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Repulsive Casimir force between Weyl semimetals Justin H. Wilson,1,2 Andrew A. Allocca,1 and Victor Galitski1,2 1Joint Quantum Institute and Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742-4111, USA 2School of Physics, Monash University, Melbourne, Victoria 3800, Australia Weyl semimetals are a class of topological materials that exhibit a bulk Hall effect due to time- reversal symmetry breaking. We show that for the idealized semi-infinite case, the Casimir force between two identical Weyl semimetals is repulsive at short range and attractive at long range. Consideringplatesoffinitethickness,wecanreducethesizeofthelong-rangeattractionevenmaking itrepulsiveforalldistanceswhenthinenough. Inthethin-filmlimit,westudytheappearanceofan 5 attractive Casimir force at shorter distances due to the longitudinal conductivity. Magnetic field, 1 thickness,andchemicalpotentialprovidetunablenobsforthiseffect,controllingtheCasimirforce: 0 whetheritisattractiveorrepulsive,themagnitudeoftheeffect,andthepositionsandexistenceof 2 a trap and antitrap. n u PACSnumbers: 04.50.-h11.15.Yc73.43.-f78.20.Jq J 1 In 1948, Casimir [1] showed that quantum fluctua- ] tions in the electromagnetic field cause a force between l l two perfectly conducting, electrically neutral objects. a This has since been extended to other materials [2, 3]. h - Throughout this time, Casimir repulsion between two s materials in vacuum has been a long sought after phe- e m nomenon [4, 5]. There are principally four categories in which repulsion can be achieved: (i) modifying the di- . t electric of the intervening medium [4, 6, 7], (ii) pairing a a m dielectricobjectandapermeableobject[5](suchaswith metamaterials[8]),(iii)usingdifferentgeometries[9–11], FIG.1. ThesetupwewillconsiderhereistwoWeylsemimet- d- and(iv)breakingtime-reversalsymmetry[12,13]. Inthis als separated by a distance a in vacuum and with distance n paper, we are concerned with Casimir repulsion in iden- between Weyl cones 2b in k space (split in the z direction). o tical time-reversal broken systems. Specifically, we will c studyhowWeylsemimetalswithtime-reversalsymmetry [ can lead to interesting transitions [18]), and a Hall effect breaking can exhibit Casimir repulsion. The key ingre- that is too strong can suppress Kerr rotation and hence 2 dient to Casimir repulsion in this paper is the existence v ofanonzerobulkHallconductanceσ =0,σ = σ lead to attraction. The latter case is an interesting phe- xy xy yx 9 [14]. (cid:54) − nomenon where “more” of a repulsive material can lead 5 to attraction. It is a general theorem that mirror symmetric objects 6 The material we are interested in is marginal in both without time-reversal symmetry breaking can only at- 7 the case of longitudinal conductance and in an over- 0 tract one another with the Casimir effect [15]. This is whelming Hall effect: Weyl semimetals [14] with the . understoodwiththeLifshitzformula[2]whereifwehave 1 Casimir setup seen in Fig. 1 and the resulting normal- two materials characterized by the two reflection matri- 0 ized Casimir pressure seen in Figs. 2–4. These materials cesR andR andseparatedbyadistanceainaparallel 5 1 2 havelinearlydispersivebandstructurescharacterizedby 1 plate geometry, we have Weyl nodes with different chiralities and characterized Xiv: Ec(a)=(cid:126)(cid:90) (2dπ2k)2 (cid:90) d2ωπ trlog[I−R1R2e−2qza], (1) bteymapcehrairtaulraenhoamvealayz[e1r9o].dcClloenangitWudeyinlaslecmoinmdeutcatlisviattyzaenrdo opticalconductivityRe[σ ] ω [20]. Additionally,they r where the trace is a matrix trace and q = √ω2+k2. xx ∝ a z exhibitabulkHalleffectexemplifiedinthedclimitbyan This