doi: 10.1038/ncomms13486 topological...

ARTICLE

Received 30 Jun 2016 | Accepted 3 Oct 2016 | Published 28 Nov 2016

Topological magnetoplasmonDafei Jin1, Ling Lu2,3, Zhong Wang4,5, Chen Fang2,3, John D. Joannopoulos3, Marin Soljacic3,

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 pþ 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.

DOI: 10.1038/ncomms13486 OPEN

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

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Since the introduction of chiral edge states from two-dimensional (2D) quantum Hall systems into photonicsystems1–8, the investigation of band topology has been

actively extended into many other 2D or three-dimensional (3D)bosonic systems9,10, including phonon11–15, magnon16,exciton17,18 and polariton19. Up to now, topological phases of2D plasmon have not been identified, in spite of some relateddiscussion20–26 and a recent proposal of Weyl plasmon27.Besides, the previous lines of work mainly focus on finitefrequencies. Little attention has been paid to near-zerofrequencies, where new symmetries and new topological statesmay emerge (as elaborated below).

Plasmon is a unique type of bosonic excitations.Microscopically, it consists of collective motion of electron–holepairs in a Coulomb interaction electron gas, whereasmacroscopically it appears as coherent electron-density oscilla-tions. Under most circumstances, it can be well described byclassical density (and velocity) fields on the hydrodynamic level.These fields, like all other classical fields, are intrinsically real-valued and respect an unbreakable particle–hole symmetry. For aparticle–hole symmetric system, its Hamiltonian H transformsunder an antiunitary particle–hole conjugate operation C viaC� 1HC ¼ �H. (Here C2 ¼ þ 1 for bosons.) This propertyensures a symmetric spectrum o qð Þ with respect to zero frequency,o qð Þ ¼ �o � qð Þ, in which q is the wavevector. The associatedwave fields are thus superpositions of complex-conjugatepairs, F r; tð Þ ¼

Rdq½F qeþ i q�r� o qð Þj jtð Þ þF�qe� i q�r� o qð Þj jtð Þ�. The

spin-0 real-scalar field governed by the Klein–Gordon equationand the spin-1 real-vector field governed by Maxwell’s equationsalso share the same feature.

Particle–hole symmetry greatly expands the classificationof topological phases, according to the results oftenfold classification28,29. For example, the 2D quantumHall phase belongs to the class-A in 2D with brokentime-reversal symmetry T . The Su–Schrieffer–Heeger modeland the recently studied phonon zero modes12,15 belongto the class-BDI in one-dimension (1D) with particle–holesymmetry C2 ¼ þ 1 and time-reversal symmetry T 2 ¼ þ 1.However, the class-D 2D topological phase with C2 ¼ þ 1 andbroken T has so far only been proposed in pþ ipsuperconductors30, which carry chiral Majorana edge states andMajorana zero modes.

In this work, we show that the historically studied2D magnetoplasmon (MP)31–41 belongs to the class-D 2Dtopological phase with unbreakable C and broken T . It isgoverned by three-component linear equations carrying a similarstructure as that of the two-band Bogoliubov–de Gennes (BdG)equations of the pþ ip topological superconductor30,42.It contains a gapped bulk spectrum around zero frequency, andpossesses gapless topological edge states and zero-frequencybound states (zero modes). Many properties of pþ ipsuperconductor can find their analogy in 2D MP. Therefore,2D MP provides the first realized example of class-D in the10-class table28,29. The experimentally observed edge MP statesdated back to 1985 (ref. 37) are in fact topologically protected,analogous to the 1D chiral Majorana edge states. Somesimulations of Majorana-like states in photonics have beenreported43–47; however, they only appear at finite frequenciesaround which the particle–hole symmetry does not rigorouslyhold, unless using high nonlinearity48. In addition, we are able toderive non-zero Chern numbers adhering to the band topology of2D MP. On the basis of this, we are able to predict a new type ofone-way edge MP states on the boundary between oppositemagnetic domains, and the existence of MP zero modes on ahollow disk. Our prediction can be experimentally verifiedin any 2D electron gas (2DEG) systems, such as charged

liquid-helium surface33, semiconductor junctions34,38,49 andgraphene50,51.

ResultsGoverning equations. We consider a 2DEG confined in the z¼ 0and r¼ xexþ yey plane. The dynamics of MPs are governed bythe linearized charge-continuity equation and the Lorentz forceequation52,53. In the frequency domain, they are

� ior r; oð Þ ¼ �rr � j r; oð Þ; ð1Þ

� ioj r; oð Þ ¼ a rð ÞE r; oð Þ�oc rð Þj r; oð Þ�ez; ð2Þwhere r r; oð Þ is the variation of 2D electron-density offequilibrium and j r; oð Þ is the induced 2D current density.a rð Þ is a space-dependent coefficient that gives a 2D locallongitudinal conductivity, s r; oð Þ ¼ ia rð Þ=o. ocðrÞ is thecyclotron frequency from a perpendicularly applied staticmagnetic field, B0(r)¼B0(r)ez. For the massive electrons chosenfor our demonstration in this paper, a rð Þ ¼ e2n0 rð Þ

m�and

oc rð Þ ¼ eB0 rð Þm�c

, in which n0(r) is the equilibrium electron-densitydistribution, m* is the effective mass and c is the speed oflight32,54.

