Raman scattering in correlated thin films as a probe of chargeless surface states Brent Perreault,1 Johannes Knolle,2 Natalia B. Perkins,1 and F. J. Burnell1 1School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 2Department of Physics, Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, U.K. (Dated: May 24, 2016) Several powerful techniques exist to detect topologically protected surface states of weakly- interacting electronic systems. In contrast, surface modes of strongly interacting systems which do not carry electric charge are much harder to detect. We propose resonant light scattering as a means of probing the chargeless surface modes of interacting quantum spin systems, and illustrate itsefficacybyaconcretecalculationforthe3DhyperhoneycombKitaevquantumspinliquidphase. 6 Weshowthatresonantscatteringisrequiredtoefficientlycoupletothismodel’ssublatticepolarized 1 surface modes, comprised of emergent Majorana fermions that result from spin fractionalization. 0 We demonstrate that the low-energy response is dominated by the surface contribution for thin 2 films, allowing identification and characterization of emergent topological band structures. y a M Introduction. One of the most striking recent develop- mentsincondensedmatterphysicshasbeenthediscovery (a) 0 that certain types of three-dimensional (3D) band struc- 2 tures harbor topologically protected surface states, which cannot be gapped out without breaking a bulk symme- ] (b) l try. Systems with such surface states include topological flatband Fermiarc e Surface - insulators [1–3], Weyl semimetals [4–9], and a number r of others [10, 11]. In these weakly-interacting systems t (c) s where the quasiparticles carry electric charge, theoreti- Weylpoint . t cal predictions have quickly led to experimental detec- a m tion: the Dirac cone surface states of 3D topological insulators were first detected several years ago [12–14] - d using high-resolution angle-resolved photoemission spec- n Fermiring troscopy (ARPES). More recently, ARPES has also de- o tected the Fermi arcs characteristic of Weyl semimetals Bulk c [ in compounds TaAs, TaP, NbAs and NbP [15–18]. FIG. 1. (c) Schematic of the Fermi ring and limiting posi- Such surface states are not restricted to weakly- 2 tionsoftheWeylpointsoftheemergentchargelessMajorana v interacting electronic systems. In fact, topological sur- fermions as κ → 0 in the bulk Brillouin Zone (BZ). (b) the 3 face states have been predicted in a number of Mott- projectionoftheWeylpointsontothesurfaceBZandthesur- 2 insulating systems where they often originate from spin faceflatband(κ=0)andFermiarc(κ>0). (a)illustratesa 6 fractionalizationinquantumspinliquids(QSLs)[19–22]. resonant light-scattering process in the anti-symmetric chan- 2 nel that can probe the surface modes. This intriguing possibility poses an experimental chal- 0 . lenge: since surface probes such as ARPES and STM 1 couple to charge, then how can such chargeless surface 0 states be detected? For bulk properties of chargeless 6 beidentifiedbyconsideringthelow-energypowerlawsin 1 topologicalstates,muchprogresshasbeenmaderecently spectraofthinfilms;and(2)thesurfacestatescontribute : in probing candidate QSLs using inelastic neutron [23– v significantly to the light-scattering response only in the 32] and Raman [33–40] scattering. Both measurements i X couple to spin degrees of freedom (d.o.f.), and hence can resonant regime [34, 54, 55]. r give signatures of fractionalized spinon excitations. Our proposal is summarized schematically in Fig. 1. a Here we study inelastic light scattering as a probe of