Spin-density-wave order in cuprates

Martino De Prato,1,2,* Andrea Pelissetto,3,† and Ettore Vicari4,‡

1Max-Planck Institut für Metallforschung, Heisenbergstrasse 3, D-70569 Stuttgart, Germany2Inst. für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, D-70569 Stuttgart, Germany

3Dipartimento di Fisica dell’Università di Roma “La Sapienza” and INFN, P.le Aldo Moro 2, I-00185 Roma, Italy4Dipartimento di Fisica dell’Università di Pisa and INFN, Largo Pontecorvo 2, I-56127 Pisa, Italy

�Received 2 May 2006; published 6 October 2006�

We study the nature of the two-dimensional quantum critical point separating two phases with and withoutlong-range spin-density-wave order, which has been recently observed in cuprate superconductors. We considerthe Landau-Ginzburg-Wilson Hamiltonian associated with the spin-density critical modes, perform a mean-field analysis of the phase diagram, and study the corresponding renormalization-group flow in two differentperturbative schemes at five and six loops, respectively. The analysis supports the existence of a stable fixedpoint in the full theory whose basin of attraction includes systems with collinear spin-density-wave order, asobserved in experiments. The stable fixed point is characterized by an enlarged O�4� � O�3� symmetry. Thecontinuous transition observed in experiments is expected to belong to this universality class. The correspond-ing critical exponents are �=0.9�2� and �=0.15�10�.

DOI: 10.1103/PhysRevB.74.144507 PACS number�s�: 74.72.�h, 75.30.Kz, 05.10.Cc, 05.70.Jk

I. INTRODUCTION

In the last few decades several aspects of cuprate super-conductors �SC’s� have been studied and many efforts havebeen spent to understand the unique and complex phase dia-gram exhibited by this class of materials; see, e.g., Ref. 1.Superconductivity in cuprates appears to be due to a mecha-nism analogous to the BCS one in ordinary superconductors.However, superconductivity is only one of the characteristicfeatures of these materials. There are many other new prop-erties that require more complex mechanisms and can beunderstood only if the interplay between BCS and additionalorder parameters is considered. For instance, at T�0,La2−�Sr�CuO4 at very low doping � is an insulator with long-range magnetic order. Increasing �, at ��0.055 an insulator-superconductor first-order transition takes place, giving riseto a superconducting state in which spins are still magneti-cally ordered.2 At ��0.14 another phase transition occurs,and for ��0.14, the material shows no magnetic order—it isparamagnetic—but is still superconducting. Neutron-scattering experiments3 suggested that this transition is con-tinuous. Moreover, in the ordered phase ��0.14, they re-vealed the presence of collinearly polarized spin-densitywaves �SDW’s� with wave vectors

K1 =2�

a�1

2− �,

1

2�, K2 =

2�

a�1

2,1

2− �� , �1.1�

where � is a function of the doping concentration and a is thelattice spacing. The wave vectors Ki are two dimensionalsince cuprates are supposed to be made of weakly interactingplanes and thus behave approximately as two-dimensionalsystems. Following Ref. 4, we assume that superconductivityis not relevant at the transition which is instead driven by theinteraction among the SDW degrees of freedom. Since T�0, one should take into account the quantum nature of thesystem. Quantum phase transitions can be studied by intro-ducing a supplementary dimension parametrized by animaginary time variable . The relevant order parameter isthe spin field which is parametrized as

Si�r,� = Re�eiK1·r1i�r,� + eiK2·r2i�r,�� , �1.2�

where ai are complex amplitudes. There are two interestinglimiting cases. The first one is when the order parameter canbe written as a�r ,�=ei�ana, which corresponds to col-linearly polarized SDW’s. The second one is when a�r ,�=na,1+ ina,2, with na,1 ·na,2=0 and �na,1 � = �na,2�, which corre-sponds to circularly polarized SDW’s. In cuprates experi-ments indicate that the ground state shows a collinearbehavior.5

The standard strategy for writing down an effectiveHamiltonian for a given physical system consists in consid-ering all polynomials of the order parameter of order lessthan or equal to 4 that are compatible with the expectedsymmetries. In the SDW-SC–to–SC phase transition the or-der parameter is the complex field ai�r ,�, with a=1,2 andi=1,2 ,3. The corresponding symmetries are the following:�i� SO�3� spin rotations, ai→Oijaj; �ii� translational sym-metry of the spin waves, ai→ei�aai; �iii� spatial inversion,

ai→ai* ; �iv� interchange of the 1̂ and 2̂ axes, 1i↔2i

and x↔y. The most general Hamiltonian with these symme-tries is4

H = d2rd��1�2 + v12��x1�2 + v2

2��y1�2 + ��2�2

+ v22��x2�2 + v1

2��y2�2 + r��1�2 + �2�2�

+u1,0

2��1�4 + �2�4� +

u2,0

2��1

2�2 + �22�2�

+ w1,0�1�2�2�2 + w2,0�1 · 2�2 + w3,0�1* · 2�2� ,

�1.3�

where v1 and v2 are parameters called SDW velocities.Terms such as a

* ·�a are forbidden by spatial inversionsymmetry and terms like ia

* ·�xa, even if permitted by allsymmetries, can be eliminated by redefining the fields asa→eiqa·ra. Hamiltonian �1.3� admits several different

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ground states depending on the values of the parameters.They are classified in Appendix A. In particular, there is thepossibility that both fields correspond to collinearly polarizedSDW’s as observed in experiments: 1=ei�1n1 and 2=ei�2n2, where the vectors n1 and n2 satisfy either n1=n2 orn1 ·n2=0.

In this paper we investigate the nature of the fixed points�FP’s� of the renormalization-group �RG� flow of the effec-tive Hamiltonian �1.3�. If a stable FP exists and its attractiondomain includes systems with collinearly polarized SDW’s,then the SDW-SC–to–SC transition may be continuous. Oth-erwise, it must be of first order. In our study, we consideronly the case v1=v2, which simplifies the analysis and allowsus to perform a high-order perturbative analysis. Therefore,we consider the theory

H = ddx��

d

����1�2 + ���2�2� + r��1�2 + �2�2�

+u1,0

2��1�4 + �2�4�+

u2,0

2��1

2�2 + �22�2�

+ w1,0�1�2�2�2 + w2,0�1 · 2�2 + w3,0�1* · 2�2� ,

�1.4�

where the field ai is a complex 2 N matrix, a=1,2, i=1, . . . ,N. The physically relevant case is N=3.

