ARTICLE Received 30 Jun 2016 | Accepted 3 Oct 2016 | Published 28 Nov 2016 DOI: 10.1038/ncomms13486 OPEN Topological magnetoplasmon Dafei Jin1, Ling Lu2,3, Zhong Wang4,5, Chen Fang2,3, John D. Joannopoulos3, Marin Soljacˇic´3, Liang Fu3 & Nicholas X. Fang1 Classical wave fields are real-valued, 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 two-dimensional (2D) magnetoplasmon, which bears gapped bulk states and gapless one-way edge states near-zero frequency, is topologically analogous to the 2D topological p þ ip superconductor with chiral Majorana edge states and zero modes. We further predict a new type of one-way edge magnetoplasmon at the interface of opposite magnetic domains, and demonstrate the existence of zero-frequency 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. 1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 2 Institute of Physics, Chinese Academy of Sciences/Beijing National Laboratory for Condensed Matter Physics, Beijing 100190, China. 3 Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA. 4 Institute for Advanced Study, Tsinghua University, Beijing 100084, China. 5 Collaborative Innovation Center of Quantum Matter, Beijing 100871, China. Correspondence and requests for materials should be addressed to L.L. (email: [email protected]) or to N.X.F. (email: [email protected]). NATURE COMMUNICATIONS | 7:13486 | DOI: 10.1038/ncomms13486 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13486 ince the introduction of chiral edge states from two- liquid-helium surface33, semiconductor junctions34,38,49 and dimensional (2D) quantum Hall systems into photonic graphene50,51. Ssystems1–8, the investigation of band topology has been actively extended into many other 2D or three-dimensional (3D) Results bosonic systems9,10, including phonon11–15, magnon16, Governing equations. We consider a 2DEG confined in the z ¼ 0 17,18 19 exciton and polariton . Up to now, topological phases of and r ¼ xex þ yey plane. The dynamics of MPs are governed by 2D plasmon have not been identified, in spite of some related the linearized charge-continuity equation and the Lorentz force discussion20–26 and a recent proposal of Weyl plasmon27. equation52,53. In the frequency domain, they are Besides, the previous lines of work mainly focus on finite or ; o ; o ; frequencies. Little attention has been paid to near-zero À i ðÞ¼Àrr r Á jrðÞ ð1Þ frequencies, where new symmetries and new topological states À iojrðÞ; o ¼ aðÞr ErðÞ; o À o ðÞr jrðÞÂ; o e ; ð2Þ may emerge (as elaborated below). c z Plasmon is a unique type of bosonic excitations. where rðÞr; o is the variation of 2D electron-density off Microscopically, it consists of collective motion of electron–hole equilibrium and jrðÞ; o is the induced 2D current density. pairs in a Coulomb interaction electron gas, whereas aðÞr is a space-dependent coefficient that gives a 2D local macroscopically it appears as coherent electron-density oscilla- longitudinal conductivity, sðÞ¼r; o iaðÞr =o. ocðrÞ is the tions. Under most circumstances, it can be well described by cyclotron frequency from a perpendicularly applied static classical density (and velocity) fields on the hydrodynamic level. magnetic field, B (r) ¼ B (r)e . For the massive electrons chosen 0 0 z 2 These fields, like all other classical fields, are intrinsically real- for our demonstration in this paper, aðÞ¼r e n0ðÞr and mà valued and respect an unbreakable particle–hole symmetry. For a o ðÞ¼r eB0ðÞr , in which n (r) is the equilibrium electron-density c mÃc 0 particle–hole symmetric system, its Hamiltonian H transforms distribution, m* is the effective mass and c is the speed of under an antiunitary particle–hole conjugate operation C via light32,54. C À 1HC ¼ À H. (Here C2 ¼þ1 for bosons.) This property ErðÞ; o in equation 2 is the electric field evaluated within the ensures a symmetric spectrum oðÞq with respect to zero frequency, z ¼ 0 plane. It is generallyR a nonlocal function of the 2D current oðÞ¼Àq oðÞÀ q , in which q is the wavevector. The associated density, ErðÞ¼; o dr0 KrðÞÁ; r0; o jrðÞ0; o , where KrðÞ; r0; o wave fields areR thus superpositions of complex-conjugate is an integration kernel determined by Maxwell’s equations. þ iðÞqÁr À jjoðÞq t à À iðÞqÁr À jjoðÞq t pairs, FðÞ¼r; t dq½F qe þFqe . The For a model system shown in Fig. 1a, and in the nonretarded spin-0 real-scalar field governed by the Klein–Gordon equation limit (E solely comes from the Coulomb interaction)52,53, the