Despite their ubiquity, a systematic classification of multifold exceptional points, n-fold spectral degeneracies (EPns), remains a significant unsolved problem. In this article, we characterize the Abelian eigenvalue topology of generic EPns and symmetry-protected EPns for arbitrary n. The former and the latter emerge in (2n-2)- and (n-1)-dimensional parameter spaces, respectively. By introducing topological invariants called resultant winding numbers, we elucidate that these EPns are stable due to topology of a map from a base space (momentum or parameter space) to a sphere defined by resultants. In a D-dimensional parameter space (D≥c), the resultant winding numbers topologically characterize (D-c)-dimensional manifolds of generic (symmetry-protected) EPns, whose codimension is c=2n-2 (c=n-1). Our framework implies fundamental doubling theorems for both generic EPns and symmetry-protected EPns in n-band models.

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 ( PT ) 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 PT -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 PT 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.

Non-Hermitian Hamiltonians, which effectively describe dissipative systems, and analogue gravity models, which simulate properties of gravitational objects, comprise seemingly different areas of current research. Here, we investigate the interplay between the two by relating parity-time-symmetric dissipative Weyl-type Hamiltonians to analogue Schwarzschild black holes emitting Hawking radiation. We show that the exceptional points of these Hamiltonians form tilted cones mimicking the behavior of the light cone of a radially infalling observer approaching a black hole horizon. We further investigate the presence of tunneling processes, reminiscent of those happening in black holes, in a concrete example model. We interpret the non-trivial result as the purely thermal contribution to analogue Hawking radiation in a Schwarzschild black hole. Assuming that our particular Hamiltonian models a photonic crystal, we discuss the concrete nature of the analogue Hawking radiation in this particular setup.

One of the unique features of non-Hermitian (NH) systems is the appearance of NH degeneracies known as exceptional points (EPs). The extensively studied defective EPs occur when the Hamiltonian becomes non-diagonalizable. Aside from this degeneracy, we show that NH systems may host two further types of non-defective degeneracies, namely, non-defective EPs and ordinary (Hermitian) nodal points. The non-defective EPs manifest themselves by i) the diagonalizability of the NH Hamiltonian at these points and ii) the non-diagonalizability of the Hamiltonian along certain intersections of these points, resulting in instabilities in the Jordan decomposition when approaching the points from certain directions. We demonstrate that certain discrete symmetries, namely parity-time, parity-particle-hole, and pseudo-Hermitian symmetry, guarantee the occurrence of both defective and non-defective EPs. We extend this list of symmetries by including the NH time-reversal symmetry in two-band systems. Two-band and four-band models exemplify our findings. Through an example, we further reveal that ordinary nodal points may coexist with defective EPs in NH models when the above symmetries are relaxed.

The advent of non-Hermitian physics has enriched the plethora of topological phases to include phenomena without Hermitian counterparts. Despite being among the most well-studied uniquely non-Hermitian features, the topological properties of multifold exceptional points, -fold spectral degeneracies (EPs) at which also the corresponding eigenvectors coalesce, were only recently revealed in terms of topological resultant winding numbers and concomitant Abelian doubling theorems. Nevertheless, a more mathematically fundamental description of EPs and their topological nature has remained an open question. To fill this void, in this article, we revisit the topological classification of EPs in generic systems and systems with local symmetries, generalize it in terms of more mathematically tractable (local) similarity relations, and extend it to include all such similarities as well as non-local symmetries. Through the resultant vector, whose components are given in terms of the resultants between the corresponding characteristic polynomial and its derivatives, the topological nature of the resultant winding number is understood in several ways: in terms of i) the tenfold classification of (Hermitian) topological matter, ii) the framework of Mayer--Vietoris sequence, and iii) the classification of vector bundles. Our work reveals the mathematical foundations on which the topological nature of EPns resides, enriches the theoretical understanding of non-Hermitian spectral features, and will therefore find great use in modern experiments within both classical and quantum physics.

We study the effect of disorder on 𝑍₂ quantum spin liquids with a Majorana Fermi line (respectively a surface in three dimensions), and we show that depending on the symmetries that are preserved on average, qualitatively different scenarios will occur. In particular, we identify the relevant non-Hermitian symmetries for which disorder will effectively split the Fermi line into two exceptional lines, with Re⁡(𝐸)=0 states filling the area in between. We demonstrate the different scenarios using both toy models as well as large-scale numerical simulations.

The entanglement spectrum is a useful tool to study topological phases of matter, and contains valuable information about the ground state of the system. Here, we study its properties for free non-Hermitian systems for both point-gapped and line-gapped phases. While the entanglement spectrum only retains part of the topological information in the former case, it is very similar to Hermitian systems in the latter. In particular, it not only mimics the topological edge modes, but also contains all the information about the polarization, even in systems that are not topological. Furthermore, we show that the Wilson loop is equivalent to the many-body polarization and that it reproduces the phase diagram for the system with open boundaries, despite being computed for a periodic system.