E r;oð Þ in equation 2 is the electric field evaluated within thez¼ 0 plane. It is generally a nonlocal function of the 2D currentdensity, E r; oð Þ ¼

Rdr0 K r; r0; oð Þ � j r0; oð Þ, where K r; r0; oð Þ

is an integration kernel determined by Maxwell’s equations.For a model system shown in Fig. 1a, and in the nonretardedlimit (E solely comes from the Coulomb interaction)52,53, theFourier transformed field–current relation can be attested to be(see Supplementary Note 1)

Ex q; oð ÞEy q; oð Þ

� �¼ 2p

ioqx qð Þq2

x qxqy

qxqy q2y

� �jx q; oð Þjy q; oð Þ

� �: ð3Þ

Here q¼ |q|, and x qð Þ ¼ 12 EA coth qdAð Þþ EB coth qdBð Þf g is a

q-dependent screening function54, in which dA and dB are thethicknesses, and EA and EB are the permittivities of the dielectricson the two sides of 2DEG. A generalization of equation 3 can bemade to include photon retardation (plasmon becomes plasmonpolariton)32,55,56 (see Supplementary Note 2). The Drude loss byelectron–phonon collision can be included in this formulation byintroducing a finite lifetime in equation 2 (ref. 54).

Bulk Hamiltonian and bulk states. We analytically solve for thecase of homogeneous bulk plasmons, where n0(r), B0(r), a rð Þ andoc rð Þ are all constants. B0 and oc can be positive or negative,depending on the direction of the magnetic field to be parallel orantiparallel to ez. We find that equations 1–3 can be casted into aHermitian eigenvalue problem. The Hamiltonian H is block-diagonalized in the momentum space, acting on a generalizedcurrent density vector J,

oJ q; oð Þ ¼ H qð ÞJ q; oð Þ; J q; oð Þ �jR q; oð ÞjD q; oð ÞjL q; oð Þ

0@

1A; ð4Þ

where jL;R q; oð Þ � 1ffiffi2p fjx q; oð Þ � ijy q; oð Þg are the left- and

right-handed chiral components of current density, and

jD q; oð Þ �ffiffiffiffiffiffiffiffi2pa

qx qð Þ

qr q; oð Þ � op qð Þ

q r q; oð Þ is a generalized

density component.

op qð Þ ¼ffiffiffiffiffiffiffiffiffiffi2paqx qð Þ

s!

upq; for small qð Þ;ffiffiffiffiffiffiffiffiffiffiffiffiffi4pa

EA þ EBq

q; for large qð Þ;

(ð5Þ

is the dispersion relation of conventional 2D bulk plasmon

without a magnetic field. up ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4padAdBEAdB þ EBdA

qis the effective

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13486

2 NATURE COMMUNICATIONS | 7:13486 | DOI: 10.1038/ncomms13486 | www.nature.com/naturecommunications


plasmon velocity originated from the screening of Coulombinteraction at long wavelengths by surrounding metals(see Fig. 1a). jD q; oð Þ asymptotically equals upr q; oð Þ at longwavelengths.

The bulk Hamiltonian H qð Þ and its long-wavelength limitread

H qð Þ ¼

þocop qð Þ

qqx � iqyffiffi

2p 0

op qð Þq

qx þ iqyffiffi2p 0

op qð Þq

qx � iqyffiffi2p

0op qð Þ

qqx þ iqyffiffi

2p �oc

0BBBBB@

1CCCCCA

!q!0

þocup qx � iqyð Þffiffi

2p 0

up qx þ iqyð Þffiffi2p 0

up qx � iqyð Þffiffi2p

0up qx þ iqyð Þffiffi

2p �oc

0BBBBBBB@

1CCCCCCCA:

ð6Þ

Strikingly, this long-wavelength three-band H qð Þ has a verysimilar structure to that of the Bogoliubov–de Gennes hamilto-nian of a 2D pþ ip topological superconductor42, which has twobulk bands. Here the odd number of bands is crucial for thetopological consequences shown below. For an even number ofbosonic bulk bands, it has been proved that the summed Chernnumber is always zero for the bands below or above the zerofrequency16.

