3.4 Exceptional points
3.4.3 Green’s function at exceptional points
Here, we want to study the form of the single-particle Green’s function at exceptional points. We will see that the singular spectrum at exceptional points leads to higher-order poles in the Green’s function. Suppose a non-Hermitian Hamiltonian ˆH exhibits an exceptional point of order n, which is also supposed to be the dimension of the Hamiltonian. At the exceptional point, the Jordan normal form can be written as
ˆ S−1H ˆˆS = EEP EEP . .. EEP + 0 1 0 . .. . .. 1 0 = EEP1 + ˆN , (3.164)
with ˆN a nilpotent matrix, which vanishes to the nthpower as ˆNn= 0. The Green’s function is given by
ˆ G(E) = 1 E − ˆH = 1 ˆ S(E − EEP− ˆN ) ˆS−1 = ˆS λ (1 − λ ˆN ) ˆ S−1. (3.165)
with λ = E−E1
EP. The inverse of 1 − λ ˆN is given by 1 (1 − λ ˆN ) = n−1 X i=0 (λ ˆN ))i, (3.166)
which can be verified as
(1 − λ ˆN ) 1 (1 − λ ˆN ) = (1 − λ ˆN ) n−1 X i=0 (λ ˆN )i = 1, (3.167)
where we use the fact (λ ˆN )n= 0. Then the Green’s function is ˆ G(E) = ˆS λ (1 − λ ˆN ) ˆ S−1 = ˆSλ + λ2N + λˆ 3Nˆ2+ · · · + λnNˆn−1 ˆS−1 = 1 E − EEP + 1 (E − EEP)2 ˆH − E EP + 1 (E − EEP)3 ˆH − E EP 2 + · · · + 1 (E − EEP)n ˆH − E EP n−1 . (3.168) Thus at an nth order exceptional point, there will be an nth order pole of the Green’s function.
Such a high-order pole leads to many interesting physical phenomena. For instance, in non-Hermitian photonic systems, it is shown that a system near an EP can exhibit a non-Lorentzian frequency response and the peak intensity of the frequency response is enhanced significantly due to the high-order pole [152, 153]. The role of Green’s functions at the exceptional point is still awaiting to be explored in the context of topological phases of matter. We expect it will bring exotic and interesting phenomena, for instance, to local density of states, and transport properties.
3.5 Conclusions
We have systematically studied non-Hermitian topological gapped systems by classifying non-Hermitian Dirac Hamiltonians with generic non-Hermitian terms. We find that non-Hermitian terms can be classi- fied into three types, according to the Clifford algebra, (i) non-Hermitian anti-commuting terms, which anti-commute with the Dirac Hamiltonian, (ii) non-Hermitian kinetic terms, and (iii) non-Hermitian mass terms. We have revealed a two-fold duality for the d dimensional Dirac models with non-Hermitian terms of the first two types, depending on whether the non-Hermiticity can be removed by non-unitary similarity transformations with either periodic or open boundary conditions. The Dirac model with type (iii) non-Hermitian terms is intrinsically non-Hermitian as it cannot be transformed to be Her- mitian regardless of boundary conditions. Our theory has been applied to understanding the band topology and topological boundary states of two representative systems of two dimensional and four dimensional topological insulator systems.
We find that the boundary condition plays a key role in classifying different non-Hermitian terms. Different to Hermitian systems, the boundary conditions not only influence the boundary states but also the bulk states in non-Hermitian systems, i.e., the band spectrum can differ drastically with different
3.5 Conclusions
boundary conditions. To study their effect, the transfer matrix method is adopted for a generic tight- binding model, from which the role of boundary conditions is clarified.
As exceptional points are characteristic features of non-Hermitian systems, their mathematical origin and physical consequences are discussed comprehensively. For the Dirac models with type (i) and (ii) non-Hermitian terms, we find there are exceptional spheres of dimension (d − 2) in the surface and bulk band structures, respectively. 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 [117, 118, 136], 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 sensitive 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 technical point of view.
4
Chapter 4Non-Hermitian topological Weyl and Dirac
semimetals
4.1 Introduction
Besides in topological gapped systems, non-trivial band topology can arise also in gapless systems, such as topological semimetals and nodal superconductors. Therefore, it is natural to extend our discussion of non-Hermitian topological band theory to gapless systems. In topological semimetals, the topological invariants are usually defined with respect to gapless points, usually in terms of an integral along a surface enclosing these points [13]. The non-trivial topological invariants in the bulk will lead to exotic topological surface states at the boundary. The low-energy effective theory that describes the physics around the gapless point usually exhibits a quantum anomaly, which results from the breakdown of classical symmetry at the quantum level. The quantum anomaly is closely related with anomalous transport properties in topological semimetals. In this chapter, instead of developing a general theory for non-Hermitian topological gapless systems, we focus on two important cases, namely, the non-Hermitian Weyl and Dirac semimetals.
