Polar Kerr effect from a time-reversal symmetry breaking unidirectional charge density wave M. Gradhand1, I. Eremin2 and J. Knolle3 1H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom 2Institut fu¨r Theoretische Physik III, Ruhr-Universit¨at Bochum, Universita¨tsstrasse 150, DE-44801 Bochum, Germany and 3T.C.M. Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom (Dated: January 5, 2015) 5 1 We analyze the Hall conductivity σxy(ω) of a charge ordered state with momentum Q=(0,2Q) 0 and calculate the intrinsic contribution to the Kerr angle ΘK using the fully reconstructed tight- 2 binding band structure for layered cuprates beyond the low energy hot spots model and particle holesymmetry. Weshowthatsuchaunidirectionalchargedensitywave(CDW),whichbreakstime n reversal symmetry as recently put forward by Wang and Chubukov [Phys. Rev. B 90, 035149 a J (2014)], leads to a nonzero polar Kerr effect as observed experimentally. In addition, we model a fluctuating CDW via a large quasiparticle damping of the order of the CDW gap and discuss pos- 2 sibleimplications for thepseudogap phase. Wecan qualitatively reproducepreviousmeasurements of underdoped cuprates but making quantitative connections to experiments is hampered by the ] n sensitivity of thepolar Kerreffect with respect to thecomplex refractive index n(ω). o c PACSnumbers: 74.70.Xa,75.10.Lp,75.30.Fv - r p u Introduction. One of the most controversial topics in superconductivity, there is another instability in the d- s the field of high-temperature superconductivity is the wave charge channel, associated with the wavevector t. origin of the so-called pseudogap phenomenon observed Qd = 2khs24. Here, khs is the momentum of one of a by various experimental techniques in the underdoped the hot spots such that 2k = (2Q,2Q) points along m hs cuprates1–3 at temperatures T∗ being larger than the one of the Brillouin Zone diagonals. Efetov, Meier, and - superconducting transition temperature, T . A large Pepin25,26 argued that the pseudogap can be the conse- d c n number of theoretical scenarios have been initially pro- quence of the competition between d-wave charge order o posed to explain the origin of the pseudogap.4 Recent (bondorder)andsuperconductivity(SC).InthiscaseT∗ c experimental studies of hole-doped cuprates, however, referstothe formationofthe combinedSC/CDWsuper- [ have indicated that the pseudogap region of the high-T vectororderparameter,butits directiongetsfixedalong c 1 cupratesisastate,whichcompeteswithsuperconductiv- the SC axis only at lower temperature TC. Sachdev and v ity, and also breaks several symmetries.5–12 In particu- collaborators27–29 proposed that magnetically-mediated 1 lar, x-ray, and neutron scattering experiments as well as interactioncanalsoyieldanattractionintheCDWchan- 3 Scanning Tunneling Microscopy (STM) have found the nel for unidirectional incoming momenta Q = (0,2Q) x 4 breaking of the lattice symmetry from C down to C or Q = (2Q,0), which has also been found in three 0 4 2 y belowthepseudogaptemperature,T∗,inseveralcuprate band models30,31. In these proposals the ground states 0 compounds5–7,10. Inaddition,recentfindingsofthepolar break C rotational symmetry of the lattice. Most re- . 4 1 Kerreffect9,11,12,andtheintra-unitcellmagneticorder13 cently Wang and Chubukov32 have shown that for large 0 pointtowardsbreakingoftime-reversalormirrorsymme- magnetic correlationlength there could be an additional 5 triesintheunderdopedcuprates. Furthermore,directin- CDW instability with both realandimaginaryorder pa- 1 : dications in favour of the static incommensurate charge- rameter at a nonzero temperature breakingalso time re- v density-wave (CDW) order with momenta Q = (2Q,0) versal symmetry. Translated into real space the latter x Xi orQy =(0,2Q)werefoundbyX-raymeasurements14–17, order has both bond and site charge density and bond nuclear magnetic resonance technique18,19 