Abstract
Classical wave fields are realvalued, ensuring the wave states at opposite frequencies and momenta to be inherently identical. Such a particle–hole symmetry can open up new possibilities for topological phenomena in classical systems. Here we show that the historically studied twodimensional (2D) magnetoplasmon, which bears gapped bulk states and gapless oneway edge states nearzero frequency, is topologically analogous to the 2D topological p+ip superconductor with chiral Majorana edge states and zero modes. We further predict a new type of oneway edge magnetoplasmon at the interface of opposite magnetic domains, and demonstrate the existence of zerofrequency modes bounded at the peripheries of a hollow disk. These findings can be readily verified in experiment, and can greatly enrich the topological phases in bosonic and classical systems.
Introduction
Since the introduction of chiral edge states from twodimensional (2D) quantum Hall systems into photonic systems^{1,2,3,4,5,6,7,8}, the investigation of band topology has been actively extended into many other 2D or threedimensional (3D) bosonic systems^{9,10}, including phonon^{11,12,13,14,15}, magnon^{16}, exciton^{17,18} and polariton^{19}. Up to now, topological phases of 2D plasmon have not been identified, in spite of some related discussion^{20,21,22,23,24,25,26} and a recent proposal of Weyl plasmon^{27}. Besides, the previous lines of work mainly focus on finite frequencies. Little attention has been paid to nearzero frequencies, where new symmetries and new topological states may emerge (as elaborated below).
Plasmon is a unique type of bosonic excitations. Microscopically, it consists of collective motion of electron–hole pairs in a Coulomb interaction electron gas, whereas macroscopically it appears as coherent electrondensity oscillations. Under most circumstances, it can be well described by classical density (and velocity) fields on the hydrodynamic level. These fields, like all other classical fields, are intrinsically realvalued and respect an unbreakable particle–hole symmetry. For a particle–hole symmetric system, its Hamiltonian transforms under an antiunitary particle–hole conjugate operation via . (Here for bosons.) This property ensures a symmetric spectrum with respect to zero frequency, , in which q is the wavevector. The associated wave fields are thus superpositions of complexconjugate pairs, . The spin0 realscalar field governed by the Klein–Gordon equation and the spin1 realvector field governed by Maxwell’s equations also share the same feature.
Particle–hole symmetry greatly expands the classification of topological phases, according to the results of tenfold classification^{28,29}. For example, the 2D quantum Hall phase belongs to the classA in 2D with broken timereversal symmetry . The Su–Schrieffer–Heeger model and the recently studied phonon zero modes^{12,15} belong to the classBDI in onedimension (1D) with particle–hole symmetry and timereversal symmetry . However, the classD 2D topological phase with and broken has so far only been proposed in p+ip superconductors^{30}, which carry chiral Majorana edge states and Majorana zero modes.
In this work, we show that the historically studied 2D magnetoplasmon (MP)^{31,32,33,34,35,36,37,38,39,40,41} belongs to the classD 2D topological phase with unbreakable and broken . It is governed by threecomponent linear equations carrying a similar structure as that of the twoband Bogoliubov–de Gennes (BdG) equations of the p+ip topological superconductor^{30,42}. It contains a gapped bulk spectrum around zero frequency, and possesses gapless topological edge states and zerofrequency bound states (zero modes). Many properties of p+ip superconductor can find their analogy in 2D MP. Therefore, 2D MP provides the first realized example of classD in the 10class table^{28,29}. The experimentally observed edge MP states dated back to 1985 (ref. 37) are in fact topologically protected, analogous to the 1D chiral Majorana edge states. Some simulations of Majoranalike states in photonics have been reported^{43,44,45,46,47}; however, they only appear at finite frequencies around which the particle–hole symmetry does not rigorously hold, unless using high nonlinearity^{48}. In addition, we are able to derive nonzero Chern numbers adhering to the band topology of 2D MP. On the basis of this, we are able to predict a new type of oneway edge MP states on the boundary between opposite magnetic domains, and the existence of MP zero modes on a hollow disk. Our prediction can be experimentally verified in any 2D electron gas (2DEG) systems, such as charged liquidhelium surface^{33}, semiconductor junctions^{34,38,49} and graphene^{50,51}.
Results
Governing equations
We consider a 2DEG confined in the z=0 and r=xe_{x}+ye_{y} plane. The dynamics of MPs are governed by the linearized chargecontinuity equation and the Lorentz force equation^{52,53}. In the frequency domain, they are
where is the variation of 2D electrondensity off equilibrium and is the induced 2D current density. is a spacedependent coefficient that gives a 2D local longitudinal conductivity, . is the cyclotron frequency from a perpendicularly applied static magnetic field, B_{0}(r)=B_{0}(r)e_{z}. For the massive electrons chosen for our demonstration in this paper, and , in which n_{0}(r) is the equilibrium electrondensity distribution, m_{*} is the effective mass and c is the speed of light^{32,54}.
in equation 2 is the electric field evaluated within the z=0 plane. It is generally a nonlocal function of the 2D current density, , where is an integration kernel determined by Maxwell’s equations. For a model system shown in Fig. 1a, and in the nonretarded limit (E solely comes from the Coulomb interaction)^{52,53}, the Fourier transformed field–current relation can be attested to be (see Supplementary Note 1)
Here q=q, and is a qdependent screening function^{54}, in which d_{A} and d_{B} are the thicknesses, and ε_{A} and ε_{B} are the permittivities of the dielectrics on the two sides of 2DEG. A generalization of equation 3 can be made to include photon retardation (plasmon becomes plasmon polariton)^{32,55,56} (see Supplementary Note 2). The Drude loss by electron–phonon collision can be included in this formulation by introducing a finite lifetime in equation 2 (ref. 54).