We can define the antiunitary particle–hole operation C andthe antiunitary time-reversal operation T here,

C ¼0 0 1

0 1 0

1 0 0

0B@

1CAK

��q!� q

; T ¼0 0 � 1

0 1 0

� 1 0 0

0B@

1CAK

��q!� q

ð7Þ

where K is the complex-conjugate operator and j q!� q standsfor a momentum flipping. As can be checked, C acts as thecomplex conjugation for all the field components jx, jy and jD,while T additionally reverses the sign for jx and jy (refer toMethods: Cartesian representation). Our Hamiltonian has C

symmetry,

CH qð ÞC� 1 ¼ �H � qð Þ; ð8Þ

but broken T symmetry by the non-zero oc.The calculated homogeneous bulk spectra, plotted in Fig. 1b,c,

are

o� qð Þ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

c þo2p qð Þ

qand o0 qð Þ ¼ 0: ð9Þ

The corresponding (unnormalized) wavefunctions are

J� qð Þ /

qx � iqyffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

c þo2p qð Þ

q� oc

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

x þ q2y

� �o2

p qð Þr

qx þ iqyffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

c þo2p qð Þ

q oc

� �

0BBBBB@

1CCCCCA /

q!012 � 1

20

12 1

2

0@

1A;

ð10Þ

J0 qð Þ /

� qx � iqyffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffio2

p qð Þq

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiq2

xþ q2y

qoc

þ qx þ iqyffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffio2

p qð Þq

0BBB@

1CCCA /q!0

010

0@

1A: ð11Þ

The positive- and negative-frequency bands are gapped by oc.The zero-frequency band represents purely rotational currents(rr � j¼ 0, rr� ja0) balanced by static charge densitydistribution.

It is worthwhile to point out that the 2D bulk plasmondiscussed here is distinctively different from the surface plasmon(or surface plasmon polariton) in a 3D boundary (seeSupplementary Note 3).

Chern number on an infinite momentum plane. Unlike theregular lattice geometries whose Brillouin zone is a torus, oursystem is invariant under continuous translation, where theunbounded wavevector plane can be mapped on a Riemannsphere30,57. As long as the Berry curvature decays faster than q� 2

as q-N, the Berry phase around the north pole (q¼N) ofRiemann sphere is zero and the Chern number C is quantized57.

� / �*�c > 0

C = +1

qd

4

2

−2

−4

qd

�c = 0

C = −1

4

2

−2

−4

0−10 −5 5 100 −10 −5 5 100

0

�c

�c

C = 0

� / �*

a

2DEG

Metal

Metal

�A

�B

dA

dB

z

xy

B0

b c

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).

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13486 ARTICLE



For our 2D MP, we can verify this analytically,

C� ¼1

2p

ZdSq � rq�

J� qð Þ½ �y � irq�

J� qð ÞJ� qð Þ½ �yJ� qð Þ

( )

¼Z

qdq1q@q

ocffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

c þo2p qð Þ

q24

35

8<:

9=;

¼ � sgn ocð Þ;

ð12Þ

C0 ¼ 0; ð13Þwhere dSq is the momentum–space surface element, and the termin the curly bracket is the Berry curvature. Since the Berrycurvature changes its sign under the particle–hole symmetry, thepositive and negative-frequency bands must have opposite Chernnumbers, and the zero-frequency band, with odd Berry curvature,must have zero Chern number.

Majorana-type one-way edge states. We are able to calculate theanalytical edge solutions in the long-wavelength limit q-0, andnumerical edge solutions for unrestricted wavelengths, as plottedin Fig. 2. When q-0, the Hamiltonian in equation 6 becomeslocal, allowing us to replace qx and qy with the operators � i@x

and � i@y and to solve for the edge states by matching boundaryconditions25,53. We consider a 1D edge situated at x¼ 0 causedby a discontinuity either in n0(x) (Configuration-I) or in B0(x)(Configuration-II).

In Configuration-I as shown in Fig. 2a, we let the x40 regionbe filled with 2DEG, up x40ð Þ ¼ up

� , while the xo0 region be

vacuum, up xo0ð Þ ¼ 0�

and the magnetic field be uniformlyapplied parallel to ez, oc xð Þ ¼ oc40ð Þ. Since we know that thechange of Chern number across the edge is DC¼±1, there mustbe a single topological edge state present in this configuration. Welook for solutions that behave like e�kxþ iqyy� iot , qx ¼ ikð Þ, inthe x40 region, where k40 is an evanescent wavenumber. The

boundary condition is jx j x¼0þ ¼ 0, meaning that the componentof the current normal to the edge must vanish. Therefore, the

edge solutions have to satisfy qyoc ¼ kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

c þ u2p q2

y � k2� �r

,

whose right-hand side is always positive. Since we have chosenoc ¼ ocj j40, qy must be positive too. The edge state is henceone-way propagating. (We discard the unphysical solution k ¼ qy

that yields a null wavefunction.) The physical solution is k ¼ ocup

,

which gives an edge state spectrum,

o 1ð Þedge qy�

¼ upqy; qy � 0�

: ð14Þ

This is a gapless state running across the zero-frequency band asshown in Fig. 2c. It is a bosonic analogue to the chiral Majoranaedge states in the topological superconductor. Its wavefunction is

