Using finite-temperature diagonalization we investigate the optical conductivity $\sigma(\omega)$ and spin-orbital dynamics in the CE phase of half-doped manganites. We find $\sigma(\omega)$ characterized by a broad spectrum with pronounced optical gap due

accountsforthecooperativeJT-interactionandde-scribestheorbital-orbital(OO)interactionbetweenfur-therMn3+neighborsmediatedbytheMn4+ionin-between[5][Fig.1(c)].Hence, jij′boringMn3+–Mn4+–Mn3+sitesalong denotesaorthreebdirectionneigh-whilethetwo-siteorbitaloperator,Tjj′,inthebasishastheformTjj′=TjzTjz′+3TjxTjx√{|x ,|z }′

2

σ(nixσ nizσ)

andTix=

1

Nω

Re

∞

dte

iωt

jx(t)jx ,(3)

wherethex-componentoftheie ξζ ij ξζ,σtijδijx currentd iξσd jζσ H.c. operatorisjx=andδij

xdenotesthex-componentofthevectorconnectingsitesiandj.WeevaluatethisKuboformulausingtheexactdiagonaliza-tiontechniquefor nitetemperaturesdevelopedbyJakliˇcandPrelovˇsek[21]forcupratesandlaterusedalsoformanganites[5,18].Inordertomaketheproblemnumer-icallytractable,wesimulatedthecorespinsbySc=1/2spinsandscaleJHtopreservethesplittingbetweenhighandlowspinstates.Thedatareportedherewascalcu-latedforJH=15t.Althoughweexploitedthetransla-tionalsymmetrytosavestorageandcomputationcalculationsweretoasmall√

time,8clusterasonlySz

tot= restrictediSzisubspaces(Si=Sic+si)canbetreatedseparately,whileasimilarsymmetryintheor-bitalsectorisabsent.

ThesubtleroleofHOOistoenhanceCEtypeor-bitalpolarizationtherebysupportingtheAFdecouplingofFM-chainsatlowT(Fig.1a),whileforsmallκ′DEdominatesinalldirectionsleadingtoaFMchargeandorbitalorderedstate[5].InFig.2weshowthefre-quencyandtemperaturedependenceofσ(ω)emergingfromgroundstateswithAF-CE[Fig.2(a)]andFM-CE[Fig.2(b)]spin-orbitalorder,respectively.Acharacteris-ticfeatureofσ(ω)isabroadabsorptionmaximumfoundfor2t ω 5twhichshiftstowardslowerfrequenciesby ω 0.5tastemperatureincreasestoT 0.2t.Furtherincreaseoftemperatureleadstothemeltingoftheorbitalorderandtheclosingofthechargegap(ω 2t)withfurthershiftofthehighfrequencybroadstructure.Theshapeofσ(ω)anditstemperatureevo-lutionissimilartotheexperimentaldata,e.g.,forthecubicLa0.5Ca0.5MnO3[13]andlayeredLa1/2Sr3/2MnO4[11]compound.Assumingt=0.4eV[18,20]one nds

2

FIG.1:(Coloronline)SketchoftheAF-CEspin,orbital,andlatticeorder.In(a)large(small)spinsindicatesiteswith(without)anegelectron,respectively.ShadingindicatesdirectionofDEcarrierpropagation.In(b)fullsymbolsrep-resentoccupiedegorbitalsandin(c)fullcirclesindicatetheJT-shiftedOions,andthearrowthedisplacementofMn4+.

0.12

(a)

0.08)

ω0.080.04

(σ0ρ0.041.01.5

0.00

0.12

)

ω0.08(σρ0

0.040.00

ω/t

FIG.2:(Coloronline)Temperaturedependenceoftheopticalconductivityσ(ω)obtainedinthephasewith:(a)AF-CEorderinthegroundstate,and(b)aFM-CEgroundstateforJAF=0.04t.Otherparameters:(a)V=0.2t,κ′=0.15t;(b)V=t,κ′=0.05t.ThespectraaregeneratedusingaLorentzianbroadeningΓ=0.2t.Inset:Mid-gapabsorptionbroadenedwithΓ=0.05t.

thepositionofabroadpeakstructureinFig.2centeredclosetotheexperimentalvalue1.2eVmeasuredinsingle-layerLa1/2Sr3/2MnO4[11].Asubtletestistheabsolutescaleofσ(ω)whichiscontrolledbythesumrulethatlinksσ(ω)andthekineticenergyofthemodel[18,22].

Forthemaximumoftheabsorption(σmax≈0.1ρ 1

ndwithρ= a/e2[18]andthelatticeconstant0)we0a～5.5 A,σmax≈500( cm) 1.Observedvaluesforσmaxare～1000( cm) 1inLa1/2Sr3/2MnO4[11,12]and～700( cm) 1incubicLa0.5Ca0.5MnO3[13].

ThemainabsorptioninFig.2describesexcitationsacrossthechargegap( corbitally 3(Vordered+2κ′)incaseofper-fectCO)whichinthestateinvolveegelectrontransfermainlybetweenbridge(groundstate)andcorner(excitedstate)siteswithinthesamezig-zagchain.Tounderstandtheroleofhoppingtitisusefultoconsideranembeddedthree-sitemodelwithanegelec-tronbeingeitheratthebridge(|B )oratoneofthecorner(|CLtionatωc ,|CR )sites.One ndstheopticaltransi-non-bonding ( c+2t2/ c,betweenbonding～||CL |CR )/

√B and