Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Duin, C.N.A. van

Zaanen, J.

Leiden University Scholary Publications

VOLUME 78, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 14 APRIL 1997andsearaty.Origin of the Quantum-Critical Transition in the Bilayer Heisenberg ModelC. N. A. van Duin and J. ZaanenInstitute Lorentz for Theoretical Physics, Leiden University, P.O.Box 9506, 2300 RA Leiden, The Netherl(Received 6 January 1997)The bilayer Heisenberg antiferromagnet is known to exhibit a quantum-critical transition at aparticular value of the interlayer coupling. Using a new type of coherent state, appropriate to thspecial order parameter structure of the bilayer, we map the problem onto the quantum nonlinesigma model. It is found that the bare coupling constant diverges at the classical transition, so thin any finite dimension the actual transition occurs inside the ordered phase of the classical theor[S0031-9007(97)02986-4]PACS numbers: 75.10.Jm, 64.60.– i, 71.27.+a, 75.45.+ jtinegiclaeaithoeorntheint-heisyThe study of nonclassical collective quantum statesmatter is a central theme of modern condensed maphysics. Despite the successes in1 1 1 dimensions, it hasproven difficult to address these matters in higher dimesions. Either the minus-sign problem intervenes (ase.g., thet-J model and frustrated spin models), or the tedency towards classical order is too strong (e.g., unfrutrated spin models). The class of bilayer Heisenberg modis special in this regard [1,2]. It is sign-free, and convincing numerical evidence exists showing that its lonwavelength behavior is governed by theOs3d quantumnonlinear sigma model (QNLS) with tunable bare couplinconstantu [3]. The relationship between the microscopmodel and its long wavelength behavior is nontriviaChubukov and Morr (CM) made the key observation thain order to construct the classical limit, the severe loc(interplanar) fluctuations have to be integrated out first [4In the resulting singlet-triplet representation, a phase trasition between a Néel state and an incompressible statfound already at the classical level. Here it is shown ththis transition does not correspond to the quantum crcal transition found in numerical studies. Because of tspecial structure of the order parameter, the standard SUs2dgeneralized spin coherent state does not suffice for the cstruction of the path integral. We introduce a novel typof coherent state which allows us to straightforwardly rcover the QNLS describing the long wavelength behaviWe find that the bare coupling constant of the field theodiverges at the classical transition. The quantum phasetransition therefore occurs well before the classical trasition can occur, and the latter is therefore in any finidimension an artifact. Our results are consistent with tindications of quantum criticality found by CM.It is convenient to consider the “bilayer” model inarbitrary dimensions, with an added magnetic field ($B),H ­ J1Xkijls$si1 ? $sj1 1 $si2 ? $sj2d1 J2Xi$si1 ? $si2 2 $B ?Xis$si1 1 $si2d , (1)where kijl runs over the bonds of twod-dimensionalhypercubes 1 and 2. The antiferromagnetically coupl0031-9007y97y78(15)y3019(4)$10.00oftern-n,-s-ls--g.t,l].n-isti-en-e-r.y-eed(J1 . 0) s ­12 Heisenberg spins$ih are coupled locallyby J2. Following CM, we first integrate out theJ2 term[4,5]. Define the sum and the difference of the spoperators,$S ­ $s1 1 $s2;$̃S ­ $s1 2 $s2 , (2)such thatH ­12J1Xkijls $Si ? $Sj 1 $̃Si ? $̃Sjd114J2Xis $S2i 2$̃S2i d 2 $B ?Xi$Si . (3)Equation (2) amounts to a transformation to a singletriplet basis. Introducing hard-core bosons creating tlocal singlet state,ayi ­1p2scyi1#cyi2" 2 cyi1"cyi2#d, and thelocal triplet byi1,0,21 (by1i ­ cyi1"cyi2" etc.), Eq. (2) can bealternatively written asSz ­ by1 b1 2 by21b21 ,S1 ­p2 sby1 b0 1 by0 b21d ,S̃z ­ 2ayb0 2 by0 a ,(4)S̃1 ­p2 sby1 a 2 ayb21d .