# XY frustrated systems: continuous exponents in discontinuous phase transitions

**ABSTRACT** XY frustrated magnets exhibit an unsual critical behavior: they display scaling laws accompanied by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they undergo weak first order phase transitions. We show that all perturbative approaches that have been used to investigate XY frustrated magnets fail to reproduce these features. Using a nonperturbative approach based on the concept of effective average action, we are able to account for this nonuniversal scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems. Comment: 11 pages, 3 figures, revised and extended version

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**ABSTRACT:**In view of its physical importance in predicting the order of chiral phase transitions in QCD and frustrated spin systems, we perform the conformal bootstrap program of $O(n)\times O(2)$-symmetric conformal field theories in $d=3$ dimensions with a special focus on $n=3$ and $4$. The existence of renormalization group fixed points with these symmetries has been controversial over years, but our conformal bootstrap program provides the non-perturbative evidence. In both $n=3$ and $4$ cases, we find singular behaviors in the bounds of scaling dimensions of operators in two different sectors, which we claim correspond to chiral and collinear fixed points, respectively. In contrast to the cases with larger values of $n$, we find no evidence for the anti-chiral fixed point. Our results indicate the possibility that the chiral phase transitions in QCD and frustrated spin systems are continuous with the critical exponents that we predict from the conformal bootstrap program.07/2014; - SourceAvailable from: Karim Essafi[Show abstract] [Hide abstract]

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**ABSTRACT:**The non-perturbative renormalization group (NPRG), in its modern form, constitutes an efficient framework to investigate the physics of systems whose long-distance behavior is dominated by strong fluctuations that are out of reach of perturbative approaches. We present here the basic principles underlying the NPRG and illustrate its power in the context of two longstanding problems of condensed matter and soft matter physics: the nature of the phase transition occuring in frustrated magnets in three dimensions and the phase diagram of polymerized phantom membranes.Modern Physics Letters B 11/2011; 25(12n13). · 0.69 Impact Factor

Page 1

arXiv:cond-mat/0107183v2 [cond-mat.stat-mech] 12 Mar 2003

XY frustrated systems: continuous exponents in discontinuous phase transitions.

M. Tissier∗

Laboratoire de Physique Th´ eorique et Mod` eles Statistiques,

Universit´ e Paris Sud, Bat. 100, 91405 Orsay Cedex, France.

B. Delamotte†and D. Mouhanna‡

Laboratoire de Physique Th´ eorique et Hautes Energies,

Universit´ es Paris VI-Pierre et Marie Curie - Paris VII-Denis Diderot,

2 Place Jussieu, 75252 Paris Cedex 05, France.

(Dated: February 1, 2008)

XY frustrated magnets exhibit an unsual critical behavior: they display scaling laws accompanied

by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they

undergo weak first order phase transitions. We show that all perturbative approaches that have been

used to investigate XY frustrated magnets fail to reproduce these features. Using a nonperturbative

approach based on the concept of effective average action, we are able to account for this nonuniversal

scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems.

PACS numbers: 75.10.Hk,11.10.Hi,11.15.Tk,64.60.-i

I.INTRODUCTION

After twenty-five years of intense activity, the physics

of XY and Heisenberg frustrated systems is still the sub-

ject of a great controversy concerning, in particular, the

nature of their phase transitions in three dimensions (see

for instance Ref.1 for a review). On the one hand, a

recent high-order perturbative calculation2,3predicts in

both cases a stable fixed point in three dimensions and,

thus, a second order phase transition. On the other hand,

a nonperturbative approach, the effective average action

method, based on a Wilson-like Exact Renormalization

Group (ERG) equation, leads to first order transitions4.

Actually, it turns out that, in the Heisenberg case, these

two theoretical approaches are almost equivalent from

the experimental viewpoint (see however Ref.5). Indeed,

within the ERG approach, the transitions are found to be

weakly of first order and characterized by very large corre-

lation lengths and pseudo-scaling associated with pseudo-

critical exponents close to the exponents obtained within

the perturbative approach. This occurence of pseudo-

scaling and quasi-universality has been explained within

ERG approaches by the presence a local minimum in the

speed of the flow4,6, related to the presence of a complex

fixed point with small imaginary parts, called pseudo-

fixed point6.

