Real-space topological localizer index to fully characterize the dislocation skin effect

The dislocation skin effect exhibits the capacity of topological defects to trap an extensive number of modes in two-dimensional non-Hermitian systems. Similar to the corresponding skin effects caused by system boundaries, this phenomenon also originates from nontrivial topology. However, finding the relationship between the dislocation skin effect and nonzero topological invariants, especially in disordered systems, can be obscure and challenging. Here, we introduce a real-space topological invariant based on the spectral localizer to characterize the skin effect on two-dimensional lattices. We demonstrate that this invariant consistently predicts the occurrence and location of both boundary and dislocation skin effects, offering a unified approach applicable to both ordered and disordered systems. Our work demonstrates a general approach that can be utilized to diagnose the topological nature of various types of skin effects, particularly in the absence of translational symmetry when momentum-space descriptions are inapplicable.

Figure 1

Panel (a) shows the spectrum of the momentum-space Hamiltonian Eq. (1) in the complex energy plane. The spectrum consists of a set of ellipses displaced relative to each other along the real axis, and shows a point gap around the origin, $E=0$. Panel (b) shows the SPD of Eq. (2) for a finite-sized system consisting of $25\times 25$ sites with OBC. For ease of visualization, $\rho$ is also shown as a varying color scale. For both panels, we use $t_y=0.4$ and $\gamma =0.3$ in units of $t_x$.

Figure 2

Panel (a) shows a sketch of the the weak Hatano-Nelson model in the presence of two dislocations. The dashed lines show two closed contours, one of which encircles the dislocation and one which does not. The path that encircles the defect ends up with a net displacement equal to the Burgers vector, here $\mathbf{B}=(B_x,B_y)=(0,1)$, shown by the thick arrow. Panels (b), (c) show the skin and anti-skin effect at the two dislocations. There is an accumulation or a depletion of $\rho$ compared to its bulk value depending on whether $B_y$ is positive or negative. In each case, the system size is $40\times 20$ sites, the distance between the two dislocations is 20 sites, $t_y=0.4$, and $\gamma =0.4$. Panel (b) shows the case of unit Burgers vectors, $\mathbf{B}=(0,\pm 1)$, whereas the dislocations in panel (c) have $\mathbf{B}=(0,\pm 2)$.

Figure 3

The local density of zero modes ${\rho }_0\left(\mathbf{r}\right)$ for the doubled Hamiltonian Eq. (3) is shown in panel (a) for PBC and in panel (c) for OBC. ${\rho }_0\left(\mathbf{r}\right)$ is the summation of the eigenvector probability carried out over only the zero-energy modes, defined using a tolerance of ${10}^{-4}$ in units of $t_x$. Correspondingly, the spectra of the two systems are shown in panels (b) and (d). In all plots, we use $t_y=\gamma =0.4$, a system size of $60\times 30$ sites, and introduce dislocations with $B_y=\pm 1$ that are 10 sites apart. In the case of PBC, there are only two zero-energy modes, each one localized at a dislocation core. With OBC, these defect modes coexist with boundary states formed by the topological end modes of each SSH chain in the stack.

Figure 4

Panel (a) shows the topological index (red) sweeping over $x_0$ for a $40\times 20$ lattice with $B_y=\pm 1$, $\gamma =0.4,t_y=0.4$. The index shows a jump equal to 1 when the origin is between the point defects. Panel (b) shows the variation of ${\nu }_L$ (red) for the same system with $B_y=\pm 2$. ${\nu }_L$ jumps by 2 between the point defects, showing the complete topological classification yielded by the localizer index. The blue lines show the respective localizer gap as a function of the position. The localizer gap is defined as the absolute value of the localizer eigenvalue closest to zero. Near the system boundaries ($x_0=0$ and 39), the localizer gap is small, such that the intermediate values between 0 and $-20$ are unstable to disorder. Deep inside the bulk, however the localizer gap is large ($\simeq 0.5$), leading to a robust invariant for both the boundary as well as the dislocation NHSE.

Figure 5

Panel (a): Average localizer index characterizing the dislocation NHSE (red), and average bulk gap of the doubled Hamiltonian (blue) as a function of disorder strength $W$. The system consists of $40\times 20$ lattice sites and contains two dislocations with $B_y=\pm 1$ that are positioned 20 lattice sites apart. We use $t_y=\gamma =0.4$ and each point is obtained by averaging over 100 independent disorder realizations. The localizer index characterizing the dislocation topology is obtained as a difference between the values of ${\nu }_L$ computed at $x_0=20.1$ and $x_0=7.1$. Panel (b) shows the SPD for a single disorder configuration at disorder strength $W=1$. The skin and antiskin peaks can be prominently seen.

Figure 6

Localizer index calculated using Eq. (6) (blue) and Eq. (A6) (red). Both approaches give the same value for the invariant for the $40\times 20$ system with a dislocation with $B_y=1$, with a significantly faster computation for the non-Hermitian invariant in Eq. (A6).