Weyl points and exceptional rings with polaritons in bulk semiconductors
WWeyl points and exceptional rings with polaritons in bulk semiconductors
R. L. Mc Guinness and P. R. Eastham
School of Physics and CRANN, Trinity College Dublin, Dublin 2, Ireland. (Dated: June 12, 2020)Weyl points are the simplest topologically-protected degeneracy in a three-dimensional dispersionrelation. The realization of Weyl semimetals in photonic crystals has allowed these singularities andtheir consequences to be explored with electromagnetic waves. However, it is difficult to achievenonlinearities in such systems. One promising approach is to use the strong-coupling of photonsand excitons, because the resulting polaritons interact through their exciton component. Yet topo-logical polaritons have only been realized in two dimensions. Here, we predict that the dispersionrelation for polaritons in three dimensions, in a bulk semiconductor with an applied magnetic field,contains Weyl points and Weyl line nodes. We show that absorption converts these Weyl pointsto Weyl exceptional rings. We conclude that bulk semiconductors are a promising system in whichto investigate topological photonics in three dimensions, and the effects of dissipation, gain, andnonlinearity.
INTRODUCTION
Degeneracies in bandstructures are a key concept atthe heart of recent developments in condensed-matterphysics and optics [1]. Two-dimensional materials suchas graphene posesss Dirac points, where the disper-sion is locally linear, which are responsible for many oftheir unique properties. In three-dimensional materialsWeyl points have been found in photonic [2] and elec-tronic [3] bandstructures, providing low-energy modelsof Weyl fermions. More generally, topological consider-ations mean that materials hosting degeneracies are thebasis for realizing topological insulators and related ef-fects such as robust edge modes. Such work is now alsobeing extended to dissipative systems, such as photonicmaterials with gain and loss, described by non-HermitianHamiltonians [4]. In this case the singularities includeexceptional points [5, 6] in parameter space, at whichboth the frequencies and lifetimes of the modes becomedegenerate. Rings of such exceptional points have beenshown to emerge from Dirac points in photonic crystals[7]. In the three-dimensional case, Weyl points can be-come Weyl exceptional rings [8], which have a quantizedChern number and a quantized Berry phase. Like theircounterparts in Hermitian systems, such non-Hermitiansingularities give rise to interesting physical effects [9],including edge modes [10], unusual transmission proper-ties, topological lasing, and Fermi arcs arising from half-integer topological charge [11].Polaritons are exciton-photon superpositions that areformed by strong light-matter coupling in semiconduc-tors [12, 13]. Their half-matter half-light nature impliesrelatively strong nonlinearities, and this feature amongothers makes them an interesting system in which tostudy topological effects. Topological phases have beenpredicted [14–19] and observed [20] for polaritons formedfrom quantum-well excitons coupled to photons confinedin microcavities. Topological lasing [21] and exceptionalpoints [22] have also been studied. However, as micro- cavities and quantum-wells are two-dimensional systems,phenomena such as Weyl points, Fermi arcs, and thethree-dimensional topological phases [1], have not beenconsidered.In this paper, we report topologically non-trivial dis-persion relations for polaritons propagating in three di-mensions. We consider a bulk semiconductor in a mag-netic field, and show that the p-type structure of thevalence band leads to intricate dispersion relations con-taining topologically-protected degeneracies. In the ab-sence of non-radiative losses there are eight sheets of thedispersion surface, which host Weyl points [2, 3, 23–25],for wavevectors along the field direction, and ring de-generacies, for wavevectors transverse to it. In the non-Hermitian case [4, 5, 10], with absorption, we show thatthe Weyl points become Weyl exceptional rings, whichcan be reached by tuning the frequency and the anglebetween the propagation direction and the applied field.These results show that bulk semiconductors could beused to study topological effects in three spatial dimen-sions. Furthermore, bulk polariton lifetimes can be longsince, unlike microcavity polaritons, they are not subjectto radiative decay. They may therefore give access to thestrongly-interacting regime of topological photonics [1].
