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Disorder can fundamentally modify the transport properties of a system. A striking example is Anderson localization, suppressing transport due to destructive interference of propagation paths. In inhomogeneous many-body systems, not all particles are localized for finite-strength disorder, and the system can become partially diffusive. Unraveling the intricate signatures of localization from such observed diffusion is a longstanding problem. Here, we experimentally study a degenerate, spin-polarized Fermi gas in a disorder potential formed by an optical speckle pattern. We record the diffusion through the disordered potential upon release from an external confining potential. We compare different methods to analyze the resulting density distributions, including a new approach to capture particle dynamics by evaluating absorption-image statistics. Using standard observables, such as diffusion exponent and coefficient, localized fraction, or localization length, we find that some show signatures for a transition to localization above a critical disorder strength, while others show a smooth crossover to a modified diffusion regime. In laterally displaced disorder, we spatially resolve different transport regimes simultaneously, which allows us to extract the subdiffusion exponent expected for weak localization. Our work emphasizes that the transition toward localization can be investigated by closely analyzing the system's diffusion, offering ways of revealing localization effects beyond the signature of exponentially decaying density distribution.

We show that the nonlinear transport of bosonic excitations in a two-dimensional honeycomb lattice of spin-orbit-coupled Rydberg atoms gives rise to disordered quantum phases which are topological and may be candidates for quantum spin liquids. As recently demonstrated by Lienhard et al. [Phys. Rev. X 10, 021031 (2020)2160-330810.1103/PhysRevX.10.021031] the spin-orbit coupling breaks time-reversal and chiral symmetries and leads to a tunable density-dependent complex hopping of spin excitations which behave as hard-core bosons. Using exact diagonalization (ED), we numerically investigate the phase diagram resulting from the competition between density-dependent and direct transport terms as well as density-density interactions. In mean-field approximation there is a phase transition from a condensate to a 120^{∘} phase when the amplitude of the complex hopping exceeds that of the direct one. In the full model a new phase emerges close to the mean-field critical point as a result of quantum correlations induced by the density dependence of the complex hopping. We show that without density-density interactions this phase is a genuine disordered one, has large spin chirality, and is characterized by a nontrivial many-body Chern number. The Chern number is found to be robust to disorder. ED simulations of small lattices with up to 30 lattice sites give indications for a nondegenerate ground state with finite spin and collective gaps and thus for a bosonic integer quantum Hall phase, protected by U(1) symmetry. On the other hand, while staying finite, the many-body gap varies substantially when different twisted boundary conditions are applied, which points to a gapless phase. For very strong negative nonlinear hopping amplitudes we find another disordered regime with vanishing spin gap. This phase also has a large spin chirality and could be a gapless spin liquid but lies outside the parameter regime experimentally accessible in the Rydberg system.