Interplay of disorder and point-gap topology: Chiral modes, localization, and non-Hermitian Anderson skin effect in one dimension

Symmetry-protected spectral topology in extended non-Hermitian quantum systems has interesting manifestations such as dynamically anomalous chiral currents and skin effect. In this paper, we study the interplay between symmetries and disorder in a paradigmatic model for spectral topology—the nonreciprocal Su-Schrieffer-Heeger model. We consider the effect of on-site perturbations (both real and purely imaginary) that explicitly break the sublattice symmetry. Such symmetry-breaking terms can retain a nontrivial spectral topology but lead to a different symmetry class. We numerically study the effect of disorder in on-site and nonreciprocal hopping terms. Using a real-space winding number that is self-averaging and quantized, we investigate the impact of disorder on the spectral topology and associated anomalous chiral modes under periodic boundary conditions. We discover a remarkable robustness of chiral current and its self-averaging nature under disorder. The value of the chiral current retains the clean system value, is independent of disorder strength, and is tracked completely by the real-space winding number for class A which has no symmetries, and class AIII, which has a sublattice symmetry. In class ${\mathrm{D}}^{†}$, which has $PT$-symmetric on-site gain and loss terms, we find that the disorder-averaged current is not robust while the winding number is robust. We study the localization physics using the inverse participation ratio and local density of states. As the disorder strength is increased, a mobility-edge phase with a finite winding appears. The abrupt vanishing of the winding number marks a transition from a partially localized to a fully localized phase. Under open boundary conditions, we similarly observe a series of transitions through skin effect–partial skin effect–no skin effect phases. Further, we study the non-Hermitian Anderson skin effect (NHASE) for different symmetry classes, where the system without skin effect develops a disorder-driven skin effect at intermediate disorder values. Remarkably, while NHASE is present for different classes, the real-space winding number shows a direct correspondence with it only when all symmetries are broken.

Figure 1

Illustrating the model. Schematic of the non-Hermitian SSH lattice model with hopping and on-site disorder. $A$ and $B$ correspond to the two sublattices of the model while $j$ is the lattice site index. $t_1$ and $t_2$ are the intra- and intercell hopping strengths, respectively. ${\delta }_{1,2}$ are the non-Hermiticity parameters making the intracell hopping nonreciprocal. ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ are the on-site potentials added to $A$ and $B$ sublattices, respectively.

Figure 2

Non-Hermitian topological phase with disorder belonging to different symmetry classes. (a) The complex space energy spectrum in the undisordered NH top phase in the presence of real on-site terms, which has a point gap at $E=0$. In (b), we have added real hopping disorder to the clean system with disorder strength ${\delta }^R=2.0$. This shows the NH top spectrum in class AIII. Panel (c) corresponds to class A, where we have added random real on-site disorder to each sublattice with disorder strength ${\mathrm{\Delta }}^R=2.0$. In (d), we have included imaginary on-site disorder of the form $\pm i{\mathrm{\Delta }}^R$ to sublattices A and B, leading to class ${\mathrm{D}}^{†}$. Despite disorder, the NH top phase under all the symmetry classes shows a point-gap topology up to a considerable disorder strength. Here we choose $n=200, t_1=1.1, \delta =1.2$. The chosen parameters correspond to the NH top phase of the disorder-free system.

Figure 3

Phase diagram with real symmetric on-site potential. Winding number as a function of parameters $t_1$ and $\delta$ for different values of on-site energy ${\mathrm{\Delta }}_o$. The critical lines of the regular NH SSH model can be seen in (a) where ${\delta }_c=\pm |1\pm t_1|$ shows a transition in winding number. As the on-site energy increases [(b)–(d)] the NH top region moves toward higher $t_1$ values invading the former H triv region. Here, $t_2=1, n=100$, and base energy $E_b=0$.

Figure 4

Phase diagram with imaginary antisymmetric on-site potential, ${\mathrm{\Delta }}_{1,2}=\pm i\mu$. (a) The winding number as a function of $\delta$ and $t_1$ for $\mu =0.5$. (b) The winding number as a function of $\mu$ and $t_1$ for $\delta =0.5$. The region with nontrivial winding number is the non-Hermitian topological phase which is extended due to the presence of $\mu$. Here $n=100$.

Figure 5

Characterizing the disorder-induced phases under PBC. (a) The winding number, $W$, as a function of disorder. (b) The number of localized states (with IPR $\ge$ 0.4) as a function of disorder, demarcating three regions corresponding to the different phases. (c) IPR as a function of $\text{Re}\left(E\right)$ for the three regions, where I, II, and III correspond to ${\mathrm{\Delta }}^R=$ 0.0, 1.0, 4.5, respectively. The inset shows the zoomed-in plot for region I (in brown). (d) LDOS for region I (${\mathrm{\Delta }}^R=0.0$), which is the extended or delocalized phase, (e) region II (${\mathrm{\Delta }}^R=1.0$), which is the mobility-edge phase, and (f) region III (${\mathrm{\Delta }}^R=4.5$), the bulk-localized phase. All IPR plots have been disorder-averaged. Winding number is self-averaging. LDOS plots (shown only up to the 40th lattice site) are for a representative disorder configuration. Here, number of lattice sites $n=400, t_1=1.0, t_2=1.0, \delta =0.3, {\mathrm{\Delta }}_o=0.0$, symmetric on-site-disorder strength ${\mathrm{\Delta }}^R$ varies from 0 to 5 in steps of 0.05.