integral generally yields an attractive force; how- axionicfieldtheory[21]whereinadditiontotheMaxwell ever, if we break time reversal symmetry, obtaining an- action, we have tisymmetric off-diagonal terms in the reflection matrix Rxy = −Ryx there is the possibility of Casimir repul- S = e2 (cid:90) d3rdtθ(r,t)(cid:15)µναβF F , (2) sion[16]. Onecandidateisatwo-dimensionalHallmate- A 32π2(cid:126)c µν αβ rial [12], and similarly, another is a topological insulator wherethesurfacestateshavebeengappedbyamagnetic where θ(r,t) = 2b r 2b t and 2b is the distance be- 0 · − field [13, 17]. A Hall conductance does not guarantee re- tween Weyl nodes in k space whereas 2b is their energy 0 pulsion;longitudinalconductancecanoverwhelmanyre- offset, e is the charge of an electron, (cid:126) is Planck’s con- pulsion from the Hall effect (although the magnetic field stant, c is the speed of light, F is the electromagnetic µν 2 fieldstrengthtensor,and(cid:15)µναβ isthefullyantisymmetric 0.27 four-tensor. InversionsymmetrybreakingWeylsemimet- als,ontheotherhand,donotexhibitadcHalleffect[22] 0.2 andthereforewillnotseetheeffectsdescribedinthispa- per. The electrodynamics of this were investigated in 0 P / Ref. [23] where the authors even comment on the possi- Pc 0.1 bility for a repulsive Casimir effect. The marginal nature of Weyl semimetals makes them prime candidates for tuning the Casimir force between Attractive 0 attractive and repulsive regimes. In constructed Weyl 0.02 Repulsive − semimetalsmadeofheterostructuresofnormalandtopo- 0 4 8 12 16 20 logicalinsulators[24]anexternalmagneticfieldcancon- σ a/c trol the Hall effect [25] and hence the repulsive effects. xy Additionally, some of the first materials that have been FIG.2. (Coloronline)ThenormalizedCasimirforcebetween predicted were pyrochlore iridates [26]; these could also two semi-infinite bulk Hall materials. Repulsion is seen for see a repulsion tunable with carrier doping or an addi- σ a/c (cid:46) 4.00. P is the distant-dependent ideal Casimir tional magnetic field. xy 0 force [1]. For σ a/c→∞, P /P →1. In a real material and experiment at finite tempera- xy c 0 tures, disorder and interactions should be taken into account, and in Weyl semimetals they lead to a finite dc conductivity [20, 24, 27]. We simulate this effect in the only depends on the ratio cqz/σxy. This dependence latter part of this paper by raising the chemical poten- has implications for the Casimir force. After changing tial in our clean system, leading to intraband transitions variables to solely qz, we can inspect the Casimir that contribute to the longitudinal conductivity (in the pressure Pc(a) = ∂Ec/∂a, and we have an expression − (cid:104) (cid:105) dc limit these are singular contributions). P = 2(cid:126)c (cid:82) dq q3 g qz ,2q a , with a function Tobegin,weconsidertwosemi-infiniteslabsofaWeyl g(c qz ,(22qπ)a2) wrizttezn outσxiny/cthe Szupplemental Material semimetal (z <0 and z >a to be precise), neglecting all σxy/c z [28] for completeness. If we then change variables to frequency dependence to the conductivities by assuming x = 2aq and normalize by Casimir’s original result for theelectromagneticresponseiscapturedbyEq.