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Abstract
We propose a vortex Chern insulator, motivated by recent experimental demonstrations on programmable arrangements of cavity polariton vortices by [S. Alyatkin et al., arXiv, ArXiv:2207.01850 (2022) [CrossRef] ] and [J. Wang et al, Natl. Sci. Rev. 10, Nwac096 (2022) [CrossRef] ]. In the absence of any external fields, time-reversal symmetry is spontaneously broken through polariton condensation into structured arrangements of localized co-rotating vortices. We characterize the response of the rotating condensate lattice by calculating the spectrum of Bogoliubov elementary excitations and observe the crossing of edge-states, of opposite vorticity, connecting bands with opposite Chern numbers. The emergent topologically nontrivial energy gap stems from inherent vortex anisotropic polariton-polariton interactions and does not require any spin-orbit coupling, external magnetic fields, or elliptically polarized pump fields.
Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Corrections
S. L. Harrison, A. Nalitov, P. G. Lagoudakis, and H. Sigurðsson, "Polariton vortex Chern insulator [Invited]: publisher’s note," Opt. Mater. Express 13, 3137-3137 (2023)https://opg.optica.org/ome/abstract.cfm?uri=ome-13-11-3137
3 October 2023: A correction was made to the last author's name.
1. Introduction
Topological Chern insulators are exotic materials with a bulk band gap but gapless edge states that experience topologically robust protection from scattering due to time reversal symmetry [1–4]. The difference between an ordinary- and a topological insulator (TI) is manifested in the famous quantum Hall effect describing quantization of edge conductance in the presence of a magnetic field [5]. The origin of which comes from twisting Berry phase of the Bloch bands that accumulates, over the Brillouin zone, in an integer topological invariant called the Chern number. Nearly two decades ago, the quantum spin Hall effect was demonstrated in HgTe quantum wells [6,7]. Since then, topological properties have been explored in other electronic structures [8–12], ultracold bosonic [13] and fermionic [14] gases, photonic systems [15–23] and recently in solid-state polaritonic devices ranging from semiconductor optical microcavities [24–32] to photonic metasurfaces with two-dimensional materials including transition metal dichalcogenides [33] and hexagonal boron nitrides [34] in the visible and infrared domains, respectively. Notably, experimental implementation of a polariton Chern insulator was done in Ref. [24] based on photonic graphene utilizing an external magnetic field and photonic spin-orbit coupling.
Polaritons are hybrid light-matter quasiparticles appearing in the strong coupling regime [35]. They possess extremely light effective mass and strong interactions making them highly suitable to explore out-of-equilibrium Bose-Einstein condensation at the interface of photonics and condensed matter, and elevated temperatures [36]. Quantum fluids are known to be a source of fascinating dynamical behaviour [37] with polariton condensates being no exception when combined with TIs [25,32,38]. Previous works have proposed that TIs can spontaneously emerge in the elementary excitation spectrum (also know as Bogoliubov excitations [39,40]) in structured polariton fluids containing nontrivial hopping phases [41,42] in analogy to the Haldane model [43]. Motivated by these past developments and the recent experimental demonstrations of polariton vortex lattices in optically generated potentials [44], lithographically patterned cavities [45], and photonic mesas [46] (as well vortex arrays in solid state lasers [47] and plasmon polariton systems [48]) we develop a tight binding theory for localized polariton vortices. We apply our model to the honeycomb lattice and demonstrate a vortex Chern insulator and outline some close analogies to the conventional spinor polariton Chern insulator [24,31]. Our findings underpin a nonlinear optical system (i.e., a quantum fluid of polaritons) hosting both chiral and topologically protected edges modes which offers perspectives on optical-based nontrivial signal processing schemes and uni-directional flow of information.
A cornerstone of our work is the unique spatial coupling mechanism between two optically trapped polariton condensates populating the $l = \pm 1$ orbital angular momentum (OAM) modes of each trap which was theoretically investigated by Cherotchenko et al. [49]. It was shown that ballistic expansion and overlapping of adjacent trapped condensates results in a dynamic dissipative coupling mechanism which depended on the relative angle between traps. Moreover, the coupling between adjacent co-rotating vortices and anti-rotating vortices could be optically engineered to be of similar strength. This is in contrast to spinor polariton fluids in which opposite-spin coupling defined by splitting of transverse electric and transverse magnetic (TE-TM) cavity photon modes is much weaker than same-spin coupling, severely limiting the size of the topological gap opening [24].