Figure 6

Disorder-induced phases under OBCs. (a) The density of states at the edge (${\rho }_{\text{edge}}$), which drops down from close to unity as the skin effect disappears. This helps distinguish between regions I, II, and III. The inset shows the change in winding number with disorder which distinguishes between regions II and III. The LDOS for (b) region I (${\mathrm{\Delta }}^R=0.0$); which is the skin phase; (c) region II (${\mathrm{\Delta }}^R=1.5$), which is the mixed phase, and (d) region III (${\mathrm{\Delta }}^R=4.5$), which is the bulk-localized phase. Here number of lattice sites, $n=400, t_1=1.0, t_2=1.0, \delta =0.3, {\mathrm{\Delta }}_o=0.0$, and symmetric on-site-disorder strength ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\mathrm{\Delta }}^R$ varies from 0 to 5 in steps of 0.05.

Figure 7

Correspondence between chiral current and winding number. (a), (c) The winding number, $W$, as a function of disorder under PBCs. (b), (d) The chiral current as a function of different kinds of disorder in the same system. (a) and (b) correspond to the effect of introducing a symmetric on-site disorder strength (class A) with ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\mathrm{\Delta }}^R$. (c) and (d) show the effect of hopping disorder (class AIII). We find that the system is more robust to on-site disorder rather than disorder in hopping and that there is a clear equivalence between winding number and chiral current with disorder. Here $n=400, t_1=1.0, t_2=1.0, \delta =0.3$, and ${\mathrm{\Delta }}_o=0.0$.

Figure 8

Disorder-induced properties under PBC in class ${\mathrm{D}}^{†}$. (a) The winding number $W$ as a function of PT-symmetric disorder for the system under PBC. (b) The chiral current as a function of the same disorder which varies from 0 to 5 in steps of 0.05. The chiral current randomly fluctuates between $-1$ and 1, which when disorder averaged becomes trivially zero (shown in the inset). (c) shows the number of localized states for the same disorder, while (d) shows the IPR at ${\mathrm{\Delta }}^R=1.0$ (region II), which has a mobility edge with respect to the imaginary part of the eigenvalues. The figure shows that for class ${\mathrm{D}}^{†}$, winding number tracks the complete localization in the system demarcating regions II and III. $N_{\text{loc}}$ helps demarcate between the extended phase (region I), intermediate mobility edge (region II), and the bulk localized phase (region III). The chiral current does not show equivalence to the winding number anymore; rather, it fluctuates between 1 and $-1$, averaging out to 0 on taking a disorder average. The IPR, contrary to classes A and AIII, shows a mobility edge as a function of $\mathrm{Im}\left(E\right)$. Here, number of lattice sites $n=400,t_1=1.0,t_2=1.0,\delta =0.3,{\mathrm{\Delta }}_o=0.0$.

Figure 9

Winding number and edge density under OBCs in class ${\mathrm{D}}^{†}$. (a) The winding number $W$ as a function of PT-symmetric disorder for the system under OBCs. (b) The density of states at the edge under the same disorder which varies from 0 to 5 in steps of 0.05. The winding number change from 1 to 0 tracks the complete localization of the states into the bulk. The edge density ${\rho }_{\text{edge}}$ shows skin states in region I, an intermediary mixed phase with partial localization and partial skin states (region II), and, finally, region III where all the states are localized in the bulk. Here, number of lattice sites $n=400,t_1=1.0,t_2=1.0,\delta =0.3,{\mathrm{\Delta }}_o=0.0$.

Figure 10

Winding number as a function of disorder from a critical line under PBC. Plot of winding number as function of on-site symmetric disorder and hopping disorder showing a $0\to 1\to 0$ transition when we sit on the critical line ${\delta }_c=|1+t_1|$. Here, parameter values $t_1=1.1,{\delta }_c=2.1,{\mathrm{\Delta }}_o=0.0$ and $n=100$.

Figure 11

The NHASE under OBCs. (a) The winding number as a function of hopping disorder strength ${\delta }^R$ and random on-site disorder strength ${\mathrm{\Delta }}^R$ which vary from 0 to 4 in steps of 0.05. (b) The density of states at the edge of the lattice as a function of both ${\delta }^R$ and ${\mathrm{\Delta }}^R$. A system with no skin effect initially develops a skin effect under the effect of disorder which subsequently disappears. Here, ${\mathrm{\Delta }}_{1,2}={\mathrm{\Delta }}^R{\omega }_{{\mathrm{\Delta }}_{1,2}}$ such that ${\omega }_{{\mathrm{\Delta }}_1}\ne {\omega }_{{\mathrm{\Delta }}_2}\Rightarrow {\mathrm{\Delta }}_1\ne {\mathrm{\Delta }}_2$. When there is no symmetry in the system, the figure shows that the winding number and density of edge states show the same variation with disorder. Here, $n=100,t_2=1.0,t_1=1.1,\delta =2.1,{\mathrm{\Delta }}_o=0.0$.

Figure 12

The winding number and NHASE under OBCs in for imaginary disorder. (a) The winding number as a function of disorder strength ${\delta }^R$ for hopping and disorder strength ${\mathrm{\Delta }}^R$ for imaginary on-site terms. Both vary from 0 to 4 in steps of 0.05. (b) The density of states at the edge of the lattice as a function of both ${\delta }^R$ and ${\mathrm{\Delta }}^R$. Here, ${\mathrm{\Delta }}_{1,2}=\pm i{\mathrm{\Delta }}^R{\omega }_{{\mathrm{\Delta }}_{1,2}}$ such that ${\omega }_{{\mathrm{\Delta }}_1}\ne {\omega }_{{\mathrm{\Delta }}_2}$. Here too, the winding number corresponds to the occurrence of anomalous edge states. The parameter values here are: $n=100,t_2=1.0,t_1=1.1,\delta =2.1,{\mathrm{\Delta }}_o=0.0$.