Theideaisthatthoughlightscatteringistypicallyabulk the chargeless topological surface states that can arise probe, when applied to sufficiently thin films the surface in strongly-interacting systems. We focus on the exam- responsescanbeobservableifthedensityofstates(DOS) ple of the Kitaev QSL on the hyperhoneycomb lattice of surface states decays more slowly with frequency than [41, 42], which is known to harbor such boundary states the bulk DOS so that the surface response dominates at [10, 21]. This model is of particular interest because the sufficiently low energies. We show that this occurs for insulating magnet β−Li IrO [43, 44] is believed to be both time-reversal (TR) symmetric and TR broken[22] 2 3 described by an effective Hamiltonian on a hyperhoney- topological phases of the hyperhoneycomb Kitaev QSL, comblatticewithdominantKitaev-typeinteractions[45– allowingdirectexperimentaldetectionofthecorrespond- 53]. Ourmainfindingsarethat(1)thesurfacemodescan ing topological surface states. 2 linear )c]-2 a g(I[ quadratic lo -4 0.2 (c) (b) -1.5I[ac]lIac]o0[ac]g.01(.002-ω1..12/J) -0.5 κκκ κ=== κκ=000 .. ==010 300.03 00000..2.2..222 I[00.1 κ κ= =0 .00.31 0 0.2 0.4 0.6 0.8 FIG.2. Thehyper√honeycomblattice,H-0. Thelatticevectors IIIII[ac][ac][ac][ac][ac]00000..1.1..111 0000 0.02.κκ2κκκκκκ κκ == ==0 ====ω == .000 004 00/00..00J.40.0.ω.003033/033J.06.6κ =0 .008..18 are a1/2 =(−1,∓ 2,0) and a3 =(−1,0,3). κκκκ κ = = == = 0 0 0ω0.0.1.1./.11J1 00000 (a) 00000 00000..2.2..222 00000..4.4..444 00000..6.6..666 00000..8.8..888 ωωωωω///JJ//JJJ 3D Kitaev model. The Kitaev Hamiltonian [56] is FIG. 3. (a) Low-energy bulk scattering intensity in the anti- symmetricchannelI forvariousvaluesoftheeffectiveper- [ac] H =J (cid:88) σασα , (1) turbation κ and (b) the log-log version. The index [ac] rep- K i j resentsthechannelantisymmetrizedover“in”and“out”po- (cid:104)ij(cid:105) α larizations in the a and c directions [58]. (c) illustrates reso- nant light-scattering processes contributing to the [ac] chan- where(cid:104)ij(cid:105) arenearest-neighbor(NN)bonds,σα arethe nel, which mimics the DOS’. α j Pauli matrices, and α = {x,y,z} specifies which com- ponents of spins interact along each of three inequiv- evant term at low-energy is alent bonds. The model is solved exactly by replac- ing the spin operator at each site j with four Majorana (cid:88) (cid:88) fermionscj andbαj viaσjα =ibαjcj [56]. IntermsofMajo- Hh =κ σiασlγσjβ =iκ u˜(cid:104)(cid:104)ij(cid:105)(cid:105)γcicj, (2) ranafermions,theHamiltonianinEq.(1)takestheform (cid:104)(cid:104)ij(cid:105)(cid:105)γ (cid:104)(cid:104)ij(cid:105)(cid:105)γ Ha KZ2=laJtt(cid:80)ice(cid:104)igj(cid:105)aαuuge(cid:104)ijfi(cid:105)αelcdicfjo,rwthheercejtMheaujo(cid:104)irja(cid:105)nαa≡s.iTbαihbeαjuf(cid:104)oijr(cid:105)mα where(il),(jl)arepairsofneighborsalongabondoftype α and β respectively, and γ is complementary. In terms commute with each other and the Hamiltonian, and are of Majorana spinons this gives a next-nearest neighbor therefore static. (NNN) hopping term with u˜ ≡ u u and In the Majorana description, the physical d.o.f are the (cid:104)(cid:104)ij(cid:105)(cid:105)γ (cid:104)il(cid:105)α (cid:104)lj(cid:105)β κ ∼ hxhyhz/∆2. On the H-0 lattice, the magnetic field fluxes of the Z gauge theory on elementary plaquettes 2 perturbation gaps out the Fermi ring, leaving a pair of of the lattice, and dispersing Majorana spinons c in the i Weyl points [22], which are fixed to the Fermi energy flux background. The ground state on a given lattice by the unbroken inversion symmetry [57]. The surface corresponds to a fixed