Weyl points and exceptional rings with polaritons in bulk semiconductors

Weyl points are the simplest topologically-protected degeneracy in a three-dimensional dispersion relation. The realization of Weyl semimetals in photonic crystals has allowed these singularities and their consequences to be explored with electromagnetic waves. However, it is difficult to achieve nonlinearities in such systems. One promising approach is to use the strong-coupling of photons and excitons, because the resulting polaritons interact through their exciton component. Yet topological polaritons have only been realized in two dimensions. Here, we predict that the dispersion relation 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 points to Weyl exceptional rings. We conclude that bulk semiconductors are a promising system in which to investigate topological photonics in three dimensions, and the effects of dissipation, gain, and nonlinearity.


INTRODUCTION
Degeneracies in bandstructures are a key concept at the heart of recent developments in condensed-matter physics and optics [1]. Two-dimensional materials such as graphene posesss Dirac points, where the dispersion is locally linear, which are responsible for many of their unique properties. In three-dimensional materials Weyl points have been found in photonic [2] and electronic [3] bandstructures, providing low-energy models of Weyl fermions. More generally, topological considerations mean that materials hosting degeneracies are the basis for realizing topological insulators and related effects such as robust edge modes. Such work is now also being extended to dissipative systems, such as photonic materials with gain and loss, described by non-Hermitian Hamiltonians [4]. In this case the singularities include exceptional points [5,6] in parameter space, at which both the frequencies and lifetimes of the modes become degenerate. Rings of such exceptional points have been shown to emerge from Dirac points in photonic crystals [7]. In the three-dimensional case, Weyl points can become Weyl exceptional rings [8], which have a quantized Chern number and a quantized Berry phase. Like their counterparts in Hermitian systems, such non-Hermitian singularities give rise to interesting physical effects [9], including edge modes [10], unusual transmission properties, topological lasing, and Fermi arcs arising from halfinteger topological charge [11].
Polaritons are exciton-photon superpositions that are formed by strong light-matter coupling in semiconductors [12,13]. Their half-matter half-light nature implies relatively strong nonlinearities, and this feature among others makes them an interesting system in which to study topological effects. Topological phases have been predicted [14][15][16][17][18][19] and observed [20] for polaritons formed from quantum-well excitons coupled to photons confined in microcavities. Topological lasing [21] and exceptional points [22] have also been studied. However, as micro-cavities and quantum-wells are two-dimensional systems, phenomena such as Weyl points, Fermi arcs, and the three-dimensional topological phases [1], have not been considered.
In this paper, we report topologically non-trivial dispersion relations for polaritons propagating in three dimensions. We consider a bulk semiconductor in a magnetic field, and show that the p-type structure of the valence band leads to intricate dispersion relations containing topologically-protected degeneracies. In the absence of non-radiative losses there are eight sheets of the dispersion surface, which host Weyl points [2,3,[23][24][25], for wavevectors along the field direction, and ring degeneracies, for wavevectors transverse to it. In the non-Hermitian case [4,5,10], with absorption, we show that the Weyl points become Weyl exceptional rings, which can be reached by tuning the frequency and the angle between the propagation direction and the applied field. These results show that bulk semiconductors could be used to study topological effects in three spatial dimensions. Furthermore, bulk polariton lifetimes can be long since, unlike microcavity polaritons, they are not subject to radiative decay. They may therefore give access to the strongly-interacting regime of topological photonics [1].

Exciton spectra
We consider polaritons formed from 1s excitons in direct band-gap zinc-blende semiconductors such as GaAs. These involve p-type valence band states with Γ 8 symmetry, and s-type conduction band states with Γ 6 symmetry. The combinations of the hole spin m h = ± 3 /2, ± 1 /2 and the electron spin m s = ± 1 /2 then give rise to eight exciton spin states, denoted |X n for n = 1, . . . 8, with energies E n .
