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Home , Quantum Hall effect

Chiral without

Justin 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 , California Institute of Technology, Pasadena, CA 91125; cDepartment of Physics, California Institute of Technology, Pasadena, CA 91125; and dCenter for Quantum 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, possess new platform for achieving a range of magneto-optical effect an- emergent properties that dramatically alter the optical response of alogs that are magnetic field free and “on demand.” Aprime metals. We predict the existence of a new class of plasmons— example is optical nonreciprocity (10), which is central for optical chiral Berry plasmons (CBPs)—for a wide range of 2D metallic device components, e.g., optical isolators and circulators. Above a systems including gapped Dirac materials. As we show, in these threshold frequency (ωth; shaded area in Fig. 1A), the single uni- materials the interplay between Berry curvature and electron– directionally propagating mode ωedge allows for chiral transport of electron interactions yields chiral plasmonic modes at zero mag- − light via coupling to CBPs. Such CBP-mediated waveguides pro- netic field. The CBP modes are confined to system boundaries, – even in the absence of topological edge states, with chirality man- vide a novel paradigm for deep subwavelength (11 13), linear, and ifested in split energy dispersions for oppositely directed magnetic field-free strong nonreciprocity, crucial for miniaturizing waves. We unveil a rich CBP phenomenology and propose setups optical components. In particular, we predict that CBPs can en- for realizing them, including in anomalous Hall metals and opti- able nonreciprocity over a large technologically important band- cally pumped 2D Dirac materials. Realization of CBPs will offer a width (terahertz to mid-IR). powerful paradigm for magnetic field-free, subwavelength optical The intrinsic chirality of CBPs starkly contrasts with that achieved nonreciprocity, in the mid-IR to terahertz range, with tunable split- via of charged particles in a magnetic field, with tings as large as tens of THz, as well as sensitive all-optical diag- important quantitative and qualitative consequences. In the latter nostics of topological bands. case, chirality arises via the and gives rise to con- ventional magnetoplasmons (14–18). There, the cyclotron fre- topological materials | interactions | nonreciprocal response | Berry curvature quency, Zωc = ZeB=m (19–21), determines the constant splitting between magnetoplasmon modes of opposite chirality, which can be ’ n electronic systems, chirality expresses the system s ability to oftheorderofafewmillielectronvolts for accessible field strengths. discriminate between forward and backward propagation of I In contrast, chirality in CBPs arises from the combined action of electronic signals along certain directions. This technologically useful and hotly sought-after property can be achieved through plasmonic self-generated electric fields and the anomalous velocity the application of external magnetic fields. However, the need of Bloch electrons, the space dual to the Lorentz force. This for strong applied fields “on-chip” brings many challenges for combination makes the CBP mode splitting directly sensitive to applications. Recently, materials exhibiting chirality in the ab- plasmon wavelength, the Berry flux, and interaction strength, in sence of a magnetic field have started to gain prominence. contrast to magnetoplasmon splittings, which only depend on † These materials include metals exhibiting anomalous (1) and magnetic field (17, 22). As a result, for short wavelengths and quantum anomalous (2–6) Hall effects, as well as nonmagnetic unscreened interactions, large splittings ZΔω of several tens to a materials pushed out of equilibrium, where, for example, a hundred millielectron volts can be achieved (Eq. 3). zero-field charge was recently demonstrated (7). In each case, zero-field chirality arises from Bloch band Berry Self-Induced Anomalous Velocity curvature, a fundamental property of Bloch eigenstates that The origin of CBPs in two dimensions can be understood from dramatically affects single-particle electronic motion and ma- the Euler