Conical intersections are features of parameter spaces where two or more energy surfaces become degenerate at one point, while staying linear in its vicinity. Two prominent examples are Hamilton’s diabolical point in biaxial crystals in optics [1] and Dirac cones in solid state’s iconic material graphene [2]. For the latter, due to the mathematical analogy with the Dirac equation for massless electrons, a microscopic degree of freedom called pseudospin was introduced [3]. Unlike the polarization-related photon spin or the intrinsic spin of electrons, this form of angular momentum is not associated with any intrinsic property of particles. Instead, it arises from the substructure of space given by the periodic potential in which the wave function resides [4].

Pseudospin quasiparticles in periodic lattices with conical intersections represent a practical test bed for observing quantum relativistic effects implied by the Dirac equation and its higher-spin versions such as Klein tunneling [5,6] or Zitterbewegung [7]. Photonic model systems such as evanescently coupled waveguides, so-called photonic lattices, allow observing a variety of classical analogs of both relativistic and non-relativistic quantum phenomena associated with the evolution of electrons in periodic potentials [8,9] due to the formal correspondence between the Schrödinger equation and the paraxial wave equation. A convenient feature of photonic lattices is that they provide direct access to the evolution of the wave function during propagation. Therefore, a natural step was to use a photonic platform to realize periodic lattices hosting conical intersections in their spectrum and to demonstrate their peculiarities by studying light propagation through them. This has already led to realizations of pseudospin-1/2 photonic graphene [10], the pseudospin-1 photonic Lieb lattice [11], and the Lieb-kagome transition lattice [12].

A challenging open problem in artificial lattice systems is the design of conical intersections with higher pseudospin values [4]. Although there have been proposals for generalized conical intersections with arbitrary pseudospin [13,14], and pseudospin-2 ones have been considered theoretically [15,16], no realistic system containing a conical intersection with pseudospin higher than one has been demonstrated until now.

Here we present a photonic chiral borophene lattice hosting a pseudospin-2 conical intersection in its band structure at the center of its Brillouin zone. We derive the five pseudospin eigenstates using both an intuitive approach and a rigorous mathematical–analytical approach and numerically study their conical diffraction during propagation through the photonic lattice. We prove the interaction of pseudospin and orbital angular momentum by directly observing topological charge conversion giving rise to optical phase vortices in the conically diffracted output light fields. Here, by topological charge, we mean the winding number of the optical wavefront around the vortex core. Our results apply to various other wave systems beyond photonics such as metamaterials [17], Bose–Einstein and polariton condensates [18,19], and importantly also to electronic wave functions in atomic borophene allotropes.

Figure 1(a) shows a sketch of the chiral borophene lattice. The lattice has a hexagonal unit cell with six lattice sites labeled A to F. This configuration has been calculated to be stable as a planar sheet of boron atoms [20,21]. This distinguishing feature may lead to the realization of pseudospin-2 conical intersections in a solid-state 2D material. The geometry of the lattice corresponds to an Archimedean tiling of the plane. More precisely, it is the (34,6) or snub hexagonal tiling that interestingly exists in two chiral variants [22]. The band structure of the chiral borophene lattice is depicted in Fig. 1(b). Intriguing features include the pseudospin-1/2 Dirac cones [23] and the partially flat band [24]. The peculiarity we are interested in for this work is the conical intersection of five bands at the degenerate Γ-point shown in the zoom-in of Fig. 1(c). This fivefold degeneracy has been shown to be protected by site-permutation symmetries and therefore to be robust to long-range isotropic interactions such as pth nearest neighbor hopping for p→∞ [25].