Bulk Hamiltonian and bulk states
We analytically solve for the case of homogeneous bulk plasmons, where n_{0}(r), B_{0}(r), and are all constants. B_{0} and can be positive or negative, depending on the direction of the magnetic field to be parallel or antiparallel to e_{z}. We find that equations 1, 2, 3 can be casted into a Hermitian eigenvalue problem. The Hamiltonian is blockdiagonalized in the momentum space, acting on a generalized current density vector J,
where are the left and righthanded chiral components of current density, and is a generalized density component.
is the dispersion relation of conventional 2D bulk plasmon without a magnetic field. is the effective plasmon velocity originated from the screening of Coulomb interaction at long wavelengths by surrounding metals (see Fig. 1a). asymptotically equals at long wavelengths.
The bulk Hamiltonian and its longwavelength limit read
Strikingly, this longwavelength threeband has a very similar structure to that of the Bogoliubov–de Gennes hamiltonian of a 2D p+ip topological superconductor^{42}, which has two bulk bands. Here the odd number of bands is crucial for the topological consequences shown below. For an even number of bosonic bulk bands, it has been proved that the summed Chern number is always zero for the bands below or above the zero frequency^{16}.
We can define the antiunitary particle–hole operation and the antiunitary timereversal operation here,
where K is the complexconjugate operator and stands for a momentum flipping. As can be checked, acts as the complex conjugation for all the field components j_{x}, j_{y} and j_{D}, while additionally reverses the sign for j_{x} and j_{y} (refer to Methods: Cartesian representation). Our Hamiltonian has symmetry,
but broken symmetry by the nonzero .
The calculated homogeneous bulk spectra, plotted in Fig. 1b,c, are
The corresponding (unnormalized) wavefunctions are
The positive and negativefrequency bands are gapped by . The zerofrequency band represents purely rotational currents (∇_{r}·j=0, ∇_{r} × j≠0) balanced by static charge density distribution.
It is worthwhile to point out that the 2D bulk plasmon discussed here is distinctively different from the surface plasmon (or surface plasmon polariton) in a 3D boundary (see Supplementary Note 3).
Chern number on an infinite momentum plane
Unlike the regular lattice geometries whose Brillouin zone is a torus, our system is invariant under continuous translation, where the unbounded wavevector plane can be mapped on a Riemann sphere^{30,57}. As long as the Berry curvature decays faster than q^{−2} as q→∞, the Berry phase around the north pole (q=∞) of Riemann sphere is zero and the Chern number C is quantized^{57}. For our 2D MP, we can verify this analytically,
where dS_{q} is the momentum–space surface element, and the term in the curly bracket is the Berry curvature. Since the Berry curvature changes its sign under the particle–hole symmetry, the positive and negativefrequency bands must have opposite Chern numbers, and the zerofrequency band, with odd Berry curvature, must have zero Chern number.
Majoranatype oneway edge states
We are able to calculate the analytical edge solutions in the longwavelength limit q→0, and numerical edge solutions for unrestricted wavelengths, as plotted in Fig. 2. When q→0, the Hamiltonian in equation 6 becomes local, allowing us to replace q_{x} and q_{y} with the operators −i∂_{x} and −i∂_{y} and to solve for the edge states by matching boundary conditions^{25,53}. We consider a 1D edge situated at x=0 caused by a discontinuity either in n_{0}(x) (ConfigurationI) or in B_{0}(x) (ConfigurationII).
In ConfigurationI as shown in Fig. 2a, we let the x>0 region be filled with 2DEG, , while the x<0 region be vacuum, and the magnetic field be uniformly applied parallel to e_{z}, . Since we know that the change of Chern number across the edge is ΔC=±1, there must be a single topological edge state present in this configuration. We look for solutions that behave like , , in the x>0 region, where is an evanescent wavenumber. The boundary condition is , meaning that the component of the current normal to the edge must vanish. Therefore, the edge solutions have to satisfy , whose righthand side is always positive. Since we have chosen , q_{y} must be positive too. The edge state is hence oneway propagating. (We discard the unphysical solution that yields a null wavefunction.) The physical solution is , which gives an edge state spectrum,
This is a gapless state running across the zerofrequency band as shown in Fig. 2c. It is a bosonic analogue to the chiral Majorana edge states in the topological superconductor. Its wavefunction is
The zerofrequency edge state at q_{y}=0 has the same finite decay length and is well localized at the edge, despite being degenerate with the flat middle band.