equations for electron density, nðr, tÞ (23) terial responses (1, 8, 9). Here we show that Berry curvature can work in concert with Significance interactions, leading to new types of collective modes in 2D to- pological metals, with nonvanishing Berry flux (i.e., net Berry curvature), F. In particular, F gives rise to chiral plasmonic A class of collective excitations is introduced, arising from the excitations—propagating charge density waves with split disper- combined action of electron interactions and Berry curvature. sion for oppositely directed modes—in the absence of a magnetic These excitations manifest as chiral plasmonic modes at zero field (Fig. 1). We refer to these collective modes as chiral Berry magnetic field. These collective modes are predicted to arise in a plasmons (CBPs). Notably, these chiral modes are localized to range of anomalous Hall metals, yielding a fundamental char- the edge of the 2D metal, even in the absence of topological edge acteristic of interacting topological metals with non-zero Berry states, and exhibit a rich phenomenology (see below). flux. Featuring a rich set of properties, these modes will enable We expect CBPs to be manifested in a wide variety of mag- versatile new tools for molding the flow of light. netic and nonmagnetic materials. The former are materials that Author contributions: J.C.W.S. and M.S.R. designed the research, performed the research, exhibit anomalous Hall effects, wherein time reversal symmetry and wrote the paper. (TRS) breaking is encoded in the Bloch band Berry curvature. The authors declare no conflict of interest. The latter include a range of readily available gapped Dirac materials, e.g., transition metal dichalcogenides and / This article is a PNAS Direct Submission. 1 hBN, wherein TRS breaking is achieved by inducing a non- To whom correspondence should be addressed. Email: [email protected]. equilibrium valley polarization (7). In both cases CBPs are char- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. acterized by clear optical signatures such as split peaks in optical 1073/pnas.1519086113/-/DCSupplemental. † ’ absorption. We note that Kohn s theorem guarantees that splittings are unchanged for parabolic bands; weak renormalizations are expected for linearly dispersing systems such as gra- From a technological perspective, CBPs in nonmagnetic ma- phene, where it has been measured to be at most of order 10% (42). In contrast, for terials are particularly appealing because they provide an entirely CBPs, 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 Downloaded by guest on September 24, 2021 Crucially, va depends directly on the self-generated electric ∂tnðr, tÞ + ∇ · vðr, tÞ = 0, [1a] field −∇ϕðrÞ. Consequently, the magnitude of the splitting is governed by the wave vector q and the strength of Coulomb ∂ p r t − n r t e∇ϕ r t = [1b] t ð , Þ ð , Þ ð , Þ 0, interactions. We emphasize, however, that CBPs are a linear Δω v r t p r t phenomenon, with the mode splitting independent of the where ð , Þ and ð , Þ are the velocity and momentum density magnitude of ϕ. As we will show, the q-dependent CBP splitting fields, ϕðrÞ is the scalar electric potential, and −e < 0 is the elec- can be large tron charge. We note that in principle, the force equation, Eq. 1b, also includes contributions arising from the stress density of the e2 electronic fluid. However, at long-wavelengths these contribu- ZΔω = Z ωedge − ωedge ≈ q [3] + − A κ Fj j, tions yield only subleading corrections to the plasmon dispersion (24). As our aim here is to clearly and most simply demonstrate the where A is a numerical prefactor of order unity that depends existence and main features of CBPs, in this work we neglect small on geometry, and we used the 2D Coulomb potential WðqÞ = corrections due to the stress density. For an alternative formulation ‡ 2πe=ðκjqjÞ with backgroundffiffiffi dielectric constant κ.Foredge in terms of currents and conductivities, see SI Appendix. p CBPs, we find A = 8 2π=9 (see below), yielding large splittings To fully specify the dynamics, Eq. 1 must be supplemented by − ZΔω ≈ 6 − 60 meV for q = 1 − 10 μm 1 (here we used F = 1, κ = 1). a set of constitutive relations that relate velocity, momentum, e2=κ 3 1 The appearance of on the right side of Eq. signals