Restoration of the non-Hermitian bulk-boundary correspondence via topological amplification

Abstract

Non-Hermitian (NH) lattice Hamiltonians display a unique kind of energy gap and extreme sensitivity to boundary conditions. Due to the NH skin effect, the separation between edge and bulk states is blurred and the (conventional) bulk-boundary correspondence is lost. Here, we restore the bulk-boundary correspondence for the most paradigmatic class of NH Hamiltonians, namely those with one complex band and without symmetries. We obtain the desired NH Hamiltonian from the (mean-field) unconditional evolution of driven-dissipative cavity arrays, in which NH terms -- in the form of non-reciprocal hopping amplitudes, gain and loss -- are explicitly modeled via coupling to (engineered and non-engineered) reservoirs. This approach removes the arbitrariness in the definition of the topological invariant, as point-gapped spectra differing by a complex-energy shift are not treated as equivalent; the origin of the complex plane provides a common reference (base point) for the evaluation of the topological invariant. This implies that topologically non-trivial Hamiltonians are only a strict subset of those with a point gap and that the NH skin effect does not have a topological origin. We analyze the NH Hamiltonians so obtained via the singular value decomposition, which allows to express the NH bulk-boundary correspondence in the following simple form: an integer value $\nu$ of the topological invariant defined in the bulk corresponds to $|\nu |$ singular vectors exponentially localized at the system edge under open boundary conditions, in which the sign of $\nu$ determines which edge. Non-trivial topology manifests as directional amplification of a coherent input with gain exponential in system size. Our work solves an outstanding problem in the theory of NH topological phases and opens up new avenues in topological photonics.

arXiv comment21 pages, 10 figures

References in corpus (78)

• Topological Insulators
• Topological insulators and superconductors
• Topological Photonics
• Edge states and topological invariants of non-Hermitian systems
• Quantum fluids of light
• Introduction to Quantum Noise, Measurement and Amplification
• Exceptional Topology of Non-Hermitian Systems
• Non-Hermitian Physics
• Localization transitions in non-Hermitian quantum mechanics
• Topological phases of non-Hermitian systems
• Symmetry and Topology in Non-Hermitian Physics
• Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
• Topological Origin of Non-Hermitian Skin Effects
• Topological Band Theory for Non-Hermitian Hamiltonians
• Anomalous edge state in a non-Hermitian lattice
• Non-Bloch Band Theory of Non-Hermitian Systems
• Topolectrical circuit realization of topological corner modes
• Observation of phononic helical edge states in a mechanical 'topological insulator'
• Correspondence between winding numbers and skin modes in non-hermitian systems
• Non-Hermitian Chern bands
• Observation of bulk boundary correspondence breakdown in topolectrical circuits
• Non-Hermitian Boundary Modes
• Efficient Light Funneling based on the non-Hermitian Skin Effect
• Non-Hermitian robust edge states in one-dimension: Anomalous localization and eigenspace condensation at exceptional points
• Topologically protected elastic waves in phononic metamaterials
• Probing topological invariants in the bulk of a non-Hermitian optical system
• Topolectrical circuits
• Vortex Pinning and Non-Hermitian Quantum Mechanics
• What is Nonreciprocity?
• Non-Hermitian skin effect and chiral damping in open quantum systems
• Exciton-polariton topological insulator
• Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory
• Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering
• Nonreciprocal Photon Transmission and Amplification via Reservoir Engineering
• Edge states and topological phases in non-Hermitian systems
• Non-reciprocal robotic metamaterials
• Phase preserving amplification near the quantum limit with a Josephson Ring Modulator
• Non-Hermitian Topological Sensors
• Reciprocal skin effect and its realization in a topolectrical circuit
• Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits
• Probing non-Hermitian Skin Effect and non-Bloch Phase Transitions
• Time Reversal Invariant Topologically Insulating Circuits
• Nonreciprocal reconfigurable microwave optomechanical circuit
• Observation of arbitrary topological windings of a non-Hermitian band
• Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence
• Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics
• Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition
• Non-Bloch band theory of non-Hermitian Hamiltonians in the symplectic class
• Tenfold Way for Quadratic Lindbladians
• Phase-dependent chiral transport and effective non-Hermitian dynamics in a bosonic Kitaev-Majorana chain
• Quantum-limited directional amplifiers with optomechanics
• Topological framework for directional amplification in driven-dissipative cavity arrays
• Interplay between long range hopping and disorder in topological systems
• Simple formulas of directional amplification from non-Bloch band theory
• Stabilization of Solitons Generated by a Supersonic Flow of Bose-Einstein Condensate Past an Obstacle
• Topological quantum fluctuations and travelling wave amplifiers
• Non-degenerate, three-wave mixing with the Josephson ring modulator
• Non-Hermitian chiral phononics through optomechanically-induced squeezing
• Quantum Non-Hermitian Topological Sensors
• Topological phase transitions and chiral inelastic transport induced by the squeezing of light
• Active topolectrical circuits
• Exceptional Topological Insulators
• Non-equilibrium stationary states of quantum non-Hermitian lattice models
• Observation of non-Hermitian topology with non-unitary dynamics of solid-state spins
• Topological Amplification in Photonic Lattices
• Realization of directional amplification in a microwave optomechanical device
• Topology by Dissipation: Majorana Bosons in Metastable Quadratic Markovian Dynamics
• An elementary rigorous proof of bulk-boundary correspondence in the generalized Su-Schrieffer-Heeger model
• Hermitian zero modes protected by nonnormality: Application of pseudospectra
• Topological lattices realized in superconducting circuit optomechanics
• Theory of the coherence of topological lasers
• Phase Transitions and Generalized Biorthogonal Polarization in Non-Hermitian Systems
• Topological input-output theory for directional amplification
• Theory of chiral edge state lasing in a two-dimensional topological system
• Bridging the gap between topological non-Hermitian physics and open quantum systems
• Point-gap topology with complete bulk-boundary correspondence in dissipative quantum systems
• Microwave measurement beyond the quantum limit with a nonreciprocal amplifier
• Topological transport in the steady state of a quantum particle with dissipation

