text: 22 Physical Review Letters 117, 216403 (2016), arXiv:1602.0a8n4d9i9m[pcounridt-imesatth.matesb-rheaalkl].particle-hole symmetry,” Phys. Rev. Lett. 117, 096802 (2016). [59] Z. Wang and S. Chakravarty, “Pairing of particle-hol[e61sy] mLumkeatsrzicFciodmkopwosskiti,e XfeiremCiohnesni,n ahnadlf-AﬁlslhedvinLaVndisahuwanath, “Non-abelian topological order on the surface of level,” Phys. Rev. B 94, 165138 (2016), arXiv:1606.00899a [3cdontdo-pmoalotg.sitcra-lels]u.perconductor from an exactly solved model,” Phys. Rev. X 3, 041016 (2013). [60] P. T. Zucker and D. E. Feldman, “Stabilization of th[e62p]arMti.clLe-ehvoinleaPnfdaﬃDa.nTohradnehr bSyonL,an“dPaaur-tliecvlee-lHmoilxeinSgymmetry and Electromagnetic Response of a Half-Filled Landau Level,” ArXiv e-prints (2016), arXiv:1612.06402 [cond-mat.mes-hall]. ∗ [27] E. H. Rezayi and F. D. M. Haldane, “Incompressible Paired Hall State, Stripe Order, and the Composite [1] Dam Thanh Son, “Is the Composite Fermion a Dirac ParFteircmle?io”nPLhiyqsu.idRPevh.aXse5in, 0H31a0lf2-F7i(ll2e0d15L)a.ndau Levels,” Phys. Rev. Lett. 84, 4685–4688 (2000). [2] B. I. Halperin, Patrick A. Lee, and Nicholas Read,[“2T8]hAeo.rCy .oBf athlreamhalafn-ﬁdllJed. KL.anJdaainu, l“eNvealt,”urPehoyfsc.oRmepvo. site fermions and the role of particle-hole symmetry: A B 47, 7312–7343 (1993). microscopic account,” Phys. Rev. B 93, 235152 (2016), arXiv:1604.03911 [cond-mat.str-el]. [3] J. K. Jain, “Composite-fermion approach for the fr[a2c9t]ioSn.aAl .quKainvteulsmonH, aDll-Heﬀ. eLcete,”, YP.hyKs.roRtoevv,. Laentdt. J6.3,Gan, “Composite-fermion hall conductance at ν = 1/2,” 199–202 (1989). Phys. Rev. B 55, 15552–15561 (1997). [4] Vadim Kalmeyer and Shou-Cheng Zhang, “Metallic p[h3a0s]eMof.thBeaqrkueasnhtlui,mMH.aMll suylslitgeamn,at aenvedn-Mde.nPo.mAin.atFoirsher, “Particle-hole symmetry and the composite Fermi ﬁlling fractions,” Phys. Rev. B 46, 9889–9892 (1992). liquid,” Phys. Rev. B 92, 165125 (2015), arXiv:1502.05404 [cond-mat.str-el]. [5] Ana Lopez and Eduardo Fradkin, “Fractional quantu[m31h]aAll.eCﬀ.ecPtoatntedr,chMer.nS-esrimbyonn,s gaanudgAe .thVeiosrhiwesa,n”aPthh,y“s.Thermoelectric Transport Signatures of Dirac Composite Rev. B 44, 5246–5262 (1991). Fermions in the Half-Filled Landau Level,” Physical Review X 6, 031026 (2016), arXiv:1512.06852 [cond- [6] Martin Greiter and Frank Wilczek, “Exact solutions and mthaet.asdtria-ebla].tic heuristic for quantum Hall states,” Nuclear Physics B 370, 577–600 (1992). [32] R. Willett, J. P. Eisenstein, H. L. Sto¨rmer, D. C. Tsui, A. C. Gossard, and J. H. English, “Observation [7] B. Rejaei and C. W. J. Beenakker, “Vector-mean-ﬁeld theoofryanofetvheen-fdraecntoiomnianlaqtouranqtuuamntHumall neuﬀmecbt,e”rPinhytsh.e fractional quantum Hall eﬀect,” Phys. Rev. Lett. 59, Rev. B 46, 15566–15569 (1992). 1776–1779 (1987). [8] Yong Baek Kim, Akira Furusaki, Xiao-Gang Wen, a[3n3d] PGartergiockryAM. oLoeree, “aGndauNgiec-hinovlaasriRanetadr,es“pNoonnseabfuelniocn- s in the fractional quantum hall eﬀect,” Nuclear Physics tions of fermions coupled to a gauge ﬁeld,” Phys. Rev. BB503,601,7931672––17399362((11999914)).. [9] B. L. Altshuler, L. B. Ioﬀe, and A. J. Millis, “Low-[e3n4e] rNgy. RpreoapderatnidesDomf fietrrmy iGonrseewn,ith“Psainirgeudlasrtaintetseroafc-fermions in two dimensions with breaking of parity and tions,” Phys. Rev. B 50, 14048–14064 (1994). time-reversal symmetries and the fractional quantum hall eﬀect,” Phys. Rev. B 61, 10267–10297 (2000). [10] S. H. Simon and B. I. Halperin, “Finite-wave-vector [e3le5c]tMroimchaagenletLiecvriens,pBonesretroafnfdraIc.tiHonaalpl eqruinan, tizaenddHBaellrnd Rosenow, “Particle-hole symmetry and the pfaﬃan states,” Phys. Rev. B 48, 17368–17387 (1993), cond-mat/st9a3t0e7,”04P8.hys. Rev. Lett. 99, 236806 (2007). [11] Ady Stern and Bertrand I. Halperin, “Singularities[3i6n] tShuengF-eSrimk iL-leiqe,uiSdhidnessecirRipytuio,nCohfetaanpaNrtaiyaallky, ﬁalnledd Matthew P. A. Fisher, “Particle-hole symmetry and the Landau level and the energy gaps of fractional quantum Hν a=ll5stqautaens,t”umPhhyas.llRsteavt.eB,”5P2h, y5s8.9R0–e5v9.0L6et(t1.99995,).236807 (2007). 2 [12] M. A. Metlitski and A. Vishwanath, “Particle-vortex[37d]uMali.tyP.oLf i2lldy,DKir.aBc.fCeromopioenr,frJo.mP.eEleiscetnrisct-emina,gLn.etNic. Pfeiﬀer, and K. W. West, “Evidence for an Anisotropic duality of 3d topological insulators,” ArXiv e-prints (20S15ta),tearoXf iTvw:1o5-0D5i.m05e1n4s2io[ncaolndE-lmecattr.osntrs-einl].High Landau Levels,” Phys. Rev. Lett. 82, 394–397 (1999). [13] Chong Wang and T. Senthil, “Half-ﬁlled Landau level,[3t8o]pAol.ogAic.aKl ionuslualkaotvo,r Msu.rfMac.esF,oagnledrt,hraened-dBim. eIn. sSihonkalolvskii, “Charge density wave in two-dimensional electron quantum spin liquids,” Phys. Rev. B 93, 085110 (2016). liquid in weak magnetic ﬁeld,” Phys. Rev. Lett. 76, 499–502 (1996). [14] S. D. Geraedts, M. P. Zaletel, R. S. K. Mong, M. A[3.9]MMet.liMts.kiF, oAgl.eVr iasnhdwaAn.atAh,. KaonudlaOko. vI,. “MLoaturguhnliicnh,liquid to charge-density-wave transition at high Landau “The half-ﬁlled Landau level: The case for Dirac comlpeovseiltse,”fePrhmyiso.nRs,e”v.SBcie5n5c,e9332562–,9139279–(2109197()2.016), arXiv:1508.04140 [cond-mat.str-el]. [40] R. Moessner and J. T. Chalker, “Exact results for interacting electrons in high Landau levels,” Phys. Rev. [15] Ganpathy Murthy and R. Shankar, “ν = 1 Landau level:BH5a4l,f-5e0m06p–ty50v1e5rs(u1s99h6a)l.f-full,” Phys. Rev. B 93, 2 085405 (2016). [41] M. I. D’yakonov and A. V. Khaetskii, “Transport cross section for small angle scattering,” Sov. Phys. [16] C. Wang and T. Senthil, “Composite Fermi liquids in theJEloTwPes7t2L, a5n9d0a(u19le9v1e)l.,” Phys. Rev. B 94, 245107 (2016), arXiv:1604.06807 [cond-mat.str-el]. [42] P. Nozi`eres and C. Lewiner, “A simple theory of the anomalous Hall eﬀect in semiconductors,” Journal [17] David F. Mross, Andrew Essin, and Jason Alicea, “CompdoesitPehDysiriqauceLi3q4u,id90s:1–P9a1r5en(t19S7t3a)t.es for Symmetric Surface Topological Order,” Phys. Rev. X 5, 011011[4(230] 1N5.).R. Cooper, B. I. Halperin, and I. M. Ruzin, “Thermoelectric response of an interacting two-dimensional [18] Chong Wang and T. Senthil, “Dual Dirac Liquid on theeSleucrtfraocne goafstihneaEqleucatnrtoinzinTgopmoaloggniectailcIﬁnesludl,a”toPrh,”ys. Rev. B 55, 2344–2359 (1997). Phys. Rev. X 5, 041031 (2015). [44] R. L. Willett, R. R. Ruel, K. W. West, and L. N. Pfeiﬀer, “Experimental demonstration of a Fermi [19] C. Wang and T. Senthil, “Time-Reversal Symmetric U(1s)uQrfuaacnetautmonSep-ihnaLlfiqﬁulliidnsg,”ofPthhyesilcoawleRstevLiaenwdaXu6l,evel,” Phys. Rev. Lett. 71, 3846–3849 (1993). 011034 (2016), arXiv:1505.03520 [cond-mat.str-el]. [45] D. Kamburov, Yang Liu, M. A. Mueed, M. Shayegan, L. N. Pfeiﬀer, K. W. West, and K. W. Baldwin, [20] M. A. Metlitski, “S-duality of u(1) gauge theory with θ =“Wπ ohnatndoent-eorrmieinnteasbltehemfaenrmifoildwsa:veApvepclitcoartioofncsotmoposite fermions?” Phys. Rev. Lett. 113, 196801 (2014). topological insulators and superconductors,” ArXiv [e4-6p]riWnt.s K(a2n01g5, )H, a. rLX.ivS:t1o5r1m0.e0r5,6L6.3 N[h.epP-ftehiﬀ].er, K. W. Baldwin, and K. W. West, “How real are composite [21] D. F. Mross, J. Alicea, and O. I. Motrunich, “Explicit Dfeerrmiviaotniosn?”ofPDhyusa.liRtyevb.eLtwetete.n71a,F3r8e5e0D–3i8ra5c3 C(1o9n9e3). and Quantum Electrodynamics in (2 +1 ) Dimens[i4o7n]s,J”. HPh. ySsmiceatl, DR.evWieewissL, Ret.teHr.sB1l1ic7k,, 0G1.6L8u0¨2tje(r2i0n1g6, )K, . von Klitzing, R. Fleischmann, R. Ketzmerick, T. Geisel, arXiv:1510.08455 [cond-mat.str-el]. and G. Weimann, “Magnetic focusing of composite fermions through arrays of cavities,” Phys. Rev. Lett. [22] N. Seiberg, T. Senthil, C. Wang, and E. Witten, “A d7u7a,li2t2y7w2–e2b27in5 (219+961).dimensions and condensed matter physics,” Annals of Physics 374, 395–433 (20[1468)], Ja.rXHi.vS:1m6e0t6,.0K1.9v8o9n[hKelpit-zthin].g, D. Weiss, and W. Wegscheider, “dc transport of composite fermions in weak [23] A. Karch and D. Tong, “Particle-Vortex Duality from 3pDerBioodsiocnpizoatteinotnia,”ls,P”hPyshiycsa.l RReevv.ieLwetXt. 860,,043513084–34541 (1998). (2016), arXiv:1606.01893 [hep-th]. [49] J. H. Smet, S. Jobst, K. von Klitzing, D. Weiss, W. Wegscheider, and V. Umansky, “Commensurate com- [24] J. Murugan and H. Nastase, “Particle-vortex duality ipnostiotpeofleorgmiciaolnsinisnuwlaetaokrspaenridodsiucpeelreccotrnodsutcattoicrsp,”otentials: Direct evidence of a periodic eﬀective magnetic ArXiv e-prints (2016), arXiv:1606.01912 [hep-th]. ﬁeld,” Phys. Rev. Lett. 83, 2620–2623 (1999). [25] S. Kachru, M. Mulligan, G. Torroba, and H. Wang,[“50B]oRso.nLiz.atWioinlleatntd, Km.irWro.r Wsymesmt, etarnyd,” LP.hNys.. PRfeeviﬀ. eDr, “Geometric resonance of composite fermion cyclotron 94, 085009 (2016), arXiv:1608.05077 [hep-th]. orbits with a ﬁctitious magnetic ﬁeld modulation,” Phys. Rev. Lett. 83, 2624–2627 (1999). [26] S. M. Girvin, “Particle-hole symmetry in the anomalo[5u1s]qSu.aDnt.uMm.HZawlel resﬀcehckte,”aPndhyRs.. RRe.vG. eBrh2a9r,d6t0s1,2“–P6o0s1i4tive magnetoresistance of composite fermion systems with (1984). a weak one-dimensional density modulation,” Phys. Rev. Lett. 83, 2616–2619 (1999). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Physical

label: reference

text: Review

text: Letters

text: symmetry,”

text: Phys.

