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 i ðÞ¼ rr r jrðÞ ð1Þ frequencies, where new symmetries and new topological states iojrðÞ; o ¼ aðÞr ErðÞ; o o ðÞr jrðÞ ; 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 m valued and respect an unbreakable particle–hole symmetry. For a o ðÞ¼r eB0ðÞr , in which n (r) is the equilibrium electron-density c m c 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 1HC ¼ 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 oðÞ¼ q oðÞ q , in which q is the wavevector. The associated density, ErðÞ¼; o dr0 KrðÞ ; 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 iðÞq r jjoðÞq t iðÞq r 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, ðÞ q; o p1ffiffi f ðÞ q; 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 ðÞ q; o 2pa rðÞ q; 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). (b) No magnetic field is present. The two circles connected by a dashed line represent the particle–hole symmetry of the dispersion curves. (c) A uniform magnetic field is applied along the positive z direction. Different branches carry different Chern numbers. The negative-frequency branch reflects the redundant degrees of freedom of the real-valued classical field and so is shaded. Here oc ¼ o , where o is a characteristic frequency defined to normalize the frequencies (see Methods: Theoretical model). plasmon velocity originated from the screening of Coulomb symmetry, interaction at long wavelengths by surrounding metals 1 (see Fig. 1a). jDðÞq; o asymptotically equals uprðÞq; o at long CHðÞCq ¼ H ðÞq ; ð8Þ wavelengths. o The bulk Hamiltonian HðÞq and its long-wavelength limit but broken T symmetry by the non-zero c. read The calculated homogeneous bulk spectra, plotted in Fig. 1b,c, 0 1 are opðÞq q iq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ o x pffiffi y 0 B c q 2 C 2 2 B C o ðÞ¼ q oc þ opðÞq and o0ðÞ¼q 0: ð9Þ B o ðÞq o ðÞq C B p qx þffiffiiqy p qx ffiffiiqy C HðÞ¼q B p 0 p C @ q 2 q 2 A The corresponding (unnormalized) wavefunctions are 0 1 opðÞq qx þffiffiiqy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 p oc q iq q 2 x pffiffi y 2 2 0 1 oc þ opðÞq oc 0 1 B 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1 1 u q iq ð6Þ B C pðÞxpffiffi y B C q!0@ 2 2 A B þ oc 0 C B 2 2 2 C 0 ; B 2 C J ðÞ/q B qx þ qy opðÞq C / B C @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 1 1 q!0B u q þ iq u q iq C q þ iq 2 2 ! B pðÞpx ffiffi y pðÞpx ffiffi y C: x pffiffi y o2 þ o2ðÞq o B 0 C 2 c p c B 2 2 C @ A upðÞqx þ iqy ð10Þ 0 pffiffi o 2 c 0 qffiffiffiffiffiffiffiffiffiffiffiffi 1 q iq Strikingly, this long-wavelength three-band HðÞq has a very x pffiffi y o2ðÞq 0 1 B qffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 p C 0 similar structure to that of the Bogoliubov–de Gennes hamilto- B C q!0@ A 42 J ðÞ/q B q2 þ q2 o C / 1 : ð11Þ nian of a 2D p þ ip topological superconductor , which has two 0 @ x qyffiffiffiffiffiffiffiffiffiffiffiffic A bulk bands. Here the odd number of bands is crucial for the qx þ iqy 0 þ pffiffi o2ðÞq topological consequences shown below. For an even number of 2 p bosonic bulk bands, it has been proved that the summed Chern number is always zero for the bands below or above the zero The positive- and negative-frequency bands are gapped by oc. 16 The zero-frequency band represents purely rotational currents frequency . a We can define the antiunitary particle–hole operation C and (rr j ¼ 0, rr j 0) balanced by static charge density the antiunitary time-reversal operation T here, distribution. 0 1 0 1 It is worthwhile to point out that the 2D bulk plasmon
001 00 1 discussed here is distinctively different from the surface plasmon B C B C @ A @ A (or surface plasmon polariton) in a 3D boundary (see C¼ 010K ; T¼ 010K Supplementary Note 3). 100 q! q 10 0 q! q ð7Þ Chern number on an infinite momentum plane. Unlike the regular lattice geometries whose Brillouin zone is a torus, our where K is the complex-conjugate operator and j q! q stands system is invariant under continuous translation, where the for a momentum flipping. As can be checked, C acts as the unbounded wavevector plane can be mapped on a Riemann 30,57 2 complex conjugation for all the field components jx, jy and jD, sphere . As long as the Berry curvature decays faster than q - while T additionally reverses the sign for jx and jy (refer to as q N, the Berry phase around the north pole (q ¼ N)of Methods: Cartesian representation). Our Hamiltonian has C Riemann sphere is zero and the Chern number C is quantized57.
