Dzyaloshinsky-Moriya-induced order in the spin-liquid phase of the S=1∕2 pyrochlore antiferromagnet

Abstract

We show that the S=1∕2 pyrochlore lattice with both Heisenberg and antisymmetric Dzyaloshinsky-Moriya (DM) interactions can order antiferromagnetically into a state with chiral symmetry, dictated by the distribution of the DM interactions. The chiral antiferromagnetic state is characterized by a small staggered magnetic moment induced by the DM interaction. An external magnetic field can also lead to characteristic field-induced ordering patterns, strongly dependent on the field direction, and generally separated by a quantum phase transition from the chiral ordered phase. The phase diagram at finite temperature is also discussed.

Full text

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arXiv:cond-mat/0410739v3 [cond-mat.str-el] 12 Jul 2005
Dzyaloshinsky-Moriya-Induced Order in the Spin-Liquid Phase of the S=1/2
Pyrochlore Antiferromagnet
Valeri N. Kotov,1, ∗Maged Elhajal,2Michael E. Zhitomirsky,3and Fr´ ed´ eric Mila1
1Institute of Theoretical Physics, Swiss Federal Institute of Technology (EPFL), 1015 Lausanne, Switzerland
2Max-Planck-Institut f¨ ur Mikrostrukturphysik, Weinberg 2, 06120 Halle, Germany
3Commissariat ` a l’Energie Atomique, DSM/DRFMC/SPSMS, 38054 Grenoble, France
We show that the S=1/2 pyrochlore lattice with both Heisenberg and antisymmetric,
Dzyaloshinsky-Moriya (DM) interactions, can order antiferromagnetically into a state with chiral
symmetry, dictated by the distribution of the DM interactions. The chiral antiferromagnetic state
is characterized by a small staggered magnetic moment induced by the DM interaction. An external
magnetic field can also lead to characteristic field-induced ordering patterns, strongly dependent on
the field direction, and generally separated by a quantum phase transition from the chiral ordered
phase. The phase diagram at finite temperature is also discussed.
I.INTRODUCTION
The behavior of many-body systems involving quan-
tum spins has been one of the central topics in recent
years since the properties of such systems are relevant to
a great variety of materials, mostly oxides. The structure
of the ground state and the various symmetry broken
phases that emerge are issues of special interest, espe-
cially in systems of low-dimensionality and/or where frus-
tration is present.1In this context the Heisenberg model
on the three-dimensional pyrochlore lattice consisting of
corner sharing tetrahedra, shown in Fig. 1(a), is in a
league of its own. The pyrochlore lattice is strongly ge-
ometrically frustrated and is relevant to numerous com-
pounds. It has been argued that no magnetic order is
present in the ground state.2,3,4The effects of various ad-
ditional interactions have also been studied, such as mag-
netoelastic couplings,5long-range dipolar interactions,6
and orbital degeneracy.7These interactions (in addition
to various anisotropies) can generally lead to bond, mag-
netic and/or orbital order, and which of them is domi-
nant depends on the details of the model relevant to the
specific class of materials.
In the present work we study a new mechanism for
magnetic order in the S=1/2 pyrochlore lattice, driven by
the Dzyaloshinsky-Moriya (DM) interactions. In the py-
rochlore such interactions are expected to be present by
symmetry. For the S=1/2 Heisenberg model on the py-
rochlore lattice it has been suggested2,4that the ground
state is dimerized (non-magnetic), but macroscopic de-
generacy still remains. For certain other lattices, such
as the 2D pyrochlore and related models,8the ground
state is a unique valence bond solid, and while the DM
interactions (if present) can lead to non-trivial order in
the ground state, such DM induced order can only occur
above a critical threshold, due to its inherent competi-
tion with the underlying dimer order.8In this work we
show that in the 3D pyrochlore antiferromagnet, where a
macroscopic degeneracy is present, the DM interactions
have a more profound effect and can lift the degeneracy,
leading to a chiral antiferromagnetic state with a small
staggered magnetic moment. In an external magnetic
field quantum transitions between weakly ordered states
with different symmetries, depending on the field direc-
tion, are possible. We determine the field-induced pat-
terns for several field orientations, generally pointing in
highly-symmetric crystal directions. The phase diagram
at finite temperature is also briefly discussed.
?
?
?
?
?
?
?
?
?
?
?
?
?
?
??
??
??
??
D
1
4
3
(a)
(b)
x
y
z
2
FIG. 1: (a) Pyrochlore lattice. (b) Distribution of DM vectors
on a single tetrahedron (four of the six shown, see text).
The spin Hamiltonian (S=1/2) is
ˆ H =
?
i,j
Ji,jSi.Sj+
?
i,j
Di,j.(Si× Sj), (1)
where Di,j are the DM vectors, to be specified later.
We start by summarizing the results for Di,j = 0, i.e.
the Heisenberg case. Our starting point is the strong-
coupling approach, similar to that of Refs. 2,4, with the
lattice divided into two interpenetrating sub-lattices, one
of them formed by “strong” tetrahedra (with exchange
J), connected by “weak” tetrahedra (exchange J′). The
“strong” tetrahedra then form a fcc lattice, as shown in
Fig. 2(a), where every site represents a tetrahedron, and
one can attempt to analyze the structure of the ground
state starting from the limit J′≪ J.
For J′= 0 the tetrahedra are disconnected, and on
a single tetrahedron the ground state is a singlet and is
twofold degenerate. We choose the two ground states as:
|s1? =
1
√3{[1,2][3,4] + [2,3][4,1]}, |s2? = {[1,2][3,4] −