the crucial density, and the electric potential. Plasmons emerge from Eq. role interactions play in Δω. as self-sustainedR collective oscillations of nðr, tÞ, with the poten- tial ϕðr, tÞ = d2r′ Wðr, r′Þδnðr′, tÞ generated by the plasmon’s Chiral Edge Berry Plasmons δn = n r t − n n density fluctuations ð , Þ 0. Here 0 is the average car- We now analyze collective motion described by Eq. 1, treating W r r′ rier density, and ð , Þ is the Coulomb interaction. the electric potential ϕ self-consistently. For an infinite bulk, As we argue, in the presence of Berry curvature, the con- applying ∂t to the continuity relation in Eq. 1a, using Eq. 2, and stitutive relations take on an anomalous character. The form substituting in the force equation, Eq. 1b, yields of the constitutive relations can be found by starting with Z ∞ the semiclassical equations of motion (8), n0e ∂«p −∂2δn = ∇2ϕ ϕ r = dr′ W r − r′ δn r′ [4] vp = + 1 p_ × ΩðpÞ, p_ = e∇ϕðrÞ, where vp = r_ is the quasiparticle t , ð Þ ð Þ ð Þ. ∂p Z m −∞ velocity and «p and ΩðpÞ = ΩðpÞ^z are the Bloch band dispersion § and Berry curvature, respectively. The corresponding velocity In arriving at Eq. 4, we used ∇ · va ∝∂x∂yϕðrÞ −∂y∂xϕðrÞ = 0. Im- density fields are found from these relations and the phase space portantly, Berry flux F is absent in Eq. 4 and has no effect on f r p t p i density iPð , R, Þ by summing over all momentum and bands f g, bulk plasmon dispersion. Indeed, decomposing into Fourier modes r t = d2p f r p t = πZ 2 iωt−iq · r Oð , Þ i O ið , , Þ ð2 Þ . This approach gives δn ∼ e and using the Coulomb interaction Wðr, r′Þ = −e=ðκjr − r′jÞ yields the usual 2D plasmon dispersion pðr, tÞ eF vðr, tÞ = + vaðr, tÞ, va = ½ð∇ϕÞ × ^z, [2]  .  m Z 2 πZ2 κμ Zωbulk μ = a q=q a = 2 n q = [5] P R q 0ð 0Þ, 0 0, 0 2 , 2 0 2 mμ e where F = i d p ΩiðpÞfi ðpÞ=ð2πZÞ is the (dimensionless) Berry flux, with f 0ðpÞ being the equilibrium band occupancy; i which remains gapless at q = 0. In contrast, bulk magnetoplas- here, we have only kept terms linear in δn and ∇ϕ. In addition mons are gapped due to cyclotron motion (15–18). to the conventional first term, which governs the behavior of Close to a boundary, the situation is dramatically altered: Berry ordinary plasmons, vðr, tÞ in Eq. 2 exhibits a self-induced anom- flux F leads to the emergence of chiral edge plasmons, 1D chiral 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, analogs of surface plasmons (20, 26).Thisbehaviorismosteasily illustrated for an infinite metallic half-plane, where nðr, tÞ and vðr, tÞ which characterizes the collective motion of the Fermi sea (25). x ≥ x < CBP chirality can be understood intuitively by examining the are generically finite for 0, but are zero for 0. Here the plas- y 4 anomalous velocity pattern set up by the plasmon’s electrostatic mon propagates as a plane wave along , with the fields in Eq. potential ϕ (a more complete treatment is given below). Due to taking the form the cross-product in Eq. 2, the anomalous velocity flow is di- ϕ r t = ϕ x eiωt−iqy δn r t = δn x eiωt−iqy [6] rected along the equipotential contour lines of ϕ (see arrow- ð , Þ qð Þ , ð , Þ qð Þ .

heads in Fig. 1 C and D). Near an edge, surface charges PHYSICS D associated with the plasmon wave produce a potential as shown The presence of the edge allows charges to accumulate (Fig. 1 ). 1a in Fig. 1D. The corresponding anomalous velocity field directs Indeed, inserting the fields defined above into Eq. yields δn r eiωt = −∇ · v r t Θ x =iω Θ x electrons into the nodal regions to the left of each region of ð Þ ½ ð , Þ ð Þ . Here we explicitly inserted ð Þ negative charge build up (i.e., excess electron density), for the to emphasize the vanishing of the velocity density outside the > metallic half-plane [ΘðxÞ = 1 for x ≥ 0 and ΘðxÞ = 0 for x < 0]. Due orientation shown and F 0. Thus, for a leftward direction of Θ x v plasmon propagation, the anomalous velocity flow assists the to the ð Þ inside the divergence above, a nonzero x flowing into the boundary induces an oscillating surface charge component of collective motion of the electronic density wave, leading to faster iωt edge δnðrÞe concentrated at the edge x = 0 (26, 27). By replacing propagation ω + . For the right-propagating mode, the anoma- δnðr, tÞ by −∇ · ½vðr, tÞΘðxÞ=iω in the integral for ϕðr, tÞ in Eq. lous flow works against the collective motion, yielding slower 4 ∂ ϕ x x = edge , we thus find a jump condition for x qð Þ at 0: propagation ω− . 