J 1ð Þedge x; qy�

/� iffiffi

2p

1þ iffiffi

2p

0@

1Ae�

ocup

xþ iqyy; x40ð Þ: ð15Þ

The zero-frequency edge state at qy¼ 0 has the same finite decaylength k ¼ oc

upand is well localized at the edge, despite being

degenerate with the flat middle band.In Configuration-II as shown in Fig. 2b, we let the 2DEG

have a constant electron density throughout the wholespace, up xð Þ ¼ up, but the magnetic field have opposite signsin the two regions (oc xð Þ ¼ þ ocj j40 for x40 andoc xð Þ ¼ � ocj jo0 for xo0). This is a novel configuration withDC¼±2 across the edge, and permits two topological edgestates. The boundary conditions are jx j x¼0� ¼ jx j x¼0þ andjD j x¼0� ¼ jD j x¼0þ , meaning that both the normal currentand density must be continuous across the edge. In ourlong-wavelength approximation, the first edge solution has thesame spectrum and wavefunction as those of Configuration-I,except for a symmetric extension of the wavefunction into thexo0 region (compare the plots of r and jy in red in Fig. 2a,b).The second edge solution satisfies k ¼ � qy , and so qy must be

� / �* � / �*

qyd

Configuration-I

Topologicaledge state

c

qyd

Configuration-II

Secondtopologicaledge state

Firsttopologicaledge state

d

Non-topologicaledge state

x

�, jy

2DEG (C = +1)Vacuum (C = 0)

Configuration-I: single domain

x

�, jy

2DEG (C = +1)2DEG (C = −1)

Configuration-II: opposite domains

x

y

z

−10 −5 5 100 −10 −5 5 100

0

4

2

−4

−2

Bulk states Bulk states

4

2

−4

−2

0

�c �c

�c �c

a

b

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.




negative to ensure k positive. This edge state is also one-way witha spectrum,

o 2ð Þedge qy�

¼ ocj j; qy � 0�

; ð16Þ

as shown in Fig. 2d. Its wavefunction is

J 2ð Þedge x; qy�

/12 � 1

20

12 1

2

0@

1Ae� qyxþ iqyy; x_0ð Þ: ð17Þ

It is antisymmetric for jy as plotted in green in Fig. 2b. Theantisymmetry plus the continuity condition for jD here make thedensity oscillations vanish identically; therefore, the secondedge state carries only current oscillations about the magneticdomain wall.

Figure 2c,d gives our numerically calculated bulk and edgespectra that are not restricted by the small-q local approximation(refer to Methods: Numerical scheme). When q is small, thenumerical results accurately agree with our analytical derivationabove. When q is large, the first edge dispersion asymptoticallyapproaches the bulk bands and eventually connects to them atq-N. The second edge dispersion drops gradually towards thezero-frequency bulk band and connects to it atq-N. It does show a (tiny) positive group velocity indicatingthe correct chiral direction, in spite of its negative phase velocity(refer to Fig. 2b,d). Physically though, the plasmon picture breaksdown in the large-q regime, where electron–hole pair productiontakes place.

Besides, we should note that there exists a non-topologicalgapped edge state in Configuration-I (see the blue curve in

Fig. 2c), which is consistent with the literature54. At longwavelengths q-0, this state resembles a bulk state bearing anexcitation gap of oc. Its edge confinement shows up only at shortwavelengths. As elaborated below, under a parameter evolutionfrom Configuration-I to II, this edge state evolves, remarkably,into the second gapless topological edge state in Configuration-II.

One-way propagation of edge states. Owing to the topologicalprotection, one-way edge states are immune to backscatteringfrom a random defect. In Fig. 3, we display two snapshots fromour real-time simulation. Edge states are excited by a point sourceoscillating at a frequency of 1

2 oc within the bulk gap. Thegenerated waves propagate only towards the right. A sharp zigzagdefect has been purposely inserted into the route of propagation.The waves then exactly follow the edge profile and insist onpropagating forward without undergoing any backscattering.The slightly visible fluctuation to the left of the point source iscompletely local (non-propagating). It is caused by the uniquelong-range Coulomb interaction in this system, different from themore familiar photonic and acoustic systems.

Figure 3a corresponds to the traditional Configuration-I, wherethe physical edge is formed by the density termination of 2DEG atvacuum. Although the one-way nature in this configuration isknown historically, its absolute robustness due to the topologicalprotection is less known and is manifested here.

Figure 3b corresponds to our new Configuration-II. Only thefirst edge state classified in Fig. 2 is excited here; the second edgestate has an energy too close to the band edge oc and hence is notexcited here. On the basis of our argument on topology above,Configuration-II permits protected one-way edge states on amagnetic domain boundary even if the 2DEG may behomogeneous. Figure 3b vividly demonstrates this scenario.Furthermore, it shows the perfect immunization to backscatteringwhen the two magnetic domains drastically penetrate each other.

Evolution of edge states. We find that the adiabatic modeevolution of the MP edge states described by a three-band modelhere is rather different from that in 2D topological insulatorsor superconductors, which can be commonly described by atwo-band or four-band model. The zero-frequency bulk bandhere plays a critical role for the appearance and disappearance ofthe topological edge states. We have investigated the evolutioncorresponding to the aforementioned two configurations shownin Fig. 2.