text: Rev.

text: Lett.

text: Z.

text: Wang

text: and

text: S.

text: Chakravarty,

text: “Pairing

text: of

text: mLumkeatsrzicFciodmkopwosskiti,e

text: XfeiremCiohnesni,n

text: ahnadlf-AﬁlslhedvinLaVndisahuwanath,

text: “Non-abelian

text: topological

text: order

text: on

text: the

text: surface

text: of

text: level,”

text: Phys.

text: Rev.

text: B

text: from

text: an

text: exactly

text: solved

text: model,”

text: Phys.

text: Rev.

text: X

text: P.

text: T.

text: Zucker

text: and

text: D.

text: E.

text: Feldman,

text: “Stabilization

text: of

text: and

text: Electromagnetic

text: Response

text: of

text: a

text: Half-Filled

text: Landau

text: Level,”

text: ArXiv

text: e-prints

text: [cond-mat.mes-hall].

text: E.

text: H.

text: Rezayi

text: and

text: F.

text: D.

text: M.

text: Haldane,

text: “Incompressible

text: Paired

text: Hall

text: State,

text: Stripe

text: Order,

text: and

text: the

text: Composite

text: Dam

text: Thanh

text: Son,

text: “Is

text: the

text: Composite

text: Fermion

text: a

text: Dirac

text: Levels,”

text: Phys.

text: Rev.

text: Lett.

text: B.

text: I.

text: Halperin,

text: Patrick

text: A.

text: Lee,

text: and

text: Nicholas

text: .oBf

text: athlreamhalafn-ﬁdllJed.

text: KL.anJdaainu,

text: l“eNvealt,”urPehoyfsc.oRmepvo.

text: site

text: fermions

text: and

text: the

text: role

text: of

text: particle-hole

text: symmetry:

text: A

text: B

text: microscopic

text: account,”

text: Phys.

text: Rev.

text: B

text: [cond-mat.str-el].

text: J.

text: K.

text: Jain,

text: “Composite-fermion

text: approach

text: for

text: the

text: .quKainvteulsmonH,

text: aDll-Heﬀ.

text: eLcete,”,

text: YP.hyKs.roRtoevv,.

text: Laentdt.

text: “Composite-fermion

text: hall

text: conductance

text: at

text: ν

text: Phys.

text: Rev.

text: B

text: Vadim

text: Kalmeyer

text: and

text: Shou-Cheng

text: Zhang,

text: “Metallic

text: suylslitgeamn,at

text: aenvedn-Mde.nPo.mAin.atFoirsher,

text: “Particle-hole

text: symmetry

text: and

text: the

text: composite

text: Fermi

text: ﬁlling

text: fractions,”

text: Phys.

text: Rev.

text: B

text: liquid,”

text: Phys.

text: Rev.

text: B

text: [cond-mat.str-el].

text: Ana

text: Lopez

text: and

text: Eduardo

text: Fradkin,

text: “Fractional

text: gaanudgAe

text: Transport

text: Signatures

text: of

text: Dirac

text: Composite

text: Rev.

text: B

text: Fermions

text: in

text: the

text: Half-Filled

text: Landau

text: Level,”

text: Physical

text: Review

text: X

text: [cond-

text: Martin

text: Greiter

text: and

text: Frank

text: Wilczek,

text: “Exact

text: solutions

text: and

text: mthaet.asdtria-ebla].tic

text: heuristic

text: for

text: quantum

text: Hall

text: states,”

text: Nuclear

text: Physics

text: B

text: R.

text: Willett,

text: J.

text: P.

text: Eisenstein,

text: H.

text: L.

text: Sto¨rmer,

text: D.

text: C.

text: Tsui,

text: A.

text: C.

text: Gossard,

text: and

text: J.

text: H.

text: English,

text: “Observation

text: B.

text: Rejaei

text: and

text: C.

text: W.

text: J.

text: Beenakker,

text: “Vector-mean-ﬁeld

text: neuﬀmecbt,e”rPinhytsh.e

text: fractional

text: quantum

text: Hall

text: eﬀect,”

text: Phys.

text: Rev.

text: Lett.

text: Rev.

text: B

text: Yong

text: Baek

text: Kim,

text: Akira

text: Furusaki,

text: Xiao-Gang

text: Wen,

text: PGartergiockryAM.

text: oLoeree,

text: s

text: in

text: the

text: fractional

text: quantum

text: hall

text: eﬀect,”

text: Nuclear

text: Physics

text: tions

text: of

text: fermions

text: coupled

text: to

text: a

text: gauge

text: ﬁeld,”

text: Phys.

text: Rev.

text: B.

text: L.

text: Altshuler,

text: L.

text: B.

text: Ioﬀe,

text: and

text: A.

text: J.

text: Millis,

text: rNgy.

text: RpreoapderatnidesDomf

text: fietrrmy

text: in

text: two

text: dimensions

text: with

text: breaking

text: of

text: parity

text: and

text: tions,”

text: Phys.

text: Rev.

text: B

text: time-reversal

text: symmetries

text: and

text: the

text: fractional

text: quantum

text: hall

text: eﬀect,”

text: Phys.

text: Rev.

text: B

text: S.

text: H.

text: Simon

text: and

text: B.

text: I.

text: Halperin,

text: “Finite-wave-vector

text: eqruinan,

text: tizaenddHBaellrnd

text: Rosenow,

text: “Particle-hole

text: symmetry

text: and

text: the

text: pfaﬃan

text: states,”

text: Phys.

text: Rev.

text: B

text: Rev.

text: Lett.

text: Ady

text: Stern

text: and

text: Bertrand

text: I.

text: Halperin,

text: tShuengF-eSrimk

text: ﬁalnledd

text: Matthew

text: P.

text: A.

text: Fisher,

text: “Particle-hole

text: symmetry

text: and

text: the

text: Landau

text: level

text: and

text: the

text: energy

text: gaps

text: of

text: fractional

text: quantum

text: Hν

text: M.

text: A.

text: Metlitski

text: and

text: A.

text: Vishwanath,

text: Pfeiﬀer,

text: and

text: K.

text: W.

text: West,

text: “Evidence

text: for

text: an

text: Anisotropic

text: duality

text: of

text: topological

text: insulators,”

text: ArXiv

text: e-prints

text: Landau

text: Levels,”

text: Phys.

text: Rev.

text: Lett.

text: Chong

text: Wang

text: and

text: T.

text: Senthil,

text: “Half-ﬁlled

text: Landau

text: ionuslualkaotvo,r

text: Msu.rfMac.esF,oagnledrt,hraened-dBim.

text: eIn.

text: sSihonkalolvskii,

text: “Charge

text: density

text: wave

text: in

text: two-dimensional

text: electron

text: quantum

text: spin

text: liquids,”

text: Phys.

text: Rev.

text: B

text: liquid

text: in

text: weak

text: magnetic

text: ﬁeld,”

text: Phys.

text: Rev.

text: Lett.

text: S.

text: D.

text: Geraedts,

text: M.

text: P.

text: Zaletel,

text: R.

text: S.

text: K.

text: Mong,

text: M.

text: oAgl.eVr

text: iasnhdwaAn.atAh,.

text: KaonudlaOko.

text: vI,.

text: “MLoaturguhnliicnh,liquid

text: to

text: charge-density-wave

text: transition

text: at

text: high

text: Landau

text: “The

text: half-ﬁlled

text: Landau

text: level:

text: The

text: case

text: for

text: Dirac

text: [cond-mat.str-el].

text: R.

text: Moessner

text: and

text: J.

text: T.

text: Chalker,

text: “Exact

text: results

text: for

text: interacting

text: electrons

text: in

text: high

text: Landau

text: levels,”

text: Phys.

text: Rev.

text: Ganpathy

text: Murthy

text: and

text: R.

text: Shankar,

text: “ν

text: Landau

text: Phys.

text: Rev.

text: B

text: M.

text: I.

text: D’yakonov

text: and

text: A.

text: V.

text: Khaetskii,

text: “Transport

text: cross

text: section

text: for

text: small

text: angle

text: scattering,”

text: Sov.

text: Phys.

text: C.

text: Wang

text: and

text: T.

text: Senthil,

text: “Composite

text: Fermi

text: liquids

text: in

text: Phys.

text: Rev.

text: B

text: [cond-mat.str-el].

text: P.

text: Nozi`eres

text: and

text: C.

text: Lewiner,

text: “A

text: simple

text: theory

text: of

text: the

text: anomalous

text: Hall

text: eﬀect

text: in

text: semiconductors,”

text: Journal

text: David

text: F.

text: Mross,

text: Andrew

text: Essin,

text: and

text: Jason

text: Alicea,

text: for

text: Symmetric

text: Surface

text: Topological

text: Order,”

text: Phys.

text: Rev.

text: X

text: Cooper,

text: B.

text: I.

text: Halperin,

text: and

text: I.

text: M.

text: Ruzin,

text: “Thermoelectric

text: response

text: of

text: an

text: interacting

text: two-dimensional

text: Chong

text: Wang

text: and

text: T.

text: Senthil,

text: “Dual

text: Dirac

text: Liquid

text: on

text: theeSleucrtfraocne

text: Rev.

text: B

text: Phys.

text: Rev.

text: X

text: R.

text: L.

text: Willett,

text: R.

text: R.

text: Ruel,

text: K.

text: W.

text: West,

text: and

text: L.

text: N.

text: Pfeiﬀer,

text: “Experimental

text: demonstration

text: of

text: a

text: Fermi

text: C.

text: Wang

text: and

text: T.

text: Senthil,

text: “Time-Reversal

text: Symmetric

text: Phys.

text: Rev.

text: Lett.

text: [cond-mat.str-el].

text: D.

text: Kamburov,

text: Yang

text: Liu,

text: M.

text: A.

text: Mueed,

text: M.

text: Shayegan,

text: L.

text: N.

text: Pfeiﬀer,

text: K.

text: W.

text: West,

text: and

text: K.

text: W.

text: Baldwin,

text: M.

text: A.

text: Metlitski,

text: “S-duality

text: of

text: gauge

text: theory

text: with

text: θ

text: =“Wπ

text: fermions?”

text: Phys.

text: Rev.

text: Lett.

text: topological

text: insulators

text: and

text: superconductors,”

text: ArXiv

text: )H,

text: a.

text: N[h.epP-ftehiﬀ].er,

text: K.

text: W.

