Topology and exceptional points of massive Dirac models with generic non-Hermitian perturbations

TL;DR

This paper systematically classifies and analyzes massive Dirac models with various non-Hermitian perturbations, revealing their exceptional points, boundary modes, and topological properties, which are relevant for photonic applications like topological lasers.

Contribution

It provides a comprehensive classification of non-Hermitian Dirac models, detailing their topological features and exceptional points, filling a gap in the understanding of experimental non-Hermitian topological systems.

Findings

Identified three types of non-Hermitian terms in Dirac models.

Determined bulk exceptional points and boundary modes for each case.

Suggested applications in photonic topological lasers and optical devices.

Abstract

Recently, there has been a lot of activity in the research field of topological non-Hermitian physics, partly driven by fundamental interests and partly driven by applications in photonics. However, despite these activities, a general classification and characterization of non-Hermitian Dirac models that describe the experimental systems is missing. Here, we present a systematic investigation of massive Dirac models on periodic lattices, perturbed by general non-Hermitian terms. We find that there are three different types of non-Hermitian terms. For each case we determine the bulk exceptional points, the boundary modes, and the band topology. Our findings serve as guiding principles for the design of applications, for example, in photonic lattices. For instance, periodic Dirac systems with non-Hermitian mass terms can be used as topological lasers. Periodic Dirac systems with non-Hermitian anti-commuting terms, on the other hand, exhibit exceptional points at the surface, whose non-trivial topology could be utilized for optical devices.

Figures

Equations

$$V^{-1}HV=H^{\prime },H^{\prime †}=H^{\prime }.$$

$$H_{\text{0}}(\mathbf{k})=\sum _{i=1}^d\sin k_i{\Gamma }_i+(M-\sum _{i=1}^d\cos k_i){\Gamma }_{d+1},$$

$$H_{\text{0}}(\tilde{\mathbf{k}})$$

$$H(\mathbf{k})=H_0(\mathbf{k})+i\lambda \Gamma ,$$

$$H(\mathbf{k})=H_0(\mathbf{k})+i\lambda {\Gamma }_j=\sum _{i=1}^d{k}_i{\Gamma }_i+m{\Gamma }_{d+1}+i\lambda {\Gamma }_j,$$

$$\begin{gathered} H^{\prime }(\tilde{\mathbf{k}})=\frac{1}{2i}(\hat{S}-{\hat{S}}^{†})\otimes {\Gamma }_j-\frac{1}{2}(\hat{S}+{\hat{S}}^{†})\otimes {\Gamma }_{d+1} \\ +\mathbb{I}\otimes (\sum _{i\ne j}\sin k_i{\Gamma }_i+\sqrt{M_k^2-{\lambda }^2}{\Gamma }_{d+1}), \end{gathered}$$

Full text

Despite these recent activities, a general framework for the study and classification of non-Hermitian Dirac models that describe the aforementioned experimental systems is lacking. In particular, a general classification of exceptional points and topological surface states in non-Hermitian systems is missing, although various attempts have been made with partial success for certain special cases Yao and Wang (2018); Lee (2016); Shen et al. (2018); Yao et al. (2018a); Martinez Alvarez et al. (2018); Lee and Thomale (2018); Yao and Wang (2018); Yao et al. (2018a); Kunst et al. (2018); Kawabata et al. (2018, 2018); Gong et al. (2018); Zhou and Lee (2019). Since most non-Hermitian experimental systems can be faithfully described by Dirac models with small non-Hermitian perturbations Bandres et al. (2018); Harari et al. (2018); Feng et al. (2014); Hodaei et al. (2014); Goldzak et al. (2018); Parto et al. (2018); Yao et al. (2018b); Zhao et al. (2018); St-Jean et al. (2017); Weimann et al. (2017); Rüter et al. (2010); Feng et al. (2017), a systematic investigation of general non-Hermitian perturbations of Dirac Hamiltonians would be particularly valuable. This would be not only of fundamental interest, but could also inform the design of new applications.

