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Genuinely two-dimensional robust crosscurrents -- which flow against thenatural direction of heat flux -- have been missing since the discovery oftheir one-dimensional counterpart. We provide a setup to realize them on acubic three-dimensional (3D) lattice hosting a Hofstadter model coupled to twoheat baths with different temperatures. We show that these currents exhibitdissipative robustness: they are stable against the presence of impurities andtilting of the gauge field in certain nonequilibrium configurations. Moreover,we find protected boundary currents with genuinely 3D robustness, i.e. they areonly stable if tunnelling can occur in all three spatial directions. The modelalso presents generic surface currents, which are robust for both bosonic andfermionic systems. We identify the underlying qualitative mechanism responsiblefor the robustness of the surface currents and the crucial role played bycertain discrete symmetries.

Robust nonequilibrium surface currents in the 3D Hofstadter model