41 research outputs found
On the symmetry breaking phenomenon
We investigate the problem of symmetry breaking in the framework of dynamical
systems with symmetry on a smooth manifold. Two cases will be analyzed: general
and Hamiltonian dynamical systems. We give sufficient conditions for symmetry
breaking in both cases
Homotopy types of stabilizers and orbits of Morse functions on surfaces
Let be a smooth compact surface, orientable or not, with boundary or
without it, either the real line or the circle , and
the group of diffeomorphisms of acting on by the rule
, where and .
Let be a Morse function and be the orbit of under this
action. We prove that for , and
except for few cases. In particular, is aspherical, provided so is .
Moreover, is an extension of a finitely generated free abelian
group with a (finite) subgroup of the group of automorphisms of the Reeb graph
of .
We also give a complete proof of the fact that the orbit is tame
Frechet submanifold of of finite codimension, and that the
projection is a principal locally trivial -fibration.Comment: 49 pages, 8 figures. This version includes the proof of the fact that
the orbits of a finite codimension of tame action of tame Lie group on tame
Frechet manifold is a tame Frechet manifold itsel
Internal Frame Dragging and a Global Analog of the Aharonov-Bohm Effect
It is shown that the breakdown of a {\it global} symmetry group to a discrete
subgroup can lead to analogues of the Aharonov-Bohm effect. At sufficiently low
momentum, the cross-section for scattering of a particle with nontrivial
charge off a global vortex is almost equal to (but definitely different from)
maximal Aharonov-Bohm scattering; the effect goes away at large momentum. The
scattering of a spin-1/2 particle off a magnetic vortex provides an amusing
experimentally realizable example.Comment: (14 pp
Biaxial nematic phases in ultracold dipolar Fermi gases
Ultracold dipolar Fermi gases represent relatively unexplored, strongly
correlated systems arising from long-range and anisotropic interactions. We
demonstrate the possibility of a spontaneous symmetry breaking biaxial phase in
these systems, which may be realized in, e.g., gases of ultracold polar
molecules or strongly magnetic atoms. This biaxial nematic phase is manifest in
a spontaneous distortion of the Fermi surface perpendicular to the axis of
polarization. We describe these dipolar interaction induced phases using Landau
Fermi liquid theory.Comment: 4 pages, 1 figure; clarifying comments added to tex