In this Rapid Communication, we present a systematic investigation of massive Dirac Hamiltonians on periodic lattices, perturbed by small non-Hermitian terms. We show that these non-Hermitian Dirac Hamiltonians can be either intrinsically or superficially non-Hermitian, depending on whether the non-Hermiticity can be removed by a similarity transformation. According to the Clifford algebra, general non-Hermitian terms can be categorized into three different types: (i) non-Hermitian terms that anti-commute with the whole Dirac Hamiltonian, (ii) kinetic non-Hermitian terms, and (iii) non-Hermitian mass terms. Remarkably, we find a two-fold duality for the first two types of non-Hermitian perturbations: Dirac models perturbed by type-(i) terms are superficially non-Hermitian with periodic boundary conditions (PBCs), but intrinsically non-Hermitian with open boundary conditions (OBCs). Vice versa, Dirac models with type-(ii) terms are intrinsically non-Hermitian with PBCs, but superficially non-Hermitian with OBCs. Interestingly, for type-(i) and type-(ii) terms the non-Hermiticity leads to $(d-2)$-dimensional exceptional spheres in the surface and bulk band structures, respectively. Type-(iii) terms, on the other hand, induce intrinsic non-Hermiticity both for OBCs and PBCs, but with a purely real surface-state spectrum and no exceptional spheres.

Intrinsic versus superficial non-Hermiticity.— We begin by discussing some general properties of non-Hermitian physics. First, we recall that in Hermitian physics only unitary transformations of the Hamiltonian are considered, because only these preserve the reality of the expectation values. In non-Hermitian physics, however, the Hamiltonian can be similarity transformed, $H\rightarrow V^{-1}HV$, by any invertible matrix $V$, which is not necessarily unitary but is required to be local. For this reason, a large class of non-Hermitian Hamiltonians $H$ can be converted into Hermitian ones by non-unitary similarity transformations, i.e.,

[TABLE]

Using this observation, we call Hamiltonians whose non-Hermiticity can or cannot be removed by the above transformation as superficially or intrinsically non-Hermitian, respectively.

For non-interacting local lattice models, which is our main focus here, $H$ is a quadratic form, whose entries are specified as $H_{(\mathbf{r}\alpha),(\mathbf{r}^{\prime}\alpha^{\prime})}$ with $\mathbf{r}$ the positions of the unit cells and $\alpha$ a label for internal degrees of freedom. Correspondingly, the similarity transformation has matrix elements $V_{(\mathbf{r}\alpha),(\mathbf{r}^{\prime}\alpha^{\prime})}$. By the locality condition, the matrix elements $H_{(\mathbf{r}\alpha),(\mathbf{r}^{\prime}\alpha^{\prime})}$ and $V_{(\mathbf{r}\alpha),(\mathbf{r}^{\prime}\alpha^{\prime})}$ are required to tend to zero sufficiently fast as $|\mathbf{r}-\mathbf{r}^{\prime}|\rightarrow\infty$. If $H$ can be converted into a Hermitian Hamiltonian by a local transformation $V$, its eigenvalues are necessarily real. Conversely, any local lattice Hamiltonian with a real spectrum is either entirely Hermitian or superficially non-Hermitian 111We note that, for instance, space-time-inversion symmetric (i.e., $\mathcal{PT}$ symmetric) spinless lattice models have always real spectra and are therefore only superficially non-Hermitian..

