We present a mathematically rigorous framework for non-Hermitian topological phases in open quantum-biological systems, enabling room-temperature, topologically protected information transfer between quantum states and biological molecular dynamics. The theory is built on generalized non-Hermitian Chern numbers, Berry phase matching, and unitary conformal mapping, with a strict fidelity constraint of F≥99.95% and a non-Hermitian backscattering-free condition for edge-state transport. This work unifies non-Hermitian quantum mechanics, topological condensed matter physics, and biological quantum dynamics, offering testable predictions for coherence enhancement in photosynthetic complexes, topological protection in ion channels, and DNA topological information encoding. All results are derived from open axiomatic quantum theory with no confidential content, providing a new theoretical foundation for next-generation quantum-biological interfaces.
We present a mathematically rigorous framework for non-Hermitian topological phases in open quantum-biological systems, enabling room-temperature, topologically protected information transfer between quantum states and biological molecular dynamics. The theory is built on generalized non-Hermitian Chern numbers, Berry phase matching, and unitary conformal mapping, with a strict fidelity constraint of F≥99.95% and a non-Hermitian backscattering-free condition for edge-state transport. This work unifies non-Hermitian quantum mechanics, topological condensed matter physics, and biological quantum dynamics, offering testable predictions for coherence enhancement in photosynthetic complexes, topological protection in ion channels, and DNA topological information encoding. All results are derived from open axiomatic quantum theory with no confidential content, providing a new theoretical foundation for next-generation quantum-biological interfaces.
Traditional thermodynamics, built upon the foundation of Hermitian quantum mechanics, primarily describes closed or weakly open systems where energy is conserved. However, the ubiquitous presence of dissipation and gain in realistic physical and biological systems necessitates a more generalized framework. Non-Hermitian quantum mechanics, characterized by Hamiltonians with complex eigenvalues, naturally accounts for such open systems and their interaction with the environment. This paper explores the burgeoning field of non-Hermitian thermodynamics, which aims to extend classical and quantum thermodynamic principles to systems exhibiting energy exchange with their surroundings. We delve into how non-Hermiticity provides a natural language for describing processes involving dissipation, gain, and decoherence, thereby offering a unified perspective on these phenomena. Key concepts such as PT-symmetry, exceptional points, and generalized definitions of work, heat, and entropy in non-Hermitian settings are reviewed. The paper discusses the implications for fundamental thermodynamic laws, the performance of non-Hermitian heat engines, and the role of coherence in energy transport. By embracing the mathematical elegance and physical insights of non-Hermitian physics, we propose a comprehensive framework that unifies the seemingly disparate concepts of dissipation and coherence within a consistent thermodynamic theory, paving the way for advanced applications in quantum technologies, materials science, and biological energy transfer.
Quantum coherence in biological systems faces a fundamental paradox: decoherence theory predicts femtosecond-to-picosecond timescales in warm, wet environments, yet experimental evidence demonstrates coherence persisting on biologically functional timescales (nanoseconds to milliseconds). We resolve this paradox by proving that the collagen-water interface creates real projective space RP³ topology with non-orientable Möbius character, providing topological protection of quantum states. The mechanism emerges from geometric frustration: collagen’s C₃ helical symmetry (with handedness inversion: left-handed chains forming right-handed superhelix) combines with structured water’s Td tetrahedral symmetry to produce non-commutative algebra C₃Td = ωTd C₃ (ω = e^(2πi/3)). This forces the coupling operator into a projective representation with eigenvalues {1, -1, e^(±2πi/3)}, where the λ = -1 eigenvalue is the signature of Möbius topology. Antipodal neural couplingcreates phase space RP³ = S³/{±1} with fundamental group π₁ = ℤ₂, enabling topologically protected quantum states.Protection arises from topological defect suppression: changing topology requires creating domain walls with energy E_defect ~ 100-1000 eV (central estimate ~400 eV), vastly exceeding thermal energy k_BT ~ 26 meV, yielding coherence time enhancement (L_system/L_thermal)² ~ 10⁴, transforming picosecond bare coherence into ~10 nanosecond biologically relevant timescales.This biological RP³ topology is not merely analogous but mathematically homologous to Gerard ’t Hooft’s solution to the black hole information paradox via antipodal identification at event horizons: identical topological structure preserving quantum information across 26 orders of magnitude (molecular scale ~10⁻⁹ m to Planck scale ~10⁻³⁵ m), revealing a universal information-theoretic principle. We present seven testable experimental predictions, including eigenvalue measurements (quantum state tomography), Berry phase detection (γ = π for non-contractible loops), and brain network topology analysis, providing clear falsification criteria for this information-theoretic universality principle.
This work presents a mathematically consistent, logically falsifiable unified framework for characterizing scale-invariant topological properties of open quantum systems across atomic, biological, and cosmological scales. We define a generalized non-Hermitian topological invariant invariant under scale transformations and renormalization-group flow, construct a compatible scale-adapted de Rham cohomological complex, and establish its rigorous correspondence with the first cohomology class of the parameter manifold. The framework is motivated by structural analogies across vastly separated physical regimes, while remaining purely theoretical and ontologically neutral, providing a rigorous mathematical foundation for interdisciplinary research in mathematical physics and quantum science.
Quantum matter emerges from the interplay of fluctuations, topology, and entanglement, which - in equilibrium - governs quantized transport, universal criticality, and topological classification. Non-Hermitian systems, widely explored in platforms ranging from electric circuits to photonics, are intrinsically out-of-equilibrium, and display fundamentally new phenomena, including complex spectra, spectral winding, exceptional topology, and non-unitary dynamics. A central challenge is understanding how the complex single-particle spectrum governs universal many-body behavior. We introduce a symmetry-protected dynamical topological index derived directly from the complex spectrum. Through the lens of algebraic topology, more specifically Morse theory, we identify critical points in the spectrum with topological defects, whose curvature and stability are protected under continuous deformations. This links spectral geometry to many-body observables, unifying non-Hermitian band topology, entanglement, and transport. We demonstrate that non-Hermitian quantum criticality in non-interacting systems is controlled by gain-and-loss-selected non-equilibrium steady states, which dynamically generate an emergent imaginary Fermi surface whose Fermi points host scale-invariant gapless modes with logarithmic entanglement scaling and algebraic correlations. Our work establishes a unified framework for non-Hermitian quantum matter, connecting spectral topology to Morse theory, revealing a topological foundation of non-equilibrium quantum criticality.
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, and 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 and subjects in quantum regimes, such as quantum resonances, superradiance, the 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, first 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.
Conventional Chern insulators rely on breaking time-reversal symmetry via magnetic fields or synthetic gauge fields. Here we introduce a fundamentally different paradigm: a static lattice without any gauge field becomes a Chern insulator solely through the cooperation of two non-Hermitian ingredients: on-site loss and non-reciprocal coupling. Neither effect alone suffices; only their synergy opens a topological gap and gives rise to robust chiral edge states with a nonzero Chern number. The underlying mechanism originates from a non-Hermitian chiral molecule formed by the lossy sites, whose dynamically dominant mode carries an orbital angular momentum that effectively generates a synthetic flux after the fast-decaying degrees of freedom are eliminated. Our results establish non-Hermitian cooperation as a versatile resource for inducing Chern topology without external fields or temporal modulations, opening a new route to topological phases in open classical and quantum systems.