Exceptional points in non-Hermitian systems have recently been shown to possess nontrivial topological properties and to give rise to many exotic physical phenomena. However, most studies thus far have focused on isolated exceptional points or one-dimensional lines of exceptional points. Here, we substantially expand the space of exceptional systems by designing two-dimensional surfaces of exceptional points, and find that symmetries are a key element to protect such exceptional surfaces. We construct them using symmetry-preserving non-Hermitian deformations of topological nodal lines, and analyze the associated symmetry, topology, and physical consequences. As a potential realization, we simulate a parity-time-symmetric 3D photonic crystal and indeed find the emergence of exceptional surfaces. Our work paves the way for future explorations of systems of exceptional points in higher dimensions, and applications in emission control and sensing.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
12 February 2019: Minor corrections were made to the citations.
In recent years, there has been growing interest in exploring novel effects in non-Hermitian systems [1–3]. This has provided great insight in fundamental science, including parity-time () symmetry [4–6] and novel topological phases [7–15], while also opening the doors to new photonic applications, such as unconventional transmission and reflection , sensing functionalities [17–21], lasers , and chiral mode transfer [23,24]. Key to many of these explorations are the special properties of exceptional points (EPs)—unique spectral degeneracies where the real and imaginary parts of two or more eigenvalues coincide and the eigenvectors coalesce [1,2]. Indeed, only realizable in non-Hermitian systems, EPs mark the boundaries of phase transitions and give rise to the unconventional optical responses mentioned above. Moreover, they possess topological properties such as a Berry phase and vorticity [7,11], and can give rise to open Fermi arcs in the bulk dispersion of systems [13,25].
However, the majority of studies thus far have focused on the properties of isolated points (0D) [13,23,24] or continuous lines (1D) [14,26–30] of EPs. Correspondingly, this has limited the types of achievable band dispersions and observable phenomena in these non-Hermitian systems. This calls for approaches to go beyond these lower-dimensional EP systems and explore higher-dimensional configurations, such as surfaces of EPs.
In this paper, we propose and analyze several models to realize exceptional surfaces. First, we examine the general conditions for EPs to occur, and find that the emergence of EP surfaces requires additional symmetry protection, indicating that EP surfaces can be understood as the generalization of Hermitian symmetry-protected nodal phases [31,32] to the non-Hermitian setting. Motivated by this, we consider a non-Hermitian deformation of topological nodal lines, and show that under certain symmetry conditions, this can give rise to a non-Hermitian system with EPs configured in a torus geometry. We show that the EP torus is characterized by separated components inside and outside the torus, as well as a quantized non-Hermitian Berry phase inherited from the Hermitian nodal line. In addition, we find that the EP surface encloses an open nodal volume—a three-dimensional generalization of bulk Fermi arcs—in which the real parts of two bands are degenerate with each other within an entire volume. This offers remarkable control of the band structure and spectral density of states (DOS) of the system, which could have interesting applications such as controlling spontaneous emission and enhancing nonlinearities in optical systems. Finally, we utilize -symmetric gain-loss modulation in a simple three-dimensional photonic crystal structure to realize EP surfaces, which can be readily implemented in experiments. Our results are of particular interest to the photonics community, where non-Hermitian terms are often naturally present in the form of gain, material loss, or radiation loss.
We start by considering the conditions to create EPs, in order to understand why 0D or 1D EP configurations are typically generated. EPs occur when the real and imaginary parts of two or more eigenvalues coalesce and the eigenspace becomes defective. We can consider a generic two-band non-Hermitian Hamiltonian
Thus, we shall generically find that two constraints need to be satisfied to create an EP. The dimensionality of the EP contour generated will then be in spatial dimension , as observed in previous experiments [13,14,23,24]. However, this also implies that additional mechanisms are required to realize EP surfaces in physical dimensions within 3.
At this stage, it is instructive to examine the case of Hermitian systems: in 3D, the robust spectral degeneracies are 0D Weyl points, and the realization of 1D nodal lines requires symmetry protection . In analogy, we also expect the 2D EP surfaces to be symmetry-protected and serve as a natural non-Hermitian generalization of symmetry-protected topological nodal phases. The symmetry conditions, which can include crystalline symmetries or more general non-Hermitian symmetries , can reduce the number of independent equations that need to be simultaneously satisfied, leading to a dimensional EP configuration in dimensions. In analogy to lower-dimensional examples [4,28], we expect symmetry to be one such symmetry sufficient to generate EP surfaces in 3D. We note that adding a constant gain/loss term will not change the spectral degeneracies, and thus the system will still host EP surfaces.