1 ∂xϕq + −∂xϕq − = ∂xWq − −∂xWq + vxj +, [7] 0 0 iω 0 0 0 ‡ See SI Appendix for discussions of: an alternative formulation of CBPs based on current R = densities and the conductivity tensor, chiral Berry plasmon dipole modes in a disk, optical W x = − e=κ ∞ dk eikx= q2 + k2 1 2 selection rules for gapped Dirac materials with inversion symmetry breaking, and photo- where qð Þ ð Þ −∞ is the 1D effective induced CBPs. Coulomb interaction. Here we decomposed Eq. 4 into plane- § 6 The Berry curvature ΩðpÞ = ∇k × Ak depends on the crystalline Bloch wavefunctions juki, waves using Eq. and integrated across the delta function, = u i∇ u ∂ Θ x = δ x where Ak h kj kj ki is the Berry connection. x ð Þ ð Þ, which accounts for the surface charge layer. The

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Fig. 1. CBPs at zero magnetic field. (A) CBPs are manifested along the edges of a 2D metal with nonvanishing Berry flux, F (Inset). Counterpropagating edge modes exhibit a split dispersion, ω ± ðqÞ, with the splitting increasing with wave vector q along the edge. Above a threshold frequency ωth (shaded region) ωedge ωbulk ωedge the fast mode + merges with bulk plasmon modes, q , see Eq. 5 and dashed line, leaving a single unidirectional mode − propagating along the edge. B A ωedge ωbulk ω C D ( ) Zoom-in of showing separation of + and q below th.( and ) 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. Near edge edges the buildup of surface charges leads to an anomalous velocity flow that assists the collective propagation for the ω + mode (see text below Eq. 2). For edge ω− , the anomalous velocity flow opposes the collective motion (see text below Eq. 2). Parameter values used: F = 1.0 and a0 = 3forA and B (Eq. 5)and jq=q0j = 0.3, F = 0.3, and a0 = 3forC and D.

parameter q corresponds to the wavevector in y (along the edge), Using Eq. 9, the boundary conditions of continuous ϕ and the and k describes variations in x (perpendicular to the edge). Cru- jump condition (Eq. 7) can be expressed compactly via the re- T 2 vx + Φ = Φ = ϕ ϕ cially, finite F in Eq. makes j0 depend on both the magni- lation S 0, with ð 0, 1Þ : tude and sign of q along the edge.{   We now find edge CBP solutions of Eq. 4 using the boundary 2   γ ωbulk condition (7) and continuity of ϕðrÞ. This problem is a nonlocal − 2 1 q q2 ~ q = pffiffiffi1 1 D = − Fsgnð Þ [10] integro-differential equation owing to the kernel WqðxÞ.We S q γ − D , 2 , 2j j 1 ω ω adopt a simplified approach, approximating WqðxÞ by a similar ~ kernel W qðxRÞ with the same area and second moment (22, 26): ~ e ∞ ikx ~ D 2 7 F~ = πe2 =κZ W qðxÞ = − dk 2qe =ðk2 + 2q2Þ. We emphasize that W qðxÞ is where was obtained using Eqs. and , and 4 F . κ −∞ ωedge extended in x and captures the long-range Coulomb behavior of Left- and right-moving plane wave modes along the edge, ± , ~ can be identified through the zero modes of S. We first note that WqðxÞ. Indeed, the Fourier transforms of WqðxÞ and W qðxÞ match ~ k=q for F = 0, solving detðSÞ = 0 yields degenerate nonchiral edge for small . This method has been used successfully to mimic edge 1=2 bulk modes with ω ± = ð2=3Þ ωq (17, 22, 26). Coulomb interactions in isolated systems (17, 28). ~ edge ~ For nonzero F, the zero mode solutions ω ± of Eq. 10 be- Crucially, the simple form of W qðxÞ above allows the integro- 4 come split, yielding chiral edge plasmons (CBPs) as shown in differential equation in Eq. to be expressed as a purely dif- edge edge Figs. 1 and 2. The modes ω + and ω− propagate as waves in ferential one with ϕqðxÞ obeying opposite directions along the edge, with faster and slower edge πe q speeds, respectively. Importantly, the frequencies ω ± depend 2 2 > 4 j j 2 2 < ∂x − 2q ϕq ðxÞ = δnqðxÞ, ∂x − 2q ϕq ðxÞ = 0, [8] both on q along the edge (Fig. 1A), as well as F (Fig. 2A). The κ ~ splitting between modes grows with q and F, because va = > < e∇ϕ × =Z 2 q ~ q ~ ωbulk where ϕq and ϕq are defined inside (x ≥ 0) and outside (x < 0) the F (Eq. ). Indeed, for small F (so that F q ), sample, respectively. Eq. 8 yields simple ϕ profiles we obtain an approximate