and the spin-1 real-vector field governed by Maxwell’s equations Fourier transformed field–current relation can be attested to be also share the same feature. (see Supplementary Note 1) Particle–hole symmetry greatly expands the classification E ðÞq; o q2 q q j ðÞq; o of topological phases, according to the results of x 2p x x y x : ¼ ioqxðÞq 2 ð3Þ tenfold classification28,29. For example, the 2D quantum EyðÞq; o qxqy qy jyðÞq; o Hall phase belongs to the class-A in 2D with broken 1 Here q ¼ |q|, and xðÞ¼q fgEA cothðÞþqdA EB cothðÞqdB is a time-reversal symmetry T . The Su–Schrieffer–Heeger model 2 54 q-dependent screening function , in which dA and dB are the and the recently studied phonon zero modes12,15 belong thicknesses, and EA and EB are the permittivities of the dielectrics to the class-BDI in one-dimension (1D) with particle–hole on the two sides of 2DEG. A generalization of equation 3 can be symmetry C2 ¼þ1 and time-reversal symmetry T 2 ¼þ1. 2 made to include photon retardation (plasmon becomes plasmon However, the class-D 2D topological phase with C ¼þ1 and polariton)32,55,56 (see Supplementary Note 2). The Drude loss by broken T has so far only been proposed in p þ ip 30 electron–phonon collision can be included in this formulation by superconductors , which carry chiral Majorana edge states and introducing a finite lifetime in equation 2 (ref. 54). Majorana zero modes. In this work, we show that the historically studied Bulk Hamiltonian and bulk states. We analytically solve for the 2D magnetoplasmon (MP)31–41 belongs to the class-D 2D case of homogeneous bulk plasmons, where n (r), B (r), aðÞr and topological phase with unbreakable C and broken T .Itis 0 0 o ðÞr are all constants. B and o can be positive or negative, governed by three-component linear equations carrying a similar c 0 c depending on the direction of the magnetic field to be parallel or structure as that of the two-band Bogoliubov–de Gennes (BdG) antiparallel to e . We find that equations 1–3 can be casted into a equations of the p þ ip topological superconductor30,42. z Hermitian eigenvalue problem. The Hamiltonian H is block- It contains a gapped bulk spectrum around zero frequency, and diagonalized in the momentum space, acting on a generalized possesses gapless topological edge states and zero-frequency current density vector J, bound states (zero modes). Many properties of p þ ip 0 1 superconductor can find their analogy in 2D MP. Therefore, j ðÞq; o @ R A 2D MP provides the first realized example of class-D in the oJqðÞ¼H; o ðÞq JqðÞ; o ; JqðÞ; o jDðÞq; o ; ð4Þ 28,29 10-class table . The experimentally observed edge MP states jLðÞq; o dated back to 1985 (ref. 37) are in fact topologically protected, ðÞq; o p1ffiffi f ðÞÆq; o ðÞgq; o analogous to the 1D chiral Majorana edge states. Some where jL;R 2 jx ijy are the left- and simulations of Majorana-like states in photonics have been right-handedqffiffiffiffiffiffiffiffi chiral components of current density, and 43–47 reported ; however, they only appear at finite frequencies j ðÞq; o 2pa rðÞq; o opðÞq rðÞq; o is a generalized around which the particle–hole symmetry does not rigorously D qxðÞq q 48 density component. hold, unless using high nonlinearity . In addition, we are able to sffiffiffiffiffiffiffiffiffiffi ( derive non-zero Chern numbers adhering to the band topology of 2paq qffiffiffiffiffiffiffiffiffiffiffiffiffiupq; ðÞfor small q ; 2D MP. On the basis of this, we are able to predict a new type of o ðÞ¼q ! ð5Þ p x 4pa q; ðÞfor large q ; one-way edge MP states on the boundary between opposite ðÞq EA þ EB magnetic domains, and the existence of MP zero modes on a hollow disk. Our prediction can be experimentally verified is the dispersion relation of conventionalqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2D bulk plasmon in any 2D electron gas (2DEG) systems, such as charged without a magnetic field. u ¼ 4padAdB is the effective p EAdB þ EBdA 2 NATURE COMMUNICATIONS | 7:13486 | DOI: 10.1038/ncomms13486 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13486 ARTICLE a bc c = 0 c > 0 / * / * Metal 4 4 C = +1 dA B0 z A 2 2 y x c 2DEG 0 0 C = 0 qd qd d −10 −50 5 10 −10 −50 5 10 B B c −2 −2 Metal C = −1 −4 −4 Figure 1 | Model system and analytically calculated homogeneous bulk spectra. (a) Schematics of the structure. (The presence of metal plates and dielectrics permits a general theoretical treatment but is not essential to the topological behaviours of 2D MP).