The proposals for realizing exotic particles through coupling of quantum Hall effect to superconductivity involve spatially non-uniform magnetic fields. As a step toward that goal, we study, both theoretically and experimentally, a system of Dirac electrons exposed to an Abrikosov flux lattice. We theoretically find that the non-uniform magnetic field causes a carrier-density–dependent reduction of the Hall conductivity. Our studies show that this reduction originates from a rather subtle effect: a levitation of the Berry curvature within Landau levels broadened by the non-uniform magnetic field. Experimentally, we measure the magneto-transport in a monolayer graphene-hexagonal boron nitride-niobium diselenide (NbSe2) heterostructure, and find a density-dependent reduction of the Hall resistivity of graphene as the temperature is lowered from above the superconducting critical temperature of NbSe2, when the magnetic field is uniform, to below, where the magnetic field bunches into an Abrikosov flux lattice.

An interplay between pairing and topological orders has been predicted to give rise to superconducting states supporting exotic emergent particles, such as Majorana particles obeying non-Abelian braid statistics. We consider a system of spin polarized electrons on a Hofstadter lattice with nearest-neighbor attractive interaction and solve the mean-field Bogoliubov–de Gennes equations in a self-consistent fashion, leading to gauge-invariant observables and a rich phase diagram as a function of the chemical potential, the magnetic field, and the interaction. As the strength of the attractive interaction is increased, the system first makes a transition from a quantum Hall phase to a skyrmion lattice phase that is fully gapped in the bulk but has topological chiral edge current, characterizing a topologically nontrivial state. This is followed by a vortex phase in which the vortices carrying Majorana modes form a lattice; the spectrum contains a low-energy Majorana band arising from the coupling between neighboring vortex-core Majorana modes but does not have chiral edge currents. For some parameters, a dimer vortex lattice occurs with no Majorana band. The experimental feasibility and the observable consequences of skyrmions as well as Majorana modes are indicated.

Finding and understanding non-Fermi liquid transport behaviors are at the core of condensed matter physics. Most of the existing studies were devoted to the monolayer Hubbard model, which is the simplest model that captures essential features of high-temperature superconductivity. Here we discover a new type of non-Fermi liquid behavior emergent in the hole-doped bilayer Hubbard model, using dynamical mean-field theory with a full consideration of the short-range interlayer electron correlation. We find that at low temperatures, the Hall coefficient has a strong nonmonotonic dependence on temperature, leading to a double or quadruple reversal of its sign depending on the doping level. At the same time, the resistivity exhibits two plateaus rather than linearity in its temperature dependence. We show that these intriguing transport behaviors stem from the formation of coherent interlayer singlets, which scatter off gapped collective modes arising from short-range interlayer antiferromagnetic fluctuations.