A characteristic feature of non-Hermitian lattice models is the existence of exceptional points in parameter space, where two or multiple eigenvectors become identical, leading to a non-diagonalizable Hamiltonian. However, it is important to note that such exceptional points are not dense in parameter space. That is, there exist arbitrarily small perturbations which remove the exceptional points, rendering the Hamiltonian diagonalizable 222It should be noted, however, that in periodic Hamiltonians exceptional points can be stable, in the sense that small perturbations cannot remove them from the BZ.. One such perturbation relevant for lattice models are the boundary conditions Kunst et al. (2018); Kunst and Dwivedi (2019), which modify the hopping amplitudes between opposite boundaries. For a general classification of non-Hermitian Hamiltonians, it is therefore essential to distinguish between different types of boundary conditions, in particular OBCs and PBCs. With PBCs and assuming translation symmetry, we can perform a Fourier transformation of Eq. (1) to obtain $H^{\prime}(\mathbf{k})=V^{-1}(\mathbf{k})H(\mathbf{k})V(\mathbf{k})$. Here, $V(\mathbf{k})$ is assumed to be local in momentum space. It is worth noting that the locality in momentum space is essentially different from that in real space. Generically, the Fourier transform of ${V}(\mathbf{k})$, $V_{\mathbf{r},\mathbf{r}^{\prime}}=\sum_{\mathbf{k}}{V}(\mathbf{k})e^{i\mathbf{k}\cdot(\mathbf{r}-\mathbf{r}^{\prime})}$, is not local in general.

Non-Hermitian Dirac Hamiltonians.— We now apply the above concepts to non-Hermitian Dirac models of the form $H=H_{0}+\lambda U$, where $H_{0}$ is a Hermitian Dirac Hamiltonian with mass $M$, and $U$ a non-Hermitian perturbation with $\lambda\ll M$. Assuming PBCs in all directions, we consider the following Hermitian Dirac Hamiltonian on the $d$-dimensional cubic lattice

[TABLE]

where $\Gamma_{\mu}$ denote the gamma matrices that satisfy $\{\Gamma_{\mu},\Gamma_{\nu}\}=2\delta_{\mu\nu}$ and $M$ is the real mass parameter. With OBCs in the $j$th direction and PBCs in all other directions, the Hamiltonian reads

[TABLE]

where $\tilde{\mathbf{k}}$ denotes the vector of all momenta except $k_{j}$, $\widehat{S}_{ij}=\delta_{i,j+1}$ is the right-translational operator, and $N_{j}$ stands for the number of layers in the $j$th direction. From the above two equations it is now clear that, according to the Clifford algebra, there exist only the three types of non-Hermitian perturbations discussed above. We will now study these individually.

Non-Hermitian anti-commuting terms.— We start with non-Hermitian terms of type (i), i.e., terms that anti-commute with the Dirac Hamiltonian $H_{0}$. Such non-Hermitian terms are possible for all Altland-Zirnbauer classes with chiral symmetry Altland and Zirnbauer (1997); Chiu et al. (2016), in which case they are given by the chiral operator $\Gamma$. With PBCs the Hamiltonian perturbed by these type-(i) terms is expressed as

[TABLE]

where $\Gamma$ is an additional gamma matrix with $\{\Gamma,{H}_{0}(\mathbf{k})\}=0$ and $\lambda$ a real parameter. The spectrum of $H({\bf k})$ is given by $E(\mathbf{k})=\pm\sqrt{d^{2}(\mathbf{k})-\lambda^{2}}$ with $d^{2}(\mathbf{k})\mathbb{I}={H}_{0}^{2}(\mathbf{k})$, which is completely real for all $\mathbf{k}$, provided that $|\lambda|$ is smaller than the energy gap of ${H}_{0}$. Thus, Hamiltonian (4) with $\lambda\ll M$ is only superficially non-Hermitian and we can remove the non-Hermitian term by a similarity transformation. The corresponding transformation matrix $V({\bf k})$ can be derived systematically by noticing that the flattened Hamiltonian $\tilde{{H}}_{0}(\mathbf{k})={H}_{0}(\mathbf{k})/d(\mathbf{k})$ and $\Gamma$ form a Clifford algebra and, thus, $i[\tilde{H}_{0}(\mathbf{k}),\Gamma]/4$ generates rotations of the plane spanned by $\tilde{H}_{0}(\mathbf{k})$ and $\Gamma$. Hence, the explicit expression of the transformation matrix is $V(\mathbf{k})=\exp[-\tfrac{i}{2}\tilde{H}_{0}(\mathbf{k})\Gamma\eta(\mathbf{k})]$, with $e^{\eta(\mathbf{k})}=\sqrt{(d(\mathbf{k})+\lambda)/(d(\mathbf{k})-\lambda)}$. From Eq. (1) it follows that the transformed Hamiltonian is ${H}^{\prime}(\mathbf{k})=\sqrt{1-\lambda^{2}/d^{2}(\mathbf{k})}{H}_{0}(\mathbf{k})$, which is manifestly Hermitian for $\lambda\ll M$.

