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Experimentally Probing Non-Hermitian Spectral Transition and Eigenstate Skewness (2501.08160v1)

Published 14 Jan 2025 in cond-mat.mes-hall and quant-ph

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.

Summary

  • The paper introduces a Green’s function-based method that directly measures complex spectral topology and eigenstate skewness in non-Hermitian acoustic systems.
  • It overcomes traditional measurement challenges by investigating how spectral responses depend on lattice geometry, boundary conditions, and macroscopic shapes.
  • Results confirm key theoretical predictions, revealing distinct spectral and eigenstate behaviors that inform future research in photonics, mechanics, and metamaterials.

Overview of the Non-Hermitian Spectral Transition and Eigenstate Skewness Study

The paper presents a novel methodological approach to probing non-Hermitian (NH) physics, particularly addressing the experimental challenges associated with observing complex-valued energy spectra and biorthogonal eigenstates in NH systems. This contribution is significant in the context of acoustics, where the interplay between gain and loss, represented by the imaginary parts of eigenenergies, manifests distinctly from Hermitian systems.

The work emphasizes the critical aspects of NH systems, such as the NH skin effect (NHSE) and eigenstate skewness. Unlike Hermitian systems, NH systems feature disparate left and right eigenstates, leading to unique phenomena absent in traditional physics models. The experimental challenges become more pronounced in higher-dimensional NH systems, where system responses are highly sensitive to lattice geometry, macroscopic shapes, and boundary conditions.

Methodology and Experimental Innovation

The researchers propose a Green's function-based method to directly measure the complex spectral topology and eigenstates of NH lattices. This method is applied using active acoustic crystals, a practical platform for observing NH physics experimentally. The implementation in two-dimensional settings under various boundary conditions and spatial configurations allows for comprehensive characterizations that were previously unattainable.

The meticulous construction of a Green's function matrix from the frequency response data of lattice sites underpins this method, overcoming the limitations of conventional techniques. The Green's function, once diagonalized, yields both the energy spectra and the left and right eigenstates, thereby facilitating a direct measure of skewness—a substantial improvement over previous real-frequency-limited approaches.

Results and Theoretical Implications

The application of the proposed method confirms pivotal theoretical predictions in NH physics. A significant observation is the finite-area complex spectral topology in higher-dimensional NH systems, in contrast to the line spectra typical of one-dimensional counterparts. The spectral topology is robust against geometric changes in nonreciprocal NH lattices but exhibits sensitivity in reciprocal NH systems—the spectral topology is notably dependent on lattice geometry and aspect ratios beyond a critical point.

Additionally, the paper investigates the NHSE in both nonreciprocal and reciprocal contexts. The findings distinctively highlight skewness between left and right eigenstates in nonreciprocal systems, while reciprocal systems exhibit nearly symmetric eigenstates with minimal skewness. These nuanced insights into NH physics accentuate the method's efficacy in revealing complex eigenstate behaviors and spectral dependencies on structural and geometrical factors.

Future Prospects and Extensions

The universality of the Green's function formalism lends itself to applications across diverse physical platforms, including photonics and mechanics, broadening the experimental accessibility of NH physics. This work sets a foundational precedent for future exploration of NH system dynamics, particularly in verifying advanced theoretical models regarding spectral graph topology, defect-induced states, and critical NHSE, among others.

Future research could advance the understanding of NH systems by exploring phenomena that remain experimentally elusive, utilizing the established methodology to test theoretical predictions further. Such endeavors may not only enrich the theoretical framework but also lead to practical applications in fields reliant on complex wave behaviors, potentially influencing the development of innovative audio-visual technologies and metamaterials.

In conclusion, the paper's methodological and experimental advancements provide a versatile approach to interrogating the spectral complexity and eigenstate behaviors of NH systems, paving the way for enriching the current theoretical landscape with empirically driven insights.

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