To evaluate the polariton spectrum we need the en-ergies and polarizations of the exciton transitions. To obtain these we diagonalize the effective Hamiltonian for the 1s excitons given in Ref. [26]. The parameters in this effective Hamiltonian are related to the underlying electron-hole exchange parameters, Luttinger parameters, and g-factors. This approach treats the valenceband anisotropy, magnetic field, and electron-hole exchange as perturbations on a spherically-symmetric electron-hole Hamiltonian [27]. The unperturbed wavefunction is of the usual hydrogenic form, with the binding energy R 0 = µe 4 /32π 2 2 0 2h2 and Bohr radius a 0 = 4π is the isotropic part of the effective mass for the valence band, related to the Luttinger parameter γ 1 , and m c is the effective mass for the conduction band. For this perturbative approach to be valid the cyclotron energy must be small compared with the exciton binding energy R 0 . We take the specific criterion given by Altarelli and Lipari [27], to define the maximum field B max of the perturbative regime. In the following we will consider the specific case of GaAs, with applied field B max in the [001] direction, using the bandstructure parameters from Ref. [28]. For the electron-hole exchange parameters [26] we take ∆ 1 = −9.61 µeV [29], and ∆ 0 = ∆ 2 = 0. The exciton spectrum computed for these parameters is shown in Fig. 1(a). As expected, the magnetic field lifts the degeneracies of the eight electron hole pair states. This splitting of the energies of the excitons will result in an anisotropic and multiply resonant optical susceptibility, and hence a direction and polarization dependent polariton dispersion.

Polariton Hamiltonian
The topological singularities of the polariton dispersion arise from the polarization dependence of the excitonphoton coupling. In the Coulomb gauge the interaction between the vector potential and the electrons, from the (2) Here the first sum is over the electrons, and the second over the photon wavevectors, k, and polarizations, s, with corresponding polarization vectors e k,s .â k,s is the photon annihilation operator, ω = c|k| the photon frequency, and V a quantization volume. Thus we have the second-quantized Hamiltonian in the subspace of the   eight 1s exciton states, |X k,n , where we have made the rotating-wave approximation.
In the envelope function approximation the matrix elements appearing in Eq. (3) are products of the matrix elements of the Bloch functions at k = 0 and the hydrogenic exciton wavefunctions χ ms,m h F n (r = 0). For the spatial part of the latter we take the unperturbed result |F n (0)| 2 ≈ 1/πa 3 0 . For the spin part χ ms,m h we note that at the field B max we are considering the Zeeman terms dominate over the electron-hole exchange. Thus the excitons are, to a good approximation, diagonal in the spin projections m s and m h . Using the standard forms for the valence band wavefunctions [30] and the Kane parameter The states |X 1 . . . |X 4 correspond to excitons with electron spin m s = 1 /2 and hole spin m h = 3 /2, 1 /2, − 1 /2, − 3 /2, respectively, while |X 5 . . . |X 8 are the corresponding states with m s = − 1 /2. Thus |X 1 and |X 8 are dark states, and the remaining transitions are either circularlypolarized in the xy plane, or linearly polarized in the z direction, as shown in Figs. 1(a) and 1(b). Using these exciton matrix elements in Eq. (3), and approximating ω ≈ E g /h in the prefactor of the coupling, we find for the exciton-photon Hamiltonian, where k = |k| and we use a Rabi splittinḡ to quantify the light-matter coupling in the material.

Polariton spectra
In the following we will consider the polariton spectrum, which we obtain from the Heisenberg equationsof-motion for the exciton and photon annihilation operators by looking for solutions with time dependence e −iωt , i.e., settingâ(t) = e −iωtâ (0) and similarly for the exciton operators. We specify the wavevector direction in terms of the polar coordinates θ, φ, with the field direction and the [001] crystal axis correponding to θ = 0. For the photon polarization we use the circularly-polarized states, e k,± = (e k,θ ± ie k,φ )/ √ 2, constructed from the linearly-polarized basis transverse to k, Since there are two photon polarizations and six excitons (discounting the irrelevant dark states), this procedure gives an eight-by-eight Hamiltonian matrix, H 8 , with elements dependent on the wavevector magnitude and direction. We determine the polariton dispersion ω(k, θ, φ) by finding the eigenvalues, ω, of H 8 numerically.