Cited by in corpus (29)

• Optomechanical realization of the bosonic Kitaev-Majorana chain
• Nonreciprocal Synchronization of Active Quantum Spins
• Diagnosing non-Hermitian Many-Body Localization and Quantum Chaos via Singular Value Decomposition
• Dissipative Spin-wave Diode and Nonreciprocal Magnonic Amplifier
• Non-reciprocal dynamics and the non-Hermitian skin effect of repulsively bound pairs
• Topological zero modes and edge symmetries of metastable Markovian bosonic systems
• The non-Hermitian skin effect: A perspective
• Self acceleration from spectral geometry in dissipative quantum-walk dynamics
• Localized states and skin effect around non-Hermitian impurities in tight-binding models
• Hidden zero modes and topology of multiband non-Hermitian systems
• The Interplay of Finite and Infinite Size Stability in Quadratic Bosonic Lindbladians
• Topological transitions in dissipatively coupled Su-Schrieffer-Heeger models
• Numerical instability of non-Hermitian Hamiltonian evolutions
• Mode-Shell correspondence, a unifying phase space theory in topological physics -- Part I: Chiral number of zero-modes
• Lindbladian versus Postselected Non-Hermitian Topology
• Restoration of non-Hermitian bulk-boundary correspondence by counterbalancing skin effect
• Topological, multi-mode amplification induced by non-reciprocal, long-range dissipative couplings
• Dissipative phase transition of interacting non-reciprocal fermions
• Unifying Anderson transitions and topological amplification in non-Hermitian chains
• Exceptional Points, Bulk-Boundary Correspondence, and Entanglement Properties for a Dimerized Hatano-Nelson Model with Staggered Potentials
• Limits of the non-Hermitian description of decay models
• Topological Amplification of the Bosonic Kitaev Chain with Non-Uniform Loss
• Infinite-component $BF$ field theory: Connection of fracton order, Toeplitz braiding, and non-Hermitian amplification
• Hard edge asymptotics of correlation functions between singular values and eigenvalues
• Emergence of Hermitian topology from non-Hermitian knots
• Non-Hermitian second-order topological insulator with point gap
• Correlation functions between singular values and eigenvalues
• Nonlinearity-driven Topology via Spontaneous Symmetry Breaking
• Bulk-boundary correspondence in topological two-dimensional non-Hermitian systems: Toeplitz operators and singular values