistent perturbation expansion for Bose-Einstein condensates
[
T. Kita: Phys. Rev. B 80 (2009) 214502
]
Assemblies of identical particles are classified into two groups
according to the statistics they obey, which exhibit completely different
properties at low temperatures.
The first category is relevant to particles with half-integer spins
called fermions. Here, no pair of particles can occupy a single one-particle state due to the "Pauli exclusion principle."
The other category corresponds to particles with integer spins
called bosons.
They are free from the exclusion principle, on the other hand, and may condense into a single
quantum state
below a certain critical temperature T_{0} to acquire remarkable "coherent" physical properties.
This phenomenon was predicted by Einstein in 1925 based on
the work by Bose on the statistics of bosons in 1924. Thus, it
is now generally known as Bose-Einstein condensation (BEC).
Whereas Einstein considered an ideal Bose gas, real assemblies of particles do have interactions between particles.
A notable example is superfluid ^{4}He, which had been the only known system
in connection with BEC for a considerable period. Here the interaction drives the system far apart from an ideal gas to the extent that Einstein's theory is no longer applicable.
This fact had raised considerable interests and efforts
to realize BEC in systems with weak interactions.
A breakthrough was finally brought about by JILA group in 1995 when they
realized BEC with an atomic alkali gas of ^{87}Rb; it triggered a rapid progress in the experiment of BEC using different atoms.
For example, we now have
widely accepted reports on the observation of quantized vortices
that may be regarded as a direct confirmation of BEC.
Since the interaction between particles is weak in these systems,
one may generally expect excellent quantitative agreements between theory and
experiment.
However, a fundamental unsolved issue had remained on the topic,
i.e., there was no reliable approximation schemes for Bose-Einstein
condensates (BECs) even at the mean-field level comparable to the Bardeen-Cooper-Schrieffer theory
of superconductivity. Thus, finite-temperature properties have not
been understood sufficiently even for weakly interacting systems.
The field-theoretic approach to condensed Bose systems was
pioneered by Bogoliubov in 1947; he predicted a gapless phonon-like excitation spectrum at long wavelengths at zero temperature.
It was followed by active studies of BEC in 1950's-60's. Among them,
Beliaev introduced Green's function technique, which was used subsequently by Hugenholtz and Pines to show
that there should be no gap in the excitation spectrum
by analyzing the structure of the simple perturbation expansion.
However, the approximate Bogoliubov theory at zero temperature has been known to encounter difficulties when extending it to finite temperatures.
To be specific, if we adopt the conventional
self-consistent Wick decomposition procedure
to the interaction term,
we end up having an unphysical energy gap
in the excitation spectrum.
Although there exists a prescription to kill the energy gap,
such a theory cannot be used to describe dynamical properties
as it violates various conservation laws.
In general, the field theory for condensed Bose systems
has not been as successful as that for other systems.
Here we have a couple of basic quantities to be renormalized self-consistently,
i.e., the condensate wave function and the
quasiparticle Green's function, for which we did not have well-established systematic self-consistent
approximations.
With these observations, I tried to construct a self-consistent approximation scheme that
simultaneously satisfies a couple of properties of the exact theory,
i.e., "gapless'' and "conserving.''
Here "gapless'' denotes that the excitation spectrum is gapless in accordance with the Hugenholtz-Pines theorem,
whereas "conserving'' means that the theory obeys various conservation laws.
It was not clear whether such a theory could ever be established within a single
approximation scheme.
For this purpose, I first derived a couple of exact identities obeyed by Green's function.
Next, I extended the Luttinger-Ward formalism of a self-consistent perturbation expansion for fermions
to write down the thermodynamic potential of BEC as a functional of the
condensate wave function and the quasiparticle Green's function.
I subsequently determined the weights of Feynman diagrams characteristic of BEC
by using the two exact identities.
One of the notable properties of this formalism is that it satisfies the Hugenholtz-Pines theorem and conservation laws simultaneously,
thus resolving the so-called conserving vs gapless dilemma.
This theory is expected to form a new theoretical basis
to study Bose-Einstein condensates at finite
temperatures.
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