A Widom determinant expansion reveals that finite-size edge states in non-Hermitian chains can be absorbed by the bulk spectrum at a critical system size due to competition between boundary-projected Green's function subsets.
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Non-Hermitian Physics
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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.
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Imaginary magnetic fields induce exceptional points in neutral meson mass spectra computed via hadronic effective Lagrangian and constituent quark models, separating real and complex eigenvalue regimes.
For any diagonalisable non-Hermitian H with real spectrum, the biorthogonal Gibbs functional satisfies positivity of ω_bi(A†A) for all A if and only if H is quasi-Hermitian.
The index of non-Hermitian Dirac operators that anticommute with a chirality operator is topologically protected when the operators are diagonalizable and elliptic.
Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.
A PT-symmetric non-Hermitian free-fermion field theory realizes logarithmic conformal field theory with central charge c=-2 via a biorthogonal Virasoro algebra construction.
Using complex scaling on a 1D scattering Hamiltonian, the authors realize an exceptional point from resonance-continuum coalescence and derive the associated Berry phase and Chern characteristic.
An exceptional-point framework for black-hole ringdown characterizes resonances near avoided crossings, demonstrates enhanced mode contributions in the time domain, and identifies the EP frequency as the physically relevant observable.
Holographic dual of PT-symmetric BCFT via imaginary scalar on EOW brane shows spontaneous PT breaking and enhanced entanglement entropy growth in quenched state.
Short-time real-time pseudo entropy obeys S_A(t,0)=S_A(0)-it ⟨K_A(H−⟨H⟩)⟩ + O(t²), with imaginary response from symmetrized covariance of H and K_A.
Kerr QNM anomalies near algebraically special frequencies arise from avoided crossings with resonant excitation and pole skipping due to quasinormal-Matsubara pole-zero cancellations.
Generalized quantum dimensions from SymTFTs classify massless and massive RG flows in pseudo-Hermitian systems and relate coset constructions to domain walls.
In asymmetric Damour-Solodukhin wormholes, reflectionless and echo modes share asymptotic spectral properties parallel to the real frequency axis with matching spacing, and reflectionless modes lie closer to the axis yielding larger echo amplitudes.
Pseudospectrum of Schwarzschild QNMs shows stability of the fundamental mode to infrared perturbations respecting asymptotics and instability of overtones to ultraviolet high-frequency perturbations.
Uniformly dissipative Rashba nanowires with skewed magnetic fields exhibit nonreciprocal conductance from interference-induced differences in transmission times between left- and right-moving electrons.
Supergravity truncations admit real-metric wormhole solutions via imaginary scalar extensions whose holographic duals are pseudo-Hermitian and PT-symmetric, interpretable as entangled brane states.
Krylov complexity saturates in the full high-temperature Caldeira-Leggett system, reproduces dissipative features when decoherence is suppressed, shows oscillations when dissipation is suppressed, and remains insensitive to decoherence onset because the Krylov basis differs from the conventional one
Lecture notes on defining scalar QFTs and assigning geometric structure to the space of possible theories.
A focused review organizing literature on quantum photonic time crystals from temporal boundaries and Floquet spectra to light-matter dynamics.
citing papers explorer
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Non-Hermitian Edge State Endocytosis
A Widom determinant expansion reveals that finite-size edge states in non-Hermitian chains can be absorbed by the bulk spectrum at a critical system size due to competition between boundary-projected Green's function subsets.
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Hadronic exceptional points
Imaginary magnetic fields induce exceptional points in neutral meson mass spectra computed via hadronic effective Lagrangian and constituent quark models, separating real and complex eigenvalue regimes.
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Kubo-Martin-Schwinger conditions for non-Hermitian systems
For any diagonalisable non-Hermitian H with real spectrum, the biorthogonal Gibbs functional satisfies positivity of ω_bi(A†A) for all A if and only if H is quasi-Hermitian.