With OBCs, on the other hand, type-(i) perturbations lead to intrinsic non-Hermiticity, provided the Dirac Hamiltonian $H_{0}$ is in the topological phase. This is because the topological boundary modes acquire complex spectra due to the non-Hermitian term $i\lambda\Gamma$, even for infinitesimally small $\lambda$. To see this, we first observe that for any eigenstate $\psi_{0}$ of $H_{0}$ with energy $E_{0}$, $\Gamma\psi_{0}$ is also an eigenstate of $H_{0}$, but with opposite energy $-E_{0}$. Applying chiral perturbation theory to $H=H_{0}+i\lambda\Gamma$, we find that since $i\lambda\Gamma$ scatters $\psi_{0}$ into $\Gamma\psi_{0}$, eigenstates of $H$ can be expressed as superpositions of $\psi_{0}$ and $\Gamma\psi_{0}$. Explicitly, we find that the eigenstates of $H$ are $\psi_{\pm}=\psi_{0}+c_{\pm}\Gamma\psi_{0}$, with $c_{\pm}=iE_{0}/\lambda\pm\sqrt{1-E_{0}^{2}/\lambda^{2}}$ and energy $E_{\pm}=\pm\sqrt{E_{0}^{2}-\lambda^{2}}$. This analysis holds in particular also for the topological boundary modes of $H_{0}$, which are massless Dirac fermions with linear dispersions. Consequently, even for arbitrarily small $\lambda$, there exists a segment in the spectrum of $H$ around $E=0$ with purely imaginary eigenergies.

To make this more explicit, we can derive a low-energy effective theory for the boundary modes, by projecting the bulk Hamiltonian onto the boundary space. Generically, the boundary theory is of the form ${H}_{b}(\tilde{\mathbf{k}})=\sum_{i\neq j}{k}_{i}\gamma^{i}+i\lambda\gamma$, where the first term describes the boundary massless Dirac fermions of ${H}_{0}$. The matrices $\gamma^{i}$ and $\gamma$ are the projections of $\Gamma_{i}$ and $\Gamma$, respectively, onto the boundary space, and satisfy $\{\gamma^{i},\gamma\}=0$. With this, we find that the boundary spectrum is $E_{b}=\pm\sqrt{|\tilde{\mathbf{k}}|^{2}-\lambda^{2}}$, and that there exists a $(d-2)$-dimensional exceptional sphere of radius $|\tilde{\mathbf{k}}|=\lambda$ in the boundary Brillouin zone, which separates eigenstates with purely real and purely complex energies from each other.

As an aside, we remark that even arbitrarily large non-Hermitian terms $i\lambda\Gamma$ cannot remove the topological surface state. The reason for this is that $i\lambda\Gamma$ is a chiral operator, which acts only within a unit cell and does not couple different sites. In other words, the expectation value of the position operator $X_{i}$ is independent of $\lambda$, i.e., $\frac{d}{d\lambda}\langle\psi^{\alpha}_{\lambda}|X_{i}|\psi^{\beta}_{\lambda}\rangle=0$ with $\langle\psi^{\alpha}_{\lambda}|$ and $|\psi^{\beta}_{\lambda}\rangle$ the left and right eigenstates of $H$, respectively.

