Experimentally Probing Non-Hermitian Spectral Transition and Eigenstate Skewness

Non-Hermitian (NH) systems exhibit intricate spectral topology arising from complex-valued eigenenergies, with positive/negative imaginary parts representing gain/loss. Unlike the orthogonal eigenstates of Hermitian systems, NH systems feature left and right eigenstates that form a biorthogonal basis and can differ significantly, showcasing pronounced skewness between them. These characteristics give rise to unique properties absent in Hermitian systems, such as the NH skin effect and ultra spectral sensitivity. However, conventional experimental techniques are inadequate for directly measuring the complex-valued spectra and left and right eigenstates – key elements for enhancing our knowledge of NH physics. This challenge is particularly acute in higher-dimensional NH systems, where the spectra and eigenstates are highly sensitive to macroscopic shapes, lattice geometry, and boundary conditions, posing greater experimental demands compared to one-dimensional systems. Here, we present a Green’s function-based method that enables the direct measurement and characterization of both complex-valued energy spectra and the left and right eigenstates in arbitrary NH lattices. Using active acoustic crystals as the experimental platform, we observe spectral transitions and eigenstate skewness in two-dimensional NH lattices under both nonreciprocal and reciprocal conditions, with varied geometries and boundary conditions. Our approach renders complex spectral topology and left eigenstates experimentally accessible and practically meaningful, providing new insights into these quantities. The results not only confirm recent theoretical predictions of higher-dimensional NH systems but also establish a universal and versatile framework for investigating complex spectral properties and NH dynamics across a wide range of physical platforms.

💡 Research Summary

The paper addresses a fundamental experimental challenge in non‑Hermitian (NH) physics: the direct measurement of complex‑valued energy spectra and the biorthogonal left‑right eigenstates that define NH systems. While conventional pump‑probe techniques rely on real‑frequency excitation and thus capture only real‑valued eigenenergies, they fail to access lossy (Im E < 0) or amplifying (Im E > 0) modes. Moreover, higher‑dimensional NH lattices exhibit extreme sensitivity of their spectra and eigenstates to macroscopic shape, lattice geometry, and boundary conditions, making existing methods insufficient.

To overcome these limitations, the authors develop a Green’s‑function‑based experimental protocol. They construct the full Green’s function matrix G(ω) = (ω − H)⁻¹ by sequentially exciting every site of a lattice with an acoustic source (loudspeaker) and recording the complex pressure response at every other site with microphones. Both amplitude and phase are measured, providing the complete complex response for all source‑detector pairs. Diagonalizing G(ω) yields eigenvalues of the form 1/(ω − Eₙ) and the corresponding left and right eigenvectors ⟨ψᴸₙ|, |ψᴿₙ⟩ of the underlying Hamiltonian H. By scanning ω and fitting each eigenvalue peak to the simple 1/(ω − Eₙ) form, the real and imaginary parts of each eigenenergy Eₙ are extracted with high precision. Simultaneously, the eigenvectors give direct access to the left and right eigenstates, allowing quantitative assessment of their skewness.

The experimental platform is an active acoustic crystal consisting of a 7 × 7 two‑dimensional lattice of speaker‑microphone pairs. Non‑reciprocal hopping amplitudes (κ⁺, κ⁻) and auxiliary couplings (κ′) are implemented electronically, enabling the realization of both non‑reciprocal (NH) and reciprocal lattices. The authors explore a variety of boundary conditions (open, periodic, and mixed) and macroscopic shapes (rectangular, parallelogram, different aspect ratios).

Key findings include:

1. Finite‑area complex spectra – In 2D NH lattices the eigenenergies populate a bounded region of the complex plane, contrasting with the line‑like spectra of 1D NH chains. This confirms theoretical predictions that higher‑dimensional NH systems exhibit richer spectral topology.

2. Hierarchical inclusion of spectra – Spectra under fully open boundary conditions (OBC) are fully contained within those under mixed OBC/PBC, which in turn are contained within fully periodic spectra. This mirrors the well‑known 1D inclusion hierarchy and demonstrates its extension to two dimensions.

3. Geometry‑independent spectral shape – Changing the lattice shape from a rectangle to a parallelogram, or varying the aspect ratio, does not significantly alter the occupied region in the complex plane. Hence the spectral topology is robust against macroscopic geometric deformations.

4. State skewness and corner skin effect – The measured left and right eigenstates are markedly different in the non‑reciprocal case. Right eigenstates localize strongly at the top‑right corner, a clear manifestation of the corner NH skin effect. The authors introduce a skewness parameter γ = 1 − |⟨ψᴸₙ|ψᴿₙ⟩|/(‖ψᴸₙ‖‖ψᴿₙ‖), obtaining γ ≈ 0.79, which quantifies the pronounced biorthogonal asymmetry.

5. Reciprocal vs non‑reciprocal behavior – In reciprocal lattices the left and right eigenstates coincide and no skin localization is observed, highlighting the essential role of non‑reciprocity in generating both spectral skewness and spatial accumulation.

The method’s strengths lie in (i) direct access to both loss and gain modes via complex‑frequency probing, (ii) inclusion of phase information for full Green’s‑function reconstruction, and (iii) automated acquisition of L² measurements (2401 for a 7 × 7 lattice), making the approach scalable. Limitations arise from the need for high signal‑to‑noise ratio and accurate phase calibration; the experiments maintain >40 dB SNR and <1° phase error, ensuring reliable extraction. The technique assumes simple first‑order poles; higher‑order poles or exceptional point coalescence would require more sophisticated analysis.

In summary, the work provides the first experimental demonstration of simultaneous complex‑energy spectroscopy and biorthogonal eigenstate imaging in higher‑dimensional NH systems. By establishing a universal Green’s‑function framework, it opens a pathway to explore NH phenomena—such as corner skin effects, geometry‑dependent spectral transitions, and eigenstate skewness—across a broad class of wave‑based platforms including photonics, mechanics, and electronics.