(2). The z resultisjustamaterialthatissolelyabulkHallmaterial perfect conductors P0 = −2(cid:126)4c0πa24 [1], we can write the with current responses given by the Hall conductivity equation for the pressure as Pc/P0 =f(σxya/c). σ = e2b/2π2(cid:126). This response can be encoded in the Withthisformulation,weplotnormalizedforceP /P xy c 0 dielectric function so that (cid:15) =(cid:15) =(cid:15) =1 and (cid:15) = asafunctionofσ a/cobtainingthesinglefunctionseen xx yy zz xy xy (cid:15) = iσ /ω. With this set up, if an incident wave k in Fig. 2. We see that for σ a/c (cid:46) 4.00 we obtain yx xy xy −hits such a material it will break up into two different repulsion whereas for σ a/c (cid:38) 4.00 we obtain attrac- xy polarizations in the material k that satisfies k = k , tion. Thus, these similar materials trap each other at a ± x± x k = k , and (k )2 = k (k σ /c). Additionally, fixeddistancesimplydependentontheHallconductivity, thy±e twoyelliptical pz±olarizatizonszD± =xy(cid:15)(ω)E are D a 4.00 . If we insert the value of σ = e2b/2π2(cid:126) tkchkzω±e<(pklσazxn±ye/σco,xfyoi/nncce)ideoe1fn∓tcheiekapz±noedl2aerw1izh=aetrieoe2±ne×s2iksis±ep.vaeNrnpoe±etsicncedenictt.hualat±rfot∝or iatnhbTtarloeatp.tihf≈i1s/σebxxy∼p/creOs(sniomn,),wtehefinndaTarTaprap∼≈O8(µ60m/)xby.quTihteisrmeaesaonns- The incident and reflected polarizations can be bro- As the distance between the materials gets long, ken upintotransverseelectric (TE)andtransverse mag- P /P 1. This behavior is markedly different from netic (TM) modes, and the reflection matrix R(ω,k) c 0 → the thin film Hall case obtained by Tse and MacDonald just connects incident to reflected (ETM,ETE)T = r r in Ref. [12] and Rodriguez-Lopez and Grushin Ref. [17]. R(ω,k)(ETM,ETE)T. As shown in the Lifshitz formula 0 0 Theyfoundasmall[two-dimensional(2D)]quantumHall Eq. (1), we need the imaginary frequency reflection ma- effect implies a quantized and repulsive Casimir force at trix. If we let ω iω and define q2 = ω2/c2+k2 +k2, → z x y long distances. In our case, we get attraction at long the reflection matrix for a semi-infinite slab of this bulk distances for a bulk Hall material independent of the Hall material is magnitude of the Hall effect. To resolve this seeming 1 (cid:18)Q σ /c Q +2q (cid:19) inconsistency, imagine a finite thickness of the bulk Hall R∞(iqz)= σxy/c Q−+−−2xqyz Q−−−+ σxy/zc , (3) mduactteirviiatlyoσf thdicdkinveersgseds,atshden the ,twano-ddiimnetnhseiocnaaslecoofna- xy → ∞ where we have defined for brevity Q = 2D quantum Hall plate with infinite Hall conductivity, (cid:114) (cid:113) ± the Casimir effect is attractive and approaches P . 0 2q ( q2+σ2 /c2 q ) (the real frequency ver- z z xy ± z To make this argument more precise, one can actually sion of R is found in the Supplemental Material [28]). findthereflectionmatrixforabulkHallsystemofthick- Inspecting∞R (iq ), we see that the reflection matrix nessdandtheresultis(derivationofR dependsonlyon z d ∞ 3 the axionic action Eq. (2) and can be found in the Sup- plemental Material [28], calculated for real frequencies) (cid:18) (cid:19) R R R (iq )= xx xy , (4) d z R R xy xx − with R = 1σxy(Q sinhQ d+ σxy coshQ d Q sinQ d σxy cosQ d)/D, (5) xx −2 c − + c + − + − − c − R = 1σxy(Q sinhQ d+2q coshQ d Q sinQ d 2q cosQ d)/D, (6) xy −2 c + + z + − − − − z − where D =(Q2 + 1σx2y)coshQ d+(2q Q + σxyQ )sinhQ d+(Q2 1σx2y)cosQ d+(2q Q σxyQ )sinQ d. (7) + 2 c2 + z + c − + −− 2 c2 − z −− c + − It can be shown that limd Rd(iqz)=R (iqz). 0.27 σxyd/c→∞ Similarly, in the limit of→d∞ 0, if we kee∞p σ2D =σ d → xy xy 0.2 constant, we obtain what was found in Ref. [12], dli→m0Rd(iqz)= 1+(σx21Dy/2c)2 (cid:18)−(σσx2x2DyDy//22cc)2 −−(σσx2x2DyDy//22cc)2(cid:19). PP/c0 0.01 Attractive σσσxxxyyyddd///ccc===022...227555 For the rest of our discussion, define R0(iqz) = Repulsive σσxxyydd//cc==01..7755 lim R (iq ) with σ2D =σ d held constant. 