While the dispersion ω(k, θ, φ) is given by the eigenvalues of H 8 , there is another approach to analyzing the topological singularities of the polariton spectrum, in terms of the function k(ω, θ, φ). This latter function provides a natural description of optics at a fixed frequency, and is related to constructs such as the refractive index surface of classical optics [34][35][36]. For example, a radial plot of k at some fixed frequency over angles gives a contour (in k-space) of the dispersion relation ω(k). The normal to such an isofrequency surface is therefore the group velocity, ∇ k ω, controlling the refraction direction at that frequency. While the two functions ω(k, θ, φ) and k(ω, θ, φ) are equivalent in the absence of dissipation, we shall see that they have some differences in its presence, and we therefore consider both representations in the following.
To obtain the dispersion in the form k(ω, θ, φ) we eliminate the exciton amplitudes from the Heisenberg equations-of-motion. This leads to a two-dimensional eigenproblem for the photon amplitudes, so that the magnitudes of the wavevectors are the eigenvalues of a two-by-two matrix, whose elements are functions of the frequency and propagation direction.
It is useful to note that this form, Eq. (7), can also be derived semiclassically, by looking for plane-wave solutions to Maxwell's equations, including the excitonic resonances via a frequency-dependent dielectric function (ω). In an optically isotropic material (ω) is a scalar, and the polariton dispersion satisfies c 2 k 2 /ω 2 = (ω) [37]. In the present case, however, the optical response is anisotropic due to the magnetic field, and we must consider the vector equation Longitudinal modes, with k E, occur if (ω) = 0. To obtain the equation for the transverse modes we take matrix elements of Eq. (8) in a basis perpendicular tô k, such as e k,± . This eliminates the zero eigenvalue of the operator k × k×, i.e. the longitudinal polariton, and gives where χ ss (ω) = e † k,s χ(ω)e k,s is the transverse part of the excitonic susceptibility and the background permittivity. We approximate the prefactors in this expression Comparing this expression with Eq. (7) we see that the final term in the latter is related to the susceptibility by The spectrum k(ω, θ, φ) can be found straightforwardly by solving the secular equation for Eq. (7), which is a quadratic in k. It may be noted that χ ss , and hence the polariton spectrum, is independent of φ. This reflects the rotational symmetry of the problem about the magnetic field (θ = 0). The combination of the form of Eq. (7) with that of the χ ss imposes an additional symmetry between the solutions at θ and those at π − θ. We may therefore set φ = 0 and consider the interval θ ∈ [0, π 2 ]. We note that whereas the secular equation for Eq. (7) is a quadratic in k, that for H 8 is an eighth-order polynomial in ω. Thus there are, in general, two wavevectors for each frequency, from the two dispersing photon modes. There are however eight frequencies for each wavevector, from those two photon modes as well as the six non-dispersing bright excitons.

Topological singularities: Hermitian case
Our primary interest is in the degeneracy structure of the magneto-exciton-polariton dispersion relation, which we first consider in the Hermitian case without dissipation. It is possible to make some observations that constrain the possible degeneracies based on the symmetry of the problem. Owing to the φ invariance of the solutions we know that propagation in the θ = 0 direction is the only configuration for which isolated degeneracies are possible. Correspondingly, if degeneracies occur at any non-zero θ they are necessarily extended degeneracies over all φ.