Let us now illustrate the above general considerations by considering as an example, ${H}_{\text{TI}}(\mathbf{k})=\sin k_{x}\Gamma_{1}+\sin k_{y}\Gamma_{2}+(M-\cos k_{x}-\cos k_{y})\Gamma_{3}+i\lambda\Gamma_{4}$, with $\Gamma_{j}$ the $4\times 4$ Dirac gamma matrices, which describes a topological superconductor in class DIII or a topological insulator in class AII Chiu et al. (2016). The energy spectra of ${H}_{\text{TI}}$ with periodic and open boundary conditions are shown in Figs. 1(a) and 1(b), respectively [see Supplemental Material (SM) Sup for details]. We observe that the bulk spectrum is purely real, while the surface spectrum is complex with two exceptional points of second order located at $k_{x}=\pm\arcsin|\lambda|$.

Non-Hermitian kinetic terms.— We proceed by considering non-Hermitian kinetic terms [i.e., terms of type (ii)] added to $H_{0}$ with PBCs. The effects of these non-Hermitian terms can be most clearly seen by studying the continuous version of Eq. (2), namely

[TABLE]

with 1$$\leq$$j$$\leq$$d and $\lambda$ real. The energy spectrum of $H(\mathbf{k})$, $E(\mathbf{k})=\pm\sqrt{\sum_{i\neq j}k_{i}^{2}+(k_{j}+i\lambda)^{2}+m^{2}}$, is complex and exhibits exceptional points on the $(d-2)$-dimensional sphere $\sum_{i\neq j}k_{i}^{2}+m^{2}=\lambda^{2}$ within the $k_{j}=0$ plane. Hence, $H({\bf k})$ with PBCs is intrinsically non-Hermitian.

To study the case of OBCs we consider $H(\mathbf{k})$ in a slab geometry with the surface perpendicular to the $j$th direction. The energy spectrum in this geometry is obtained from ${H}(\tilde{\mathbf{k}},-i\partial_{j})$, i.e., by replacing $k_{j}$ by $-i\partial_{j}$ in Eq. (5). Then, it is obvious that the non-Hermitian term $i\lambda\Gamma_{j}$ can be removed by the similarity transformation $V=e^{\lambda x_{j}}$. That is, $e^{-\lambda x_{j}}H(\tilde{\mathbf{k}},-i\partial_{j})e^{\lambda x_{j}}={H}_{0}(\tilde{\mathbf{k}},-i\partial_{j})$, which is manifestly Hermitian. Accordingly, $H(\tilde{\mathbf{k}},-i\partial_{j})$ has a real spectrum and its eigenstates are related to those of $H_{0}(\tilde{\mathbf{k}},-i\partial_{j})$ by $\psi(x_{j},\tilde{\mathbf{k}})=e^{\lambda x_{j}}\psi_{0}(x_{j},\tilde{\mathbf{k}})$. We conclude that continuous Dirac models perturbed by non-Hermitian kinetic terms are superficially non-Hermitian with OBCs, but intrinsically non-Hermitian with PBCs. The same holds true for lattice Dirac models.

To exemplify this, we consider the lattice Dirac model of Eq. (Topology and exceptional points of massive Dirac models with generic non-Hermitian perturbations) perturbed by the non-Hermitian term $i\lambda\Gamma_{j}$, i.e., $H(\tilde{\mathbf{k}})=H_{0}(\tilde{\mathbf{k}})+\mathbb{I}\otimes i\lambda\Gamma_{j}$. This Hamiltonian can be transformed to (see SM Sup for details)

[TABLE]

by the similarity transformation $V=\operatorname{diag}\{1,\alpha,\cdots,\alpha^{N_{j}-1}\}\otimes\left[(1+\alpha)\mathbb{I}+i(1-\alpha)\Gamma_{j}\Gamma_{d+1}\right].$ Here $\alpha=\sqrt{(M_{k}-\lambda)/(M_{k}+\lambda)}$ and $M_{k}=M-\sum_{i\neq j}\cos k_{i}$. Equation (6) is manifestly Hermitian for $\lambda\ll M$. As a concrete example, we set in Eq. (6) $d=2$ with $\Gamma_{i}$ the Dirac gamma matrices, which describes a two-dimensional topological insulator. The energy spectra for this case with periodic and open boundary conditions are shown in Figs. 1(c) and 1(d), respectively.

