When a gas of electrons is confined to two dimensions, application of a strong magnetic field may lead to startling phenomena such as emergence of electron pairing. According to a theory this manifests itself as appearance of the fractional quantum Hall effect with a quantized conductivity at an unusual half-integer ν = 5 2Landau level filling. Here we show that similar electron pairing may occur in quantum dots where the gas of electrons is trapped by external electric potentials into small quantum Hall droplets. However, we also find theoretical and experimental evidence that, depending on the shape of the external potential, the paired electron state can break down, which leads to a fragmentation of charge and spin densities into incompressible domains. The fragmentation of the quantum Hall states could be an issue in the proposed experiments that aim to probe for non-abelian quasi-particle characteristics of the ν = 5 2 quantum Hall state.

We consider predicting systemic financial crises one to five years ahead using recurrent neural networks. The prediction performance is evaluated with the Jorda-Schularick-Taylor dataset, which includes the crisis dates and relevant macroeconomic series of 17 countries over the period 1870-2016. Previous literature has found simple neural network architectures to be useful in predicting systemic financial crises. We show that such predictions can be significantly improved by making use of recurrent neural network architectures, especially suited for dealing with time series input. The results remain robust after extensive sensitivity analysis.

We apply reduced density-matrix functional theory to the parabolically confined quantum Hall droplet in the spin-frozen strong magnetic field regime. One-body reduced density matrix functional method performs remarkably well in obtaining ground states, energies, and observables derivable from the one-body reduced density matrix for a wide range of system sizes. At the strongly correlated regime, the results go well beyond what can be obtained with the density functional theory. However, some of the detailed properties of the system, such as the edge Green's function, are not produced correctly unless we use the much heavier two-body reduced density matrix method.Comment: 13 pages, 7 figure