We first study the interaction between two topological edgestates propagating along the opposite directions on two 2DEGedges in the same magnetic field. The result is shown in Fig. 4a.When we gradually narrow the spacer, we let the equilibriumdensity in the spacer to gradually change from 0 to the bulkdensity n0. This allows the two topological edge states to coupleand end up with a bulk configuration. Instead of opening a gapand entering the top band (which would happen in theconventional two-band system), the two topological edge stateshere (marked in red) fall down into the flat middle band. Inaddition, there are two non-topological edge states (marked inblue, and refer to Fig. 2c) in this configuration. They movecompletely into the bulk during this process.

We then study the interaction between two topological edgestates propagating along the same direction on the two 2DEGedges under opposite magnetic fields. The result is shown inFig. 4b. When we gradually narrow the spacer, we find that one ofthe edge states (marked in red) goes slightly upwards, while theother falls downwards into the flat middle band. In themeanwhile, one of the non-topological edge states (marked inblue) detaches from the top band, continuously bends downwards

Domain boundary

Density boundary

b

a Configuration-I: single domain

Configuration-II: opposite domains

+1

−1

0

Point source

Point source

Vacuum

2DEG

2DEG

2DEG

x y

z

x yz

Figure 3 | Snapshots of one-way propagating topological edge states

immune to backscattering at a sharp zigzag defect. Here the cyclotron

frequency oc ¼ 4o�, which corresponds to B0¼0.9 T. The driving

frequency for the point source is 12 oc, which corresponds to f¼ 12.7 GHz.

(Refer to Methods: Theoretical model). (a) Configuration-I: a 2DEG is in

contact with vacuum under a single-domain magnetic field.

(b) Configuration-II: a 2DEG is uniformly filled in the whole space under an

opposite-domain magnetic field. Note that the profile is plotted for the

electric scalar potential, which is non-zero in the vacuum region. The

difference between a and b on the wavelengths is consistent with the

slightly different dispersion slope of the topological edge states between

Configuration-I and II. (See online Supplementary Movies 1 and 2 for real-

time evolution).




until it reaches the frequency oc, where it transits into the secondtopological edge state (marked in green). According to ourprevious argument, this edge state connects to the middle band atthe momentum infinity. At all times, there are two gapless edgestates. This is consistent with the preserved difference of Chernnumbers (DC¼±2) between the left part and right part of2DEG.

MP on a hollow disk. For infinitely long edges as discussedabove, the first topological edge state when ky-0 merges into thezero-frequency middle band, as can be seen in Fig. 2c,d. Impor-tantly, we want to quest whether such zero-frequency modespreserve (not being gapped) when we wrap a long edge into asmall circle. We want to investigate the behaviours of 2D MP on,for instance, a hollow disk geometry depicted in Fig. 5a.

We write the low-energy long-wavelength Hamiltonian in real-space polar coordinates and still use the chiral representationjL;R r; f; oð Þ � 1ffiffi

2p fjrðr; f; oÞ � ijf r; f; oð Þge� if,

jD r; f; oð Þ � upr r; f; oð Þ. The eigen-equation is

þocupe� ij

iffiffi2p @r � i

r @f�

0

upeþ if

iffiffi2p @r þ i

r @f�

0 upe� if

iffiffi2p @r � i

r @f�

0 upeþ if

iffiffi2p @r þ i

r @f�

�oc

0BBB@

1CCCA

jR r; f; oð ÞjD r; f; oð ÞjL r; f; oð Þ

0B@

1CA

¼ o

jR r; f; oð ÞjD r; f; oð ÞjL r; f; oð Þ

0B@

1CA: ð18Þ

The azimuthal symmetry ensures the eigensolutions of jr, jfand jD to pick up a common factor einf, where n is the integer ofdiscretized angular momentum. This low-energy long-wavelengthproblem can be solved semi-analytically. In all situations, jr and jf

are derived quantities from jD,

jr r; n; oð Þ ¼ � iup

o2�o2c

ocnrþo@r

h ijD r; n; oð Þ; ð19Þ

jf r; n; oð Þ ¼ up

o2�o2c

onrþoc@r

h ijD r; n; oð Þ: ð20Þ

Depending on the energy to be above or below the bandgap,we have

jD r; n; oð Þ ¼AnJn krrð ÞþBnYn krrð Þ; o2oo2

c

� ;

CnKn krrð Þ; 0oo2oo2c

� :

8<: ð21Þ

where kr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2�o2

c

p=up, k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffio2

c �o2p

=up, Jn krrð Þ andYn krrð Þ are the n-th order Bessel and Neumann functions, andKn krrð Þ is the modified Bessel function. An, Bn and Cn arecoefficients. The no-normal-current boundary condition

qy qy qy

qy qy qy

� � �

� � �

a

b

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.

�, j�

ra

Zero mode

� / �*

−10 100−5 5

4

2

−2

ν

Discretizededge states

B0

�

a

Bulk states

ba

−4

Figure 5 | Topological zero mode. (a) Illustration of a hollow disk geometry

and zero-mode profile attached to the inner circular edge. (b) Spectrum of

bulk states and edge states versus the discrete angular momentum

n ¼ 0; � 1; � 2; . . . .




jr r; n; oð Þ j r¼a¼ 0 determines the allowed eigenfrequencies ofor every given n. In this approximate model, the solutions abovethe bandgap o2oo2

c are all continuous bulk-like and within thebandgap 0oo2oo2

c are all discrete edge-like, as shown in Fig. 5b.