text: Baldwin,

text: and

text: K.

text: W.

text: West,

text: “How

text: real

text: are

text: composite

text: D.

text: F.

text: Mross,

text: J.

text: Alicea,

text: and

text: O.

text: I.

text: Motrunich,

text: “Explicit

text: and

text: Quantum

text: Electrodynamics

text: in

text: HPh.

text: ySsmiceatl,

text: DR.evWieewissL,

text: )K,

text: von

text: Klitzing,

text: R.

text: Fleischmann,

text: R.

text: Ketzmerick,

text: T.

text: Geisel,

text: [cond-mat.str-el].

text: and

text: G.

text: Weimann,

text: “Magnetic

text: focusing

text: of

text: composite

text: fermions

text: through

text: arrays

text: of

text: cavities,”

text: Phys.

text: Rev.

text: Lett.

text: N.

text: Seiberg,

text: T.

text: Senthil,

text: C.

text: Wang,

text: and

text: E.

text: Witten,

text: “A

text: and

text: condensed

text: matter

text: physics,”

text: Annals

text: of

text: Physics

text: D.

text: Weiss,

text: and

text: W.

text: Wegscheider,

text: “dc

text: transport

text: of

text: composite

text: fermions

text: in

text: weak

text: A.

text: Karch

text: and

text: D.

text: Tong,

text: “Particle-Vortex

text: Duality

text: from

text: RReevv.ieLwetXt.

text: [hep-th].

text: J.

text: H.

text: Smet,

text: S.

text: Jobst,

text: K.

text: von

text: Klitzing,

text: D.

text: Weiss,

text: W.

text: Wegscheider,

text: and

text: V.

text: Umansky,

text: “Commensurate

text: com-

text: J.

text: Murugan

text: and

text: H.

text: Nastase,

text: “Particle-vortex

text: duality

text: Direct

text: evidence

text: of

text: a

text: periodic

text: eﬀective

text: magnetic

text: ArXiv

text: e-prints

text: [hep-th].

text: ﬁeld,”

text: Phys.

text: Rev.

text: Lett.

text: S.

text: Kachru,

text: M.

text: Mulligan,

text: G.

text: Torroba,

text: and

text: H.

text: Km.irWro.r

text: Wsymesmt,

text: etarnyd,”

text: LP.hNys..

text: PRfeeviﬀ.

text: eDr,

text: “Geometric

text: resonance

text: of

text: composite

text: fermion

text: cyclotron

text: [hep-th].

text: orbits

text: with

text: a

text: ﬁctitious

text: magnetic

text: ﬁeld

text: modulation,”

text: Phys.

text: Rev.

text: Lett.

text: S.

text: M.

text: Girvin,

text: “Particle-hole

text: symmetry

text: in

text: the

text: resﬀcehckte,”aPndhyRs..

text: RRe.vG.

text: magnetoresistance

text: of

text: composite

text: fermion

text: systems

text: with

text: a

text: weak

text: one-dimensional

text: density

text: modulation,”

text: Phys.

text: Rev.

text: Lett.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 where Since, to leading order, J0 = nme∗τ E, the δJ term leads to a contribution to σcf of the form − ∂ xy r = Arg C . (43) k ∂k k − 1 σsj = . (50) There are two contributions to the shift of the average xy 8π position. The ﬁrst is seen when we consider a momentum | − | (cid:29) 2. Scattering rate in a composite fermion liquid in the k in the scattered wave, with k k ∆, so that 0 presence of an electric ﬁeld C0 = 0. Then, to lowest order, C may be replaced by k k C1, and the phase is equal to the phase of C1. Using (23) k Eq. (50) is half of the amount we need for PH for U, we ﬁnd that C1 has an extra argument, beyond the − symmetry. The second half is a consequence of contribution from e iεkt, arising from the complex valu∼e having a liquid of composite fermions, in which of U (cid:48). This extra argument has the form Arg C1 − kk · × k an applied electric ﬁeld aﬀects the occupation of q2/[2zˆ (k k )], and it leads to an extra displacement 0 momentum states. While the scattering rate from of the center of the scattered wave packet by an amount (cid:48) momentum k to momentum k is symmetric with × × (cid:48) · −zˆ k respect to the sign of (k k ) zˆ for a single compos- δr(1) = . (44) ite fermion in the absence of an electric ﬁeld, the k 2k2 F situation is more complicated in the presence of The second contribution to the average displacement both a liquid of composite fermions and an elec- comes from weight that has been asymmetrically re- tric ﬁeld. In that case, due to the electric ﬁeld moved from the incident part of the wave packet, where the side-jump is associated with a change of the k is close to k0. Here there is an interference between C0 com· posite fermion’s kinetic energy by an amount and C2. If one assumes that V (q) is vanishing for q = 0, eE δrq. The eﬀect of this change of energy on then one ﬁnds that(cid:88)the contribution from this term is the transport is best understood by means of the given by Boltzmann equation. For dc transport in the pres- ence of impurities th(cid:88)e equation reads | | × δr = C1 2(zˆ k)/(2k2 ). (45) 0 k F k · ∇ − − (cid:48) F f = W (cid:48)(f(k) f(k )), (51) k k,k Summing the two contributions we ﬁnd that the net dis- k(cid:48) placement (“side jump”) associated with a particle that scatters from a direction k0 into direction k = k0 + q where depends on the transferred momentum, and is given by × | | · − −(zˆ q) W (cid:48) = 2π V (cid:48) 2δ((cid:15) + F δr (cid:15) (cid:48)) (52) δr = . (46) k,k k,k k q k q (2k2 ) F This side jump contr(cid:88)ibutes directly to the total current Here f is the Fermi-Dirac distribution, F = eE 0 through a net charge displacement per unit time is the force acting on the composite fermions, (cid:15) is the energy, and V is the disordered potential. The δJ = f(k)Wk,k(cid:48)δrk(cid:48)−k, (47) δ-function expresses the change of the kinetic en- (cid:48) k,k ergy incurred by the scattered electron, a change which is our main focus here. where f(k) is the occupation probability for a state of As customary, linear response to F is analyzed momentum k, and W k,k (cid:80) by settin−g f to be f on the left-hand side of (51) isthetransitionprobability[seeEq. (52)below].Wecanexpresstheside 0 · and by writing f(k) = f +f = f + ∂f0 u v on the jumpcontributionδJ in terms of the current in the 0 1 0 ∂(cid:15) k ∗ right-hand side. The transfer of energy aﬀects absence of that contribution, J = f(k)k/m . 0 k the expansion of the distribution functions on the Using Eq. (46) for the displacement, and noticing right hand side. Speciﬁcally we have, that the transport scattering rate is given by (cid:80) (cid:88) (cid:104) (cid:105) (cid:80) − (cid:48) 1 ≡ W (cid:48)(1 − kˆ · kˆ(cid:48)) (48) k(cid:48) Wk,k(cid:48)(f(k) f(k )) = τ k(cid:48) k,k − ∂∂f(cid:15)0 u · k(cid:48) Wk,k(cid:48)(v(k) − v(k(cid:48))) + ∂∂f(cid:15)0 Wk,k(cid:48)F · δrq . we can simplify Eq. (47) to We now make use of the deﬁnition of the trans- ∗ − m × port scattering rate (48) to write the Boltzmann δJ = J zˆ. (49) 2τk2 0 equation as F ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: where

text: Since,

text: to

text: leading

text: order,

text: nme∗τ

text: E,

text: the

text: δJ

text: term

text: leads

text: to

text: a

text: contribution

text: to

text: σcf

text: of

text: the

text: form

text: xy

text: r

text: Arg

text: C

text: k

text: ∂k

text: k

text: σsj

text: There

text: are

text: two

text: contributions

text: to

text: the

text: shift

text: of

text: the

text: average

text: xy

text: 8π

text: position.

text: The

text: ﬁrst

text: is

text: seen

text: when

text: we

text: consider

text: a

text: momentum

label: section

text: Scattering

text: rate

text: in

text: a

text: composite

text: fermion

text: liquid

text: in

text: the

text: k

text: in

text: the

text: scattered

text: wave,

text: with

text: k

text: k

text: so

text: that

text: presence

text: of

text: an

text: electric

text: ﬁeld

text: Then,

text: to

text: lowest

text: order,

text: C

text: may

text: be

text: replaced

text: by

text: k

text: k

text: and

text: the

text: phase

text: is

text: equal

text: to

text: the

text: phase

text: of

text: Using

text: k

text: Eq.

text: is

text: half

text: of

text: the

text: amount

text: we

text: need

text: for

text: PH

text: for

text: U,

text: we

text: ﬁnd

text: that

text: has

text: an

text: extra

text: argument,

text: beyond

text: the

text: symmetry.

text: The

text: second

text: half

text: is

text: a

text: consequence

text: of

text: contribution

text: from

text: e

text: iεkt,

text: arising

text: from

text: the

text: complex

text: valu∼e

text: having

text: a

text: liquid

text: of

text: composite

text: fermions,

text: in

text: which

text: of

text: U

text: This

text: extra

text: argument

text: has

text: the

text: form

text: Arg

text: kk

text: k

text: an

text: applied

text: electric

text: ﬁeld

text: aﬀects

text: the

text: occupation

text: of

text: (k

text: k

text: and

text: it

text: leads

text: to

text: an

text: extra

text: displacement

text: momentum

text: states.

text: While

text: the

text: scattering

text: rate

text: from

text: of

text: the

text: center

text: of

text: the

text: scattered

text: wave

text: packet

text: by

text: an

text: amount

text: momentum

text: k

text: to

text: momentum

text: k

text: is

text: symmetric

text: with

text: −zˆ

text: k

text: respect

text: to

text: the

text: sign

text: of

text: (k

text: k

text: zˆ

text: for

text: a

text: single

text: compos-

text: ite

text: fermion

text: in

text: the

text: absence

text: of

text: an

text: electric

text: ﬁeld,

text: the

text: k

text: F

text: situation

text: is

text: more

text: complicated

text: in

text: the

text: presence

text: of

text: The

text: second

text: contribution

text: to

text: the

text: average

text: displacement

text: both

text: a

text: liquid

text: of

text: composite

text: fermions

text: and

text: an

text: elec-

text: comes

text: from

text: weight

text: that

text: has

text: been

text: asymmetrically

text: re-

text: tric

text: ﬁeld.

text: In

text: that

text: case,

text: due

text: to

text: the

text: electric

text: ﬁeld

text: moved

text: from

text: the

text: incident

text: part

text: of

text: the

text: wave

text: packet,

text: where

text: the

text: side-jump

text: is

text: associated

text: with

text: a

text: change

text: of

text: the

text: k

text: is

text: close

text: to

text: Here

text: there

text: is

text: an

text: interference

text: between

text: com·

text: posite

text: fermion’s

text: kinetic

text: energy

text: by

text: an

text: amount

text: and

text: If

text: one

text: assumes

text: that

text: V

text: (q)

text: is

text: vanishing

text: for

text: q

text: eE

text: δrq.

text: The

text: eﬀect

text: of

text: this

text: change

text: of

text: energy

text: on

text: then

text: one

text: ﬁnds

text: contribution

text: from

text: this

text: term

text: is

text: the

text: transport

text: is

text: best

text: understood

text: by

text: means

text: of

text: the

text: given

text: by

text: Boltzmann

text: equation.