and in exper- current modulations. r a iments on the sound velocity in a magnetic field20. In In this Communication, we follow Ref. 32 and analyze these systems 2Q refers to the distance between the so- the Hall conductivity which arises from such a unidirec- called hot spots on the Fermi surface, i.e. points where tionaltime-reversalsymmetrybreakingCDWphase. We the magnetic Brillouine Zones intersects the Fermi sur- do not attempt to answer the question whether or not face (FS). It was also argued that the charge order has predominantly d-waveform factor10 and that the forma- this state truly minimizes the free energy of a partic- ular model but concentrate on the experimental conse- tion of Fermi surface arcs coincides with the formation of CDW order21. quences once the state is formed. Our main objective is to study the intrinsic contributions of the full lattice Possible explanations of the charge order and related band structure to the polar Kerr effect, which is well es- othersymmetrybreakinginclude loop-currentorder22 or tablished experimentally in the pseudogap phase9,11,12. d-density-wave (current) order23. Within the so-called Its microscopic origin is still poorly understood and un- spin fluctuation scenario, which supports also d−wave der current debate. Two original proposals33,34 to ex- 2 i) ii) iii) π π π the hot spots. Theory. Our starting point is the CDW mean-field Hamiltonian 0 0 0 Hˆ = ε c† c +∆Qc† c +h.c. (1) k kσ kσ k kσ k+2Q Xk,σ n o -π -π -π -π 0 π -π 0 π -π 0 π with ∆Q = ∆ cos(k −k )cosQ − k 1 x 0 y i∆ sin(k −k )sinQ and2Q = π andk =0.9π. We FIG.1. ThespectralfunctionI(k,ω=0)isshowninthefull 2 x 0 y y 5 0 use the parameters of the tight-binding energy disper- Brillouin zone for three different cases (from left to right): i) normal state, ∆1 =∆2 =0 and small quasiparticle damping sion from Ref.9 to obtain a commensurate CDW. Note Γ=10 meV; ii) fluctuating CDW state, ∆1 =∆2 =50 meV that the above order parameter satisfies the following ∗ modelled by a large damping Γ = 50 meV; iii) long range symmetry relations ∆Q = ∆−Q and transforms as ordered CDW state, ∆1 =∆2 =50 meV (Γ=10 meV). ∗ (cid:16) k(cid:17) k ∆−Q = ∆Q under time reversal symmetry. The −k −k (cid:16) (cid:17) centre of mass momentum of the CDW is at k such 0 that k = (±k ,±Q ). This stripe-type order breaks hs 0 y 0.00 both the rotational and the time-reversal symmetries. d)] -2.00 RIme[[σσxy((ωω))]] S10in×ce10∆Qkm+a2tQrix6=wi∆thQkaliltmiusltnipecleesssNar2yQtoudpiatgoonNali=ze1a0 2-e/(h 2.0×10-2 xy that closes the periodic Brillouin zoney(BZ). Since the 3 -10 -4.00 1.5×10-2 CDW order parameter has two components there is the σ [xy -6.00 51..00××1100--32 binrteearkeisntigngonployssriboitlaittyio3n2atlhaatnadctoimmpeosrietveeIrssianlgslyikmemoredtreyr 0.0 forms above T at which long range translational -8.00 -5.0×10-3 0.6 0.8 order (LRO) gCetDsWbroken. In the following we do not 0.00 0.10 0.20 0.30 0.40 0.50 explicitly calculate such a fluctuating CDW but we -hω [eV] mimick the effect by a largequasiparticledamping,Γ, of the order of the CDW gap itself, ∆≈Γ. 0.50 Wefirstinvestigatethedirectfeedbackofchargeorder RIme[[σσxy((ωω))]] on the electronic structure which have been shownto be xy intimately connected21. Angle resolved photo emmision 0.00 d)] spectroscopy (ARPES) measures directly the electronic -32-σ [10e/(hxy -0.50 5112....0050××××11110000----3222 sbpeeccatrlcaullfautnedctfioronmI(tkhe,ωb)ar=e GImreheGnˆs0(fkun,ωct)−iio11n,1: which can -1.00 0.0 Gˆ0(k,ω)= ω+iΓ−Hˆ(k) . (2) -5.0×10-3 (cid:16) (cid:17) -1.0×10-2 0.6 0.8 -1.50 We have omitted the spin label, Hˆ(k) is the 10 × 10 0.00 0.10 0.20 0.30 0.40 0.50 0.60 HamiltonianmatrixandΓisourphenomenologicalquasi- -hω [eV] particle damping. The results are shown in Fig. 1 for three different sets of parameters: (a) normal state FS, FIG. 2. The real and imaginary part of theoptical Hall con- (b) the fluctuating CDW state modeled