In Fig. 2 we plot the dispersion of the transverse modes, obtained from the secular equation for Eq. (7). The three panels refer to propagation along the field, θ = 0, at a small angle to it, θ = π/8, and perpendicular to it, θ = π/2. The polarization of the modes is shown by the coloring. Energy is measured relative to the bandgap E g and in units of the exciton Rydberg energy. The wavector is measured relative to k 0 = √ (E g −R 0 )/hc, which is the wavevector at which the bare linear photonic dispersion would cross an unperturbed exciton. The wavevector is measured in units of the inverse exciton Bohr radius.
The spectrum for propagation along the z-axis is shown in Fig. 2(a). In this case the two z-polarized excitons, X 3 and X 6 , do not couple to light, and there are only six modes in the transverse spectrum. The other excitons are circularly polarized, and each circular polarization of light mixes with the two excitons of that polarization. This gives rise to a spectrum with a lower, intermediate, and upper branch for each circular polarization. The lower branch begins at low energy as a purely photonic, linearly dispersing, mode, which anticrosses with the lower energy exciton, asymptoting horizontally to approach that exciton energy at large k. Above that energy there is an intermediate branch, which initially has an imaginary k as it lies in the polaritonic (longitudinaltransverse) gap [13]. This mode then becomes a propagating polariton with k = 0 at the gap edge, and then approaches the higher exciton energy at large k. Above this there is an upper branch, which again begins as a solution with imaginary k, before becoming a propagating solution, and finally approaching the photon dispersion at large k. Fig. 2(b) shows the spectrum at a small angle to the z-axis. Comparing this spectrum with that in Fig. 2(a) we see that there are now eight branches, because the z-polarized excitons now couple to light. Moreover we find that this spectrum is gapped, with avoided crossings which orginate from the degeneracies at θ = 0. The two degeneracies in Fig. 2(a) between the different circular modes have split. In addition we see that the three intersections between the z-polarized excitons and the transverse modes at θ = 0 have split, to organize into the two additional transverse branches at θ = 0. There is in fact a fourth intersection of this nature, involving the highest energy z-polarized exciton, but it is with an evanescent mode at an imaginary k. Figure 3 shows the dispersion relation near to some of these singularities. One of the degeneracies between the two circular polarization modes seen in Fig. 2(a) lies between the second and third sheets in Fig. 3, counting from low energies. It can be seen to split linearly in k z but quadratically in k x -and, therefore, also in k y , due to the rotational symmetry. The same structure appears for the other crossing of the circular modes at θ = 0 (not shown). Such a dispersion implies these degeneracies have zero topological charge. However, Fig. 3 also shows three crossings involving the z-polarized excitons, which can be seen to disperse linearly in k x , and therefore k y , as well as k z . They are thus isolated Weyl points of the polariton dispersion relation, and are topologically non-trivial. Two Weyl points are clearly visible, between the third and fourth sheets, and between the fourth and fifth sheets. A third lies between the second and third sheets to the right of the quadratic degeneracy in the figure, but is hidden by the perspective. The final, fourth, degeneracy between a transverse polariton and a z-polarized exciton at θ = 0, lying in one of the polaritonic gaps, is also a Weyl point, but at an imaginary k.
The dispersion perpendicular to the applied field (θ = π/2) is shown in Fig. 2(c). As indicated by the coloring, in this case the transverse modes are purely linearly polarized, along the polar vectors e θ = −z and e φ . Considering this geometry we identify two further degeneracies in the polariton dispersion, where modes with these two polarizations cross. These are both extended ring degeneracies, due to the φ invariance of the system.
We see that there are, in total, eight distinct degeneracies of the polariton dispersion relation in the region 0 ≤ θ ≤ π/2. Of these eight degeneracies six are isolated degeneracies occurring in the θ = 0 direction and two are extended degeneracies occurring in the θ = π/2 plane. The six isolated degeneracies divide into four Weyl points, one of which is at an imaginary k, and two topologically trivial degeneracies with a mixed quadratic/linear dispersion.