In ConfigurationII as shown in Fig. 2b, we let the 2DEG have a constant electron density throughout the whole space, , but the magnetic field have opposite signs in the two regions ( for x>0 and for x<0). This is a novel configuration with ΔC=±2 across the edge, and permits two topological edge states. The boundary conditions are and , meaning that both the normal current and density must be continuous across the edge. In our longwavelength approximation, the first edge solution has the same spectrum and wavefunction as those of ConfigurationI, except for a symmetric extension of the wavefunction into the x<0 region (compare the plots of ρ and j_{y} in red in Fig. 2a,b). The second edge solution satisfies , and so q_{y} must be negative to ensure positive. This edge state is also oneway with a spectrum,
as shown in Fig. 2d. Its wavefunction is
It is antisymmetric for j_{y} as plotted in green in Fig. 2b. The antisymmetry plus the continuity condition for j_{D} here make the density oscillations vanish identically; therefore, the second edge state carries only current oscillations about the magnetic domain wall.
Figure 2c,d gives our numerically calculated bulk and edge spectra that are not restricted by the smallq local approximation (refer to Methods: Numerical scheme). When q is small, the numerical results accurately agree with our analytical derivation above. When q is large, the first edge dispersion asymptotically approaches the bulk bands and eventually connects to them at q→∞. The second edge dispersion drops gradually towards the zerofrequency bulk band and connects to it at q→∞. It does show a (tiny) positive group velocity indicating the correct chiral direction, in spite of its negative phase velocity (refer to Fig. 2b,d). Physically though, the plasmon picture breaks down in the largeq regime, where electron–hole pair production takes place.
Besides, we should note that there exists a nontopological gapped edge state in ConfigurationI (see the blue curve in Fig. 2c), which is consistent with the literature^{54}. At long wavelengths q→0, this state resembles a bulk state bearing an excitation gap of . Its edge confinement shows up only at short wavelengths. As elaborated below, under a parameter evolution from ConfigurationI to II, this edge state evolves, remarkably, into the second gapless topological edge state in ConfigurationII.
Oneway propagation of edge states
Owing to the topological protection, oneway edge states are immune to backscattering from a random defect. In Fig. 3, we display two snapshots from our realtime simulation. Edge states are excited by a point source oscillating at a frequency of within the bulk gap. The generated waves propagate only towards the right. A sharp zigzag defect has been purposely inserted into the route of propagation. The waves then exactly follow the edge profile and insist on propagating forward without undergoing any backscattering. The slightly visible fluctuation to the left of the point source is completely local (nonpropagating). It is caused by the unique longrange Coulomb interaction in this system, different from the more familiar photonic and acoustic systems.
Figure 3a corresponds to the traditional ConfigurationI, where the physical edge is formed by the density termination of 2DEG at vacuum. Although the oneway nature in this configuration is known historically, its absolute robustness due to the topological protection is less known and is manifested here.
Figure 3b corresponds to our new ConfigurationII. Only the first edge state classified in Fig. 2 is excited here; the second edge state has an energy too close to the band edge and hence is not excited here. On the basis of our argument on topology above, ConfigurationII permits protected oneway edge states on a magnetic domain boundary even if the 2DEG may be homogeneous. Figure 3b vividly demonstrates this scenario. Furthermore, it shows the perfect immunization to backscattering when the two magnetic domains drastically penetrate each other.
Evolution of edge states
We find that the adiabatic mode evolution of the MP edge states described by a threeband model here is rather different from that in 2D topological insulators or superconductors, which can be commonly described by a twoband or fourband model. The zerofrequency bulk band here plays a critical role for the appearance and disappearance of the topological edge states. We have investigated the evolution corresponding to the aforementioned two configurations shown in Fig. 2.
We first study the interaction between two topological edge states propagating along the opposite directions on two 2DEG edges in the same magnetic field. The result is shown in Fig. 4a. When we gradually narrow the spacer, we let the equilibrium density in the spacer to gradually change from 0 to the bulk density n_{0}. This allows the two topological edge states to couple and end up with a bulk configuration. Instead of opening a gap and entering the top band (which would happen in the conventional twoband system), the two topological edge states here (marked in red) fall down into the flat middle band. In addition, there are two nontopological edge states (marked in blue, and refer to Fig. 2c) in this configuration. They move completely into the bulk during this process.
We then study the interaction between two topological edge states propagating along the same direction on the two 2DEG edges under opposite magnetic fields. The result is shown in Fig. 4b. When we gradually narrow the spacer, we find that one of the edge states (marked in red) goes slightly upwards, while the other falls downwards into the flat middle band. In the meanwhile, one of the nontopological edge states (marked in blue) detaches from the top band, continuously bends downwards until it reaches the frequency , where it transits into the second topological edge state (marked in green). According to our previous argument, this edge state connects to the middle band at the momentum infinity. At all times, there are two gapless edge states. This is consistent with the preserved difference of Chern numbers (ΔC=±2) between the left part and right part of 2DEG.