$S describesS ­ 1 spins, while $̃S is related to fluctuationsfrom triplets to singlets. These operators form anos4ddynamical algebra,fSa , Sbg ­ i´abcSc, (5)fS̃a , S̃bg ­ i´abcSc, (6)fSa , S̃bg ­ i´abcS̃c. (7)At the J2 ­ 0 point (two decoupled layers) the problemhas anOs4d global invariance, which is broken for anyfinite J2, leaving only an invariance under the SUs2dsubgroup Eq. (5). The unconventional aspect of thproblem is that for positiveJ2 the spontaneous symmetrbreaking involves the generatorsS̃. TheJ2 . 0 classical© 1997 The American Physical Society 3019VOLUME 78, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 14 APRIL 1997i-e,tureeiff-heentter-saddle-point of CM is easily seen to correspond to thvacuum amplitudes (z ­ 2d),$̃V ­1NøXis21di $̃Si¿­vuut1 2 J22J21 z2QsJ1z 2 J2dn̂ ,(8)nA ­1NøXiayi ai¿­12µ1 1J2J1z∂QsJ1z 2 J2d1 QsJ2 2 J1zd , (9)wheren̂ is a vector on the unit sphere. The condensatioof $̃S [Eq. (8)] and the existence of a mean singlet densi[Eq. (9)] is a direct ramification of the explicit symmetrybreaking due to the interplanar coupling.$̃V is still avector order parameter, because$̃S transforms as a vectorunder $S. It is therefore a Néel order state, albeit one wita variable local moment size, which implies that its lonwavelength behavior should be described by the QNLSOn the classical level it is found thatnA is nonzero forall positiveJ2, while$̃V vanishes continuously atJ2 ­ J1z,wherenA becomes equal to one. This is the lowest ordresult found by CM. Regarding its formal status, it is easiseen that this classical theorybecomes exact in infinitedimensions[5,6]. The energy fluctuations disappear in thilimit: DEyE ~ 1ypNd. In addition, we note that$̃V alsoexists ins2 1 1dD, at least in the vicinity of the quantumcritical point: the correlation functions in terms of$s1 and$s2go to zero at the transition with their ratios fixed accordinto Eq. (8) [3].What is wrong with the assertion that this transitioand the quantum-critical transition ins2 1 1dD are thesame? The transition in infinite dimensions is acl ssicaltransition. In terms of the singlet-triplet basis, the quantum fluctuations disappear at the lattice cutoff and thermfluctuations dominate at any finite temperature. The nmerical study shows quantum criticality [3]: at zero temperature, the quantum fluctuations are scale independIn the remainder we will show that this classical theorbecomes pathological in the neighborhood of the classictransition.Coherent state path integrals offer a convenient framwork to study quantum order parameter fluctuations [7Because of the special status of the order parameEq. (8), the usual generalized spin coherent states dosuffice. Our key result is the discovery of a special coheent state for this type of order parameter structure. Neto the general requirements of normalizability and the eistence of the identity, it should be demanded from coheent states that they reproduce all properties of the classisector. Besides reproducing Eqs. (8) and (9), they shoalso allow for an$S derived vacuum expectation value,$V ­1NøXi$Si¿. (10)We find that the following coherent state satisfies all therequirements:3020entyhg.erlysgn-alu--ent.yale-].ternotr-xtx-r-caluldsejVṼl ­ eifSzeiuSyeiu2Sxeic S̃yjxl , (11)with the reference state,jxl ­ scosxay 1 sinxby0 