An experimental photonic lattice realization of chiral borophene would rely on an array of evanescently coupled waveguides arranged according to the lattice geometry shown in Fig. 1(a). The waveguides could be created by either femtosecond direct laser writing [26] or optical induction in a photorefractive medium [10]. In this study, we use a numerical approach that has proven to agree very well with experimental realizations [11]. In our photonic waveguide model, we apply a tight-binding approximation as a discrete model describing the evanescent coupling between lattice sites. Considering only nearest neighbor coupling, we obtain the following k-space [k=(kx,ky)] Hamiltonian: H^k=t(01e−ia1ke−ia2ke−ia2k1101e−ia2keia3keia3keia1k101eia3keia1keia2keia2k101eia1keia2ke−ia3ke−ia3k1011e−ia3ke−ia1ke−ia1k10),(1)where the lattice vectors are given by a1=(d/2)(3,1), a2=d(0,1), and a3=a1−a2, d is the lattice constant, and t is the coupling strength. Setting d=t=1 without restriction of generality, the eigenvalues of H^ give the spectrum β(k) shown in Fig. 1(b). In our case, the band structure with its propagation constants β(k) (which describes the rate of phase evolution in the propagation direction) represents a diffraction relation describing the spatial evolution dynamics of photonic wave functions in the lattice. In an atomic borophene lattice, this corresponds to an energy spectrum describing the temporal evolution of the electronic wave function.

At the singular Γ-point, a single-band approximation fails. It is, however, possible to understand complex multi-band effects by introducing the pseudospin as an analog to a real spin. To obtain the five pseudospin eigenstates describing the conical intersection of the chiral borophene lattice, we can proceed analytically and Taylor expand H^ around the singular point [27]. A detailed analytic derivation is presented in Appendix C. Here, we showcase how to derive the pseudospin eigenstates intuitively.

We start with exciting six Γ-points at the centers of the six Brillouin zones surrounding the first one. The resulting interference of six plane waves is known to give rise to a family of discrete nondiffracting beams [28]. For six plane waves with a specific phase relation, resembling a discrete phase vortex, the nondiffracting fields are periodic with a sixfold symmetry. Five of these cases lead to the desired pseudospin eigenstates as illustrated for the pseudospin eigenstate with |ψms=−2⟩ in Fig. 2. For differently charged discrete phase vortices of the six plane waves, we obtain the other four eigenstates (see Appendix B). In the sublattice basis, the normalized eigenstates finally read $$\begin{gathered} |ψ−2⟩=16(+1ei13πei23π−1ei43πei53π)T, \\ |ψ−1⟩=16(+1ei23πei43π+1ei23πei43π)T, \\ |ψ0⟩=16(+1−1+1−1+1−1)T, \\ |ψ+1⟩=16(+1ei43πei23π+1ei43πei23π)T, \\ |ψ+2⟩=16(+1ei53πei43π−1ei23πei13π)T, \end{gathered}$$(2)where the six entries represent the complex amplitudes at lattice sites A to F. |ψ−2,−1,0,+1,+2⟩ are all eigenvectors of H^(0,0) for the eigenvalue β=−1. Together with the eigenvector of the sixth band, which has a different eigenvalue or propagation constant of β=5, |ψβ=5⟩=(11⋯1)T, they form an orthonormal orbital angular momentum basis for the unit cell. Our intuitive derivation allows obtaining the five pseudospin eigenstates, although what is still missing is the assignment to the correct pseudospin value ranging from ms=−2 to ms=+2. We obtain this directly from the analytical derivation presented in Appendix C. However, again an intuitive explanation arises from comparing the phase distributions of the different pseudospin eigenstates. As can be seen in Fig. 3, the order relies on a difference in the topological charges of the microscopic optical phase vortices internal to the unit cell luc: when we increase the pseudospin by unity from |ψ−2⟩ to |ψ−1⟩, the vorticity is also increased from luc=+1 to luc=+2. The same happens when going from |ψ−1⟩ to |ψ0⟩ with luc=+2 increasing to luc=+3. |ψ0⟩ plays a crucial role in this process as the discreteness of its internal topological charge can be regarded to have either a positive or negative value of luc=±3. As a result, from |ψ0⟩ to |ψ+1⟩, we again have a unitary increase from luc=−3 to luc=−2. For the last transition from |ψ+1⟩ to |ψ+2⟩, the relation is once again valid changing from luc=−2 to luc=−1. At first glance, this ordering may appear a bit counterintuitive. In particular, one might ask why there is no 1:1 correspondence between pseudospin and the internal topological charge in the form ms=luc. The answer is that the state having luc=0 has a propagation constant β≠−1, and due to its not being part of the conical intersection, it does not couple to the other states. This prevents the transition luc=−1→luc=0→luc=+1, which would be necessary to go from |ψ−1⟩, over |ψ0⟩ to |ψ+1⟩ in the case of a direct correspondence between pseudospin and internal topological charge.

