41 research outputs found

    On the symmetry breaking phenomenon

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    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

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    Let MM be a smooth compact surface, orientable or not, with boundary or without it, PP either the real line R1R^1 or the circle S1S^1, and Diff(M)Diff(M) the group of diffeomorphisms of MM acting on C∞(M,P)C^{\infty}(M,P) by the rule h⋅f↦f∘h−1h\cdot f\mapsto f \circ h^{-1}, where h∈Diff(M)h\in Diff(M) and f∈C∞(M,P)f \in C^{\infty}(M,P). Let f:M→Pf:M \to P be a Morse function and O(f)O(f) be the orbit of ff under this action. We prove that πkO(f)=πkM\pi_k O(f)=\pi_k M for k≥3k\geq 3, and π2O(f)=0\pi_2 O(f)=0 except for few cases. In particular, O(f)O(f) is aspherical, provided so is MM. Moreover, π1O(f)\pi_1 O(f) is an extension of a finitely generated free abelian group with a (finite) subgroup of the group of automorphisms of the Reeb graph of ff. We also give a complete proof of the fact that the orbit O(f)O(f) is tame Frechet submanifold of C∞(M,P)C^{\infty}(M,P) of finite codimension, and that the projection Diff(M)→O(f)Diff(M) \to O(f) is a principal locally trivial S(f)S(f)-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

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    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 Z2\Z_2 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

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    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
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