Motivated by the fact that both Hermitian nodal lines and EP surfaces can be protected by symmetry, we now consider symmetry-preserving non-Hermitian deformations to Hermitian nodal lines. This approach thus directly makes clear the connection between EP surfaces—examples of non-Hermitian nodal phases—and their Hermitian counterparts. Compared with previous work that has studied non-Hermitian nodal lines [34,35], the symmetry preservation plays a key role in generating EP surfaces as opposed to lines of EPs.
First, we consider a minimal two-band model describing an EP surface, with the Hamiltonian given by
For illustration purposes, we choose and . In the Hermitian limit , the spectral degeneracies of the system clearly show two nodal rings [Fig. 1(a)]. Once a non-Hermitian perturbation is included, each nodal ring splits into a torus of EPs [Fig. 1(b)]. The non-Hermitian perturbation is specifically chosen to respect the combined symmetry of the nodal line, where and , being the complex-conjugation operation. This guarantees that in the non-Hermitian setting, the only symmetry-admissible perturbation to the nodal line Hamiltonian is proportional to , thus resulting in an EP torus (we do not impose the symmetry on the shift proportional to identity, as it does not affect the spectral degeneracies). On the other hand, if we choose a perturbation that breaks the symmetry, e.g., instead of , then the conditions for EPs to occur become , , which describe four EP rings instead of EP surfaces [34,35], as shown in Fig. 1(c).
The physics discussed here can be easily realized in microwave experiments, such as the system employed in Ref. , consisting of a metallic-mesh 3D photonic crystal. Nodal chains have been discovered in this simple structure, where the nodal lines are protected by symmetry and the chain crossing point is additionally protected by a mirror symmetry.
By adding non-Hermitian perturbations that preserve the symmetry, as shown in Fig. 2(a), each nodal line can be split into an EP surface. The deformation to the chain crossing point, however, can be more complicated, and in general may lead to novel EP geometries (see Supplement 1 for details). Although the presence of symmetry-breaking terms will gap out the nodal chain crossing point [Figs. 2(b) and 2(c)], a non-Hermitian term that is larger in magnitude can restore the connectedness of the feature [Fig. 2(d)], even if the mirror symmetry of the system is broken.
We now analyze the topological properties associated with the EP torus. For a Hamiltonian with symmetry, we can choose a basis such that the symmetry corresponds to complex-conjugation, in which case the Hamiltonian must be real, and complex eigenvalues come in complex-conjugate pairs. Therefore, when changing a Hamiltonian in which all eigenvalues are real into one with complex-conjugate pairs, the continuity of eigenvalues requires a degeneracy of eigenvalues on the real line to appear in between, corresponding to an EP. Thus, the inside (complex eigenvalues) and outside (real eigenvalues) of the torus belong to two disconnected branches, and there must exist a continuous surface of EPs in between them. This is further characterized by the discriminant of the characteristic polynomial of the Hamiltonian (see Supplement 1 for details).
In addition, we find that the generalization of the Hermitian Berry phase protects the EP surface to form a continuous torus. Since the Hamiltonian still respects symmetry in the non-Hermitian case, we shall find that the Berry phase , defined via right eigenvectors on both sides, is still quantized to 0 or (mod ), as long as the integration path chosen is in the -unbroken phase , where all eigenvalues are real (see Supplement 1 for details). Note that by continuity, the region outside the EP torus, which was originally gapped in the Hermitian limit, will belong to the -unbroken phase and have a quantized Berry phase.
For the preceding model in Eq. (2), we calculate the Berry phase along different paths outside the torus using the normalized eigenvectors. For a path not linking with the EP torus [Fig. 1(b), path b], the Berry phase is 0; whereas, along a path linking with the EP torus [Fig. 1(b), path a], we find that the Berry phase is . This is consistent with the Berry phase for the Hermitian nodal line, indicating that the topological invariant is inherited by the non-Hermitian system, and the topological protection against the EP torus breaking apart remains.
