Quantized surface magnetism and higher-order topology: Application to the Hopf insulator

Abstract

We identify topological aspects of the subextensive magnetic moment contributed by the surfaces of a three-dimensional crystallite -- assumed to be insulating in the bulk as well as on all surface facets, with trivial Chern invariants in the bulk. The geometric component of this subextensive moment is given by its derivative with respect to the chemical potential, at zero temperature and zero field, per unit surface area, and hence corresponds to the surface magnetic compressibility. The sum of the surface compressibilities contributed by two opposite facets of a cube-shaped crystallite is quantized to an integer multiple of the fundamental constant $e/hc$; this integer is in one-to-one correspondence with the net chirality of hinge modes on the surface of the crystallite, manifesting a link with higher-order topology. The contribution by a single facet to the magnetic compressibility need not be quantized to integers; however, symmetry and/or Hilbert-space constraints can fix the single-facet compressibility to half-integer multiples of $e/hc$, as will be exemplified by the Hopf insulator.

arXiv comment11+5 pages, 7 figures

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• ${\mathbb{Z}}_2$ Spin Hopf Insulator: Helical Hinge States and Returning Thouless Pump
• Extrinsic and Intrinsic Nonlinear Hall Effects across Berry-Dipole Transitions
• Real Hopf Insulator
• Towards a theory of surface orbital magnetization
• Protected Gapless Edge States In Trivial Topology
• Berry-dipole Semimetals
• Scattering theory of delicate topological insulators
• Flat Bands in Three-dimensional Lattice Models with Non-trivial Hopf Index
• Non-Hermitian Hopf insulators
• Disorder-induced delocalization and reentrance in a Chern-Hopf insulator