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

TL;DR

This paper explores the topological properties of surface magnetism in three-dimensional insulators, revealing quantized surface magnetic compressibility linked to higher-order topology, exemplified by the Hopf insulator.

Contribution

It introduces a topological characterization of surface magnetic moments and their quantization, connecting surface properties to higher-order topological invariants.

Findings

Sum of surface compressibilities is quantized and linked to hinge mode chirality.

Single-facet magnetic compressibility can be half-integer quantized under symmetry constraints.

Hopf insulator exemplifies the quantized surface magnetic response.

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 $e/h c$; 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.