XY frustrated magnets are rather different from this

point of view since their nonperturbative RG flows dis-

play neither a fixed point nor a minimum.

in this article that they nevertheless generically exhibit

large correlation lengths at the transition and thus,

pseudo-scaling, but now without quasi-universality. More

precisely, we show that quantities like correlation length

and magnetization behave as powers of the reduced tem-

perature on several decades. A central aspect of our ap-

proach is that, although the RG flow displays neither a

fixed point nor a minimum, it remains sufficiently slow

in a large domain in coupling constant space to produce

We show

generically large correlation lengths and scaling behav-

iors. We argue that our approach allows to account for

the striking properties of the XY frustrated magnets like

the XY Stacked Triangular Antiferromagnets (STA) such

as CsMnBr3, CsNiCl3, CsMnI3, CsCuCl3, as well as XY

helimagnets such as Ho, Dy and Tb, which display scal-

ing at the transition without any evidence of universal-

ity. Our conclusions are in marked contrast with those

drawn from the perturbative approach of Pelissetto et al.

2,3which leads to predict a second order phase transition

for XY frustrated magnets.

II.

DISTANCE EFFECTIVE HAMILTONIAN

THE STA MODEL AND ITS LONG

The prototype of XY frustrated systems is given by the

STA model. It consists of spins located on the sites of

stacked planar triangular lattices. Its hamiltonian reads:

H =

?

?ij?

Jij?Si.?Sj

(1)

where the?Si are two-component vectors and the sum

runs on all pairs of nearest neighbors. The spins inter-

act antiferromagnetically inside the planes and either fer-

romagnetically or antiferromagnetically between planes,

the nature of this last interaction being irrelevant to

the long distance physics. Due to the intra-plane an-

tiferromagnetic interactions the system is geometrically

frustrated and the spins exhibit a 120◦structure in the

ground state (see FIG. 1.a). As H is invariant under

rotation, other ground states can be built by rotating

simultaneously all the spins.

Let us describe the symmetry breaking scheme of

the STA model in the continuum limit.

temperature phase, the hamiltonian (1) is invariant un-

der the SO(2)×Z Z2group acting in the spin space and the

In the high-

Page 2

2

?

??

??

?

?

?

?

?

?

?

?

?

?

?

?

?

?

?

FIG. 1: The ground state configurations a) of the spins on the

triangular lattice and b) of the order parameter made of two

orthonormal vectors. The three-dimensional structure of the

ground state is obtained by piling these planar configurations.

O(2) group associated to the symmetries of the triangu-

lar lattice43. In the low-temperature phase, the residual

symmetries are given by the group O(2)diagwhich is a

combination of the group acting in spin space and of the

lattice group. The symmetry breaking scheme is given

by7,8:

G = O(2) × SO(2) × Z Z2→ H = O(2)diag(2)

and thus consists in a fully broken SO(2)×Z Z2group. The

Z Z2degrees of freedom are known as chirality variables.

Due to the 120◦structure, the local magnetization,

defined on each elementary plaquette as:

?Σ =?S1+?S2+?S3

(3)

vanishes in the ground state and cannot constitute the

order parameter. In fact, as in the case of colinear antifer-

romagnets, one has to build the analogue of a staggered

magnetization. It is given by a pair of two-component

vectors?φ1and?φ2— defined at the center x of each el-

ementary cell of the triangular lattice — that are or-

thonormal in the ground state7,8,9(see FIG. 1.b). They

can be conveniently gathered into a square matrix:

Φ(x) = (?φ1(x),?φ2(x)) . (4)

Once the model is formulated in terms of the order

parameter, the interaction, originally antiferromagnetic,

becomes ferromagnetic. It is thus trivial to derive the

effective low-energy hamiltonian relevant to the study of

the critical physics which writes:

H = −J

2

?

ddxTr?∂tΦ(x).∂Φ(x)?

(5)

where

It is convenient to consider, in the following, a general-

ization of the models (1) and (5) to N-component spins.

The order parameter consists in this case in a N ×2 ma-

trix and the symmetry-breaking scheme is thus given by

O(N)×O(2) → O(N−2)×O(2)diag. Frustrated magnets

thus correspond to a symmetry breaking scheme isomor-

phic to O(N) → O(N−2) that radically differs from that

of the usual vectorial model which is O(N) → O(N −1).

The matrix nature of the order parameter together with

the symmetry breaking scheme led naturally in the 70’s

to the hypothesis of a new universality class7,8,9— the

“chiral” universality class — gathering all materials sup-

posed to be described by the hamiltonian (1): STA and

helimagnets. As we now show, examining the current

state of the experimental and numerical data, there is,

in fact, no clear indication of universality in the critical

behavior of XY frustrated magnets.

tΦ denotes the transpose of Φ.