text: For

text: dc

text: transport

text: in

text: the

text: pres-

text: ence

text: of

text: impurities

text: equation

text: reads

text: δr

text: k

text: F

text: k

text: F

text: f

text: W

text: f(k

text: k

text: k,k

text: Summing

text: the

text: two

text: contributions

text: we

text: ﬁnd

text: that

text: the

text: net

text: dis-

text: placement

text: (“side

text: jump”)

text: associated

text: with

text: a

text: particle

text: that

text: scatters

text: from

text: a

text: direction

text: into

text: direction

text: k

text: q

text: where

text: depends

text: on

text: the

text: transferred

text: momentum,

text: and

text: is

text: given

text: by

text: −(zˆ

text: q)

text: W

text: 2π

text: V

text: F

text: δr

text: δr

text: k,k

text: k,k

text: k

text: q

text: k

text: q

text: F

text: This

text: side

text: jump

text: directly

text: to

text: the

text: total

text: current

text: Here

text: f

text: is

text: the

text: Fermi-Dirac

text: distribution,

text: F

text: eE

text: through

text: a

text: net

text: charge

text: displacement

text: per

text: unit

text: time

text: is

text: the

text: force

text: acting

text: on

text: the

text: composite

text: fermions,

text: is

text: the

text: energy,

text: and

text: V

text: is

text: the

text: disordered

text: potential.

text: The

text: δJ

text: δ-function

text: expresses

text: the

text: change

text: of

text: the

text: kinetic

text: en-

text: k,k

text: ergy

text: incurred

text: by

text: the

text: scattered

text: electron,

text: a

text: change

text: which

text: is

text: our

text: main

text: focus

text: here.

text: where

text: f(k)

text: is

text: the

text: occupation

text: probability

text: for

text: a

text: state

text: of

text: As

text: customary,

text: linear

text: response

text: to

text: F

text: is

text: analyzed

text: momentum

text: k,

text: and

text: W

text: k,k

text: by

text: settin−g

text: f

text: to

text: be

text: f

text: on

text: the

text: left-hand

text: side

text: of

text: isthetransitionprobability[seeEq.

text: and

text: by

text: writing

text: f(k)

text: f

text: +f

text: f

text: u

text: v

text: on

text: the

text: jumpcontributionδJ

text: in

text: terms

text: of

text: the

text: current

text: in

text: the

text: k

text: right-hand

text: side.

text: The

text: transfer

text: of

text: energy

text: aﬀects

text: absence

text: of

text: that

text: contribution,

text: J

text: f(k)k/m

text: k

text: the

text: expansion

text: of

text: the

text: distribution

text: functions

text: on

text: the

text: Using

text: Eq.

text: for

text: the

text: displacement,

text: and

text: noticing

text: right

text: hand

text: side.

text: Speciﬁcally

text: we

text: have,

text: that

text: the

text: transport

text: scattering

text: rate

text: is

text: given

text: by

text: W

text: kˆ

text: f(k

text: τ

text: k,k

text: u

text: δrq

text: we

text: can

text: simplify

text: Eq.

text: to

text: We

text: now

text: make

text: use

text: of

text: the

text: deﬁnition

text: of

text: the

text: trans-

text: m

text: port

text: scattering

text: rate

text: to

text: write

text: the

text: Boltzmann

text: δJ

text: J

text: zˆ.

text: equation

text: as

text: F

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 (cid:18) (cid:19) the mean free path for elastic scattering by im- × · purities. This restriction becomes more severe as · − zˆ k ∂f u v ∂f F v 0 = k 0 , (53) the sample becomes more ideal. k 2k2 τ ∂(cid:15) τ ∂(cid:15) F which amounts to (cid:18) (cid:19) 2. Thermopower − × The heat current jQ = jE µjN induced by a ﬁeld · − zˆ k ∂f E f1(k) = τF vk 2k2 τ ∂(cid:15)0 . (54) cf applied to the CE Fs is described by a response F function, jQ = Lcf cf, assuming that the temper- µ µν ν ature is a constant. For a non-interacting Fermi As this expression shows, in the limit of a small gas, at low temperatures, expanding around the scattering rate 1/τ the shift of the Fermi sea that Fermi level leads to the general result results from the application of the electric ﬁeld is primarily parallel to the electric ﬁeld, but in- cludes also a small term perpendicular to the π2k2 T2 dΣcf (cid:82) (cid:82) Lcf = B µν . (56) ﬁeld. This term contributes to the Hall conduc- µν 3e dE (cid:82) tivity. The current is J = dkf (k)v , with dk = ∗ 1 k Since t−he Hall conductivity of the CFs is ﬁxed (2mπ)2 d(cid:15) dθ. The angular integral gives π for both to σxcfy = e2/2h, requiring dΣcxfy/dE = 0, then components of the current (each component from a diﬀerent term), leading to σcf = kFvFτ , and − xx 4π δσxcfy = 81π. This contribution to the Hall con- Lcµfν = Lcxfxδµν. (57) ductivity adds to the side-jump contribution cal- culated in the previous subsection, with the sum − of the two being 1 . This (diagonal) result is of the form required 4π by PH symmetry, as discussed in [31], so that σcf D. Thermopower and thermal transport µν and Lcf are each characterized by a single non- µν universal quantity, σcf = σcf and Lcf = Lcf . 1. General considerations xx yy xx yy To construct the thermoelectric response ten- (cid:126) sor for the electrons (not the CFs), one must take In this subsection, we again restore and the account of the fact that the electric ﬁeld that cou- electron charge e. The formulas are correct for ples to the electrons is either sign of e, provided that the product of the electron charge and the z-component of the ex- E ternal magnetic ﬁeld is positive. For eB < 0, the E = cf + ρˆCSj (58) x and y axes should be interchanged. The thermoelectric and thermal responses for the CFs can be obtained from standard results where j is the current of either electrons or CFs for non-interacting fermions, based on interpret- and ing the CF conductivity in terms of an energy- dependent conduc(cid:90)tivity Σcf(cid:18)(ε) th(cid:19)rough (cid:126) µν ≡ − ρˆCS 4π (cid:15)ˆ. (59) e2 −∂f σcf = Σcf (ε) dε, (55) µν µν ∂ε The response tensors for the electrons are read- ily found to be with f the Fermi distribution. We explore the consequences, making use only of the fact that − (cid:126) − σcf = (e2/4nπ ), independent of the Fermi en- σˆ = σˆcf(1 + ρˆCSσˆcf) 1 (60) xy ergy, and hence that dΣcxfy/dε = 0. Here we focus Lˆ = Lˆcf(1 + ρˆCSσˆcf)−1 . (61) on the ν = 1/2 state with n = 1. Although observations of thermal eﬀects re- quire that the temperature should not be too With our speciﬁc forms of σˆcf and Lˆcf, these be- small, the calculations here also assume that the come (cid:20) (cid:21) temperature should not be too high. In particu- lar, we assume that the temperature is suﬃciently low that the mean free path for inelastic scatter- e2 e2 σˆ = (cid:126) (cid:15)ˆ+ (cid:126) , (62) ing of composite fermions is large compared to 4π 4π σcf xx ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: the

text: mean

text: free

text: path

text: for

text: elastic

text: scattering

text: by

text: im-

text: purities.

text: This

text: restriction

text: becomes

text: more

text: severe

text: as

text: zˆ

text: k

text: ∂f

text: u

text: v

text: ∂f

text: F

text: v

text: k

text: the

text: sample

text: becomes

text: more

text: ideal.

text: k

text: τ

text: τ

text: F

text: which

text: amounts

text: to

text: Thermopower

text: The

text: heat

text: current

text: jQ

text: jE

text: µjN

text: induced

text: by

text: a

text: ﬁeld

text: zˆ

text: k

text: ∂f

text: E

text: τF

text: vk

text: τ

text: cf

text: applied

text: to

text: the

text: CE

text: Fs

text: is

text: described

text: by

text: a

text: response

text: F

text: function,

text: jQ

text: Lcf

text: cf,

text: assuming

text: that

text: the

text: temper-

text: µ

text: µν

text: ν

text: ature

text: is

text: a

text: constant.

text: For

text: a

text: non-interacting

text: Fermi

text: As

text: this

text: expression

text: shows,

text: in

text: the

text: limit

text: of

text: a

text: small

text: gas,

text: at

text: low

text: temperatures,

text: expanding

text: around

text: the

text: scattering

text: rate

text: the

text: shift

text: of

text: the

text: Fermi

text: sea

text: that

text: Fermi

text: level

text: leads

text: to

text: the

text: general

text: result

text: results

text: from

text: the

text: application

text: of

text: the

text: electric

text: ﬁeld

text: is

text: primarily

text: parallel

text: to

text: the

text: electric

text: ﬁeld,

text: but

text: in-

text: cludes

text: also

text: a

text: small

text: term

text: perpendicular

text: to

text: the

text: dΣcf

text: Lcf

text: B

text: µν

text: ﬁeld.

text: This

text: term

text: contributes

text: to

text: the

text: Hall

text: conduc-

text: µν

text: dE

text: tivity.

text: The

text: current

text: is

text: J

text: dkf

text: (k)v

text: with

text: dk

text: k

text: Since

text: t−he

text: Hall

text: conductivity

text: of

text: the

text: CFs

text: is

text: ﬁxed

text: dθ.

text: The

text: angular

text: integral

text: gives

text: π

text: for

text: both

text: to

text: σxcfy

text: requiring

text: dΣcxfy/dE

text: then

text: components

text: of

text: the

text: current

text: (each

text: component

text: from

text: a

text: diﬀerent

text: term),

text: leading

text: to

text: σcf

text: kFvFτ

text: and

text: xx

text: 4π

text: δσxcfy

text: This

text: contribution

text: to

text: the

text: Hall

text: con-

text: Lcµfν

text: Lcxfxδµν.

text: ductivity

text: adds

text: to

text: the

text: side-jump

text: contribution

text: cal-

text: culated

text: in

text: the

text: previous

text: subsection,

text: with

text: the

text: sum

text: of

text: the

text: two

text: being

text: This

text: (diagonal)

text: result

text: is

text: of

text: the

text: form

text: required

text: 4π

text: by

text: PH

text: symmetry,

text: as

text: discussed

text: in

text: so

text: that

text: σcf

text: D.

text: Thermopower

text: and

text: thermal

text: transport

text: µν

text: and

text: Lcf

text: are

text: each

text: characterized

text: by

text: a

text: single

text: non-

text: µν

text: universal

text: quantity,

text: σcf

text: σcf

text: and

text: Lcf

text: Lcf

text: General

text: considerations

text: xx

text: yy

text: xx

text: yy

text: To

text: construct

text: the

text: thermoelectric

text: response

text: ten-

text: sor

text: for

text: the

text: electrons

text: (not

text: the

text: CFs),

text: one

text: must

text: take

text: In

text: this

text: subsection,

text: we

text: again

text: restore

text: and

text: the

text: account

text: of

text: the

text: fact

text: that

text: the

text: electric

text: ﬁeld

text: that

text: cou-

text: electron

text: charge

text: e.

text: The

text: formulas

text: are

text: correct

text: for

text: ples

text: to

text: the

text: electrons

text: is

text: either

text: sign

text: of

text: e,

text: provided

text: that

text: the

text: product

text: of

text: the

text: electron

text: charge

text: and

text: the

text: z-component

text: of

text: the

text: ex-

text: E

text: ternal

text: magnetic

text: ﬁeld

text: is

text: positive.

text: For

text: eB

text: the

text: E

text: cf

text: ρˆCSj

text: x

text: and

text: y

text: axes

text: should

text: be

text: interchanged.

text: The

text: thermoelectric

text: and

text: thermal

text: responses

text: for

text: the

text: CFs

text: can

text: be

text: obtained

text: from

text: standard

text: results

text: where

text: j

text: is

text: the

text: current

text: of

text: either

text: electrons

text: or

text: CFs

text: for

text: non-interacting

text: fermions,

text: based

text: on

text: interpret-

text: and

text: ing

text: the

text: CF

text: conductivity

text: in

text: terms

text: of

text: an

text: energy-

text: dependent

text: µν

text: ρˆCS

text: 4π

text: −∂f

text: σcf

text: Σcf

text: (ε)

text: dε,

text: µν

text: µν

text: ∂ε

text: The

text: response

text: tensors

text: for

text: the

text: electrons

text: are

text: read-

text: ily

text: found

text: to

text: be

text: with

text: f

text: the

text: Fermi

text: distribution.

