Beyond the Bloch Line: Seeing Spectral Distortion in Momentum Space

Author: Denis Avetisyan

Researchers directly observe how energy bands warp in complex momentum space within a non-Hermitian system, opening new avenues for manipulating light and matter.

Under specific parameter settings (δ1=0.31π, δ2=0, η=0.25π, γ=0.057π), transmission spectra and corresponding complex eigenenergies—measured at varying potentials (μ=0, -0.1, -0.23, -0.48) and normalized between 0 and 1—demonstrate a strong correlation between experimental results and theoretical predictions, particularly at μGBZ=-0.23, which resides on the GBZ and reflects the eigenspectrum under open boundary conditions.

This study experimentally validates theoretical predictions of spectral deformation in non-Hermitian systems using a programmable photonic platform to probe non-Bloch band theory.

While non-Hermitian physics predicts exotic spectral properties and topological phenomena, direct experimental observation of these effects in momentum space has remained a significant challenge. Here we report on ‘Observation of Non-Hermitian Spectral Deformation in Complex Momentum Space’, presenting a programmable photonic platform to visualize and characterize spectral deformation within a non-Hermitian system. By encoding complex momenta in the phase and amplitude of light, we reconstruct the eigenspectrum and reveal key features like exceptional points and open-boundary spectra. This work not only validates theoretical predictions of non-Bloch band theory but also establishes a versatile approach to exploring complex momentum space – what new insights will this platform unlock regarding the interplay between non-Hermiticity and topological states of matter?

Beyond the Conventional: Exploring the Limits of Band Theory

Traditional band theory, built upon the Bloch theorem, provides a robust framework for understanding crystalline materials. However, its reliance on reciprocity and Hermiticity limits its predictive power when applied to contemporary systems exhibiting broken symmetry. These limitations are increasingly relevant in areas like topological materials and non-Hermitian optics, necessitating new theoretical frameworks.

Experimental and theoretical transmission spectra, analyzed at μ = 0, -0.03, -0.06, and -0.09, demonstrate long-range coupling effects, with complex eigenenergies extracted from the spectra and compared to theoretical results, as indicated by the gray curves representing open boundary conditions.

The exploration of non-Hermitian systems introduces spectral deformation, leading to phenomena like the non-Hermitian skin effect. Understanding these effects requires a departure from conventional band theory and embracing the richness of non-Hermitian physics.

Mapping the Complex Landscape of Non-Bloch Bands

Describing non-Bloch bands necessitates moving beyond reciprocal space to complex momentum space. Investigating these bands is experimentally challenging, as standard spectroscopic techniques are ill-equipped to probe features in complex momentum space. Complex-momentum-resolved spectroscopy directly maps these spectral features, confirming the predictions of non-Bloch band theory.

Simulating Non-Hermitian Physics with Light

Researchers have demonstrated a novel approach to simulating the non-Hermitian Su-Schrieffer-Heeger (SSH) model using the orbital angular momentum (OAM) of photons. This spatial light modulator-based technique allows for investigation of non-Hermitian physics in a spatially resolved manner. The SSH model parameters were tuned to engineer spectral deformation and observe exceptional points within a synthetic dimension, offering direct control over topological properties.

Characterizing Spectral Deformations with Mathematical Rigor

Characterizing non-Hermitian spectral features requires advanced analytical tools. The Ronkin function provides a framework for describing the complex potential governing these systems, enabling precise parameter determination. Quantitative comparison of distinct systems is facilitated by the Wasserstein metric, revealing subtle differences in their underlying physics. Investigation of spectral self-intersections led to the identification of the Generalized Brillouin Zone (GBZ) at -0.23, enhancing understanding of non-Bloch bands.

The Ronkin function landscape, derived from both experimental data and theoretical calculations with parameters δ1 = 0.31π, δ2 = 0, η = 0.25π, and γ = 0.057π, reveals consistent behavior, as confirmed by the comparison of experimental points and theoretical curves at fixed energies of E = 0 and E = 0.74π.

Extending the Boundaries of Non-Hermitian Physics

The Amoeba formulation extends the description of non-Bloch bands beyond traditional Bloch theory, particularly in higher dimensions. This allows for a more complete characterization of band topology and the emergence of novel electronic states. This platform is readily generalizable to explore complex non-Hermitian Hamiltonians and a wider range of topological phases, potentially leading to robust and dissipationless devices. Combining this spectroscopic technique with novel materials promises breakthroughs in photonics, quantum information processing, and materials science.

The pursuit of understanding spectral deformation in complex momentum space, as demonstrated by this research, echoes a fundamental principle of elegant design. It reveals how even within seemingly abstract systems—like non-Hermitian physics—underlying order and predictability prevail. Louis de Broglie once stated, “Every man of science must believe that any fact can be explained.” This belief is clearly evidenced in the careful validation of theoretical predictions through programmable photonic platforms. The ability to directly probe non-Bloch physics isn’t merely about observation; it’s about revealing the inherent beauty in the relationship between theory and experiment, a harmony achieved through rigorous exploration and a commitment to clarity. The work highlights that true insight emerges not from complexity, but from the reduction of phenomena to their essential components.

What’s Next?

The direct observation of spectral deformation in complex momentum space, as demonstrated by this work, feels less like an arrival and more like a carefully considered opening of a door. The elegance of using a programmable photonic platform to visualize non-Bloch physics is undeniable; however, it simultaneously highlights the inherent limitations of translating abstract mathematical spaces into tangible physical systems. One cannot help but wonder if the observed deformations are merely artifacts of the chosen implementation, whispers of the underlying machinery rather than pure expressions of the theory.

Future investigations should prioritize exploring the robustness of these spectral features under varying degrees of non-Hermiticity and system complexity. The current study serves as a compelling proof-of-concept, but the true test lies in extending these observations to systems exhibiting more intricate topological properties. A particularly intriguing avenue lies in investigating the interplay between spectral deformation and many-body effects – can collective phenomena further sculpt the complex momentum landscape, or will they wash away the delicate features now becoming visible?

Ultimately, the pursuit of non-Hermitian physics is not simply about adding complexity to existing models; it is about fundamentally rethinking the relationship between symmetry, topology, and the very nature of physical reality. The aesthetic appeal of these complex momentum spaces—the swirling patterns and unusual symmetries—should not be dismissed. Such visual cues often hint at deeper, underlying principles waiting to be unveiled, a reminder that a beautiful theory is often a truthful one.

Original article: https://arxiv.org/pdf/2511.06844.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/