Chiral plasmons without magnetic fieldJustin C. W. Songa,b,c,1 and Mark S. Rudnerd

aWalter Burke Institute of Theoretical Physics, California Institute of Technology, Pasadena, CA 91125; bInstitute of Quantum Information and Matter,California Institute of Technology, Pasadena, CA 91125; cDepartment of Physics, California Institute of Technology, Pasadena, CA 91125; and dCenter forQuantum Devices and Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark

Edited by Allan H MacDonald, The University of Texas at Austin, Austin, TX, and approved February 19, 2016 (received for review September 24, 2015)

Plasmons, the collective oscillations of interacting electrons, possessemergent properties that dramatically alter the optical response ofmetals. We predict the existence of a new class of plasmons—chiral Berry plasmons (CBPs)—for a wide range of 2D metallicsystems including gapped Dirac materials. As we show, in thesematerials the interplay between Berry curvature and electron–electron interactions yields chiral plasmonic modes at zero mag-netic field. The CBP modes are confined to system boundaries,even in the absence of topological edge states, with chirality man-ifested in split energy dispersions for oppositely directed plasmonwaves. We unveil a rich CBP phenomenology and propose setupsfor realizing them, including in anomalous Hall metals and opti-cally pumped 2D Dirac materials. Realization of CBPs will offer apowerful paradigm for magnetic field-free, subwavelength opticalnonreciprocity, in the mid-IR to terahertz range, with tunable split-tings as large as tens of THz, as well as sensitive all-optical diag-nostics of topological bands.

topological materials | interactions | nonreciprocal response | Berry curvature

In electronic systems, chirality expresses the system’s ability todiscriminate between forward and backward propagation of

electronic signals along certain directions. This technologicallyuseful and hotly sought-after property can be achieved throughthe application of external magnetic fields. However, the needfor strong applied fields “on-chip” brings many challenges forapplications. Recently, materials exhibiting chirality in the ab-sence of a magnetic field have started to gain prominence.These materials include metals exhibiting anomalous (1) andquantum anomalous (2–6) Hall effects, as well as nonmagneticmaterials pushed out of equilibrium, where, for example, azero-field charge Hall effect was recently demonstrated (7). Ineach case, zero-field chirality arises from Bloch band Berrycurvature, a fundamental property of Bloch eigenstates thatdramatically affects single-particle electronic motion and ma-terial responses (1, 8, 9).Here we show that Berry curvature can work in concert with

interactions, leading to new types of collective modes in 2D to-pological metals, with nonvanishing Berry flux (i.e., net Berrycurvature), F . In particular, F gives rise to chiral plasmonicexcitations—propagating charge density waves with split disper-sion for oppositely directed modes—in the absence of a magneticfield (Fig. 1). We refer to these collective modes as chiral Berryplasmons (CBPs). Notably, these chiral modes are localized tothe edge of the 2D metal, even in the absence of topological edgestates, and exhibit a rich phenomenology (see below).We expect CBPs to be manifested in a wide variety of mag-

netic and nonmagnetic materials. The former are materials thatexhibit anomalous Hall effects, wherein time reversal symmetry(TRS) breaking is encoded in the Bloch band Berry curvature.The latter include a range of readily available gapped Diracmaterials, e.g., transition metal dichalcogenides and graphene/hBN, wherein TRS breaking is achieved by inducing a non-equilibrium valley polarization (7). In both cases CBPs are char-acterized by clear optical signatures such as split peaks in opticalabsorption.From a technological perspective, CBPs in nonmagnetic ma-

terials are particularly appealing because they provide an entirely

new platform for achieving a range of magneto-optical effect an-alogs that are magnetic field free and “on demand.” A primeexample is optical nonreciprocity (10), which is central for opticaldevice components, e.g., optical isolators and circulators. Above athreshold frequency (ωth; shaded area in Fig. 1A), the single uni-directionally propagating mode ωedge

