Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Duin, C.N.A. van

Zaanen, J.

Leiden University Scholary Publications

VOLUME 80, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 16 FEBRUARY 1998ndson-earreonsnceCharge- Versus Spin-Driven Stripe Order: Role of Transversal Spin FluctuationsC. N. A. van Duin and J. ZaanenInstitute Lorentz for Theoretical Physics, Leiden University, P.O.B. 9506, 2300 RA Leiden, The Netherla(Received 18 July 1997)The separation of the charge- and spin-ordering temperatures of the stripe phase in cuprate supercductors has been used to argue that the striped phase is charge driven. Scaling analysis of a nonlinsigma model shows that the effect of spatial anisotropy on the transversal spin fluctuations is much modrastic at finite temperatures than at zero temperature. These results suggest that the spin fluctuatiprohibit the spin system to condense at the charge-ordering temperature, despite a possible dominaof charge-spin coupling in the longitudinal channel. [S0031-9007(97)05247-2]PACS numbers: 74.20.Mn, 71.27.+a, 74.72.–heby age,eddelf 2mng-tesesszedhe.edizedThe observation of a novel type of electronic ordercuprate superconductors and other doped antiferromaghas attracted considerable attention recently. In this strphase, the carriers are confined to lines which are atsame time Ising domain walls in the Néel backgroun[1]. Substantial evidence exists that dynamical stricorrelations persist in the normal- and superconductstates of the cuprates [2].A further characterization of the fluctuation modesthe stripe phase is needed. In this regard, the finite teperature evolution of the static stripe phase might offeclue. Both in cuprates [1] and in nickelates [3], the charorders at a higher temperature than the spin, and btransitions appear to be of second order. Zachar, Emeand Kivelson [4] argue on the basis of a Landau frenergy that the stripe instability ischarge driven:if thecoupling between the charge and longitudinal spin mowould dominate, charge and spin would order simultanously in a first order transition. This is a mean-field analsis, and fluctuations can change the picture drasticaFor instance, at length scales larger than the interstrdistance the spin system remaining after the chargeordered is just a quantum Heisenberg antiferromagne2 1 1 dimensions which cannot order at finite tempertures according to the Mermin-Wagner theorem. Zachet al. argue that the orientational (“transversal”) fluctuations of the spin system can be neglected at the tempetures of interest, because it appears that the spin sysleft behind after the charge has ordered is not racally different from the antiferromagnet in the half-filledcuprates, exhibiting a Néel temperature of order 300an order of magnitude larger than that in the stripe phaAn important constraint is that theT ­ 0 staggeredmagnetization in the stripe phase appears to be comrable to that at half filling [2]. If the transversal fluctuations are responsible for the charge and spin transitioit has to be demonstrated that the additional thermfluctuations due to the presence of stripes have a mgreater effect on the Néel state than theT ­ 0 quan-tum fluctuations. To investigate this, we consider thsimplest possible source of stripe induced spin disord0031-9007y98y80(7)y1513(4)$15.00innetsipethedpeingofm-r ageothry,eedee-y-lly.ipehast ina-ar-ra-temdi-K,se.pa--ns,alucheer.Following Castro Neto and Hone (CH) [5] we assumthat the exchange coupling between spins separatedcharge stripe is weaker than the interdomain exchanso that the collective spin fluctuations are describby a spatially anisotropicOs3d quantum nonlinear sigma(AQNLS) model. From our scaling analysis we finthat a moderate anisotropy (a factor of,4 difference inspin wave velocities) can explain a reduction of the Nétemperature by an order of magnitude, while theT ­ 0staggered magnetization is reduced only by a factor ofrom its isotropic value. The reason can be inferred frothe crossover diagram (Fig. 1). As a function of increasianisotropy, theT ­ 0 transition between the renormalized classical (RC) and quantum disordered (QD) stascales to smaller coupling constant, but the dimensionltemperature associated with the crossover renormaliFIG. 1. Crossover diagram for the anisotropic QNLS. Tlines are fora ­ 1, 0.4, 0.1, and 0.025 from top to bottomThe end points of the quantum-critical to quantum-disorderlines map ontosg1, t1d ­ s8p , 2pd. Notice that whent0 be-comes larger than the crossover temperature from renormalclassical to quantum critical atg0 ­ 0 one dimensional fluctu-ations are dominating for all values ofg0.