We first perform a standard analysis close to fourdimensions,6 computing the RG functions in powers of � 4−d. A one-loop analysis indicates that a stable FP existsonly for N�42.8. Apparently, this result casts doubts on theexistence of a stable FP in three dimensions. However, inthree dimensions there may exist FP’s that are absent for��1. This is indeed what happens in the Ginzburg-Landaumodel of superconductors, in which a complex scalar fieldcouples to a gauge field7 and in O�2� � O�n�-symmetricmodels.8,9 Thus, a more careful investigation of the RG flowin three dimensions calls for strictly three-dimensional per-turbative schemes. For this purpose we consider two field-theoretical perturbative approaches: the minimal-subtractionscheme without � expansion10 �in the following we will in-dicate it as the 3d-MS scheme� and the massive zero-momentum �MZM� renormalization scheme.11 The use oftwo different schemes is crucial, since the comparison of thecorresponding results provides a nontrivial check on the re-liability of our conclusions. In the 3d-MS scheme one con-siders the massless �critical� theory in dimensionalregularization,12 determines the RG functions from the diver-gences appearing in the perturbative expansion of the corre-lation functions, and finally sets � 4−d=1without expand-ing in powers of � �this scheme therefore differs from thestandard � expansion6�. In the MZM scheme one considersinstead the three-dimensional massive theory in the disor-dered �high-temperature� phase. We compute the � functionsto five loops in the 3d-MS scheme and to six loops in theMZM scheme. We use a symbolic manipulation program thatgenerates the diagrams �approximately 1000 at 6 loops� andcomputes their symmetry and group factors and the compi-

lation of Feynman integrals of Refs. 13 and 14. The five-loopMS and six-loop MZM series of the model �1.4� are reportedin Ref. 15. The perturbative expansions are then resummedusing the known large-order behavior.

The perturbative analysis of the RG flow in the full theoryis not sufficiently stable to provide reliable results. There-fore, we have focused on the stability of the FP’s that occurin specific submodels of Hamiltonian �1.4�. The analysis ofthe perturbative series indicates the stability of the O�4�� O�3� collinear FP that occurs in the model with w1,0

=u1,0−u2,0 and w2,0=w3,0=u2,0�0. Its basin of attraction in-cludes systems with collinear SDW’s. Therefore, we expectthe continuous transition observed experimentally in cu-prates to belong to this universality class. This implies aneffective enlargement of the symmetry at the transition point.The corresponding critical exponents are

� = 0.9�2�, � = 0.15�10� . �1.5�

The paper is organized as follows. In Sec. II we discussthe possible ordered phases that occur in model �1.3� in themean-field approximation. Details are given in Appendix A.In Sec. III we discuss the FP structure close to four dimen-sions in the standard � expansion. Section IV contains themain results of this work. We consider three different sub-models �Sec. IV A� and then investigate the stability proper-ties of the FP’s occurring in each of them �Secs. IV B–IV D�.Conclusions are presented in Sec. V. In Appendixes B and Cwe give some technical details.

II. MEAN-FIELD ANALYSIS

The phase diagram of Hamiltonian �1.3� can be studied inthe mean-field approximation. Due to the large number ofcouplings, the analysis is quite complex. We have limited ourconsiderations to the case N�3. We summarize here theresults that are derived in Appendix A. For r�0 the systemis disordered and 1=2=0. For r=0 a continuous phasetransition occurs followed by a magnetized phase with r�0. The nature of the ordered phase depends on the valuesof the quartic parameters. The analysis reported in AppendixA shows that there are seven possibilities.

�1� 1 is a collinear SDW �1=ei�1n, n real� while2=0.

�2� 1 is a circularly polarized SDW �1=ei�1�n1+ in2�,n1 and n2 real, �n1 � = �n2�, n1 ·n2=0� while 2=0.

�3� 1 and 2 correspond to collinear SDW’s with thesame axis and amplitude: 1=ei�1n, 2=ei�2n, n real.

�4� 1 and 2 correspond to collinear SDW’s with or-thogonal axes and same amplitude: 1=ei�1n1, 2=ei�2n2,n1, n2 real, n1 ·n2=0, �n1 � = �n2�.

�5� 1 and 2 are circularly polarized SDW’s with thesame rotation plane and amplitude: 1=ei�1�n1+ in2�, 2

=ei�2�n1+ in2�, n1 and n2 real, �n1 � = �n2�, n1 ·n2=0.�6� 1 is a collinear SDW and 2 is a circularly polarized

SDW. The rotation plane of 2 is orthogonal to the axis of1. Explicitly, 1=ei�1n1, 2=ei�2�n2+ in3�, ni real, �n2 �= �n3�, ni ·n j =0.

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�7� 1 and 2 are elliptically polarized SDW’s withdifferent rotation planes but with the same amplitude,�1 � = �2�.

For cuprates the relevant solutions are �3� and �4�. Neces-sary conditions to obtain �3� are w2,0+w3,0�0 and

w1,0 + w2,0 + w3,0 − u1,0 � u2,0

� Min�u1,0 − w1,0 − w2,0 − w3,0,− w2,0,− w3,0� ,

�2.1�

while �4� requires w2,0+w3,0�0 and

w1,0 − u1,0 � u2,0 � Min�u1,0 − w1,0,w2,0,w3,0� . �2.2�

These conditions are not sufficient, since for some values ofthe parameters satisfying �2.1� and �2.2� the ordered phase isgiven by solutions �6� or �7�. Note that the sign of u2,0 is notthe relevant parameter that selects the collinear SDW’samong all possible solutions.

It is interesting to note that the mean-field solution pre-dicts either 1 �2 or 1�2 in the case of collinearSDW’s. This result is easy to understand. If both fields cor-respond to collinear SDW’s, then one can take 1 and 2real. In this case the only term of the Hamiltonian that con-tains a scalar product of the two fields is �w2,0+w3,0� �1 ·2�2, which forces the two fields to be either parallelor orthogonal, depending on the sign of w2,0+w3,0. Note thatthis also holds if we add additional higher-order terms to theHamiltonian, as long as the transition is continuous. Indeed,for a continuous transition a→0 at the transition �a=0 inthe disordered phase� and thus higher-order terms do notplay any role. On the other hand, this relation may not bevalid if the transition is of first order. Also the coupling to thecharge-density waves �CDW’s� that are present incuprates4,16 does not change this conclusion, since theycouple to the scalars a

2, �a�2.Solutions �3� and �4� also satisfy �1 � = �2�. This prop-

erty does not necessarily hold if we take into account theCDW’s �see Refs. 16 and 4 for an extensive discussion�.Indeed, let �1 and �2 be the complex amplitudes of theCDW’s coupled, respectively, to 1

2 and 22. In the absence

of the CDW-SDW coupling, for some values of the CDWHamiltonian parameters, the ordered solution corresponds to��1 � �0, �2=0. If now the CDW-SDW coupling is included,one may obtain a ground state with ��1 � � ��2 � �0 and�1 � � �2 � �0.