− allows for chiral transport oflight via coupling to CBPs. Such CBP-mediated waveguides pro-vide a novel paradigm for deep subwavelength (11–13), linear, andmagnetic field-free strong nonreciprocity, crucial for miniaturizingoptical components. In particular, we predict that CBPs can en-able nonreciprocity over a large technologically important band-width (terahertz to mid-IR).The intrinsic chirality of CBPs starkly contrasts with that achieved

via cyclotron motion of charged particles in a magnetic field, withimportant quantitative and qualitative consequences. In the lattercase, chirality arises via the Lorentz force and gives rise to con-ventional magnetoplasmons (14–18). There, the cyclotron fre-quency, Zωc = ZeB=m (19–21), determines the constant splittingbetween magnetoplasmon modes of opposite chirality, which can beof the order of a few millielectron volts for accessible field strengths.In contrast, chirality in CBPs arises from the combined action ofplasmonic self-generated electric fields and the anomalous velocityof Bloch electrons, the phase space dual to the Lorentz force. Thiscombination makes the CBP mode splitting directly sensitive toplasmon wavelength, the Berry flux, and interaction strength, incontrast to magnetoplasmon splittings, which only depend onmagnetic field (17, 22).† As a result, for short wavelengths andunscreened interactions, large splittings ZΔω of several tens to ahundred millielectron volts can be achieved (Eq. 3).

Self-Induced Anomalous VelocityThe origin of CBPs in two dimensions can be understood fromthe Euler equations for electron density, nðr, tÞ (23)

Significance

A class of collective excitations is introduced, arising from thecombined action of electron interactions and Berry curvature.These excitations manifest as chiral plasmonic modes at zeromagnetic field. These collective modes are predicted to arise in arange of anomalous Hall metals, yielding a fundamental char-acteristic of interacting topological metals with non-zero Berryflux. Featuring a rich set of properties, these modes will enableversatile new tools for molding the flow of light.

Author contributions: J.C.W.S. and M.S.R. designed the research, performed the research,and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: justin.song.cw@gmail.com.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1519086113/-/DCSupplemental.

†We note that Kohn’s theorem guarantees that splittings are unchanged for parabolicbands; weak renormalizations are expected for linearly dispersing systems such as gra-phene, where it has been measured to be at most of order 10% (42). In contrast, forCBPs, the splitting grows directly proportional with interaction strength (Eq. 3).

4658–4663 | PNAS | April 26, 2016 | vol. 113 | no. 17 www.pnas.org/cgi/doi/10.1073/pnas.1519086113

∂tnðr, tÞ+∇ · vðr, tÞ= 0, [1a]

∂tpðr, tÞ− nðr, tÞ  e∇ϕðr, tÞ= 0, [1b]

where vðr, tÞ and pðr, tÞ are the velocity and momentum densityfields, ϕðrÞ is the scalar electric potential, and −e< 0 is the elec-tron charge. We note that in principle, the force equation, Eq. 1b,also includes contributions arising from the stress density of theelectronic fluid. However, at long-wavelengths these contribu-tions yield only subleading corrections to the plasmon dispersion(24). As our aim here is to clearly and most simply demonstrate theexistence and main features of CBPs, in this work we neglect smallcorrections due to the stress density. For an alternative formulationin terms of currents and conductivities, see SI Appendix.‡

To fully specify the dynamics, Eq. 1 must be supplemented bya set of constitutive relations that relate velocity, momentum,density, and the electric potential. Plasmons emerge from Eq. 1as self-sustained collective oscillations of nðr, tÞ, with the poten-tial ϕðr, tÞ= R

d2r′ W ðr, r′Þδnðr′, tÞ generated by the plasmon’sdensity fluctuations δn= nðr, tÞ− n0. Here n0 is the average car-rier density, and W ðr, r′Þ is the Coulomb interaction.As we argue, in the presence of Berry curvature, the con-

stitutive relations take on an anomalous character. The formof the constitutive relations can be found by starting withthe quasiparticle semiclassical equations of motion (8),vp =

