# Classification of massive Dirac models with generic non-Hermitian perturbations

###### Abstract

We present a systematic investigation of -dimensional massive Dirac models perturbed by three different types of non-Hermitian terms: (i) non-Hermitian terms that anti-commute with the Dirac Hamiltonian, (ii) non-Hermitian kinetic terms, and (iii) non-Hermitian mass terms. We show that these perturbations render the Hamiltonian either intrinsically or superficially non-Hermitian, depending on whether the non-Hermiticity can be removed by non-unitary similarity transformations. A two-fold duality is revealed for the first two types of non-Hermitian perturbations: With open boundary conditions non-Hermitian terms of type (i) give rise to intrinsic non-Hermiticity, while terms of type (ii) lead to superficial non-Hermiticity. Vice versa, with periodic boundary conditions type-(i) perturbations induce superficial non-Hermiticity, while type-(ii) perturbations generate intrinsic non-Hermiticity. Importantly, for the type-(i) and type-(ii) terms the intrinsic non-Hermiticity manifests itself by exceptional spheres of dimension in the surface and bulk band structures, respectively. Type-(iii) perturbations, in contrast, render the Hamiltonian always intrinsically non-Hermitian, independent of the boundary condition, but do not induce exceptional spheres in the band structure. For each of the three perturbations we study the band topology, discuss the topological surface states, and briefly mention the relevance for potential applications.

###### pacs:

The fields of non-Hermitian physics and topological materials have recently intertwined to create the new research direction of non-Hermitian topological phases. As a result of the joint efforts from both fields, fascinating new discoveries have been made, both at the fundamental level and with respect to applications Chen et al. (2017); Bandres et al. (2018); Harari et al. (2018); Feng et al. (2014); Hodaei et al. (2014); Goldzak et al. (2018); Weimann et al. (2017); Cerjan et al. (2018a); Leykam et al. (2017); Pan et al. (2018); Gong et al. (2018a); Lee (2016); Martinez Alvarez et al. (2018); Yao and Wang (2018); Lee and Thomale (2018); Shen et al. (2018); Yao et al. (2018a); Kunst et al. (2018); Kawabata et al. (2018a); Zhen et al. (2015); Zhou et al. (2018); Xu et al. (2017). For instance, topological exceptional points have been found in one-dimensional non-Hermitian lattices Lee (2016); Martinez Alvarez et al. (2018); Yao and Wang (2018); Lee and Thomale (2018) and in non-Hermitian Chern insulators Shen et al. (2018); Yao et al. (2018a); Kawabata et al. (2018a); Kunst et al. (2018). Exceptional rings and bulk Fermi arcs have been discovered in non-Hermitian topological semimetals Zhen et al. (2015); Zhou et al. (2018); Xu et al. (2017); Papaj et al. (2018); CarlstrÃ¶m and Bergholtz (2018); Yang and Hu (2018); Cerjan et al. (2018b). At these exceptional points and rings, two or more eigenstates become identical and self-orthogonal, leading to a defective Hamiltonian with nontrivial Jordan normal form Heiss (2012). These exceptional manifolds have many interesting applications, e.g., enhanced sensitivity of microcavity sensors Chen et al. (2017), single-mode lasing of photonic devices Bandres et al. (2018); Harari et al. (2018); Feng et al. (2014); Hodaei et al. (2014), and stopping of light in coupled optical waveguides Goldzak et al. (2018). Furthermore, it has been shown that non-Hermitian topological Hamiltonians provide useful descriptions of strongly correlated materials in the presence of disorder or dissipation Zyuzin and Zyuzin (2018); Kozii and Fu (2017); Yoshida et al. (2018); LourenÃ§o et al. (2018, 2018); Nakagawa et al. (2018); Kawabata et al. (2018b); Avila et al. (2018); Shen and Fu (2018); Harrison (2018). This has given new insights into the Majorana physics of semiconductor-superconductor nanowires Avila et al. (2018) and into the quantum oscillations of SmB Shen and Fu (2018); Harrison (2018).