by ∆1 = ∆2 = ductivityasafunctionoffrequency. Theimpuritybroadening 50 meV and a large damping Γ = 50 meV, (c) the LRO isΓ=10meVandΓ=50meVintheupperandlowerpanel, CDW state with the same set of ∆ but with a smaller i respectively. Theinsetsshowtherelevantfrequencies(0.8eV) damping Γ=10 meV. at which experimentswere performed so far9,11,12,39. The formation of the CDW gaps out states at the hot spots k which together with the suppressed spectral hs weightoutside the AFM BZ leads to the observedFermi plain the nonzero Kerr signal based on inversion sym- arcstructureofthespectralfunction. Inaccordancewith metry breaking were retracted35,36 and in a number of ARPES experiments16 the wave vector connecting the articles the necessity of time reversal symmetry break- tips of the arc is larger than the wave vector connecting ing was emphasized.35–38 Note that our present study is the hot spots of the normal state FS. The effect appears complementary to a very recent work37 which, however, naturallyinourmean-fielddescriptioninwhichafullre- analyzed only the extrinsic contributions to the Polar gion around the hot spots is gapped out by the CDW Kerr effect in a similar setting but restricted to an ef- order. However, we have chosen the momentum of our fectivelowenergydescriptionkeepingonlystatesaround CDW to match the vector connecting two hot spots32, 3 which could likely turn out to be different when investi- siccontribution37,41–44,wherethelattercanbeexpressed gating the origin of the ordered state beyond weak cou- entirelyintermsofthe electronicstructureoftheunper- pling. As this question is beyond the scope of our study turbed crystal.45 In contrast, the extrinsic contribution we concentrate on the experimental consequences of the ismediatedviaimpurityscattering.37,43,46–48 Fundamen- formedCDWphase. Ourfollowingresultsdonotdepend tally,the systemneedsto breakmirrorandtime reversal sensitively on the precise value of the ordering vector. symmetry to exhibit a finite intrinsic or extrinsic Hall NextwestudythepolarKerreffectwhichisquantified conductivity.37,49 Furthermore, in order to create an in- by the Kerr angle θ . It can be expressed in terms of trinsic contribution multiple bands have to be consid- K the Hall conductivity σ (ω) and the complex refractive ered. In a single band system the effect would vanish xy index n(ω) as40 by definition.50,51 In the present case of a charge den- sity wave, which breaks time reversal and C rotational 4 1 σxy(ω) symmetry, the gap parameter is providing all conditions θ = Im , (3) K ǫ ω (cid:20)n(ω)(n2(ω)−1)(cid:21) intrinsically and creates an effective multi-band system 0 in the reduced Brillouin zone. where for a complex refractive index both the real and imaginarypartoftheHallconductivitycontribute. Gen- Ingeneral,the antisymmetric partofthe Hallconduc- erally, the Hall conductivity has an extrinsic and intrin- tivity can be expressed as51 e2~ σxy(ω)= Vm2 f(En(k)[1−f(En′(k))]× nX,n′,k Im u†n′(k)Hkxun(k)u†n(k)Hkyun′(k)−un′(k)Hkyun(k)u†n(k)Hkxun′(k) × h i , (4) (En′(k)−En(k))2−(~ω+iΓ)2 where u†(k) = (u1(k),u2(k),...,u10(k)) are the eigen- operator Hx,y is the k derivative with respect to k or n n n n k x functionsofthekdependentHamiltonmatrixHˆ(k). The ky, respectively. For the tight-binding model under con- siderationthe importantmatrix elements canbe written as u1 (k) † ∇ ǫ ∇ ∆Q ... u1(k) n′ kx,y k kx,y k n hn′k|Hkx,y|nki= u2n′...(k)   ∇kx,y(...∆Qk)∗ ∇kx,y...ǫk−2Q ......  u2n...