∂«p∂p +

1Z_p×ΩðpÞ, _p= e∇ϕðrÞ, where vp = _r is the quasiparticle

velocity and «p and ΩðpÞ=ΩðpÞz are the Bloch band dispersionand Berry curvature, respectively.§ The corresponding velocitydensity fields are found from these relations and the phase spacedensity fiðr, p, tÞ by summing over all momentum p and bands fig,Oðr, tÞ=P

i

Rd2pOfiðr, p, tÞ=ð2πZÞ2. This approach gives

vðr, tÞ=pðr, tÞm

+ vaðr, tÞ,   va = eFZ

½ð∇ϕÞ× z�, [2]

where F =P

i

Rd2p ΩiðpÞf 0i ðpÞ=ð2πZÞ2 is the (dimensionless)

Berry flux, with f 0i ðpÞ being the equilibrium band occupancy;here, we have only kept terms linear in δn and ∇ϕ. In additionto the conventional first term, which governs the behavior ofordinary plasmons, vðr, tÞ in Eq. 2 exhibits a self-induced anom-alous velocity component va that yields chirality as shown in Fig.1. Note that the mass m appearing in Eq. 2 is the plasmon mass,which characterizes the collective motion of the Fermi sea (25).CBP chirality can be understood intuitively by examining the

anomalous velocity pattern set up by the plasmon’s electrostaticpotential ϕ (a more complete treatment is given below). Due tothe cross-product in Eq. 2, the anomalous velocity flow is di-rected along the equipotential contour lines of ϕ (see arrow-heads in Fig. 1 C and D). Near an edge, surface chargesassociated with the plasmon wave produce a potential as shownin Fig. 1D. The corresponding anomalous velocity field directselectrons into the nodal regions to the left of each region ofnegative charge build up (i.e., excess electron density), for theorientation shown and F > 0. Thus, for a leftward direction ofplasmon propagation, the anomalous velocity flow assists thecollective motion of the electronic density wave, leading to fasterpropagation ωedge

+ . For the right-propagating mode, the anoma-lous flow works against the collective motion, yielding slowerpropagation ωedge

− .

Crucially, va depends directly on the self-generated electricfield −∇ϕðrÞ. Consequently, the magnitude of the splitting isgoverned by the wave vector q and the strength of Coulombinteractions. We emphasize, however, that CBPs are a linearphenomenon, with the mode splitting Δω independent of themagnitude of ϕ. As we will show, the q-dependent CBP splittingcan be large

ZΔω= Z�ωedge+ −ωedge

−�≈A e2

κFjqj, [3]

where A is a numerical prefactor of order unity that dependson geometry, and we used the 2D Coulomb potential W ðqÞ=2πe=ðκjqjÞ with background dielectric constant κ. For edgeCBPs, we find A= 8

ffiffiffi2

pπ=9 (see below), yielding large splittings

ZΔω≈ 6− 60 meV for q= 1− 10  μm−1 (here we used F = 1, κ= 1).The appearance of e2=κ on the right side of Eq. 3 signals the crucialrole interactions play in Δω.

Chiral Edge Berry PlasmonsWe now analyze collective motion described by Eq. 1, treatingthe electric potential ϕ self-consistently. For an infinite bulk,applying ∂t to the continuity relation in Eq. 1a, using Eq. 2, andsubstituting in the force equation, Eq. 1b, yields

−∂2t δn=n0em

∇2ϕ,   ϕðrÞ=Z ∞

−∞dr′ W ðr− r′Þδnðr′Þ. [4]

In arriving at Eq. 4, we used ∇ · va ∝∂x∂yϕðrÞ−∂y∂xϕðrÞ= 0. Im-portantly, Berry flux F is absent in Eq. 4 and has no effect onbulk plasmon dispersion. Indeed, decomposing into Fourier modesδn∼ eiωt−iq · r and using the Coulomb interaction W ðr, r′Þ=−e=ðκjr− r′jÞ yields the usual 2D plasmon dispersion

�Zωbulk

q

.μ�2

= a0ðq=q0Þ, a0 =2πZ2

mμn0, q0 =

κμ

e2, [5]

which remains gapless at q= 0. In contrast, bulk magnetoplas-mons are gapped due to cyclotron motion (15–18).Close to a boundary, the situation is dramatically altered: Berry

flux F leads to the emergence of chiral edge plasmons, 1D chiralanalogs of surface plasmons (20, 26). This behavior is most easilyillustrated for an infinite metallic half-plane, where nðr, tÞ and vðr, tÞare generically finite for x≥ 0, but are zero for x< 0. Here the plas-mon propagates as a plane wave along y, with the fields in Eq. 4taking the form