dispersion for edge CBPs as pffiffiffi ωedge ≈ = 1=2ωbulk ± q ~ = + q2 ~ 2 < γ x > −γ x ± ð2 3Þ q 2j jF 9 Oð F Þ.Asaresult,weob- ϕ x = ϕ e 0 ϕ x = ϕ e 1 [9] q ð Þ 0 , q ð Þ 1 , tain the q-dependent Δω in Eq. 3. This behavior sharply contrasts pffiffiffi pffiffiffi with that of magnetoplasmons, which have a q-independent γ = q γ = q ωbulk 2 − ω2 = ωbulk 2 − where 0 2j j,and 1 2j j f½ð q Þ ½2ð q Þ – ω2 1=2 4 7 splitting given by the cyclotron frequency (17, 19 21, 22, 26). As g . The latter was obtained from Eqs. and by eliminat- a result, far larger splittings, arising from interactions, can be ing δnq. achieved for CBPs. edge Interestingly, for large enough q and/or F, the ω + mode A B A { (blue line in Fig. 1 and and 2 ) merges with the bulk The jump discontinuity boundary condition in Eq. 7 depends on the perpendicular ve- ωbulk locity density at the boundary, vx ðx = 0Þ. We use the subscript 0+ in the text to emphasize plasmon mode q (dashed line). As this mode merges with the that the velocity density should be evaluated on the metallic side, x ≥ 0. bulk, its potential profile ceases to be localized along the edge.

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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 with q =q = B γ−1 ωedge ωedge ωedge j j 0 0.3. ( ) CBP edge-confinement length, 1 ,of + and − (Eq. 9) for increasing F. For large F, the confinement length of the + mode diverges, edge indicating that it joins the bulk, whereas the ω− mode becomes more confined to the edge. (Inset) Electric profile shown for jFj = 0.3, exhibiting con- edge finement of ω ± to the edge.

edge iωt The disappearance of the ω + mode is shown by a diverging With an a.c. probing electric field ∼ E e , the COM equations γ−1 B confinement length of the electric potential, 1 (Fig. 2 ). In of motion are edge contrast, ω− stays far from the bulk dispersion, yielding a potential 2 2 iωt ω2m (and electric field) tightly confined to the edge. The threshold ∂t fxg + ω fxg + ωa∂tfyg = −eEx e F edge 0 ω = 0 [12] ω ω 2 2 iωt a . th above which + merges with the bulk can be obtained ∂t fyg + ω fyg − ωa∂tfxg = −eEy e , n Z bulk 0 0 from Eq. 10. Setting ω = ωq in Eq. 10 yields the threshold frequency Here ω0ðdÞ is the bare plasmon frequency in a disk of diameter d, in the absence of Berry curvature. Â Ã j = en ∂ x 2 n −2 12 Writing the as 0 tf g, we invert the Z n0 0 cm 10 Zωth = pffiffiffi = 18.3 meV, [11] COM equations of motion to obtain the optical absorption (real m m m = × ‡ 2 jFj ð ½ e 0.03Þ jFj part of the longitudinal conductivity). As shown in Fig. 3B,we find a split peak structure with the dipolar CBP peaks given by with me the free electron mass. For scale we consider a plasmon the poles of Eq. 12: mass m ∼ 0.03 me, as measured in graphene (25). sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω q ωedge Conservation of and along the edge protect the − mode ω2 ω disk a a 9F from coupling to bulk 2D plasmons. Scattering processes that ω = ω2 + ± ZΔω ≡ Zωp ≈ [13] ± 0 , F d μ meV, relax q contribute to propagation losses. However, the tight edge 4 2 ½ m confinement of the ωedge mode suppresses its electric field in the − Δω = ωdisk − ωdisk = ω Zω = ω2m=n bulk regions (Fig. 2B, Inset), suppressing its coupling to bulk where + − a and p 0 0.Ontherightside, ω2 ≈ πe2n q =m q ≈ =d ω we estimated 0 2 0j j ,withj j 1 (approximating the plasmons. Therefore, above the threshold th (gray region in Fig. # κ = A ωedge lowest lying plasmonic excitation in a disk). Here we used 1. Im- 1 ), the single, well-defined, − mode propagates unidirec- Δω d tionally along the edge. When hybridized with light, it will allow portantly, depends on the disk size, , a unique property of CBPs. The tunable optical absorption split peak structure (via F and for strong nonreciprocal propagation of CBP- without d magnetic field (see below). ) in the absence of an applied magnetic field gives a clear ex- perimental signature of CBPs. In plotting Fig. 3B, we included 2 2 Experimental Signatures of CBPs the damping rate phenomenologically via ∂t → ∂t + Γ∂t, yielding a Γ Strong plasmon-mediated light–matter interactions (11–13) make Lorentzian lineshape with its half-width determined by . Split Δω J Γ ωdisk optics an ideal means of probing/controlling CBPs. cou- peaks are clearly visible when , yielding peaks at ± . ZΓ ∼ pling to plasmons with gapless dispersion (e.g., 2D plasmons, and To give a sense of scale, we note a typical value few meV