Despite these recent activities, a general framework for the study and the complete classification of non-Hermitian topological phases is still absent. In particular, the formulation of bulk topological invariants, the associated bulk-boundary correspondence, and the role of boundary conditions are still unclear for non-Hermitian topological phases, although various attempts have been made with partial successes 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. (2018a, b); Gong et al. (2018b); Zhou and Lee (2018). Since most non-Hermitian experimental systems can be faithfully captured by Dirac Hamiltonians 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); RÃ¼ter et al. (2010); Feng et al. (2017), a systematic investigation of non-Hermitian Dirac models would be particularly valuable. This would be not only of fundamental interest, but could also inform the design of new applications.

In this Letter, we present a systematic investigation of -dimensional massive Dirac Hamiltonians perturbed by small non-Hermitian terms. We show that these can be either intrinsically or superficially non-Hermitian, depending on whether the non-Hermiticity can be removed by a similarity transformation with open or periodic boundary conditions. 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 -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.

General formalism.— 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, , by any invertible matrix , which is not necessarily unitary but is required to be local. For this reason, a large class of non-Hermitian Hamiltonians can be converted into Hermitian ones by non-unitary similarity transformations, i.e.,

(1) |

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, is a quadratic form, whose
entries are specified as with the positions of the unit cells and a label for internal degrees of freedom. Correspondingly, the similarity transformation has matrix elements .
By the locality condition, the matrix elements and
are required to tend to zero sufficiently fast as .
If can be converted into a Hermitian Hamiltonian by a local transformation , its eigenvalues are necessarily real.
Conversely, any local lattice Hamiltonian with real spectrum is either entirely Hermitian or superficially non-Hermitian
^{1}^{1}1We note that, for instance, space-time symmetric (i.e., 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 one or multiple eigenvalues become identical, leading to a non-diagonalizable Hamiltonian. However, it is important to note that such exceptional points are not dense in parameter space. I.e., there exist arbitrarily small perturbations which remove the exceptional points, rendering the Hamiltonian diagonalizable. One such perturbation relevant for lattice models are the boundary conditions Kunst et al. (2018); Kunst and Dwivedi (2018), 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 . Here, 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 , , is not local in general.

Non-Hermitian Dirac Hamiltonians.— We now apply the above concepts to non-Hermitian Dirac models of the form , where is a Hermitian Dirac Hamiltonian with mass , and a non-Hermitian perturbation with . Assuming PBCs in all directions, we consider the following Hermitian Dirac Hamiltonian on the -dimensional cubic lattice

(2) |

where denote the gamma matrices that satisfy and is the real mass parameter. With OBCs in the th direction and PBCs in all other directions, the Hamiltonian reads

where denotes the vector of all momenta except , is the right-translational operator, and stands for the number of layers in the 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 in the introductions. We will now study these individually.

Non-Hermitian terms of type (i).— We start with non-Hermitian terms that anti-commute with the Dirac Hamiltonian . 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 . With PBCs the Hamiltonian perturbed by these type-(i) terms is expressed as

(4) |

where is an additional gamma matrix with and a real parameter. The spectrum of is given by with , which is completely real for all , provided that is smaller than the energy gap of . Thus, Hamiltonian (4) with is only superficially non-Hermitian and we can remove the non-Hermitian term by a similarity transformation. The corresponding transformation matrix can be derived systematically by noticing that the flattened Hamiltonian and form a Clifford algebra and, thus, generates rotations of the plane spanned by and . Hence, the explicit expression of the transformation matrix is , with . From Eq. (1) it follows that the transformed Hamiltonian is , which is manifestly Hermitian for .