Majorana-type bound state: zero mode. We are particularlyinterested in the limit o! 0. According to equations 19 and 21,the boundary condition can be satisfied if and only if n ¼ 0,which completely annihilates the radial current, jr(r, 0, 0)¼ 0. Thezero-mode profile is given by

jD r; 0; 0ð Þ ¼ A0K0ocj jup

r

� � e�

ocj jup

rffiffiffiffiffiffiffiffiffiocj jup

rq ; ð22Þ

jf r; 0; 0ð Þ ¼ A0sgn ocð ÞK1ocj jup

r

� � sgn ocð Þe�

ocj jvp

rffiffiffiffiffiffiffiffiffiocj jup

rq :

jD describes a static charge distribution; jf describes a d.c. cir-culating current. The electric field generated by jD balances outthe Lorentz force due to jf and B0. The overall amplitude andphase in A0 can be arbitrary, but jD and jf are phase-locked toeach other by a ratio of sgn(oc), that is, the sign of magnetic field.In the chiral representation, with the arbitrary phase ignored, thezero-mode wavefunction behaves in the below manner:

Jzeromode rð Þ /� iffiffi

2p e� if

sgnðocÞþ iffiffi

2p eþ if

0B@

1CA e� ocj j

uprffiffiffiffiffiffiffiffiffi

ocj jup

rq ; r4að Þ: ð23Þ

We see that the Majorana-type topological edge state withinthe bandgap indeed preserve as o! 0 and n ¼ 0, on thefinite-sized inner edge of a hollow disk. It is easy to verify thatshrinking the hole radius will only reduce the number of discreteedge states at a finite n, but not kill the zero mode. Theseobservations are consistent with the ky-0 limit of the infinitestraight edge case (see Fig. 2c,d). Nevertheless, we shouldcautiously note that the zero mode here is not isolated.It degenerates with a large number of other zero-frequencymodes in the middle band (see Supplementary Note 4).

The guaranteed existence of zero mode in a hollow diskgeometry for 2D MP is reminiscent of the Majorana zero modesin the two-band pþ ip topological superconductor30. In the lattercase, a p flux inside the vortex core is needed to balance theantiperiodic boundary condition in the f direction. Bycomparison, our three-band Hamiltonian here hosts zero modewith periodic boundary condition, that is the wavefunction inequation 23 returns to itself when f rotates 2p. More generally,similar kind of discretized edge states and zero modes must existon a noncircular geometry as well, and can be on both the innerand outer edges, if the geometry is finite. For a finite geometry,the inner and outer edge zero modes must come out in pairs. Ifone closes the inner boundary and turns the hollow disk into asolid disk, then both modes must vanish. Otherwise, the vortexcurrents associated with both zero modes are singular at thecentre, and are unphysical.

DiscussionAlthough the configuration of 2D MP looks almost identical tothat of the quantum Hall effect (QHE), there are fundamentaldifferences39. QHE deals with electron (fermionic) transport,whereas 2D MP deals with collective electron-density oscillations(bosonic excitations). In QHE, the bulk is electronicallyinsulating, whereas in 2D MP, the bulk is electronicallyconducting. QHE experiments usually require a high magneticfield (\5 T) and a high electron density (\1011 cm� 2). By

comparison, MP experiments normally does not require a highmagnetic field or a high electron density. Moreover, the bandgapin QHE is for electron transport, while the bandgap for 2D MP isfor electron-density oscillations.

2D MP belongs to the topological class-D, which is differentfrom the quantum Hall class-A. The topological edge states in MPare not governed by the traditional bulk-edge correspondence ofthe quantum Hall states. The traditional rule states that thenumber of gapless edge states inside a gap is equal to the sum ofall the bulk Chern numbers from the energy zero up to the gap. Inour topological MP, the lowest bulk band (zero-frequency flatband) contributes zero Chern number. A generalized bulk-edgecorrespondence must be established by considering the particle–hole symmetry that extends the spectrum into the ‘redundant’negative-frequency regime. This accounts for the Berry fluxexchanged across the zero frequency. Only by doing so, the sumof all the bulk Chern numbers, spanning both the positive andnegative-frequency regimes, can be correctly kept zero.

Experiments for 2D plasmon were performed by pioneeringresearchers dated back to 1970s on the charged liquid heliumsurface33 and in the semiconductor inversion layer34. 2D MP wasfirst observed in the semiconductor inversion layer in 1977(ref. 35) and edge MP was first observed on the liquid heliumsurface in 1985 (ref. 37) by measuring radiofrequency absorptionpeaks. The nonreciprocity of edge MP was verified insemiconductor heterojunctions at microwave frequencies40.Recently, similar experiments have been performed on 2Dmaterials58–61 and on the surface of a topological insulator62.However, all the previous studies on edge MPs correspond toConfiguration-I in our paper, where the physical edge is formedbetween a 2DEG and vacuum under a uniform magnetic field.

