ON-LINE QLunch: Christopher Cedzich
Title: Singular-continuous Cantor spectrum for quantum walks in magnetic fields
Describing particles in external electromagnetic fields is a basic task of quantum mechanics. The standard scheme for this is known as "minimal coupling", and consists of replacing the momentum operators in the Hamiltonian by modified ones with an added vector potential. In lattice systems it is not so clear how to do this, because there are no momenta.
Moreover, when time is also discrete, as in quantum walk systems, there is no Hamiltonian, only a unitary step operator. In this talk I present a unified framework of gauge theory for such discrete systems, keeping a close analogy to the continuum case. In particular, I show how to implement minimal coupling in a way that automatically guarantees unitary dynamics. The scheme works in any lattice dimension, for any number of internal degrees of freedom, for walks that allow jumps to a finite neighborhood rather than to nearest neighbours, is naturally gauge invariant, and prepares possible extensions to non-abelian gauge groups.
As examples I will briefly review quantum walks in electric fields and relate their spectral and dynamical properties to the rationality of the respective field. Moreover, I will sketch the proof of singular continuous Cantor spectrum for quantum walks in two spatial dimensions with irrational magnetic field.
C. Cedzich, T. Rybár, A. H. Werner, A. Alberti, M. Genske, and R. F. Werner.
Propagation of quantum walks in electric fields. Phys. Rev. Lett., 111:160601, 2013.
C. Cedzich, T. Geib, A. H. Werner, and R. F. Werner.
Quantum walks in external gauge fields. J. Math. Phys., 60(1):012107, 2019.
C. Cedzich and A. H. Werner.
Anderson localization for electric quantum walks and skew-shift CMV matrices.
C. Cedzich, J. Fillman, T. Geib, and A. H. Werner.
Singular continuous Cantor spectrum for magnetic quantum walks. Lett.
Math. Phys., 2019.