text: We

text: explore

text: the

text: consequences,

text: making

text: use

text: only

text: of

text: the

text: fact

text: that

text: σcf

text: independent

text: of

text: the

text: Fermi

text: en-

text: σˆ

text: ρˆCSσˆcf)

text: xy

text: ergy,

text: and

text: hence

text: that

text: dΣcxfy/dε

text: Here

text: we

text: focus

text: Lˆ

text: on

text: the

text: ν

text: state

text: with

text: n

text: Although

text: observations

text: of

text: thermal

text: eﬀects

text: re-

text: quire

text: that

text: the

text: temperature

text: should

text: not

text: be

text: too

text: With

text: our

text: speciﬁc

text: forms

text: of

text: σˆcf

text: and

text: Lˆcf,

text: these

text: be-

text: small,

text: the

text: calculations

text: here

text: also

text: assume

text: that

text: the

text: come

text: temperature

text: should

text: not

text: be

text: too

text: high.

text: In

text: particu-

text: lar,

text: we

text: assume

text: that

text: the

text: temperature

text: is

text: suﬃciently

text: low

text: that

text: the

text: mean

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text: inelastic

text: scatter-

text: σˆ

text: ing

text: of

text: composite

text: fermions

text: is

text: large

text: compared

text: to

text: 4π

text: 4π

text: σcf

text: xx

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 As discussed in [34], the computational cost of matched- data, thus paving a natural path to realizing real-time multi- ﬁltering searches increases signiﬁcantly when targeting GW messenger astrophysics with a uniﬁed framework. sources that span a higher dimensional parameter space. In As NR continues to shed light into the physics of GW contrast, when using deep learning, all the intensive computa- sources[6], we will rely on an extensive exploitation of HPC tion is diverted to the one-time training stage, after which the resources to obtain NR waveforms to train our DNN algo- datasets can be discarded, i.e., the size of template banks that rithm. At the same time, we are using HPC facilities to describe the GW signals we search for present no limitation carry out large scale parameter sweeps to ﬁnd optimal DNNs when using deep learning. Indeed, it is preferable to use large for GW detection and parameter estimation. The approach datasets of GW signals for the one-time training stage to cover we discuss here employs recent advances in artiﬁcial intelli- as deep a parameter space as possible. With existing compu- gence algorithms, by computer scientists and industries, for tational resources on supercomputers such as Blue Waters, we accelerating scientiﬁc discovery by enhancing the use of tra- estimate that it would possible to ﬁnish training the DNNs on ditional HPC resources, while allowing us to exploit emerg- templates across 10 or more dimensions of parameters within ing hardware architectures such as deep-learning-optimized a few weeks. Graphics Processing Units (GPUs) [44], Application-Speciﬁc The main objective in developing Deep Filtering is Integrated Circuits (ASICs) [45], Field-Programmable Gate to enhance existing, low latency GW detection algorithms to Arrays (FPGAs) [46], quantum computers [47] and brain- enable deeper and faster GW searches. We envision using like neuromorphic chips [48]. This approach may provide Deep Filtering to identify and rapidly constrain the as- the needed platform to address common challenges on large trophysical parameters of GW transients. This real-time anal- scale data analytics on disparate ﬁelds of research to effec- ysis would then be followed up by existing LIGO pipelines tively consolidate different windows of observation into the focusing on a narrow region of GWs’ higher dimensional pa- Universe. rameter space. A targeted search of this nature will signiﬁ- This article is organized as follows: Section provides a cantly reduce the size of multi-dimensional template banks, comprehensive overview of artiﬁcial neural networks and enabling the use of established matched-ﬁltering searches at deep learning, particularly focusing on convolutional neural a fraction of their computational cost to quantify the signiﬁ- networks in the context of time-series signal processing. In cance of new GW detections. This approach would combine Section , we describe our assumptions, datasets, and proce- the best of two approaches: the scalable, multidimensional dure to construct the DNN-based GW analysis pipeline. We nature of neural networks with the sophistication of LIGO de- report the results of our analysis in Section . In Section , tection pipelines. To accomplish this, we are working with the we discuss its immediate applications, and their implications developers of PyCBC [29] to implement Deep Filtering for GW astrophysics missions, along with scope for improve- as a module to increase the depth and speed of this pipeline. ments. We summarize our ﬁndings and outline its broader The results we present in this article conﬁrm that DNNs are impact in Section . ideal tools for future GW analysis. We have found that DNN II. DEEP NEURAL NETWORKS are able to interpolate between waveform templates, in a sim- ilar manner to Gaussian Process Regression (GPR) 2, and to In this section we provide a brief overview of the main generalize to new classes of signals beyond the templates used concepts of deep learning, including machine learning, arti- for training. Furthermore, our DNNs can be evaluated faster ﬁcial neural networks, and convolutional neural networks in than real-time with a single CPU, and very intensive searches the context of time-series signal processing. over a broader range of signals can be easily carried out with The vast majority of algorithms are designed with a spe- one dedicated GPU. The intelligent nature of deep learning ciﬁc task in mind. They require extensive modiﬁcations be- would allow automated learning of persistent and transient fore they can be re-used for any other task. The term machine characteristics of noises inherent to the detectors, while in- learning refers to a special class of algorithms that can learn corporating real-time data quality information. This analy- from examples to solve new problems without being explic- sis, combined with recent work to understand and characterize itly re-programmed. This enables cross-domain applications aLIGO non-Gaussian noise transients [42, 43], strongly sug- of the same algorithm by training it with different data [49]. gests that it is feasible to create a single efﬁcient pipeline to More importantly, some of these algorithms are able to tackle perform all tasks—identifying the presence or absence of GW problems which humans can solve intuitively but ﬁnd difﬁcult signals, classifying noise transients, and reconstructing the as- to explain using well-deﬁned rules, hence they are often called trophysical properties of detected GW sources. Furthermore, “artiﬁcial intelligence” [49]. since this technique can be applied to other types of raw time- The two main categories of machine learning are supervised series data, similar DNNs can be used to process telescope and unsupervised learning. In supervised learning, the algo- rithm learns from some data that is correctly labeled, while unsupervised learning algorithms have to make sense of un- structured and unlabeled data [50]. We will be focusing on an 2 GPR [39–41] is a statistical tool that can serve as a probabilistic interpola- application of supervised learning in this work, where we use tion algorithm providing information about the training set of NR simula- labeled data obtained from physics simulations to train an al- tions needed to accurately describe a given parameter-space and generates interpolated waveforms that match NR counterparts above any given accu- gorithm to detect signals embedded in noise and also estimate racy. multiple parameters of the source. ##LTLine##

text: As

text: discussed

text: in

text: the

text: computational

text: cost

text: of

text: matched-

text: data,

text: thus

text: paving

text: a

text: natural

text: path

text: to

text: realizing

text: real-time

text: multi-

text: ﬁltering

text: searches

text: increases

text: signiﬁcantly

text: when

text: targeting

text: GW

text: messenger

text: astrophysics

text: with

text: a

text: uniﬁed

text: framework.

text: sources

text: that

text: span

text: a

text: higher

text: dimensional

text: parameter

text: space.

text: In

text: As

text: NR

text: continues

text: to

text: shed

text: light

text: into

text: the

text: physics

text: of

text: GW

text: contrast,

text: when

text: using

text: deep

text: learning,

text: all

text: the

text: intensive

text: computa-

text: we

text: will

text: rely

text: on

text: an

text: extensive

text: exploitation

text: of

text: HPC

text: tion

text: is

text: diverted

text: to

text: the

text: one-time

text: training

text: stage,

text: after

text: which

text: the

text: resources

text: to

text: obtain

text: NR

text: waveforms

text: to

text: train

text: our

text: DNN

text: algo-

text: datasets

text: can

text: be

text: discarded,

text: i.e.,

text: the

text: size

text: of

text: template

text: banks

text: that

text: rithm.

text: At

text: the

text: same

text: time,

text: we

text: are

text: using

text: HPC

text: facilities

text: to

text: describe

text: the

text: GW

text: signals

text: we

text: search

text: for

text: present

text: no

text: limitation

text: carry

text: out

text: large

text: scale

text: parameter

text: sweeps

text: to

text: ﬁnd

text: optimal

text: DNNs

text: when

text: using

text: deep

text: learning.

text: Indeed,

text: it

text: is

text: preferable

text: to

text: use

text: large

text: for

text: GW

text: detection

text: and

text: parameter

text: estimation.

text: The

text: approach

text: datasets

text: of

text: GW

text: signals

text: for

text: the

text: one-time

text: training

text: stage

text: to

text: cover

text: we

text: discuss

text: here

text: employs

text: recent

text: advances

text: in

text: artiﬁcial

text: intelli-

text: as

text: deep

text: a

text: parameter

text: space

text: as

text: possible.

text: With

text: existing

text: compu-

text: gence

text: algorithms,

text: by

text: computer

text: scientists

text: and

text: industries,

text: for

text: tational

text: resources

text: on

text: supercomputers

text: such

text: as

text: Blue

text: Waters,

text: we

text: accelerating

text: scientiﬁc

text: discovery

text: by

text: enhancing

text: the

text: use

text: of

text: tra-

text: estimate

text: that

text: it

text: would

text: possible

text: to

text: ﬁnish

text: training

text: the

text: DNNs

text: on

text: ditional

text: HPC

text: resources,

text: while

text: allowing

text: us

text: to

text: exploit

text: emerg-

text: templates

text: across

text: or

text: more

text: dimensions

text: of

text: parameters

text: within

text: ing

text: hardware

text: architectures

text: such

text: as

text: deep-learning-optimized

text: a

text: few

text: weeks.

text: Graphics

text: Processing

text: Units

text: (GPUs)

text: Application-Speciﬁc

text: The

text: main

text: objective

text: in

text: developing

text: Deep

text: Filtering

text: is

text: Integrated

text: Circuits

text: (ASICs)

text: Field-Programmable

text: Gate

text: to

text: enhance

text: existing,

text: low

text: latency

text: GW

text: detection

text: algorithms

text: to

text: Arrays

text: (FPGAs)

text: quantum

text: computers

text: and

text: brain-

text: enable

text: deeper

text: and

text: faster

text: GW

text: searches.

text: We

text: envision

text: using

text: like

text: neuromorphic

text: chips

text: This

text: approach

text: may

text: provide

text: Deep

text: Filtering

text: to

text: identify

text: and

text: rapidly

text: constrain

text: the

text: as-

text: the

text: needed

text: platform

text: to

text: address

text: common

text: challenges

text: on

text: large

text: trophysical

text: parameters

text: of

text: GW

text: transients.

text: This

text: real-time

text: anal-

text: scale

text: data

text: analytics

text: on

text: disparate

text: ﬁelds

text: of

text: research

text: to

text: effec-

text: ysis

text: would

text: then

text: be

text: followed

text: up

text: by

text: existing

text: LIGO

text: pipelines

text: tively

text: consolidate

text: different

text: windows

text: of

text: observation

text: into

text: the

text: focusing

text: on

text: a

text: narrow

text: region

text: of

text: GWs’

text: higher

text: dimensional

text: pa-

text: Universe.

text: rameter

text: space.

text: A

text: targeted

text: search

text: of

text: this

text: nature

text: will

text: signiﬁ-

text: This

text: article

text: is

text: organized

text: as

text: follows:

text: Section

text: provides

text: a

text: cantly

text: reduce

text: the

text: size

text: of

text: multi-dimensional

text: template

text: banks,

text: comprehensive

text: overview

text: of

text: artiﬁcial

text: neural

text: networks

text: and

text: enabling

text: the

text: use

text: of

text: established

text: matched-ﬁltering

text: searches

text: at

text: deep

text: learning,

text: particularly

text: focusing

text: on

text: convolutional

text: neural

text: a

text: fraction

text: of

text: their

text: computational

text: cost

text: to

text: quantify

text: the

text: signiﬁ-

text: networks

text: in

text: the

text: context

text: of

text: time-series

text: signal

text: processing.

text: In

text: cance

text: of

text: new

text: GW

text: detections.

