Sensitivity to Lattice Structure in the Mesoscopic-Loop Models of Planar Systems

Some planar systems display universal magnetic behavior at low temperatures, e.g. either superconductivity or quantization of the Hall conductance. These are the phenomena that traditionally attracted the most interest among researchers. At the same time, nanotechnology inspires us to examine complementary, albeit still strongly quantum, regimes of operation, in which electronic properties depend on the specifics of the material. In particular, one is compelled to address the problem of efficient extraction of material-specific information from essentially quantum systems. As demonstrated in my recent article, mesoscopic feedback’, a certain extension of the Schrödinger equation—namely, an extension via the so-called mesoscopic loop—leads to a PDE model of systems operating in the quantum Hall regime. In the present article, we construct lattice-type solutions of the mesoscopic loop model with periodic potential. The method of construction is geometric, and leads to continuous strong solutions. At the same time, the result brings to light a constraint that the quantum Hall phenomenology imposes on the parameters of the model. This suggests that the model has the capacity to transcribe material-specific information inherent in the spatially distributed Hall conductance data outside the FQHE regime.