(k) , (5)      where the main assumption is that all matrix elements bles the Berry curvature and is connected via Kramers of the position operatorvanish.51 With that at hand the Kronig transformation to the imaginary part both have Hall conductivity can be evaluated and the results are to vanish54,55. The argument can be extended to finite shown in Fig. 2 for different broadening accounting for frequencies as well. Only for a complex order parameter random impurity scattering. theconsideredmodelwillexhibitafiniteBerrycurvature For the considered model Hamiltonian, the antisym- i.e. afinite opticalHallconductivity. Inthefollowingwe metricpartoftheHallconductivitywillvanishincaseof will discuss the numerical results for the time reversal arealorderparameternotbreakingtimereversalsymme- symmetry breaking order parameter in detail. try. Hence,thebreakingofC symmetryisnotsufficient Results. Fig.2 summarizesthe the realandimaginary 4 to induce a Kerr signal. The reason is straightforward partofthe antisymmetricHallconductivity(Eq.(4)) for andfollowsfromargumentsusedfortheexistenceornon- the considered gap function and two distinct impurity existence of the Berry curvature of a generic parameter- scattering amplitudes. The signal is already nonzero at dependent Hamiltonian. In case of a real Hamiltonian a frequency of the order parameter ∆ = ∆ = 5 meV 1 2 the wave functions can be chosen to be real, leading to which is distinct from the results for Sr RuO .50,51 For 2 4 a vanishing Berry curvature for all k points, except for thelatter,thesignalappearedattheinterorbitalcoupling degeneracies in the electronic bands.52,53 Since the real strength. Nevertheless,bothresultsarenotincontradic- partof the optical conductivity atzero frequency resem- tion to each other but in the present case the finding is 4 0.0 V) 1 e -1.0 8 0. = 0.8 ω -2.0 ωσ=0.8eV)/(0,xy 00..46 (∆(TRIm)e/∆[[σσ(0xxyy))((*ωω*))2]] θµ (x) [ rad)] ---543...000 ΓΓ==00..0051eeVV T, 0.2 σ(xy -6.0 0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 -7.0 T/T 0.00 0.05 0.10 0.15 0.20 CDW x 50.0 FIG.3. Thenormalizedrealandimaginarypartoftheoptical 45.0 Hallconductivityasafunctionoftemperature(Γ=10meV). 40.0 The temperature dependence of the gap is modeled mean- field-likeusing thefunction ∆(T)=p1−(T/TCDW)3. d)] 3305..00 ΓΓ==00..0051eeVV a µ r 25.0 merely an artefact of the fact that the multi-band struc- θ(x) [ 20.0 tureandthesymmetrybreakingarebothinducedbythe 15.0 gap function. Furthermore, the signal is two orders of 10.0 magnitude largerthan for the realistic band structure of 5.0 Sr2RuO4.51Itresultsfromtheorderparameterbeingone 0.0 order of magnitude larger since it enters the conductiv- 0.00 0.05 0.10 0.15 0.20 ity quadratically.50 In the inset of Fig. 2 we present the x frequency region where the Kerr effect experiments on Sr RuO andthecuprateswereperformed.9,11,12,39Here, FIG.4. TheKerrangleasafunctionofdopingx. Thedoping 2 4 ismodelledwithalineardependenceofthegaponthedoping the signal drops by three orders of magnitude and the as ∆(x) = ∆(x = 0)∗(1−x/0.18)56. For the upper panel imaginarypartalmostvanishes. IncontrasttoSr RuO , 2 4 weassumed therefractive indextobepurereal (n=1.692) in in the cuprates it might be feasible to perform corre- accordance to Ref.56. In the lower panel we assumed a com- sponding experiments at lower frequencies due to the plexrefractiveindex(takenfromRef.57)as(n=1.692-i*0.403), significantly higher energy scales (e.g. critical temper- leadingtoadominanceoftherealpartoftheopticalHallcon- atures) and the much larger predicted signal. Extending ductivity. Note the change of sign between the two different the measurements to a larger frequency range would al- assumptions. lowformorequantitativecomparisonbetweentheoryand experiments. To model the temperature dependence of the optical fractive index57 as n = 1.692−i∗0.403 (lower panel of conductivity we perform additional calculations assum- Fig. 4) leads to the dominance of the real part of the ing that the gap function follows a mean-field like be- optical conductivity in the Kerr angle. The significantly haviour as ∆(T) = 1−(T/T )3. The result is larger Kerr angle is furthermore almost independent of CDW shown in Fig. (3), forpthe optical frequency of 0.8 eV. theimpuritybroadening. Nevertheless,theobservedsign The temperature dependence of the conductivity follows change of the Kerr angle between the two approxima- the square of the order parameter as expected in such a tionshighlightstheimportanceofanaccurateknowledge simple model case.50 of the frequency dependent complex refractive index for Finally, we performed doping dependent calculations qualitative but especially quantitative predictions. Fur- fortheopticalKerreffectexploitingtheresultsforEq.