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We study a three-dimensional chiral second order topological insulator (SOTI)subject to a magnetic field. Via its gauge field, the applied magnetic fieldinfluences the electronic motion on the lattice, and via the Zeeman effect, thefield influences the electronic spin. We compare two approaches to the problem:an effective surface theory, and a full lattice calculation. The surface theorypredicts a massive Dirac spectrum on each of the gapped surfaces, giving riseto Landau levels once the surfaces are pierced by magnetic flux. The surfacetheory qualitatively agrees with our lattice calculations, accuratelypredicting the surface gap as well as the spin and orbital components of thestates at the edges of the surface Dirac bands. In the context of the latticetheory, we calculate the spectrum with and without magnetic field and find adeviation from the surface theory when a gauge field is applied. The energy ofthe lowest-lying Landau level is found closer to zero than is predicted by thesurface theory, which leads to an observable magnetotransport signature: insidethe surface gap, there exist different energy regions where either one or twochiral hinge modes propagate in either direction, quantizing the differentialconductance to either one or two conductance quanta.

Second-order topological insulator under strong magnetic field: Landau levels, Zeeman effect, and magnetotransport