The Weyl points are degeneracies between the zpolarized excitons and the xy-polarized polaritons at θ = 0. To see why this gives a Weyl point, with a linear dispersion, we note that the coupling between such modes near to the degeneracy is proportional to sin(θ) [see Eqs. (4,5,6)]. Thus moving away from θ = 0 there is a splitting of the degeneracy which is linear in sin θ ≈ θ and hence linear in k x (or k y ). Formally, the Hamiltonian for the two modes near the degeneracy takes the form which, with sin(θ) ≈ k x /k 0 (for k y =0), gives a linear dispersion in k x and k z −k 0 at the degeneracy ω 0 = ω x . Ω is the strength of the coupling between the z-polarized exciton and the polariton, involving the amplitude, in the polariton, of the e θ -polarized photon. k 0 is the wavevector at the degeneracy, ω 0 = ω x = c k 0 the frequency, and c the velocity. Notably, these Weyl points lie at the critical tilt between a type-I and a type-II point [38], and as such are the three-dimensional (Weyl) generalization of the recently achieved type-III Dirac point [39].

Exceptional points in the dispersion relation
In the presence of damping the Hamiltonian, H 8 , becomes non-Hermitian, and the polariton dispersion, ω(k, θ, φ), can contain rings of exceptional points arising from the Weyl points described above. This can be seen by considering the local Hamiltonian, Eq. (11), for one of the Weyl points. In the presence of damping we have ω 0,x → ω 0,x + iγ 0,x , and the frequency difference of the two coupled modes, at the bare resonance, becomes Ω 2 sin 2 θ − (γ 0 − γ x ) 2 . Thus the real (imaginary) parts of the spectrum split for angles greater (less) than θ = arcsin |γ 0 − γ x |/Ω. This occurs at all φ, so that we have a ring of exceptional points, where both the real and imaginary parts of the polariton energies are degenerate, and the Hamiltonian matrix is defective.
In Fig. 4 we show an exceptional point of this type, in the spectrum of the full Hamiltonian, H 8 . Figs. 4(a) and (b) show the real and imaginary parts, respectively, of the energies, and the expected local structure around an exceptional point [6] is clearly visible. From the analysis above the critical angle for the exceptional point depends on the difference in the damping constants, so this should be non-zero for the structure to be observable. Thus to demonstrate the effect we have introduced damping γ 3 = 0.01R 0 /h for the X 3 exciton only. We anticipated observable exceptional point structures in the case where the excitons all have one damping rate, and the photons another, but did not find this numerically. Spin-dependent exciton damping may be necessary for a clear observation of these structures.

Exceptional points in the isofrequency surface
We now consider the effect of damping in terms of the complex-valued wavevector k(E, θ, φ) = k(E, θ) as a function of the real-valued energy and propagation direction. This may be compared with the treatment above, where we considered the complex-valued energy as a function of the real-valued wavevector. At a particular energy the two-sheeted function k(E, θ, φ) is an isofrequency surface of the dispersion relation, whose normals give the ray directions [34]. More generally, real energies correspond to monochromatic continuous-wave excitation, and the function k(E, θ) describes the propagation of polaritons under such conditions. As we shall see, the isofrequency surface can have rings of exceptional points (at particular real energies), similar to those in the dispersion relation (at particular real wavevectors). Figure 5 shows the complex-valued wavevector function obtained using the parameters of Fig. 2(a) with damping γ = 0.015R 0 /h for the excitons. Here and throughout this subsection we consider an equal damping rate of all the excitons, γ n = γ. As can be seen, the damping blurs the distinction between the lower, in- termediate and upper polariton branches, joining them together for each polarization. In the imaginary parts of the wavevectors, i.e. the absorption coefficients, we can see the micro-structure of the individual excitonic resonances and their associated oscillator strengths. We see that there are energies where the real parts of the wavevectors for the two polarizations are degenerate, and there are, also, energies where the absorption coefficients are degenerate. These are not exceptional points, however, as the degeneracies in the real and imaginary parts of k occur at different energies. The exceptional points of the wavevector function k(E, θ) are values of E and θ where, simultaneously, the real parts of k and the imaginary parts of k for each polarization are degenerate. To identify these degeneracies we consider the characteristic