2. Results
2.1 Model of localized and coupled polariton condensate vortices
Exciton-polariton condensates are conventionally described in the mean field picture using the generalized Gross-Pitaevskii equation coupled to an equation describing the dynamics of an exciton reservoir [35]. Instead of solving a 2+1 dimensional nonlinear partial differential equation, we project the condensate order parameter onto a truncated basis of the first-excited angular harmonics $l = \pm 1$ localized within each trap in the lattice. This gives $2 N_l$ coupled ordinary differential equations where $N_l$ is the number of lattice sites [see Supplement 1 for derivation],
Here, $\psi _{n,\pm }$ is the phase and amplitude of the $n$th condensate component with OAM $l=\pm 1$, $J$ and $\mathcal {J}$ are the "tunneling" rates between co-rotating and counter-rotating vortices, $\alpha$ corresponds to the repulsive polariton-polariton interactions, the negative imaginary term represents a gain saturation mechanism in the adiabatic exciton-reservoir limit [35], and $p$ is the combined non-resonant optical pumping rate and cavity losses. From here on we scale time and other parameters in units of $J^{-1}$.
The sum runs over nearest neighbours and $\Theta _{n,m}$ is the angle of the link between two condensates in separate traps [49] [i.e., angle of the edges connecting lattice sites in Fig. 1(a)]. The double-angle dependence of the coupling $\mathcal {J}$ between spatially separate counter-rotating vortices is analogous to a photonic spin-orbit coupling (SOC) mechanism in patterned cavities with TE-TM splitting [50]. In fact, this turns out to be one of the needed ingredients to obtain topological gap opening in spinor polariton Chern insulators [24,31,32,51]. The directionally dependent coupling in (1) not only offers perspectives on using vortices to simulate spinor polariton TIs, but also spin transport phenomena under strong SOC ($J<\mathcal {J}$) in artificial polariton lattices [52].
Fig. 1. (a) Stripe honeycomb lattice consisting of two intercalated triangular lattices with $A$ and $B$ lattices sites shown in blue and yellow respectively. $\boldsymbol {a}_1$ and $\boldsymbol {a}_2$ are the lattice unit vectors, and $\boldsymbol {\delta }_i$, $i = \left \{1,2,3\right \}$ are the nearest-neighbour translations. A break in the lattice between chain 3 and $N$ is represented by $//$, where $N$ is even in this schematic. (b) Momentum space of the bulk lattice with both Cartesian and reciprocal lattice coordinate systems indicated. The reciprocal lattice vectors are shown by $\boldsymbol {b}_{1,2}$ in red, mapping out a diamond-shaped Brillouin zone, in addition to the hexagonal Brillouin zone shown by the grey hexagon. (c) Schematic showing the profile of the trapped vortex (or antivortex) $|\psi _{\pm }|^2$ (coloured profile). (d) Schematic of a polariton condensate vortex lattice injected and contained within ring-shaped optical traps (red) in a honeycomb structure (black dashed lines).
To observe the emergence of topologically protected edge states in the Bogoliubov elementary excitations of the polariton vortex fluid we will work with a honeycomb lattice geometry, $\Theta _{n,m} \in \{\pi, \pm \pi /3\}$, hosting Dirac points. To open the gap at those points, time reversal symmetry must be broken [1]. Typically, the gap can be opened for honeycomb polaritons through direct methods such as applying a normal-incident magnetic field or an elliptically polarized excitation source resulting in either real or effective Zeeman fine-structure splitting of spinor polaritons [24,31]. However, for truly emergent TIs, the breaking of time-reversal symmetry is spontaneous corresponding to a many-body effect [32,42] such as condensation into a macroscopic coherent state which can possess nontrivial chiral phase and density profile [44].
2.2 Honeycomb lattice
The honeycomb lattice is made up of a two-site unit cell. Each site in the unit cell is labelled $A$ and $B$ (shown by blue and yellow spots in Fig. 1), which when expanded following lattice vectors $\boldsymbol {a}_1$ and $\boldsymbol {a}_2$ makes a structure of many tessellating hexagons with a nearest neighbour separation distance of $a$. Here, the real space lattice vectors are defined as:
The hexagonal structure of the honeycomb lattice leads to a hexagonal structure in its Brillouin zone. The Dirac points are located at $\boldsymbol {K}$ and $\boldsymbol {K}^\prime$, as shown in Fig. 1, whose positions in momentum space are:
The lattice strip consists of $N$ zigzag chains [see box in Fig. 1(a)] along the $x$-axis and infinite along the $y$ axis. The index of each chain is labelled by a red number.