d jvacl . (12)Equation (11) looks conventional. It refers to the varous rotations related to theOs4d symmetry. The noveltyis Eq. (12): instead of the usual maximum weight statthis nonexact state underlies the order parameter strucEqs. (8) and (9), with$̃V chosen along thez axis, whilex isfixed by the explicit symmetry breaking interaction,J2.The freedom implied by Eq. (11) might at first instancappear as redundant. However, it turns out that the stness in the temporal direction is caused entirely by tfluctuations from $̃V into the $V direction, and the fourangles appearing in Eq. (12) take care of the independrotations of $̃V and $V. Explicitly, c parametrizes a rota-tion from $̃V to $V' $̃V ( $S ? $̃S ­ 0). The rotation of $̃V inthe plane perpendicular to$V is parametrized byu2. Thisis the only free rotation left to$̃V in a magnetic field. uandf fix the direction of $V.We obtain the following expressions for the vacuumamplitudes with respect to this coherent statenA ­ cos2 x cos2 c , (13)$V ­ sin2x sincs2 cosu cosf, 2 cosu sinf, sinud ,(14)$̃V ­ sin2x coscfcosu2ûsu, fd 2 sinu2f̂sfdg , (15)where û and f̂ are the local unit vectors in theuandf directions,û ­ ssinu cosf, sinu sinf, cosud andf̂ ­ ssinf, 2 cosf, 0d. The identity becomes1 ­Zdms $V, $̃Vd jVṼl kVṼj­2p4Z py20dxZ py20dcZ 2p0du2 (16)3Z 2p0dfZ py22py2du cosujVṼl kVṼj .By taking expectation values with regard tojVṼl(classical limit), we find theOs3d invariant version of themean-field theory of Chubukov and Morr. Minimizationf the classical energy with regard to the coherent staangles yieldscos2x0 ­J2J1z1 O sB2d , (17)sinc0 ­BJ1zsJ1z 2 J2J1z 1 J21 O sB2d , (18)with u and f fixed such that $V points in the directionof the magnetic field. We recover the classical ordedisorder transition atJ2 ­ J1z, where bothṼ and theinduced magnetizationV vanish according to Eqs. (14)and (15).VOLUME 78, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 14 APRIL 1997arteeycet,rleedinalethssst,e-ent.gThe derivation of the path integral is standard [7Using the Trotter formula, the evolution operator iimaginary time is written as (Nt is the number of timeslices,dt the imaginary time interval,Ntdt ­ b),Z ­ limNt!`dt !0Tr tNtYl­1e2dtPiHi . (19)Inserting the identity (16) at every intermediary time anexpandingZ to lowest order indt,Z ­ limNt !`dt !0ZD mNtYl­1YifkVṼstl , id j VṼstl11, idl2 dtkVṼstl , idjHijVṼstl , idlg , (20)where the integration measureD m is given byQNtl­1 dmsh $Vlj, h$̃Vljd, while htlj is the set of interme-diary times in the imaginary time interval [0, b]. Thekinetic term in the action follows from the first terminside the square brackets,YikVṼstl , id jVṼstl11, idl ­ 1 1 idtFstld 1 O sd2t d ,(21)withF ­Xisin2xi sincissinul≠tfi 1 ≠tu2id­ 2Xisin2xi $Oi ? ≠t$̃OiÕi3$̃OiÕi, (22)where $O ­ $Vy sin2x and $̃O ­ $̃Vy sin2x, so O2 1Õ2 ­ 1.The potential energy is (B ­ 0),V ­J12Xki,jlsin2xi sin2xjs $Oi ? $Oj 1 $̃Oi ? $̃Ojd114J2Xis1 2 4Õ2i cos2 xid . (23)Taking the time continuum limit, the path integrabecomesZ ­RD me2SM , with the real-time actionSM ­Z T0dx0f2Fsx0d 1 V sx0dg . (24)To derive the long wavelength theory,$O and $̃O areseparated into a slowly varying order-parameter part aa rapidly fluctuating part which will be integrated outThe fluctuations inx are massive because of the explicsymmetry breaking, and can be neglected. We are lwith$̃Oi ­ his $̃mi 1 a $̃Lkid 1 a $̃L'i , (25)$Oi ­ $mi 1 a $Li . (26)The (staggered) fluctuation$̃Lki is parallel to the orderparameter$̃mi (hi ­ 61 depending on the sublattice).$Lhas a component along$m, but is perpendicular to$̃m].ndlnd.iteftbecause of the constraint$Oi ?