flux configuration, which is flux flat bands are reduced to surface Fermi-arcs connecting free for the H-0 lattice [41, 53, 57]. theprojectionoftheWeylpointsinthesurfaceBrillouin In this flux-free configuration, the Hamiltonian is zone (BZ) [5] (see Fig. 1). quadratic in the Majorana spinons {ci}. Diagonaliza- Raman scattering. The key features of the bulk tionleadstoabandstructurewithtwodistinctbandson and surface Majorana spinon bands described above can the H-0 lattice, where the modes at zero energy form a be detected using inelastic photon spectroscopy. To es- one-dimensional Fermi ring shown schematically by the tablish this, we first review some important aspects of bluelineinFig.1(c). Withopenboundaries, surfaceflat the derivation of the Raman operator in Mott insula- bands occur within the projection of the Fermi ring onto tors [34, 54, 55] to show that the magnetic field has a the surface BZ (see Fig. 1(b)) [21]. This band structure negligible effect, and clarify the resonant processes of in- andtheassociatedsurfacestatesareprotectedbyacom- terest. bination C = IT of inversion and TR symmetry, which Most inelastic light scatting of low-energy magnetism isasublatticesymmetrywithintheMajoranaspinonde- has been in the regime of Raman spectroscopy [59, scription [21]. The gapless surface modes are sublattice 60], which probes excitations ranging 1–100 meV (10– polarized,andhenceprotectedfromback-scattering[10]. 1000cm−1) [60, 61]. However, given the expected Ki- Applying a magnetic field H = (cid:80) hαSα, where taevexchange-interactionscale(J)inthehoneycombiri- h j,α j {hx,hy,hz} are all nonzero, destroys the Fermi ring by dates of around 2 – 4 meV [46, 62, 63], the energy scales breaking TR, and therefore the sublattice symmetry C. discussed here are in the regime of Brillouin scattering: If h is much smaller than the flux gap ∆, the low-energy 0.01–1 meV (0.1–10 cm−1) [60, 61], which differs from Hamiltonian retains the zero-flux ground-state flux con- Raman only by the spectrometer. We continue to refer figuration, and the first non-vanishing TR symmetry- to ‘Raman’ operators and spectra although each applies breaking terms appear at third order in h [56]. The rel- for both experiments. 3 At zero temperature the Raman response can be writ- For NN Kitaev interactions, the LF vertex does not ten as a correlation function of scattering operators couple to fluxes, probing only Majorana spinon band structures [37]. However, because the NN spin-exchange (cid:90) I(ω)=2π dωeiωt(cid:104)R(t)†R(0)(cid:105), (3) processesthatenterinto(7)involvebothsublattices,the LF vertex cannot couple to the sublattice-polarized sur- whereω =ω −ω isthetotalenergytransferredfrom face flat bands. Moreover, even if the sublattice sym- in out metry is broken with a small magnetic field, the surface the in- and out-going photons to the system, and in the following we assume that ω (cid:28) ω . For a Mott- states are still approximately polarized, and the LF ver- in(out) tex couples only very weakly to the surface states. insulator, the Raman operator is Inordertodetectthegaplesssurfacemodes, aRaman R=−PH(cid:15)∗out(H −iη)−1H(cid:15)inP, (4) operator should contain terms coupling two Majorana t t spinons on the same sublattice. Such a term can ap- whereP istheprojectorontostateswithafixedelectron pear by tuning the photon frequency resonant with the occupancy per site, (cid:15)in and (cid:15)out are the incoming and Mott gap so that t/(U − ωin) is no longer very small, outgoing photon polarization vectors, respectively, and and higher-order terms in the expansion of