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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-

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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

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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

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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,

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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−

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6

dκ

dt= − (d − 2 + η)κ + 4vd

+ω

λl2+d

dλ

dt=(d − 4 + 2η)λ + vd

?1

2ld

01(0,0,κω) + (N − 2)ld

10(0,0,0) +3

2ld

10(κλ,0,0) +

?

1 + 2µ

λ

?

ld

10(κµ,0,0)+

01(0,0,κω)

?

?

(12a)

2λ2(N − 2)ld

20(0,0,0) + λ2ld

02(0,0,κω) + 9λ2ld

20(κλ,0,0) + 2(λ + 2µ)2ld

20(κµ,0,0)+

+ 4λωl2+d

02(0,0,κω) + 4ω2l4+d

02(0,0,κω)

?

(12b)

dµ

dt=(d − 4 + 2η)µ − 2vdµ

?

−2

κld

01(0,0,κω) +3(2λ + µ)

κ(µ − λ)ld

10(κλ,0,0) +

8λ + µ

κ(λ − µ)ld

10(κµ,0,0)+

+ µld

11(κµ,0,κω) + µ(N − 2)ld

20(0,0,0)

?

(12c)

η = −d lnZ

dt

= 2vd

dκ

?

(4 − d)κωld

01(0,0,κω) + 2κ2ω2l2+d

02(0,0,κω) + 2md

02(0,0,κω) − 4md

11(0,0,κω)+

+ 2(−2 + d)κωld

10(0,0,0) + 2md

20(0,0,κω) + 2κ2λ2md

2,2(κλ,0,0) + 4κ2µ2md

?

ld

2,2(κµ,0,0)+

+ 4κωnd

02(0,0,κω) − 8κωnd

11(0,0,κω) + 4κωnd

20(0,0,κω)

(12d)

dω

dt=(d − 2 + 2η)ω +4vd

− 3κωl2+d

dκ2

?

κω

?(4 − d)

2

ld

01(0,0,κω) +(d − 16)

2

01(κλ,0,κω) + κωl2+d

02(0,0,κω)−

02(κλ,0,κω) + (d − 2)ld

10(0,0,0) − (d − 8)ld

10(κλ,0,0) + 8κλld

11(κλ,0,κω) + 2κωl2+d

20(κµ,0,0)

+ 2κω(N − 2)l2+d

20(0,0,0)

?

+ md

02(0,0,κω) − md

02(κλ,0,κω) − 2md

11(0,0,κω) + 2md

11(κλ,0,κω)+

(12e)

+ md

20(0,0,κω) − md

20(κλ,0,κω) + κ2λ2md

22(κλ,0,0) + 2κ2µ2md

22(κµ,0,0) + 2κωnd

02(0,0,κω)−

− 4κωnd

02(κλ,0,κω) − 4κωnd

11(0,0,κω) + 8κωnd

11(κλ,0,κω) + 2κωnd

20(0,0,κω) − 4κωnd

20(κλ,0,κω)

?

In these equations appear the dimensionless functions

n1,n2,md

they govern the decoupling of the massive modes enter-

ing in the action (11). They encode the nonperturbative

content of the flow equations (12). They are complicated

integrals over momenta and are explicitly given in the

Appendix.

ld

n1,n2,nd

n1,n2, called threshold functions, since

VI.RESULTS AND PHYSICAL DISCUSSION

A.Checks of the method

We have first proceeded to all possible checks of our

method by comparing our results with all available data

obtained within the different pertubative approaches.

Our method fulfills all these checks.

1) Around d = 4, we have checked that, in the limit

of small coupling constant, our equations degenerate in

those obtained from the weak coupling expansion at one

loop9,22,23:

dλ

dt= (d − 4)λ +

dµ

dt= (d − 4)µ +

1

16π2

1

16π2

?4λµ + 4µ2+ λ2(N + 4)?

?6λµ + Nµ2?.

(13)

This can be easily verified considering the asymptotic

expressions of the threshold functions that are given in

the Appendix.

2) Also, around d = 2, performing a low-temperature

expansion — corresponding to a large κ expansion — of

our equations and making the change of variables:

?η1= 2πκ

η2= 4πκ(1 + κω)

(14)

one recovers the β-functions found in the framework of

the nonlinear σ model at one loop36:

β1= −(d − 2)η1+ N − 2 −η2

2η1

β2= −(d − 2)η2+N − 2

2

?η2

η1

?2

.