text: This

text: approach

text: would

text: combine

text: Section

text: we

text: describe

text: our

text: assumptions,

text: datasets,

text: and

text: proce-

text: the

text: best

text: of

text: two

text: approaches:

text: the

text: scalable,

text: multidimensional

text: dure

text: to

text: construct

text: the

text: DNN-based

text: GW

text: analysis

text: pipeline.

text: We

text: nature

text: of

text: neural

text: networks

text: with

text: the

text: sophistication

text: of

text: LIGO

text: de-

text: report

text: the

text: results

text: of

text: our

text: analysis

text: in

text: Section

text: In

text: Section

text: tection

text: pipelines.

text: To

text: accomplish

text: this,

text: we

text: are

text: working

text: with

text: the

text: we

text: discuss

text: its

text: immediate

text: applications,

text: and

text: their

text: implications

text: developers

text: of

text: PyCBC

text: to

text: implement

text: Deep

text: Filtering

text: for

text: GW

text: astrophysics

text: missions,

text: along

text: with

text: scope

text: for

text: improve-

text: as

text: a

text: module

text: to

text: increase

text: the

text: depth

text: and

text: speed

text: of

text: this

text: pipeline.

text: ments.

text: We

text: summarize

text: our

text: ﬁndings

text: and

text: outline

text: its

text: broader

text: The

text: results

text: we

text: present

text: in

text: this

text: article

text: conﬁrm

text: that

text: DNNs

text: are

text: impact

text: in

text: Section

text: ideal

text: tools

text: for

text: future

text: GW

text: analysis.

text: We

text: have

text: found

text: that

text: DNN

text: II.

text: DEEP

text: NEURAL

text: NETWORKS

text: are

text: able

text: to

text: interpolate

text: between

text: waveform

text: templates,

text: in

text: a

text: sim-

text: ilar

text: manner

text: to

text: Gaussian

text: Process

text: Regression

text: (GPR)

text: and

text: to

text: In

text: this

text: section

text: we

text: provide

text: a

text: brief

text: overview

text: of

text: the

text: main

text: generalize

text: to

text: new

text: classes

text: of

text: signals

text: beyond

text: the

text: templates

text: used

text: concepts

text: of

text: deep

text: learning,

text: including

text: machine

text: learning,

text: arti-

text: for

text: training.

text: Furthermore,

text: our

text: DNNs

text: can

text: be

text: evaluated

text: faster

text: ﬁcial

text: neural

text: networks,

text: and

text: convolutional

text: neural

text: networks

text: in

text: than

text: real-time

text: with

text: a

text: single

text: CPU,

text: and

text: very

text: intensive

text: searches

text: the

text: context

text: of

text: time-series

text: signal

text: processing.

text: over

text: a

text: broader

text: range

text: of

text: signals

text: can

text: be

text: easily

text: carried

text: out

text: with

text: The

text: vast

text: majority

text: of

text: algorithms

text: are

text: designed

text: with

text: a

text: spe-

text: one

text: dedicated

text: GPU.

text: The

text: intelligent

text: nature

text: of

text: deep

text: learning

text: ciﬁc

text: task

text: in

text: mind.

text: They

text: require

text: extensive

text: modiﬁcations

text: be-

text: would

text: allow

text: automated

text: learning

text: of

text: persistent

text: and

text: transient

text: fore

text: they

text: can

text: be

text: re-used

text: for

text: any

text: other

text: task.

text: The

text: term

text: machine

text: characteristics

text: of

text: noises

text: inherent

text: to

text: the

text: detectors,

text: while

text: in-

text: learning

text: refers

text: to

text: a

text: special

text: class

text: of

text: algorithms

text: that

text: can

text: learn

text: corporating

text: real-time

text: data

text: quality

text: information.

text: This

text: analy-

text: from

text: examples

text: to

text: solve

text: new

text: problems

text: without

text: being

text: explic-

text: sis,

text: combined

text: with

text: recent

text: work

text: to

text: understand

text: and

text: characterize

text: itly

text: re-programmed.

text: This

text: enables

text: cross-domain

text: applications

text: aLIGO

text: non-Gaussian

text: noise

text: transients

text: strongly

text: sug-

text: of

text: the

text: same

text: algorithm

text: by

text: training

text: it

text: with

text: different

text: data

text: gests

text: that

text: it

text: is

text: feasible

text: to

text: create

text: a

text: single

text: efﬁcient

text: pipeline

text: to

text: More

text: importantly,

text: some

text: of

text: these

text: algorithms

text: are

text: able

text: to

text: tackle

text: perform

text: all

text: tasks—identifying

text: the

text: presence

text: or

text: absence

text: of

text: GW

text: problems

text: which

text: humans

text: can

text: solve

text: intuitively

text: but

text: ﬁnd

text: difﬁcult

text: signals,

text: classifying

text: noise

text: transients,

text: and

text: reconstructing

text: the

text: as-

text: to

text: explain

text: using

text: well-deﬁned

text: rules,

text: hence

text: they

text: are

text: often

text: called

text: trophysical

text: properties

text: of

text: detected

text: GW

text: sources.

text: Furthermore,

text: “artiﬁcial

text: intelligence”

text: since

text: this

text: technique

text: can

text: be

text: applied

text: to

text: other

text: types

text: of

text: raw

text: time-

text: The

text: two

text: main

text: categories

text: of

text: machine

text: learning

text: are

text: supervised

text: series

text: data,

text: similar

text: DNNs

text: can

text: be

text: used

text: to

text: process

text: telescope

text: and

text: unsupervised

text: learning.

text: In

text: supervised

text: learning,

text: the

text: algo-

text: rithm

text: learns

text: from

text: some

text: data

text: that

text: is

text: correctly

text: labeled,

text: while

text: unsupervised

text: learning

text: algorithms

text: have

text: to

text: make

text: sense

text: of

text: un-

text: structured

text: and

text: unlabeled

text: data

text: We

text: will

text: be

text: focusing

text: on

text: an

text: GPR

text: is

text: a

text: statistical

text: tool

text: that

text: can

text: serve

text: as

text: a

text: probabilistic

text: interpola-

text: application

text: of

text: supervised

text: learning

text: in

text: this

text: work,

text: where

text: we

text: use

text: tion

text: algorithm

text: providing

text: information

text: about

text: the

text: training

text: set

text: of

text: NR

text: simula-

text: labeled

text: data

text: obtained

text: from

text: physics

text: simulations

text: to

text: train

text: an

text: al-

text: tions

text: needed

text: to

text: accurately

text: describe

text: a

text: given

text: parameter-space

text: and

text: generates

text: interpolated

text: waveforms

text: that

text: match

text: NR

text: counterparts

text: above

text: any

text: given

text: accu-

text: gorithm

text: to

text: detect

text: signals

text: embedded

text: in

text: noise

text: and

text: also

text: estimate

text: racy.

text: multiple

text: parameters

text: of

text: the

text: source.

text: ##LTLine##


text: × (cid:54) Next, entries in a and a , were changed as in the 2 2 if W and H really do break the form, or A = WH, the 0 1 case. As long as the entries were O(1) (as opposed to O((cid:15)) algorithm fails the test on this input. Otherwise, it passes. or O(1)), the behavior of the algorithms was similar. Note that even widely accepted algorithms do fail these tests (cid:15) × × Finally, A larger than 3 3 were examined. Several different occasionally, especially with matrices larger than 8 8, so it’s × × sizes of matrices were tested, ranging from 4 4 to 20 20, advisable to perform the test many times to get a more accurate always keeping A, W, and H square, with positive entries idea of an algorithm’s performance. only on the main diagonal and the superdiagonal. The experi- V. CONCLUSION ments followed the same general pattern; nonzero entries in W This paper proposes an approach to the problem of testing and H appeared only on the main diagonal and superdiagonal. NMF algorithms by running the algorithms on simple input × × Using similar logic to the 2 2 and 3 3 cases, it can be shown that can produce an exact non-negative factorization, and that these are the only exact solutions. However, in practice, perturbations of such input. In particular, square matrices as the matrices get larger, exceptions to this pattern become with O(1) entries on the main diagonal and O((cid:15)) entries more common, particularly in Algorithm 3. The general rule on the superdiagonal are proposed, because they have exact seems to mostly hold (over half the time) until A becomes solutions that can enumerated mathematically, or because they × around 20 20. Note, however, that because the run-time of are perturbations of matrices with exact solutions. the algorithms are cubic in the size of the matrix, at best, the The test cases have been used as input on three known NMF sample size for large matrices is small. algorithms that represent a variety of algorithms, and all of IV. PROPOSED TESTS FOR NMF ALGORITHMS them behaved similarly, which suggests testable, quantiﬁable Since all three algorithms, which cover a variety of ap- behaviors that many NMF algorithms share. These test cases proaches to NMF, had a lot in common in their solutions, offer one approach for testing candidate NMF implentations it is propose that these inputs A could be used as a test case to help determine whether it behaves as it should. of an NMF algorithm implementation. In this section, it is ACKNOWLEDGMENT proposed how such test cases could be executed. The authors would like to thank Dr. Alan Edelman for The test begins with input of the form providing and overseeing this research opportunity, and Dr. Vijay Gadepally for his advice and expertise. A = A0 + (cid:15)A1 = diag(a0) + (cid:15)diag(a1, 1) (13) REFERENCES [1] Berry Browne Langville Pauca and Plemmons, Algorithms and applica- × tions for approximate nonnegative matrix factorization, Computational ×A is square, and preferably somewhere between 3 3 and Statistics and Data Analysis 52 (2007), 155–173. 8 8 in size, although bigger inputs may be useful as well. [2] Rainer Gemulla, Erik Nijkamp, Peter J. Haas, and Yannis Sismanis, The entries should vary between tests. Each test should start Large-scale matrix factorization with distributed stochastic gradient de- scent, Proceedings of the 17th ACM SIGKDD International Conference by using (cid:15) = 0 so that A is diagonal. The results of this on Knowledge Discovery and Data Mining (New York, NY, USA), KDD test should have W and H monomial - only one nonzero ’11, ACM, 2011, pp. 69–77. element in each row and column. Ignore entries that are below [3] John Gilbert, personal communication, Sep 2015. − [4] N. Guan, D. Tao, Z. Luo, and B. Yuan, Nenmf: An optimal gradient O(10 10), for the entirety of testing, as any such entries are method for nonnegative matrix factorization, IEEE Transactions on Signal negligible. Processing 60 (2012), no. 6, 2882–2898. If W or H is not monomial, or if the product WH is [5] Daniel D. Lee and H. Sebastian Seung, Algorithms for non-negative matrix factorization, Advances in Neural Information Processing Systems not equal to A to within a negligible margin of error, the 13 (T. K. Leen, T. G. Dietterich, and V. Tresp, eds.), MIT Press, 2001, algorithm fails the test. Otherwise, the generated solution can pp. 556–562. be used to ﬁnd the permutation matrix P that makes WP and [6] Suvrit Sra and Inderjit S. Dhillon, Generalized nonnegative matrix ap- − proximations with bregman divergences, Advances in Neural Information P 1H diagonal by replacing the nonzero entries of H with Processing Systems 18 (Y. Weiss, B. Scho¨lkopf, and J. C. Platt, eds.), 1’s. Since A = WH is diagonal, WP is also diagonal, and MIT Press, 2006, pp. 283–290. − − since I = P 1P is diagonal, so is P 1H. Knowing P will [7] Wenwu Wang, Instantaneous vs. convolutive non-negative matrix factor- ization: Models, algorithms and applications, Machine Audition: Prin- make the rest of the testing much simpler since it is easier to ciples, Algorithms and Systems: Principles, Algorithms and Systems identify whether a solution is of the form given above when (2010), 353. it is not permuted. Next, run the test again using a positive value for (cid:15); (cid:15) = − 10 3 seems to work well, although using a variety of (cid:15) is also recommended. Make sure to use the same random seeds that were used in the (cid:15) = 0 test to produce corresponding output. Then check that the W and H given by the algorithm are − such that WP and P 1H have nonzero entries only on the two diagonals that they are supposed to. If this doesn’t hold, changing (cid:15) might have changed which permutation returns W and H to the proper form, so check again; this happens more commonly among larger matrices than smaller ones. However, ##LTLine##

font: default

text: Next,

text: entries

text: in

text: a

text: and

text: a

text: were

text: changed

text: as

text: in

text: the

text: if

text: W

text: and

text: H

text: really

text: do

text: break

text: the

text: form,

text: or

text: A

text: WH,

text: the

text: case.

