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; - [Show abstract] [Hide abstract]
ABSTRACT: Two models of classic XY antiferromagnets in three dimensions are studied by Monte Carlo simulation: the model on a simple cubic lattice with two extra intralayer exchanges and the model on a stackedtriangular lattice with an extra interlayer exchange. In suggested models, the order parameters are magnetization and two chiral parameters. A transition corresponds to breaking ℤ2 ⊗ ℤ2 ⊗ SO(2) symmetry. A distinct first order transition is found in both models.Journal of Experimental and Theoretical Physics 113(4). · 0.92 Impact Factor - [Show abstract] [Hide abstract]
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). · 0.48 Impact Factor
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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-
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−