ϕðr, tÞ=ϕqðxÞeiωt−iqy, δnðr, tÞ= δnqðxÞeiωt−iqy. [6]

The presence of the edge allows charges to accumulate (Fig. 1D).Indeed, inserting the fields defined above into Eq. 1a yieldsδnðrÞeiωt =−∇ · ½vðr, tÞΘðxÞ�=iω. Here we explicitly inserted ΘðxÞto emphasize the vanishing of the velocity density outside themetallic half-plane [ΘðxÞ= 1 for x≥ 0 and ΘðxÞ= 0 for x< 0]. Dueto the ΘðxÞ inside the divergence above, a nonzero vx flowing intothe boundary induces an oscillating surface charge component ofδnðrÞeiωt concentrated at the edge x= 0 (26, 27). By replacingδnðr, tÞ by −∇ · ½vðr, tÞΘðxÞ�=iω in the integral for ϕðr, tÞ in Eq.4, we thus find a jump condition for ∂xϕqðxÞ at x= 0:

∂xϕq

��0+−∂xϕq

��0−=

1iω

�∂xWq

��0− − ∂xWq

��0+�vxj0+, [7]

where WqðxÞ=−ðe=κÞ R∞−∞ dk  eikx=

��q2 + k2��1=2 is the 1D effective

Coulomb interaction. Here we decomposed Eq. 4 into plane-waves using Eq. 6 and integrated across the delta function,∂xΘðxÞ= δðxÞ, which accounts for the surface charge layer. The

‡See SI Appendix for discussions of: an alternative formulation of CBPs based on currentdensities and the conductivity tensor, chiral Berry plasmon dipole modes in a disk, opticalselection rules for gapped Dirac materials with inversion symmetry breaking, and photo-induced CBPs.

§The Berry curvature ΩðpÞ=∇k ×Ak depends on the crystalline Bloch wavefunctions juki,where Ak = hukji∇k juki is the Berry connection.

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parameter q corresponds to the wavevector in y (along the edge),and k describes variations in x (perpendicular to the edge). Cru-cially, finite F in Eq. 2 makes vxj0+ depend on both the magni-tude and sign of q along the edge.{

We now find edge CBP solutions of Eq. 4 using the boundarycondition (7) and continuity of ϕðrÞ. This problem is a nonlocalintegro-differential equation owing to the kernel WqðxÞ. Weadopt a simplified approach, approximating WqðxÞ by a similarkernel ~WqðxÞ with the same area and second moment (22, 26):~WqðxÞ=−e

κ

R∞−∞ dk  2qeikx=ðk2 + 2q2Þ. We emphasize that ~WqðxÞ is

extended in x and captures the long-range Coulomb behavior ofWqðxÞ. Indeed, the Fourier transforms of WqðxÞ and ~WqðxÞmatchfor small k=q. This method has been used successfully to mimicCoulomb interactions in isolated systems (17, 28).Crucially, the simple form of ~WqðxÞ above allows the integro-

differential equation in Eq. 4 to be expressed as a purely dif-ferential one with ϕqðxÞ obeying

�∂2x − 2q2

�ϕ>q ðxÞ=

4πejqjκ

δnqðxÞ,�∂2x − 2q2

�ϕ<q ðxÞ= 0, [8]

where ϕ>q and ϕ<

q are defined inside (x≥ 0) and outside (x< 0) thesample, respectively. Eq. 8 yields simple ϕ profiles

ϕ<q ðxÞ=ϕ0   e

γ0x, ϕ>q ðxÞ=ϕ1 e

−γ1x, [9]

where γ0 =ffiffiffi2

p jqj, and γ1 =ffiffiffi2

p jqj f½ðωbulkq Þ2 −ω2�=½2ðωbulk

q Þ2 −ω2�g1=2. The latter was obtained from Eqs. 4 and 7 by eliminat-ing δnq.