Eq. (6) con- H(cid:15) is the electron/photon vertex for the polarization (cid:15) tribute intermediate electron hops. Two such resonant t given by hopping processes, involving three different sites and an NNN electronic hop, are illustrated in Fig. 3(c); such (cid:18) (cid:19) Ht(cid:15) = (cid:126)iec (cid:88) (dn,n(cid:48) ·(cid:15))Tnγ,,nγ(cid:48)(cid:48)a†n,γan(cid:48),γ(cid:48). (5) processes can lead to the desired low-energy term: n,n(cid:48),γ,γ(cid:48) (cid:88) R =iκ(cid:48) σασγσβ ×A res i l j ilj HereTnγ,,nγ(cid:48)(cid:48) describestheelectronichoppingmatrix,dn,n(cid:48) (cid:28)ij(cid:29)γ iisstthheesapnantiihaillavteicotnoropfreormatloarttfoicreasniteelnecttorosnitaetns(cid:48)i,taenndwanit,hγ =−κ(cid:48) (cid:88) u˜(cid:104)(cid:104)ij(cid:105)(cid:105)γcicj ×Ailj (8) (cid:104)(cid:104)ij(cid:105)(cid:105)γ γ running over spin and orbitals. ThefullHamiltonianintheresolvent(−H+iη)−1 can whereκ(cid:48) containstheelectron-photoncouplingandspin- be written as H = Ht+HU, where Ht is the electronic exchange constants. There are other three-spin terms hopping Hamiltonian, and for convenience we define the with α,β, and γ permuted in Eq. (8) that create fluxes interactiontermHU describingtheon-siteelectroninter- andarethereforeunimportantatlowenergies. Thecom- actions,suchasCoulombrepulsionandHund’scoupling, putationoftheresonantlight-scatteringmatrixelements relative to the initial photon energy: HU = Hint −ωin. follows Refs. 55 and 34; details for the iridates will be The resolvent (−H+iη)−1 can be formally expanded to presented elsewhere. give Unlike the LF scattering vertex, the polarization-dependent factor A = R=PHt(cid:15)∗out(cid:2)HU−1+HU−1HtHU−1+...(cid:3)Ht(cid:15)inP, (6) [((cid:15)in·dji)((cid:15)∗out·dil)−((cid:15)∗out·dji)((cid:15)in·dil)i]l,jisonlynon- zero in polarization channels that are anti-symmetric in (we dropped −iη). In the presence of a magnetic field, the exchange of in and out polarizations. This requires theresolventinEq.(6)hasanadditionalsmallparameter the use of polarization channels that do not appear with proportional to h/U: only the LF operator Eq. (7). We focus on one of these, [H +H +H ]−1 =H−1(cid:2)1+H H−1+H H−1+...(cid:3). the anti-symmetric [ac] combination of the two-photon U t h U t U h U processes (illustrated in Fig. 1), in which one photon Hence, in the regime h(cid:28)t we can neglect the magnetic has polarization along a and another along c. Due to field during the scattering process. the low symmetry of the model, isolating this channel If t/(U − ω ) is small, electron hopping is strongly requires observation of several directly-observable spec- in suppressed, and the derivation of the Raman operator tra. However, in the resonant regime the anti-symmetric proceeds as it does for a spin-exchange Hamiltonian. part is expected to dominate the response in a few The lowest-order terms contributing to R are linear in directly-observable channels, such as Iab [58]. Because t/(U−ω )andhavethewell-knownLoudon-Fleury(LF) of the form of Eq. (8), the computation of low-energy in form [64] Raman spectra for this system amounts to evaluating four-spinon dynamic correlators of a quadratic fermionic (cid:88) R= (dn,n(cid:48) ·(cid:15)in)(dn,n(cid:48) ·(cid:15)∗out)Hnα,,nβ(cid:48)σnασnβ(cid:48), (7) Hamiltonian [37, 40]. Results. The bulk light-scattering intensity at low fre- n,n(cid:48);α,β quencyisshownfordifferentvaluesofκinFigs.3(a)and where Hα,β defines the generic spin-exchange Hamilto- 3(b). The scattering intensity is almost linear at κ = 0 n,n(cid:48) nian on the bonds (cid:104)n,n(cid:48)(cid:105), which we take