Geometric and nongeometric contributions to the surface anomalous Hall conductivity

Rauch, Tomáš; Olsen, Thomas; Vanderbilt, David; Souza, Ivo

doi:10.1103/physrevb.98.115108

Cited by 29 publications

(35 citation statements)

References 22 publications

(63 reference statements)

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“…In this exact relation, the ambiguity modulo 2π in θ is consistent with a freedom to prepare insulating surfaces with values of σ surf AHC differing by the quantum e 2 /h, e.g., by changing the Chern number of some surface bands, or of adding or deleting a surface layer with a nonzero Chern number. 22,[35][36][37][38] If all surfaces adopt the same branch choice -i.e., the same value of σ surf AHC -then the sample as a whole exhibits a true magnetoelectric response of −α CS , where the quantized part of the response has been absorbed into the branch choice for α CS . This phenomenon is a higher-dimensional analog of the modern theory of electric polarization, 39 where the 2π ambiguity of the Berry phase reflects the inability to define the bulk polarization, or to predict the bound charge density of an insulating surface, except modulo a quantum.…”

Section: A Axion Coupling In the Bloch Representationmentioning

confidence: 99%

Axion coupling in the hybrid Wannier representation

2020

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Many magnetic point-group symmetries induce a topological classification on crystalline insulators, dividing them into those that have a nonzero quantized Chern-Simons magnetoelectric coupling ("axion-odd" or "topological"), and those that do not ("axion-even" or "trivial"). For time-reversal or inversion symmetries, the resulting topological state is usually denoted as a "strong topological insulator" or an "axion insulator" respectively, but many other symmetries can also protect this "axion Z2" index. Topological states are often insightfully characterized by choosing one crystallographic direction of interest, and inspecting the hybrid Wannier (or equivalently, the non-Abelian Wilson-loop) band structure, considered as a function of the two-dimensional Brillouin zone in the orthogonal directions. Here, we systematically classify the axion-quantizing symmetries, and explore the implications of such symmetries on the Wannier band structure. Conversely, we clarify the conditions under which the axion Z2 index can be deduced from the Wannier band structure. In particular, we identify cases in which a counting of Dirac touchings between Wannier bands, or a calculation of the Chern number of certain Wannier bands, or both, allows for a direct determination of the axion Z2 index. We also discuss when such symmetries impose a "flow" on the Wannier bands, such that they are always glued to higher and lower bands by degeneracies somewhere in the projected Brillouin zone, and the related question of when the corresponding surfaces can remain gapped, thus exhibiting a half-quantized surface anomalous Hall conductivity. Our formal arguments are confirmed and illustrated in the context of tight-binding models for several paradigmatic axion-odd symmetries including time reversal, inversion, simple mirror, and glide mirror symmetries.

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Section: A Axion Coupling In the Bloch Representationmentioning

confidence: 99%

Axion coupling in the hybrid Wannier representation

2020

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“…between the AHC of a gapped surface and the bulk axion coupling [10,20]. Once a specific branch has been chosen for θ , a unique integer n can be assigned to each surface, and for n to change the surface gap must close and reopen.…”

Section: B Surface Topological Transitions and Surface Anomalous Halmentioning

confidence: 99%

“…We have calculated the surface AHC according to Refs. [20,22] for slabs of different thicknesses (7, 13, and 19 cells across y, and 7, 9, and 11 cells across z). The extrapolated results are plotted in Fig.…”

Section: B Surface Topological Transitions and Surface Anomalous Halmentioning

confidence: 99%

Gapless hinge states from adiabatic pumping of axion coupling

et al. 2020

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We demonstrate that chiral hinge modes naturally emerge in insulating crystals undergoing a slow cyclic evolution that changes the Chern-Simons axion angle θ by 2π . This happens when the surface (not just the bulk) returns to its initial state at the end of the cycle, in which case it must pass through a metallic state to dispose of the excess quantum of surface anomalous Hall conductivity pumped from the bulk. If two adjacent surfaces become metallic at different points along the cycle, there is an interval in which they are in topologically distinct insulating states, with chiral modes propagating along the connecting hinge. We illustrate these ideas for a tight-binding model consisting of coupled layers of the Haldane model with alternating parameters. The surface topology is determined in a slab geometry using two different markers, surface anomalous Hall conductivity and surface-localized charge pumping (flow of surface-localized Wannier bands), and we find that both correctly predict the appearance of gapless hinge modes in a rod geometry. When viewing the axion pump as a fourdimensional crystal with one synthetic dimension, the hinge modes trace Fermi arcs in the Brillouin zone of the two-dimensional hinge connecting a pair of three-dimensional surfaces of the four-dimensional crystal.

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“…again in full analogy to the LAHC in a slab geometry 12 . Using Eq.…”

Section: A Periodic Boundary Conditionsmentioning

confidence: 99%

“…11 the authors project the velocity-operator on the Wannier functions which are the basis of their model. These are rather simple approaches, because usually different local projections are possible and they are not necessarily equivalent 12 . The authors of Ref.…”

Section: Introductionmentioning

confidence: 99%