METHODExciton spectra
We consider polaritons formed from 1s excitons in di-rect band-gap zinc-blende semiconductors such as GaAs.These involve p-type valence band states with Γ symme-try, and s-type conduction band states with Γ symme-try. The combinations of the hole spin m h = ± / , ± / and the electron spin m s = ± / then give rise to eightexciton spin states, denoted | X n (cid:105) for n = 1 , . . . , withenergies E n .To evaluate the polariton spectrum we need the en- a r X i v : . [ c ond - m a t . m e s - h a ll ] J un ergies and polarizations of the exciton transitions. Toobtain these we diagonalize the effective Hamiltonianfor the 1s excitons given in Ref. [26]. The parametersin this effective Hamiltonian are related to the underly-ing electron-hole exchange parameters, Luttinger param-eters, and g-factors. This approach treats the valence-band anisotropy, magnetic field, and electron-hole ex-change as perturbations on a spherically-symmetricelectron-hole Hamiltonian [27]. The unperturbed wave-function is of the usual hydrogenic form, with the bind-ing energy R = µe / π (cid:15) (cid:15) ¯ h and Bohr radius a =4 π(cid:15) (cid:15) ¯ h /µe , where µ − = m − c + m − v . m v = m /γ is the isotropic part of the effective mass for the valenceband, related to the Luttinger parameter γ , and m c isthe effective mass for the conduction band. For this per-turbative approach to be valid the cyclotron energy mustbe small compared with the exciton binding energy R .We take the specific criterion given by Altarelli and Li-pari [27], γ = ¯ hω c R = e ¯ hB µR ≤ . , (1)to define the maximum field B max of the perturbativeregime. In the following we will consider the specificcase of GaAs, with applied field B max in the [001] direc-tion, using the bandstructure parameters from Ref. [28].For the electron-hole exchange parameters [26] we take ∆ = − . µ eV [29], and ∆ = ∆ = 0 . The exci-ton spectrum computed for these parameters is shownin Fig. 1(a). As expected, the magnetic field lifts thedegeneracies of the eight electron hole pair states. Thissplitting of the energies of the excitons will result in ananisotropic and multiply resonant optical susceptibility,and hence a direction and polarization dependent polari-ton dispersion. Polariton Hamiltonian
The topological singularities of the polariton dispersionarise from the polarization dependence of the exciton-photon coupling. In the Coulomb gauge the interactionbetween the vector potential and the electrons, from theHamiltonian (cid:80) i [ˆ p i + e ˆ A (ˆ r i )] / (2 m ) , is ˆ H ep = em (cid:88) i (cid:88) k ,s (cid:114) ¯ h (cid:15) ωV (cid:2) ˆ a k ,s e k ,s e i k . ˆ r i + H . c . (cid:3) · ˆ p i . (2)Here the first sum is over the electrons, and the sec-ond over the photon wavevectors, k , and polarizations, s , with corresponding polarization vectors e k ,s . ˆ a k ,s isthe photon annihilation operator, ω = c | k | the photonfrequency, and V a quantization volume. Thus we havethe second-quantized Hamiltonian in the subspace of the (a) (b) - - - - - - z ( T ) E - E g ( m e V ) σ + σ + σ - σ - X X X X X X z z3/2 1/2 1/2-1/2-1/2 -3/2m h m s FIG. 1. (a) Calculated 1s exciton energies relative to thebandgap for GaAs with a magnetic field B z ∈ [0 , B max ] in the [001] direction. The line coloring indicates the polarizationof each transition: right-circular (red/ σ − ) and left-circular(blue/ σ + ), with the field in the xy plane, or linear in thez direction (green/z). The black curves are the spin-2 darkexcitons. (b) Polarization and spin structure of the excitontransitions, in terms of the hole spin, m h , and electron spin, m s . eight 1s exciton states, | X k ,n (cid:105) , ˆ H xp = em (cid:88) k ,s,n (cid:114) ¯ h (cid:15) ωV (cid:2) ˆ X † k ,n ˆ a k ,s e k ,s · (cid:104) X k ,n | ˆ p | (cid:105) + H . c . (cid:3) , (3)where we have made the rotating-wave approximation.In the envelope function approximation the matrix el-ements appearing in Eq. (3) are products of the matrixelements of the Bloch functions at k = 0 and the hydro-genic exciton wavefunctions χ m s ,m h F n ( r = 0) . For thespatial part of the latter we take the unperturbed result | F n (0) | ≈ /πa . For the spin part χ m s ,m h we note thatat the field B max we are considering the Zeeman termsdominate over the electron-hole exchange. Thus the ex-citons are, to a good approximation, diagonal in the spinprojections m s and m h . Using the standard forms for thevalence band wavefunctions [30] and the Kane parameter P [31–33] we then have (cid:104) X n | ˆ p | (cid:105) = √ V mF ∗ n (0) | P | v n / ¯ h ,where v = v = and v = 1 √ i v = (cid:114) i v = 1 √ − i v = 1 √ i v = (cid:114) i v = 1 √ − i . (4)The states | X (cid:105) . . . | X (cid:105) correspond to excitons with elec-tron spin m s = / and hole spin m h = / , / , − / , − / ,respectively, while | X (cid:105) . . . | X (cid:105) are the correspondingstates with m s = − / . Thus | X (cid:105) and | X (cid:105) are darkstates, and the remaining transitions are either circularly-polarized in the xy plane, or linearly polarized in the zdirection, as shown in Figs. 1(a) and 1(b).Using these exciton matrix elements in Eq. (3), andapproximating ω ≈ E g / ¯ h in the prefactor of the coupling,we find for the exciton-photon Hamiltonian, ˆ H = (cid:88) k (cid:26) (cid:88) s ¯ hck √ (cid:15) ˆ a † k ,s ˆ a k ,s + (cid:88) n E n ˆ X † k ,n ˆ X k ,n + ¯ h Ω (cid:88) s,n ( e k ,s · v n ) ˆ X † k ,n ˆ a k ,s + H . c . (cid:27) , (5)where k = | k | and we use a Rabi splitting ¯ h Ω = (cid:115) e | P | (cid:15) (cid:15)E g πa to quantify the light-matter coupling in the material. Polariton spectra
In the following we will consider the polariton spec-trum, which we obtain from the Heisenberg equations-of-motion for the exciton and photon annihilation opera-tors by looking for solutions with time dependence e − iωt ,i.e., setting ˆ a ( t ) = e − iωt ˆ a (0) and similarly for the exci-ton operators. We specify the wavevector direction interms of the polar coordinates θ, φ , with the field direc-tion and the [001] crystal axis correponding to θ = 0 . Forthe photon polarization we use the circularly-polarizedstates, e k , ± = ( e k ,θ ± i e k ,φ ) / √ , constructed from thelinearly-polarized basis transverse to k , e k ,θ = cos θ cos φ cos θ sin φ − sin θ , e k ,φ = − sin φ cos φ . (6)Since there are two photon polarizations and six excitons(discounting the irrelevant dark states), this proceduregives an eight-by-eight Hamiltonian matrix, H , with el-ements dependent on the wavevector magnitude and di-rection. We determine the polariton dispersion ω ( k, θ, φ ) by finding the eigenvalues, ω , of H numerically.While the dispersion ω ( k, θ, φ ) is given by the eigen-values of H , there is another approach to analyzingthe topological singularities of the polariton spectrum, interms of the function k ( ω, θ, φ ) . This latter function pro-vides a natural description of optics at a fixed frequency,and is related to constructs such as the refractive indexsurface of classical optics [34–36]. For example, a radialplot of k at some fixed frequency over angles gives a con-tour (in k-space) of the dispersion relation ω ( k ) . Thenormal to such an isofrequency surface is therefore the group velocity, ∇ k ω , controlling the refraction directionat that frequency. While the two functions ω ( k, θ, φ ) and k ( ω, θ, φ ) are equivalent in the absence of dissipation, weshall see that they have some differences in its presence,and we therefore consider both representations in the fol-lowing.To obtain the dispersion in the form k ( ω, θ, φ ) weeliminate the exciton amplitudes from the Heisenbergequations-of-motion. This leads to a two-dimensionaleigenproblem for the photon amplitudes, (cid:20) ( ω − ck/ √ (cid:15) ) δ ss (cid:48) − Ω (cid:88) n ( e ∗ k ,s · v ∗ n ) . ( e k ,s (cid:48) · v n ) ω − E n / ¯ h (cid:21) ˆ a k ,s (cid:48) (0) = 0 , (7)so that the magnitudes of the wavevectors are the eigen-values of a two-by-two matrix, whose elements are func-tions of the frequency and propagation direction.It is useful to note that this form, Eq. (7), can alsobe derived semiclassically, by looking for plane-wave so-lutions to Maxwell’s equations, including the excitonicresonances via a frequency-dependent dielectric function (cid:15) ( ω ) . In an optically isotropic material (cid:15) ( ω ) is a scalar,and the polariton dispersion satisfies c k /ω = (cid:15) ( ω ) [37]. In the present case, however, the optical responseis anisotropic due to the magnetic field, and we mustconsider the vector equation − k × k × E = ω c (cid:15) ( ω ) E . (8)Longitudinal modes, with k (cid:107) E , occur if (cid:15) ( ω ) = 0 . Toobtain the equation for the transverse modes we takematrix elements of Eq. (8) in a basis perpendicular to ˆ k , such as e k , ± . This eliminates the zero eigenvalue ofthe operator k × k × , i.e. the longitudinal polariton, andgives (cid:20) ( ω − c k /(cid:15) ) δ ss (cid:48) + ω (cid:15) χ ss (cid:48) (cid:21) E k ,s (cid:48) = 0 , (9)where χ ss (cid:48) ( ω ) = e † k ,s χ ( ω ) e k ,s (cid:48) is the transverse part ofthe excitonic susceptibility and (cid:15) the background permit-tivity. We approximate the prefactors in this expressionas ( ω − c k /(cid:15) ) ≈ ω ( ω − ck/ √ (cid:15) ) and ω /(cid:15) ≈ ωE g / ¯ h(cid:15) .Comparing this expression with Eq. (7) we see that thefinal term in the latter is related to the susceptibility by χ ss (cid:48) = − Ω ¯ h(cid:15) E g (cid:88) n ( e ∗ k ,s · v ∗ n ) . ( e k ,s (cid:48) · v n ) ω − E n / ¯ h . (10)The spectrum k ( ω, θ, φ ) can be found straightforwardlyby solving the secular equation for Eq. (7), which is aquadratic in k . It may be noted that χ ss (cid:48) , and hence thepolariton spectrum, is independent of φ . This reflects therotational symmetry of the problem about the magneticfield ( θ = 0 ). The combination of the form of Eq. (7)with that of the χ ss (cid:48) imposes an additional symmetrybetween the solutions at θ and those at π − θ . We maytherefore set φ = 0 and consider the interval θ ∈ [0 , π ] .We note that whereas the secular equation for Eq. (7) is aquadratic in k , that for H is an eighth-order polynomialin ω . Thus there are, in general, two wavevectors for eachfrequency, from the two dispersing photon modes. Thereare however eight frequencies for each wavevector, fromthose two photon modes as well as the six non-dispersingbright excitons. RESULTSTopological singularities: Hermitian case
Our primary interest is in the degeneracy structure ofthe magneto-exciton-polariton dispersion relation, whichwe first consider in the Hermitian case without dissipa-tion. It is possible to make some observations that con-strain the possible degeneracies based on the symmetryof the problem. Owing to the φ invariance of the solu-tions we know that propagation in the θ = 0 directionis the only configuration for which isolated degeneraciesare possible. Correspondingly, if degeneracies occur atany non-zero θ they are necessarily extended degenera-cies over all φ .In Fig. 2 we plot the dispersion of the transverse modes,obtained from the secular equation for Eq. (7). Thethree panels refer to propagation along the field, θ = 0 ,at a small angle to it, θ = π/ , and perpendicular to it, θ = π/ . The polarization of the modes is shown by thecoloring. Energy is measured relative to the bandgap E g and in units of the exciton Rydberg energy. The wavectoris measured relative to k = √ (cid:15) ( E g − R ) / ¯ hc , which is thewavevector at which the bare linear photonic dispersionwould cross an unperturbed exciton. The wavevector ismeasured in units of the inverse exciton Bohr radius.The spectrum for propagation along the z-axis is shownin Fig. 2(a). In this case the two z-polarized excitons, X and X , do not couple to light, and there are only sixmodes in the transverse spectrum. The other excitonsare circularly polarized, and each circular polarizationof light mixes with the two excitons of that polariza-tion. This gives rise to a spectrum with a lower, inter-mediate, and upper branch for each circular polarization.The lower branch begins at low energy as a purely pho-tonic, linearly dispersing, mode, which anticrosses withthe lower energy exciton, asymptoting horizontally to ap-proach that exciton energy at large k . Above that energythere is an intermediate branch, which initially has animaginary k as it lies in the polaritonic (longitudinal-transverse) gap [13]. This mode then becomes a propa-gating polariton with k = 0 at the gap edge, and thenapproaches the higher exciton energy at large k . Above -1.1-1-0.9-0.8-0.7(a) θ =0 E - E g ( R ) -1.1-1-0.9-0.8-0.7(b) θ = π /8 E - E g ( R ) -1-0.5 0 0.5 1 ( I + - I - ) /I -1.1-1-0.9-0.8-0.7-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4(c) θ = π /2 E - E g ( R ) k-k (a ) -1-0.5 0 0.5 1 ( I θ - I ϕ ) /I FIG. 2. (a) Solid lines: polariton dispersion relations for GaAsin a magnetic field for θ = 0 . The coloring shows the degree ofcircular polarization. Dotted lines: energies of the circularly-polarized excitons. Dashed lines: energies of the z-polarizedexcitons. (b) Polariton dispersion for θ = π/ . (c) Polaritondispersion for θ = π/ . The coloring here shows the degreeof linear polarization. Energies are relative to the bandgap inunits of the exciton Rydberg, and wavevectors relative to k in units of the exciton Bohr radius. this there is an upper branch, which again begins as a so-lution with imaginary k , before becoming a propagatingsolution, and finally approaching the photon dispersionat large k .Fig. 2(b) shows the spectrum at a small angle to thez-axis. Comparing this spectrum with that in Fig. 2(a)we see that there are now eight branches, because thez-polarized excitons now couple to light. Moreover wefind that this spectrum is gapped, with avoided crossingswhich orginate