$̃Oi ­ 0. As we alreadyindicated, despite the fact that the order-parameter pof $O is zero in the absence of a magnetic field, thfluctuations in this quantity are actually producing thstiffness in the time direction and should be carefullintegrated out. We expand to second order in the latticonstanta, which will be taken to zero at the end ofthe calculation. Using the constraintO2i 1 Õ2i ­ 1, thefluctuation $̃Lki is eliminated from the action. Differentfrom the single-layer system, two canting fields resul$̃L'i and $Li, which have to be integrated out. The formedoes not influence the long wavelength behavior, whithe latter is responsible for the kinetic term in the effectivaction.After expanding ina and eliminating $̃Lk, the kineticterm becomesF ­ 2Xisin2xiam̃2i$Li ? ≠0 $̃mi 3 $̃mi 1 stagg. terms.(27)Using $̃mi 2 $̃mj . a≠i!j $̃mi , it can be seen that thestaggered terms give contributions which are of thirorder in a. The expression forF is identical to that forthe single-layer system, apart from the factor sin2x andthe absence of a topological term.Within the limitationsof the semiclassical expansion, the above derivation isprinciple valid for any dimension, including the1 1 1dimensional two leg spin ladder systems. The usuargument for the irrelevance of topological terms in thessystems are based on the proximity of Néel order on bochains separately: the topological terms in the two rowcancel each other. Here we find that this holds regardleof the strength of the local fluctuations. We notice thaaccording to Haldane’s conjecture [8], the spectrum of thtwo leg ladder has to be gapped for anyJ2 fi 0.The potential term is written in the formV ­ J1Xki,jlsin2xi sin2xj3∑a24s≠i!j $̃mid2 1 a2sL2i 1 L̃2'id 2 2∏1 J2Xia2L2i cos2 xi 2J24Xis1 2 4 cos2 xid .(28)In the continuum limit (a ! 0), the summations over sitesare replaced by integrations over space,Pi ! a2dRddx.TheO s1d term in Eq. (28), corresponding with the meanfield energy for the bilayer model, acquires a largprefactora2d and can be integrated by steepest desceThis yields the mean-field expression forx, Eq. (17).After integrating over the fluctuations$L and $̃L' werecover the effective action responsible for the lonwavelength fluctuations, which is theOs3d QNLS,SM ­12Zdd11x∑x's≠0 $̃md2 2 rsdXa­1s≠a $̃md2∏.(29)3021VOLUME 78, NUMBER 15 P H Y S I C A L R E V I E W L E T T E R S 14 APRIL 1997entsnsndhnc-ctmwti-to-al:at-o-onshss:lgeds,.eAlthough the form of Eq. (29) is dictated by symmetrythe parameters appearing in the effective theory havequite different meaning in terms of the microscopic modthan is the case in single layer problems. Taking thsaddle-point values, the perpendicular susceptibility athe spin stiffness become, respectively,x' ­ a2d J1z 2 J2J21 z2, (30)rs ­ a22d J12µ1 2J22J21 z2∂. (31)Both the susceptibility and the spin stiffness vanishthe classical transition atJ2 ­ zJ1. The spin-wave ve-locity ys ­prsyx' remains finite at the transition, andno divergencies occur on the Gaussian level [4]. The sbility of the classical state against quantum melting ihowever, controlled by the dimensionless coupling costantu ­ a12dyprsx', which is found todivergeat theclassical transition asu , 1yJp2 , whereJp2 is the reducedinterlayer couplingJp2 ­ sJ1z 2 J2dyJ2. In any finitedimension, theOs3d QNLS quantum critical transitionoccurs at a finite value of the coupling constant andfollows that the long wavelength fluctuations destroy thṼ type Néel order before the classical critical point ireached. Accordingly, the quantum critical transition