0.12 σxyd/c=1.25 d 0 d z xy xy − W→ith the correct limits identified, we first notice that 0 4 8 12 16 20 wecanwriteRd asafunctionofonlytwovariablesRd = σxya/c R (cq /σ ,σ2D/c). Thus,bysimilarargumentstowhat d z xy xy we had for the semi-infinite case, the Casimir pressure FIG.3. (Coloronline)AplotofthenormalizedCasimirforce P =P f(σ a/c,σ2D/c). forvariousthicknessesofabulkHallmaterial(idealizedWeyl c 0 xy xy The limiting cases can be understood now by con- semimetal). Itbeginsslightlyrepulsiveforsmallσxyd/c,and sidering first Eq. (1). The exponential constrains asthisincreases,itbecomesmorerepulsiveuntilitreachesthe q 1/a and since technically the thin-film limit is maximum for a thin film material (the dashed line) at which z olitmhqe∼zrd→w0orRdds(,iqtzh)e=thRin0(fiilqmz),liwmeithiasveaptphlaictadb/leaw→he0n. wIne pσxoyin=tite2ibn/c2reπa2s(cid:126)esisttohtehbeuslekmHi-ainllfirneistpeolnimsei.t.FPig0u=re−42t(cid:126)4ac0πka2e4s,ianntdo are considering d a. The opposite limit is just when account more material properties. (cid:28) q d , and by similar arguments, that means d a z → ∞ (cid:29) is when the semi-infinite case applies. Both limits leave σ a/c and σ d/c unaffected (although in the thin film only clearly connects our case to the previously known xy xy case σ a drops out whereas in the semi-infinite case thin-film result, but also provides a theoretical justifi- xy σ d has the same limit as q d ). cation for considering a thin-film limit d a with a xTyhe→∞thin film limit czan→∞be evaluated two-dimensional conductance σµνd. (cid:28) exactly [12] and has the form PTF = Until now the plates have been idealized. Using the P 90 Re Li (cid:2)(σ2D/c)2/(σ2D/c+2i)2(cid:3) wherec Li is thin-film limit illustrated above as a reference allows us th0eπ4poly{log4arithxmy of degrxeye 4. Note}that this func4tion to easily consider some of the effects of taking into ac- has a minimum value of PTF 0.117P representing count the full frequency response of the material. Thus, how repulsive we can get. cFor≈lar−ge enou0gh σ2D/c, the we pick a σxyd that is reasonably in the repulsive regime xy force does become attractive—corresponding roughly (foralldistances)inordertoanalyzetheeffectsofinclud- to when (σ2D/c)2 > σ2D/c (i.e. when Kerr rotation is ing some of the lowest-order frequency dependence into xy xy suppressed). the conductivities. We will mainly be interested in the The cross-over between these regimes can be seen in effects of virtual vacuum transitions that are low in en- Fig. 3. As σ d/c is increased, the Casimir energy ap- ergy, which correspond to plates that are far apart from xy proachesthesemi-infinitecase. However,foranyfinited, one another. Thus, we will use the low-energy chiral each curve asymptotically approaches its thin-film value Hamiltonian for a pair of Weyl nodes, (and never goes lower than the minimum value repre- H = (cid:126)v σ (k b), (8) sented by the dashed horizontal line in Fig. 3). This not W F ± · ± 4 3 2PP(10)/−c0 022 b=b0=.5b(02=.π4b()0=2/.π3n(0)m2/.2πn()m2/πn)m/nm 2PP(10)/−c0 0.0247 vvFFvFv=F==6=31××2181×10×0551m01m50/5/msms//ss 2PP(10)/−c0 0.60122 µ=0µm=eV10µm=eV1µ7m=e2V5meV − − − 0.1 1 2 3 4 0.1 1 2 3 4 0.1 1 2 3 4 a(µm) a(µm) a(µm) (a) (b) (c) FIG. 4. (Color online) The Casimir force for a thin film Weyl semimetal taking into account low-energy virtual transitions in the band structure. An antitrap develops when the longitudinal conductivity overwhelms the Hall conductivity. In (a) we compare different values of b (or equivalently, changing the Hall effect), in (b) we compare different v ’s (the larger v , the F F smaller σ is), and in (c) we turn on a finite chemical potential which causes attraction at very long distances (and hence xx a trap). Even small chemical potentials have this property, but the trap is quite far out. Unless the parameter is varying, a =1nm, d=20nm, b=0.3(2π/a ), Λ=2π/a , v =6×105m/s, and µ=0. 