from the degeneracies at θ = 0 . The twodegeneracies in Fig. 2(a) between the different circularmodes have split. In addition we see that the three inter-sections between the z-polarized excitons and the trans-verse modes at θ = 0 have split, to organize into the twoadditional transverse branches at θ (cid:54) = 0 . There is in fact FIG. 3. Polariton dispersion relation for GaAs in a magneticfield, with k y = 0 , centered around propagation along themagnetic field direction, z. Two Weyl points, with a lineardispersion in all directions, and a band touching with mixedquadratic/linear dispersion, can be clearly seen. a fourth intersection of this nature, involving the highestenergy z-polarized exciton, but it is with an evanescentmode at an imaginary k .Figure 3 shows the dispersion relation near to someof these singularities. One of the degeneracies betweenthe two circular polarization modes seen in Fig. 2(a) liesbetween the second and third sheets in Fig. 3, count-ing from low energies. It can be seen to split linearlyin k z but quadratically in k x – and, therefore, also in k y , due to the rotational symmetry. The same structureappears for the other crossing of the circular modes at θ = 0 (not shown). Such a dispersion implies these de-generacies have zero topological charge. However, Fig. 3also shows three crossings involving the z-polarized ex-citons, which can be seen to disperse linearly in k x , andtherefore k y , as well as k z . They are thus isolated Weylpoints of the polariton dispersion relation, and are topo-logically non-trivial. Two Weyl points are clearly visible,between the third and fourth sheets, and between thefourth and fifth sheets. A third lies between the secondand third sheets to the right of the quadratic degener-acy in the figure, but is hidden by the perspective. Thefinal, fourth, degeneracy between a transverse polaritonand a z-polarized exciton at θ = 0 , lying in one of the po-laritonic gaps, is also a Weyl point, but at an imaginary k . The dispersion perpendicular to the applied field ( θ = π/ ) is shown in Fig. 2(c). As indicated by the color-ing, in this case the transverse modes are purely linearlypolarized, along the polar vectors e θ = − z and e φ . Con-sidering this geometry we identify two further degenera-cies in the polariton dispersion, where modes with thesetwo polarizations cross. These are both extended ringdegeneracies, due to the φ invariance of the system.We see that there are, in total, eight distinct degen-eracies of the polariton dispersion relation in the re-gion ≤ θ ≤ π/ . Of these eight degeneracies sixare isolated degeneracies occurring in the θ = 0 direc-tion and two are extended degeneracies occurring in the θ = π/ plane. The six isolated degeneracies divide intofour Weyl points, one of which is at an imaginary k ,and two topologically trivial degeneracies with a mixedquadratic/linear dispersion.The Weyl points are degeneracies between the z-polarized excitons and the xy-polarized polaritons at θ = 0 . To see why this gives a Weyl point, with a lin-ear dispersion, we note that the coupling between suchmodes near to the degeneracy is proportional to sin( θ ) [see Eqs. (4,5,6)]. Thus moving away from θ = 0 there isa splitting of the degeneracy which is linear in sin θ ≈ θ and hence linear in k x (or k y ). Formally, the Hamiltonianfor the two modes near the degeneracy takes the form (cid:18) c (cid:48) ( k z − k ) + ω Ω sin( θ ) /
2Ω sin( θ ) / ω x (cid:19) (11)which, with sin( θ ) ≈ k x /k (for k y =0), gives a linear dis-persion in k x and k z − k at the degeneracy ω = ω x . Ω isthe strength of the coupling between the z-polarized ex-citon and the polariton, involving the amplitude, in thepolariton, of the e θ -polarized photon. k is the wavevec-tor at the degeneracy, ω = ω x = c (cid:48) k the frequency, and c (cid:48) the velocity. Notably, these Weyl points lie at the crit-ical tilt between a type-I and a type-II point [38], and assuch are the three-dimensional (Weyl) generalization ofthe recently achieved type-III Dirac point [39]. Exceptional points in the dispersion relation