We start with the Lagrangian of L = ¯ψiγµ∂µ− m
ψ for the Dirac fields denoted by ψ in d spatial dimensions. Here ¯ψ = ψ†γ0, and µ = 0, 1, · · · , d labels the time and space dimensions. The d+1 gamma
matrices satisfy the Clifford algebra of {γµ, γν} = 2gµνwith g = diag(+1, −1, · · · ). The Dirac equation,
which was originally proposed by P.A.M. Dirac in 1928, can be obtained asiγµ∂µ− m
ψ = 0 from the Lagrangian by the Euler–Lagrange equation. This Dirac Lagrangian is adopted in high-energy physics as it is Lorentz-invariant in that under Lorentz transformations of Λ = exp(iθµνSµν) with
Sµν = 4i[γµ, γν], the Lagrangian remain the same. In 1929, Weyl noticed that in odd spatial dimensions, the Dirac equation can be further reduced by constructing an Hermitian matrix γ5 = ikγ0γ1· · · γdwith
k = (d − 1)/2. γ5 is guaranteed to commute with the kinetic terms of γ0γµ, which means that in the massless case (m = 0), they can be diagonalized simultaneously. Thus the chiral components of γ5ψ± = ±ψ± can be identified, and the Dirac Lagrangian can be further reduced. In three spatial
dimensions, by choosing the chiral representation of γ0= I ⊗ τxand γi= σi⊗ iτy, the Langrangian can
be decomposed as L = L++ L−= ψ†+σ¯µi∂µψ++ ψ−†σµi∂µψ−with σµ= (1, σi) and ¯σµ= (1, −σi). The
[154]. However, the Weyl fermions are hard to spot in high-energy physics, as the Lorentz invariance cannot be kept for the individual Weyl Lagrangians.
In condensed matter physics, it is possible to realize the Weyl fermions due to less restrictive sym- metry requirements and relatively low energy scale compared with high-energy physics. The Weyl fermion was first experimentally observed in Weyl semimetals [17]. In Weyl semimetals, the valance bands touch conductance bands and they form linear dispersing cones around the touching points. Excitations around these points can be described by Weyl Hamiltonians. This kind of Weyl fermions behaves like a monopole of Berry curvature in reciprocal space, and has a monopole charge of ±1, corresponding to the chirality of Weyl fermions. The non-trivial topology of the bulk manifests itself at the surface as exotic Fermi arc states. The Weyl semimetals also serve as fertile ground to discover new physics. Indeed, research on Weyl semimetals has lead to various fascinating phenomena, such as chiral magnetic effect and anomalous Hall effects [60, 63, 64, 72, 155–158]. In this chapter, we introduce different non-Hermitian potentials into Weyl semimetals. We pay special attention to the boundary conditions, as they play a significant role in non-Hermitian systems. We find there are two kinds of non-Hermitian potentials, the non-Hermitian kinetic terms and mass terms. For each kind of the non-Hermitian terms, we focus on their topological properties and topological surface states subject to different slab geometries.
We will further discuss the non-Hermitian topological Dirac semimetals in three dimensions. In Hermitian Dirac materials, around band degeneracy points (Dirac points), low energy quasiparticle excitations that can be described by Dirac equation in relativistic quantum mechanics have been spot- ted [159–162]. Beyond fundamental interest and practical value in itself, Dirac points play a key role as the bridge between different topological phases in the topological band theory [10, 13, 163]. The investigation of non-Hermitian Dirac semimetals therefore will pave the way for research on different non-Hermitian topological phases. We will especially focus on PT symmetric-non-Hermitian Dirac semimetals. This is of fundamental interest, as a recent revolutionary development in quantum me- chanics is using the PT symmetry, which guarantees real spectra in the PT -unbroken case, as an alternative to Hermitictiy [32, 33, 164? , 165]. Though experimentally challenging at first, it is later discovered that photonics offers an ideal platform for PT -symmetric quantum theory, where non- Hermitian components are naturally introduced by optical radiation and loss/gain [95, 96, 98, 99, 143]. For PT -symmetric Dirac semimetals, we find there are three kinds of PT -symmetric non-Hermitian potentials. For each of them, we will discuss their spectra under differernt boundary conditions, the bulk topology, and the bulk boundary correspondence.