III. RG FLOW CLOSE TO FOUR DIMENSIONS

The RG flow close to four dimensions can be investigatedperturbatively in � 4−d. In the MS scheme the one-loop �functions are

�u1= − �u1 + �N + 4�u1

2 + 4u1u2 + 4u22 + Nw1

2 + w22 + w3

2

+ 2w1w2 + 2w1w3,

�u2= − �u2 + 6u1u2 + Nu2

2 + 2w2w3,

�w1= − �w1 + 2w1

2 + w22 + w3

2 + 2�N + 1�u1w1 + 4u2w1

+ 2u1w2 + 2u1w3,

�w2= − �w2 + Nw2

2 + 2u1w2 + 4u2w3 + 4w1w2 + 2w2w3,

�w3= − �w3 + Nw3

2 + 2u1w3 + 4u2w2 + 4w1w3 + 2w2w3,

�3.1�

where ui and wi are the renormalized quartic couplings cor-responding to the quartic Hamiltonian parameters ui,0and wi,0. They are normalized so that, at the tree level,g=g0�−� /Ad, where g and g0 label the renormalizedand Hamiltonian parameters, respectively, and Ad 2d−1�d/2��d /2�. The FP’s of the RG flow are the commonzeros of the � functions. For N=3 there are four FP’s whilefor N=2 there are seven FP’s: they are all unstable. Only forN�42.8 does a stable FP exist. It has u2=w2=w3 �for N→� we obtain u1=u2=w2=w3=� /N, w1=0�, so that at theFP the symmetry becomes O�4� � O�N�. This FP is the chiralFP that occurs in O�M� � O�N� models in the large-N limit.17

In order to determine the behavior in three dimensions,one should extend the computation to higher order in � anddetermine the function Nc���=42.8+O��� such that the chiralFP point identified above exists for N�Nc��� and is nolonger present for smaller values of N. We have not pursuedthis approach for several reasons. First, the analogous five-loop computation that was performed in the O�N� � O�2�model17–19 was not able to explain the correct physics ofthese models for N=2,3 �see Sec. IID in Ref. 8�. Moreover,this calculation is only concerned with the stable FP that ispresent for 0���1 �in the present case the chiral O�4�� O�N� FP�, while in d=3 the stable FP may be different, anunstable or even a new FP. The analysis that will be pre-sented in the next section favors this last possibility.

IV. SUBMODELS AND THEIR STABILITY

The three-dimensional properties of the RG flow are de-termined by its FP’s. Some of them can be identified byconsidering particular cases in which some of the quarticparameters vanish. The corresponding FP’s are also FP’s ofthe general theory. In this section, we identify some of themand then determine their stability with respect to the com-plete theory.

A. Some particular cases

For particular values of the couplings Hamiltonian �1.4�reduces to that of simpler models. Three cases have alreadybeen extensively studied in the literature.20

�1� For w1,0=w2,0=w3,0=0 there is no interaction betweenthe two SDW’s and Hamiltonian �1.4� reduces to that of twoidentical decoupled O�2� � O�N�-symmetric models. Thegeneral O�m� � O�n�-symmetric model is defined by theHamiltonian density18,20

1

2�ai��

�

����ai�2 + r�ai2 � +

g1,0

4! ��ai

�ai2 �2

+g2,0

4! ��i,j ��a

�ai�aj�2− ��

ai

�ai2 �2� , �4.1�

where �ai is a real n m matrix field �a=1, . . . ,n and i

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=1, . . . ,m�. Hamiltonian �4.1� is obtained from Eq. �1.4� bysetting ai=�1i

�a�+ i�2i�a� and

u1,0 = g1,0/3 − g2,0/6, u2,0 = g2,0/6, w1,0 = w2,0 = w3,0 = 0.

�4.2�

The properties of O�2� � O�N� models are reviewed in Refs.8, 18, 20, and 21. In three dimensions perturbative calcula-tions have been performed within the MZM scheme to sixloops22,23 and within the 3d-MS scheme to five loops8 �theseries are also reported in Ref. 15�. Their analyses indicatethe presence of a stable chiral FP with attraction domain inthe region g2,0�0 for all values of N �only for N=6 is theevidence less clear, since the MZM analysis does not appar-ently support it�. For N=2, these conclusions have been re-cently confirmed by a Monte Carlo simulation.8 A stable col-linear FP for g2�0 exists for N�4.9,24 Apart from thecollinear FP for N=2, these FP’s do not exist close to fourdimensions. For N=2 the collinear FP is equivalent to an XYFP and corresponds to g1

*=gXY* , g2

*=−gXY* , where gXY

* is theFP value of the renormalized coupling in the O�2� �4 model.

�2� For w1,0=u1,0−u2,0 and w2,0=w3,0=u2,0, Hamiltonian�1.4� reduces to �4.1� with m=4 and n=N. The correspon-dence is given by

1i =�1i + i�2i

�2, 2i =

�3i + i�4i

�2, �4.3�

where �ei is a 4 N matrix and

g1,0 = 3�u1,0 + u2,0�, g2,0 = 6u2,0. �4.4�

We have already discussed the FP’s of the O�4� � O�2�theory. The O�4� � O�3� theory does not present stable FP’sfor g2�0.25 Analyses of the available six-loop series in theMZM scheme and five-loop series in the 3d-MS scheme in-dicate the presence of a stable collinear FP for g2�0.26 ThisFP does not exist close to four dimensions.

�3� For u2,0=w2,0=w3,0=0 we obtain the mn model withn=2 and m=2N. The so-called mn model is defined by theHamiltonian density20,27

1

2�ai��

�

����ai�2 + r�ai2 � +

g1,0

4! ��ai

�ai2 �2

+g2,0

4! �aij

�ai2 �aj

2 ,

�4.5�

where �ai is a real n m matrix—i.e., a=1, . . . ,n and i=1, . . . ,m. The correspondence is obtained by setting

ai = ��ai + i�a,i+N�/�2, g1,0 = 3w1,0, g2,0 = 3�u1,0 − w1,0� .

�4.6�

A stable FP is the O�m� FP with g1=0 and g2=gO�m�* , where

gO�m�* is the FP value of the renormalized coupling in the

O�m�-symmetric vector model. In three dimensions theanalysis of five- and six-loop series28 indicates the presenceof a second stable FP with g2�0 for n=2 and m=2, 3, and 4.The perturbative series of the mn model are also reported inRef. 15.

Beside these three models, there are two other submodelsfor which no results are available.

�a� For w2,0=w3,0=0 we obtain two chiral models coupledby an energy-energy term. Note that in this model the RGflow does not cross the planes u2=0 and w1=0.

�b� For w2,0=w3,0=w0 we obtain a model with an addi-tional U�1� symmetry: 1→1

*, 2→2*. In this model the

RG flow does not cross the plane w=0.Finally, note an additional symmetry of Hamiltonian

�1.4�. It is invariant under 1→1*, 2→2, w2,0→w3,0,

and w3,0→w2,0, while the other couplings are unchanged.This implies that the RG flow in the space of renormalizedcouplings does not cross the plane w2=w3 and that, for anyFP with w2�w3, there is an equivalent one with w2�w3. Inparticular, we can limit our considerations to w2�w3.

In order to study the RG flow of the theory one can startby discussing the stability in the full theory of the FP’s of themodels �1�, �2�, and �3� discussed above.

For N=2 and N=3, the only cases we consider, model �1�has two FP’s.

�1a� The chiral FP, in which g1=g1,ch* and g2=g2,ch

* ;correspondingly u1

*=g1,ch* /3−g2,ch

* /6, u2*=g2,ch

* /6�0, w1*

=w2*=w3

*=0.�1b� The collinear FP, in which g1=g1,cl

* and g2=g2,cl* ;

correspondingly u1*=g1,cl

* /3−g2,cl* /6, u2

*=g2,cl* /6�0, w1

*=w2*

=w3*=0.Here gi,ch and gi,cl are the chiral and collinear FP’s of the

O�2� � O�N� theory. The analogous FPs are present in model�2�.