0 0 0 F wherev istheFermivelocityandbisthepositionofthe shitzformulatorunfrom0toΛ—anapproximationvalid F of Weyl node in k space. The exact band structure will for a c Λ 1. (cid:29) vF − be important as the plates get closer although weighting First, we see that we get an anti-trap for these at ap- will still be larger on the lower-energy modes. proximately 650nm, and if we increase b as in Fig. 4a To the conductivities, we fix kz and calculate two- (with, say, an applied magnetic field), it not only moves dimensional conductivities using the Kubo-Greenwood closertozeroseparation,buttheoverallrepulsivebehav- formulation (see the Supplemental Material [28] for de- ior can be enhanced. On the other hand, if we increase tails)thenintegratetheresultingexpressionoverkz with vF as we see in Fig. 4b, we see the region of attraction is a symmetric cutoff σ (iω) = (cid:82)Λ dkzσ˜ (iω;k ) [29] suppressed,buttheoverallrepulsivebehavioratlongdis- µν Λ 2π µν z where σ˜ (iω;k ) is the two-dim−ensional conductivity tances is maintained. Modifying Λ will have effects simi- µν z with kz fixed. We evaluate this at an imaginary fre- lar to modifying vF, but since it appears logarithmically, quency to aid the Casimir calculations. it would need to change by orders of magnitude to give Weperformthisprocedureatfinitechemicalpotential appreciable changes (a simple plot for this is provided µ and throw out terms that go to zero when the cutoff in the Supplemental Material [28] but is not relevant for Λ . This yields the conductivities, the discussion here). This antitrap effect is occurring at →∞ short distances when higher order band effects also play a role, but any other effects will contribute to the longi- σ (iω)= e2 (cid:2)5ω+2ωlog(cid:0)vFΛ(cid:1) tudinal conductivity in such a way that an anti-trap will xx 12π2(cid:126)v 3 ω F appear. (cid:16) (cid:17)(cid:105) +4(cid:126)µ22ω −ωlog 1+ (cid:126)42µω22 , (9) Interestingly, when we introduce a finite chemical potential as we see in Fig. 4c, in addition to the anti-trap and σ (iω) = e2b is unchanged at this order. Due wegetatshorterdistances,westarttoseeatrapatmuch xy 2π2(cid:126) longer distances appear. This is not surprising since at to the linear dispersion of the Weyl nodes, we have a zero frequency there is a divergent longitudinal conduc- logarithmiccutoffdependence. Notethatrotatingtoreal tivity. Thus,weknowthatatlongdistances,theCasimir frequencies we get the correct result for two Weyl nodes force must be attractive, but by modifying the Hall ef- for Re[σ (ω)] [20], and a result with the appropriate xx fect, we have an intermediate regime of repulsion. A logarithmic divergence for Im[σ (ω)] [30]. This can be xx similar effect would occur if we took finite temperatures understoodintermsofchargerenormalizationduetothe ordisordercorrectionstothelongitudinalconductivities. band structure, but for ease of our purposes we let Λ 1/a where a is the lattice spacing. For our plots w∼e Considering the form of the conductance in terms of 0 0 choose a lattice spacing of a0 = 1nm, a thickness of the fine-structure constant σxyd/c = α2πbd, we see that d=20nm,b=0.3(2π/a ),Λ=2π/a ,v =6 105m/s, bd controls the strength of the repulsion in the thin-film 0 0 F and µ=0 unless its the parameter we are vary×ing. limit. Without longitudinal conductance, the repulsive Now, one can use one of two equivalent ways of calcu- regime roughly corresponds to when (σxyd/c)2 (cid:46)σxyd/c lating the Casimir energy: the reflection matrix as given or equivalently 2bd (cid:46) 1. The longitudinal conductance π α in Ref. [12], or using a microscopic analysis to find the introducesvFintothescheme,relevantphotonshaveω photon dressed RPA current-current correlators [28]. In c/a,andthusitbecomesimportantforσxxd/c∼αvcFad ≈(cid:38) order to avoid an unphysical negative σ (iω) as well as O(1) (neglecting constants) which both emphasizes that xx for consistency, we cutoff the photon energies in the Lif- v controls the longitudinal conductance’s contribution F 5 to the Casimir effect and that the term is suppressed at termediatedistances. Recentlythefirstexperimentalob- longer distances. servation of Weyl semimetals [31, 32] provided optimism We have shown here how Weyl semimetals can ex- that these theoretical materials could be a reality. The hibit a tunable repulsive Casimir force (with, for in- marginal nature of these materials could be useful for stance, magnetic-field tuning b) and how it can depend controlling the Casimir force between attractive and re- on the thickness of the material. In the thin-film limit, pulsive regimes. we showed how the semimetallic nature of these mate- This work was supported by the DOE-BES (Grant rials can work to create attraction at shorter distance No. DESC0001911) (A.A.A. and V.G.), the JQI-PFC scalesandhowafinitelongitudinalconductivitywillcre- (J.H.W.), and the Simons Foundation. We thank Liang ate long-distance attraction along with repulsion at in- Wu and Mehdi Kargarian for discussions. [1] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 [19] V. Aji, Phys. Rev. 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AXIONIC ELECTRODYNAMICS In the main text, we mention the axionic term which appears in the action alongside the usual Maxwell action e2 (cid:90) S = d3rdtθ(r,t)(cid:15)µναβF F , (10) A 32π2(cid:126)c µν αβ whereeistheelectriccharge,(cid:126)isPlanck’sconstant,cisthespeedoflight,F =∂ A ∂ A istheelectromagnetic µν µ ν ν µ − field tensor, (cid:15)µναβ is the fully antisymmetric 4-tensor, and θ(r,t)=2b r 2b t is the axionic field. 0 · − 6 For ourpurposes, we will set b=bzˆ and b =0. If we only apply S for z >0, there is no resulting surface current 0 A (i.e. this is the surface without Fermi arcs), and the current response is e2b jx(r)= Ey(r), (11) 2π2 e2b jy(r)= Ex(r). (12) −2π2 Now, we solve Maxwell’s equations in the bulk Hall system after taking the Fourier transform k E= iσ zˆ B, (13) xy · − · k B=ωB, (14) · k E=ωB, (15) × k B=iσ zˆ E ωE. (16) xy × × − One can define a frequency-dependent dielectric permitivity (cid:15)(ω) to be   1 iσ /ω 0 xy (cid:15)(ω)= iσxy/ω 1 0 (17) − 0 0 1 in which case, Maxwell’s equations can be recast as a single equation for the electric displacement D=(cid:15)(ω)E, [k k k2I](cid:15) 1(ω)D= ω2D. (18) − ⊗ − − The determinant of this matrix equation yields the frequencies that a wave vector k can have. In our case, we obtain (cid:113) ω2 =k2+ 1σ2 σ k2+ 1σ2 . (19) ± 2 xy± xy z 4 xy There is a polarization associated with each of these frequencies which we can obtain from Eq. (18) using k D = 0 · (no free charge). We choose e =yˆ kˆ and e =yˆ as our basis for the polarizations (assuming k =0 without loss 1 2 y × of generality). The resulting (unnormalized) polarizations are ω (cid:18)(cid:113) (cid:19) D1,2 = k± kz2+ 14σx2y∓ 12σxy e1±ikze2. (20) Notice that these polarizations are elliptical. To find the reflection of a wave off a half space filled with this material, k , k , and ω must remain the same on x y either side of the material, but k can change, and matching both sides of the dispersion Eq. (19), we obtain simply z that for an incident wave with wave-vector q=(q ,q ,q ), the transmitted wave has x y z (k )2 =q (q σ ). (21) z± z z± xy Each of these can be associated with a (unnormalized) polarization as (assuming q =0 without loss of generality) y D = ω (q σ )e ik e . (22) ± k± z± xy 1∓ z± 2 Atthispoint,wenotesomeinterestingelectromagneticpropertiesappearinghere. Eq.