Robust nonequilibrium surface currents in the three-dimensional Hofstadter model

Genuinely two-dimensional robust crosscurrents---which flow against the natural direction of heat flux--- have been missing since discovery their one-dimensional counterpart. We provide a setup to realize them on cubic three-dimensional (3D) lattice hosting Hofstadter model coupled two baths with different temperatures. show that these currents exhibit dissipative robustness; they are stable presence impurities and tilting gauge field in certain nonequilibrium configurations. Moreover, we find protected boundary genuinely 3D robustness, i.e., only if tunneling can occur all three spatial directions. The also presents generic surface currents, which for both bosonic fermionic systems. identify underlying qualitative mechanism responsible robustness crucial role played by discrete symmetries.