The generation of EP surfaces also has important consequences for the bulk dispersion. Similar to the bulk Fermi arcs that terminate at discrete EPs [7,13,25], the internal volume of the EP surface will have a pair of bands that are completely degenerate in the real part of their eigenvalues, which we call an open nodal volume. The non-Hermitian term then provides remarkable control of the spectral DOS of the system: as we continuously increase the non-Hermitian perturbation from 0 to a large value, the DOS near the nodal line frequency will be tuned from a relatively small value, due to the 1D nature of the nodal line, to a large value, due to the 3D volumetric factor of the open nodal volume. We calculate the spectral DOS for the model Eq. (2), incorporating the linewidth as a Lorentzian broadening, such that the calculated values are directly relevant to physical consequences such as spontaneous emission enhancement. In view of experiments, we have added a constant loss term to make the system passive. The results are illustrated in Fig. 1(d), where the spectral DOS near the nodal line clearly increases as the strength of the non-Hermitian perturbation is increased.
In physical realizations of nodal lines and the corresponding EP surfaces, a nodal line may not be completely flat in frequency, which will cause some reduction of the accessible range of DOS. However, the structure shown in Fig. 2(a) has been shown to have a remarkably uniform nodal line frequency in the Hermitian limit, with less than 1% variations across the whole Brillouin zone. This makes it a promising system to observe the EP torus discussed here, as well as to investigate the evolution of DOS with the strength of the non-Hermitian term. In addition, the system can be engineered to possess a non-Hermitian particle-hole symmetry , which can protect the nodal volume to be completely flat. The continuous controllability of DOS via non-Hermitian modulation may enable spontaneous emission engineering, while the higher dimension of the EP surface may lead to interesting opportunities in sensing [17–21]. The EP surface also enables a -superprism effect , in which small changes in incident wave angle can dramatically change the gain/loss response.
We now consider a concrete realization of EP surfaces in a -symmetric photonic crystal, obtained from gain-loss modulation in a regular 3D dielectric photonic crystal. As shown in Fig. 3(a), the Hermitian photonic crystal consists of dielectric cubes arranged in a cubic lattice. The photonic band dispersion in a 2D cut at a generic value is shown in Fig. 3(b), where along the axis, each colored band consists of two degenerate bands with different polarizations. By doubling the unit cell size in one spatial direction, and applying loss on one site and gain on the next, we form a -symmetric photonic crystal with a supercell size twice that of the original system.
In Figs. 3(c) and 3(e), we show the same 2D cut of the real and imaginary parts of the band structure when the -symmetric non-Hermitian perturbation is applied. We find that the band structure is separated into two regions: one where the real part of multiple bands coalesce and one where the imaginary parts coalesce. The surface separating the two regions consists of two completely degenerate bands, and thus corresponds to a surface of EPs. By extracting the locations at which this occurs, we find in Fig. 3(d) that there are indeed two EP surfaces in the band structure, originating from different polarizations.
While we chose a relatively large non-Hermitian term to emphasize the effects, we note that due to the initial band-folding degeneracy, a threshold-less transition is realized , and any finite non-Hermitian modulation is sufficient to produce an EP surface. The resulting band structure can be significantly modified by the non-Hermitian terms, showing relatively flat real part dispersions in the direction perpendicular to the band-folding line [Fig. 3(c)] and sharp changes in the gain/loss response as the -vector is tuned [Fig. 3(e)]. In addition, a constant shift in complex energy will not affect the EP surface, so the same design can also be implemented in purely passive systems.
In this work, we have proposed various methods to realize EP surfaces; analyzed their symmetry, topology, and applications; and discussed straightforward avenues to their experimental implementation in photonic and microwave systems. We have also shown that the bulk dispersion of these systems is drastically modified by including symmetry-preserving non-Hermitian terms, giving rise to a highly tunable DOS.
While we have focused on systems with symmetry, the analysis can be readily generalized to other types of symmetries, including general non-Hermitian symmetry classes [33,38,39] and crystalline symmetries. Similarly, it may be interesting to consider other types of invariants, particularly those unique to non-Hermitian systems. In addition, one could study the associated bulk-boundary correspondence, where unusual phenomena such as the non-Hermitian skin effect emerge [9,15].
National Science Foundation (NSF) (1838412, DMR-1720530); Air Force Office of Scientific Research (AFOSR) (FA9550-18-1-0133).
We acknowledge helpful discussions with M. Soljačić, M. Lukin, D. Borgnia, L. Chen, Y. Fu, and N. Rivera. Note added: after completing this project, we became aware of related contributions by Okugawa and Yokoyama  and Budich et al. .
See Supplement 1 for supporting content.
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