equation for Eq. (7), which is a quadratic in k. The exceptional points are the zeros of the discriminant of this quadratic. They can be found by plotting the zero contours of its real and imaginary parts, and looking for their crossings. This is shown for two different values of the damping rate in Fig. 6. Figure 6(a) shows the situation for a small damping rate, γ = 0.0005R 0 /h, while Fig. 6(b) considers a larger value, γ = 0.008R 0 /h. We see that there are exceptional points, which originate from the degeneracies of the polariton dispersion in the absence of damping. As damping is introduced the degeneracies of the polariton dispersion move in the (θ, E) plane. The richest structure in terms of degeneracies is at very low damping, as in Fig. 6(a), where we see there are six exceptional points in the region 0 ≤ θ ≤ π/2. Since all of these points occur at non-zero θ they correspond to rings of exceptional points in the isofrequency surface, at certain energies, owing to the φ-independence of the solutions. As the damping is increased the exceptional points annihilate and their number reduces, as can be seen in Fig. 6(b), where there are only now two exceptional points. Figures 4(c) and (d) show the real and imaginary parts of the complex wavevector, as functions of the real energy E and angle θ, in the region containing the two exceptional points of Fig. 6(b). The two exceptional points are joined by a line degeneracy in the real parts of the wavevector, which is clearly visible in Fig. 4(c). The structure around each exceptional point may be compared with an exceptional point in the complex energy (Figs. 4(a), (b)). We again have the expected general form, i.e., line degeneracies in each of the real and imaginary parts, which meet at the exceptional point. The overall structure may, also, be compared with that described by Berry and Dennis [35] for frequencyindependent absorbing dielectrics, for which the complex function k(θ, φ) contains degeneracies in particular wavevector directions. These exceptional points define the 'singular axes' of the crystal. They are points in the space of wavevector direction, but occur at all frequencies. The degeneracies of the complex-valued wavevector described here are, instead, extended in the space of wavevector direction (forming rings), but occur only at specific frequencies.

CONCLUSION
The strong-coupling of light to excitons in a magnetic field gives rise to topologically non-trivial dispersion relations ω(k), and wavevector surfaces k(E), for polaritons in bulk zinc-blende semiconductors. The complex degeneracy structure of the dispersion provides a route to realizing topological effects for polaritons in three dimensions, going beyond previous work in two dimensional systems [14][15][16][17][18][19][20][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 of dissipation the Weyl points become rings of exceptional points, which generalize the corresponding Dirac exceptional rings of two-dimensional dissipative systems [7]. A realization of Weyl exceptional rings in cold atomic gases has recently been proposed [8]; the present work shows that a different realization in semiconductors may be possible.
Topological bands, Weyl points, and surface states (Fermi arcs) have recently been revealed in transmission experiments [2,11,24,25,40] on photonic bandstructures [41]. The topological dispersion relations described here, and their consequences, will also give signatures in transmission, as well as reflectivity, experiments. Furthermore, in the polariton system one can pump incoherently above the polariton branches, allowing them to be studied in photoluminescence. This also creates the possibility of exploring the impact of gain on the topological bands and surface modes, and creating a polaritonic topological laser based on surface states. Perhaps the most promising way, however, in which the polariton system goes beyond existing photonic topological materials is through the presence of large nonlinearities, giving it potential for realizing topological strong-interaction effects using light.
There are, however, several disadvantages and difficulties that would need to be overcome to study topological polaritons in three dimensions in practice. One is that the scale of the effects considered here, in frequency/energy, is rather small, so that low temperatures, clean materials, and high-resolution spectroscopy would be required. Another is the well-known difficulty of analysing experiments on polaritons in bulk materials, due to the non-trivial boundary conditions [42]. It would be important to understand how to couple effectively to and from the polaritons, at wavevectors near to the topological singularities.