The gap equation for spin-polarized fermions
Abraham Freiji, Christian Hainzl, and Robert Seiringer
The authors study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. This is the continuation of recent studies where the BCS gap equation in the balanced case for systems with general pair interaction potential V was investigated. The authors find that for cosh (δμ/T)⩽2, with T the temperature and δμ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously. The authors derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.
Proof of rounding by quenched disorder of first order transitions in low-dimensional quantum systems
Michael Aizenman, Rafael L. Greenblatt, and Joel L. Lebowitz
The authors attempt to prove that for quantum lattice systems in d⩽2 dimensions the addition of quenched disorder rounds any first order phase transition in the corresponding conjugate order parameter, both at positive temperatures and at T=0. For systems with continuous symmetry the authors extend the proof up to d⩽4 dimensions. The authors achieve the extension of the proof to quantum systems by carrying out the analysis at the level of thermodynamic quantities rather than equilibrium states.
Vortex dynamics in R4
Banavara N. Shashikanth
The author studies the vortex dynamics of Euler's equations for a constant density fluid flow in R4 with special focus on singular Dirac delta distributions of the vorticity supported on two-dimensional surfaces (membranes). The self-induced velocity field of a membrane has a logarithmic divergence. A regularization done via the LIA then shows that the regularized velocity field is proportional to the mean curvature vector field of the membrane rotated by 90° in the plane of normals. The author also presents a Hamiltonian structure for the regularized self-induced motion of the membrane. The dynamics of the four-form ω ∧ ω is examined and it is shown that Ertel's vorticity theorem in R3 , for the constant density case, is a special case of this dynamics.
Composite parameterization and Haar measure for all unitary and special unitary groups
Christoph Spengler, Marcus Huber, and Beatrix C. Hiesmayr
The stochastisation hypothesis and the spacing of planetary systems