text: As

text: long

text: as

text: the

text: entries

text: were

text: (as

text: opposed

text: to

text: algorithm

text: fails

text: the

text: test

text: on

text: this

text: input.

text: Otherwise,

text: it

text: passes.

text: or

text: the

text: behavior

text: of

text: the

text: algorithms

text: was

text: similar.

text: Note

text: that

text: even

text: widely

text: accepted

text: algorithms

text: do

text: fail

text: these

text: tests

text: Finally,

text: A

text: larger

text: than

text: were

text: examined.

text: Several

text: different

text: occasionally,

text: especially

text: with

text: matrices

text: larger

text: than

text: so

text: it’s

text: sizes

text: of

text: matrices

text: were

text: tested,

text: ranging

text: from

text: to

text: advisable

text: to

text: perform

text: the

text: test

text: many

text: times

text: to

text: get

text: a

text: more

text: accurate

text: always

text: keeping

text: A,

text: W,

text: and

text: H

text: square,

text: with

text: positive

text: entries

text: idea

text: of

text: an

text: algorithm’s

text: performance.

text: only

text: on

text: the

text: main

text: diagonal

text: and

text: the

text: superdiagonal.

text: The

text: experi-

text: V.

text: CONCLUSION

text: ments

text: followed

text: the

text: same

text: general

text: pattern;

text: nonzero

text: entries

text: in

text: W

text: This

text: paper

text: proposes

text: an

text: approach

text: to

text: the

text: problem

text: of

text: testing

text: and

text: H

text: appeared

text: only

text: on

text: the

text: main

text: diagonal

text: and

text: superdiagonal.

text: NMF

text: algorithms

text: by

text: running

text: the

text: algorithms

text: on

text: simple

text: input

text: Using

text: similar

text: logic

text: to

text: the

text: and

text: cases,

text: it

text: can

text: be

text: shown

text: that

text: can

text: produce

text: an

text: exact

text: non-negative

text: factorization,

text: and

text: that

text: these

text: are

text: the

text: only

text: exact

text: solutions.

text: However,

text: in

text: practice,

text: perturbations

text: of

text: such

text: input.

text: In

text: particular,

text: square

text: matrices

text: as

text: the

text: matrices

text: get

text: larger,

text: exceptions

text: to

text: this

text: pattern

text: become

text: with

text: entries

text: on

text: the

text: main

text: diagonal

text: and

text: entries

text: more

text: common,

text: particularly

text: in

text: Algorithm

text: The

text: general

text: rule

text: on

text: the

text: superdiagonal

text: are

text: proposed,

text: because

text: they

text: have

text: exact

text: seems

text: to

text: mostly

text: hold

text: (over

text: half

text: the

text: time)

text: until

text: A

text: becomes

text: solutions

text: that

text: can

text: enumerated

text: mathematically,

text: or

text: because

text: they

text: around

text: Note,

text: however,

text: that

text: because

text: the

text: run-time

text: of

text: are

text: perturbations

text: of

text: matrices

text: with

text: exact

text: solutions.

text: the

text: algorithms

text: are

text: cubic

text: in

text: the

text: size

text: of

text: the

text: matrix,

text: at

text: best,

text: the

text: The

text: test

text: cases

text: have

text: been

text: used

text: as

text: input

text: on

text: three

text: known

text: NMF

text: sample

text: size

text: for

text: large

text: matrices

text: is

text: small.

text: algorithms

text: that

text: represent

text: a

text: variety

text: of

text: algorithms,

text: and

text: all

text: of

text: IV.

text: PROPOSED

text: TESTS

text: FOR

text: NMF

text: ALGORITHMS

text: them

text: behaved

text: similarly,

text: which

text: suggests

text: testable,

text: quantiﬁable

text: Since

text: all

text: three

text: algorithms,

text: which

text: cover

text: a

text: variety

text: of

text: ap-

text: behaviors

text: that

text: many

text: NMF

text: algorithms

text: share.

text: These

text: test

text: cases

text: proaches

text: to

text: NMF,

text: had

text: a

text: lot

text: in

text: common

text: in

text: their

text: solutions,

text: offer

text: one

text: approach

text: for

text: testing

text: candidate

text: NMF

text: implentations

text: it

text: is

text: propose

text: that

text: these

text: inputs

text: A

text: could

text: be

text: used

text: as

text: a

text: test

text: case

text: to

text: help

text: determine

text: whether

text: it

text: behaves

text: as

text: it

text: should.

text: of

text: an

text: NMF

text: algorithm

text: implementation.

text: In

text: this

text: section,

text: it

text: is

text: ACKNOWLEDGMENT

text: proposed

text: how

text: such

text: test

text: cases

text: could

text: be

text: executed.

text: The

text: authors

text: would

text: like

text: to

text: thank

text: Dr.

text: Alan

text: Edelman

text: for

text: The

text: test

text: begins

text: with

text: input

text: of

text: the

text: form

text: providing

text: and

text: overseeing

text: this

text: research

text: opportunity,

text: and

text: Dr.

text: Vijay

text: Gadepally

text: for

text: his

text: advice

text: and

text: expertise.

text: A

text: REFERENCES

text: Berry

text: Browne

text: Langville

text: Pauca

text: and

text: Plemmons,

text: Algorithms

text: and

text: applica-

text: tions

text: for

text: approximate

text: nonnegative

text: matrix

text: factorization,

text: Computational

text: ×A

text: is

text: square,

text: and

text: preferably

text: somewhere

text: between

text: and

text: Statistics

text: and

text: Data

text: Analysis

text: in

text: size,

text: although

text: bigger

text: inputs

text: may

text: be

text: useful

text: as

text: well.

text: Rainer

text: Gemulla,

text: Erik

text: Nijkamp,

text: Peter

text: J.

text: Haas,

text: and

text: Yannis

text: Sismanis,

text: The

text: entries

text: should

text: vary

text: between

text: tests.

text: Each

text: test

text: should

text: start

text: Large-scale

text: matrix

text: factorization

text: with

text: distributed

text: stochastic

text: gradient

text: de-

text: scent,

text: Proceedings

text: of

text: the

text: ACM

text: SIGKDD

text: International

text: Conference

text: by

text: using

text: so

text: that

text: A

text: is

text: diagonal.

text: The

text: results

text: of

text: this

text: on

text: Knowledge

text: Discovery

text: and

text: Data

text: Mining

text: (New

text: York,

text: NY,

text: USA),

text: KDD

text: test

text: should

text: have

text: W

text: and

text: H

text: monomial

text: only

text: one

text: nonzero

text: ACM,

text: pp.

text: element

text: in

text: each

text: row

text: and

text: column.

text: Ignore

text: entries

text: that

text: are

text: below

text: John

text: Gilbert,

text: personal

text: communication,

text: Sep

text: N.

text: Guan,

text: D.

text: Tao,

text: Z.

text: Luo,

text: and

text: B.

text: Yuan,

text: Nenmf:

text: An

text: optimal

text: gradient

text: for

text: the

text: entirety

text: of

text: testing,

text: as

text: any

text: such

text: entries

text: are

text: method

text: for

text: nonnegative

text: matrix

text: factorization,

text: IEEE

text: Transactions

text: on

text: Signal

text: negligible.

text: Processing

text: no.

text: If

text: W

text: or

text: H

text: is

text: not

text: monomial,

text: or

text: if

text: the

text: product

text: WH

text: is

text: Daniel

text: D.

text: Lee

text: and

text: H.

text: Sebastian

text: Seung,

text: Algorithms

text: for

text: non-negative

text: matrix

text: factorization,

text: Advances

text: in

text: Neural

text: Information

text: Processing

text: Systems

text: not

text: equal

text: to

text: A

text: to

text: within

text: a

text: negligible

text: margin

text: of

text: error,

text: the

text: (T.

text: K.

text: Leen,

text: T.

text: G.

text: Dietterich,

text: and

text: V.

text: Tresp,

text: eds.),

text: MIT

text: Press,

text: algorithm

text: fails

text: the

text: test.

text: Otherwise,

text: the

text: generated

text: solution

text: can

text: pp.

text: be

text: used

text: to

text: ﬁnd

text: the

text: permutation

text: matrix

text: P

text: that

text: makes

text: WP

text: and

text: Suvrit

text: Sra

text: and

text: Inderjit

text: S.

text: Dhillon,

text: Generalized

text: nonnegative

text: matrix

text: ap-

text: proximations

text: with

text: bregman

text: divergences,

text: Advances

text: in

text: Neural

text: Information

text: P

text: diagonal

text: by

text: replacing

text: the

text: nonzero

text: entries

text: of

text: H

text: with

text: Processing

text: Systems

text: (Y.

text: Weiss,

text: B.

text: Scho¨lkopf,

text: and

text: J.

text: C.

text: Platt,

text: eds.),

text: Since

text: A

text: WH

text: is

text: diagonal,

text: WP

text: is

text: also

text: diagonal,

text: and

text: MIT

text: Press,

text: pp.

text: since

text: I

text: P

text: is

text: diagonal,

text: so

text: is

text: P

text: Knowing

text: P

text: will

text: Wenwu

text: Wang,

text: Instantaneous

text: vs.

text: convolutive

text: non-negative

text: matrix

text: factor-

text: ization:

text: Models,

text: algorithms

text: and

text: applications,

text: Machine

text: Audition:

text: Prin-

text: make

text: the

text: rest

text: of

text: the

text: testing

text: much

text: simpler

text: since

text: it

text: is

text: easier

text: to

text: ciples,

text: Algorithms

text: and

text: Systems:

text: Principles,

text: Algorithms

text: and

text: Systems

text: identify

text: whether

text: a

text: solution

text: is

text: of

text: the

text: form

text: given

text: above

text: when

text: it

text: is

text: not

text: permuted.

text: Next,

text: run

text: the

text: test

text: again

text: using

text: a

text: positive

text: value

text: for

text: seems

text: to

text: work

text: well,

text: although

text: using

text: a

text: variety

text: of

text: is

text: also

text: recommended.

text: Make

text: sure

text: to

text: use

text: the

text: same

text: random

text: seeds

text: that

text: were

text: used

text: in

text: the

text: test

text: to

text: produce

text: corresponding

text: output.

text: Then

text: check

text: that

text: the

text: W

text: and

text: H

text: given

text: by

text: the

text: algorithm

text: are

text: such

text: that

text: WP

text: and

text: P

text: have

text: nonzero

text: entries

text: only

text: on

text: the

text: two

text: diagonals

text: that

text: they

text: are

text: supposed

text: to.

text: If

text: this

text: doesn’t

text: hold,

text: changing

text: might

text: have

text: changed

text: which

text: permutation

text: returns

text: W

text: and

text: H

text: to

text: the

text: proper

text: form,

text: so

text: check

text: again;

text: this

text: happens

text: more

text: commonly

text: among

text: larger

text: matrices

text: than

text: smaller

text: ones.

