arXiv:cond-mat/0107183v2 [cond-mat.stat-mech] 12 Mar 2003
XY frustrated systems: continuous exponents in discontinuous phase transitions.
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
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-
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
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.
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:
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-
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
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:
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
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
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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