(21)impliesthatthismaterialis birefringent,andforanincidentwave(atanyangle)withq < σ ,onlyone(elliptical)polarizationevenpropagates z xy | | into the material while the other is an evanescent wave – independent of the angle of incidence. A. Reflection coefficient of semi-infinite bulk Now, to obtain the reflection matrix, we call our incident wave E with wave-vector q, our reflected wave E 0 r with wave-vector q = (q ,q , q ), as well as E = (cid:15) 1(ω)D with wave vectors k for the two polarizations it is r x y z − transmitted into. The relevant−Maxwell equation±s at the inter±face between vacuum±and the bulk Hall material are then given by (E +E E E ) zˆ=0, 0 r + − − − × (23) (q E +q E k E k E ) zˆ=0. 0 r r + + × × − × − −× − × 7 We can break up the polarization of the incident and reflected waves into transverse electric (TE) and transverse magnetic (TM), and the reflection matrix is defined such that (cid:18) (cid:19) (cid:18) (cid:19) ETM ETM r =R(ω,q) 0 . (24) ETE ETE r 0 Solving for this matrix, we obtain (cid:18) (cid:19) 1 σ +k k+ i(2q k k+) R (ω,q)= xy z−− z z− z−− z . (25) ∞ σxy −i(2qz−kz−−kz+) σxy+kz−−kz+ This can then be rotated to imaginary frequencies and the result is in the main text. B. Reflection coefficient of thickness d sample Ifwehaveamaterialofthicknessd,thenweneedadditionalmatchingconditionsduetoMaxwell’sequationsatthe other interface. This requires restricting our action Eq. (2) to 0<z <d. To solve this, we just need to add another set of matching conditions. In addition to the incident E and reflected E waves, we now have forward moving Weyl 0 r ptroalnasrmizaitttieodnswEav↑±ewEithwikt↑±h w=av(kex-v,ekcyt,okrz±t)h,ebsaacmkweaarsdsthmeotvrainngsmWitetyeldpko.larizations E↓± with k↓± = (kx,ky,−kz±), and a t The resulting matching conditions are (E0+Er−E↑+−E↑−−E↓+−E↓−)×zˆ=0, (k×E0+kr×Er−k↑+×E↑+−k↑−×E↑−−k↓+×E↓+−k↓−×E↓−)×zˆ=0, (E↑+eikz+d+E↑−eikz−d+E↓+e−ikz+d+E↓−e−ikz−d−Eteikzd)×zˆ=0, (26) (k↑+×E↑+eikz+d+k↑−×E↑−eikz−d +k↓+×E↓+e−ikz+d+k↓−×E↓−e−ikz−d−k×Eteikzd)×zˆ=0. These equations can still be solved and the result is (cid:18) (cid:19) R R R (ω,q)= xx xy , (27) d R R xy xx − where R =σ sin(k d)[ik+cos(k+d)+σ sin(dk+)] ik cos(k d)sin(k+d) /D, (28) xx xy{ z− z z xy z − z− z− z } R =σ k cos(k d)sin(k+d)+sin(k d)[k+cos(k+d) 2ik sin(k+d)] /D, (29) xy xy{ z− z− z z− z z − z z } D =[2ik cos(k d)+(2k σ )sin(k d)][2ik+cos(k+d)+(2k +σ )sin(k+d)]. (30) z− z− z− xy z− z z z xy z And again, this can be rotated to imaginary frequencies to obtain the result in the main text. And as stated in the text, the various limits (semi-infinite to thin film limits) apply to this reflection matrix. II. CALCULATING THE CONDUCTIVITIES In order to calculate the conductivities in the clean limit we consider the Hamiltonian near a Weyl node H = (cid:126)v σ (k b). (31) W F ± · ± For simplicity we set (cid:126) = 1 = v unless otherwise specified. We consider a pair of these nodes, calculating the F quantities separately for each node and adding them together (which will just introduce a factor of two for both σ xx and σ ). xy Tofindtheconductivities,wefindacompletebasisofstateswhichareeasilyfoundbydiagonalizingthetwo-by-two matrix H in momentum space (label them f ). The relevant matrix elements are then (choosing the negative W k | ±(cid:105) 8 sign for the Hamiltonian) (k b)k +i(cid:15)k z x y fk+ σx fk = (cid:112)− , (32) (cid:104) | | −(cid:105) (cid:15) (cid:15)2 (k b)2 z − − (k b)k i(cid:15)k z y x fk+ σy fk = (cid:112)− − , (33) (cid:104) | | −(cid:105) (cid:15) (cid:15)2 (k b)2 z − − k x f σ f = , (34) k x k (cid:104) ±| | ±(cid:105) ± (cid:15) (cid:113) where (cid:15)= k2+k2+(k b)2. x y z− We then fix k and use the Kubo-Greenwood