III. THE EXPERIMENTAL AND NUMERICAL

CONTEXT

A. The experimental situation

Two kinds of materials are supposed to undergo a

phase transition corresponding to the symmetry breaking

scheme described above: the STA — CsMnBr3, CsNiCl3,

CsMnI3, CsCuCl3— (see Ref.10 for RbMnBr3) and the

helimagnets: Ho, Dy and Tb. The corresponding critical

exponents are given in Table I.

Note first that, concerning all these data, only one er-

ror bar is quoted in the literature, which merges sys-

tematic and statistical errors. We start by making the

hypothesis that these error bars have a purely statistical

origin. Under this assumption, we have computed the

— weighted — average values of the exponents and their

error bars. This is the meaning of the numbers we give in

the following. This hypothesis is however too na¨ ıve, and

we have checked that, if we attribute a large part of the

error bars quoted in Table I to systematic bias — typi-

cally 0.1 for β and 0.2 for ν —, our conclusions still hold.

We also make the standard assumptions that the mea-

sured exponents govern the leading scaling behavior, i.e.

the determination of the critical exponents is not signif-

icantly affected by corrections to scaling. This is gener-

ically assumed in magnetic materials where corrections

to scaling are never needed to reproduce the theoretical

results in the range of reduced temperature reachable in

experiments44. This is different for fluids where the scal-

ing domain can be very large. Moreover, since the error

Page 3

3

CsMnBr3

α=0.39(9), 0.40(5), 0.44(5)

β=0.21(1), 0.21(2), 0.22(2), 0.24(2), 0.25(1)

γ=1.01(8), 1.10(5); ν=0.54(3), 0.57(3)

α=0.342(5), 0.37(6), 0.37(8); β=0.243(5) CsNiCl3

CsMnI3

CsCuCl3

Tb

Ho

α=0.34(6)

α=0.35(5); β=0.23(2), 0.24(2), 0.25(2)

α=0.20(3); β=0.21(2), 0.23(4); ν=0.53

β=0.30(10), 0.37(10), 0.39(3),

0.39(2), 0.39(4), 0.39(4), 0.41(4)

γ=1.14(10), 1.24(15); ν=0.54(4), 0.57(4)

β=0.38(2), 0.39(1), 0.39+0.04

γ=1.05(7); ν=0.57(5)

α= 0.34(6), 0.43(10), 0.46(10)

β= 0.24(2), 0.253(10); γ=1.03(4), 1.13(5)

ν=0.48(2), 0.50(1), 0.54(2)

α=0.29(9); β=0.31(2); γ=1.10(4); ν=0.57(3)

Dy

−0.02

STA

Monte Carlo

six-loop

TABLE I: Critical exponents of the XY frustrated models,

from Refs.2,11-15 and references therein. For CsCuCl3 the

transition has been found of first order and the exponents

mentioned here hold only for a reduced temperature larger

that 5.10−3(see Ref.16).

bars in frustrated systems are much larger than in the

usual ferromagnetic systems — by a factor five to ten,

see Table I — neglecting corrections to scaling should

not bias significantly our analysis.

Under these assumptions we can analyze the data. We

find that there are three striking facts:

i) there are two groups of incompatible exponents. The

average value of β, the best measured exponent, for

CsMnBr3, CsNiCl3and Tb — called group 1 — is given

by β ∼ 0.23. It is incompatible with that of Ho and Dy

— group 2 — which is β ∼ 0.39 (see Table I for details).

Note that for CsCuCl3, whose exponents are compatible

with those of group 1, the transition has been found to

be very weakly of first order16.

ii) the exponents vary much from compound to compound

in group 1. For instance, the values of α for CsNiCl3and

CsMnBr3are only marginally compatible.

iii) the anomalous dimension η is significantly negative

for group 1. For CsMnBr3, the value of η determined

by the scaling relation η = 2β/ν − 1 with β = 0.227(6)

and ν = 0.555(21) is η = −0.182(38). The inclusion of

the data coming from CsNiCl3and Tb does not change

qualitatively this conclusion.