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For our 2D MP, we can verify this analytically, boundary condition is jx j x¼0 þ ¼ 0, meaning that the component ()ÀÁ of the current normal to the edge must vanish. Therefore, the Z y rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ½ J ðÞq irq J ðÞq C ¼ dSq rq o ¼ k o2 þ u2 2 k2 2p y edge solutions have to satisfy qy c c p qy , 8 2 ½ J ðÞq39J ðÞq Z < = whose right-hand side is always positive. Since we have chosen 1 o o ¼ jjo 40, q must be positive too. The edge state is hence @ 4qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffic 5 c c y ¼ qdq: q ; ð12Þ one-way propagating. (We discard the unphysical solution k ¼ q q o2 þ o2ðÞq y c p that yields a null wavefunction.) The physical solution is k ¼ oc, up ¼ sgnðÞoc ; which gives an edge state spectrum, ÀÁ ÀÁ ðÞ1 o qy ¼ upqy; qy 0 : ð14Þ C0 ¼ 0; ð13Þ edge where dSq is the momentum–space surface element, and the term This is a gapless state running across the zero-frequency band as in the curly bracket is the Berry curvature. Since the Berry shown in Fig. 2c. It is a bosonic analogue to the chiral Majorana curvature changes its sign under the particle–hole symmetry, the edge states in the topological superconductor. Its wavefunction is 0 1 positive and negative-frequency bands must have opposite Chern piffiffi ÀÁ 2 oc numbers, and the zero-frequency band, with odd Berry curvature, ðÞ1 @ A x þ iqyy J x; q / 1 e up ; ðÞx40 : ð15Þ must have zero Chern number. edge y þ piffiffi 2
Majorana-type one-way edge states. We are able to calculate the The zero-frequency edge state at qy ¼ 0 has the same finite decay - oc analytical edge solutions in the long-wavelength limit q 0, and length k ¼ u and is well localized at the edge, despite being numerical edge solutions for unrestricted wavelengths, as plotted p - degenerate with the flat middle band. in Fig. 2. When q 0, the Hamiltonian in equation 6 becomes In Configuration-II as shown in Fig. 2b, we let the 2DEG local, allowing us to replace qx and qy with the operators i@x have a constant electron density throughout the whole and i@y and to solve for the edge states by matching boundary space, u ðÞ¼x u , but the magnetic field have opposite signs conditions25,53. We consider a 1D edge situated at x ¼ 0 caused p p in the two regions (ocðÞ¼þx jjoc 40 for x40 and by a discontinuity either in n0(x) (Configuration-I) or in B0(x) ocðÞ¼ x jjoc o0 for xo0). This is a novel configuration with (Configuration-II). DC ¼ ±2 across the edge, and permits two topological edge In Configuration-IÀÁ as shown in Fig. 2a, we let the x40 region states. The boundary conditions are jx j x¼0 ¼ jx j x¼0 þ and be filled with 2DEG, u ðÞ¼x40 u , while the x 0 region be ÀÁp p o jD j x¼0 ¼ jD j x¼0 þ , meaning that both the normal current vacuum, upðÞ¼xo0 0 and the magnetic field be uniformly and density must be continuous across the edge. In our applied parallel to ez, ðÞocðÞ¼x oc40 . Since we know that the long-wavelength approximation, the first edge solution has the change of Chern number across the edge is DC ¼ ±1, there must same spectrum and wavefunction as those of Configuration-I, be a single topological edge state present in this configuration. We except for a symmetric extension of the wavefunction into the kx þ iqyy iot look for solutions that behave like e , ðÞqx ¼ ik ,in xo0 region (compare the plots of r and jy in red in Fig. 2a,b). the x40 region, where k40 is an evanescent wavenumber. The The second edge solution satisfies k ¼ qy, and so qy must be
a c d Configuration-I: single domain Configuration-I Configuration-II , j / / y * * 4 4
x Bulk states Bulk states
Second 2 topological 2 Non-topological edge state y edge state First Vacuum (C = 0) 2DEG (C = +1) c topological c Topological x edge state z edge state 0 0 qyd qyd b −10 −50 5 10 −10 −50 5 10 Configuration-II: opposite domains , jy c c