formula for the intra- and inter-band transitions separately to obtain z two-dimensional conductivities. Thus, σ˜inter(iω;k )= e2 (cid:88) nkγ −nkγ(cid:48) (cid:104)fkγ|jµ|fkγ(cid:48)(cid:105)(cid:104)fkγ(cid:48)|jν|fkγ(cid:105), (35) µν z i k,γ=γ(cid:48) (cid:15)kγ −(cid:15)kγ(cid:48) iω+(cid:15)kγ −(cid:15)kγ(cid:48) (cid:54) where n is the occupation in that band of that momentum and j = σ are the single particle current operators. kγ µ µ For intra-band quantities with n =θ(µ (cid:15) ) k k ± ∓ σ˜intra(iω;k )= e2 (cid:88)δ(µ (cid:15) )(cid:104)fk+|jµ|fk+(cid:105)(cid:104)fk+|jν|fk+(cid:105), (36) µν z − i − k,+ iω k assuming only the upper-band for simplicity and without loss of generality (due to particle-hole symmetry). Adding these contributions together at finite chemical potential yields [12] e2 (cid:20)(cid:18) 4(k b)2(cid:19) i (cid:18)2∆ iω(cid:19) ∆(cid:21) z σ˜ (iω;k )= 1 − log − + , (37) xx z 4π − ω2 4 2∆+iω ω e2 (cid:20)k b i (cid:18)2∆ iω(cid:19)(cid:21) z σ˜ (iω;k )= − log − , (38) xy z 2π ω 2 2∆+iω where ∆=max k b, µ . With these quantities we can then use the cutoff procedure explained in [29] z {| − | | |} (cid:90) Λ dk z σ (iω)= σ˜ (iω;k ). (39) µν µν z 2π Λ − Through which we obtain (throwing away terms that go to zero as the cutoff increases to infinity and multiplying by two for the two nodes and bringing back in the constants (cid:126) and v ) F σ (iω)= e2 (cid:104)5ω+2ωlog(cid:0)vFΛ(cid:1)+4 µ2 ωlog(cid:16)1+ 4µ2 (cid:17)(cid:105), (40) xx 12π2(cid:126)v 3 ω (cid:126)2ω − (cid:126)2ω2 F e2b σ (iω)= . (41) xy 2π2(cid:126) These quantites are what we use in the next section as input for the Casimir Force. III. CASIMIR FORCE CALCULATION A. Differentiating the Casimir energy In the main text, we find the Casimir pressure by taking the derivative of the energy P = ∂E /∂a. This leads to c c − the expression for force found in the text for two semi-infinite Weyl plates 2(cid:126)c (cid:90) (cid:104) (cid:105) P = dq q3 g qz ,2q a , (42) c (2π)2 z z σxy/c z 9 4 Λ=5π/nm Λ=4π/nm ) 2− 0 2 Λ=3π/nm 1 ( P0 / Λ=2π/nm Pc 0 0.7 − 0.1 1 2 3 4 a(µm) FIG. 5. The Casimir energy for the conductivities as defined in the main text with b=0.3(2π)nm−1, v =6×105m/s, and F µ=0. The cutoff is varied here. and the function g(u,v) is defined by R (u)2 R2 (u) [R (u)2+R2 (u)]2e v g(u,v)= 4 xx − xy − xx xy − , (43) − 2[R (u)2 R2 (u)] [R (u)2+R2 (u)]2e v xx − xy − xx xy − where R (u) and R (u) are the matrix elements of R (icq /σ ) (Eq. (3) in the main text), xx xy z xy ∞ (cid:113) (cid:112) R (u)= 2u( u2+1 1) 1, (44) xx − − (cid:113) (cid:112) R (u)=2u 2u( u2+1+1). (45) xy − B. Two-dimensional plates calculation Another well-known approach completely equivalent to the Lifshitz formula comes directly from quantum field theory. To put it briefly, if we have a conductivity like we calculated above and write it as a response function Π(iω) such that σ(iω)= Π(iω)/ω (46) − And calculate the RPA response function considering photons “skimming” along the surface of our material Π(cid:101)(iω,q)=[I Π(iω)D(iω,q,0)]−1Π(iω), (47) − with the photon propagator   √q2+ω2 D(iω,q,z)= 2ω02 01 e−√q2+ω2|z|, (48) 2√q2+ω2 then the Casimir energy takes the form Ec(a)= π1 (cid:90) (2dπ2q)2 (cid:90) ∞dωtrlog(cid:104)I−Π(cid:101)1(iω,q)D(iω,q,a)Π(cid:101)2(iω,q)D(iω,q,a)(cid:105), (49) 0 For our particular case of the conductivities introduced in Section II, we need to also cutoff the frequencies in this integral to go from 0 to v Λ for consistency. We expect higher energy virtual photons to not play a large role. F We also show the result we obtain from varying the cutoff in Fig. 5. Note that it does not affect the effect much unless it varies by orders of magnitude.