© 1998 The American Physical Society 1513VOLUME 80, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 16 FEBRUARY 1998eersgrelclassical to quantum critical scales down much fasteAlternatively, we find that the behavior found byChakravarty, Nelson, and Halperin (CNH) [6] for thecorrelation length in the RC regime of the isotropic modecan be directly generalized to the anisotropic case: the epression for the classicalnisotropicmodel remains validwhen the bare stiffness is replaced by the renormalizstiffness. It is suspected that this holds more generalIf so, the strong disordering influence of temperaturecompared to the quantum fluctuation might be generwhatever the disordering influence of the stripes is, it eerts it in an effectively three dimensional classical systeat zero temperature and in a two dimensional systemthe finite temperature renormalized classical regime.It is assumed that the Néel order parameter fluctuatioin the charge ordered stripe phase are governed byAQNLS model [5,7],SAQNLS ­12g0Z u0dtZd2x3µas≠xn̂d2 1 s≠yn̂d2 121 1 as≠tn̂d2∂.(1)where the bare coupling constantg0 and the spin-wavevelocity c are those of the isotropic system, whileaparametrizes the anisotropy. In the classical limit, this dscribes spin waves with velocitycysad ­ cps1 1 ady2andcxsad ­pa cysad in the y andx directions, respec-tively. The slab thickness in the imaginary time direction u is given bybh̄cL, whereL is the cutoff of ourspherical Brillouin zone. This model is derived by taking the naive continuum limit of a Heisenberg model witexchange couplingsJ and aJ in the y and x directions,respectively.The renormalization of this model has received somattention recently [5,8]. We adopt here a variation othe procedure as proposed by Affleck [8]. The centrobservation is that this model contains two ultraviolecutoffs. As a ramification of the anisotropy, the highesmomentum states in thex direction will have an energyEmaxx which is a factor ofpa smaller than that of thehighest momentum states in they direction. Therefore,the initial renormalization flow fromEmaxy down toEmaxxis governed by one dimensional fluctuations. AtEmaxx theresulting model can be rescaled to become isotropic, albwith “bare” parameters which are dressed up by the odimensional high energy fluctuations.Keeping the full model Eq. (1), the one dimensionafluctuations are integrated out (using momentum-sh1514r.lx-edly.asic:x-minnsane---henaltteitnelellrenormalization [6]) by neglecting the dispersions in thx direction entirely. This causes the anisotropy parameta to become a running variable as well, which is alwayrelevant. When the renormalizeda ­ 1, the model hasbecome isotropic, albeit with renormalized bare couplinconstants.Writing n̂ ­ s $p, sd, wheres is the component of̂nin the direction of ordering, we expand to one-loop ordein $p. Subsequently, we Fourier transform the$p fieldsaccording to$ps $x, td ­X̀n­2`Z d2ks2pd2$ps $k, ndeı$k?$x2ıvnt , (2)wherevn ­ 2pnyu are the Matsubara frequencies. Themomentak are rescaled withL to become dimensionless.Separating the fields according to$ps $k, nd ­($p.s$k, nd; e2l , jkyj , 1 ,$p,s$k, nd; 0 , jky j , e2l ,(3)where l is small, we integrate out the fieldsp., usinga square Brillouin zone for convenience. Rescalingky,p,, u, g, anda, we find that the model scales to largera (smaller anisotropy). We obtain the following flowequations:a ­ a0e2l , (4)≠g≠l­ 2a1 1 ag 1 g2I , (5)≠t≠l­ t 1 tgI , (6)whereI ­p1 1 a4p2 p2Z 121dkxcoths u2q11a2pak2x 1 1dpak2x 1 1,(7)and where t is the dimensionless temperature,t0 ­kBTyr0s . From Eqs. (5) and (6), we find for the slabthicknessu ­ gytu ­ u0s1 1 a01 1 ae2l . (8)From Eq. (4) it follows thata ­ 1 corresponds withl ­ l1 ­ 2 lnpa0. The bare coupling constant (atT ­0) and bare slab thickness of the effective isotropic modfollow by integrating Eqs. (5) and (6) down tol1 fg1 ­gsl1d, t1 ­ tsl1dg,g1 ­ g0¡ "s21 1 a02g02p2harsinhspa0 dypa0 1 lns1 1p1 1 a0d 2 lnfpa0 s1 1p2d2gj#. (9)g1yt1 ­ sg0yt0dqa0s1 1 a0dy2 . (10)Except for these altered bare quantities, the isotropic model is analyzed in the standard way [6].VOLUME 80, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 16 FEBRUARY 1998Puttingg1 ­ gc ­ 4p and solvingg0, we find the critical bare coupling for the anisotropic modelgcsa0d ­ 4ps21 1 a0¡ ∑1 12pharsinhspa0 dypa0 1 lns1 1p1 1 a0d 2 lnfpa0 s1 1p2d2gj∏. (11)isey-erede,e-e.f--We find this result to be the same within a couple opercents