(4) thermore, we would like to point out that in general the in combination with Eq. (3). We follow earlier work56 Hall conductivity is a sum of intrinsic and extrinsic con- andmodelthedopingdependencewithaphenomenolog- tributions. Ourcalculationsrelyontheintrinsiceffectin ical function ∆ (x) = ∆ (x = 0)∗(1−x/0.18). For contrast to Ref. 37 which focused on the extrinsic ones. 1/2 1/2 therefractiveindexweconsidertwoseparateapproxima- Although, any conclusion about the dominance of either tions. Inthefirstcase(upperpanelofFig.4)weassumed onecontributionisambiguous(especiallyatelevatedfre- it to be purely real as n=1.692 according to Ref.56. In quencies) the intrinsic effect will be dominant for rather that case the imaginary part of the optical conductiv- impure samplesbecauseofthe impurity scalingbehavior ity is dominating the signal, which is not only small but of the extrinsic mechanism (skew scattering).44,58 stronglydependsontheconsideredimpuritybroadening. Despite the mentioned obstacles preventing a deter- Ontheotherhand,assumingamoregeneralcomplexre- mination of quantitatively reliable results our calculated 5 Hall angle is several orders of magnitude larger than doped cuprates which allows us to numerically calculate in the superconducting state of Sr RuO which is in the full frequency range of the optical conductivity. Our 2 4 qualitative agreement to the experimental findings of results show that for a reliable quantitative comparison Ref. 9, 11, and 12. Most strikingly, our results in the ofexperimentaldataanddifferenttheoreticalscenariosa ordered phase (Γ = 10 meV) with a purely real refrac- precise knowledge of the (in general complex) refractive tive index nicely reproduce the correct order of mag- index is called for. Together with further measurements nitude of the polar Kerr angle in all three cuprates at lower frequencies, most desirable at the order of the YBa Cu O 12, Pb Bi Sr La CuO 9, and spectralgapitself,ithasthepotentialtoturnPolarKerr 2 3 6+x 0.55 1.5 1.6 0.4 6+δ La Ba CuO 11 which are all of the order of θ ≈ effect measurements into a more powerful tool for eluci- 1.875 0.125 4 K 1µrad. dating the complicated many-body state of underdoped Conclusion. Insummary,wehaveshownthatthe uni- cuprates. directional CDW order parameter recently proposed by WangandChubukov32 to accountfortime reversalsym- We thank A. Chubukov, K.B. Efetov and P.A. Volkov metry breaking in the pseudogap phase, is capable to forhelpfuldiscussions. M.G.acknowledgesfinancialsup- explain the polar Kerr effect measurements semi quanti- portfromthe LeverhulmeTrustvia anEarlyCareerRe- tatively. We haveexplicitly evaluatedthe antisymmetric search Fellowship (ECF-2013-538). The work of J.K. is part of the conductivity tensor for a model band struc- supported by a fellowship within the Postdoc-Program ture describing the generic Fermi surface of the under- of the German Academic Exchange Service (DAAD). 1 H. Alloul, T. Ohno, and P. Mendels, Phys. Rev.Lett. 63, 10 K. Fujita, M. H. Hamidian, S. D. Edkins, C. K. Kim, 1700 (1989). Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, H. 2 T. Timusk and B. Statt, Rep.Prog. Phys.62, 61 (1999). Eisaki, S.-i. Uchida, A. Allais, M. J. Lawler, E.-A. 3 A. Kaminski, T. Kondo, T. Takeuchi, and Genda Gu, Kim, S. Sachdev, and J. C. Seamus Davis, PNAS, Philosophical Magazine 1, (2014). doi:10.1073/pnas.1406297111. 4 D. A. Ivanov, P. A. Lee, and X.-G. Wen, Phys. Rev. 11 H. Karapetyan, M. Hu¨cker, G. D. Gu, J. M. 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