MP on a hollow disk
For infinitely long edges as discussed above, the first topological edge state when k_{y}→0 merges into the zerofrequency middle band, as can be seen in Fig. 2c,d. Importantly, we want to quest whether such zerofrequency modes preserve (not being gapped) when we wrap a long edge into a small circle. We want to investigate the behaviours of 2D MP on, for instance, a hollow disk geometry depicted in Fig. 5a.
We write the lowenergy longwavelength Hamiltonian in realspace polar coordinates and still use the chiral representation , . The eigenequation is
The azimuthal symmetry ensures the eigensolutions of j_{r}, and j_{D} to pick up a common factor , where is the integer of discretized angular momentum. This lowenergy longwavelength problem can be solved semianalytically. In all situations, j_{r} and are derived quantities from j_{D},
Depending on the energy to be above or below the bandgap, we have
where , , and are the th order Bessel and Neumann functions, and is the modified Bessel function. , and are coefficients. The nonormalcurrent boundary condition determines the allowed eigenfrequencies for every given . In this approximate model, the solutions above the bandgap are all continuous bulklike and within the bandgap are all discrete edgelike, as shown in Fig. 5b.
Majoranatype bound state: zero mode
We are particularly interested in the limit . According to equations 19 and 21, the boundary condition can be satisfied if and only if , which completely annihilates the radial current, j_{r}(r, 0, 0)=0. The zeromode profile is given by
j_{D} describes a static charge distribution; j_{ϕ} describes a d.c. circulating current. The electric field generated by j_{D} balances out the Lorentz force due to j_{ϕ} and B_{0}. The overall amplitude and phase in can be arbitrary, but j_{D} and j_{ϕ} are phaselocked to each other by a ratio of sgn(ω_{c}), that is, the sign of magnetic field. In the chiral representation, with the arbitrary phase ignored, the zeromode wavefunction behaves in the below manner:
We see that the Majoranatype topological edge state within the bandgap indeed preserve as and , on the finitesized inner edge of a hollow disk. It is easy to verify that shrinking the hole radius will only reduce the number of discrete edge states at a finite , but not kill the zero mode. These observations are consistent with the k_{y}→0 limit of the infinite straight edge case (see Fig. 2c,d). Nevertheless, we should cautiously note that the zero mode here is not isolated. It degenerates with a large number of other zerofrequency modes in the middle band (see Supplementary Note 4).
The guaranteed existence of zero mode in a hollow disk geometry for 2D MP is reminiscent of the Majorana zero modes in the twoband p+ip topological superconductor^{30}. In the latter case, a flux inside the vortex core is needed to balance the antiperiodic boundary condition in the direction. By comparison, our threeband Hamiltonian here hosts zero mode with periodic boundary condition, that is the wavefunction in equation 23 returns to itself when ϕ rotates 2π. More generally, similar kind of discretized edge states and zero modes must exist on a noncircular geometry as well, and can be on both the inner and outer edges, if the geometry is finite. For a finite geometry, the inner and outer edge zero modes must come out in pairs. If one closes the inner boundary and turns the hollow disk into a solid disk, then both modes must vanish. Otherwise, the vortex currents associated with both zero modes are singular at the centre, and are unphysical.
Discussion
Although the configuration of 2D MP looks almost identical to that of the quantum Hall effect (QHE), there are fundamental differences^{39}. QHE deals with electron (fermionic) transport, whereas 2D MP deals with collective electrondensity oscillations (bosonic excitations). In QHE, the bulk is electronically insulating, whereas in 2D MP, the bulk is electronically conducting. QHE experiments usually require a high magnetic field (≳5 T) and a high electron density (≳10^{11 }cm^{−2}). By comparison, MP experiments normally does not require a high magnetic field or a high electron density. Moreover, the bandgap in QHE is for electron transport, while the bandgap for 2D MP is for electrondensity oscillations.
2D MP belongs to the topological classD, which is different from the quantum Hall classA. The topological edge states in MP are not governed by the traditional bulkedge correspondence of the quantum Hall states. The traditional rule states that the number of gapless edge states inside a gap is equal to the sum of all the bulk Chern numbers from the energy zero up to the gap. In our topological MP, the lowest bulk band (zerofrequency flat band) contributes zero Chern number. A generalized bulkedge correspondence must be established by considering the particle–hole symmetry that extends the spectrum into the ‘redundant’ negativefrequency regime. This accounts for the Berry flux exchanged across the zero frequency. Only by doing so, the sum of all the bulk Chern numbers, spanning both the positive and negativefrequency regimes, can be correctly kept zero.
Experiments for 2D plasmon were performed by pioneering researchers dated back to 1970s on the charged liquid helium surface^{33} and in the semiconductor inversion layer^{34}. 2D MP was first observed in the semiconductor inversion layer in 1977 (ref. 35) and edge MP was first observed on the liquid helium surface in 1985 (ref. 37) by measuring radiofrequency absorption peaks. The nonreciprocity of edge MP was verified in semiconductor heterojunctions at microwave frequencies^{40}. Recently, similar experiments have been performed on 2D materials^{58,59,60,61} and on the surface of a topological insulator^{62}. However, all the previous studies on edge MPs correspond to ConfigurationI in our paper, where the physical edge is formed between a 2DEG and vacuum under a uniform magnetic field.
