Scanning-tunnelling potentiometry measurements by the Brar experimental group reveal magnetic-field-dependent voltage patterns around a circular graphene p–n junction. Motivated by this collaboration, I will discuss a related theoretical question: what determines the effective transport size of a smooth density perturbation? Our theoretical work shows that local Ohm–Hall transport can produce an effective obstacle radius R_no ~ a α^(−1/β) ≫ a, where a characterizes the perturbation, β > 2 describes its density tail, and α = σ_xx/|σ_xy| ≪ 1. This amplification arises from competition between Hall circulation and dissipative leakage, without invoking viscosity. The resulting far-field dipole also poses an inverse problem: its magnitude alone cannot separate the perturbation size, the density profile and transport coefficients, while its orientation supplies complementary information. Returning to the graphene experiment, I will discuss annular features whose separation approximately follows the cyclotron diameter, as predicted by a kinetic calculation with a specified boundary. A different question about what experiments reveal arises in correlated electron systems: can electron transfer miss the lowest charged state? Motivated by the one-third-filled charge crystal proposed for twisted MoTe₂, I studied a twelve-site, four-carrier extended Hubbard model by exact diagonalization. In this model, the lowest vacancy state retains a reversed magnetic moment up to a field about fifteen times larger than that needed to align the background. Because carrier number in each valley is conserved, removing one carrier from the aligned background cannot create this state, so it is absent from the direct removal spectrum. This raises three questions: does the reversed moment remain bound to the vacancy in larger crystals, can the charge crystal modify this binding beyond the known triangular-lattice magnetic-polaron mechanism, and can weak valley mixing make the lower state visible? A related question is whether magnetic rearrangement changes the ordering and electron-transfer weight of fractionally charged excitations in MoTe₂.

We perform a general-relativistic analysis of proton superconductivity in magnetized neutron stars by solving the coupled Einstein-Maxwell equations. The study incorporates microscopically derived pairing gaps based on many-body nuclear interactions within realistic equations of state that include two- and three-body forces and relativistic mean-field models. We examine how magnetic field geometry (toroidal and poloidal) and stellar density (hence mass) affect the topology of type-I and type-II superconducting and normal regions. The results confirm the absence of S-wave superconductivity in the inner core for realistic-mass neutron stars, consistent with observational constraints, and reveal distinct three-dimensional nonsuperconducting domains: torus-like for toroidal and prolate for poloidal fields, which are not captured in one-dimensional models. These configurations provide new insight into the interplay between dense-matter microphysics and global magneto-hydrostatic stellar structure. This has many implications including possible strong inner magnetic field in a neutron star leading to magnetic deformation and hence continuous gravitational-wave emission as an observable from millisecond pulsars.