Using Eq. 9, the boundary conditions of continuous ϕ and thejump condition (Eq. 7) can be expressed compactly via the re-lation SΦ= 0, with Φ= ðϕ0,ϕ1ÞT:

S =�

1 −1ffiffiffi2

p jqj γ1 −D

, D=

2γ1�ωbulkq

�2

ω2 −q2 ~F sgnðqÞ

ω, [10]

where D was obtained using Eqs. 2 and 7, and ~F = 4πe2F=κZ.Left- and right-moving plane wave modes along the edge, ωedge

± ,can be identified through the zero modes of S. We first note thatfor ~F = 0, solving detðSÞ= 0 yields degenerate nonchiral edgemodes with ωedge

± = ð2=3Þ1=2ωbulkq (17, 22, 26).

For nonzero ~F , the zero mode solutions ωedge± of Eq. 10 be-

come split, yielding chiral edge plasmons (CBPs) as shown inFigs. 1 and 2. The modes ωedge

+ and ωedge− propagate as waves in

opposite directions along the edge, with faster and slowerspeeds, respectively. Importantly, the frequencies ωedge

± dependboth on q along the edge (Fig. 1A), as well as F (Fig. 2A). Thesplitting between modes grows with q and ~F , because va =e∇ϕ×F=Z (Eq. 2). Indeed, for small q ~F (so that q ~F � ωbulk

q ),we obtain an approximate dispersion for edge CBPs as

ωedge± ≈ ð2=3Þ1=2ωbulk

q ±ffiffiffi2

p jqj ~F=9+Oðq2 ~F 2Þ. As a result, we ob-tain the q-dependent Δω in Eq. 3. This behavior sharply contrastswith that of magnetoplasmons, which have a q-independentsplitting given by the cyclotron frequency (17, 19–21, 22, 26). Asa result, far larger splittings, arising from interactions, can beachieved for CBPs.Interestingly, for large enough q and/or F , the ωedge

+ mode(blue line in Fig. 1 A and B and 2A) merges with the bulkplasmon mode ωbulk

q (dashed line). As this mode merges with thebulk, its potential profile ceases to be localized along the edge.

A

C

B

D

Fig. 1. CBPs at zero magnetic field. (A) CBPs are manifested along the edges of a 2D metal with nonvanishing Berry flux, F (Inset). Counterpropagatingmodes exhibit a split dispersion, ωedge

± ðqÞ, with the splitting increasing with wave vector q along the edge. Above a threshold frequency ωth (shaded region)the fast mode ωedge

+ merges with bulk plasmon modes, ωbulkq , see Eq. 5 and dashed line, leaving a single unidirectional mode ωedge

− propagating along the edge.(B) Zoom-in of A showing separation of ωedge

+ and ωbulkq below ωth. (C and D) Electric potential for bulk and edge plasmon modes. For bulk modes, the

anomalous velocity field for electrons (indicated by arrowheads) runs parallel to the wave fronts and does not affect the speed of collective propagation. Nearedges the buildup of surface charges leads to an anomalous velocity flow that assists the collective propagation for the ωedge

+ mode (see text below Eq. 2). Forωedge− , the anomalous velocity flow opposes the collective motion (see text below Eq. 2). Parameter values used: F = 1.0 and a0 = 3 for A and B (Eq. 5) and

jq=q0j= 0.3, F = 0.3, and a0 = 3 for C and D.

{The jump discontinuity boundary condition in Eq. 7 depends on the perpendicular ve-locity density at the boundary, vxðx = 0Þ. We use the subscript 0+ in the text to emphasizethat the velocity density should be evaluated on the metallic side, x ≥0.