as the pure andclosetoquadraticatκ=0.03J [65]directlyreflecting Kitaev model. power law changes in the Majorana spinon DOS antici- 4 DOS Raman (a) 0 (b) (e) 0 (f) DOS00..12 κ=0 LLLL====481362 og(DOS)-5 I[ac]0.5 κ=0 log(I)[ac]---321 0 L=64 l 0 0 0.5 1 -3 -2 -1 0 0 0.2 0.4 0.6 -2 -1 0 ω/J log(ω/J) ω/J log(ω/J) (c) 0 (d) (g) 0 (h) S0.2 κ=0.03J LL==48 OS) 0.5 κ=0.03J )ac] DO0.1 LLL===136624 log(D-5 I[ac] log(I[-5 0 0 0 0.5 1 -3 -2 -1 0 0 0.2 0.4 0.6 -2 -1 0 ω/J log(ω/J) ω/J log(ω/J) FIG. 4. The low-energy DOS is plotted for (a) κ = 0 and (c) κ = 0.03J for different slab widths L, measured in unit cells in the a direction. The low-energy peaks [plateaus] in the DOS are due to the surface flat bands in (a) [surface Fermi-arcs 1 in (c)]. (b) and (d) show the corresponding log-log plots illustrating the crossover between power laws describing the surface contribution to ones describing the bulk. (e), (f), (g), and (h) are the same plots for the resonant scattering intensity I in [ac] the antisymmetric [ac] channel. The suppression of low-frequency modes in (g) compared to (c) is due to suppression by the scattering matrix elements. pated in Ref. [22]. surface Fermi arcs is only detectable at energy scales be- To study the Brillouin/Raman scattering response of low κ. We plot the DOS for κ=0.03J in Figs. 4(c) and the surface modes, we consider systems that are infinite (d). This value is below the local flux gap of ∆=0.13J in two directions but have a finite number L of unit cells [32,53,57]puttingitintheperturbativeregime. Wefind along the stacking direction a . This is one of several thattheflatbandpeaksaresignificantlysuppressedupon 1 cutting planes with similar flat bands. However, cutting introducing the magnetic field, although the low-energy planes whose normal vector lies in the plane of b and power-laws in this case are not visible at the numerical a −a −a do not receive a projection from the interior resolution used of about 0.03J. Similar behavior is ob- 3 1 2 of the bulk Fermi ring and therefore are not required to served in the Raman intensity shown in Figs. 4(g) and host a flat band. (h), up to some additional suppression of the flat-band peak due to matrix-element effects. Figs. 4(a) and (b) show the low-frequency behavior of the DOS in the unperturbed model (κ = 0) in a lin- Experimentally, we expect that the low-energy peaks ear and log scale, respectively. The low-energy peak in and power laws in the absence of a magnetic field should the DOS seen for different slab thicknesses indicates the be discernible by Brillouin scattering for a film of thick- presence of the flat band. The position of this peak is ness 20 or 30 unit cells or less, corresponding to 100 to not strictly at zero frequency due to the top and bottom 300nm in β−Li2IrO3. surface modes hybridizing, shifting it to higher energies. Discussion. We have shown that low-frequency res- The peak drifts towards zero frequency for larger L, but onant light-scattering is a powerful probe of the QSL its height relative to the rest of the spectrum decreases state, and can reveal both the bulk and surface Majo- due to decreasing surface-to-bulk ratio. At low energies rana spinon DOS in the 3D Kitaev QSL. For the hy- ourresultsareconsistentwiththepowerlaw−1expected perhoneycomb lattice, we find a bulk signature of the for the surface states, in contrast to the linear power law T-broken Weyl spin liquid that arises upon perturbing seenabovethecrossoverfrequencyω . Figs. 4(e)and(f) the Kitaev Hamiltonian with a weak magnetic field, as c showthescatteringintensityinthe[ac]channelforκ=0, well as