Abstract
We show that two-dimensional band insulators, with vanishing bulk polarization, obey bulk-and-edge to corner charge correspondence stating that the knowledge of the bulk and the two corresponding ribbon band structures uniquely determines the fractional part of the corner charge irrespective of the corner termination. Moreover, physical observables related to macroscopic charge density of a terminated crystal can be obtained by representing the crystal as collection of polarized edge regions with polarizations $\vec P^\text{edge}_\alpha$, where the integer $\alpha$ enumerates the edges. We introduce a particular manner of cutting a crystal, dubbed "Wannier cut", which allows us to compute $\vec P^\text{edge}_\alpha$. We find that $\vec P^\text{edge}_\alpha$ consists of two pieces: the bulk piece expressed via quadrupole tensor of the bulk Wannier functions' charge density, and the edge piece corresponding to the Wannier edge polarization---the polarization of the edge subsystem obtained by Wannier cut. For a crystal with $n$ edges, out of $2n$ independent components of $\vec P^\text{edge}_\alpha$, only $2n-1$ are independent of the choice of Wannier cut and correspond to physical observables: corner charges and edge dipoles.
Highlights
While the bulk description of solid-state materials is generally available, the description close to the material’s boundaries is often not accessible
To name a few examples, the bulk electrical polarization of an insulator predicts a fractional part of the end charge [1,2,3,4,5], the bulk orbital magnetization [6,7] predicts persistent current circulating along the boundary, bulk geometric orbital magnetization [8] predicts a fractional part of the time-averaged edge current circulating along the boundary of a periodically, adiabatically driven insulator, and the bulk magnetoelectric polarizability of a three-dimensional insulator predicts a fractional part of the surface charge density resulting from the application of an external magnetic field [9,10]
We prove that corner charges and edge dipoles of band insulators can be obtained by representing a terminated crystal as a collection of edge regions with polarization Pαedge, where α enumerates the edges
Summary
While the bulk description of solid-state materials is generally available, the description close to the material’s boundaries (termination) is often not accessible. In their pioneering work, Benalcazar et al proposed the model [25] that in the presence of fourfold rotation symmetry [29] exhibits quantized corner charge that is given by topological invariant dubbed “bulk quadrupole moment” qxy, qxy = Qc − Pxedge − Pyedge mod e,. Subsequent works [30,31] by two independent groups proposed an expression that was meant to predict qxy in the absence of the symmetry constraints, using the bulk Hamiltonian as its sole input These findings were supported by calculations on several tight-binding models [30,31].
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