4660 | www.pnas.org/cgi/doi/10.1073/pnas.1519086113 Song and Rudner

The disappearance of the ωedge+ mode is shown by a diverging

confinement length of the electric potential, γ−11 (Fig. 2B). Incontrast, ωedge

− stays far from the bulk dispersion, yielding a potential(and electric field) tightly confined to the edge. The thresholdωth above which ωedge

+ merges with the bulk can be obtainedfrom Eq. 10. Setting ω=ωbulk

q in Eq. 10 yields the thresholdfrequency

Zωth =Z2n0ffiffiffi2

pmjF j= 18.3 

n0cm−2��1012

ðm½me�=0.03Þ× jF j meV, [11]

with me the free electron mass. For scale we consider a plasmonmass m∼ 0.03 me, as measured in graphene (25).Conservation of ω and q along the edge protect the ωedge

− modefrom coupling to bulk 2D plasmons. Scattering processes thatrelax q contribute to propagation losses. However, the tight edgeconfinement of the ωedge

− mode suppresses its electric field in thebulk regions (Fig. 2B, Inset), suppressing its coupling to bulkplasmons. Therefore, above the threshold ωth (gray region in Fig.1A), the single, well-defined, ωedge

− mode propagates unidirec-tionally along the edge. When hybridized with light, it will allowfor strong nonreciprocal propagation of CBP-polaritons withoutmagnetic field (see below).

Experimental Signatures of CBPsStrong plasmon-mediated light–matter interactions (11–13) makeoptics an ideal means of probing/controlling CBPs. Photon cou-pling to plasmons with gapless dispersion (e.g., 2D plasmons, andCBPs here) can be achieved through strategies such as gratings,and prism geometries (11). Observing unidirectional (nonrecip-rocal) propagation in such setups can reveal the existence ofCBPs. For demonstration, we detail an alternative experimentalprobe: CBP-photon coupling in finite geometries, such as metallicdisks, where dipolar plasmonic modes can dominate optical ab-sorption (11, 14).In metallic disks with finite F , CBPs manifest as clockwise/anti-

clockwise moving plasmonic dipole modes (Fig. 3A). These modescan be described via a simple oscillator model for the motion of thedipolar CBP center of mass (COM) coordinate,‡ fxg, where f · gdenotes the COM average. Here fvag≈F   fe∇ϕg× z (green arrow)gives rise to an intrinsic angular frequency ωa of plasmons in a disk(orange arrow), which adds to (subtracts from) the plasmonfrequency ω0 to produce nondegenerate anticlockwise (clock-wise) rotating modes (Fig. 3A and SI Appendix). Here we used Fpointing to positive z. A bosonic analog for ultracold atomicgases is discussed in ref. 29.

With an a.c. probing electric field ∼E  eiωt, the COM equationsof motion are

∂2t fxg+ω20fxg+ωa∂tfyg=−eEx   eiωt

∂2t fyg+ω20fyg−ωa∂tfxg=−eEy   eiωt,

  ωa =Fω2

0mn0Z

. [12]

Here ω0ðdÞ is the bare plasmon frequency in a disk of diameter d,in the absence of Berry curvature.Writing the current density as j= en0   ∂tfxg, we invert the

COM equations of motion to obtain the optical absorption (realpart of the longitudinal conductivity).‡ As shown in Fig. 3B, wefind a split peak structure with the dipolar CBP peaks given bythe poles of Eq. 12:

ωdisk± =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω20 +

ω2a

4

s±ωa

2,   ZΔω≡FZωp ≈

9Fd½μm� meV, [13]

where Δω=ωdisk+ −ωdisk

− =ωa and Zωp =ω20m=n0. On the right side,

we estimated ω20 ≈ 2πe2n0jqj=m, with jqj≈ 1=d (approximating the

lowest lying plasmonic excitation in a disk).# Here we used κ= 1. Im-portantly, Δω depends on the disk size, d, a unique property of CBPs.The tunable optical absorption split peak structure (via F and

d) in the absence of an applied magnetic field gives a clear ex-perimental signature of CBPs. In plotting Fig. 3B, we includedthe damping rate phenomenologically via ∂2t → ∂2t +Γ∂t, yielding aLorentzian lineshape with its half-width determined by Γ. Splitpeaks are clearly visible when ΔωJΓ, yielding peaks at ωdisk

± .To give a sense of scale, we note a typical value ZΓ∼ few meV(30), where ZΓ≈ 4 meV was measured in graphene disks. UsingΓ=ωp = 0.25 and taking F = 1, clearly resolved ωdisk

± peaks can beresolved for disk sizes dK 1  μm (Fig. 3B).