as the outcome of large-N mean-field theory [5],while the difference originates in an inaccuracy in oucalculation related to the switch from the square (atE .Exmax) to the spherical Brillouin zone of the effectivelyisotropic model.For a0 ­ 1, the one-loop crossover lines between thquantum-critical (QC) and the RCyQD regime are givenby t ­ 62ps1 2 gy4pd. Taking sg1, t1d to lie on theselines and iterating the flow equations backwards, wobtain the crossover diagram for the anisotropic modshown in Fig. 1. Note that the anisotropy has a strongeffect on thet dependence of the RC to QC line than oits g dependence. This already indicates that theT ­ 0properties will be less affected by the anisotropy thathose at finite temperatures.The one-loop mapping to an isotropic QNLS providea simple way of calculating the correlation length inthe anisotropic model. Noting that the correlation lengin the y direction scales asj ­ j0e2l under Eq. (3),it immediately follows thatjsg0, t0d ­ el1 jisotrsg1, t1d.Inserting the one-loop expression forjisotr in the RCregime [6] and using Eqs. (9) and (10) [the use of thT ­ 0 expression forg1 Eq. (9) is a good approximationif g1yt1 ¿ 1],jsg0, t0d ­0.9pa0g12t1exp∑µ1 2g14p∂ ¡t1∏. 0.9g02t0s1 1 a02expfpa0 rss0dykBT g ,(12)where the renormalizedT ­ 0 stiffness is given byrss0d ­ r0sµ1 2g0gcsa0d∂. (13)Equations (12) and (13) are our central result. It showthat the correlation length in the renormalized classicregime has a twofoldexponential dependence on theanisotropy, both originating in the high frequency ondimensional fluctuations. As already pointed out by C[5], the anisotropy causesgc to decrease (e.g., Fig. 1),leading to a reduction ofj at a given temperature.However, we find an additionalpa in the exponentwhich has been overlooked by CH, although it is includein the paper of Wang [9]. This is the specific way inwhich the greater effect of the thermal fluctuations, whicwe noted earlier, shows up in the renormalized classicregime. In fact, it shows that the basic invention of CNH[6] is straightforwardly extended to the anisotropic casThe correlation length is given by the expression for thclassical system, and quantum mechanics enters onlythe form of a redefinition of the stiffness. Howeverfor the classical correlation length expression one shoufreeel,ernnsthesaleHdhale.ein,lduse the one for theanisotropic classical model. Usingthe same procedure as for the quantum model, iteasy to demonstrate that the correlation length of thanisotropic classicalOs3d model in 2D behaves asj ,expspa0 r0s ykBT d, and this explains the occurrence of theadditionalpa0 factor [10].The finiteness of the Néel temperature is caused bsmall intraplanar spin anisotropies and interplanar couplings. Keimeret al. [11] have shown that in La2CuO4the former dominate, and these can be lumped togethin a single termaeff which plays the role of an effectivestaggered field. The Néel temperature can be estimatby comparing the thermal energykBTN to the energy costof flipping all spins in a region the size of the correlationlength in the presence of the effective staggered field.kBTN sad . JaeffµjsTN , adaMsM0∂2. (14)Because it is not expected that stripes will influence thspin anisotropies strongly, we can use the estimate foraeffas determined for the half-filled system:aeff ­ 6.5 31024 [11]. For our estimate ofTN , we will use spin-wave results for the renormalized stiffness, susceptibilityand spin-wave velocity [12]. ForS ­ 1y2, they areh̄c ­ 0.5897p8 Ja, x's0d ­ 0.514h̄2y8Ja2, and rs ­c2x's0d. The bare coupling constant is obtained fromg0y4p ­ 1ys1 1 4px'cyh̄Ld [6], which yields g0 ­9.107 for La ­ 2pp. We notice that the one-loop resultfor the prefactor is not correct, but this factor is not veryimportant as far as the reduction of the Néel temperaturis concerned.Since ourT ­ 0 results coincide with those obtainedby CH [5], we use their expression for the zero temperature staggered magnetization [13],MssadMss1d­s1 2 g0ygcsad1 2 g0y4p, (15)and its anisotropy dependence is shown together with thresults for the Néel temperature in the inset in Fig. 2To illustrate the effects of a differentaeff in the stripephase (e.g.,J' may be much reduced due to frustration)we have also plotted the results foraeffsa , 1d ­10aeff sa ­ 1d (upper dashed line) and foraeffsa ,1d ­ 0.1aeff sa ­ 1d (lower dashed line). In Fig. 2TNis plotted versusMs. As expected, the dependence ofTN on anisotropy is considerably stronger than that oMs. A reduction ofMs by a factor of 2 due to a spin-wave anisotropy