Non-Hermitian Physics (2006.01837v1)

Abstract: A review is given on the foundations and applications of non-Hermitian classical and quantum physics. First, key theorems and central concepts in non-Hermitian linear algebra, including Jordan normal form, biorthogonality, exceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated in a pedagogical and mathematically coherent manner. Building on these, we provide an overview of how diverse classical systems, ranging from photonics, mechanics, electrical circuits, acoustics to active matter, can be used to simulate non-Hermitian wave physics. In particular, we discuss rich and unique phenomena found therein, such as unidirectional invisibility, enhanced sensitivity, topological energy transfer, coherent perfect absorption, single-mode lasing, and robust biological transport. We then explain in detail how non-Hermitian operators emerge as an effective description of open quantum systems on the basis of the Feshbach projection approach and the quantum trajectory approach. We discuss their applications to physical systems relevant to a variety of fields, including atomic, molecular and optical physics, mesoscopic physics, and nuclear physics with emphasis on prominent phenomena/subjects in quantum regimes, such as quantum resonances, superradiance, continuous quantum Zeno effect, quantum critical phenomena, Dirac spectra in quantum chromodynamics, and nonunitary conformal field theories. Finally, we introduce the notion of band topology in complex spectra of non-Hermitian systems and present their classifications by providing the proof, firstly given by this review in a complete manner, as well as a number of instructive examples. Other topics related to non-Hermitian physics, including nonreciprocal transport, speed limits, nonunitary quantum walk, are also reviewed.

• The paper establishes a theoretical framework for non-Hermitian physics by detailing spectral singularities and exceptional points that govern system dynamics.
• The paper demonstrates practical applications in classical systems like photonics and mechanics, showcasing phenomena such as unidirectional invisibility and topologically protected modes.
• The paper explores quantum and many-body systems by modeling open quantum dynamics and phase transitions using non-Hermitian Hamiltonians.

Insights into Non-Hermitian Physics: A Review of Key Concepts, Applications, and Future Directions

Non-Hermitian physics represents a burgeoning domain of paper within both classical and quantum physics. This review encapsulates the theoretical underpinnings and potential applications of non-Hermitian systems, encompassing the foundational mathematical constructs and extending to complex phenomena observed in experimental realizations. The central thesis lies in the unification of diverse physical systems—across photonics, mechanics, quantum optics, and many-body physics—under the umbrella of non-Hermitian dynamics, characterized by phenomena such as exceptional points, spectral singularities, and non-Hermitian-induced topology.

Fundamental Theorems and Concepts

The review begins by systematically laying out the mathematical formalisms foundational to non-Hermitian physics. These include the spectral decomposition of non-Hermitian matrices, the stability of singular value spectra, and the Jordan normal form. The discussion highlights the inherent sensitivity of non-Hermitian systems, particularly near exceptional points (EPs) where eigenvalues and eigenvectors coalesce, leading to branch point singularities. These singularities underpin the rich phenomenology associated with non-Hermitian dynamics.

Applications in Classical Systems

In classical systems, photonics emerges as a key area where non-Hermitian effects are both theoretically intriguing and practically implementable. Here, the similarities between the Schrödinger equation and Maxwell's equations in non-Hermitian media catalyze advancements such as unidirectional invisibility, coherent perfect absorption, and PT-symmetry-induced wave manipulation. Mechanical systems, including those exhibiting friction and active feedback, also fall within the scope of non-Hermitian physics, demonstrating phenomena such as dynamical nonreciprocity and topologically protected failure modes in metamaterials.

Quantum Physics and Open Systems

The review rigorously explores the role of non-Hermitian physics within quantum systems, particularly in the context of open quantum systems where the Feshbach projection approach yields effective non-Hermitian operators. Such operators adeptly model quantum resonances and superradiance phenomena in mesoscopic and atomic systems. Additionally, the quantum trajectory approach forms the crux of analyzing measurement-induced nonunitary dynamics, offering insights into the real-time evolution of quantum states under continuous observation.

Non-Hermitian Many-Body Physics

Quantum many-body physics presents a frontier where non-Hermitian Hamiltonians elucidate critical phenomena, such as the quantum Zeno effect, in systems undergoing decay or continuous measurement. The review discusses the phase transitions and criticality inherent in non-Hermitian many-body systems, emphasizing the theoretical and practical importance of these findings in mapping out the broader landscape of non-conserved quantum dynamics.

Implications and Future Perspectives

The implications of non-Hermitian physics are manifold. The ability to manipulate wave phenomena across a broad spectrum of systems opens new avenues for technological applications, ranging from precision sensors harnessing enhanced sensitivities at EPs to robust communication pathways in photonic crystals. Theoretical advancements could spearhead further exploration into nontrivial topological states beyond Hermitian paradigms, including the potential violation of conventional bulk-boundary correspondences.

Future Directions in Non-Hermitian Physics

Looking ahead, the review suggests burgeoning areas for investigation including the detailed mapping of non-Hermitian topological phases, the synthesis of nonclassical light in PT-symmetric optical systems, and the integration of non-Hermitian dynamics in quantum information processing. The interplay between non-Hermiticity and strong interactions remains a largely uncharted territory, holding promise for unveiling exotic states of quantum matter.

In conclusion, non-Hermitian physics not only presents an intriguing theoretical challenge but also a practical pathway to novel technologies and deeper comprehension of complex systems, driving future research at the interface of physics, engineering, and beyond.