Our prediction of new oneway edge MP in ConfigurationII has not been reported, despite there have been similar studies in the QHE systems^{63,64}. Our prediction can, in principle, be verified in any 2DEG system as long as a pair of opposite magnetic fields is applied across. The magnetic domain boundary can be experimentally created by, for example, two concentric solenoids of opposite currents or ferromagnetic materials. The gapless oneway edge modes can be mapped out by microwave bulk transmission and by Fourier transforming nearfield scans along the edges, as demonstrated in ref. 65 for a topological photonic system. In Fig. 6, we provide a schematic experimental design. A semiconductor heterojunction serves as the platform for 2DEG. A ferromagnetic film is grown on the top and is polarized into up and down domains. For the circular domain boundary as sketched, discretized oneway edge states can travel around. A meandering coplanar waveguide can be fabricated on the back of the substrate. One can then send in and receive microwave signals at a frequency below , and in the meanwhile monitor the characteristic absorption, which signifies the excitation of oneway edge plasmons.
Our proposed zerofrequency bound modes feature localized charge accumulation (inducing a static electric field) and circulating current (inducing a static magnetic field). These effects are in principle measurable when the zero mode is actuated by charge or current injection.
In summary, we have revealed the salient topological nature of 2D MP as the first realization of the classD topological phase. The predicted new Majoranatype oneway edge states and zero modes can be verified experimentally. Our work opens a new dimension for topological bosons via introducing the bosonic phase protected by the particle–hole symmetry. We anticipate similar realizations for other bosonic particles^{66} and discoveries of new topological phases, after combining the particle–hole symmetry with other symmetries^{67}, for example, the timereversal and many kinds of spatial symmetries^{10,68}.
Methods
Theoretical model
Our theoretical model of 2D MP is based on a hydrodynamic formalism. In equilibrium, the 2D electron number density is n_{0}(r), which is determined by the device structure, doping concentration and gating condition^{38}. Off equilibrium, the density changes to n(r, t). For 2D MP, we are only concerned with the small deviation of 2D charge density, ρ(r, t)=−e{n(r,t)−n_{0}(r)}, and the induced 2D current density up to the linear order, j(r, t)=−en_{0}(r)v(r,t), where v(r, t) is the local velocity of electron gas restricted to move in the z=0 plane only. For massive electrons, the conductivity coefficient is and the cyclotron frequency is (refs 32, 54). For massless electrons in, for example, graphene, they are generalized to and , where is the Fermi velocity and E_{F} is the Fermi energy^{69,70}. Hence, our theory works for both massive and massless electrons.
We consider a model system shown in Fig. 1a. With different choices of the material constants and structural parameters, it can lead to different practical 2DEG systems, such as charged liquidhelium surface, metal–insulator–semiconductor junction, topgated graphene transistor, and so on. In Fig. 1a, the 2DEG lies in the z=0 plane. The 0<z<d_{A} region is filled with an insulator (or vacuum) of the dielectric constant ε_{A}. The −d_{B}<z<0 region is filled with another insulator (or semiconductor) of the dielectric constant ε_{B}. The z>d_{A} and z<−d_{B} regions are filled with perfect metals whose dielectric constant is ε_{M}=−∞ in the (radio to microwave) frequency range of MP. Encapsulating the system by two metals mimics the experimental configurations with top and bottom electrodes, and in the meanwhile, cutoffs the theoretically infinitely longranged Coulomb interaction.
For the results shown in the main article, we simply choose ε_{A}=ε_{B}=ε_{0}=1 (vacuum permittivity), m_{*}=m_{0} (bare electron mass), d_{A}=d_{B}=d=1 μm and n_{0}=1 μm^{−2}=10^{8 }cm^{−2}. We can thus obtain a characteristic frequency, , (that is, ), which is used to normalize the frequencies. For the case of cyclotron frequency , we have B_{0}=0.22 T.
Cartesian representation
In the main article we have used the chiral representation for the bulk Hamiltonian and generalized current vector, in order to demonstrate the similarity between our 2D MP and the 2D p+ip topological superconductor. However, it is sometimes more convenient to adopt the traditional Cartesian representation, especially when we solve for the edge modes and apply boundary conditions on the edge. In the Cartesian representation, the bulk Hamiltonian equation reads
where the Hamiltonian in this representation reads
The particle–hole and timereversal operators in this Cartesian representation are
Clearly, is just a complex conjugation on the matrix or vector elements (combined with a momentum reversal q→−q here, when written on the singleparticle momentum bases).
Numerical scheme
To numerically solve the nonuniform edge state problem, which is unrestricted by the longwavelength approximation, we expand the governing equations 1 and 2 in the main article into plane waves and use the form of interaction in the momentum space equation 3. We can derive a matrix equation in the Cartesian representation,
The submatrices , , , , , and the subcolumn vectors , , all have the dimension of the number of plane waves used for expansion.