a signature of topological surface states (both where we find that the behavior of the resonant Raman with and without T-breaking) in thin films. Our main intensity reflects the behavior of the DOS, as expected. results are that (1) the Weyl spin liquid can be distin- When κ (cid:54)= 0 most of the surface states are gapped, guishedfromtheparentT-symmetricstatebythepower leaving only a surface Fermi arc at very low-energies, law governing the low-frequency response; (2) the sym- whose contribution to the DOS tends to a constant at metrypropertiesoftheusualLFvertexdonotallowitto zero frequency, in contrast with the quadratic power law coupletosub-latticepolarizedlow-energysurfacemodes; seen above ω . 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The low symmetry of the hyperhoneycomb model in a magnetic field makes this case more involved and we therefore present here an example set of observations that lead to the [ac] channel discussed in the main text. The result applies for any antisymmetric channel [αβ] with no symmetry assumptions except that α and β be perpendicular, although the more general case follows from this one trivially. 7 Near resonance the light-scattering operator is a 3×3 matrix in cubic polarizations α,β =a,b,c [40]. R=(cid:15) R (cid:15)∗ , (9) in,α αβ out,β wherethecomplexconjugationoccursonlyforphoton-creationandisimportantonlyforpolarizationswithacircular component. Weallowourselvestouselabelsforothervectorsinplaceofa,b,ctorepresentRinpolarizationchannels √ √ other than these. One example is the coordinates rotated by 45 degrees: aˆ(cid:48) = (aˆ+cˆ)/ 2 and cˆ(cid:48) = (−aˆ+cˆ)/ 2. These can be decomposed in terms of the cubic coordinates as 2Ra(cid:48)a(cid:48) =Raa+Rcc+2R(ac) (10) 2Rc(cid:48)c(cid:48) =Raa+Rcc−2R(ac), (11) forinstance, where2R =R +R and2R =R −R . UsingEq.(3)fromthemaintextwecanuseEq.(10) (ac) ac ca [ac] ac ca to decompose the spectra in the rotated coordinates terms of ones in the cubic coordinates leading to 4Ia(cid:48)a(cid:48) =Iaa+Icc+2Iaa,cc+4Iaa,(ac)+4Icc,(ac)+4I(ac), (12) for example, where the mixed polarizations I = 2π(cid:82) dωeiωt(cid:104)[R†(t)R (0)+R† (t)R (0)]/2(cid:105) are not directly ob- A,B A B B A servableandmustbeinferredfromothermeasurements. Weadditionallydefinetheleftandrightpolarizationvectors √ √ 2rˆ=aˆ+icˆand 2ˆl=aˆ−icˆ. Note that this assumes that both incoming and outgoing light travels in the direction normal to the a−c plane. One can infer the desired spectra using only linearly polarized light by making measure- ments along additional non-orthogonal directions, but we do not report on those results here since back-scattering is the typical experimental setup and allows for cleaner results. Using decompositions such as Eq. (12), as well as the more trivial relation I +I = 2(I +I ), the desired ac ca (ac) [ac] polarization combination can be related to the observable ones by 4I[ac] =(Iac+Ica)−(Ia(cid:48)a(cid:48) +Ic(cid:48)c(cid:48))+(Irr+Ill). (13) If we have a symmetry of the Hamiltonian taking c → −c, as in the low-energy theory considered in the main text, or a→−a, then we have that Ia(cid:48)a(cid:48) =Ic(cid:48)c(cid:48) and Irr =Ill so that one need only measure four independent spectra. Finally, we note that inpractice one mayalready be able to see the low-energy behavior describedin the main text in directly-observable spectra for which the symmetric bulk spectrum is suppressed at low energies. Such is the case for the spectrum I , for which the symmetric part I was shown to have a vanishing bulk response below 4J in ab (ab) the absence of a magnetic field [40].