(15)

Page 7

7

This shows that our method allows to recover the per-

turbative results both near d = 2 and d = 4. This is not a

surprise since, as well known, Eq.(9) has a one-loop struc-

ture. However, for frustrated magnets, and contrary to

the O(N) case, the matching with the results of the NLσ

model is not trivial since it requires to incorporate the

non trivial current term which is irrelevant around d = 4,

and thus absent in a LGW approach.

3) Our method also matches with the leading order

results of the 1/N expansion of ν and η. In the O(N) ×

O(2) model these exponents have also been computed at

order 1/N2in d = 3. They are given by24:

We have computed ν and η for a large range of values

of N to compare our results with those obtained within

the 1/N expansion. We meet an excellent agreement —

better than 1% — for ν already for N > 10 where the

1/N expansion is reliable.

4) For N = 6, a Monte Carlo simulation has been per-

formed on the STA model37. A second order phase tran-

sition has been found without ambiguity with exponents

given in Table II. Our results compare very well with

the Monte Carlo data. Although the case N = 6 does

not correspond to any physical system this constitutes a

success of our approach from a methodological point of

view. Let us recall that, in this case, the six loop weak

coupling calculation is not converged.

ν = 1 −16

π2

1

N−

1

N−

64

3π4

?56

1

N2+ O(1/N3) .

π2−640

3π4

?

1

N2+ O(1/N3)

η =

4

π2

(16)

α

-0.100(33) 0.359(14) 1.383(36) 0.700(11) 0.025(20)

P.W. -0.1210.372 1.377

βγνη

MC

0.7070.053

TABLE II: Critical exponents for the N = 6 STA system. The

first row corresponds to Monte Carlo data37and the second

(P.W.) to the present work.

5) In agreement with the weak coupling perturbative

results, we have found by varying N in a given dimension

d that, below a critical value Nc(d) of N, no fixed point

exists4(see FIG. 2). Our value of Nc(d) agrees within ten

percents for all dimensions with the values obtained from

Eq.(7). In d = 3 we have found Nc(3) = 5.1 which almost

coincides with that obtained using Eq.(7) that leads to

Nc(3) = 5.3.

All these checks show the consistency between our

computation and most of the previous theoretical ap-

proaches. It is in contradiction with the six loop calcu-

lation of Pelissetto et al.. Note that the result Nc(3) > 3

does not exclude a priori the existence of a fixed point

not analytically related — in N and ǫ = 4 − d — to

those found above Nc(d). This is, in fact, the position

advocated by Pelissetto et al.24. However, we have nu-

merically searched for this fixed point with our equations

without success. This implies three possibilities. The

?

?

?

? ??? ?????? ???? ???

??

??

??

??

??

??

??

?

?

?

?

FIG. 2: The full line represents the curve Nc(d) obtained by

the three-loop result improved by the constraint Nc(2) = 2,

Eq.(7). The crosses represent our calculation.

first one is that our method is not able — in principle

— to find this kind of fixed point. Let us emphasize

that, although it is not possible to exclude this case, it

is improbable that a method which recovers all previous

results in a convincing way, misses such a fixed point.

Another possibility is that it has a so small basin of at-

traction that we have systematically missed it in our nu-

merical investigation of our equations. This would mean

that it probably does not play any role in the physics of

frustrated magnets. The transitions should therefore be

of first order for almost all real systems. It would then

be very difficult to explain the occurence of generic weak

first order phase transitions. Finally, there remains the

possiblity that the fixed point does not exist at all which

now appears as the most probable solution.

B.Scaling without fixed and pseudo-fixed point

The hypothesis of the absence of fixed point imme-

diately raises the theoretical challenge to explain the

occurence of scaling in absence of a fixed point.

Heisenberg spins, this question has been addressed by

Zumbach38using a local potential approximation (LPA)

of the Polchinski equation39and by the present authors

beyond LPA in the framework of the effective average

action method4. It has been realized that, when N is

lowered from N > Nc(3) to N < Nc(3), although the

stable fixed point disappears, there is no major change

in the RG flow.

This can be understood by considering a domain E of

initial conditions of the flow corresponding to all systems

we are interested in — STA, STAR, VN,2, real materials,

etc — and studying the RG trajectories starting in E.