CBPs in Anomalous Hall MaterialsWe now discuss materials where CBPs can be realized. Wepredict that metallic systems with nonvanishing F will genericallyhost CBPs. Finite F requires broken time reversal symmetry andmay arise in magnetically ordered systems or out-of-equilibriumnonmagnetic systems (see below). The former includes mag-netically doped semiconducting quantum wells (see ref. 31,where F ≈ 1=2 was predicted) and topological insulators (3–6).

A B

Fig. 2. CBPs at the boundary of a half plane. (A) The CBP frequency splitting increases with increasing Berry flux, F. Parameter values are as in Fig. 1, but withjqj=q0 = 0.3. (B) CBP edge-confinement length, γ−11 , of ωedge

+ and ωedge− (Eq. 9) for increasing F. For large F, the confinement length of the ωedge

+ mode diverges,indicating that it joins the bulk, whereas the ωedge

− mode becomes more confined to the edge. (Inset) Electric profile shown for jF j= 0.3, exhibiting con-finement of ωedge

± to the edge.

#Comparing this estimate to plasmon frequencies obtained in graphene disks (30), weobtain zero field plasmon frequencies in the disk geometry to within a factor of unity.Using parameters reported in ref. 30, our estimate yields Zω0 ≈

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πe2n0=ðmdÞp

≈ 24 meVfor a disk of d= 3  μm and reported doping μ=−0.54  eV. Ref. 30 observed a zero fieldplasmon resonance at ω0 = 130  cm−1 = 16 meV.

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As a concrete example, we examine the magnetically doped to-pological insulator chromium doped thin-film (Bi,Sb)2 Te3, whichwas recently experimentally realized (5). When moderately dopedwith electrons or holes, we estimate that F ∼ 1 can be achievedbased on the measured anomalous Hall conductivity σB=0xy K e2=h(5). These values yield a large splitting ZΔω≈ 10− 100 meV forshort wavelength plasmons (Eqs. 3 and 13). When probed in thedisk geometry, we predict this system will yield two split absorptionpeaks in the absence of magnetic field.

On-Demand CBPs in Nonmagnetic MaterialsIntriguingly, finite F can also be achieved in nonmagnetic ma-terials, without an applied magnetic field. This class includes, forexample, gapped Dirac materials where inversion symmetry isbroken. In these, a valley polarization (Fig. 3C, Inset) can beinduced by circularly polarized light (32), yielding an anomalousHall effect that has recently been observed (7).Can CBPs exist in photoexcited systems? To analyze this, we

focus on nominally time reversal invariant gapped Dirac mate-rials. We model the valley dependent F as (33) FK ,K′ = τzNsf~n1=2=½2ðnK ,K′ + ~nÞ1=2�g  sgn Δ, where nK ,K′ are the valley carrier densi-ties, τz =∓ for the K ,K′ valleys, Ns = 2 is the spin degeneracy, and~n=Δ2=4πv2FZ

2 gives a characteristic density scale. The bandgapis 2jΔj. When nK = nK′ is in equilibrium, the total flux F =FK +F ′

K vanishes.Interestingly, when the system is pushed out of equilibrium,

e.g., by circularly polarized light, the populations in the valleysmay become imbalanced, nK ≠ nK′ (32) (Fig. 3C, Inset).‡ As aresult, the net Berry flux for the entire electronic system (sum-med over both valleys) is nonzero, yielding F pe ≠ 0. We analyzethe collective modes of the photoexcited electron-hole system inthe disk geometry following Eq. 12, accounting for the mutuallyattracting electron and hole populations and using F =F pe. This

analysis yields two chiral CBP modes: ωel−h± =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2a + 2ω2