The elements of submatrices are
and
where and with are the Fourier transform of and . For the system with edges along the y axis, the expansion only needs to be done in the x direction while keeping as good quantum numbers. Typically, we use 128 plane waves to do the expansion, and the standard eigensolver to obtain all the eigenfrequencies and eigenvectors.
The 2D realspace calculation for Fig. 3 is performed by a homemade finitedifference timedomain code. The field quantities are defined on a square grid. Equations 1 and 2 are adopted for the realtime evolution after replacing with ∂_{t}. A pointsource term of a given driving frequency is added to the righthand side of equation 2.
Data availability
All relevant data are available from the authors on request.
Additional information
How to cite this article: Jin, D. et al. Topological magnetoplasmon. Nat. Commun. 7, 13486 doi: 10.1038/ncomms13486 (2016).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.
Haldane, F. D. M. & Raghu, S. Possible realization of directional optical waveguides in photonic crystals with broken timereversal symmetry. Phys. Rev. Lett. 100, 013904 (2008).
 2.
Wang, Z., Chong, Y., Joannopoulos, J. D. & Soljacic, M. Observation of unidirectional backscatteringimmune topological electromagnetic states. Nature 461, 772–775 (2009).
 3.
Hafezi, M., Demler, E. A., Lukin, M. D. & Taylor, J. M. Robust optical delay lines with topological protection. Nat. Phys. 7, 907–912 (2011).
 4.
Fang, K., Yu, Z. & Fan, S. Realizing effective magnetic field for photons by controlling the phase of dynamic modulation. Nat. Photon 6, 782–787 (2012).
 5.
Khanikaev, A. B. et al. Photonic topological insulators. Nat. Mater. 12, 233–239 (2013).
 6.
Rechtsman, M. C. et al. Photonic floquet topological insulators. Nature 496, 196–200 (2013).
 7.
Lu, L., Joannopoulos, J. D. & Soljacic, M. Topological photonics. Nat. Photon 8, 821–829 (2014).
 8.
Zhang, C. et al. Observation of the AdlerBellJackiw chiral anomaly in a Weyl semimetal. Preprint at https://arxiv.org/abs/1503.02630 (2015).
 9.
Lu, L. et al. Experimental observation of Weyl points. Science 349, 622–624 (2015).
 10.
Lu, L. et al. Symmetryprotected topological photonic crystal in three dimensions. Nat. Phys. 12, 337–340 (2016).
 11.
Prodan, E. & Prodan, C. Topological phonon modes and their role in dynamic instability of microtubules. Phys. Rev. Lett. 103, 248101 (2009).
 12.
Kane, C. L. & Lubensky, T. C. Topological boundary modes in isostatic lattices. Nat. Phys. 10, 39 (2014).
 13.
Paulose, J., Chen, B. G.G. & Vitelli, V. Topological modes bound to dislocations in mechanical metamaterials. Nat. Phys. 11, 153–156 (2015).
 14.
Wang, P., Lu, L. & Bertoldi, K. Topological phononic crystals with oneway elastic edge waves. Phys. Rev. Lett. 115, 104302 (2015).
 15.
Huber, S. D. Topological mechanics. Nat. Phys. 12, 621–623 (2016).
 16.
Shindou, R., Matsumoto, R., Murakami, S. & Ohe, J.i. Topological chiral magnonic edge mode in a magnonic crystal. Phys. Rev. B 87, 174427 (2013).
 17.
YuenZhou, J., Saikin, S. K., Yao, N. Y. & AspuruGuzik, A. Topologically protected excitons in porphyrin thin films. Nat. Mater. 13, 1026–1032 (2014).
 18.
YuenZhou, J. et al. Plexciton dirac points and topological modes. Nat. Commun. 7, 11783 (2016).
 19.
Karzig, T., Bardyn, C.E., Lindner, N. H. & Refael, G. Topological polaritons. Phys. Rev. X 5, 031001 (2015).
 20.
Ling, C. W., Xiao, M., Chan, C. T., Yu, S. F. & Fung, K. H. Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles. Opt. Express 23, 2021–2031 (2015).
 21.
Cheng, Q., Pan, Y., Wang, Q., Li, T. & Zhu, S. Topologically protected interface mode in plasmonic waveguide arrays. Laser Photon. Rev. 9, 392–398 (2015).
 22.
Sinev, I. S. et al. Mapping plasmonic topological states at the nanoscale. Nanoscale 7, 11904–11908 (2015).
 23.
Yu, Z., Veronis, G., Wang, Z. & Fan, S. Oneway electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal. Phys. Rev. Lett. 100, 023902 (2008).
 24.
Gao, W. et al. Topological photonic phase in chiral hyperbolic metamaterials. Phys. Rev. Lett. 114, 037402 (2015).
 25.
Song, J. C. & Rudner, M. S. Chiral plasmons without magnetic field. Proc. Natl Acad. Sci USA 113, 4658–4663 (2016).
 26.
Kumar, A. et al. Chiral plasmon in gapped Dirac systems. Phys. Rev. B 93, 041413(R) (2016).
 27.
