[Paper Review] Anomalous Localization and Mobility Edges in Non-Hermitian Quasicrystals with Disordered Imaginary Gauge Fields

Guolin Nan, Zhijian Li+2 more|arXiv (Cornell University)|Jan 21, 2026

Quasicrystal Structures and Properties0 citations

TL;DR

This paper analyzes anomalous localization in a 1D non-Hermitian AAH-like chain with a Bernoulli disordered imaginary gauge field, showing ENHSE vs localization transitions, a novel mobility edge with next-nearest-neighbor hopping, and dynamical probes.

ABSTRACT

We study anomalous localization in a one-dimensional non-Hermitian quasicrystal with a spatially disordered imaginary gauge field. The system is a generalized Aubry-André-Harper (AAH) chain with asymmetric nearest- and next-nearest-neighbor hoppings generated by a Bernoulli imaginary gauge field and a quasiperiodic onsite potential. In the standard non-Hermitian AAH limit, the system undergoes a transition from a fully erratic non-Hermitian skin effect (ENHSE) phase to a fully localized phase. We show that the fractal dimension cannot distinguish these phases, whereas the Lyapunov exponent and center-of-mass fluctuations provide sharp diagnostics. This transition is accompanied by a complex-to-real spectral change under periodic boundary conditions and a topological change of the spectral winding number. With next-nearest-neighbor hopping, we uncover an anomalous mobility edge separating Anderson-localized states from ENHSE states, rather than extended states. This mobility edge is captured by an energy-dependent winding number that vanishes in the localized regime. Finally, we propose a dynamical probe based on wave-packet expansion: for typical disorder realizations, the dynamics shows winding-controlled drift and disorder-selected pinning or boundary-wrapping recurrence, while disorder averaging restores Hermitian-like transport. These results offer practical spectral, topological, and dynamical diagnostics of anomalous localization and mobility edges in non-Hermitian quasicrystals.

Motivation & Objective

• Motivate understanding of localization phenomena in non-Hermitian quasicrystals with spatially disordered imaginary gauge fields.
• Characterize transitions between erratic non-Hermitian skin-like states and conventional localization.
• Explore how mobility edges arise and how they interact with spectral topology in the presence of next-nearest-neighbor hopping.
• Propose dynamical probes to detect anomalous localization and transport properties.
• Link spectral, topological, and dynamical diagnostics to practical identification of ENHSE versus localized phases.

Proposed method

• Model a 1D non-Hermitian quasicrystal with NN and NNN hoppings generated by a Bernoulli imaginary gauge field and a quasiperiodic onsite potential.
• Use a generalized Aubry-André-Harper (AAH) framework with J1, J2, and disordered h_j to create asymmetric hoppings.
• Compute fractal dimension D from the IPR to assess state localization.
• Compute Lyapunov exponent gamma_n for each eigenstate to distinguish ENHSE from localization.
• Evaluate the center-of-mass fluctuation S across eigenstates as a disorder- and size-sensitive diagnostic.
• Analyze spectral topology via a winding number w by threading flux and comparing PBC and OBC spectra, including energy-dependent winding with reference energies E_B.

Figure 1: Representative eigenstate profiles $|\psi_{j}^{(n)}|$ of the non-Hermitian AAH model [Eq. ( 1 )] with $J_{2}=0$ and $N=300$ for a fixed imaginary gauge-field realization. Ten randomly chosen eigenstates are shown under PBCs [(a) $\lambda=1$ ; (b) $\lambda=3$ ] and OBCs [(c) $\lambda=1$ ; (

Experimental results

Research questions

• RQ1 How does a spatially disordered imaginary gauge field affect localization in a 1D quasi-periodic chain?
• RQ2 Can standard fractal dimensions distinguish ENHSE from conventional localization, or are other diagnostics required?
• RQ3 What is the role of next-nearest-neighbor hopping in creating mobility edges and how do these edges relate to spectral topology?
• RQ4 How do spectral reality and winding number correlate with localization transitions under periodic boundary conditions?
• RQ5 Can dynamical wave-packet evolution reveal signatures of anomalous localization and mobility edges, beyond spectral diagnostics?

Key findings

• In the J2=0 limit, the system displays a transition at lambda_c=2 from a fully ENHSE phase to a fully localized phase, with fractal dimension D_n→0 in both regimes but Lyapunov exponent gamma changing from 0 to finite.
• Fractal dimension fails to distinguish ENHSE from conventional localization; center-of-mass fluctuations S distinguish between ENHSE (S=O(1)) and localized (S=O(N)).
• Spectral reality emerges for lambda>2 under PBCs, cooccurring with the localization transition and a topological change of the spectral winding number from nontrivial to trivial.
• With weak NNN hopping (J2≠0), an anomalous mobility edge appears, separating Anderson-localized states from ENHSE-type states, rather than extended from localized states; the edge follows the Hermitian E_c(lambda) criterion.
• The mobility edge is accompanied by energy-selective complex-to-real spectral transitions under PBCs and an energy-dependent winding number that becomes trivial across E_c.
• Dynamical wave-packet evolution shows winding-controlled drift and disorder-selected pinning or boundary-wrapping recurrences; disorder averaging restores Hermitian-like transport.

Figure 2: Mean fractal dimension $\overline{D}$ versus $1/\ln N$ at (a) $\lambda=1$ and (b) $\lambda=3$ , extrapolating to $\overline{D}\to 0$ in both phases. (c) Mean Lyapunov exponent $\overline{\gamma}$ as a function of $\lambda$ for different $N$ , showing $\overline{\gamma}\to 0$ for $\lambda<2