ZΓ ≈ PHYSICS CBPs here) can be achieved through strategies such as gratings, (30), where 4 meV was measured in graphene disks. Using Γ=ω = = ωdisk and prism geometries (11). Observing unidirectional (nonrecip- p 0.25 and taking F 1, clearly resolved ± peaks can be rocal) propagation in such setups can reveal the existence of resolved for disk sizes d K 1 μm (Fig. 3B). CBPs. For demonstration, we detail an alternative experimental CBPs in Anomalous Hall Materials probe: CBP-photon coupling in finite geometries, such as metallic disks, where dipolar plasmonic modes can dominate optical ab- We now discuss materials where CBPs can be realized. We sorption (11, 14). predict that metallic systems with nonvanishing F will generically In metallic disks with finite F, CBPs manifest as clockwise/anti- host CBPs. Finite F requires broken time reversal symmetry and clockwise moving plasmonic dipole modes (Fig. 3A). These modes may arise in magnetically ordered systems or out-of-equilibrium can be described via a simple oscillator model for the motion of the nonmagnetic systems (see below). The former includes mag- ‡ dipolar CBP center of mass (COM) coordinate, fxg,wheref · g netically doped semiconducting quantum wells (see ref. 31, ≈ = – denotes the COM average. Here fvag ≈ F fe∇ϕg × ^z (green arrow) where F 1 2 was predicted) and topological insulators (3 6). gives rise to an intrinsic angular frequency ωa of plasmons in a disk (orange arrow), which adds to (subtracts from) the plasmon # frequency ω0 to produce nondegenerate anticlockwise (clock- Comparing this estimate to plasmon frequencies obtained in graphene disks (30), we A SI Appendix obtain zero field plasmon frequencies in the disk geometry top withinffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a factor of unity. wise) rotating modes (Fig. 3 and ). Here we used F Zω ≈ πe2n = md ≈ ^z Using parameters reported in ref. 30, our estimate yields 0 2 0 ð Þ 24 meV pointing to positive . A bosonic analog for ultracold atomic for a disk of d = 3 μm and reported doping μ = −0.54 eV. Ref. 30 observed a zero field −1 is discussed in ref. 29. plasmon resonance at ω0 = 130 cm = 16 meV.

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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 absorption split 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 F produce 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, ~ 2 2 2 and nK′ = 0. The characteristic density is n = Δ =4πvF Z .(Inset) Valley polarization may be induced by above-gap circularly polarized light.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As a concrete example, we examine the magnetically doped to- − analysis yields two chiral CBP modes: ωel h = ω2 + 2ω2 ± ωa (SI pological chromium doped thin-film (Bi,Sb) Te ,which ± a 0 2 3 Appendix ω 13 was recently experimentally realized (5). When moderately doped ). Estimating 0 as in the text below Eq. ,weobtainthe el−h with electrons or holes, we estimate that F ∼ 1 can be achieved ω ± curves in Fig. 3C with ωp given by Eq. 13. The frequencies of B=0 2 based on the measured anomalous Hall conductivity σxy K e =h both modes vanish at zero photoexcited carrier density, npe.The (5). These values yield a large splitting ZΔω ≈ 10 − 100 meV for mode splitting increases with npe and can reach sizable values for 3 13 short wavelength plasmons (Eqs. and ). When probed in the npe ≥ n~,reflectingF pe pumping. disk geometry, we predict this system will yield two split absorption Large splittings require npe J n~. The large gaps Δ J 1 eV of peaks in the absence of magnetic field. many transition metal dichalcogenides (7, 34, 35) yield large characteristic densities requirements, n~ J 1013 cm−2. In contrast, On-Demand CBPs in Nonmagnetic Materials other gapped Dirac materials such as G/hBN (36–38) and dual- Intriguingly, finite F can also be achieved in nonmagnetic ma- gated bilayer graphene (39) possess Δ ≈ 10 − 200 meV, yielding terials, without an applied magnetic field. This class includes, for significantly smaller and more favorable n~. These materials ‡ example, gapped Dirac materials where inversion