�2a� The chiral FP, in which g1=g1,ch* and g2=g2,ch

* ;correspondingly u1

*=g1,ch* /3−g2,ch

* /6, u2*=g2,ch

* /6�0, w1*=u1

*

−u2*, w2

*=w3*=u2

*; It does not exist for N=3. This is the FPthat is relevant for N�Nc����42.8+O��� close to fourdimensions.

�2b� The collinear FP, in which g1=g1,cl* and g2=g2,cl

* ;correspondingly u1

*=g1,cl* /3−g2,cl

* /6, u2*=g2,cl

* /6�0, w1*=u1

*

−u2*, w2

*=w3*=u2

*. It exists for both N=2 and N=3.Here gi,ch and gi,cl are the chiral and collinear FP’s of the

O�4� � O�N� theory. Finally, the mn theory gives two FP’s.�3a� The O�2N� FP. This is unstable in the full theory,

being already unstable in model �1�.�3b� The mn FP g1=g1,mn, g2=g2,mn; correspondingly

u1*=g1,mn

* /3+g2,mn* /3, w1

*=g1,mn* /3, u2

*=w2*=w3

*=0. It existsonly for N=2.

In the following we study the stability of these FP’s in thecomplete theory �1.4�. For this purpose, using the � func-tions of the general theory we have computed the stabilitymatrices of the FP’s at six and five loops, respectively, in theMZM and 3d-MS schemes. The perturbative series havebeen resummed by using the conformal-mapping method de-scribed, e.g., in Ref. 29. For a FP belonging to a submodel,the large-order behavior needed for the conformal-mappingsummation is the same as that characterizing all series of thatsubmodel. For all submodels we consider, the large-orderbehavior is already known.22,25,30

B. Stability of the decoupled O„2…‹O„N… fixed points

We want to establish the stability properties of the decou-pled O�2� � O�N� FP’s �1a� and �1b� in the complete theory�1.4�. For this purpose we need the RG dimensions of the

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operators present in Hamiltonian �1.4� that break the symme-try of model �1�—i.e., of the operators associated with thequartic couplings wi. It is useful to rewrite them as

w1,0�1�2�2�2 + w2,0�1 · 2�2 + w3,0�1* · 2�2

= W00P00 + W11P11 + W02P02, �4.7�

where

W00 = w1,0 +1

N�w2,0 + w3,0�, W11 = − w2,0 + w3,0,

W02 = w2,0 + w3,0, �4.8�

P00 O00�1�O00

�2�, P11 O11,ij�1� O11,ij

�2� , P02 �ij

O02,ij�1� O02,ij

�2� ,

�4.9�

and, using the correspondence �4.2�,

O00�a� = �

ei

�ei�a��ei

�a�,

O11,ij�a� = �1i

�a��2j�a� − �1j

�a��2i�a�,

O02,ij�a� = �

e

�ei�a��ej

�a� −1

N�ij�

ek

�ek�a��ek

�a�. �4.10�

The quadratic operator Oml�a� transforms as a spin-m and a

spin-l representation with respect to the O�2� and O�N�groups, respectively. Since P00, P11, and P02 belong to dif-ferent irreducible representations, they do not mix under RGtransformations at the decoupled O�2� � O�N� FP’s. TheirRG dimensions Yml can be derived from the RG dimensionsyml of the quadratic operators Oml

�a� at the O�2� � O�N� FP,using the relation

Yml = 2yml − 3. �4.11�

The quadratic term O00�a� corresponds to the energy operator

and thus y00=1/� and Y00=� /�, where � and � are thespecific-heat and correlation-length critical exponents of thegiven O�2� � O�N� FP. The RG dimensions y11 and y02 werecomputed in Ref. 24 �there, they are named y1 and y3, re-spectively�.

At the chiral FP �1a� we obtain:31

Y00 = 0.3�3� , Y11 = 1.6�3� , Y02 = 0.04�8� for N = 3,

Y00 = 0.2�3� , Y11 = 1.9�4� , Y02 = − 0.4�2� for N = 2.

At the collinear FP �1b� we obtain31

Y00 = 0.3�2� , Y11 = − 0.6�2� , Y02 = 1.0�3�for N = 3,

Y00 = − 0.2182�8� , Y11 = − 0.022�8� , Y02 = 0.9240�11�for N = 2.

These results show that the decoupled O�2� � O�N� FP’s areunstable in the complete theory �1.4� for both N=3,2.

It is also interesting to discuss submodels �a� and �b� men-tioned in Sec. IV A. In model �a� one should only consider

P00. The numerical results apparently indicate that the FP’sare always unstable �but with the present errors, we cannotreally exclude the opposite possibility�, except in one case.For N=2, the collinear FP is stable. In model �b� one shouldconsider P00 and P02. For N=2,3, all FP’s are unstable.

C. Stability of the O„4…‹O„N… FP’s

Here we investigate the stability of FP’s �2a� �it does notexist for N=3� and �2b�. For this purpose we must computethe RG dimensions of the perturbations of the O�4� � O�N�model appearing in the complete theory. This is done in Ap-pendix B. There are two relevant operators with RG dimen-sions Y1 and Y2. The corresponding perturbative series arereported in Appendix B. They are analyzed using theconformal-mapping method.29,32 The errors we will reporttakes into account the variation of the estimates when chang-ing the resummation parameters b ,� defined in Ref. 32—weuse b=3, . . . ,18 and �=0, . . . ,4—and the uncertainty of theFP coordinates.

The analysis of the six-loop series in the MZM schemeand of the five-loop 3d-MS series gives the following resultsat the collinear FP �2b� �Ref. 26�:

Y1 = − 0.4�4� , Y2 = − 0.95�7� , for N = 2 �MZM� ,

Y1 = − 0.6�9� , Y2 = − 1.2�1.0� , for N = 2 �3d-MS� ,

Y1 = − 1.5�1.2� , Y2 = − 0.42�10� , for N = 3 �MZM� ,

Y1 = − 0.8�1.5� , Y2 = − 0.1�2� , for N = 3 �3d-MS� .

�4.12�

For N=3 the MZM and 3d-MS results are consistent andapparently indicate that Y1 and Y2 are negative, though withsomewhat large errors. A better understanding of the rel-evance of the two operators can be obtained from Fig. 1,where we give the distributions of the estimates of Y1 and Y2

FIG. 1. �Color online� Distribution of the results for Y1 �left�and Y2 �right� obtained by varying the resummation parameters �and b as a function of the number of loops in the MZM and 3d-MS schemes. Here N=3.

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obtained by varying the parameters � and b. For Y1 low-order calculations predict Y1�0. However, as the number ofloops increases, Y1 decreases. The six-loop MZM results in-dicate that Y1�0, a result that is also supported by the trendobserved in the 3d-MS results. As for Y2, the MZM resultsclearly indicate Y2�0. However, this is not fully confirmedby the 3d-MS results. Though they give Y2�0, there is atrend towards larger values of Y2. Overall, these results sup-port the stability of the collinear FP �2b� in the completetheory for N=3.

Similar conclusions hold for N=2. For completeness, wereport here the corresponding critical exponents:34

� = 0.71�7� , � = 0.12�1� , for N = 2 �MZM� ,

� = 0.76�10� , � = 0.11�6� , for N = 2 �3d-MS� ,

� = 0.89�16� , � = 0.18�3� , for N = 3 �MZM� ,

� = 0.88�22� , � = 0.10�10� , for N = 3 �3d-MS� .