2.3 "Ferromagnetic" vortex solution
In order to spontaneously break time reversal symmetry in the honeycomb lattice the condensate needs to form into a solution with net OAM. There exist two particularly simple fixed point solutions to (1) under periodic boundary conditions written,
Here, $n_A$ and $n_B$ refer to sites on sublattice $A,B$. These two solutions are referred to as the ferromagnetic (FM) solutions since every site in the lattice is equally populating only the $l$ vortex component while the other $-l$ component is zero. Physically, each site has a condensate with equal OAM pointing in the same direction with all nearest neighbours either in-phase ($\psi _{n_A,l} = \psi _{n_B,l}$) or anti-phase ($\psi _{n_A,l} = -\psi _{n_B,l}$). To demonstrate the stability of each solution, we numerically solve (1) and show in Fig. 2 two example simulations using (5) (for the case of $l=+1$) with some additional random noise as an initial condition. Red and blue overlaid curves correspond to $l = \pm 1$ populations, respectively, in a $36$-site lattice with periodic boundary conditions in both $x$ and $y$ directions. The in-phase solution in Fig. 2(a) is found to be stable over a wide range of powers whereas the anti-phase solution in Fig. 2(b) is stable only at low powers (here, only up to $p\sim 0.01$). The much greater stability of the former is related to the relative sign between $\alpha$ and $J$. Repulsive polariton-polariton interactions $\alpha >0$ generally favour condensation into minimum of the polariton free energy with the in-phase solution (5) corresponding to the honeycomb $\Gamma$-point dispersion minimum. A more detailed stability analysis of these nonlinear Bloch solutions is beyond the scope of the current study.
Fig. 2. Numerical simulations showing convergence towards the two $l = +1$ fixed points solutions of (5). For the in-phase solution in (a) we set $p=1$ and for the anti-phase solution in (b) we set $p=0.01$. The left insets in each panel show schematically a portion of the vortex lattice in real space. The periodic rainbow colorscale denotes the vortex phase. The right insets show schematically where the condensate solutions would be localized in momentum space. Other parameters are $\mathcal {J}=1$ and $\alpha =4$.
In experiment, the symmetry between the $l = \pm 1$ vortices is spontaneously broken through stochastic scattering of particles that seed the condensate and trigger bosonic stimulation [44]. Methods of controlling the rotation direction of the vorticity involve cavity-photon spin-orbit coupling [53], time-sensitive optical pulsing [54], or optical stirring [55,56]. Alternatively, a large-scale stable vortex lattice could be obtained more conveniently by introducing a dissipative coupling mechanisms between nearest neighbours $\text {Im}{(J,\mathcal {J})} \neq 0$ [49]. A tunable loss-dependent coupling mechanism can enhance the stability and robustness of desirable fixed point solutions (i.e., increase their phase space attractor) as shown using metasurfaces in degenerate laser cavities [47] or structured optical pumps in cavity polariton condensates [44].
2.4 Bogoliubov elementary excitations (bogolons)
2.4.1 Linearization of the condensate equations of motion
In this section we linearize the coupled condensate equations of motion (1) and derive an eigenvalue problem for the spectrum of the Bogoliubov elementary excitations (also referred to as bogolons [39,40]). We then analyse the bogolon spectrum, around the aforementioned FM solutions, an look for evidence of topologically protected edge states. The perturbed condensate solution, in momentum space, can be written
Plugging Eq. (7) into the condensate equations of motion (Eq. (1)) and keeping only terms linear in $\delta \psi _{l}$ results in an $8\times 8$ eigenvalue problem $\omega \boldsymbol {\delta \psi }=\mathbf {M}\boldsymbol {\delta \psi }$ in momentum space in the basis $\boldsymbol {\delta \psi }=[u_{A,+},v_{A,+},u_{A,-},v_{A,-},u_{B,+},v_{B,+},u_{B,-},v_{B,-}]^\mathrm {T}$. The bogolon operator $\mathbf {M}$ describes the evolution of the perturbation (analogous to the Jacobian for coupled ordinary differential equations) and can be written neatly in a $2\times 2$ block form,
The diagonal blocks depend on the condensate solution and are decomposed into the following parts,
The first term describes the gain of the system. The second and third terms describe the real and imaginary contribution to the bogolon energies from the nonlinear terms in Eq. (1), respectively:
Notice that the Bogolon matrix (Eq. (10)) is the same for the in-phase and the anti-phase solutions (Eq. (5)) because all entries are either quadratic in $\psi _{\mu,l}$ or become zero.
