Numerical verification of random phase-and-amplitude formalism of weak turbulence

The Random Phase and Amplitude Formalism (RPA) has significantly extended the scope of weak turbulence studies. Because RPA does not assume any proximity to the Gaussianity in the wavenumber space, it can predict, for example, how the fluctuation of the complex amplitude of each wave mode grows through nonlinear interactions with other modes, and how it approaches the Gaussianity. Thus, RPA has a great potential capability, but its validity has been assessed neither numerically nor experimentally. We compare the theoretical predictions given by RPA with the results of direct numerical simulation (DNS) for a three-wave Hamiltonian system, thereby assess the validity of RPA. The predictions of RPA agree quite well with the results of DNS in all the aspects of statistical characteristics of mode amplitudes studied here

Interplay between Superconductivity and Ferromagnetism on a Topological Insulator

We study theoretically proximity-induced superconductivity and ferromagnetism on the surface of a topological insulator. In particular, we investigate how the Andreev-bound states are influenced by the interplay between these phenomena, taking also into account the possibility of unconventional pairing. We find a qualitative difference in the excitation spectrum when comparing spin-singlet and spin-triplet pairing, leading to non-gapped excitations in the latter case. The formation of surface-states and their dependence on the magnetization orientation is investigated, and it is found that these states are Majorana fermions in the $d_{xy}$-wave case in stark contrast to the topologically trivial high-$T_c$ cuprates. The signature of such states in the conductance spectra is studied, and we also compute the supercurrent which flows on the surface of the topological insulator when a Josephson junction is deposited on top of it. It is found that the current exhibits an anomalous current-phase relation when the region separating the superconducting banks is ferromagnetic, and we also show that in contrast to the metallic case the exchange field in such a scenario does not induce 0-$\pi$ oscillations in the critical current. Similarly to the high-$T_c$ cuprates, the presence of zero-energy surface states on the topological surface leads to a strong low-temperature enhancement of the critical current.Comment: 12 pages, 11 figures