�4.13�

For N=2 we also study the stability of the chiral FP �2a�.Using the results of Refs. 8 and 23 for the FP we have33

Y1 = − 0.03�7� , Y2 = 0.9�2� , �MZM� ,

Y1 = − 0.2�3� , Y2 = 0.73�15� , �3d-MS� .

�4.14�

The chiral FP is clearly unstable.Finally, note that the same discussion also applies to sub-

model �b�, since the stability of the FP depends on the sameoperators with RG dimensions Y1 and Y2. For submodel �a�one should only consider Y1. In this case also the chiral FP�2b� might be stable.

D. Stability of the mn FP for N=2

Here we investigate the stability of FP �3b� for N=2 �itdoes not exist for N=3�. For this purpose we must computethe RG dimensions of the perturbations of the mn FP appear-ing in the complete theory. This is done in Appendix C.There are two relevant operators with RG dimensions Y1 andY2. The analysis of the perturbative series in the MZMscheme gives

Y1 = − 4.0�2.6�, Y2 = − 0.6�2� . �4.15�

The results in the 3d-MS scheme are very imprecise, al-though negative values for Y1 and Y2 seem to be favored.There results indicate, although with limited confidence, thatthe mn FP present for N=2 may be stable in the completetheory.

V. CONCLUSIONS

In this paper we have studied the quantum phase transi-tion that occurs in two-dimensional systems that exhibit anordered phase with SDW order. The effective Hamiltonian ofthe relevant critical modes ai is given in Eq. �1.3�. A de-tailed mean-field analysis shows that in some parameter re-gion Hamiltonian �1.3� has a continuous transition separatinga spin-disordered phase from an ordered phase characterized

by two collinearly polarized SDW’s. There are two differentpossibilities for the axes of these SDW’s: either 1 �2 or1�2. We have then investigated the role of fluctuationsin a simplified model in which the two SDW’s have the samevelocity. For this purpose we have generated six-loop pertur-bative series in the MZM scheme and five-loop series indimensional regularization with minimal subtraction. The se-ries are reported in Ref. 15. Close to four dimensions, ananalytic �-expansion calculation shows no presence of stableFP’s. However, past experience indicates that a FP may existin three dimensions and be absent for ��1. Therefore, wehave considered two strictly three-dimensional schemes. Wehave analyzed the stability of some FP’s that belong toknown submodels. The analysis of the perturbative seriessupports the stability of the O�4� � O�3� collinear FP. Theanalyses of the MZM and 3d-MS expansions do not providesufficiently stable results for the RG flow in the full theory—i.e., in the space of its five quartic couplings. In particular,they do not allow us to draw any definite conclusion on theexistence of other stable FP’s. In any case, even without theanalysis of the full flow, simple considerations �reported inAppendix A� show that systems with collinear SDW’s withthe same axis �the mean-field solution �3� reported in Sec. II�are in the attraction domain of the O�4� � O�3� collinear FP.

It should be remarked that our RG analysis is only validfor v1=v2. In order to extend the results to the generic casev1�v2 one should also consider the operator

Ov = ��x1�2 − ��y1�2 − ��x2�2 + ��y2�2 �5.1�

and determine its RG dimension yv at the O�4� � O�3� col-linear FP. If yv�0, the previous conclusions are unchanged.On the other hand, if yv�0, the O�4� � O�3� collinear FP isunstable with respect to the perturbation Ov. In this case thetransition may be of first order or continuous depending onthe existence and attraction domain of a stable FP with v1�v2. Note that, from a practical point of view, our results areof interest even if yv�0. Indeed, one expects the SDW ve-locities v1 and v2 to be close in magnitude, of the order ofthe spin-wave velocity of the Néel state of the undopedinsulator.4 Therefore, the RG flow always starts very close tothe stable FP of the theory with v1=v2, and thus the criticalbehavior is controlled by this FP except in a narrow intervalaround the critical doping.

Experiments indicate that the SDW-SC–to–SC transitionis continuous and is associated with collinear SDW’s. It isthus natural to conjecture that its critical behavior is con-trolled by the O�4� � O�3� collinear FP, since this FP is stablein model �1.4� and its basin of attraction includes systemswith collinear SDW’s. The corresponding critical exponentsare then predicted to be �=0.9�2�, �=0.15�10�.

ACKNOWLEDGMENT

M.D.P. acknowledges that part of this work was done atthe University of Roma Tre as part of his Ph.D. thesis.

APPENDIX A: GROUND-STATE CONFIGURATIONS

In this appendix we compute the possible ground-stateconfigurations of Hamiltonian �1.4�, which allow us to iden-

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tify the possible symmetry-breaking patterns. We considertranslation-invariant configurations and the space-independent Hamiltonian density

H�1,2� = r��1�2 + �2�2� + H4�1,2� , �A1�

where H4 is the part of the Hamiltonian that is quartic in thefields. Since H4�0 for stability, for r�0 the ground statealways corresponds to 1=2=0. For r�0, 1=2=0 is alocal maximum of H and thus the ground state is nontrivial.The value r=0 corresponds to a second-order transition pointin the mean-field approximation. In order to determine theground states for r�0, we will first determine all stationarypoints of H; the ground state is the one with the lowest

energy. Note that, if ̄1, ̄2 is a stationary point, then

H�̄1,̄2� =r

2��̄1�2 + �̄2�2� = − H4�̄1,̄2� . �A2�

This relation is quite general. Indeed, assume H to be of theform

H = �ij

rij�i� j + �ijkl

gijkl�i� j�k�l. �A3�

Then

H =1

4�i

�i�H

��i+

1

2�ij

rij�i� j

=1

2�i

�i�H

��i− �

ijkl

gijkl�i� j�k�l. �A4�

On a stationary solution, the derivative vanishes, provingEq. �A2�.

The calculation of the ground states also allows us todetermine the stability domain of the Hamiltonian. Indeed, apoint in the coupling space does not belong to the stabilitydomain if there is a field such that H4�0. H4 being homo-geneous, it is not restrictive to consider only fields such that�1�2+ �2�2=1. Thus, the determination of the minima of H4is equivalent to the determination of the minima of �H−r�where r is now interpreted as a Lagrange multiplier. Equa-tion �A2� shows that H4 can be negative only for r�0. Thus,the stability domain of H4 is obtained by determining thestationary points of H for r positive.