The off-diagonal blocks $\mathbf {J}_{\boldsymbol {k}}$ describe coupling between the sublattices [see Supplement 1],
Here, $\Theta _n$ is the momentum space equivalent of $\Theta _{n,m}$ in Eq. (1). The angles between nearest neighbours are $\Theta _0 = \pi$, $\Theta _1 = \pi /3$ and $\Theta _2 = -\pi /3$.
2.4.2 PT symmetry phase transition
One feature of the bogolon operator (Eq. (8)) is its inherent non-Hermitian structure as can be seen from just the matrix describing polariton interactions (Eq. (10)). Non-Hermitian topological physics has seen a lot of interest in the recent years [57] with enriched classification of topological phases of matter [58], exceptional points [59], and the non-Hermitian skin effect [60]. However, for the purpose of this study, we will consider a FM vortex lattice in which the parameters are chosen such that the bogolon dispersion is purely real-valued up to a constant imaginary factor and with separable bulk bands (i.e., $\omega _n(\mathbf {k}) \neq \omega _m(\mathbf {k})$ for all bands $m\neq n$). This then precludes the need for specialized treatment to classify non-Hermitian topological phases in the considered vortex lattice in terms of exceptional points and spectral winding numbers [59], which will become a subject of a later study.
It is worth noting that non-Hermitian linear operators with a completely real spectrum are closely related to parity-time (PT) symmetric operators (or more generally pseudo-Hermitian operators). Such operators have also gathered a lot of attention recently, especially in photonic systems with tailored gain and loss landscapes [61]. In order to establish this PT-symmetry boundary in the bogolon spectra, we calculate the standard deviation of the imaginary parts of the eigenvalues from Eq. (8), $\sqrt {\langle \text {Im}{(\omega _{n,\mathbf {k}})^2}\rangle - \langle \text {Im}{(\omega _{n,\mathbf {k}})}\rangle ^2}$ over all bands in the first Brillouin zone. Zero deviation corresponds to a purely real spectrum (up to a constant imaginary factor) which marks the PT-symmetric phase. Figure 3 shows the results in the $\alpha$-$\mathcal {J}$ plane for the case of a FM vortex solution and $p=1$. A clear boundary can be distinguished separating the zeros of the deviation measure from the rest. In this study, we focus on results within the PT-symmetric phase and leave the topological properties of the bogolons in the PT-broken phase for a future study.
2.4.3 Bogolon dispersion relation
Plugging the FM solution (Eq. (5)) into our Bogoliubov operators (Eqs. (10) and (11)) we proceed to diagonalize Eq. (8) for a $N=50$ honeycomb lattice strip (periodic boundaries along the $y$ axis). The bogolon dispersion relation for two different values of $\mathcal {J}$ is shown in Fig. 4(a,b) where we observe gap opening with four and two chiral edge states, respectively. The blue and red colors indicate localization of the bogolon on the left and right edges respectively. The transition from four to two edge states is a topological transition in which the Chern numbers of the upper and lower bands change from $C_n=\pm 2$ to $C_n = \pm 1$ as we verify in the next section. Physically, in the absence of the condensate $\psi _{n,l}=0$ additional Dirac points appear at the $M$-point of the lattice for a critical value of $\mathcal {J}=1/2$ causing a change in the dispersion topology [31].
Fig. 4. Positive energy bogolon dispersion relation obtained from diagonalizing (8) for both the (green line) infinite and (blue/black/red) $N=50$ finite honeycomb strip. The blue-red colours represent localisation of the eigenstates on the left and right lattice edges, respectively. In all plots, $k_x = 2\pi /3a$, and $\alpha p = 4$.