One finds that there exists a domain D ⊆ E such that

all trajectories emerging from D are attracted towards a

small domain R in which the flow is very slow. Since the

flow is very slow in R the RG time spent in this region is

long and, thus, the correlation lengths of the systems in

D are very large. One therefore partly recovers scaling

for systems in D (that aborts only for very small reduced

For

Page 8

8

temperatures). Moreover, the smallness of R ensures the

existence of (pseudo-)universality. Consequently R mim-

ics a true fixed point.

This idea has been formalized through the concept

of pseudo-fixed point, corresponding to the point in

R where the flow is the slowest, the minimum of the

flow38. At this point it has been possible to compute

(pseudo-)critical exponents characterizing the pseudo-

scaling (and pseudo-universality) encountered in Heisen-

berg frustrated spin systems4,38.

Within our approach, we confirm the existence, for val-

ues of N just below Nc(3), of a minimum of the flow lead-

ing indeed to pseudo-scaling and quasi-universality (see

Ref.4 and, for details, Ref.5). However, when N is low-

ered, the minimum of the flow is less and less pronounced

and, for some value of N between 2 and 3, it completely

disappears. Since (pseudo-)scaling is observed in exper-

iments and numerical simulations in XY systems this

means that the minimum of the flow does not consti-

tute the ultimate explanation of scaling in absence of a

fixed point. At this stage, one reaches the limits of the

notion of minimum of the flow as the quantity playing

the role of (pseudo-)fixed point. First, it darkens the im-

portant fact that the notion relevant to scaling is not the

existence of a minimum of the flow but that of a whole

region R in which the flow is slow, i.e. the β functions

are small. Put it differently, the existence of a minimum

of the flow does not guarantee that the flow is slow, i.e.

that the correlation length is large compared with the

lattice spacing. Reciprocally, it can happen that the RG

flow is slow, the correlation length being large, and that

scaling occurs even in absence of a minimum. Second,

reducing R to a point, one rules out the possibility to

test the violation of universality. Clearly, the degree of

universality is related to the size of R. The smaller R

the more universal is the behavior.

Qualitatively — thanks to continuity arguments — one

expects that, for N close to Nc(3), all systems in E exhibit

pseudo-scaling — D = E — and R is almost pointlike so

that the transitions are extremely weakly of first order

and universality almost holds. When N is sufficiently

decreased, two phenomena occur. First, D gets smaller

than E and thus the transitions become of strong first or-

der for all systems that belong to E but not to D. Second,

R gets wider and thus a whole spectrum of exponents

is observed and pseudo-universality breaks down. Note

that these two phenomena are not obligatorily simulta-

neous so that, for intermediate values of N, scaling still

holds while pseudo-universality is already significantly vi-

olated. These two phenomena are observed numerically

in the Heisenberg and XY systems: STA, STAR and

VN,2models. For N = 3, all these models display scal-

ing but with β exponents that are almost incompatible

[βSTA=0.285(11),40βSTAR=0.221(9) and βV3,2=0.193(4)

(see Ref.41)] which means that one probably starts leav-

ing the pseudo-universal regime. For N = 2, scaling is

only observed for STA — β = 0.24(2) here — while the

transitions for STAR and V2,2 are found to be of first

order.

From a theoretical point of view, the Heisenberg case

has been already treated in Ref.4. We now concentrate

on the XY case.

C. Our results

In practice, we numerically integrate the flow equa-

tions (12) and compute physical quantities like correla-

tion length and magnetization as a function of the re-

duced temperature tr= (T − Tc)/Tc. Since one expects

the behavior of frustrated magnets to be nonuniversal,

one should study each system independently of the oth-

ers. Thus, ideally, one should consider as initial condi-

tions of the RG flow all the microscopic parameters char-

acterizing a specific lattice system. This program would

require to identify and to deal with an infinite number of

coupling constants. This remains a theoretical challenge.

Rather than doing this, we have chosen to address the

question of scaling in absence of a fixed point indepen-

dently of a given microscopic system, using for Γka finite

ansatz similar to, but richer than, Eq.(11).

Since our truncations forbid us to relate precisely

the thermodynamical quantities to the microscopic cou-

plings, we have chosen, as initial conditions, the simplest

temperature dependence of the parameters at the scale

k = Λ. This consists in fixing all coupling constants to

temperature-independent values and in taking the usual

ansatz for the quadratic term of Γk:

κk=Λ= a + bT(17)

where a and b are parameters that we have varied to test

the robustness of our conclusions. For each temperature

we have integrated the flow equations and deduced the tr-

dependence of the physical quantities around the critical

temperature.