In order to determine the minima, we can use the symme-try of the Hamiltonian. Using the O�N� symmetry we canalways write

Re 1 = �a,0, . . . �, Im 1 = �b,c,0, . . . � . �A5�

Then, by using the U�1� symmetry we can also fix b=0.Indeed, we first perform an O�2� rotation on the first twocomponents:

�1� = �1cos � + �2sin �, �2� = − �1sin � + �2cos � ,

�A6�

where � is either Re 1 or Im 1. Then, we apply a U�1�rotation 1�=ei�1�. If we choose

tan 2� =2bc

a2 + b2 − c2 , tan � =a sin �

b sin � − c cos �, �A7�

the transformed field has the form �A5� with b=0. Once 1has been fixed, we can use O�N−2� and U�1� rotations towrite

2 = �d + ie,g + if ,l + ih,im,0, . . . � . �A8�

If N�3, one can use U�1� rotations to set l=0.The analysis of the minima is nontrivial due to the com-

plexity of the stationarity equations. We have only consid-ered the case N�3 that is relevant experimentally. Otherground states are present for N�4. We found seven relevantminima �only five of them occur for N=2�:

�1� a2=−r /u12, H=−r2 / �2u12�,�2� a2=c2=−r /u1 , H=−r2 / �2u1�,�3� a2=d2=−r / �u12+w1+w+� , H=−r2 / �u12+w1+w+�,�4� a2= f2=−r / �u12+w1� , H=−r2 / �u12+w1�,�5a��5b� a2=c2=d2= f2=−r / �2u1+2w1+w+±w−�,H=−2r2 / �2u1+2w1+w+±w−�,�6� �N�3� c2=−r�u1−w1� /�6 , d2=h2=−r�u12−w1� /

�2�6� , H=−r2�u1+u12−2w1� / �2�6� , �6=u1u12−w12,

�7� �N�3, w±�0� a2=d2�0, c , f ,h�0, ad / �cf�=w+ /w−, a2+c2=d2+ f2+h2, with energy H=−r2�u2�w2

+w3�+w2w3� /�7, with �7= �u12+w1�w2w3+w+u2�u1+w1�.Alternatively, if we define the four vectors t1=Re 1, t2=Im 1, t3=Re 2, t4=Im 2, and tij = ti · tj, the solution canbe characterized more geometrically as follows: t11= t22= t33= t44=H / �2r�, t12= t34, t13= t24, and t14= t23, with

t122 = r2w2

2w32/�7

2, t132 = r2u2

2w22/�7

2, t142 = r2u2

2w32/�7

2.

�A9�

Whenever a component is not explicitly written, it vanishes.Moreover, we defined u12 u1+u2 , w+ w2+w3, and w− w2−w3 and we simplified the notation, writing u1 insteadof u1,0, etc. Beside the seven solutions reported above, forw−�0 we also found stationary points with e=g=h=0 and

df

ac=

− 4u22 − w−

2 + w+2 ± ��4u2

2 + w−2 − w+

2�2 − 16u22w−

2

4u2w−.

�A10�

A numerical analysis indicates that they are never absoluteminima of the Hamiltonian, and thus they are never relevantfor the ground-state calculation. For this reason, these solu-tions have not been included above. The computation of allstationary points is quite straightforward, except for solution�7�. We shall now briefly sketch how it is derived. Assumethat e=g=0 and a ,c ,d , f ,h�0 and define Ea= �1/a��H /�a,etc. Then,

f

c�Ef − Eh� = − adw− + cfw+ = 0. �A11�

Using this relation, we can rewrite Ea, Ec, Ed, and Ef aslinear equations in a2, c2, d2, f2, and h2. Considering alsoa2d2w−

2 =c2f2w+2, which follows from Eq. �A11�, we obtain a

system of equations that allows us to determine all compo-nents.

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Given the list of solutions, we can determine the stabilitydomain of the Hamiltonian. Using solutions �1�–�5�, we ob-tain the necessary conditions

u1 � 0, u12 � 0, u12 + w1 � 0, u12 + w1 + w+ � 0,

u1 + w1 +1

2�w+ ± w−� � 0. �A12�

These conditions are sufficient for N=2. For N�3 we mustalso consider solutions �6� and �7�. Solution �6� gives thenecessary condition

w1 � − �u1u12. �A13�

Numerically, we find that solution �7� is also relevant forstability, although we have not been able to write down aneasy condition.

For cuprates the relevant solutions are �3� and �4�. In viewof the possibility that the O�4� � O�3� FP is stable it is im-portant to understand to which ground state of the O�4�� O�3� Hamiltonian �4.1� they correspond. For generic n andm, m�n, model �4.1� is stable for g1,0�0 and ng1,0− �n−1�g2,0�0 and has two ground states depending on the signof g2,0: for g2,0�0 the ground state is chiral, while for g2,0�0 the ground state is collinear.18 The corresponding ener-gies are

H = −3nr2

2�ng1,0 − �n − 1�g2,0��chiral� ,

H = −3r2

2g1,0�collinear� .

�A14�

Using u12=g1,0 /3, u2=g2,0 /6, w1= �g1,0−g2,0� /3, and w+

=g2,0 /3, w−=0, we immediately see that for m=4 and n=2and n=3 solutions �1� and �3� correspond to the collinearcase. For n=3 solutions �6� and �7� correspond to the chiralcase, while solutions �2�, �4�, and �5� correspond to a station-ary state that is never a ground state in the chiral theory. Forn=2, instead, solutions �2�, �4�, and �5� are those correspond-ing to the chiral case. This result is relevant to identify theattraction domain of the O�4� � O�3� collinear FP in the fulltheory. Indeed, it shows that the attraction domain of this FPincludes systems whose ground state is given by solutions�1� and �3� �and therefore two collinearly polarized SDW’s�.Nothing can be said about the other solutions: in this case ananalysis of the RG flow of the full theory is needed.

APPENDIX B: RENORMALIZATION-GROUPDIMENSIONS OF THE PERTURBATIONS AT THE

O„4…‹O„N… FIXED POINTS

We need to classify the operators that break

O�4� � O�N� → �U�1� � U�1�� � O�N�

� �SO�2� � SO�2�� � O�N� . �B1�

This is essentially discussed in Ref. 24. There are, however,two differences: first, we have only SO�2� symmetry, insteadof O�2� symmetry; second, there is an additional exchange

symmetry that forbids the appearance of spin-2 operators. Inthe notations of Ref. 24 �M and N of Ref. 24 correspond to Nand 4, respectively� we define

P1 O1133�4,4� + O1144

�4,4� + O2233�4,4� + O2244

�4,4�, �B2�

P2 O1313�4,r� + O1414

�4,r� + O2323�4,r� + O2424

�4,r� ,

P3 O1234�4,r� ,

where �ei is the real field defined in Eq. �4.3�. Note that P3would be forbidden if we had O�2� invariance instead ofSO�2� invariance. Moreover, P2 and P3 correspond to differ-ent components of the same operator, so that they have thesame RG dimension.

Hamiltonian �1.4� can then be written as

H = ddx�a

1

2����a�2 + �a

2� + t1��a

�a2�2

+ t2�a,b

���a · �b�2 − �a2�b

2� + t3P1 + t4P2 + t5P3,

�B3�

where

t1 =1

24�2u1 + 2u2 + w1 + w2 + w3� ,

t2 =1

36�u1 + 4u2 − w1 + 2w2 + 2w3� ,

t3 =1

12�− u1 − u2 + w1 + w2 + w3� ,

t4 =1

12�− 2u1 + 4u2 + 2w1 − w2 − w3� ,

t5 =1

2�w3 − w2� . �B4�

Since all operators are irreducible with respect to O�4�� O�N� transformations, if the couplings belong to theO�4� � O�N� theory, the stability matrix defined with respectto the couplings ti has the form

� =��11 �21 0 0 0

�21 �22 0 0 0

0 0 �1 0 0

0 0 0 �2 0

0 0 0 0 �2

� . �B5�

Here Y1=−�1 is the RG dimension of Oabcd�4,4� and Y2=−�2 is

the RG dimension of Oabcd�4,r�.

