Topology and Localizations in a 2D Su-Schrieffer-Heeger Model with Domain Walls, Quasi-periodic Disorder and Periodic Hopping Modulations

TL;DR

This work analyzes a two-dimensional SSH model with periodic hopping modulations, domain-wall defects, and Aubry–André–type quasiperiodic disorder to explore topology and localization. By computing Zak phases, edge/corner spectra, and defect-induced states, it reveals corner-localized bound states in the continuum (BICs), multiple in-gap edge/corner modes, and a filling-anomaly–driven bulk–boundary correspondence in a 2D HOTI context. Domain walls yield zero-energy modes localized at corners or along DW lines, while radial DW distributions produce center-localized ZES with symmetry-driven localization. Quasiperiodic disorder introduces conventional localization and, for specific disorder implementations, reentrant localization with two mobility edges, and anisotropic hopping can erase bulk ZES in favor of boundary modes, offering tunable routes for topological quantum information processing.

Abstract

A two dimensional (2D) Su-Schrieffer-Heeger (SSH) model with topological defects like domain walls (DW) / vortices or quasi-periodic disorders is a perfect blend for investigating topology and localization of quantum states. In a 2D SSH model, zero energy states (ZES) lie within the dispersion continuum for both periodic and open boundaries. We consider two different distribution of DWs of which the first one shows the bound states in continuum (BIC) to populate at the corners (producing higher order topological modes) or the DW center while the second one, with a vortex like radially symmetric distribution of hopping modulations, shows localizations along the DWs and the edges. The topological yet gapped in-gap states, with nonzero Zak phases, show an opposite trend with localizations at the edges and along the DWs in the first case as opposed to localizations at the DW center in the second case. Investigation on the effect of on-site quasiperiodic disorders manifests the usual tendency of the states to localize. However, reentrant localization behavior is also captured for judicious choice of the disordered term, making this the first reported example of its kind in a 2D system. Furthermore, while varying the hopping periodicity, we discover that anisotropic hopping modulations along x and y directions within the lattice produces significant changes in topological features where bulk ZES get exhausted leaving only topological boundary modes at zero energies. We also discuss the fate of these states in presence of the DWs. All these analysis depicting topological/localization features of varied kinds can become very useful in the field of quantum computation and information processing.

Figures (10)

• Figure 1: (a) Typical band structure of a $2D$ SSH model under PBC as a function of $k_{x}$ and $k_{y}$ away from the QPT. (b) $L-2$ number of points with $|k_x|=|k_y|$ within the BZ in a $L\times L$ square lattice (with $L=8$) where doubly degenerate zero energy modes appear. (c) Dispersions from a $48\times48$$2D$ SSH model under OBC of which (d) and (e) demonstrate the $L$ and $L+2$ number of ZES in typical trivial and topological phase respectively.
• Figure 2: Zero energy corner localized BIC's [(a),(c)] and BIC's mixed with bulk states [(b),(d)] in a $2D$ SSH model in a $48\times 48$ square lattice for $\Delta/t$ = - 0.9 (a-b) and -0.3 (c-d) respectively. (e-g) show nonzero energy in-gap states for $\Delta/t=-0.5$ with localization along (e) $x$-edges, (f) $y$-edges and (g) both $x$ and $y$ edges respectively.
• Figure 3: (a) Cartoon for distribution of hopping modulations ($\delta_i,\delta_j$) in a $24^2$ lattice with $i_0=j_0=12$ and $\xi/a=8$. Rest of the figures correspond to $72\times72$ lattices with domain walls at $i_0=j_0=36$ and for $\xi/a=1$. (b) Low energy spectra (with $L$ number of ZES) in a 2D SSH model as a function of domain wall amplitude $d_{0}$. (c,d) show ZES with corner peaks and DW peaks respectively and (e,f) show two typical in-gap states for $d_0=-0.9$.
• Figure 4: Density plots for $|\psi|^2$ corresponding to ZES localized at DW for DW position (a) $i_0=j_0=18$, (b) $i_0=54,~j_0=18$, (c) $i_0=30,~j_0=42$ on a $72\times72$ lattice. (d,e) show two typical nonzero energy in-gap states with $i_0=j_0=18$. We consider $d_{0}=0.9$ and $\xi/a=1$ in all the plots.
• Figure 5: (a) Cartoon for distribution of hopping modulations ($\delta_i,\delta_j$) in a $24^2$ lattice with $r_0=(\frac{L-1}{2},\frac{L-1}{2})a$ and $\xi/a=4$. Density plots for $|\psi|^2$ corresponding to two typical in-gap states (b,c) and a ZES (d) for $\xi/a=2$ on a $72\times72$ lattice for $d_{0}=1.2$. (d,e) show typical ZES for (e) $\xi/a=4$ and (f) $\xi/a=6$.