Let us now review the main results obtained by the

integration of the RG flow.

1) The integration of the flow equations leads generi-

cally to good power laws in reduced temperature for the

magnetization, the correlation length (see FIG. 3) and

the susceptibility. We are thus able to extract pseudo-

critical exponents varying typically, for β between 0.25

and 0.38 and for ν between 0.47 and 0.58. This phe-

nonomenon holds for a wide domain D in coupling con-

stant space.

2) We easily find initial conditions leading to pseudo-

exponents close to those of group 2: β = 0.38, ν = 0.58

and γ = 1.13 (see Ho and Dy in Table I for comparison).

We have checked that the previous result is quite stable

to both variations of the microscopic parameters and to

a change of the T-dependence of the microscopic cou-

pling constants. This is in agreement with the stability

of β in group 2. In the region of parameters leading to

this behavior, correlation lengths as large as 5000 lattice

spacings are found.

Page 9

9

???

??

??

?

?

???

??

?

?

?

?

?

? ?????????? ????????

?

????

????

????

????

????

????

????

????

????

??

???

??

??

?

?

???

??

?

?

?

?

?

? ???? ?? ?????? ???? ??

???

???

?

???

???

???

???

?

FIG. 3: Log-log plots of the magnetization m and of the corre-

lation length ξ for N = 2 as functions of the reduced temper-

ature tr for parameters corresponding to materials of group

1. The straight lines correspond to the best power law fit of

the data. The power-law behavior observed breaks down for

small tr.

3) We also easily find initial conditions leading to

β ≃ 0.25, corresponding to group 1. The power laws

then hold on smaller range of temperature and the criti-

cal exponents are more sensitive to the determination of

Tcand to the initial conditions, in agreement with point

ii) of Section III-A. For such values of β, we find a range

of values of ν — 0.47 ≤ ν ≤ 0.49 — which is somewhat

below the value found for CsMnBr3(see Table I). Also,

the corresponding η deduced from the scaling relation

η = 2β/ν − 1 is always positive and, at best, zero. Fi-

nally, it is very interesting to notice that when we find β

of order 0.25 (group 1) we also find correlation lengths at

the transition of the order of a few hundreds lattice spac-

ings which coincide rather well with the first size where a

direct evidence of a first order transition has been seen in

Monte Carlo RG simulations (lattice sizes around 100)21.

4) For β ≃ 0.3 we find critical exponents in good agree-

ment with those obtained by the six-loop calculation of

Ref.2 (see Table I). For instance, for β = 0.33 we find

ν ≃ 0.56 and γ ≃ 1.07.

VII.CONCLUSION

On the basis of their specific symmetry breaking

scheme it has been proposed7,8,9that the critical physics

of XY frustrated systems in three dimensions could un-

dergo a second order phase transition characterized by

critical exponents associated with a new universality

class. We have given convincing arguments that rather

favor the occurence of — generically weak — first order

transitions for all XY frustrated magnets with a spread-

ing of (pseudo-)critical exponents. This is supported by

experimental and numerical results that do not agree

with a second order behavior. Moreover we have shown,

using a nonperturbative approach, that this generic but

nonuniversal scaling finds a natural explanation in terms

of slowness and “geometry” of the flow. Our approach

appears to explain the main puzzling features of XY frus-

trated magnets.

We now propose several tests to confirm our approach.

On the experimental side, more accurate determinations

of critical exponents could lead to a definitive answer on

the nature of the transition, at least if no drastically new

physics emerges (as it could be the case for Helimagnets).

In particular, it is important to check that η is negative

for CsMnBr3, and therefore to refine the determination

of ν. It would also be of utmost interest to have more

precise determinations of α in CsMnBr3, CsNiCl3 and

CsMnI3which are, up to now, only marginally compat-

ible. In case they are different, this would corroborate

the lack of universality that we predict. The existence

of a continuous spectrum of critical exponents could be

directly tested by simulating, for example, a family of

models, extrapolating continuously from STA to STAR.

