Zak phase and bulk-boundary correspondence in a generalized Dirac-Kronig-Penney model

TL;DR

This paper explores the topological properties of a generalized Dirac-Kronig-Penney model, analyzing Zak phases and bulk-boundary correspondence across different symmetry classes, revealing nuanced behavior in continuum systems.

Contribution

It introduces a continuum 1D model capable of representing multiple topological classes and analyzes Zak phase quantization and edge state emergence in this context.

Findings

Zak phase is quantized in classes AIII and BDI

Zak phase is non-quantized in class D, challenging its topological role

Edge states are detected by Zak phase in AIII and BDI, but with sensitivity to boundary conditions

Abstract

We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.