Gao, W. et al. Plasmon Weyl degeneracies in magnetized plasma. Nat. Commun. 7, 12435 (2016).
 28.
Kitaev, A. Periodic table for topological insulators and superconductors. AIP Conf. Proc. 1134, 22 (2009).
 29.
Schnyder, A. P., Ryu, S., Furusaki, A. & Ludwig, A. W. W. Classification of topological insulators and superconductors in three spatial dimensions. Phys. Rev. B 78, 195125 (2008).
 30.
Read, N. & Green, D. Paired states of fermions in two dimensions with breaking of parity and timereversal symmetries and the fractional quantum hall effect. Phys. Rev. B 61, 10267 (2000).
 31.
Stern, F. Polarizability of a twodimensional electron gas. Phys. Rev. Lett. 18, 546 (1967).
 32.
Chiu, K. W. & Quinn, J. J. Plasma oscillations of a twodimensional electron gas in a strong magnetic field. Phys. Rev. B 9, 4724 (1974).
 33.
Grimes, C. C. & Adams, G. Observation of twodimensional plasmons and electronripplon scattering in a sheet of electrons on liquid helium. Phys. Rev. Lett. 36, 145 (1976).
 34.
Allen, S. J., Tsui, D. C. & Logan, R. A. Observation of the twodimensional plasmon in silicon inversion layers. Phys. Rev. Lett. 38, 980 (1977).
 35.
Theis, T. N., Kotthaus, J. P. & Stiles, P. J. Twodimensional magnetoplasmon in the silicon inversion layer. Solid State Commun. 24, 273 (1977).
 36.
Glattli, D., Andrei, E., Deville, G., Poitrenaud, J. & Williams, F. Dynamical hall effect in a twodimensional classical plasma. Phys. Rev. Lett. 54, 1710 (1985).
 37.
Mast, D. B., Dahm, A. J. & Fetter, A. L. Observation of bulk and edge magnetoplasmons in a twodimensional electron fluid. Phys. Rev. Lett. 54, 1706 (1985).
 38.
Ando, T., Fowler, A. B. & Stern, F. Electronic properties of twodimensional systems. Rev. Mod. Phys. 54, 437 (1982).
 39.
Andrei, E., Glattli, D., Williams, F. & Heiblum, M. Low frequency collective excitations in the quantumhall system. Surf. Sci. 196, 501–506 (1988).
 40.
Ashoori, R. C., Stormer, H. L., Pfeiffer, L. N., Baldwin, K. W. & West, K. Edge magnetoplasmons in the time domain. Phys. Rev. B 45, 3894 (1992).
 41.
Aleiner, I. L. & Glazman, L. I. Novel edge excitations of twodimensional electron liquid in a magnetic field. Phys. Rev. Lett. 72, 2935 (1994).
 42.
Bernevig, B. A. Topological Insulators and Topological Superconductors Princeton University Press (2013).
 43.
Tan, W., Chen, L., Ji, X. & Lin, H.Q. Photonic simulation of topological superconductor edge state and zeroenergy mode at a vortex. Sci. Rep. 4, 7381 (2014).
 44.
RodriguezLara, B. M. & MoyaCessa, H. M. Optical simulation of Majorana physics. Phys. Rev. A 89, 015803 (2014).
 45.
Keil, R. et al. Optical simulation of charge conservation violation and Majorana dynamics. Optica 2, 454–459 (2015).
 46.
Poddubny, A., Miroshnichenko, A., Slobozhanyuk, A. & Kivshar, Y. Topological majorana states in zigzag chains of plasmonic nanoparticles. ACS Photon. 1, 101–105 (2014).
 47.
Iadecola, T., Schuster, T. & Chamon, C. Nonabelian braiding of light. Phys. Rev. Lett. 117, 073901 (2016).
 48.
Peano, V., Houde, M., Brendel, C., Marquardt, F. & Clerk, A. A. Topological phase transitions and chiral inelastic transport induced by the squeezing of light. Nat. Commun. 7, 10779 (2016).
 49.
Stone, K. et al. Photovoltaic oscillations due to edgemagnetoplasmon modes in a very highmobility twodimensional electron gas. Phys. Rev. B 76, 153306 (2007).
 50.
Jablan, M., Buljan, H. & Soljacic, M. Plasmonics in graphene at infrared frequencies. Phys. Rev. B 80, 245435 (2009).
 51.
Koppens, F. H. L., Chang, D. E. & Garcia de Abajo, F. J. Graphene plasmonics: a platform for strong lightmatter interactions. Nano Lett. 11, 3370–3377 (2011).
 52.
Fetter, A. L. Edge magnetoplasmons in a bounded twodimensional electron fluid. Phys. Rev. B 32, 7676 (1985).
 53.
Fetter, A. L. Edge magnetoplasmons in a twodimensional electron fluid confined to a halfplane. Phys. Rev. B 33, 3717 (1986).
 54.
Volkov, V. A. & Mikhailov, S. A. Edge magnetoplasmons low frequency weakly damped excitations. Sov. Phys. JETP 67, 1639 (1988).
 55.
