Joint BCAM-UPV/EHU Analysis and PDE seminar: Continuum limit for a discrete Hodge-Dirac operator on square lattices

In this talk, we examine the limit of Schrödinger and Dirac operators defined on the n-dimensional square lattice hℤn​ as happroaches 0. We review the positive results obtained for Schrödinger operators and the negative ones for the Dirac case, which is related with the so-called "fermion doubling". Then we introduce a discrete Hodge-Dirac operator, which is different from the usual discretization of the Dirac operator. To be able to define this operator, we begin by establishing an alternative framework for a higher-dimensional discrete differential calculus. We express our operator as a differential operator acting on discrete forms and obtain the limit to the continuous standard Hodge-Dirac operator. In the final part, we discuss the relation of the discrete Hodge-Dirac operator with other alternatives aimed at overcoming the "fermion doubling".