Non-Hermitian second-order topological insulator with point gap

Abstract

The zero-mode corner states in the gap of two-dimensional non-Hermitian Su-Schrieffer-Heeger model are robust to infinitesimal perturbations that preserve chiral symmetry. However, we demonstrate that this general belief is no longer valid in large-sized systems. To reveal the higher-order topology of non-Hermitian systems, we establish a correspondence between the stable zero-mode singular states and the topologically protected corner states of energy spectrum in the thermodynamic limit. Within this framework, the number of zero-mode singular values is directly linked to the number of mid-gap corner states. The winding numbers in real space can be defined to count the number of stable zero-mode singular states. Our results formulate a bulk-boundary correspondence for both static and Floquet non-Hermitian systems, where topology arises intrinsically from the non-Hermiticity, even without symmetries.

Figures (4)

• Figure 1: Schematics of the 2D non-Hermitian Su-Schrieffer-Heeger model on a square lattice. The box indicates the unit cell.
• Figure 2: (a) The minimum modulus of the energy eigenvalues, Min$[|E|]$ under open boundary conditions as a function of the horizontal inter-cell hopping amplitude $v_{x}$ (green solid curve). The orange dashed curve shows the corresponding winding number $\mathcal{V}_{2D}$. The system size is $1000\times 1000$ lattice sites, with parameters $w_{x}=1$, $\gamma _{x}=1.5$, $w_{y}=0$, $\gamma _{y}=9$, and $v_{y}=6v_{x}$. When $\left\vert v_{x}\right\vert >0.75$, the system enters a higher-order topological phase, signaled by Min$[|E|]\rightarrow 0$, indicating the emergence of topologically protected corner states. (b) The minimum energy modulus (pink dashed curve) and the weighted inverse participation ratio (wipr, sky-blue solid curve) as a function of disorder strength $d$. The system size is $60\times 60$, with fixed parameters $v_{x}=-1.5$, $w_{x}=1$, $\gamma _{x}=1.5$, $w_{y}=0$, $\gamma _{y}=9$, and $v_{y}=-9$.
• Figure 3: (a) The smallest singular value (Min$[s]$) of the 2D non-Hermitian SSH model as a function of the horizontal inter-cell hopping amplitude $v_x$. The red solid curve corresponds to the clean system ($d = 0$), while the magenta dashed curve shows the result with weak disorder of strength $d = 0.05$ that preserves sublattice symmetry; the orange dashed curve represents the case where both $H_x$ and $H_y$ are perturbed by a fully random and symmetry-breaking disorder of strength $d' = 0.05$. Remarkably, the zero-mode of Min$[s]$ persists, indicating that the bulk-boundary correspondence remains intact. (b) Exponential decay of the smallest singular value with increasing system size in either the $x$- or $y$-direction ($L_x$ or $L_y$), confirming its correspondence to a zero-mode. The inset displays the spatial probability distribution $|\psi_{x,y}|^2$ of the associated corner state, localized at the system's corner. (c) Winding number $\mathcal{V}$ of the bulk Hamiltonian as a function of $v_x$. In panels (a) and (c), the system size is $L_x=L_y=40$; all panels share the same parameters: $w_x = 1$, $w_y = 0$, $\gamma_x = 1.5$, $\gamma_y = 9$, and $v_y = 6v_x$.
• Figure 4: (a-b) The smallest singular values Min$[s_{\mp}]$ of (a) $U(T) - \mathbb{I}$ and (b) $U(T) + \mathbb{I}$ as functions of the horizontal inter-cell hopping amplitude $v_x$. The zero-mode singular values in (a) and (b) signal the presence of topological $0$-mode and $\pi$-mode states, respectively. (c-d) Corresponding topological invariants $\mathcal{V}_-$ and $\mathcal{V}_+$ as functions of $v_x$, which characterize the $0$- and $\pi$-quasienergy topological phases. Parameters: system size $L_x = L_y = 40$; $w_x = 1$, $w_y = 0$, $\gamma_x = 1.5$, $\gamma_y = 10.5$, $v_y = 7v_x$; full driving period $T = 0.6$, the duration of the first segment within each period $T_1 = 0.3$, and the intensity ratio factor between the second and the first stage $q_x=q_y = 0.2$.