To confirm that chiral borophene indeed hosts a pseudospin-2 conical intersection in its band structure and to validate the derived pseudospin eigenstates, we perform numerical experiments of light propagation in the lattice (see Appendix A for details on the numerical methods). The simulations are based on the paraxial wave equation and are carried out via a standard pseudo-spectral split-step propagation method [29]. The numerical parameters are chosen to be within the experimental reach and match those in previously reported experiments in laser-written photonic lattices with a refractive index contrast of the waveguides Δn=1.3×10−3, wavelength of λ=532nm, and nearest neighbor waveguide separation of Λ=18μm [30,31].

We excite the lattice with a light field given by the pseudospin eigenstate |ψ−2⟩ multiplied by a Gaussian envelope with FWHM=120μm, as shown in Figs. 4(a) and 4(b). In k-space, this corresponds to a Gaussian instead of a point-like excitation at the conical intersection [Figs. 4(e) and 4(f)]. After propagation in the lattice for the distance z=7.12cm, which corresponds to four coupling lengths Lc for the chosen parameters, we clearly identify the conical diffraction in the output field as shown in Figs. 4(c) and 4(d). Exactly at the Γ-point, the Bloch bands become degenerate, and thus a plane wave input state with an arbitrary pseudospin would be invariant under propagation. However, a finite-sized input beam will excite a range of wave vectors in the vicinity of the Γ-point. Away from the Γ-point, the degeneracy is lifted, and thus, in general, eigenstates of the pseudospin will not coincide with the Bloch wave eigenstates of the Hamiltonian. Therefore there will be a coupling between different pseudospin eigenstates during propagation at a rate determined by the splitting of the Bloch waves’ propagation constants. For this coupling between different pseudospin eigenstates to take place, there has to be compensation for the difference in their pseudospin values based on the conservation of total angular momentum. This compensation leads to topological charge conversion in the form of optical vortices in the macroscopic phase profiles of the output fields for the different pseudospin eigenstates. If we excite the lattice with a pseudospin state msin, the topological charges of the vortices present in the decomposed output fields with pseudospin msout follow the relation l=msin−msout.(3)

Accordingly, the output field in Figs. 4(c) and 4(d) is composed of a superposition of |ψ−2,−1,0,+1,+2⟩ with phase vortices of topological charges l=0,−1,−2,−3,−4, respectively. To confirm this hypothesis, we look at the spectral components by performing a Fourier transform to the fields. In the Fourier transform, we can see multiple spectral components at the centers of the higher-order Brillouin zones. While the higher components come from the waveguide structure and the degree of localization of the waveguide modes, we are interested in the symmetry properties that are fully captured by the six components at the centers of the second Brillouin zone. While the input [Figs. 4(e) and 4(f)] is composed of Gaussian spots at the center of the Brillouin zones, the output [Figs. 4(g) and 4(h)] is composed of more complex spots that we can identify as a superposition of Laguerre–Gaussian modes LG0,0+LG0,−1+LG0,−2+LG0,−3+LG0,−4. This is clearly seen by comparing one of the spectral components in the output [Figs. 4(i) and 4(j)] with an ideal superposition of the Laguerre–Gaussian modes [Figs. 4(k) and 4(l)]. Crucially, both fields display a quadruply charged optical phase vortex peculiar for a pseudospin-2 conical intersection resulting from the conservation of angular momentum going from the pseudospin state |ψ−2⟩ in the input to |ψ+2⟩ in the output.