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2
[2,3][4,1]}, where [k,l] denotes a singlet formed by the
nearest-neighbor spins k and l, labeled as in Fig. 1(b). In
the pseudo-spin T = 1/2 representation, so that Tz= 1/2
corresponds to |s1? and Tz= −1/2 corresponds to |s2?,
one finds that third order is the lowest one contributing to
the effective inter-tetrahedron Hamiltonian in the singlet
sub-space:9
ˆ Heff=J′3
8J2
?
ˆ H(2)
eff+ˆ H(3)
eff
?
+ Const., (2)
where
ˆ H(2)
eff=
?
?i,j?
?
Ωx
ijTx
iTx
j+ Ωz
ijTz
iTz
j+ Ωxz
ij(Tx
iTz
j+Tz
iTx
j)
?
(3)
,
ˆ H(3)
eff=
?
(i,j,k)
?1
3Tz
iTz
jTz
k−Tz
iTx
jTx
k+Tz
i
√3(Tx
jTz
k−Tz
jTx
k)
?
(4)
In the two-body part we have defined
Ωx
Ωz
Ωxz
03= Ωx
23= Ωz
23= Ωxz
12= 1/2,Ωz
01= Ωz
01= −Ωxz
03= Ωz
13= 1/3,
02= −Ωxz
12= −1/6,
02= Ωz
13= 1/(2√3).(5)
All remaining Ωij = 0, i < j.
i,j refer to the fcc lattice made of individual tetra-
hedra, Fig. 2(a), and it is sufficient to know the in-
teractions on one ”supertetrahedron”, shown in green
(containing the sites 0,1,2,3).
teraction the indexes run over the values: (i,j,k) =
{(3,2,1),(1,0,3),(2,3,0),(0,1,2)}.
On a mean-field level the ground state ofˆ Heffis defined
by the following averages:
1? = −√3/4, ?Tz
?Tx
?Tx
?Tx
This means that while a dimerization pattern sets in on
sites 1,2,3, the pseudospins on the “0” sites, shown in blue
in Fig. 2(a) remain “free”, i.e. there is no fixed dimer
pattern on those sites and consequently a macroscopic
degeneracy remains.4
One should certainly keep in mind that the strong-
coupling approach breaks artificially the lattice symme-
try and while one hopes that the structure of the ground
state is correct even in the isotropic limit J′= J, it is
very difficult to assess this by other means (e.g. exact
diagonalizations) at the present time. Nevertheless this
approach is expected to provide reliable description of the
ground state properties as long as the relevant physics re-
mains in the singlet subspace, i.e. the triplet modes stay
high in energy and no magnetic order is generated, as
might be the case for the pyrochlore antiferromagnet due
The site indexes
In the three-body in-
?Tx
1? = 1/4,
?Tz
?Tz
?Tz
2? =√3/4,
3? = 0,
0? = 0,
2? = 1/4,
3? = −1/2,
0? = 0.(6)
to the strong frustration. Fluctuations around the mean-
field solution, Eq. (6), can lift the degeneracy, leading to
unique dimer order. However the corresponding degen-
eracy lifting energy scale is very small,4of the order of
10−3β, β ≡ J′3/(48J2). A unique (singlet) ground state
is also produced if one starts the expansion from a larger
cluster of 16 sites, with an ordering energy scale (energy
gain) of 10−2J, extrapolated to the limit where all cou-
plings are equal.10
In what follows we will take the mean-field solution
as a starting point and discuss a physical mechanism,
based on the presence of interactions beyond Heisenberg
exchange, that can lead to the lifting of degeneracy and
consequently to (magnetic) order in the ground state.
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?????? ??????
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??????????
??????????
(a)
(b)
<S>
0
1
2
3
1
2
3
0
1
3
2
0
FIG. 2: (Color online.) (a) Fcc lattice of tetrahedra (tetrahe-
dron = dot) with interactions J′between them. (b) Antifer-
romagnetic chiral order on the blue (dark gray) tetrahedra,
with magnetic moment |?S?| ∼˜D, induced by the DM inter-
actions. On the gray tetrahedra (labeled as 1,2,3) the order
has the same symmetry, but is much weaker |?S?| ∼˜D3≪˜D,
Eq. (9), and is not shown.
II.
INDUCED BY THE DZYALOSHINSKY-MORIYA
INTERACTIONS
CHIRAL ANTIFERROMAGNETIC ORDER
Now we consider the effect of the DM interactions11,12
on the ground state properties.
hedron the DM vectors are distributed as shown in
Fig. 1(b), or explicitly:
D13 =
D
√2(−1,−1,0), D43=D
D14=
magnitude of the (all equal) DM vectors. The directions
of the DM vectors respect the pyrochlore lattice sym-
metry and thus the DM interactions are expected to be
always present in the system.13Since Dijoriginate from
the spin-orbit coupling,11,12we have D ≪ J,J′, and typi-
cally the values of the DM interactions are several percent
of the Heisenberg couplings. There are two DM distri-
bution patterns that are equally acceptable on symme-
try grounds - the one shown in Fig. 1(b), and one with
all directions of the vectors Dijreversed (Dij→ −Dij).
On a single tetra-
D
√2(−1,1,0), D24 =
√2(0,−1,−1),
√2(1,0,−1). Here D is the
√2(0,−1,1), D12=D
D
D
√2(1,0,1), D23=