Our prediction of new one-way edge MP in Configuration-IIhas not been reported, despite there have been similar studies inthe QHE systems63,64. Our prediction can, in principle, beverified in any 2DEG system as long as a pair of opposite

Ferromagnetic film

Coplanar waveguide

Domain boundary

2DEG

B

Top view

One-wayedge plasmon

Semiconductorheterojunction

Figure 6 | Schematics of a proposed experimental design based on the

2DEG in a semiconductor heterojunction. Microwave transmitted on the

coplanar waveguide can excite one-way edge plasmons at the boundary of

opposite magnetic domains. Characteristic resonant absorption can be

observed.




magnetic fields is applied across. The magnetic domain boundarycan be experimentally created by, for example, two concentricsolenoids of opposite currents or ferromagnetic materials. Thegapless one-way edge modes can be mapped out by microwavebulk transmission and by Fourier transforming near-field scansalong the edges, as demonstrated in ref. 65 for a topologicalphotonic system. In Fig. 6, we provide a schematic experimentaldesign. A semiconductor heterojunction serves as the platform for2DEG. A ferromagnetic film is grown on the top and is polarizedinto up- and down domains. For the circular domain boundary assketched, discretized one-way edge states can travel around.A meandering coplanar waveguide can be fabricated on the backof the substrate. One can then send in and receive microwavesignals at a frequency below oc, and in the meanwhile monitorthe characteristic absorption, which signifies the excitation ofone-way edge plasmons.

Our proposed zero-frequency bound modes feature localizedcharge accumulation (inducing a static electric field) andcirculating current (inducing a static magnetic field). Theseeffects are in principle measurable when the zero mode isactuated by charge or current injection.

In summary, we have revealed the salient topological nature of2D MP as the first realization of the class-D topological phase.The predicted new Majorana-type one-way edge states and zeromodes can be verified experimentally. Our work opens a newdimension for topological bosons via introducing the bosonicphase protected by the particle–hole symmetry. We anticipatesimilar realizations for other bosonic particles66 and discoveriesof new topological phases, after combining the particle–holesymmetry with other symmetries67, for example, the time-reversal and many kinds of spatial symmetries10,68.

MethodsTheoretical model. Our theoretical model of 2D MP is based on a hydrodynamicformalism. In equilibrium, the 2D electron number density is n0(r), which isdetermined by the device structure, doping concentration and gating condition38.Off equilibrium, the density changes to n(r, t). For 2D MP, we are only concernedwith the small deviation of 2D charge density, r(r, t)¼ � e{n(r,t)� n0(r)}, and theinduced 2D current density up to the linear order, j(r, t)¼ � en0(r)v(r,t), wherev(r, t) is the local velocity of electron gas restricted to move in the z¼ 0 plane only.

For massive electrons, the conductivity coefficient is a rð Þ ¼ e2 n0 rð Þm?

and the cyclotron

frequency is oc rð Þ ¼ eB0 rð Þm�c (refs 32,54). For massless electrons in, for example,

graphene, they are generalized to a rð Þ ¼ e2uF

ffiffiffiffiffiffiffiffin0 rð Þp‘ffiffipp and oc rð Þ ¼ eu2

F B0 rð ÞcEF

, where uF is

the Fermi velocity and EF is the Fermi energy69,70. Hence, our theory works forboth massive and massless electrons.

We consider a model system shown in Fig. 1a. With different choices of thematerial constants and structural parameters, it can lead to different practical2DEG systems, such as charged liquid-helium surface, metal–insulator–semiconductor junction, top-gated graphene transistor, and so on. In Fig. 1a, the2DEG lies in the z¼ 0 plane. The 0ozodA region is filled with an insulator(or vacuum) of the dielectric constant EA. The � dBozo0 region is filled withanother insulator (or semiconductor) of the dielectric constant EB. The z4dA andzo� dB regions are filled with perfect metals whose dielectric constant isEM¼ �N in the (radio to microwave) frequency range of MP. Encapsulating thesystem by two metals mimics the experimental configurations with top and bottomelectrodes, and in the meanwhile, cutoffs the theoretically infinitely long-rangedCoulomb interaction.

For the results shown in the main article, we simply choose EA¼ EB¼ E0¼ 1(vacuum permittivity), m*¼m0 (bare electron mass), dA¼ dB¼ d¼ 1 mm andn0¼ 1 mm� 2¼ 108 cm� 2. We can thus obtain a characteristic frequency,

o? ¼ffiffiffiffiffiffiffiffiffiffi2pe2 n0

m0d

q¼ 3:99�1010 s� 1, (that is, f? ¼ o?

2p ¼ 6:35 GHz), which is used to

normalize the frequencies. For the case of cyclotron frequency oc ¼ o? , we haveB0¼ 0.22 T.