On the theoretical side, it would be of interest to

push the derivative expansion to refine the value of η for

group 1, which is, as usual, overestimated32. This would

allow to reproduce its observed negative value. More-

over, it would be interesting to clarify the discrepancy

between the nonperturbative approach and the six loop

result. The ability of our approach to reproduce exactly

the whole set of exponents found by Pelissetto et al. sug-

gests that their fixed point does not correspond to a true

fixed point but, in fact, to a region where the flow is very

slow. One can thus question the convergence of the per-

turbative result which is not Borel summable. A detailed

analysis of this problem of convergence could reveal that

the real fixed point found in the perturbative approach

is, actually, a complex one.

Finally, the major characteristics of XY-frustrated

magnets i.e. the existence of scaling laws with contin-

uously varying exponents are probably encountered in

other physical contexts, generically systems with a crit-

ical value Ncof the number of components of the order

parameter, separating a true second order behavior and

a na¨ ıvely first order one.

Acknowledgments

We thank P. Schwemling for useful discussions. Labo-

ratoire de Physique Th´ eorique et Hautes Energies, Uni-

versit´ e Paris VI Pierre et Marie Curie — Paris VI Denis

Page 10

10

Diderot — is a laboratoire associ´ e au CNRS UMR 7589.

Laboratoire de Physique Th´ eorique et Mod` eles Statis-

tiques, Universit´ e Paris Sud, is a laboratoire associ´ e au

CNRS UMR 8626.

APPENDIX A: THRESHOLD FUNCTIONS

We discuss in this appendix the different threshold

functions ld

n1,n2, md

n1,n2and nd

n1,n2appearing in the flow

equations (12).

The threshold functions are defined as:

ld

n1,n2(w1,w2,a) = −1

2

?∞

?∞

?∞

0

dy yd/2−1˜∂t

?

?

?

1

(P1(y) + w1)n1(P2(y) + w2)n2

y(∂yP1(y))2

(P1(y) + w1)n1(P2(y) + w2)n2

y∂yP1(y)

(P1(y) + w1)n1(P2(y) + w2)n2

?

?

?

, (A1a)

md

n1,n2(w1,w2,a) = −1

2

0

dy yd/2−1˜∂t

, (A1b)

nd

n1,n2(w1,w2,a) = −1

2

0

dy yd/2−1˜∂t

(A1c)

with:

P1(y) = y(1 + r(y) + a)

P2(y) = y(1 + r(y)) .

(A2)

(A3)

In all the previous expressions y is a dimensionless quan-

tity: y = q2/k2where q is the momentum variable over

which the integral in Eq.(9) is performed. As for r(y), it

corresponds to the dimensionless renormalized infrared

cut-off:

r(y) =Rk(q2)

Zq2

=Rk(yk2)

Zk2y

. (A4)

In Eqs.A1a–A1c,˜∂tmeans that only the t-dependence

of the function Rkis to be considered and not that of the

coupling constants. Therefore one has:

˜∂tPi(y) =∂Rk

∂t

∂

∂RkPi(y) (A5)

= −y(ηr(y) + 2yr′(y)). (A6)

Now the threshold functions can be expressed as explicit

integrals if we compute the operation of˜∂t.

end, it is interesting to notice the equality: ∂y˜∂tPi(y) =

˜∂t∂yPi(y), so that:

To this

˜∂t∂yr = −η(r + yr′) − 2y(2r′+ yr′′) (A7)

We then get:

ld

n1,n2(w1,w2,a) = −1

2

?∞

?∞

0

dy yd/2

ηr + 2yr′

(P1(y) + w1)n1(P2(y) + w2)n2

?

n1

P1(y) + w1

+

n2

P2(y) + w2

?

(A8)

nd

n1,n2(w1,w2,a) = −1

2

0

dy yd/2

1

(P1(y) + w1)n1(P2(y) + w2)n2

?

y(1 + a + r + yr′)(ηr + 2yr′)×

?

n1

P1(y) + w1

+

n2

P2(y) + w2

?

− η(r + yr′) − 2y(2r′+ yr′′)

?

(A9)

md

n1,n2(w1,w2,a) = −1

2

?∞

0

dy yd/2

1 + a + r + yr′

(P1(y) + w1)n1(P2(y) + w2)n2

?

y(1 + a + r + yr′)(ηr + 2yr′)×

?

n1

P1(y) + w1

+

n2

P2(y) + w2

?

− 2η(r + yr′) − 4y(2r′+ yr′′)

?

.

(A10)

Page 11

11

∗Electronic address: tissier@lpthe.jussieu.fr

†Electronic address: delamotte@lpthe.jussieu.fr

‡Electronic address: mouhanna@lpthe.jussieu.fr

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