symmetry is possess valley-selective optical selection rules (32), and present broken. In these, a valley polarization (Fig. 3C, Inset) can be an ideal venue to achieve maximal F = 1 (and large Δω), even induced by circularly polarized light (32), yielding an anomalous with relatively weak pump fluence.jj Hall effect that has recently been observed (7). A further promising strategy to achieve large CBP splittings is Can CBPs exist in photoexcited systems? To analyze this, we to stack m layers of gapped Dirac materials on top of each other, focus on nominally time reversal invariant gapped Dirac mate- with no tunnel coupling between the layers. Stacking achieves 1=2 rials. We model the valley dependent F as (33) F K,K′ = τzNsfn~ = i 1=2 ( ) larger photo-excited carrier densities due to the increased nK K′ + n~ Δ nK K′ ½2ð , Þ g sgn , where , are the valley carrier densi- absorption and (2) a larger maximal Berry flux (when npe n~) τz = ∓ K K′ Ns = ties, for the , valleys, 2 is the degeneracy, and and, hence, larger CBP splittings; the maximal Berry flux is F max = n~ = Δ2= πv2 Z2 4 F gives a characteristic density scale. The bandgap mF single. In such a structure, the long-range Coulomb interaction is 2jΔj.WhennK = nK′ is in equilibrium, the total flux F = allows the photoexcited densities in different layers to oscillate + ′ F K F K vanishes. collectively. Interestingly, when the system is pushed out of equilibrium, e.g., by circularly polarized light, the populations in the valleys Conclusion ‡ may become imbalanced, nK ≠ nK′ (32) (Fig. 3C, Inset). As a CBPs are robust collective excitations of metallic systems, arising result, the net Berry flux for the entire electronic system (sum- from two simple ingredients: Berry flux and interactions. Our med over both valleys) is nonzero, yielding F pe ≠ 0. We analyze the collective modes of the photoexcited electron-hole system in 12 the disk geometry following Eq. , accounting for the mutually jjNote that the (neutral) that may form in some large-gap materials feel zero net attracting electron and hole populations and using F = F pe. This Berry curvature (43) and to first approximation do not contribute to CBPs.

4662 | www.pnas.org/cgi/doi/10.1073/pnas.1519086113 Song and Rudner Downloaded by guest on September 24, 2021 analysis indicates that CBPs survive for both weak and strong we propose, CBP mediated unidirectional waveguides can be re- interactions. As a result, we conclude that CBPs are generic in alized in readily available nonmagnetic materials (e.g., the van der metallic anomalous Hall phases, including out-of-equilibrium Waals material class). CBP-based nonreciprocity, if realized ex- states where finite F emerges from driving (e.g., in optically pumped perimentally, stands to play a vital role in the miniaturization of valley polarized gapped Dirac materials). Indeed, thisfeatureallows optical components that are magnetic field free. CBPs to be used for all-optical diagnostics of anomalous Hall and topological phases, as well as pumped or periodically driven Note Added in Proof. During the review of our manuscript, a systems (40, 41). Optical probes of the latter are particularly relatedworkonCBPsinphotoexcitedgappedDiracsystems appealing because transport measurements require contacts that appeared (44). often complicate and destroy the novel electron distributions and coherences of driven systems. ACKNOWLEDGMENTS. We thank J. Alicea, J. Eisenstein, M. Kats, L. Levitov, Perhaps the most appealing prospect is coupling CBPs with light M. Schecter, and B. Skinner for useful conversations. J.C.W.S. acknowledges for subwavelength and strong nonreciprocal propagation. Exhibiting support from the Walter Burke Institute for Theoretical Physics as part of a q Burke fellowship at California Institute of Technology. M.S.R. acknowledges asinglechiralmodeatlarge (large splitting and large frequency), support from the Villum Foundation and the People Programme (Marie Curie precisely where plasmons give large compression of optical mode Actions) of the European Union’s Seventh Framework Programme (FP7/2007- volume, hybrid CBP-polaritons are strongly nonreciprocal (10). As 2013) under Research Executive Agency Grant PIIF-GA-2013-627838.

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