−2 −2 x
−4 −4 2DEG (C = −1) 2DEG (C = +1)
Figure 2 | Schematics of the system configurations and numerically calculated bulk and edge spectra. (a) Configuration-I: A 2DEG is in contact with vacuum under a uniform magnetic field along the positive z direction. (b) Configuration-II: A 2DEG is uniformly filled in the whole space but two opposite magnetic fields are applied along the positive and negative z direction. The edge state profiles of density and current components are shown as well. For Configuration-II, the two edge states have opposite symmetries. (c) The spectra corresponding to Configuration-I. (d) The spectra corresponding to Configuration-II. The negative-frequency part reflects the redundant degrees of freedom of the real-valued classical field and so is shaded.
4 NATURE COMMUNICATIONS | 7:13486 | DOI: 10.1038/ncomms13486 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13486 ARTICLE negative to ensure k positive. This edge state is also one-way with Fig. 2c), which is consistent with the literature54. At long a spectrum, wavelengths q-0, this state resembles a bulk state bearing an ÀÁ ÀÁ ðÞ2 excitation gap of oc. Its edge confinement shows up only at short oedge qy ¼ jjoc ; qy 0 ; ð16Þ wavelengths. As elaborated below, under a parameter evolution from Configuration-I to II, this edge state evolves, remarkably, as shown in Fig. 2d. Its wavefunction is 0 1 into the second gapless topological edge state in Configuration-II. 1 1 ÀÁ 2 2 ðÞ2 ; @ A qyx þ iqyy; _ : Jedge x qy / 0 e ðÞx 0 ð17Þ One-way propagation of edge states. Owing to the topological 1 1 2 2 protection, one-way edge states are immune to backscattering from a random defect. In Fig. 3, we display two snapshots from It is antisymmetric for jy as plotted in green in Fig. 2b. The antisymmetry plus the continuity condition for j here make the our real-time simulation. Edge states are excited by a point source D 1 o density oscillations vanish identically; therefore, the second oscillating at a frequency of 2 c within the bulk gap. The edge state carries only current oscillations about the magnetic generated waves propagate only towards the right. A sharp zigzag domain wall. defect has been purposely inserted into the route of propagation. Figure 2c,d gives our numerically calculated bulk and edge The waves then exactly follow the edge profile and insist on spectra that are not restricted by the small-q local approximation propagating forward without undergoing any backscattering. (refer to Methods: Numerical scheme). When q is small, the The slightly visible fluctuation to the left of the point source is numerical results accurately agree with our analytical derivation completely local (non-propagating). It is caused by the unique above. When q is large, the first edge dispersion asymptotically long-range Coulomb interaction in this system, different from the approaches the bulk bands and eventually connects to them at more familiar photonic and acoustic systems. q-N. The second edge dispersion drops gradually towards the Figure 3a corresponds to the traditional Configuration-I, where zero-frequency bulk band and connects to it at the physical edge is formed by the density termination of 2DEG at q-N. It does show a (tiny) positive group velocity indicating vacuum. Although the one-way nature in this configuration is the correct chiral direction, in spite of its negative phase velocity known historically, its absolute robustness due to the topological (refer to Fig. 2b,d). Physically though, the plasmon picture breaks protection is less known and is manifested here. down in the large-q regime, where electron–hole pair production Figure 3b corresponds to our new