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3
These two cases were named, respectively, ”indirect” and
”direct” in Ref. 13. The reader is referred to that paper
for more details on Moriya’s rules as applied to the py-
rochlore lattice. In the extreme quantum case of S=1/2,
and within our approach, we have found that the two
allowed (by symmetry) DM distributions lead to qualita-
tively the same physics (see discussion following Eq. (8)).
Following the strong-coupling approach outlined above
for the purely Heisenberg case, we have to determine how
the singlet ground states |s1?,|s2? on a single tetrahedron
are modified by the presence of D. Since the DM in-
teractions break the spin rotational invariance, they ad-
mix triplets to the two ground states, not lifting their
degeneracy.8We will also be interested in effects in the
presence of an external magnetic field, and in this case
the field (in combination with the DM interactions) also
mixes certain triplet states with |s1?,|s2?. In order to
determine the additional contributions toˆ Heff, it is con-
venient to express the spin operators on a singlet tetrahe-
dron, labeled as in Fig. 1(b), in terms of the pseudospin
operators. For magnetic field H =
1 − 3 bond), assuming D ≪ J and H ≪ J, we obtain
(defining the rescaled quantities˜D,˜H along the way):
H
√2(1,1,0) (along the
H =
H
√2(1,1,0);
˜D ≡ D/J,
˜D˜H
√3Tx, Sx
˜D˜H
√3Tx, Sy
˜H ≡ H/J,
2,4= ∓2˜D
2,4= ±2˜D
2,4= −2˜D
Sx
1,3= ∓2˜D
1,3= ∓2˜D
√6Ty−
√6Ty+
˜D˜H
√3Tx
˜D˜H
√3Tx
˜D˜H
√3Tx(7)
Sy
√6Ty+
2˜D
√6Ty∓˜D˜HTz, Sz
√6Ty−
Sz
1,3=
√6Ty±
The notation Si,j simply combines in one line the for-
mulas for both Si and Sj, where i and j label sites on
a tetrahedron (as defined in Fig. 1(b)), while the left in-
dex in Si,j corresponds to the upper sign on the right
hand side, and the right index - to the lower sign. The
formulas Eq. (7) are obtained by using the ground state
wave-functions, written explicitly in Ref. 8 (Eqs. (2,5)),
to lowest order in˜D and˜D˜H. For magnetic field in the
z direction, the corresponding expressions are given in
Appendix A.
First we analyze the case of zero magnetic field (H =
0).Taking into account the connections between the
tetrahedra (green bonds in Fig. 2(a)), and using Eq. (7),
we obtain an additional interaction term, so that the full
effective Hamiltonianˆ H(DM)
eff
becomes
ˆ H(DM)
eff
=ˆ Heff− J′˜D22
3
?
i<j
Ty
iTy
j,(8)
whereˆ Heffis the part originating from the Heisenberg
exchanges, Eq. (2). The above result is obtained in low-
est, first order in J′. While extra terms of the same power
J′(D′/J)2also arise from the DM interactions D′on the
inter-tetrahedral bonds, we find that they only give a
small renormalization of the energy scale J′3/(8J2) in
Eq. (2) and are, therefore, neglected.