Several conclusions follow from the analysis of the

data. From point i), it appears that materials that are

supposed to belong to the same universality class dif-

fer as for their critical behavior. There are essentially

three ways to explain this. In the first one, the two sets

of exponents correspond to two true second order phase

transitions, each one being described by a fixed point. In

the second, one set corresponds to a true second order

transition and the other to pseudo-critical exponents as-

sociated to weakly first order transitions. In the third,

all transitions are weakly of first order.

The first scenario can be ruled out since η is negative

for group 1 (point iii)) while it cannot be so in a second

order phase transition when the underlying field theory

is a Ginzburg-Landau ϕ4-like theory17, as it is the case

here7. The transition undergone by CsMnBr3, CsNiCl3

and Tb is therefore very likely not continuous but weakly

of first order. This would explain the lack of universality

for the exponents of group 1 (point ii)).

In the second scenario, the materials of group 2 un-

dergo a second order phase transition — η is found pos-

itive there — while those of group 1 all undergo weakly

first order transitions with pseudo-scaling and pseudo-

critical exponents. Note that although this scenario can-

not be excluded, it is quite unnatural in terms of the

usual picture of a second order phase transition. Indeed,

it would imply a fine-tuning of the microscopic coupling

constants — i.e. of the initial conditions of the flow —

for the materials of group 1 in such a way that they lie

out of, but very close to, the border of the basin of at-

traction of the fixed point governing the critical behavior

of materials of group 2.

The third scenario, that of generic weak first order

behaviors for the two groups of materials, seems even

more unnatural, at least in the usual explanation of weak

first order phase transitions.

Actually, we shall provide arguments in favor of this

last scenario. Also, as we shall see in the framework

of the effective average action, the generic character of

pseudo-scaling in this scenario has a natural explanation,

not relying on the concept of fixed point. Then, no fine-

tuning of parameters is required to explain generically

weak first order behaviors in frustrated systems.

B.The numerical situation

There are no convincing numerical data concerning he-

limagnets. For the STA system, three different versions

have been simulated:

1) the STA itself18,19,20.

2) The STAR (Ref.15) — with R for rigid — which

consists in a STA where the fluctuations of the spins

around their ground state 120◦structure have been

frozen.This is realized by imposing the rigidity con-

straint?Σ =?S1+?S2+?S3= 0 at all temperatures.

3) A discretized version of the hamiltonian (5), called

the Stiefel V2,2model15. There, one considers a system

of dihedrals interacting ferromagnetically, which is rep-

resented on FIG. 1.b.

At this stage, we emphasize that the rigidity constraint

?Σ = 0 which is imposed in the STAR, as well as the

formal manipulations leading to the Stiefel V2,2 model

affects only the massive — non critical — modes. Thus,

all the STA, STAR and Stiefel models have the same

critical modes, the same symmetries and thus the same

order parameter. One thus could expect a priori that

they all exhibit the same critical behavior.

For the STA system, scaling laws are found18,19,20

so that a second order behavior could be inferred.

The STAR and V2,2 models both undergo first order

Page 4

4

transitions15. Therefore, by changing microscopic details

to go from the STA model to the STAR or V2,2models,

the nature of transitions appears to change drastically.

This situation indicates that if STA undergoes a genuine

second order phase transition, the critical behavior of

frustrated magnets in general is characterized by a low

degree of universality, a conclusion already drawn from

the experimental situation.

With these behaviors one is brought back to the two

last scenarios proposed in the previous section: i) the be-

havior of the STA system is controlled by a fixed point

while the STAR and Stiefel models lie outside its basin of

attraction ii) all systems undergo first order phase tran-

sitions.

In fact, as shown in Ref.15, using the two scaling rela-

tions η = 2β/ν−1 and η = 2−γ/ν, η is found to be neg-

ative in STA systems — although less significantly than

in experiments — for all simulations where these calcu-

lations can be performed. One can thus suspect a (weak)

first order behavior even for the STA system. This hy-

pothesis is strengthened by a recent work of Itakura who

has employed Monte Carlo RG techniques in order to in-

vestigate the critical behavior of both the STA system

and the Stiefel model21. Using systems with lattice sizes

up to 126×144×126, he has provided evidences for weak

first order behaviors.

Let us draw first conclusions from the experimental

and numerical situations.It appears that the critical

physics of frustrated magnets cannot be explained in

terms of a single — universal — second order phase tran-

sition. A careful analysis of the experimental and numer-

ical data seems to indicate that a whole class of materi-

als undergo (weak) first order phase transitions. At this

stage, no conclusion can be drawn about the existence

or absence of a true fixed point controlling the physics of

some realizations of frustrated magnets. To clarify this

issue, we now present the theoretical situation.