In the presence of damping the Hamiltonian, H ,becomes non-Hermitian, and the polariton dispersion, ω ( k, θ, φ ) , can contain rings of exceptional points aris-ing from the Weyl points described above. This canbe seen by considering the local Hamiltonian, Eq. (11),for one of the Weyl points. In the presence of dampingwe have ω ,x → ω ,x + iγ ,x , and the frequency differ-ence of the two coupled modes, at the bare resonance,becomes (cid:113) Ω sin θ − ( γ − γ x ) . Thus the real (imagi-nary) parts of the spectrum split for angles greater (less)than θ = arcsin | γ − γ x | / Ω . This occurs at all φ , so thatwe have a ring of exceptional points, where both the realand imaginary parts of the polariton energies are degen-erate, and the Hamiltonian matrix is defective.In Fig. 4 we show an exceptional point of this type, inthe spectrum of the full Hamiltonian, H . Figs. 4(a) and(b) show the real and imaginary parts, respectively, ofthe energies, and the expected local structure around anexceptional point [6] is clearly visible. From the anal-ysis above the critical angle for the exceptional pointdepends on the difference in the damping constants, sothis should be non-zero for the structure to be observ-able. Thus to demonstrate the effect we have introduceddamping γ = 0 . R / ¯ h for the X exciton only. We an-ticipated observable exceptional point structures in the (a) (b)(c) (d) FIG. 4. Exceptional points in the polariton dispersion (toppanels) and wavevector surface (bottom panels). (a, b): Realand imaginary parts of the polariton energies E ( k, θ ) , for areal wavevector of magnitude k , at propagation direction θ .Only one exciton, X , is damped, with rate γ = 0 . R / ¯ h .(c, d): Real and imaginary parts of the polariton wavevector k ( E, θ ) , for a real energy E . All the excitons have an equaldamping rate γ = 0 . R / ¯ h . case where the excitons all have one damping rate, andthe photons another, but did not find this numerically.Spin-dependent exciton damping may be necessary for aclear observation of these structures. Exceptional points in the isofrequency surface
We now consider the effect of damping in terms ofthe complex-valued wavevector k ( E, θ, φ ) = k ( E, θ ) as afunction of the real-valued energy and propagation direc-tion. This may be compared with the treatment above,where we considered the complex-valued energy as a func-tion of the real-valued wavevector. At a particular energythe two-sheeted function k ( E, θ, φ ) is an isofrequency sur-face of the dispersion relation, whose normals give the raydirections [34]. More generally, real energies correspondto monochromatic continuous-wave excitation, and thefunction k ( E, θ ) describes the propagation of polaritonsunder such conditions. As we shall see, the isofrequencysurface can have rings of exceptional points (at particularreal energies), similar to those in the dispersion relation(at particular real wavevectors).Figure 5 shows the complex-valued wavevector func-tion obtained using the parameters of Fig. 2(a) withdamping γ = 0 . R / ¯ h for the excitons. Here andthroughout this subsection we consider an equal damp-ing rate of all the excitons, γ n = γ . As can be seen,the damping blurs the distinction between the lower, in- -1-0.9-0.8 -0.2 -0.1 0 0.1 0 0.1 0.2 0.3 E - E g ( R ) k-k (a ) α (a ) -1-0.5 0 0.5 1 ( I + - I - ) /I FIG. 5. Polariton dispersion relation at real energies (verticalaxis) and complex wavevectors (horizontal axes), for propa-gation at θ = 0 . The curves on the left side/top axis showthe imaginary part of the complex wavevector, i.e. the ab-sorption coefficient, and those on the right/bottom axis showthe real part. Coloring shows the circular polarization of themodes. Horizontal lines show the energies of the circularly-polarized excitons. All excitons have an equal damping rate γ = 0 . R / ¯ h . (a) (b)
0. 0.25 0.5 0.75 1. - - - θ ( π ) E - E g ( R )
0. 0.25 0.5 0.75 1. - - - θ ( π ) E - E g ( R ) FIG. 6. Exceptional points of the complex wavevector k ( E, θ ) for real energies, for two values of the exciton damping: γ =0 . R / ¯ h (a), and γ = 0 . R / ¯ h (b). The blue (orange)curves are the zero contours of the real (imaginary) part ofthe discriminant for the characteristic equation determining k . The points show the locations of the exceptional points. termediate and upper polariton branches, joining themtogether for each polarization. In the imaginary partsof the wavevectors, i.e. the absorption coefficients, wecan see the micro-structure of the individual excitonicresonances and their associated oscillator strengths. Wesee that there are energies where the real parts of thewavevectors for the two polarizations are degenerate, andthere are, also, energies where the absorption coefficientsare degenerate. These are not exceptional points, how-ever, as the degeneracies in the real and imaginary partsof k occur at different energies.The exceptional points of the wavevector function k ( E, θ ) are values of E and θ where, simultaneously, thereal parts of k and the imaginary parts of k for each po-larization are degenerate. To identify these degeneracieswe