Imaginary Gauge-steerable Edge Modes In Non-Hermitian Aubry-André-Harper Model

TL;DR

This work addresses how boundary-localized in-gap modes of a quasiperiodic non-Hermitian AAH chain respond to a spatially fluctuating imaginary gauge field with zero mean. By constructing an exact site-dependent nonunitary gauge transformation, the authors map the non-Hermitian system under open boundaries to the Hermitian AAH model, preserving the spectrum while imprinting a realization-dependent random-walk envelope on the eigenfunctions. They uncover two in-gap edge modes: one remains boundary-pinned, while the other is gauge-steerable, its localization center controlled solely by the gauge realization. A biorthogonal-weight–based local gain protocol is proposed to selectively amplify the steerable mode from a bulk wave packet, robust to the specific gauge realization, highlighting a route to static and dynamic control of in-gap states in non-Hermitian quasiperiodic lattices with potential experimental realization.

Abstract

We investigate a non-Hermitian Aubry-André-Harper lattice exhibiting quasiperiodicity, featuring an imaginary gauge field that varies spatially but averages to zero. In the presence of open boundary conditions, this system is precisely mapped, through a nonunitary gauge transformation, to the Hermitian AAH model with balanced hopping terms. The mapping leaves the spectrum unchanged but reshapes each eigenfunction by a realization-dependent random-walk envelope. In a parameter regime where the Hermitian counterpart hosts spectrally isolated in-gap boundary modes, we identify two such modes with sharply different responses to the envelope: one stays anchored at the boundary, while the other is controllable via the gauge, allowing its peak intensity to be relocated solely by altering the gauge setup without modifying the associated eigenenergy. Additionally, we demonstrate that this steerable mode can be preferentially enhanced and generated from an initial bulk wavefunction by introducing mild site-specific amplification at a location determined exclusively from the Hermitian model using the biorthogonal function. These findings offer pathways for both static and dynamic manipulation of spatially adjustable in-gap states in quasiperiodic non-Hermitian lattices.

Figures (6)

• Figure 1: Schematic of the AAH model with disordered imaginary gauge field. Blue (red) arrows indicate $J_R(n)$ [$J_L(n)$]. The gray bars depict the onsite potential $V_n$ on each site, and the dashed curve is a guide to the quasiperiodic modulation profile.
• Figure 2: Static properties and phase structure for fixed $\Delta=0.4$, $\beta=(\sqrt{5}-1)/2$ and $\varphi=\pi/5$ hereafter for calculations. (a) Phase diagram in the $(J,\lambda)$ plane, in which $N=1000$, color coded by $S_I$. The dashed line $\lambda_c=2J$ marks the self-dual transition. The blue star and green circle indicate the parameter sets $(J,\lambda)=(1,1)$ and $(1,3)$, respectively. (b) Cumulative height $X_n$ (black, left axis) and averaged intensity $I_n$ (colored curves, right axis) for the two representative parameter sets marked in (a). (c),(d) Eigenstate density in the ENHSE regime $(J,\lambda)=(1,1)$ and in the AAH-localized regime $(J,\lambda)=(1,3)$, respectively.
• Figure 3: Spectral and real-space structure at the parameter set indicated by the blue star marker in Fig. \ref{['fig:fig1']}. (a) Open-boundary spectrum in the plane $(\mathrm{Re}\,E,|E|)$, with color representing the Lyapunov exponent. Zoomed windows Z1, Z2, and Z3 mark the bulk state BS.1 and the in-gap states ES.2 and ES.3. In (b) and (c) eigenstates are ordered by increasing $\mathrm{Re}\,E$. (b) Three-dimensional plot of the Hermitian AAH densities $|\phi_n|^2$ versus eigenstate index and site index. (c) Corresponding plot of the ENHSE--AAH densities $|\psi_n|^2$ obtained from the same eigenvalues via $\psi_n=e^{X_n}\phi_n$. In (b) and (c) the states BS.1, ES.2, and ES.3 are highlighted. Each eigenstate is normalized to unit norm.
• Figure 4: Finite-size diagnostics of the three reference eigenstates in the Hermitian counterpart at $J=\lambda=1$, $\beta=(\sqrt{5}-1)/2$, and $\varphi=\pi/5$. (a) Eigenenergies in the gap hosting ES.2 as a function of the system size $N$, with the tracked ES.2 branch highlighted. (b) Same as (a) but for the gap hosting ES.3. (c) Boundary weight $P_{\mathrm{edge}}$ of BS.1, ES.2, and ES.3 versus $N$ evaluated with a fixed boundary window length $n_b=100$. (d) Boundary weight $P_{\mathrm{edge}}$ of BS.1, ES.2, and ES.3 versus the boundary window size $n_b$ at $N=40000$.
• Figure 5: Real-space probability densities $|\psi_n|^2$ of the in-gap state ES.2 for five independent realizations of the Bernoulli sequence $\{s_n\}$. All microscopic parameters are identical to those used for ES.2 in Fig. \ref{['fig:edge_profiles']}, only the random sequence $\{s_n\}$ is changed. Each curve corresponds to one realization.