To verify that the gap is complete in the bulk we diagonalize Eq. (8) for periodic boundaries along both $x$ and $y$. The corresponding "infinite lattice" bulk dispersion is shown with green solid lines which indeed do not cross. By varying the cross-vortex tunneling strength $\mathcal {J}$ we find a peak gap size of around $E_\text {gap} \approx 3$ on an interval of reasonable condensate interaction energies $4 \leq \alpha p \leq 20$ [see Supplemental Material]. Considering a typical polariton tunneling strength of $J \sim 100$ $\mu$eV in patterned inorganic polariton cavities [24] the gap size we report for the bogolons could reach values around $E_g \sim 300$ $\mu$eV which is well within resolution of angle-resolved spectrometer techniques [40]. We stress that all the topological features in the positive energy bogolon branches are symmetrically present in the negative energy branches, and that only measurement of the former is sufficient to verify nontrivial gap opening. For completeness, in the Supplement 1 we show that our results qualitatively, and nearly quantitatively, match the conventional spinor polariton TI which utilizes TE-TM splitting and a real external magnetic field [24,31,51].
The presence of chiral edge states in the dispersion implies that bogolons experience the FM vortex lattice as an effective out-of-plane magnetic field that breaks time-reversal symmetry with consequent nontrivial topological gap opening. This effect can be appreciated by scrutinizing the energy shift coming from just a single isolated condensate onto the bogolons. For the case of a single isolated condensate (i.e., $\mathbf {J}_\mathbf {k} = 0_{4\times 4}$) containing a $l=+1$ vortex the real-part of the bogolon spectrum is only determined by Eq. (10). Since there is no lattice in this case we now conveniently work in a smaller basis of $[u_+, v_+, u_-, v_-]^\text {T}$,
The off-diagonal terms mix together $l=+1$ positive and negative energy bogolons ($|u_+\rangle$ and $|v_+\rangle$) while $l=-1$ bogolons ($|u_-\rangle$ and $|v_-\rangle$) are still good eigenstates. The eigenvalues of (14) are $\omega _{1,2} = \pm \sqrt {3} \alpha p$ and $\omega _{3,4} = \pm 2\alpha p$ and the corresponding eigenvectors,
Therefore, the vortex splits the energies of $l = \pm 1$ bogolons by amount $\Delta = |\omega _{3,4} - \omega _{1,2}| = \alpha p (2-\sqrt {3})$ similar to a spin in a magnetic field. When the condensate vorticity reverses the splitting also reverses. The manifestation of this splitting in a honeycomb lattice is the removal of the band degeneracies around the Dirac points corresponding to a topologically nontrivial gap opening.
2.4.4 Chern numbers
To further verify that the edge states observed in Fig. 4 are topologically nontrivial in nature we calculate the Berry curvature and the associated Chern numbers of the upper and lower bulk bogolon bands. For band $n$, with normalised eigenvector $\vert \phi _{n} (\boldsymbol {k}) \rangle$, the Berry curvature in two dimensions is,
The Chern number of the $n$th band can be found from integrating over the Berry curvature within the first Brillouin zone,
In order to calculate the Berry curvature and the associated Chern number of the lattice bands on a discrete $\mathbf {k}$-space grid we use the approach of Fukui et al. [62].
We show results only for the positive energy branches of the bogolon dispersion since the negative branches (also referred to as ghost branches) are just reflected about $\omega =0$. Counting the bands from top to bottom, the cumulative Berry curvature and Chern numbers for the positive energy “conduction” and “valence” bands are written $\mathbf {B}_{c(v)} = \mathbf {B}_{1(3)} + \mathbf {B}_{2(4)}$ and $C_{c(v)} = C_{1(3)} + C_{2(4)}$ respectively. The results are shown in Fig. 5 for $\mathcal {J}=0.1$ and $\mathcal {J}=1.0$. For weaker opposite-vortex coupling strengths a clear trigonal warping in the Berry curvature, localised around the $\boldsymbol {K}$ and $\boldsymbol {K}^\prime$ points, can be observed which has been reported before for spinor polariton TIs [25]. From Eq. (18) we obtain band Chern numbers of $C_c = -2$ and $C_v = +2$ which, when the bulk-boundary correspondence is invoked [1], results in four topologically protected edge states crossing the gap as seen in Fig. 4(a). At higher opposite-vortex coupling strengths the system undergoes a topological transition where additional Dirac cones form and annihilate in pairs, reducing the Chern number from $\pm 2$ to $\pm 1$. Our calculations of the band Chern numbers verify that the bogolon gap opening is an emergent topologically nontrivial transition induced by the vortex lattice [41,42].
