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]
The fundamental building blocks in band theory are band representations -bands whose infinitelynumbered Wannier functions are generated (by action of a space group) from a finite number of symmetric Wannier functions centered on a point in space. This work aims to simplify questions on a multi-rank band representation by splitting it into unit-rank bands, via the following crystallographic splitting theorem: being a rank-N band representation is equivalent to being splittable into a finite sum of bands indexed by {1, 2, . . . , N }, such that each band is spanned by a single, analytic Bloch function of k, and any symmetry in the space group acts by permuting {1, 2, . . . , N }. We prove this theorem for all band representations (of crystallographic space groups) whose Wannier functions transform in the integer-spin representation; in the half-integer-spin case, the only exceptions to the theorem exist for three-spatial-dimensional space groups with cubic point groups. Applying this theorem, we develop computationally efficient methods to determine whether a given energy band (of a tight-binding or Schrödinger-type Hamiltonian) is a band representation, and, if so, how to numerically construct the corresponding symmetric Wannier functions. Thus we prove that rotation-symmetric topological insulators in Wigner-Dyson class AI are fragile, meaning that the obstruction to symmetric Wannier functions can be removed by addition of band representations to the filled-band subspace. An implication of fragility is that its boundary states, while robustly covering the bulk energy gap in finite-rank tight-binding models, can be destabilized if the Hilbert space is expanded to include all symmetry-allowed representations. These fragile insulators have photonic analogs that we identify; in particular, we prove that an existing photonic crystal built by Yihao Yang et al. [Nature 565, 622 (2019)] is fragile topological with removable boundary states, which disproves a widespread perception of 'topologically-protected' boundary states in timereversal-invariant, gapped photonic/phononic crystals. As a final application of our theorem, we derive various symmetry obstructions on the Wannier functions of topological insulators; for certain space groups, these obstructions are proven to be equivalent to the nontrivial holonomy of Bloch functions.CONTENTS 46 G. Tightly-bound BRs and the existence of the symmetric tight-binding limit 47 1. G-vector bundles and tight-binding lattice models 47 2. BRs and tightly-bound BRs as G-vector bundles 48 3. Existence of symmetric tight-binding limit 48 H. Lemma for Zak phases of tightly-bound band representations 48 I. Proof of localization obstruction lemma 49 References 50
For any symmorphic magnetic space group G, it is proven that topological band insulators with vanishing first Chern numbers cannot have a groundstate composed of a single, energetically-isolated band. This no-go statement implies that the minimal dimension of tight-binding Hamiltonians for such topological insulators is four if the groundstate is stable to addition of trivial bands, and three if the groundstate is unstable. A sure-fire recipe is provided to design models for Chern and unstable topological insulators by splitting elementary band representations; this recipe, combined with recently-constructed Bilbao tables on such representations, can be systematized for mass identification of topological materials. All results follow from our theorem which applies to any single, isolated energy band of a G-symmetric Schrödinger-type or tight-binding Hamiltonian: for such bands, being topologically trivial is equivalent to being a band representation of G.
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