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At low temperature, the transverse transport of
particles subjected
to an external magnetic field is quantized. This
quantum Hall effect
is elegantly described at the single-particle level
in terms of
topological invariants called the Chern numbers.
In this work, one presents several methods that allow
the numerical
and analytical computation of these Chern numbers for
a huge variety of physical systems.
In particular, one applies these methods to ultracoldatoms trapped
in optical lattices and subjected to external Abelian
and non-Abelian gauge
fields. Then one investigates the
spectral
properties of quantum graphs and establishes the
quantization of
the Hall conductivity for these systems.
Finally, one explores the Mott quantum phase
transition in optical
lattices subjected to external gauge fields.
Furthermore, one shows
that vortices are created in the vicinity of the Mott
regime.
This work presents powerful tools for the study of
quantum
transport properties in both condensed matter and
cold atoms
systems, and
should therefore be useful to modern theoretical
physicists.
particles subjected
to an external magnetic field is quantized. This
quantum Hall effect
is elegantly described at the single-particle level
in terms of
topological invariants called the Chern numbers.
In this work, one presents several methods that allow
the numerical
and analytical computation of these Chern numbers for
a huge variety of physical systems.
In particular, one applies these methods to ultracoldatoms trapped
in optical lattices and subjected to external Abelian
and non-Abelian gauge
fields. Then one investigates the
spectral
properties of quantum graphs and establishes the
quantization of
the Hall conductivity for these systems.
Finally, one explores the Mott quantum phase
transition in optical
lattices subjected to external gauge fields.
Furthermore, one shows
that vortices are created in the vicinity of the Mott
regime.
This work presents powerful tools for the study of
quantum
transport properties in both condensed matter and
cold atoms
systems, and
should therefore be useful to modern theoretical
physicists.