Experimentally Probing Non-Hermitian Spectral Transition and Eigenstate Skewness

Abstract

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.

Figures (4)

• Figure 1: Challenges of probing energy spectra and eigenstates in NH lattices.a, Schematic of probed time-domain signals when the lattice is excited by a source with a single-frequency sine signal. For a Hermitian system, where all states have zero imaginary part of their energies ($\Im(E)=0$), the source excitation produces stable signals, enabling reliable extraction of real-valued energy spectra and eigenstates. For a NH system, states with $\Im(E)<0$ exhibit signal decay due to loss, while those with $\Im(E)>0$ experience excessive amplification due to gain, challenging conventional experimental techniques and preventing access to complex-valued energy spectra and eigenstates. b, 1D NH lattice exhibiting the NHSE. The panels from top to bottom display complex-valued energy spectra, skewed distributions of two representative eigenstates, and the schematic of the lattice structure. c--e, Higher-dimensional NH lattice exhibiting (c) corner NHSE and (d, e) geometry-dependent NHSE. The panels from top to bottom in c, d display spectra, eigenstates, and the lattice schematic. In the corner NHSE (c), the left and right eigenstates are highly skewed, while in geometry-dependent NHSE (d, e), the eigenstates are identical. The geometry-dependent NHSE appears in certain lattice configurations, such as rectangles (Geometry 1 in d), but vanishes in other shapes, such as parallelograms (Geometry 2 in e). Compared to 1D NH lattices, higher-dimensional systems exhibit more intricate spectral topology and eigenstate behaviors, amplifying experimental challenges.
• Figure 2: Illustration of the Green's function-based method for directly measuring the complex energy spectra and left and right eigenstates of NH systems.a, Sketch of a $7\times 7$ nonreciprocal NH lattice with OBCs along the $x$-direction and PBCs along the $y$-direction (See Supplementary Materials for details on implementations of other boundary condition configurations). A source is sequentially excited along the white path, and the frequency responses of all sites are measured by detectors to obtain the full Green's function. The bottom-left inset provides a detailed schematic of the nonreciprocal NH lattice structure. The Bloch Hamiltonian of the system is $H_\mathrm{NR}(\vb{k}) = \omega_0+\kappa_+ \mathrm{e}^{-\mathrm{i}k_x} + \kappa_+ \mathrm{e}^{-\mathrm{i}k_y} - \kappa_- \mathrm{e}^{\mathrm{i}k_x} - \kappa_- \mathrm{e}^{\mathrm{i}k_y} + 4\kappa' \cos k_x \cos k_y$Wang2024AmoebaFormulationNonBloch. Parameters used in experiments are $\omega_0/(2\uppi) = 1040\,\mathrm{Hz} - 6\mathrm{i}\,\mathrm{Hz}, \kappa_+/(2\uppi)=2.72\,\mathrm{Hz}, \kappa_-/(2\uppi) = 0.48\,\mathrm{Hz}$ and $\kappa'/(2\uppi)=0.64\,\mathrm{Hz}$. b, Experimentally measured Green's functions of the lattice shown in $\textbf{a}$, showing (left) amplitude and (right) phase at four representative frequencies: 1038 Hz, 1040 Hz, 1042 Hz, and 1044 Hz. c, Magnitudes of diagonalized Green's functions obtained from the results in b. (d, e) Experimental results: ($\textbf{d}$) energy spectrum and (e) left and right eigenstates derived from c.
• Figure 3: Experimental observation of spectral shrink and state skewness in nonreciprocal NH lattices. Parameters used in experiments are $\omega_0/(2\uppi) = 1040\,\mathrm{Hz} - 6\mathrm{i}\,\mathrm{Hz}, \kappa_+/(2\uppi)=2.72\,\mathrm{Hz}, \kappa_-/(2\uppi) = 0.48\,\mathrm{Hz}$ and $\kappa'/(2\uppi)=0.64\,\mathrm{Hz}$. a, Schematic representation of the lattice structure under different boundary conditions. (b, c) Complex energy spectra: Left panel, theoretical predictions using the tight-binding model; right panel, experimental results. The gray shaded region represents the thermodynamic limit with PBCs in both $x$ and $y$ directions. b, Comparison of energy spectra for different boundary conditions with the lattice size of $L_x = L_y=7$. c, Comparison of energy spectra for OBCs in different geometries. (d, e) Left and (f, g) right eigenstates, comparing (d, f) rectangular and (e, g) parallelogram lattices ($L_x=L_y=7$).
• Figure 4: Experimental observation of spectral sensitivity transitions in reciprocal NH lattices.a, Schematic representation of the lattice structure. The Bloch Hamiltonian of this system is $H_\mathrm{R}(\vb{k}) =\omega_0+ 2 \kappa_x \cos k_x+ 2 \kappa_y \cos(k_x+k_y)$Zhang2024EdgeTheoryNonHermitian. Parameters used in experiments are $\omega_0/(2\uppi) = 1040\,\mathrm{Hz} - 6\mathrm{i}\,\mathrm{Hz}, \kappa_x/(2\uppi)=2.25\mathrm{i}\,\mathrm{Hz}$ and $\kappa_y/(2\uppi)=4.5\,\mathrm{Hz}$. b, Definition of the spectral shift $\Delta$, quantifying the deviation of the spectrum of the rectangular lattice under full OBCs. c, Sensitivity diagram from the tight-binding model for various lattices sizes. d, Experimental results and theoretical predictions of the spectral shift as a function of selected aspect ratio configurations, also annotated in c. e, f Experimental results and theoretical predictions of energy spectra for different aspect ratios, showing transitions in spectral topology. (g) Energy spectra of parallelogram lattices ($L_x=L_y=7$) obtained experimentally. g--k Experimental results of (h, j) left and (i, k) right eigenstates, comparing (h, i) rectangular and (j, k) parallelogram lattices ($L_x=L_y=7$), demonstrating geometry-dependent spectral and eigenstate behaviors.