Configuration-II. Only the takes place. first edge state classified in Fig. 2 is excited here; the second edge Besides, we should note that there exists a non-topological state has an energy too close to the band edge oc and hence is not gapped edge state in Configuration-I (see the blue curve in excited here. On the basis of our argument on topology above, Configuration-II permits protected one-way edge states on a magnetic domain boundary even if the 2DEG may be a Configuration-I: single domain homogeneous. Figure 3b vividly demonstrates this scenario. Furthermore, it shows the perfect immunization to backscattering Vacuum when the two magnetic domains drastically penetrate each other. +1 Point source Density boundary z Evolution of edge states. We find that the adiabatic mode x y evolution of the MP edge states described by a three-band model here is rather different from that in 2D topological insulators 2DEG or superconductors, which can be commonly described by a two-band or four-band model. The zero-frequency bulk band here plays a critical role for the appearance and disappearance of b 0 Configuration-II: opposite domains the topological edge states. We have investigated the evolution corresponding to the aforementioned two configurations shown in Fig. 2. 2DEG We first study the interaction between two topological edge Point source Domain boundary states propagating along the opposite directions on two 2DEG z edges in the same magnetic field. The result is shown in Fig. 4a. x y When we gradually narrow the spacer, we let the equilibrium −1 density in the spacer to gradually change from 0 to the bulk 2DEG density n0. 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 Figure 3 | Snapshots of one-way propagating topological edge states conventional two-band system), the two topological edge states immune to backscattering at a sharp zigzag defect. Here the cyclotron here (marked in red) fall down into the flat middle band. In frequency oc ¼ 4o , which corresponds to B0 ¼ 0.9 T. The driving addition, there are two non-topological edge states (marked in 1 frequency for the point source is 2 oc, which corresponds to f ¼ 12.7 GHz. blue, and refer to Fig. 2c) in this configuration. They move (Refer to Methods: Theoretical model). (a) Configuration-I: a 2DEG is in completely into the bulk during this process. contact with vacuum under a single-domain magnetic field. We then study the interaction between two topological edge (b) Configuration-II: a 2DEG is uniformly filled in the whole space under an states propagating along the same direction on the two 2DEG opposite-domain magnetic field. Note that the profile is plotted for the edges under opposite magnetic fields. The result is shown in electric scalar potential, which is non-zero in the vacuum region. The Fig. 4b. When we gradually narrow the spacer, we find that one of difference between a and b on the wavelengths is consistent with the the edge states (marked in red) goes slightly upwards, while the slightly different dispersion slope of the topological edge states between other falls downwards into the flat middle band. In the Configuration-I and II. (See online Supplementary Movies 1 and 2 for real- meanwhile, one of the non-topological edge states (marked in time evolution). blue) detaches from the top band, continuously bends downwards
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a
qy qy qy
b
qy qy qy
Figure 4 | Evolution of the edge states when two pieces of 2DEG are brought into contact with each other. The density spacer is gradually filled to the bulk value as the distance shrinks to zero. (a) The magnetic field is uniformly applied along the positive z direction in the whole space. (b) The magnetic field has opposite directions between the left part and right part of 2DEG.