We would like to also point out that in general the
coupling constant in
ˆ H(DM)
eff
D → −D. To verify this requires a calculation of the next
to leading order in˜D in Eq. (7). We have found that, in
the case of zero field H = 0, the next order present is˜D2,
and one has to substitute in all formulas˜D →˜D−
Consequently the same substitution has to be made in the
coefficient −J′˜D2 2
the Ty
jstructure of the interaction is not affected by
increasing the strength of˜D ≪ 1, and from now on we
will work with the leading order in˜D. Therefore the
physics (ground state structure) associated with the two
DM distribution patterns will be the same. This conclu-
sion might be connected with the fact that we have kept
only the lowest non-trivial order in the coupling J′in the
effective Hamiltonian - we have used this as our guiding
principle as the difficulties associated with the derivation
and analysis of higher orders seem insurmountable.
We have performed mean-field calculations of the
Hamiltonian defined by Eqs. (2,3,4,8) in the unit cell of
Fig. 2(a), as represented by the four sites connected by
green lines. The results can be particularly simply sum-
marized in the limit˜D ≪ 1, which is also the case of
physical relevance. It is physically clear that ferromag-
netic order in the Ty
icomponent is generated on the “0”
sites, since no order in the Tx,z
der) was present on those sites without DM interactions
(on mean-field level), Eq. (6). Indeed we find ?Ty
while for the other sites we have, to lowest non-trivial or-
der in D, ?Ty
is then clear that a non-zero average of the operator Ty
corresponds to a finite moment in the ground state, with
magnitude |?S?i| =˜D√2?Ty
is not symmetric under
3
4√2˜D2.
3in Eq. (8). Most importantly however
iTy
i
components (dimer or-
0? = 1/2,
i? ≈ 1.8(D/J′)2,i = 1,2,3. From Eq. (7) it
i
i?. To summarize:
?Ty
0? = 1/2, ?Ty
˜D
√2,i = 0; |?S?i|≈3.6
i? ≈ 1.8D2
J′2,i = 1,2,3
˜DD2
⇒
|?S?i| =
√2
J′2,i = 1,2,3. (9)
Here |?S?i| stands for the magnitude of the moment on
each site of pyrochlore lattice, belonging to a tetrahe-
dron labeled by the index i. From (7) it follows that
the moments point out of the cube’s center (the cube is
defined in Fig. 1(b)), leading to formation of sublattices
and the order shown in Fig. 2(b). Since from Eq. (9)
|?S?i|/|?S?0| ∼ (D/J′)2≪ 1, i = 1,2,3, we have ne-
glected the magnetic order on those tetrahedra.
The antiferromagnetic order of Fig. 2(b) corresponds
to non-zero scalar chirality χ = ?Sm· (Sn× Sl)? ?= 0,
where m,n,l are any three spins on a given tetrahedron.
The Ising symmetry Ty
ken in the ground state, which in terms of real spins
corresponds to the time-reversal symmetry broken state
of Fig. 2(b). In this state the two ground state wave
i→ −Ty
iis spontaneously bro-