IV.THE THEORETICAL SITUATION

The early RG studies of the STA and helimagnets —

and its generalization to N-component spins — was per-

formed in a double expansion in coupling constant and in

ǫ = 4−d on the Ginzburg-Landau-Wilson (GLW) version

of the model in Refs.7,9,22,23. It has appeared that, for

a given dimension d, there exists a critical number of spin

component, called Nc(d), above which the transition is

of second order and below which it is of first order. Nat-

urally, a great theoretical challenge in the study of frus-

trated magnets, has been the determination of Nc(d). Its

value has been determined within perturbative computa-

tion at three-loop order24:

Nc(4 − ǫ) = 21.8 − 23.4ǫ+ 7.09ǫ2+ O(ǫ3) . (6)

Unfortunately, this series is not well behaved since the co-

efficients are not decreasing fast. It has been conjectured

by Pelissetto et al.24that Nc(2) = 2. Using this con-

jecture, these authors have reexpressed (6) in the form:

Nc(4−ǫ) = 2+(2−ǫ)(9.9−6.77ǫ+0.16ǫ2)+O(ǫ3) . (7)

The coefficients of this expression are now rapidly de-

creasing so that it can be used to estimate Nc(d). For

d = 3 it provides Nc(3) = 5.3 and leads to the conclu-

sion that the transition is of first order in the relevant

Heisenberg and XY cases.

In agreement with this result, the perturbative ap-

proaches performed at three loops, either in 4 − ǫ or

directly in three dimensions, lead to a first order phase

transition for XY systems with a Nc(3) given respectively

by Nc(3) = 3.91 (Ref.25) and Nc(3) = 3.39 (Ref.26).

However, according to the authors, these computations

are not well converged. It is only recently that a six-loop

calculation has been performed2,3directly in three di-

mensions which is claimed to be converged in the Heisen-

berg and XY cases. Note that, for values of N between

N ≃ 5 and N ≃ 7, the resummation procedures do not

lead to converged results, forbidding the authors to com-

pute Nc(3) in this way. For N = 2 and N = 3 a fixed

point is found. The exponents associated to the N = 2

case are given in Table I. Note that γ and ν compare

reasonably well with the experimental data of group 1.

However, as we already stressed, the existence of a fixed

point implies η > 0 — η = 0.08 in Ref.2 — and is thus

incompatible with the negative value of η found for the

group 1. Moreover, the value β = 0.31(2) found in Ref.2

is far — 4 standard deviations — from the average exper-

imental value β = 0.237(6) for group 1 and also far — 3.7

standard deviations — from that obtained from group 2,

Ho and Dy: β = 0.388(7). It is thus incompatible with

the two sets of experimental values. This point strongly

suggests that the six-loop fixed point neither describes

the physics of materials belonging to group 2 that, in

the simplest hypothesis, should also undergo a first order

phase transition.

The preceding discussion does not rule out the exis-

tence of the fixed point found in Ref.2. This just shows

that, if it exists, it must have a very small basin of attrac-

tion, and that the initial conditions corresponding to the

STA and helimagnets lie out of it. In fact, as we argue in

the following, this fixed point probably does not exist at

all so that we expect that all transitions are of (possibly

very weak) first order.

V.THE EXACT RG APPROACHES

There exists an alternative theoretical approach to the

perturbative RG calculations which explains well, qual-

itatively and to some extent quantitatively, all the pre-

ceding facts. It relies on the Wilsonian RG approach to

critical phenomena, based on the concept of block spins

and scale dependent effective theories27,28. Although it

has been originally formulated in terms of hamiltonians,

Page 5

5

its most recent and successful implementation involves

the effective (average) action29,30,31. In the same way as

in the original Wilsonian approach, one constructs an ef-

fective action, noted Γk, that only includes high-energy

fluctuations — with momenta q2> k2— of the micro-

scopic system. At the lattice scale k = Λ = a−1, Γk

corresponds to the classical Hamiltonian H since no fluc-

tuation has been taken into account. When the running

scale k is lowered, Γkincludes more and more low-energy

fluctuations. Finally, when the running scale is lowered

to k = 0, all fluctuations have been integrated out and

one recovers the usual effective action or Gibbs free en-

ergy Γ. To summarize one has:

Γk=Λ= H

Γk=0= Γ .

