Bulk-interface correspondences for one-dimensional topological materials with inversion symmetry

Abstract

The interface between two materials described by spectrally gapped Hamiltonians is expected to host an in-gap interface mode, whenever a certain topological invariant changes across the interface. We provide a precise statement of this bulk-interface correspondence and its rigorous justification. The correspondence applies to continuum and lattice models of interfaces between one-dimensional materials with inversion symmetry, with dislocation models being of particular interest. For continuum models, the analysis of the parity of the ‘edge’ Bloch modes is the key component in our argument, while for the lattice models, the relative Zak phase and index theory are.

Surface impedance and topologically protected interface modes in one-dimensional phononic crystals

“…Note added: After we submitted this manuscript, we became aware of reference [18], which reaches similar conclusions. In particular, the monotonicity of the impedance with frequency is also shown, although restricted to the Helmholtz equation (2.1) and not the generalization of equation (5.1).…”

“…A second approach consists in focusing on specific models, where the correspondence can be established by direct calculations [17]. Moreover, several works have exploited the concept of impedance [16][17][18][19], which allows one to relate the existence of interface modes to bulk properties. This approach is elegant and offers a new point of view.…”

Topological features of Bloch impedance

“…3) Given that T (x) = [ψ 10 (x), ψ 01 (x)] T , direct differentiation yields [det T (x)] = 0 and, since T (0) is the identity matrix, det T (x) = 1 [7,8]. Alternatively, this result follows from the Abel-Liouville-Jacobi-Ostrogradskii identity for the Wronskian of a linear system (see [13], sect.…”

“…To this end, we, following refs. [5,7,8], adopt an impedance-centric view of the topological properties of waves in periodic (a) E-mail: igor@uakron.edu (corresponding author)…”

Topological edge states of 1D chains and index theory

“…Notice that the bulk Hamiltonians in these three examples are unitarily related to each other by a change of convention in the unit cell labelling and/or grading V + ↔ V − . These conventions are implicitly specified via boundary conditions, and the intrinsic meaning of the "bulk winding number invariant" is actually quite subtle, see [12,13].…”