With OBCs, on the other hand, type-(i) perturbations lead to intrinsic non-Hermiticity, provided the Dirac Hamiltonian is in the topological phase. This is because the topological boundary modes acquire complex spectra due to the non-Hermitian term , even for infinitesimally small . To see this, we first observe that for any eigenstate of with energy , is also an eigenstate of , but with opposite energy . Applying chiral perturbation theory to , we find that since scatters into , eigenstates of can be expresses as superpositions of and . Explicitly, we find that the eigenstates of are , with and energy . This analysis holds in particular also for the topological boundary modes of , which are massless Dirac fermions with linear dispersions. Consequently, even for arbitrarily small , there exists a segment in the spectrum of around 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 , where the first term describes the boundary massless Dirac fermions of . The matrices and are the projections of and , respectively, onto the boundary space, and satisfy . With this, we find that the boundary spectrum is , and that there exists a -dimensional exceptional sphere of radius 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 cannot remove the topological surface state. The reason for this is that 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 is independent of , i.e., with and the left and right eigenstates of , respectively.

Let us now illustrate the above general considerations by considering as an example, , with the 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 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 .

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

(5) |

with and real. The energy spectrum of , , is complex and exhibits exceptional points on the -dimensional sphere within the plane. Hence, with PBCs is intrinsically non-Hermitian.

To study the case of OBCs we consider in a slab geometry with surface perpendicular to the th direction. The energy spectrum in this geometry is obtained from , i.e., by replacing by in Eq. (5). Then, it is obvious that the non-Hermitian term can be removed by the similarity transformation . That is, , which is manifestly Hermitian. Accordingly, has real spectrum and its eigenstates are related to those of by . 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. (Classification of massive Dirac models with generic non-Hermitian perturbations) perturbed by the non-Hermitian term , i.e., . This Hamiltonian can be transformed to (see SM Sup () for details)

(6) |

by the similarity transformation Here and . Eq. (6) is manifestly Hermitian for . As a concrete example, we set in Eq. (6) with 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 (d), respectively.

Non-Hermitian terms of type (iii).— Finally, we examine the effects of non-Hermitian mass terms. For that purpose, we add to Eqs. (2) or (Classification of massive Dirac models with generic non-Hermitian perturbations), which is equivalent to assuming that the mass 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. (Classification of massive Dirac models with generic non-Hermitian perturbations) perturbed by . I.e., we solve with the ansatz for the left eigenvector of the boundary mode, , where is a spinor, labels the latice sites along the th direction, and is a scalar with (see SM Sup () for details). By solving this Schrödinger equation we find that and that the boundary mode is an eigenstate of with eigenvalue . Hence, the projector onto the boundary space is given by and the effective boundary Hamiltonian is obtained by , with satisfying . Since anti-commutes with , the non-Hermitian perturbation vanishes under the projection. Thus, the effective boundary Hamiltonian becomes , with , whose spectrum is manifestly real. We note that while the effective boundary Hamiltonian is not altered by the non-Hermitian mass term , the range of in which the boundary modes exists is changed to .

To illustrate these general considerations, we consider as an example the two-dimensional topological insulator with a non-Hermitian mass term, i.e., . 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 , whose spectrum is purely real.

Conclusions and Discussions.— In summary, we systematically investigated non-Hermitian massive Dirac models with three different types of non-Hermitian terms. We find that there is a two-fold duality for the first two types of terms, which lead either to superficial or intrinsic non-Hermiticity, depending on the boundary conditions. Moreover, the first and second type of terms give rise to exceptional points in the surface and bulk band structures, respectively. Terms of the third type, however, always induce intrinsic non-Hermiticity, but do not produce exceptional points. Our findings can be used as guiding principles for the design of applications in, e.g., photonic devices. For example, our analysis shows that single mode lasing Bandres et al. (2018); Harari et al. (2018); Feng et al. (2014); Hodaei et al. (2014), which utilizes bulk exceptional points, is only possible in Dirac models perturbed by the second type of non-Hermitian terms. Sensors, on the other hand, which make use of surface exceptional points, can be designed using Dirac models with the first type of non-Hermitian terms. We emphasize, that the exceptional points are extremely sensitivity to the boundary conditions, as they are bare (i.e., not dense) in parameter space. That is, infinitesimally small perturbations can change them into regular points, a fact that can be exploited for sensor applications. This important fact deserves further investigations, both from a fundamental and applications point of view.

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See supplemental See supplemental See supplemental