(8)

Note also that the original hamiltonian depends on the

original spins while the effective action — at k = 0 —

is a function of the order parameter. At an intermediate

scale k, Γkis a function of an average order parameter at

scale k noted φk(q) — or more simply, φ(q) — that only

includes the fluctuations with momenta q2> k2. Thus

Γkhas the meaning of a free energy at scale k.

At a generic intermediate scale k, Γk is given as the

solution of an exact equation that governs its evolution

with the running scale32:

∂Γk[φ]

∂t

=1

2Tr

??

Γ(2)

k[φ] + Rk

?−1∂Rk

∂t

?

, (9)

where t = ln(k/Λ) and the trace has to be understood as

a momentum integral as well as a summation over inter-

nal indices. In Eq.(9), Γ(2)

k[φ] is the exact field-dependent

inverse propagator i.e. the second functional derivative

of Γk. The quantity Rkis the infrared cut-off which sup-

presses the propagation of modes with momenta q2< k2.

A convenient cut-off, that realizes the constraints (8), is

provided by33:

Rk(q2) = Z(k2− q2)θ(k2− q2) (10)

where Z is the k-dependent field renormalization.

Ideally, in order to relate the thermodynamical quan-

titites to the microscopic ones, one should integrate the

flow equation starting from k = Λ with the microscopic

hamiltonian H as an initial condition and decrease k

down to zero. However, Eq.(9) is too complicated to be

solved exactly and one has to perform approximations.

To render Eq.(9) manageable, one truncates the effective

action Γk[φ] to deal with a finite number of coupling con-

stants. The most natural truncation, well suited to the

study of the long distance physics of a field theory, is to

perform a derivative expansion32of Γk[φ]. This consists

in writing an ansatz for Γk as a series in powers of ∂φ.

The physical motivation for such an expansion is that

since the anomalous dimension η is small, terms with

high numbers of derivatives should not drastically affect

the physics.

Actually, another truncation is performed. It consists

in expanding the potential, which involves all powers of

the O(N) × O(2) invariants built out of?φ1 and?φ2, in

powers of the fields. This kind of approximation allows

to transform the functional equation (9) into a set of or-

dinary coupled differential equations for the coefficients

of the expansion. It has been shown during the last ten

years that low order approximations in the field expan-

sion give very good results (see Ref.34 for a review and

Ref.35 for an exhaustive bibliography).

such truncation is4:

The simplest

Γk=

?

ddx

?

Z

2Tr?∂tΦ.∂Φ?+ω

+λ

4

4(?φ1.∂?φ2−?φ2.∂?φ1)2+

?ρ

2− κ

?2

+µ

4τ

?

.

(11)

Let us first discuss the different quantities involved in

this expression. One recalls that Φ is the N × 2 ma-

trix gathering the N-component vectors?φ1and?φ2(see

Eq.(4)). There are two independent O(N) × O(2) in-

variants given by ρ = TrtΦΦ and τ =

1

4(TrtΦΦ)2. The set {κ,λ,µ,Z,ω} denotes the scale-

dependent coupling constants which parametrize the

model at this order of the truncation. The first quan-

tity in Eq.(11) corresponds to the standard kinetic term

while the third and fourth correspond to the potential

part.Actually, apart from the second term — called

the current term —, Γkin Eq.(11) looks very much like

the usual Landau-Ginzburg-Wilson action used to study

perturbatively the critical physics of the O(N) × O(2)

model, up to trivial reparametrizations. There is however

a fundamental difference since we do not use Γk within

a weak-coupling perturbative approach. This allows the

presence of the current term which corresponds to a non-

standard kinetic term. This term is irrelevant by power

counting around four dimensions since it is quartic in the

fields and quadratic in derivatives. However its presence

is necessary around two dimensions to recover the results

of the low-temperature approach of the nonlinear sigma

(NLσ) model since it contributes to the field renormal-

ization of the Goldstone modes. Being not constrained

by the usual power counting we include this term in our

ansatz. Note also that we have considered much richer

truncations than that given by Eq.(11) by putting all the

terms up to Φ10and by adding all terms with four fields

and two derivatives. This has allowed us to check the

stability of our results with respect to the field expan-

sion.

We do not provide the details of the computation. The

general technique is given in several publications and its

implementation on the specific O(N) × O(2) model will

be given in a forthcoming article5. The β functions for

the different coupling constants entering in (11) are given

by:

1

2Tr(tΦΦ)2−