Page 4

4
functions |Φ? and |Ψ? (see (A1)) form linear combina-
tions in the ”chiral” sector: α|Φ? + iβ|Ψ?, where α,β
are real coefficients (α2+ β2= 1). This combination
is ferromagnetically repeated on every Tyordered tetra-
hedron. A straightforward calculation shows that both
?Ty
Fig. 2(a)) from the formation of the ordered state is
∆E = ?ˆ H(DM)
eff
we have just discussed is in competition with other mech-
anisms for lifting of the degeneracy that could originate
from the Heisenberg interactions themselves (e.g. fluctu-
ations beyond the mean-field), typically also leading to
very small energy scales.4
i? ∝ αβ, χ ∝ αβ. The energy gain (per site of
?−?ˆ Heff? ≈ −1.8J′˜D2(D/J′)2. The order
III.
EXTERNAL MAGNETIC FIELDS IN THE
PRESENCE OF DM INTERACTIONS
MAGNETIC ORDER INDUCED BY
In the presence of an external magnetic field other pos-
sibilities for lifting of the degeneracy exist. We will con-
sider three symmetric field orientations, for which the
results are particularly transparent. The magnetic field
generally leads to splitting of the ground states, which in
the language of the pseudospins produces an on-site “ef-
fective magnetic field” h in the pseudospin z direction.
The effective Hamiltonian has the form
ˆ H(H)
eff=ˆ H(DM)
eff
+ h
?
i
Tz
i+ δˆ H(H)
eff.(10)
We consider fields in the (1,1,0) and (0,0,1) directions,
as well as comment on the case (1,1,1), where the axes
are defined in Fig. 1(b). Using the wave-functions in a
field we obtain (see Eq. (A3)):
h =



1
2D2H2/J3, H =
−D2H2/J3, H = H(0,0,1)
0,
H =
H
√2(1,1,0)
H
√3(1,1,1)
(11)
δˆ H(H)
eff
ing from the various combinations in Eq. (7) once the
tetrahedra are coupled, and also producing terms of or-
der D2H2. These terms are cumbersome and are not
explicitly written, but their effect is taken into account
in the (numerical) mean-field implementation within the
unit cell of Fig. 2(a). A further discussion appears in
Appendix B.
We will mostly discuss the two cases with h ?= 0. Then
the on-site h term in Eq. (10) is responsible for the main
effect, namely competition between order in the Tz
dospin component and order in the “chiral” Ty
nent favored by Eq. (8). Therefore the physics is that
of the transverse field Ising model (although in our case
the unit cell is larger). It is also clear that the mentioned
competition is most effective on the “0” (blue) sites, while
the non-zero averages of Tz,x
i
much affected by the presence of small D and H. We have
in (10) represents lattice contributions, originat-
ipseu-
icompo-
on the other sites are not
1
2
1
H
H
H=0
c
0<H<HH>H
[1,1,0]
c
:
0
(a)(b)(c)
4
4
2
213
3
3
π/2
FIG. 3: (Color online.) (a,b,c) Evolution of magnetic order
on the blue (dark gray) tetrahedra of Fig. 2(b) in an external
magnetic field in the (1,1,0) direction.
induced order on the rest of the tetrahedra. The tetrahedra
are labeled 0,1,2,3 as in Fig. 2(a,b). Blue (dark gray) arrows
in (a,b) correspond to moments |?S?| ∼˜D, while the red (light
gray) arrows on the upper row and (b,c) are the field-induced
moments |?S?| ∼˜D˜ H.
Upper row: field-
found that a quantum transition takes place between a
state with ?Ty
The result for˜D ≪ 1 can be written in an explicit way,
and we have for the field H =
0? ?= 0,H < Hc and ?Ty
0? = 0,H ≥ Hc.
H
√2(1,1,0)
?Ty
0?2=1
4

