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

TL;DR

This work introduces a continuum 1D Dirac–Kronig–Penney model with a $ ext{U}(2)$-valued array of point interactions to realize multiple AZC symmetry classes in one dimension. By deriving a spectral function and computing Zak phases for insulating bands, the authors reveal that Zak-phase quantization occurs in chiral classes BDI and AIII but not in class D, highlighting a nuanced role for the Zak phase in continuum systems. They also analyze the bulk–boundary correspondence by truncating the chain and examining edge states under various boundary conditions, showing that the BBC is highly sensitive to edge position and boundary choices, and that the bulk Zak phase acts as a relative boundary index rather than a universal topological marker in this setting. The results emphasize the crucial influence of boundary conditions in continuum topological matter and point to future work on disorder, higher-spin extensions, and experimental realizations of continuum topological phases.

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.

Figures (11)

• Figure 1: Schematic representation of the gDKP model. At each lattice point $n\in \mathbb{Z}$ there is a point interaction realizing the $\mathrm{U}(2)$ boundary condition $\Psi_-(n)=U\Psi_+(n)$.
• Figure 2: The physical system described by the fiber Hamiltonian $h_U(k)$ is a ring with two point interactions at antipodal points. Here, the left and right point interactions implement respectively the Bloch--Floquet condition $\Psi(\tfrac{1}{2})=\mathrm{e}^{\mathrm{i}\mkern1mu k}\Psi(-\tfrac{1}{2})$ and the $\mathrm{U}(2)$ coupling condition $\Psi_-(0)=U\Psi_+(0)$.
• Figure 3: Energy bands close to zero for the couplings $U_C(\theta)$ in class D, with $m=1$. The energy gaps never close, therefore the set $\mathcal{U}_{\text{D}}$ consists of a single connected region with no topological phase transitions.
• Figure 4: Zak phase $Z_{U_C(\theta),n}$ for the class D, with $n=0$ (red), $n=\pm1$ (blue) and $n=\pm 2$ (green); some values of $Z_{U_C(\theta),0}$ have been shifted by $2\pi$ in order to show a continuous curve.
• Figure 5: Energy bands close to zero for the couplings $U_{CS}(\theta)$ in class BDI, with $m=1$. The central energy gap closes for $\theta=\pm\theta_m$, whereas all the other gaps close for $\theta=\pm\tfrac{\pi}{2}$. The set $\mathcal{U}_{\text{BDI}}$ is thus partitioned into four connected regions (represented by alternating colors in the right diagram) where no gap of the corresponding Hamiltonian closes.