0

q±ωa (SI

Appendix). Estimating ω0 as in the text below Eq. 13, we obtain theωel−h

± curves in Fig. 3C with ωp given by Eq. 13. The frequencies ofboth modes vanish at zero photoexcited carrier density, npe. Themode splitting increases with npe and can reach sizable values fornpe ≥ ~n, reflecting F pe pumping.Large splittings require npe J ~n. The large gaps ΔJ 1  eV of

many transition metal dichalcogenides (7, 34, 35) yield largecharacteristic densities requirements, ~nJ 1013   cm−2. In contrast,other gapped Dirac materials such as G/hBN (36–38) and dual-gated bilayer graphene (39) possess Δ≈ 10− 200 meV, yieldingsignificantly smaller and more favorable ~n. These materialspossess valley-selective optical selection rules (32),‡ and presentan ideal venue to achieve maximal F = 1 (and large Δω), evenwith relatively weak pump fluence.jj

A further promising strategy to achieve large CBP splittings isto stack m layers of gapped Dirac materials on top of each other,with no tunnel coupling between the layers. Stacking achieves(i) larger photo-excited carrier densities due to the increasedabsorption and (2) a larger maximal Berry flux (when npe � ~n)and, hence, larger CBP splittings; the maximal Berry flux is Fmax =mF single. In such a structure, the long-range Coulomb interactionallows the photoexcited densities in different layers to oscillatecollectively.

ConclusionCBPs are robust collective excitations of metallic systems, arisingfrom two simple ingredients: Berry flux and interactions. Our

A B

C

Fig. 3. CBPs in a disk and in valley polarized gapped Dirac systems. (A) Illustration of CBPs in a disk, showing anticlockwise/clockwise rotating dipole modes.The mode splitting arises from the intrinsic angular frequency, ωa, induced by the combination of F and the self-induced electric field. (B) Optical absorptionsplit peaks for dipolar CBPs in a disk (A) obtained by inverting Eq. 12 for F = 0, 0.5, 1.0 (dashed black, purple, orange). Smaller disk sizes and/or larger Fproduce larger splittings (Eq. 13). Parameter values: ω0=ω* = 1 and Γ=ω* = 0.25. (C) CBP dipole modes for valley polarized gapped Dirac systems, with nK =npe,and nK′ = 0. The characteristic density is ~n=Δ2=4πv2F Z

2. (Inset) Valley polarization may be induced by above-gap circularly polarized light.

jjNote that the (neutral) excitons that may form in some large-gap materials feel zero netBerry curvature (43) and to first approximation do not contribute to CBPs.

4662 | www.pnas.org/cgi/doi/10.1073/pnas.1519086113 Song and Rudner

analysis indicates that CBPs survive for both weak and stronginteractions. As a result, we conclude that CBPs are generic inmetallic anomalous Hall phases, including out-of-equilibriumstates where finite F emerges from driving (e.g., in optically pumpedvalley polarized gapped Dirac materials). Indeed, this feature allowsCBPs to be used for all-optical diagnostics of anomalous Halland topological phases, as well as pumped or periodically drivensystems (40, 41). Optical probes of the latter are particularlyappealing because transport measurements require contacts thatoften complicate and destroy the novel electron distributions andcoherences of driven systems.Perhaps the most appealing prospect is coupling CBPs with light

for subwavelength and strong nonreciprocal propagation. Exhibitinga single chiral mode at large q (large splitting and large frequency),precisely where plasmons give large compression of optical modevolume, hybrid CBP-polaritons are strongly nonreciprocal (10). As

we propose, CBP mediated unidirectional waveguides can be re-alized in readily available nonmagnetic materials (e.g., the van derWaals material class). CBP-based nonreciprocity, if realized ex-perimentally, stands to play a vital role in the miniaturization ofoptical components that are magnetic field free.

Note Added in Proof. During the review of our manuscript, arelated work on CBPs in photoexcited gapped Dirac systemsappeared (44).

ACKNOWLEDGMENTS. We thank J. Alicea, J. Eisenstein, M. Kats, L. Levitov,M. Schecter, and B. Skinner for useful conversations. J.C.W.S. acknowledgessupport from the Walter Burke Institute for Theoretical Physics as part of aBurke fellowship at California Institute of Technology. M.S.R. acknowledgessupport from the Villum Foundation and the People Programme (Marie CurieActions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under Research Executive Agency Grant PIIF-GA-2013-627838.

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