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Atiyah--Singer Index Theorem for Non-Hermitian Dirac Operators
The index of non-Hermitian Dirac operators that anticommute with a chirality operator is topologically protected when the operators are diagonalizable and elliptic.
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Exact WKB analysis of inverted triple-well: resonance, PT-symmetry breaking, and resurgence
Exact WKB analysis produces median-summed spectra and an algebraic equation for the exceptional point of PT-symmetry breaking in the inverted triple-well system.
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Non-Hermitian free-fermion critical systems and logarithmic conformal field theory
A PT-symmetric non-Hermitian free-fermion field theory realizes logarithmic conformal field theory with central charge c=-2 via a biorthogonal Virasoro algebra construction.
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Geometric phase from encircling an exceptional point of a quantum resonance in the complex-scaling method
Using complex scaling on a 1D scattering Hamiltonian, the authors realize an exceptional point from resonance-continuum coalescence and derive the associated Berry phase and Chern characteristic.
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Exceptional Points and Resonance in Black Hole Ringdown
An exceptional-point framework for black-hole ringdown characterizes resonances near avoided crossings, demonstrates enhanced mode contributions in the time domain, and identifies the EP frequency as the physically relevant observable.
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Holographic Dual of PT Symmetric BCFT
Holographic dual of PT-symmetric BCFT via imaginary scalar on EOW brane shows spontaneous PT breaking and enhanced entanglement entropy growth in quenched state.
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Real-time pseudo entropy and modular-Hamiltonian correlations
Short-time real-time pseudo entropy obeys S_A(t,0)=S_A(0)-it ⟨K_A(H−⟨H⟩)⟩ + O(t²), with imaginary response from symmetrized covariance of H and K_A.
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Pole Skipping, Avoided Crossing, and Resonant Excitation in Kerr Quasinormal Modes near Algebraically Special Frequencies
Kerr QNM anomalies near algebraically special frequencies arise from avoided crossings with resonant excitation and pole skipping due to quasinormal-Matsubara pole-zero cancellations.
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Generalizing quantum dimensions: Symmetry-based classification of local pseudo-Hermitian systems and the corresponding domain walls
Generalized quantum dimensions from SymTFTs classify massless and massive RG flows in pseudo-Hermitian systems and relate coset constructions to domain walls.
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Reflectionless and echo modes in asymmetric Damour-Solodukhin wormholes
In asymmetric Damour-Solodukhin wormholes, reflectionless and echo modes share asymptotic spectral properties parallel to the real frequency axis with matching spacing, and reflectionless modes lie closer to the axis yielding larger echo amplitudes.
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Pseudospectrum and black hole quasi-normal mode (in)stability
Pseudospectrum of Schwarzschild QNMs shows stability of the fundamental mode to infrared perturbations respecting asymptotics and instability of overtones to ultraviolet high-frequency perturbations.
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Nonreciprocal conductance in uniformly dissipative devices
Uniformly dissipative Rashba nanowires with skewed magnetic fields exhibit nonreciprocal conductance from interference-induced differences in transmission times between left- and right-moving electrons.
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Supergravity flows, wormholes and their pseudo-Hermitian holographic duals
Supergravity truncations admit real-metric wormhole solutions via imaginary scalar extensions whose holographic duals are pseudo-Hermitian and PT-symmetric, interpretable as entangled brane states.
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Krylov Complexity for Open Quantum System: Dissipation and Decoherence
Krylov complexity saturates in the full high-temperature Caldeira-Leggett system, reproduces dissipative features when decoherence is suppressed, shows oscillations when dissipation is suppressed, and remains insensitive to decoherence onset because the Krylov basis differs from the conventional one
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Scalar Quantum Fields: Theory Space and its Geometry
Lecture notes on defining scalar QFTs and assigning geometric structure to the space of possible theories.
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Quantum Photonic Time Crystals: From Temporal Boundaries to Floquet Light-Matter Interactions
A focused review organizing literature on quantum photonic time crystals from temporal boundaries and Floquet spectra to light-matter dynamics.
- Black hole spectroscopy: from theory to experiment