Topological Bloch oscillations

Abstract

Bloch oscillations originate from the translational symmetry of crystals. These oscillations occur with a fundamental period that a semiclassical wavepacket takes to traverse a Brillouin-zone loop. We introduce a new type of Bloch oscillations whose periodicity is an integer (µ>1) multiple of the fundamental period. The period multiplier µ is a topological invariant protected by the space groups of crystals, which include more than just translational symmetries. For example, µ divides n for crystals with an n-fold rotational or screw symmetry; with a reflection, inversion or glide symmetry, µ equals two. We identify the commonality underlying all period-multiplied oscillations: the multi-band Berry-Zak phases, which encode the holonomy of adiabatic transport of Bloch functions in quasimomentum space, differ pairwise by integer multiples of 2π/µ. For a class of multi-band subspaces whose projected-position operators commute, period multiplication has a complementary explanation through the real space distribution of Wannier functions. This complementarity follows from a one-to-one correspondence between Berry-Zak phases and the centers of Wannier functions. A Wannier description of period multiplication does not always exist, as we exemplify with band subspaces with either a nonzero Chern number or Z2 Kane-Mele topological order. In the former case, we present general constraints between Berry-Zak phases and Chern numbers, as well as introduce a recipe to construct nontrivial Chern bands -by splitting elementary band representations. To help identify band subspaces with µ>1, a general theorem is presented that outputs Zak phases that are symmetry-protected to integer multiples of 2π/n, given the point-group symmetry representation of any gapped band subspace. A cold-atomic experiment that has observed period-multiplied Bloch oscillations is discussed, and directions are provided for future experiments.CONTENTS arXiv:1708.02943v4 [cond-mat.other]

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