of,pa , 1y4 order is accompanied bya suppression ofTN by roughly an order of magnitude.In the above we relate different experimentally accessible quantities (spatial and spin anisotropies, Néel temperature,T ­ 0 staggered order, correlation length) and1515VOLUME 80, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 16 FEBRUARY 1998--y).i-.,-.t/FIG. 2. The Néel temperature versus the zero temperatstaggered magnetization, with the anisotropy as implicit paraeter. Both quantities are normalized with respect to their valin the isotropic system. The upper/lower dashed line givesT̄Nfor aeff (Néel stabilizing field) a factor of 10 larger/smaller thain the isotropic system. Inset:̄TN andM̄s as a function of theanisotropy parametera.further [14,15] experimentation is needed to unambigously demonstrate that spatial anisotropy is the causethe low spin-ordering temperature. If the fluctuation bhavior in the RC regime is indeed as general as suggesby the present analysis, other sources of stripe inducspin order could have similar consequences. For instanlocal charge deficiencies in the stripes caused by quencdisorder would give rise to unscreened (by charge) piecof domain walls. Such stripe defects are like the dipoldefects discussed by Aharonyet al. [16], and their frus-trating effect is expected to be disproportionally strongat finite temperature than at zero temperature.Above all, the present analysis shows that a Landmean-field analysis falls short as a description for the thmodynamic behavior of the stripe phase because ofimportance of fluctuations. Furthermore, thermodynamidoes not offer an unambiguous guidance regarding thecroscopy (frustrated phase separation [17] versus “holotype mechanisms [18]). Here we have focused on ttransversal spin fluctuations, and given that there is amevidence for a pronounced slowing down of the spin dnamics at the charge-ordering temperature, these undoedly play an important role. It is noted that recent resupoint at a similarly important role of fluctuations in thecharge sector [19].1516urem-uenu-ofe-tededce,hedesarerauer-thecsmi-n”hepley-ubt-ltsWe thank V. J. Emery, B. I. Halperin, S. A. Kivelson,and W. van Saarloos for stimulating discussions. Financial support was provided by the Foundation of Fundamental Research on Matter (FOM), which is sponsored bThe Netherlands Organization of Pure Research (NWOJ. Z. acknowledges support by the Dutch Academy of Scences (KNAW).[1] J. M. Tranquadaet al., Nature (London)375, 561 (1995).[2] J. M. Tranquadaet al., Phys. Rev. Lett.78, 338 (1997);J. M. Tranquada, cond-mat/9706261.[3] V. Sachanet al., Phys. Rev. B54, 12 318 (1996).[4] O. Zachar, S. A. Kivelson, and V. J. Emery, cond-mat/9702055.[5] A. H. Castro Neto and D. Hone, Phys. Rev. Lett.76, 2165(1996); cond-mat/9701042.[6] S. Chakravarty, B. I. Halperin, and D. R. Nelson, Phys.Rev. Lett.60, 1057 (1988); Phys. Rev. B39, 2344 (1989).[7] A. Parola, S. Sorella, and Q. F. Zhong, Phys. Rev. Lett71, 4393 (1993).[8] I. Affleck and B. I. Halperin, J. Phys. A29, 2627(1996).[9] Ziqiang Wang, Phys. Rev. Lett.78, 126 (1997).[10] Following CNH [6], Eqs. (12) and (13) can also beobtained by integrating out all quantum fluctuations in therenormalized classical region to one-loop order. Thisyields a 2D anisotropic classical nonlinear sigma modelwith effective dimensionless temperature1t0­1kBT"rss0d 1kBT2ppa0ln√h̄cLkBTs1 1 a02!1 O sT2d#,where gcsa0d is now precisely the critical couplingobtained by CH. Momentum-shell renormalization of this2D model again yields Eq. (12).[11] B. Keimeret al., Phys. Rev. B46, 14 034 (1992).[12] Jun-ichi Igarahsi, Phys. Rev. B46, 10 763 (1992).[13] Because of the neglect of topological terms, this expression yields a lower bound for the staggered magnetization[14] J. M. Tranquada, P. Wochner, and D. J. Buttrey, cond-ma9612007.[15] V. Kataevet al., Phys. Rev. B55, R3394 (1997).[16] A. Aharonyet al., Phys. Rev. Lett.60, 1330 (1988).[17] U. Löw et al., Phys. Rev. Lett.72, 1918 (1994).[18] J. Zaanen and O. Gunnarsson, Phys. Rev. B40, 7391(1989); J. Zaanen and A. M. Oles´, Ann. Phys. (Leipzig)5, 224 (1996); C. Nayak and F. Wilczek, Int. J. Mod.Phys. B10, 2125 (1996); S. R. White and D. J. Scalapino,cond-mat/9705218.[19] S.-H. Lee and S.-W. Cheong, cond-mat/9706110.

Charge- versus spin-driven stripe order: Role of transversal spin fluctuations

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Charge- versus spin-driven stripe order: Role of transversal spin fluctuations

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