These results already confirm the central message of this work: demonstrating the existence of a pseudospin-2 conical intersection with five conically diffracting eigenstates in the linear spectrum of a chiral borophene lattice and the generation of highly charged optical vortices. Going beyond the spectral analysis, we give in the following a more detailed picture of the propagation dynamics in our photonic lattice close to the spectral singular point. To this aim, we decompose the output light field into the respective pseudospin components. This significantly simplifies the output phase profiles and allows the underlying mechanisms to be better elucidated. We carry out the decomposition by projecting the output field shown in Figs. 4(c) and 4(d), unit cell by unit cell, onto the pseudospin eigenstates of Eq. (2) [11]. For each unit cell, we obtain five complex values representing the amplitude and phase of the respective pseudospin eigenstate (see Appendix A for details). We represent those values as hexagonal pixels in Fig. 5 for the |ψ−2⟩ output (for the remaining cases see Appendix E). In the phase profiles of the projections, we obtain optical phase vortices with topological charges following the relation l=msin−msout. Of particular interest is the l=−4 phase vortex that arises when projecting onto the pseudospin state |ψ+2⟩, since it is characteristic for a pseudospin-2 conical intersection. The phase vortices appear due to conservation of total angular momentum as the pseudospin value increases stepwise from ms=−2 to ms=+2 during propagation. To reveal the dynamics of this process, we numerically solve the beam propagation in the photonic lattice according to the coupled differential equations of a discrete tight-binding model. As shown further in Appendix D, the tight-binding simulations match the continuous model ones extremely well. We then decompose the output field during propagation in the chiral borophene photonic lattice for different z values. For each step, we calculate the probability amplitude of the total field for each pseudospin eigenstate and for the eigenstate of the sixth band |ψβ=5⟩. Thus, we obtain curves of projection percentages with respect to the z propagation as depicted in Fig. 6. At z=0, the lattice is excited with a light field primarily in the |ψ−2⟩ state. There are also minor components in |ψ−1⟩ and |ψβ=5⟩ due to the finite size of the Gaussian envelope. During propagation, we observe the share of the field in |ψ−2⟩ decrease, while sequentially the shares in |ψ−1⟩, |ψ0⟩, |ψ+1⟩, and |ψ+2⟩ increase. This shows that the pseudospin gradually increases in the order presented in Fig. 3 as it converts from ms=−2 to ms=+2, and that there is no coupling via the state of the sixth band |ψβ=5⟩ with luc=0. This clearly confirms that phase vortices form to compensate for the difference in internal topological charge luc between the different pseudospin eigenstates. In this picture, the explanation of pseudospin–orbit interaction appears natural. Both are forms of the orbital angular momentum of light: the first is microscopic and internal to the unit cell, while the latter is macroscopic in the form of optical phase vortices in the total conically diffracted output field.

In conclusion, we have studied a novel type of pseudospin-2 state in a photonic chiral borophene lattice at its fivefold conical intersection. We have numerically studied conical diffraction with topological charge conversion leading to the formation of optical phase vortices with topological charge values as high as l=±4. We are able to unveil this conversion as being the result of pseudospin–orbit interaction and conservation of total angular momentum. Moreover, it has been shown that the underlying mechanism is of topological origin due to a nontrivial Berry phase winding and therefore, also persists in systems where angular momentum is not conserved [32]. Together with the fact that our numerical studies were carried out in a photonic analog of an atomic borophene allotrope, this paves the way for harnessing the unique properties of pseudospin-2 conical intersections in photonic applications such as pseudospin coupling and the generation of nano-scale higher-charged optical vortices [33], or scale-invariant lasing [34]. Furthermore, the existence of two chiral variants of our borophene lattice combined with their pseudospin-2 conical intersections could provide additional interesting opportunities, e.g., in bilayer borophene stacking [23,35] or chiral topological photonics [36].

Acknowledgment. We gratefully acknowledge support from the Open Access Publication Fund of the University of Münster.