text: However,

text: ##LTLine##


text: 12 FIG. 6. Comparison of conduction band, c , of BBLG for bias 1 values of a = 0, 50, 150, and 200 meV. The dotted black line shows the band structure of MLG for comparison. At higher FIG. 8. Normalized conduction electron density distribution bias we see the presence of the ’sombrero’ feature. in k-space for a bias of a=150 meV at (a) the initial thermal conditions, (b) at time t=1.55 ps , (c) at t=1.75 ps, (d) and at t=2.00 ps for a 50 kV/cm incident ﬁeld. White lines indicate the position of the Dirac point. conduction band is driven to the left of the Dirac point by the incident ﬁeld. What is interesting, is that the popula- tion has been ’split’ in two by the sombrero feature; with half the carriers going above the feature, and half below. Fig. 8c shows the distribution at t=1.75 ps , at what is essentially its farthest position to the left of the Dirac point. We can see that as the carriers are driven passed FIG. 7. Normalized intraband current density of BBLG in the sombrero feature, they begin to merge again on the response to the 1 THz pulse with amplitude of 50 kV/cm for far side of it. Finally, in Fig. 8d we see the distribution a number of diﬀerent bias values, a = 0, 50, 150 and 200 meV. Intraband current density normalized to incident ﬁeld ampli- at t=2.00 ps, when the carriers are moving back towards tude is shown.The dotted black line shows intraband current the Dirac point, and are in the process of forming the ring density of MLG under identical conditions. structure once again. The distribution then settles back into thermal equilibrium (Fig. 8a) once the pulse leaves the system. Once again, we can determine the eﬀect this ative current density in MLG at this carrier density is ’splitting’ of the carrier density has on the nonlinear be- approximately 400 µS. It is also quite easy to see the havior by looking at the the normalized time-dependent presence of distortion in the intraband current density, reﬂected ﬁelds for the diﬀerent bias values, shown in Fig. for each of the bias values. The shape of this distortion 9a, along with the spectral responses normalized to the appears to be similar for all biases considered. One might peak amplitude at the fundamental frequency (1 THz), expect the distortions at the higher bias values (150 & shown in Fig. 9b. 200 meV) to be of a diﬀerent nature than those at lower Looking at ﬁrst the reﬂected ﬁeld, we see that it is biases. This expectation is based on the sombrero fea- almost 180 degrees out of phase with the intraband cur- ture being present for large bias values. Certainly the rent density; which is expected since the interband cur- motion of the electrons in the conduction band - which rent is negligible in this case. The maximum amplitude is the basis for our intraband current density - should be occurs at a bias of a=0 meV, i.e when no external bias is aﬀected by this feature. present. The value obtained is approximately 3.5 kV/cm We can visualize the eﬀect the sombrero feature has on for an incident ﬁeld of 50 kV/cm. We can compare this the intraband motion via a plot of the electron density in with the maximum value of 4.0 kV/cm found in MLG the conduction band, as shown in Fig. 8. For this simu- at the same incident ﬁeld amplitude. In terms of the lation we used the bias value of a=150 meV, with which visual distortions in the reﬂected ﬁeld, the sombrero fea- we are clearly able to see the sombrero feature present ture surprisingly had little eﬀect, other than to decrease when the carriers are in equilibrium in Fig. 8a. Here we the overall maximum amplitude. The shape of the distor- see that the density of carriers in the conduction band tions is relatively the same for all bias values (and even appears as a ’Fermi ring’ in k-space. We are interested MLG). in what happens to the Fermi ring during the interaction We can determine the nonlinear behavior by examining with the incident THz ﬁeld. In Fig. 8b we see the density the spectral composition of the reﬂected ﬁelds for these at approx t=1.55 ps, after half of the incident pulse has bias values. This is shown in Fig. 9b. It is immediately passed. In this instant the distribution of carriers in the clear from this plot that the maximum third harmonic, ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: Comparison

label: caption

text: of

text: conduction

text: band,

text: c

text: of

text: BBLG

text: for

text: bias

text: values

text: of

text: a

text: and

text: meV.

text: The

text: dotted

text: black

text: line

text: shows

text: the

text: band

text: structure

text: of

text: MLG

text: for

text: comparison.

text: At

text: higher

text: FIG.

text: Normalized

text: conduction

text: electron

text: density

text: distribution

text: bias

text: we

text: see

text: the

text: presence

text: of

text: the

text: ’sombrero’

text: feature.

text: in

text: k-space

text: for

text: a

text: bias

text: of

text: meV

text: at

text: (a)

text: the

text: initial

text: thermal

text: conditions,

text: (b)

text: at

text: time

text: ps

text: (c)

text: at

text: ps,

text: (d)

text: and

text: at

text: ps

text: for

text: a

text: kV/cm

text: incident

text: ﬁeld.

text: White

text: lines

text: indicate

text: the

text: position

text: of

text: the

text: Dirac

text: point.

text: conduction

text: band

text: is

text: driven

text: to

text: the

text: left

text: of

text: the

text: Dirac

text: point

text: by

text: the

text: incident

text: ﬁeld.

text: What

text: is

text: interesting,

text: is

text: that

text: the

text: popula-

text: tion

text: has

text: been

text: ’split’

text: in

text: two

text: by

text: the

text: sombrero

text: feature;

text: with

text: half

text: the

text: carriers

text: going

text: above

text: the

text: feature,

text: and

text: half

text: below.

text: Fig.

text: shows

text: the

text: distribution

text: at

text: ps

text: at

text: what

text: is

text: essentially

text: its

text: farthest

text: position

text: to

text: the

text: left

text: of

text: the

text: Dirac

text: point.

text: We

text: can

text: see

text: that

text: as

text: the

text: carriers

text: are

text: driven

text: passed

text: FIG.

text: Normalized

text: intraband

text: current

text: density

text: of

text: BBLG

text: in

text: the

text: sombrero

text: feature,

text: they

text: begin

text: to

text: merge

text: again

text: on

text: the

text: response

text: to

text: the

text: THz

text: pulse

text: with

text: amplitude

text: of

text: kV/cm

text: for

text: far

text: side

text: of

text: it.

text: Finally,

text: in

text: Fig.

text: we

text: see

text: the

text: distribution

text: a

text: number

text: of

text: diﬀerent

text: bias

text: values,

text: a

text: and

text: meV.

text: Intraband

text: current

text: density

text: normalized

text: to

text: incident

text: ﬁeld

text: ampli-

text: at

text: ps,

text: when

text: the

text: carriers

text: are

text: moving

text: back

text: towards

text: tude

text: is

text: shown.The

text: dotted

text: black

text: line

text: shows

text: intraband

text: current

text: the

text: Dirac

text: point,

text: and

text: are

text: in

text: the

text: process

text: of

text: forming

text: the

text: ring

text: density

text: of

text: MLG

text: under

text: identical

text: conditions.

text: structure

text: once

text: again.

text: The

text: distribution

text: then

text: settles

text: back

text: into

text: thermal

text: equilibrium

text: (Fig.

text: once

text: the

text: pulse

text: leaves

text: the

text: system.

text: Once

text: again,

text: we

text: can

text: determine

text: the

text: eﬀect

text: this

text: ative

text: current

text: density

text: in

text: MLG

text: at

text: this

text: carrier

text: density

text: is

text: ’splitting’

text: of

text: the

text: carrier

text: density

text: has

text: on

text: the

text: nonlinear

text: be-

text: approximately

text: µS.

text: It

text: is

text: also

text: quite

text: easy

text: to

text: see

text: the

text: havior

text: by

text: looking

text: at

text: the

text: the

text: normalized

text: time-dependent

text: presence

text: of

text: distortion

text: in

text: the

text: intraband

text: current

text: density,

text: reﬂected

text: ﬁelds

text: for

text: the

text: diﬀerent

text: bias

text: values,

text: shown

text: in

text: Fig.

text: for

text: each

text: of

text: the

text: bias

text: values.

text: The

text: shape

text: of

text: this

text: distortion

text: along

text: with

text: the

text: spectral

text: responses

text: normalized

text: to

text: the

text: appears

text: to

text: be

text: similar

text: for

text: all

text: biases

text: considered.

text: One

text: might

text: peak

text: amplitude

text: at

text: the

text: fundamental

text: frequency

text: THz),

text: expect

text: the

text: distortions

text: at

text: the

text: higher

text: bias

text: values

text: shown

text: in

text: Fig.

text: meV)

text: to

text: be

text: of

text: a

text: diﬀerent

text: nature

text: than

text: those

text: at

text: lower

text: Looking

text: at

text: ﬁrst

text: the

text: reﬂected

text: ﬁeld,

text: we

text: see

text: that

text: it

text: is

text: biases.

text: This

text: expectation

text: is

text: based

text: on

text: the

text: sombrero

text: fea-

text: almost

text: degrees

text: out

text: of

text: phase

text: with

text: the

text: intraband

text: cur-

text: ture

text: being

text: present

text: for

text: large

text: bias

text: values.

text: Certainly

text: the

text: rent

text: density;

text: which

text: is

text: expected

text: since

text: the

text: interband

text: cur-

text: motion

text: of

text: the

text: electrons

text: in

text: the

text: conduction

text: band

text: which

text: rent

text: is

text: negligible

text: in

text: this

text: case.

text: The

text: maximum

text: amplitude

text: is

text: the

text: basis

text: for

text: our

text: intraband

text: current

text: density

text: should

text: be

text: occurs

text: at

text: a

text: bias

text: of

text: meV,

text: i.e

text: when

text: no

text: external

text: bias

text: is

text: aﬀected

text: by

text: this

text: feature.

text: present.

text: The

text: value

text: obtained

text: is

text: approximately

text: kV/cm

text: We

text: can

text: visualize

text: the

text: eﬀect

text: the

text: sombrero

text: feature

text: has

text: on

text: for

text: an

text: incident

text: ﬁeld

text: of

text: kV/cm.

text: We

text: can

text: compare

text: this

text: the

text: intraband

text: motion

text: via

text: a

text: plot

text: of

text: the

text: electron

text: density

text: in

text: with

text: the

text: maximum

text: value

text: of

text: kV/cm

text: found

text: in

text: MLG

text: the

text: conduction

text: band,

text: as

text: shown

text: in

text: Fig.

text: For

text: this

text: simu-

text: at

text: the

text: same

text: incident

text: ﬁeld

text: amplitude.

text: In

text: terms

text: of

text: the

text: lation

text: we

text: used

text: the

text: bias

text: value

text: of

text: meV,

text: with

text: which

text: visual

text: distortions

text: in

text: the

text: reﬂected

text: ﬁeld,

text: the

text: sombrero

text: fea-

text: we

text: are

text: clearly

text: able

text: to

text: see

text: the

text: sombrero

text: feature

text: present

text: ture

text: surprisingly

text: had

text: little

text: eﬀect,

text: other

text: than

text: to

text: decrease

text: when

text: the

text: carriers

text: are

text: in

text: equilibrium

text: in

text: Fig.

text: Here

text: we

text: the

text: overall

text: maximum

text: amplitude.

text: The

text: shape

text: of

text: the

text: distor-

text: see

text: that

text: the

text: density

text: of

text: carriers

text: in

text: the

text: conduction

text: band

text: tions

text: is

text: relatively

text: the

text: same

text: for

text: all

text: bias

text: values

text: (and

text: even

text: appears

text: as

text: a

text: ’Fermi

text: ring’

text: in

text: k-space.

text: We

text: are

text: interested

text: MLG).

text: in

text: what

text: happens

text: to

text: the

text: Fermi

text: ring

text: during

text: the

text: interaction

text: We

text: can

text: determine

text: the

text: nonlinear

text: behavior

text: by

text: examining

text: with

text: the

text: incident

text: THz

text: ﬁeld.

text: In

text: Fig.

text: we

text: see

text: the

text: density

text: the

text: spectral

text: composition

text: of

text: the

text: reﬂected

text: ﬁelds

text: for

text: these

text: at

text: approx

text: ps,

text: after

text: half

text: of

text: the

text: incident

text: pulse

text: has

text: bias

text: values.

text: This

text: is

text: shown

text: in

text: Fig.

text: It

text: is

text: immediately

text: passed.

text: In

text: this

text: instant

text: the

text: distribution

text: of

text: carriers

text: in

text: the

text: clear

text: from

text: this

text: plot

text: that

text: the

text: maximum

text: third

text: harmonic,

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##