othe bilayer model is of theOs3d QNLS kind, and the clas-sical transition exists only in infinite dimensions.This theory is even quantitatively reasonable. One loorenormalization theory for the QNLS in2 1 1 dimen-sions puts the critical coupling atup ­ 4p [9]. Usingthe saddle-point values for the spin stiffness and suscetibility [Eq. (31)], we find the quantum transition to oc-cur at Jc2 yJ1 ­ 3.3. Given that1yS-like corrections areneglected [5], the agreement with the value of 2.5–2.6 otained from quantum Monte Carlo [3] and series expasions [2] is reasonable.In summary, we have clarified the origin of the quantum critical transition of the bilayer Heisenberg problemThe key aspect is that the order parameter structure ascovered by Chubukov and Morr [4] is unusual. Althougthis order parameter is macroscopically of the usualOs3dvector kind, and therefore described by theOs3d quantumnonlinear sigma model, its microscopic status is uncoventional. The operators acquiring a vacuum amplitud(S̃) are not the ones expressing the global SUs2d invari-3022,aledata-,-itefpp-b---.is--eance of the problem. This kind of order parameter struture arises naturally in the present context and we expeit to be quite common in the general context of quantumagnetism [10]. Our main result is the discovery of a netype of spin coherent state which allows for the requanzation of such order parameter structures. As appliedthe bilayer problem, the novelty is that in any finite dimension the classical theory becomes highly pathologicthe bare coupling constant of the field theory divergesthe classical transition, explaining why the quantum transition obeysOs3d QNLS universality.We acknowledge very helpful discussions withA. Chubukov and S. Sachdev. Financial support was prvided by the Foundation of Fundamental ResearchMatter (FOM), which is sponsored by the NetherlandOrganization for the Advancement of Pure Researc(NWO), and by the Dutch Academy of Science(KNAW).[1] T. Matsuda and K. Hida, J. Phys. Soc. Jpn.59, 2223(1990); A. J. Millis and H. Monien, Phys. Rev. Lett.70,2810 (1993); Phys. Rev. B50, 167 606 (1994).[2] K. Hida, J. Phys. Soc. Jpn.61, 1013 (1992).[3] A. W. Sandvik and D. J. Scalapino, Phys. Rev. Lett.72,2777 (1994).[4] A. V. Chubukov and D. K. Morr, Phys. Rev. B52, 3521(1995).[5] C. N. A. van Duin and J. Zaanen (to be published).[6] The same holds for infinite ranged intraplanar interactionC. Gros, W. Wenzel, and J. Richter, Europhys. Lett.32,747 (1995). It is noted that the spectrum of physicaexcitations is in this case gapped. The mode-softenintransition discussed by Gros et al. occurs actually in ththermodynamically irrelevant thin spectrum. E. Lieb anD. Mattis, J. Math. Phys.3, 749 (1962); T. A. Kaplan, W.von der Linden, and P. Horsch, Phys. Rev. B42, 4663(1990).[7] E. Fradkin,Field Theories of Condensed Matter System(Addison-Wesley, Reading, MA, 1991); A. AuerbachInteracting Electrons and Quantum Magnetism(Springer-Verlag, New York, 1994).[8] F. D. M. Haldane, Phys. Rev. Lett.50, 1153 (1983).[9] S. Chakravarty, B. I. Halperin, and D. R. Nelson, PhysRev. Lett.60, 1057 (1988); Phys. Rev. B39, 2344 (1989);A. V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B49,11 919 (1994).[10] For instance, the so-called singlet-triplet models: P. Fuldand I. Peschel, Adv. Phys.21, 1 (1972).

Origin of the quantum-critical transition in the bilayer Heisenberg model

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Origin of the quantum-critical transition in the bilayer Heisenberg model

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