Cartesian representation. In the main article we have used the chiral repre-sentation for the bulk Hamiltonian and generalized current vector, in order todemonstrate the similarity between our 2D MP and the 2D pþ ip topologicalsuperconductor. However, it is sometimes more convenient to adopt the traditionalCartesian representation, especially when we solve for the edge modes and applyboundary conditions on the edge. In the Cartesian representation, the bulk

Hamiltonian equation reads

oJ q; oð Þ ¼ H qð ÞJ q; oð Þ; J q; oð Þ �jx q; oð ÞjD q; oð Þjy q; oð Þ

0@

1A;

where the Hamiltonian in this representation reads

H qð Þ ¼0 op qð Þ

q qx � ioc

op qð Þq qx 0 op qð Þ

q qy

þ iocop qð Þ

q qy 0

0BB@

1CCA

!q!00 upqx � ioc

upqx 0 upqy

þ ioc upqy 0

0@

1A: ð24Þ

The particle–hole and time-reversal operators in this Cartesian representationare

C ¼1 0 00 1 00 0 1

0@

1AK

��q!� q

; T ¼� 1 0 00 1 00 0 � 1

0@

1AK

��q!� q

: ð25Þ

Clearly, C is just a complex conjugation on the matrix or vector elements(combined with a momentum reversal q-� q here, when written on the single-particle momentum bases).

Numerical scheme. To numerically solve the nonuniform edge state problem,which is unrestricted by the long-wavelength approximation, we expand the gov-erning equations 1 and 2 in the main article into plane waves and use the form ofinteraction in the momentum space equation 3. We can derive a matrix equation inthe Cartesian representation,

oJx q; oð ÞJD q; oð ÞJy q; oð Þ

0@

1A ¼ 0 Ux q; q0ð Þ � iW q; q0ð Þ

Qx q; q0ð Þ 0 Qy q; q0ð Þþ iW q; q0ð Þ Uy q; q0ð Þ 0

0@

1A Jx q; oð Þ

JD q; oð ÞJy q; oð Þ

0@

1A :

ð26ÞThe submatrices Ux q; q0ð Þ, Uy q; q0ð Þ, Qx q; q0ð Þ, Qy q; q0ð Þ, W q; q0ð Þ, and the

subcolumn vectors Jx q0ð Þ, JD q0ð Þ, Jy q0ð Þ all have the dimension of the number ofplane waves used for expansion.

The elements of submatrices are

Ux q; q0ð Þ ¼ 2p~a q� q0ð Þ q0xq0x q0ð Þ ; ð27Þ

Uy q; q0ð Þ ¼ 2p~a q� q0ð Þq0y

q0x q0ð Þ ; ð28Þ

Qx q; q0ð Þ ¼ dq; q0q0x ; ð29Þ

Qy q; q0ð Þ ¼ dq; q0q0y ; ð30Þ

and

W q; q0ð Þ ¼ oc q� q0ð Þ: ð31Þwhere ~a Dqð Þ and oc Dqð Þ with Dq ¼ q� q0 are the Fourier transform of a rð Þ andoc rð Þ. For the system with edges along the y axis, the expansion only needs to bedone in the x direction while keeping qy ¼ q0y as good quantum numbers.Typically, we use 128 plane waves to do the expansion, and the standard eigen-solver to obtain all the eigenfrequencies and eigenvectors.

The 2D real-space calculation for Fig. 3 is performed by a homemade finite-difference time-domain code. The field quantities are defined on a square grid.Equations 1 and 2 are adopted for the real-time evolution after replacing � io withqt. A point-source term of a given driving frequency is added to the right-hand sideof equation 2.

Data availability. All relevant data are available from the authors on request.

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AcknowledgementsWe would like to thank Duncan Haldane and Aris Alexandradinata for discussing thereality 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 sup-ported by AFOSR MURI under Award No. FA9550-12-1-0488. L.L. was supported inpart by the MRSEC Program of the NSF under Award No. DMR-1419807, the NationalThousand-Young-Talents Program of China, the National 973 Program of China (Nos2013CB632704 and 2013CB922404) and the National Natural Science Foundation ofChina (No. 11434017). L.L. and C.F. were supported in part by the Ministry of Scienceand Technology of China under Grant No. 2016YFA0302400. Z.W. was supported byNSFC under Grant No. 11674189. C.F. and L.F. were supported by the DOE Office ofBasic Energy Sciences, Division of Materials Sciences and Engineering under Award No.DE-SC0010526. J.D.J. was supported in part by the U.S. ARO through the ISN, underContract No. W911NF-13-D-0001. M.S. and L.L. were supported in part by the MITS3TEC EFRC of DOE under Grant No. DE-SC0001299.



http://arxiv.org/abs/1503.02630


Author contributionsD.J. and L.L. conceived the concept and wrote the manuscript. D.J. performed theanalytical and numerical calculation. Z.W., C.F. and L.F. joined effective discussion andprovided pivotal advices. J.D.J. and M.S. offered additional guidance and proofread themanuscript. L.L. and N.X.F. directed the project.

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How to cite this article: Jin, D. et al. Topological magnetoplasmon. Nat. Commun. 7,13486 doi: 10.1038/ncomms13486 (2016).

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