Homotopy, Symmetry, and Non-Hermitian Band Topology

Abstract

Non-Hermitian matrices are ubiquitous in the description of nature ranging from classical dissipative systems, including optical, electrical, and mechanical metamaterials, to scattering of waves and open quantum many-body systems. Seminal line-gap and point-gap classifications of non-Hermitian systems using K-theory have deepened the understanding of many physical phenomena. However, ample systems remain beyond this description; reference points and lines do not in general distinguish whether multiple non-Hermitian bands exhibit intriguing exceptional points, spectral braids and crossings. To address this we consider two different notions: non-Hermitian band gaps and separation gaps that crucially encompass a broad class of multi-band scenarios, enabling the description of generic band structures with symmetries. With these concepts, we provide a unified and comprehensive classification of both gapped and nodal systems in the presence of physically relevant parity-time ($\mathcal{P}\mathcal{T}$) and pseudo-Hermitian symmetries using homotopy theory. This uncovers new stable topology stemming from both eigenvalues and wave functions, and remarkably also implies distinct fragile topological phases. In particular, we reveal different Abelian and non-Abelian phases in $\mathcal{P}\mathcal{T}$-symmetric systems, described by frame and braid topology. The corresponding invariants are robust to symmetry-preserving perturbations that do not induce (exceptional) degeneracy, and they also predict the deformation rules of nodal phases. We further demonstrate that spontaneous $\mathcal{P}\mathcal{T}$ symmetry breaking is captured by Chern-Euler and Chern-Stiefel-Whitney descriptions, a fingerprint of unprecedented non-Hermitian topology previously overlooked. These results open the door for theoretical and experimental exploration of a rich variety of novel topological phenomena in a wide range of physical platforms.

44 pages, 12 figures, published version

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Cited by in corpus (25)

• Scale-tailored localization and its observation in non-Hermitian electrical circuits
• Tutorial: Classifying Photonic Topology Using the Spectral Localizer and Numerical $K$-Theory
• Braiding Topology of Non-Hermitian Open-Boundary Bands
• Real-space topological invariant for time-quasiperiodic Majoranas
• Versatile Braiding of Non-Hermitian Topological Edge States
• Braids and Higher-order Exceptional Points from the Interplay Between Lossy Defects and Topological Boundary States
• Lindbladian versus Postselected Non-Hermitian Topology
• Exceptional magic angles in non-Hermitian twisted bilayer graphene
• Quantum Sensing with Driven-Dissipative Su-Schrieffer-Heeger Lattices
• Winding Topology of Multifold Exceptional Points
• Observation of Braid-Protected Unpaired Exceptional Points
• Exceptional deficiency of non-Hermitian systems
• Dynamic transition of the density-matrix topology under parity-time symmetry
• Measuring topological invariants of even-dimensional line-gapped non-Hermitian systems through quench dynamics
• Non-Hermitian Hopf insulators
• Computation and Verification of Spectra for Non-Hermitian Systems
• Higher-order Topological Knots and the classification of non-Hermitian lattices under $C_n$ symmetry
• Non-Hermitian wave-packet dynamics and its realization within a non-Hermitian chiral cavity
• Anatomy of Higher-Order Non-Hermitian Skin and Boundary Modes
• Thermodynamics of analogue black holes in a non-Hermitian tight-binding model
• Covariant formulation of the Berry connection in non-Hermitian systems
• Supercharging exceptional points: Full-spectrum pairwise coalescence in non-Hermitian systems
• Exceptional topology on nonorientable manifolds
• Emergence of Imaginary Time Crystals in the non-Hermitian Su-Schrieffer-Heeger model
• Geometric origin of self-intersection points in non-Hermitian energy spectra