consider the characteristic equation for Eq. (7), whichis a quadratic in k . The exceptional points are the zerosof the discriminant of this quadratic. They can be foundby plotting the zero contours of its real and imaginaryparts, and looking for their crossings. This is shown fortwo different values of the damping rate in Fig. 6.Figure 6(a) shows the situation for a small dampingrate, γ = 0 . R / ¯ h , while Fig. 6(b) considers a largervalue, γ = 0 . R / ¯ h . We see that there are exceptionalpoints, which originate from the degeneracies of the po-lariton dispersion in the absence of damping. As dampingis introduced the degeneracies of the polariton dispersionmove in the ( θ, E ) plane. The richest structure in termsof degeneracies is at very low damping, as in Fig. 6(a),where we see there are six exceptional points in the re-gion ≤ θ ≤ π/ . Since all of these points occur atnon-zero θ they correspond to rings of exceptional pointsin the isofrequency surface, at certain energies, owing tothe φ -independence of the solutions. As the dampingis increased the exceptional points annihilate and theirnumber reduces, as can be seen in Fig. 6(b), where thereare only now two exceptional points.Figures 4(c) and (d) show the real and imaginaryparts of the complex wavevector, as functions of thereal energy E and angle θ , in the region containingthe two exceptional points of Fig. 6(b). The two ex-ceptional points are joined by a line degeneracy in thereal parts of the wavevector, which is clearly visible inFig. 4(c). The structure around each exceptional pointmay be compared with an exceptional point in the com-plex energy (Figs. 4(a), (b)). We again have the ex-pected general form, i.e., line degeneracies in each of thereal and imaginary parts, which meet at the exceptionalpoint. The overall structure may, also, be compared withthat described by Berry and Dennis [35] for frequency-independent absorbing dielectrics, for which the com-plex function k ( θ, φ ) contains degeneracies in particularwavevector directions. These exceptional points definethe ‘singular axes’ of the crystal. They are points in thespace of wavevector direction, but occur at all frequen-cies. The degeneracies of the complex-valued wavevec-tor described here are, instead, extended in the space ofwavevector direction (forming rings), but occur only atspecific frequencies. CONCLUSION
The strong-coupling of light to excitons in a magneticfield gives rise to topologically non-trivial dispersion re-lations ω ( k ) , and wavevector surfaces k ( E ) , for polari-tons in bulk zinc-blende semiconductors. The complexdegeneracy structure of the dispersion provides a routeto realizing topological effects for polaritons in three di-mensions, going beyond previous work in two dimensionalsystems [14–21] such as microcavities. In the absence of dissipation the polariton dispersion contains Weyl points,for propagation along the field, and ring degeneracies,for propagation perpendicular to it. In the presence ofdissipation the Weyl points become rings of exceptionalpoints, which generalize the corresponding Dirac excep-tional rings of two-dimensional dissipative systems [7].A realization of Weyl exceptional rings in cold atomicgases has recently been proposed [8]; the present workshows that a different realization in semiconductors maybe possible.Topological bands, Weyl points, and surface states(Fermi arcs) have recently been revealed in transmissionexperiments [2, 11, 24, 25, 40] on photonic bandstruc-tures [41]. The topological dispersion relations describedhere, and their consequences, will also give signatures intransmission, as well as reflectivity, experiments. Fur-thermore, in the polariton system one can pump inco-herently above the polariton branches, allowing them tobe studied in photoluminescence. This also creates thepossibility of exploring the impact of gain on the topo-logical bands and surface modes, and creating a polari-tonic topological laser based on surface states. Perhapsthe most promising way, however, in which the polaritonsystem goes beyond existing photonic topological materi-als is through the presence of large nonlinearities, givingit potential for realizing topological strong-interaction ef-fects using light.There are, however, several disadvantages and diffi-culties that would need to be overcome to study topo-logical polaritons in three dimensions in practice. Oneis that the scale of the effects considered here, in fre-quency/energy, is rather small, so that low tempera-tures, clean materials, and high-resolution spectroscopywould be required. Another is the well-known difficultyof analysing experiments on polaritons in bulk materi-als, due to the non-trivial boundary conditions [42]. Itwould be important to understand how to couple effec-tively to and from the polaritons, at wavevectors near tothe topological singularities. ACKNOWLEDGMENTS
