Coexistence of Topological Anderson Insulator and Multifractal Critical Phase in a Non-Hermitian Quasicrystal

Abstract

The interplay of topology, disorder, and non-Hermiticity gives rise to phenomena beyond the conventional classification of quantum phases. We propose a one-dimensional non-Hermitian Su-Schrieffer-Heeger model with quasiperiodically modulated nonreciprocal intracell hopping. We show that quasiperiodic modulation can substantially enhance topological robustness and, remarkably, induce a non-Hermitian topological Anderson insulator (TAI) phase. Beyond the topological transition, increasing nonreciprocity drives a cascade of localization transitions in which all bulk eigenstates evolve from extended to multifractal critical and ultimately to localized states. Strikingly, the extended-to-critical transition coincides exactly with a real-complex spectral transition. We establish complete phase diagrams and derive exact analytical boundaries for both topological and localization transitions, uncovering an unanticipated coexistence of TAI and multifractal critical phases. Finally, we propose a feasible implementation in topolectrical circuits. Our results reveal a new paradigm for the cooperative effects of topology, quasiperiodicity, and non-Hermiticity.

Figures (4)

• Figure 1: (Color online) (a) Real and (b) imaginary parts of the eigenenergies under periodic boundary conditions as a function of the nonreciprocity strength $\lambda$. The black and blue dashed lines mark the critical values $\lambda=v$ and $\lambda=2w$, separating the extended, multifractal critical, and localized regimes. The color scale indicates the fractal dimension $\mathrm{FD}$ of the corresponding eigenstates. (c) Energy spectrum $|E|$ under open boundary conditions. (d)–(e) Representative spatial profiles of eigenstates in the extended, critical, and localized phases, respectively. (f) Zero-energy edge modes in the topological Anderson insulating phase. Parameters: $v=1.2$, $w=1$, and system size $L=2N=1220$.
• Figure 2: (Color online) (a) Phase diagram in the $v-\lambda$ plane showing the topological Anderson insulating (TAI) and trivial phases. The color scale represents the real-space winding number $W$. Red dashed and solid lines denote the analytical phase boundaries given by Eq. (\ref{['TopoPhase']}). (b) Phase diagram of the extended, multifractal critical, and localized regimes determined from the Lyapunov exponent and fractal dimension. The black dashed line indicates the cut at $v=1.2$.
• Figure 3: (Color online) (a) Mean inverse participation ratio (MIPR) and (b) mean fractal dimension (MFD) as functions of the nonreciprocity strength $\lambda$ for different system sizes. The black dot-dashed and blue dashed lines mark the analytical phase boundaries separating the extended–critical and critical–localized regimes, respectively. Parameters: $v=1.2$ and $w=1$.
• Figure 4: (Color online) Schematic of the topolectrical circuit realizing the non-Hermitian quasicrystal. The numbered black nodes represent lattice sites of the SSH chain. Capacitive couplings implement the Hermitian hopping terms, while the element highlighted in red denotes a negative-impedance converter with current inversion (INIC), which generates direction-dependent (nonreciprocal) intracell hopping. The effective impedances $Z_j$ and $Z_j^\prime$ are controlled by the orientation of the INIC. The lower panel illustrates the internal structure of the INIC, whose impedance depends on the direction of the current flow.