Fig. 5. Berry curvature summed over (counting from the highest energy band) (a,c) the first pair and (b,d) the second pair of bands of the infinite honeycomb vortex lattice following from Bogoliubov elementary excitation with (a,b) $\mathcal {J} = 0.1$ and (c,d) $\mathcal {J} = 1.0$. The Brillouin zone follows the red diamond in Fig. 1(c). Here, $\alpha p = 4$.
2.5 Numerical simulations
To demonstrate the chiral response of the bogolons at the edges of the FM vortex honeycomb strip we perform numerical simulations of (1) under resonant injection. Here, we choose the more stable in-phase condensate FM solution from Eq. (5) as an initial condition with $l=+1$ vorticity. We choose parameters corresponding to the dispersion shown in Fig. 4(b) and set $p=0.1$ and $\alpha p = 4$, and $\mathcal {J}=1$. Numerically solving the equations of motion (Eq. (1)) in time we observe that the FM solution remains stable in the finite geometry and only experiences slight density modulations at the edges of the $N=10$ honeycomb strip (not shown). We note that do not include any spatial disorder in our simulations for simplicity. The estimated gap size (see Section 2.4.3) exceeds the $\sim 10$ $\mu$eV disorder levels in strain compensated planar microcavities [63] and thus should survive in high quality cavities.
We then resonantly drive the antivortex bogolons ($l=-1$) at two lattice sites located at opposite edges of the strip (see black arrows in Fig. 6). The resonant excitation corresponds to a term $+P_n e^{-i \tilde {\omega } t}$ appended to the right hand side of (1). The tightly localized resonant drives at the opposite lattice sites correspond to a wide excitation in momentum space which overlaps with the edge modes [blue and red lines in Fig. 4(b)]. We choose a laser frequency value of $\tilde {\omega }=7.5$ which lies at the midgap energy in Fig. 4(b). The simulation reveals that the antivortex perturbations (or injections) propagate in opposite directions at the opposite edges with weak penetration into the bulk, as expected in Chern insulators. This underlines the chiral behaviour of the system due to the influence of the background $\psi _+$ vortex lattice (not shown). We point out that the perturbations are attenuated, with a lifetime $\tau _\text {bog} = 1/p$, which is the reason they decay along the strip. This means that the nonresonant pump power $p$ can be adjusted to change the propagation length of the injected signal but at the cost of a weaker condensate background and a smaller gap.
Fig. 6. Simulation using Eq. (1) showing the chiral dynamics of the bogolons. Here, the $l=+1$ vortex polaritons have condensed into an in-phase FM vortex lattice (not shown) with the $l=-1$ antivortex component (shown) mostly empty. The antivortex component $\vert \psi _{-}(t)\vert ^2$ is then perturbed with a weak normally incident ($\mathbf {k}=0$) cw resonant injection at the indicated sites displaying clear chiral bogolon response. Here the parameters are: $\alpha = 40$, $p = 0.1$, $\mathcal {J} = 1$, $P = 0.1$, and $\tilde {\omega } = 7.5$.
3. Conclusions
We have designed a tight-binding model describing the coupling between localized polariton vortices and identify analogies with conventional spin-orbit-coupled spinor polaritons [50]. Our model offers a path towards simulating spinor polariton phenomena [52] with tunable nonlinearities and anistropic coupling strengths. In particular, we investigate the paradigmatic honeycomb lattice [24,31], possessing Dirac points, and show that when all vortices start rotating in the same direction time-reversal symmetry is spontaneously broken for the condensate Bogoliubov elementary excitations. A gap opens around the lattice Dirac points with the emergence of chiral topologically protected edge states bridging bulk bands of opposite Chern numbers. We note that the driven-dissipative part of our equations plays a purely supportive role in the topological gap opening of bogolons, implying that our findings can be extended to other types of structured quantum fluids or nonlinear lasers with spontaneously broken symmetries. Our findings can be extended to Lieb and Kagome geometries and generalized to higher order vortex modes and could play a role in optical based nonlinear topological signal processing schemes with uni-directional flow of vortical information.
Funding
Icelandic Research Fund (239552-051); Horizon 2020 Framework Programme (899141, 964770); Russian Science Foundation (22-12-00144); Engineering and Physical Sciences Research Council (EP/T517859/1).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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