Time-Reversal Symmetry in Artificial Topological Graphene Rings

Abstract

We study an artificial graphene ring described by the low-energy two-valley Dirac Hamiltonian with a uniform mass term. When the ring is threaded by an Aharonov-Bohm flux, the problem reduces to a second-order radial equation solved by Bessel functions. We give the exact secular equation and, for narrow rings, closed expressions for the spectrum and the persistent current. We then add a uniform magnetic field perpendicular to the plane. The radial problem is solved with confluent hypergeometric functions, and the persistent current and magnetization are written as derivatives of a secular determinant from which the spurious roots have been removed. Besides the usual orbital coupling, the field enters through a valley-dependent term that comes from the Dirac structure and lifts the valley degeneracy. A weak-field expansion, accurate up to corrections of order B^4, gives this valley splitting explicitly and separates the diamagnetic and paramagnetic contributions to the susceptibility. Finally, we work out what time reversal implies for the spectrum. Physical time reversal, which exchanges the valleys, leads to an exact relation between levels at opposite fields. The effective single-valley time reversal, on the other hand, is broken by the mass term and by the boundary conditions, yet leaves no trace in the zero-field spectrum. Numerical results for the spectrum, the persistent current, and the magnetization illustrate the analysis.