Kukushkin, I. V. et al. Observation of retardation effects in the spectrum of twodimensional plasmons. Phys. Rev. Lett. 90, 156801 (2003).
 56.
Deng, H.Y. & Wakabayashi, K. Retardation effects on plasma waves in graphene, topological insulators, and quantum wires. Phys. Rev. B 92, 045434 (2015).
 57.
Silveirinha, M. G. Chern invariants for continuous media. Phys. Rev. B 92, 125153 (2015).
 58.
Crassee, I. et al. Intrinsic terahertz plasmons and magnetoplasmons in large scale monolayer graphene. Nano Lett. 12, 2470–2474 (2012).
 59.
Yan, H. et al. Infrared spectroscopy of tunable Dirac terahertz magnetoplasmons in graphene. Nano Lett. 12, 3766–3771 (2012).
 60.
Petkoviic, I. et al. Carrier drift velocity and edge magnetoplasmons in graphene. Phys. Rev. Lett. 110, 016801 (2013).
 61.
Kumada, N. et al. Resonant edge magnetoplasmons and their decay in graphene. Phys. Rev. Lett. 113, 266601 (2014).
 62.
Autore, M. et al. Observation of magnetoplasmons in bi_{2}se_{3} topological insulator. ACS Photon. 2, 1231–1235 (2015).
 63.
Reijniers, J. & Peeters, F. M. Snake orbits and related magnetic edge states. J. Phys. Conds. Matter 12, 9771C9786 (2000).
 64.
Ye, P. D. et al. Electrons in a periodic magnetic field induced by a regular array of micromagnets. Phys. Rev. Lett. 74, 3014–3016 (1995).
 65.
Skirlo, S. A. et al. Experimental observation of large chern numbers in photonic crystals. Phys. Rev. Lett. 115, 253901 (2015).
 66.
Chen, W.J., Zhang, Z.Q., Dong, J.W. & Chan, C. T. Symmetryprotected transport in a pseudospinpolarized waveguide. Nat. Commun. 6, 8183 (2015).
 67.
Chiu, C.K., Teo, J. C., Schnyder, A. P. & Ryu, S. Classification of topological quantum matter with symmetries. Rev. Mod. Phys. 88, 035005 (2015).
 68.
Fu, L. Topological crystalline insulators. Phys. Rev. Lett. 106, 106802 (2011).
 69.
Wang, W., Apell, P. & Kinaret, J. Edge plasmons in graphene nanostructures. Phys. Rev. B 84, 085423 (2011).
 70.
Sounas, D. L. & Caloz, C. Edge surface modes in magnetically biased chemically doped graphene strips. Appl. Phys. Lett. 99, 231902 (2011).
Acknowledgements
We would like to thank Duncan Haldane and Aris Alexandradinata for discussing the reality condition. We are also thankful to Patrick A. Lee, Xiaogang Wen, Frank Wilczek, Yang Qi, Junwei Liu and Fan Wang for helpful discussions. D.J. and N.X.F. were supported by AFOSR MURI under Award No. FA95501210488. L.L. was supported in part by the MRSEC Program of the NSF under Award No. DMR1419807, the National ThousandYoungTalents Program of China, the National 973 Program of China (Nos 2013CB632704 and 2013CB922404) and the National Natural Science Foundation of China (No. 11434017). L.L. and C.F. were supported in part by the Ministry of Science and Technology of China under Grant No. 2016YFA0302400. Z.W. was supported by NSFC under Grant No. 11674189. C.F. and L.F. were supported by the DOE Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award No. DESC0010526. J.D.J. was supported in part by the U.S. ARO through the ISN, under Contract No. W911NF13D0001. M.S. and L.L. were supported in part by the MIT S3TEC EFRC of DOE under Grant No. DESC0001299.
Author information
Affiliations
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
 Dafei Jin
 & Nicholas X. Fang
Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China
 Ling Lu
 & Chen Fang
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
 Ling Lu
 , Chen Fang
 , John D. Joannopoulos
 , Marin Soljačić
 & Liang Fu
Institute for Advanced Study, Tsinghua University, Beijing 100084, China
 Zhong Wang
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
 Zhong Wang
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Contributions
D.J. and L.L. conceived the concept and wrote the manuscript. D.J. performed the analytical and numerical calculation. Z.W., C.F. and L.F. joined effective discussion and provided pivotal advices. J.D.J. and M.S. offered additional guidance and proofread the manuscript. L.L. and N.X.F. directed the project.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Ling Lu or Nicholas X. Fang.
Supplementary information
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Supplementary Information
Supplementary Notes 14 and Supplementary References
Videos
 1.
Supplementary Movie 1
Realtime movie for oneway propagating topological edge states that are immune to backscattering at a sharp zigzag defect. The 2DEG is in contact with vacuum under a singledomain magnetic field.
 2.
Supplementary Movie 2
Realtime movie for oneway propagating topological edge states that are immune to backscattering at a sharp zigzag defect. The 2DEG is uniformly filled in the whole space under an oppositedomain magnetic field.
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Further reading

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Plasmon propagation pushed to the limit
Nature (2018)
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