We acknowledge support from the Irish ResearchCouncil through Award No. GOIPG/2015/3570 andthe Science Foundation Ireland through Award No.15/IACA/3402. [1] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi,L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zil-berberg, and I. Carusotto, Rev. Mod. Phys. , 015006(2019).[2] L. Lu, Z. Wang, D. Ye, L. Ran, L. Fu, J. D. Joannopoulos,and M. Soljačić, Science , 622 (2015). [3] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane,G. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez,B. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin,S. Jia, and M. Z. Hasan, Science , 613 (2015).[4] H. Shen, B. Zhen, and L. Fu, Phys. Rev. Lett. ,146402 (2018).[5] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H.Musslimani, S. Rotter, and D. N. Christodoulides, Nat.Phys. , 11 (2018).[6] W. D. Heiss, J. Phys. A , 444016 (2012).[7] B. Zhen, C. W. Hsu, Y. Igarashi, L. Lu, I. Kaminer,A. Pick, S.-L. Chua, J. D. Joannopoulos, and M. Soljačić,Nature , 354 (2015).[8] Y. Xu, S.-T. Wang, and L.-M. Duan, Phys. Rev. Lett. , 045701 (2017).[9] J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl,A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moi-seyev, and S. Rotter, Nature , 76 (2016).[10] D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong, andF. Nori, Phys. Rev. Lett. , 040401 (2017).[11] H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson,L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen,Science , 1009 (2018).[12] A. Kavokin, J. J. Baumberg, G. Malpuech, and F. P.Laussy, Microcavities (Oxford University Press, 2017).[13] P. Yu and M. Cardona,
Fundamentals of Semiconductors:Physics and Materials Properties (Springer, 1995).[14] R. Ge, W. Broer, and T. C. H. Liew, Phys. Rev. B ,195305 (2018).[15] C.-E. Bardyn, T. Karzig, G. Refael, and T. C. H. Liew,Phys. Rev. B , 161413(R) (2015).[16] A. V. Nalitov, D. D. Solnyshkov, and G. Malpuech, Phys.Rev. Lett. , 116401 (2015).[17] C. Li, F. Ye, X. Chen, Y. V. Kartashov, A. Ferrando,L. Torner, and D. V. Skryabin, Phys. Rev. B ,081103(R) (2018).[18] K. Yi and T. Karzig, Phys. Rev. B , 104303 (2016).[19] T. Karzig, C.-E. Bardyn, N. H. Lindner, and G. Refael,Phys. Rev. X , 031001 (2015).[20] S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler,R. Ge, M. A. Bandres, M. Emmerling, L. Worschech,T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling,Nature , 552 (2018). [21] Y. V. Kartashov and D. V. Skryabin, Phys. Rev. Lett. , 083902 (2019).[22] T. Gao, E. Estrecho, K. Bliokh, T. Liew, M. Fraser,S. Brodbeck, M. Kamp, C. Schneider, S. Höfling, Y. Ya-mamoto, et al. , Nature , 554 (2015).[23] F. Li, X. Huang, J. Lu, J. Ma, and Z. Liu, Nat. Phys. , 30 (2018).[24] J. Noh, S. Huang, D. Leykam, Y. D. Chong, K. P. Chen,and M. C. Rechtsman, Nat. Phys. , 611 (2017).[25] B. Yang, Q. Guo, B. Tremain, R. Liu, L. E. Barr, Q. Yan,W. Gao, H. Liu, Y. Xiang, J. Chen, C. Fang, A. Hibbins,L. Lu, and S. Zhang, Science , 1013 (2018).[26] K. Cho, S. Suga, W. Dreybrodt, and F. Willmann, Phys.Rev. B , 1512 (1975).[27] M. Altarelli and N. O. Lipari, Phys. Rev. B , 3798(1973).[28] R. Winkler, Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems (Springer, 2003).[29] H. Fu, L.-W. Wang, and A. Zunger, Phys. Rev. B ,5568 (1999).[30] K. Cho, Phys. Rev. B , 4463 (1976).[31] P. Basu, Theory of Optical Processes in Semiconductors (Clarendon Press, 1997).[32] R. Willardson and A. Beer,
Semiconductors andSemimetals , Vol. 3 (Academic Press, 1967).[33] L. Świerkowski, Il Nuovo Cimento B , 340 (1975).[34] M. Born and E. Wolf, Principles of Optics (CambridgeUniversity Press, 1959).[35] M. V. Berry and M. R. Dennis, Proc. Royal Soc. A ,1261 (2003).[36] K. E. Ballantine, J. F. Donegan, and P. R. Eastham,Phys. Rev. A , 013803 (2014).[37] H. Haug and S. Koch, Quantum Theory of the Opticaland Electronic Properties of Semiconductors (World Sci-entific Publishing, 2004).[38] Z. Jalali-Mola and S. A. Jafari, Phys. Rev. B , 205413(2019).[39] M. Milićević, G. Montambaux, T. Ozawa, O. Ja-madi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet,A. Harouri, J. Bloch, and A. Amo, Phys. Rev. X ,031010 (2019).[40] W.-J. Chen, M. Xiao, and C. T. Chan, Nat. Commun. , 13038 (2016).[41] L. Lu, L. Fu, J. D. Joannopoulos, and M. Soljačić, Nat.Photonics , 294 (2013).[42] K. Henneberger, Phys. Rev. Lett.80