Non-Hermitian mass terms.— Finally, we examine the effects of non-Hermitian mass terms, i.e., non-Hermitian terms of type (iii). For that purpose, we add $i\lambda\Gamma_{d+1}$ to Eqs. (2) or (Topology and exceptional points of massive Dirac models with generic non-Hermitian perturbations), which is equivalent to assuming that the mass $M$ is complex. Hence, the energy spectrum is always complex independent of the boundary conditions (see SM Sup ). Thus, massive Dirac models perturbed by non-Hermitian mass terms are intrinsically non-Hermitian, both for open and periodic boundary conditions. Furthermore, we find that there are no exceptional points, not in the bulk and not in the surface band structure. Indeed, remarkably, topological boundary modes are unaffected by type-(iii) perturbations and keep their purely real energy spectra.

To demonstrate this explicitly, we solve for the boundary modes of Eq. (Topology and exceptional points of massive Dirac models with generic non-Hermitian perturbations) perturbed by $i\lambda\Gamma_{d+1}$. That is, we solve $(H_{\text{0}}(\tilde{\mathbf{k}})+\mathbb{I}_{N_{j}}\otimes i\lambda\Gamma_{d+1})|\psi_{\tilde{\mathbf{k}}}\rangle=E|\psi_{\tilde{\mathbf{k}}}\rangle$ with $|\psi_{\tilde{\mathbf{k}}}\rangle$ the ansatz for the right eigenvector of the boundary mode, $|\psi_{\tilde{\mathbf{k}}}\rangle=\sum_{i=1}^{N_{j}}\beta^{i}|i\rangle\otimes|\xi_{\tilde{\mathbf{k}}}\rangle$, where $|\xi_{\tilde{\mathbf{k}}}\rangle$ is a spinor, $i$ labels the latice sites along the $j$th direction, and $\beta$ is a scalar with $|\beta|<1$ (see SM Sup for details). By solving this Schrödinger equation we find that $\beta=M-\sum_{i}\cos\tilde{k}_{i}+i\lambda$ and that the boundary mode is an eigenstate of $i\Gamma_{d+1}\Gamma_{j}$ with eigenvalue $+1$. Hence, the projector $P$ onto the boundary space is given by $P=(1+i\Gamma_{d+1}\Gamma_{j})/2$ and the effective boundary Hamiltonian is obtained by ${H}_{b}(\tilde{\mathbf{k}})=P({H}_{0}(\tilde{\mathbf{k}})+i\lambda\Gamma_{d+1})P$, with $\tilde{\mathbf{k}}$ satisfying $|\beta|=|M-\sum_{i}\cos\tilde{k}_{i}+i\lambda|<1$. Since $\Gamma_{d+1}$ anti-commutes with $\Gamma_{j}$, the non-Hermitian perturbation $i\lambda\Gamma_{d+1}$ vanishes under the projection. Thus, the effective boundary Hamiltonian becomes ${H}_{b}(\tilde{\mathbf{k}})=\sum_{i\neq j}\sin k_{i}\gamma^{i}$, with $\gamma^{i}=P\Gamma_{i}P$, whose spectrum is manifestly real. We note that while the effective boundary Hamiltonian is not altered by the non-Hermitian mass term $i\lambda\Gamma_{d+1}$, the range of $\mathbf{\tilde{k}}$ in which the boundary modes exists is changed to $|M-\sum_{i}\cos\tilde{k}_{i}+i\lambda|<1$.

