Stochasticity-induced non-Hermitian skin criticality

Abstract

Typically, scaling up the size of a system does not change the shape of its energy spectrum, other than making it denser. Exceptions, however, occur in the new phenomenon of non-Hermitian skin criticality, where closely competing generalized Brillouin zone (GBZ) solutions for non-Hermitian state accumulation give rise to anomalously scaling complex spectra. In this work, we discover that such non-Hermitian criticality can generically emerge from stochasticity in the lattice bond orientation, a surprising phenomenon only possible in 2D or beyond. Marked by system size-dependent amplification rate, it can be physically traced to the proliferation of feedback loops arising from excess local non-Hermitian skin effect (NHSE) accumulation induced by structural disorder. While weak disorder weakens the amplification as intuitively anticipated, stronger disorder enigmatically strengthens the amplification almost universally, scaling distinctly from conventional critical system. By representing cascades of local excess NHSE as ensembles of effectively coupled chains, we analytically derived a critical GBZ that predicts how state amplification scales with the system size and disorder strength, highly consistent with empirical observations. Our new mechanism for disordered-facilitated amplification applies generically to structurally perturbed non-Hermitian lattices with broken reciprocity, and would likely find applications in non-Hermitian sensing through various experimentally mature meta-material platforms.

Figures (4)

• Figure 1: (a) Randon on-site energies or bond strengths lead to phenomena with Hermitian analogs, and cannot give rise to non-Hermitian skin criticality by themselves. (b) But non-Hermitian criticality can occur due to competitive feedback mechanisms from structural disorder. Randomly perturbing atoms with positional standard deviation $\sigma$ leads to stochastically modified asymmetric left/right inter-atomic hoppings (green/yellow) [Eq. \ref{['HamNH']}]. Local inhomogeneities among them yield effective feedback loops. (c,d) In both 1D and 2D, the spectrum $E$ of crystalline $H$ [Eq. \ref{['HamNH']}] depends very little on the system size $N$, as is typical in most known models. (e,f) With structural disorder $\sigma=0.3$, the amplification rate Im$(E)$ in 2D but not 1D increases significantly with $N$ (blue to orange), signaling the prospect of non-Hermitian skin criticality (see discussions surrounding Fig. \ref{['Fig4']}). $\kappa=0.15$ for all plots.
• Figure 2: Locality and profile of eigenstates under structural disorder. All spectra correspond an instance of $H$ [Eq. \ref{['HamNH']}] with $N=20$, colored by IPR [Eq. \ref{['EqIPR']}]. (a) Eigenstates are extended Bloch waves (black) in the crystalline Hermitian case. (b) Non-Hermiticity ($\kappa=0.4$) introduces various degrees of skin localization (red to yellow). (c,d) Disorder ($\sigma=0.3$) introduces Anderson-localized states outside of the original "skin-dominated" range of Re$(E)$ (pale blue), and also suppresses the amplification rate Im$(E)$ when non-Hermitian. (e) Switching from OBCs to PBCs makes almost all eigenstates in the skin-dominated range extended (black), but leaves Anderson-localized states untouched. (f,g) Representative eigenstates for the spectra in (d,e). Saliently, only skin-dominated eigenstates (f-3,f-4) respond to the NHSE and are relevant to non-Hermitian skin criticality.
• Figure 3: Anomalous scaling of amplification rate max(Im$(E)$). (a,c) Averaged max(Im$(E)$) over 400 disorder instances, at different non-Hermiticities $\kappa$. At small disorder $\sigma$, the amplification drops sharply as structural disorder impedes the NHSE. But larger $\sigma$ gives rise to proliferating feedback loops that lead to increasing max(Im$(E)$) with system size $N$. (b,d) Corresponding $N$-dependent amplification enhancement ratios $A$ [Eq. \ref{['ANN0']}], which can exceed 60% at large $\sigma$.
• Figure 4: Feedback mechanism for stochasticity-induced scaling of amplification. (a,b) Illustrative eigenstates and their disorder-perturbed hopping weights across orbitals (blue disks) with the largest occupancies. A local feedback loop exists only in (a), where Im$(E)\neq 0$. Larger disks reflect larger state occupancies $\left|\psi_{i\alpha}\right|^{2}$. Arrows indicate net hopping directions, with color intensity representing effective hopping strength $G = \sqrt{t_{\alpha,\beta} t_{\alpha,\beta}^{'}}$ and line width representing actual transition amplitude $W = G \sqrt{\left|\psi_{i\alpha} \psi_{j\beta} \right|}$. (c1-c3) Cascades of locally asymmetric hoppings form effective NHSE chains. Sufficiently far cascades with nontrivial hopping asymmetry contrast $\Delta \gamma$ can be modeled as a pair of antagonistic weakly-coupled chains $\tilde{\mathcal{H}}$ [Eq. \ref{['EqbA']}] exhibiting length $L$-dependent amplification rate. (d) $L$-dependent GBZ of $\tilde{\mathcal{H}}$, with excellent agreement between analytic [Eq. \ref{['Eq4']}] and numerical results from diagonalizing Eq. \ref{['EqbA']}. (e) The corresponding amplification rate max(Im$(E)$) increases similarly with $L$ and $\Delta\gamma$, consistent with that in the full stochastic model [Fig. \ref{['Fig3']}]. Here, $t=\sqrt{t_{B}/t_{A}}$, $t_{A/B}$ represents the amplitudes of nearest-neighbor hopping to the left/right.