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In the MZM scheme for N=2 we find

�1 = 1 − �0.238732u1 + 0.31831u2� + �0.0324838u12 + 0.0614494u1u2 + 0.0342428u2

2� − �0.00570145u13 + 0.016556u1

2u2

+ 0.0218393u1u22 + 0.00451785u2

3� + �0.00156398u14 + 0.0052904u1

3u2 + 0.0097641u12u2

2 + 0.00663506u1u23

+ 0.000911563u24� − �0.00045218u1

5 + 0.00188296u14u2 + 0.00448727u1

3u22 + 0.00487778u1

2u23 + 0.00209129u1u2

4

+ 0.000250855u25� + �0.000158666u1

6 + 0.000747711u15u2 + 0.00212573u1

4u22 + 0.00311938u1

3u23 + 0.00230887u1

2u24

+ 0.000759722u1u25 + 0.0000778366u2

6� , �B6�

�2 = 1 − �0.238732u1 − 0.159155u2� + �0.0324838u12 − 0.0145415u1u2 − 0.0164178u2

2� − �0.00570145u13 − 0.00426028u1

2u2

− 0.00222856u1u22 + 0.00294388u2

3� + �0.00156398u14 − 0.000683968u1

3u2 + 9.94721 · 10−6u12u2

2 + 0.000950042u1u23

− 0.000152361u24� − �0.00045218u1

5 − 0.000214037u14u2 + 0.00011481u1

3u22 + 0.000428198u1

2u23 − 0.0000869867u1u2

4

− 0.0000326009u25� + �0.000158666u1

6 − 0.000038529u15u2 + 0.000128979u1

4u22 + 0.000233286u1

3u23 − 0.0000225715u1

2u24

− 0.0000164528u1u25 + 8.9658 10−6u2

6� . �B7�

For N=3 we obtain:

�1 = 1 − �0.238732u1 + 0.397887u2� + �0.0378782u12 + 0.0687202u1u2 + 0.0413963u2

2� − �0.00623158u13 + 0.0193945u1

2u2

+ 0.0290399u1u22 + 0.00434652u2

3� + �0.0020362u14 + 0.00594275u1

3u2 + 0.0126485u12u2

2 + 0.00940724u1u23

+ 0.000356974u24� − �0.000572885u1

5 + 0.00220348u14u2 + 0.00583754u1

3u22 + 0.0069323u1

2u23 + 0.00263869u1u2

4

+ 0.000106569u25� + �0.000227195u1

6 + 0.000881969u15u2 + 0.00277192u1

4u22 + 0.00439833u1

3u23 + 0.00335526u1

2u24

+ 0.000841582u1u25 + 0.0000352387u2

6� , �B8�

�2 = 1 − �0.238732u1 − 0.0795775u2� + �0.0378782u12 − 0.00727073u1u2 − 0.0314283u2

2� − �0.00623158u13 − 0.00255553u1

2u2

− 0.00424767u1u22 + 0.00445854u2

3� + �0.0020362u14 − 0.000329962u1

3u2 − 0.000353201u12u2

2 + 0.00115746u1u23

− 0.000132244u24� − �0.000572885u1

5 − 0.000151507u14u2 + 0.000153088u1

3u22 + 0.000587068u1

2u23 − 0.0000150878u1u2

4

− 0.0000901012u25� + �0.000227195u1

6 − 0.0000200211u15u2 + 0.000205186u1

4u22 + 0.000321211u1

3u23 + 0.000031431u1

2u24

− 0.0000663864u1u25 + 0.0000193321u2

6� . �B9�

In the 3d-MS we find, for N=2,

�1 = 1 − �6u1 + 8u2� + �30.5u12 + 58u1u2 + 32u2

2� − �327.297u13 + 928.74u1

2u2 + 1185.79u1u22 + 258.397u2

3� + �5835.31u14

+ 20132.4u13u2 + 35648.3u1

2u22 + 23296.2u1u2

3 + 4377.86u24� − �123668u1

5 + 506531u14u2 + 1.1389 106u1

3u22 + 1.1852

106u12u2

3 + 552355u1u24 + 85949u2

5� , �B10�

�2 = 1 − �6u1 − 4u2� + �30.5u12 − 14u1u2 − 16u2

2� − �327.297u13 − 219.798u1

2u2 − 163.596u1u22 + 103.301u2

3� + �5835.31u14

− 3001.7u13u2 − 2298.63u1

2u22 + 1415.85u1u2

3 + 198.484u24� − �123668u1

5 − 48887.2u14u2 − 24466.2u1

3u22 + 34690.3u1

2u23

− 12396.4u1u24 − 4256.31u2

5� . �B11�

And finally, for N=3,

�1 = 1 − �6u1 + 10u2� + �35.5u12 + 65u1u2 + 38.5u2

2� − �369.646u13 + 1082.64u1

2u2 + 1548.49u1u22 + 251.598u2

3� + �7381.82u14

+ 23673.0u13u2 + 45677.7u1

2u22 + 31394.4u1u2

3 + 3819.3u24� − �169602u1

5 + 615152u14u2 + 1.48093 106u1

3u22 + 1.61998

106u12u2

3 + 719990u1u24 + 76742u2

5� , �B12�

�2 = 1 − �6u1 − 2u2� + �35.5u12 − 7u1u2 − 30.5u2

2� − �369.646u13 − 130.399u1

2u2 − 306.317u1u22 + 164.3u2

3� + �7381.82u14

− 1861.44u13u2 − 4739.46u1

2u22 + 2138.26u1u2

3 + 581.765u24� − �169602u1

5 − 33233.6u14u2 − 55928.8u1

3u22 + 53320.3u1

2u23

− 13607.5u1u24 − 6933.24u2

5� . �B13�

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APPENDIX C: RENORMALIZATION-GROUPDIMENSIONS OF THE PERTURBATIONS

AT THE mn FIXED POINTS

The analysis of the perturbations at the mn FP is quitesimple. In our case m=2N, n=2 and the relevant symmetrygroup is O�2N�, which is broken by the terms proportional tou2, w2, and w3. If �ai is the field defined in Eq. �4.6�, a=1,2, i=1, . . .2N, we define the following spin-2 and spin-4operators that transform irreducibly under O�2N�:

Va,i,j�2� �ai�aj −

1

2N�ij�a

2, �C1�

Va,i,j,k,l�4� �ai�aj�ak�al −

1

2�N + 2��a

2��ij�ak�al + 5 perm�

+1

4�N + 1��N + 2���a

2�2��ij�kl + �ik� jl + �il� jk� ,

�C2�

where �a2 �i�ai

2 . Then, the relevant operators are

P1 �a=1

2

���=0

1

�ij=1

N

Va,i+�N,i+�N,j+�N,j+�N�4� , �C3�

P2 ���=0

1

�ij=1

N

V1,i+�N,j+�N�2� V2,i+�N,j+�N

�2� , �C4�

P3 �����=0

1

�ij=1

N

������V1,i+�N,j+�N�2� V2,i+�N,j+�N

�2� , �C5�

where �01=−�10=1 and �00=�11=0. These operators give riseto different breakings of O�2N�:

O�2N�→P1

�O�N� � O�2�� � �O�N� � O�2��→P2

O�N� � �O�2�

� O�2��→P3

O�N� � S�O�2� � O�2�� . �C6�

In terms of P1, P2, and P3 Hamiltonian �1.4� can then bewritten as

H = ddx �a

1

2����a�2 + �a

2� + t1�a

��a2�2

+ t2�12�2

2 + t3P1 + t4P2 + t5P3, �C7�

where

t1 =u1

2+

u2

N + 1, t2 = w1 +

1

N�w2 + w3�, t3 = u2,

t4 = w2 + w3, t5 = w3 − w2. �C8�

Since all operators are irreducible with respect to O�2N�transformations, if the couplings belong to the mn theory, thestability matrix defined with respect to the couplings ti hasthe form

� =��11 �21 0 0 0

�21 �22 0 0 0

0 0 �1 0 0

0 0 0 �2 0

0 0 0 0 �2

� . �C9�

Note that two eigenvalues are degenerate, since P2 and P3are different components of the same irreducible operatorV1,i,j

�2� V2,k,l�2� . The corresponding RG dimensions are Y1=−�1,

Y2=−�2.

*Electronic address: Deprato@mf.mpg.de†Electronic address: Andrea.Pelissetto@roma1.infn.it‡Electronic address: Ettore.Vicari@df.unipi.it1 S. Sachdev, Rev. Mod. Phys. 75, 913 �2003�.2 S. Wakimoto, G. Shirane, Y. Endoh, K. Hirota, S. Ueki, K. Ya-

mada, R. J. Birgeneau, M. A. Kastner, Y. S. Lee, P. M. Gehring,and S. H. Lee, Phys. Rev. B 60, R769 �1999�; S. Wakimoto, R.J. Birgeneau, Y. S. Lee, and G. Shirane, ibid. 63, 172501 �2001�.

3 G. Aeppli, T. E. Mason, S. M. Hayden, H. A. Mook, and J. Kulda,Science 278, 1432 �1997�.

4 Y. Zhang, E. Demler, and S. Sachdev, Phys. Rev. B 66, 094501�2002�.

5 Y. S. Lee, R. J. Birgeneau, M. A. Kastner, Y. Endoh, S.Wakimoto, K. Yamada, R. W. Erwin, S.-H. Lee, and G. Shirane,Phys. Rev. B 60, 3643 �1999�.

6 K. G. Wilson and M. E. Fisher, Phys. Rev. Lett. 28, 240 �1972�.7 See, for example, S. Mo, J. Hove, and A. Sudbø, Phys. Rev. B

65, 104501 �2002�, and references therein.8 P. Calabrese, P. Parruccini, A. Pelissetto, and E. Vicari, Phys. Rev.

B 70, 174439 �2004�.

9 M. De Prato, A. Pelissetto, and E. Vicari, Phys. Rev. B 70,214519 �2004�.

10 R. Schloms and V. Dohm, Nucl. Phys. B 328, 639 �1989�.11 G. Parisi, J. Stat. Phys. 23, 49 �1980�.12 G. ’t Hooft, and M. J. G. Veltman, Nucl. Phys. B 44, 189 �1972�.13 B. G. Nickel, D. I. Meiron, and G. A. Baker, Jr.,Guelph Univer-

sity Report, 1977 �unpublished�.14 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of

�4-Theories �World Scientific, Singapore, 2001�.15 See EPAPS Document, No. E-PRBMDO-74-057634 for the five-

loop MS and six-loop MZM series of model �1.4�, of theO�N� � O�M� model �4.1�, and of the mn model �4.5�. Thisdocument can be reached via a direct link in the online article’sHTML reference section or via the EPAPS homepage �http://www.aip.org/pubservs/epaps.html�.

16 O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. B 57,1422 �1998�.

17 A. Pelissetto, P. Rossi, and E. Vicari, Nucl. Phys. B 607, 605�2001�.

18 H. Kawamura, J. Phys.: Condens. Matter 10, 4707 �1998�.

DE PRATO, PELISSETTO, AND VICARI PHYSICAL REVIEW B 74, 144507 �2006�

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19 P. Calabrese and P. Parruccini, Nucl. Phys. B 679, 568 �2004�.20 A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 �2002�.21 B. Delamotte, D. Mouhanna, and M. Tissier, Phys. Rev. B 69,

134413 �2004�.22 A. Pelissetto, P. Rossi, and E. Vicari, Phys. Rev. B 63, 140414�R�

�2001�.23 P. Calabrese, P. Parruccini, and A. I. Sokolov, Phys. Rev. B 66,

180403 �2002�; , 68, 094415 �2003�.24 P. Calabrese, A. Pelissetto, and E. Vicari, Nucl. Phys. B 709, 550

�2005�.25 P. Parruccini, Phys. Rev. B 68, 104415 �2003�.26 For N=2 the collinear FP is �Ref. 24� u1=6.3�4�, u2=−8.3�3�

�MZM� and u1=0.33�3�, u2=−0.30�2� �3d-MS�. For N=3 thecollinear FP is u1=4.3�5�, u2=−7.5�6� �MZM� and u1=0.23�5�,u2=−0.23�3� �3d-MS�.

27 A. Aharony, In Phase Transitions and Critical Phenomena, editedby C. Domb and M. S. Green �Academic, New York, 1976�, Vol.6.

28 A. Pelissetto and E. Vicari, Condens. Matter Phys. 8, 87 �2005�.29 J. Zinn-Justin, Quantum Field Theory and Critical Phenomena,

4th ed. �Clarendon Press, Oxford, 2001�.

30 A. Pelissetto and E. Vicari, Phys. Rev. B 62, 6393 �2000�.31 At the chiral FP we use �Refs. 8 and 24� �=0.60�8�, y11=2.3�2�,

and y02=1.52�6� �N=3� and �=0.63�9�, y11=2.45�25�, and y02

=1.30�15� �N=2�. At the collinear FP, for N=3 we use �Refs. 9and 24� �=0.60�5�, y11=1.20�15�, and y02=2.0�2�. For N=2, weuse the mapping with the XY model �Refs. 18 and 24� and theresults of M. Campostrini, M. Hasenbusch, A. Pelissetto, and E.Vicari, Phys. Rev. B74 �2006� �in press�,�=0.6717�1�, y11

=1/�=1.4888�2�, and y02=2−�=1.9619�2�.32 J. M. Carmona, A. Pelissetto, and E. Vicari, Phys. Rev. B 61,

15136 �2000�.33 We use u1=−1.1�5�, u2=6.1�4� �MZM� and u1=0.03�2�, u2

=0.215�13� �3d-MS�.34 It is interesting to note that the exponents of the O�4� � O�2�

model are also relevant for quantum chromodynamics �thetheory of strong interactions� with two quarks at finite tempera-ture if the anomaly contribution at the critical point is small�F. Basile, A. Pelissetto, and E. Vicari, POS �LAT2005� 199;hep-lat/0509018 �unpublished�; J. High Energy Phys. 02, 044�2005�; A. Butti, A. Pelissetto, and E. Vicari, ibid. 08, 029�2003��.

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