To illustrate these general considerations, we consider as an example the two-dimensional topological insulator with a non-Hermitian mass term, i.e., ${H}_{\text{TI}}(\mathbf{k})=\sin k_{x}\Gamma_{1}+\sin k_{y}\Gamma_{2}+(M-\cos k_{x}-\cos k_{y})\Gamma_{3}+i\lambda\Gamma_{3}$. As shown in Fig. 2, the spectrum of this non-Hermitian Hamiltonian is complex both with open and periodic boundary conditions. With OBCs there appear surface states within the region $|M-\cos k_{x}+i\lambda|<1$, whose spectrum is purely real.

Discussion.— In summary, we systematically investigated $d$-dimensional massive Dirac models on periodic lattices perturbed by general non-Hermitian terms. We find that there are three different types of non-Hermitian terms: (i) non-Hermitian anti-commuting terms, (ii) non-Hermitian kinetic terms, and (iii) non-Hermitian mass terms. Interestingly, we find a two-fold duality for the first two types of non-Hermitian perturbations: With open boundary conditions non-Hermitian anti-commuting terms give rise to intrinsic non-Hermiticity, while non-Hermitian kinetic terms lead to superficial non-Hermiticity. Vice versa, with periodic boundary conditions non-Hermitian anti-commuting terms induce superficial non-Hermiticity, while non-Hermitian kinetic terms generate intrinsic non-Hermiticity. Importantly, for the anti-commuting and kinetic terms the intrinsic non-Hermiticity manifests itself by exceptional points in the surface and bulk band structures, respectively. Non-Hermitian mass terms, in contrast, render the Hamiltonian always intrinsically non-Hermitian, independent of the boundary condition, but do not induce exceptional points in the band structure.

Our findings can be used as guiding principles for the design of applications in, e.g., photonic cavity arrays. For example, our analysis shows that topological Dirac systems perturbed by non-Hermitian mass terms exhibit robust surface states with a purely real spectrum [see Fig. 2(c)]. In photonic cavity arrays, these surface states provide robust channels for light propagation, which are protected against perturbations and disorder. Importantly, this property can be exploited for the design of efficient laser systems that are immune to disorder. That is, optically pumping the boundary of the cavity array induces single-mode lasing in the surface states, with high slope efficiency. This was recently demonstrated, in two-dimensional optical arrays of microresonators Bahari et al. (2017); Harari et al. (2018); Bandres et al. (2018). It follows from our analysis that these phenomena occur in a broader class of photonic band structures, namely in any topological Dirac system Chiu et al. (2016) perturbed by non-Hermitian mass terms.

Another possible application are photonic devices that utilize the exceptional points formed by the surface states of Dirac systems with non-Hermitian anti-commuting terms [see Fig. 1(b)]. While in non-periodic systems the creation of exceptional points requires fine tuning, the surface exceptional points of the discussed lattice Dirac systems are required to exist by topology. Moreover, these surface exceptional points are protected against noise and disorder and cannot be removed by symmetry preserving perturbations (see SM Sup for details). They could potentially be used for surface sensing.

We close by discussing several interesting directions for future studies. First, our approach can be generalized in a straightforward manner to non-periodic Dirac models and to periodic gapless Dirac models, i.e., to non-Hermitian periodic lattices with semi-metallic band structures. Second, it would be interesting to study the role of symmetries, specifically, how the symmetries constrain the form of the non-Hermitian terms. Third, it would be of great value to derive a general and exhaustive classification of exceptional points based on symmetry and topology Budich et al. (2019). This is particularly important in view of the numerous applications of exceptional points in photonic devices.

Acknowledgments.— This work is supported by the NSFC (Grant No. 11874201), the Fundamental Research Funds for the Central Universities (Grant No. 0204/14380119), and the GRF of Hong Kong (Grant No. HKU173057/17P).