until it reaches the frequency o , where it transits into the second c a b topological edge state (marked in green). According to our / * previous argument, this edge state connects to the middle band at 4 the momentum infinity. At all times, there are two gapless edge Bulk states states. This is consistent with the preserved difference of Chern numbers (DC ¼ ±2) between the left part and right part of a 2 2DEG. B0 Discretized Zero mode edge states MP on a hollow disk. For infinitely long edges as discussed ν - above, the first topological edge state when ky 0 merges into the −10−50 5 10 zero-frequency middle band, as can be seen in Fig. 2c,d. Impor- , j tantly, we want to quest whether such zero-frequency modes preserve (not being gapped) when we wrap a long edge into a −2 r small circle. We want to investigate the behaviours of 2D MP on, a for instance, a hollow disk geometry depicted in Fig. 5a. We write the low-energy long-wavelength Hamiltonian in real- −4 space polar coordinates and still use the chiral representation Figure 5 | Topological zero mode. (a) Illustration of a hollow disk geometry j ðÞ r; f; o p1ffiffi fj ðr; f; oÞ ij ðÞgr; f; o e if, L;R 2 r f and zero-mode profile attached to the inner circular edge. (b) Spectrum of j ðÞr; f; o u rðÞr; f; o . The eigen-equation is D p bulk states and edge states versus the discrete angular momentum 0 ÀÁ 1 n ¼ 0; 1; 2; .... u e ij ppffiffi i þ oc @r r @f 0 B ÀÁi 2 ÀÁC B u þ if u if C are derived quantities from jD, B pepffiffi @ þ i @ 0 pepffiffi @ i @ C @ i 2 r r f i 2 r r f A hi ÀÁ iup n u e þ if j ðÞ¼r; n; o o þ o@ j ðÞr; n; o ; ð19Þ ppffiffi @ þ i @ o r o2 o2 c r D 0 i 2 r r f c c r 0 1 hi j ðÞr; f; o u n B R C ; n; o p o o @ ; n; o : jfðÞ¼r 2 2 þ c r jDðÞr ð20Þ @ jDðÞr; f; o A o oc r ; f; o 0 jLðÞr 1 Depending on the energy to be above or below the bandgap, we have jRðÞr; f; o 8 ÀÁ B C < A J ðÞþBk r Y ðÞk r ; o2 o2 ; ¼ o@ jDðÞr; f; o A: ð18Þ n n r n n r o c jDðÞ¼r; n; o : ÀÁð21Þ jLðÞr; f; o 2 2 CnKnðÞk r ; 0oo oo : pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c 2 2 2 2 The azimuthal symmetry ensures the eigensolutions of jr, jf where kr ¼ o oc =up, k ¼ oc o =up,JnðÞkrr and inf and jD to pick up a common factor e , where n is the integer of YnðÞkrr are the n-th order Bessel and Neumann functions, and discretized angular momentum. This low-energy long-wavelength KnðÞkrr is the modified Bessel function. An, Bn and Cn are problem can be solved semi-analytically. In all situations, jr and jf coefficients. The no-normal-current boundary condition
6 NATURE COMMUNICATIONS | 7:13486 | DOI: 10.1038/ncomms13486 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13486 ARTICLE
jrðÞjr; n; o r¼a¼ 0 determines the allowed eigenfrequencies o comparison, MP experiments normally does not require a high for every given n. In this approximate model, the solutions above magnetic field or a high electron density. Moreover, the bandgap 2 2 the bandgap o ooc are all continuous bulk-like and within the in QHE is for electron transport, while the bandgap for 2D MP is 2 2 bandgap 0oo ooc are all discrete edge-like, as shown in Fig. 5b. for electron-density oscillations. 2D MP belongs to the topological class-D, which is different Majorana-type bound state: zero mode. We are particularly from the quantum Hall class-A. The topological edge states in MP interested in the limit o ! 0. According to equations 19 and 21, are not governed by the traditional bulk-edge correspondence of the boundary condition can be satisfied if and only if n ¼ 0, the quantum Hall states. The traditional rule states that the which completely annihilates the radial current, jr(r,0,0)¼ 0. The number of gapless edge states inside a gap is equal to the sum of zero-mode profile is given by all the bulk Chern numbers from the energy zero up to the gap. In jjoc our topological MP, the lowest bulk band (zero-frequency flat u r jjoc qe ffiffiffiffiffiffiffiffiffip band) contributes zero Chern number. A generalized bulk-edge jDðÞ¼Ar; 0; 0 0K0 r ; ð22Þ up jjoc r correspondence must be established by considering the particle– up hole symmetry that extends the spectrum into the ‘redundant’