1 −
?
˜H
˜Hc
?4
,
˜H ≤˜Hc≈ 5.3
?
J
J′˜D (12)
?Tz
0?2= 1/4 − ?Ty
0?2
(13)
(and ?Tz
moments for given values of ?Tx,y,z
can be determined directly from Eq. (7). On the “0”
(blue) sites this leads to evolution of the magnetic order
as shown in Fig. 3(a,b,c). For H = 0 there is only chiral
order (blue arrows) with moment |?S?| ∼˜D, changing,
for H > 0 into a combination of chiral and field induced
order (red arrows) with |?S?| ∼˜D˜H. Gradually, as H ap-
proaches Hcthe chiral order diminishes (Eq. (12)), leav-
ing for H > Hconly the field-induced component, equal
to |?S?| =˜D˜H|?Tz
On the tetrahedra 1,2,3 labeled as in Fig. 2(a,b) there
is virtually no evolution as a function of the field, and
the order is determined by Eq. (7) with ?Tix,z? fixed by
the Heisenberg exchanges, see Eq. (6). This leads to the
magnetic moments (proportional to˜D˜H) shown in Fig. 3,
upper row. On tetrahedra 1 and 2 the spins point along
0? < 0 since h > 0). The values of the spin
i
? on a tetrahedron
0?| =˜D˜H/2, H > Hc.

Page 5

5
the internal diagonals of the cube perpendicular to the
field. Dimerization is also present in the ground state
(bolder lines = stronger bonds) and co-exists with the
magnetic order.
H
H
1
2
3
3
H=0
c
0<H<H H>H
c
:
0
(a)(b) (c)
2
4
1
123
[0,0,1]
4
FIG. 4:
ternal magnetic field in the (0,0,1) direction.
gray) arrows in (a) correspond to the DM induced order with
|?S?| ∼˜D, red (light gray) arrows on the upper row and (c)
correspond to the field-induced component |?S?| ∼˜D˜H, and
black arrows in (b) are a mixture of the two.
(Color online.) Same as Fig. 3, but for an ex-
Blue (dark
A similar quantum transition takes place for magnetic
field in the z direction H = H(0,0,1). In this case the
formulas (A2) from Appendix A have to be used, and the
field-induced order is shown in Fig. 4. The critical field is
also somewhat smaller in this case˜ Hc≈ 3.8(J/J′)1/2˜ D,
mainly due to the fact that h is larger by a factor of 2, see
Eq. (11). For other, less symmetric field directions, the
form of the effective Hamiltonian, and consequently the
field-induced patterns can be quite complex. Finally, in
the case of a field H ∼ (1,1,1), when h = 0, the quantum
transition described above does not take place, and the
chiral order of Fig. 3(a) essentially does not evolve. In
this case the various additional terms similar to the ones
described in Appendix B may lead to small, sub-leading
deviations from the perfect chiral sate.
In addition to the field-induced ordered patterns of
Fig. 3 and Fig. 4, determined mostly by the inter-
tetrahedral interactions, a single tetrahedron with DM
interactions also possesses a finite moment in the direc-
tion of the field,8meaning that the spins in Fig. 3 and
Fig. 4 would also tend to tilt in that direction. However
the moment along the field is proportional to˜D2˜H, as can
be deduced from the fact that the ground state energy
varies as˜D2˜H2from (A3). Consequently this component
has not been taken into account in Eqs. (7,A2), valid to
lowest order in˜D,˜H. Finally, we emphasize that while
we have assumed˜D,˜H to be small, the ratio˜D/˜H can
be arbitrary, meaning that the quantum transitions in a
field are within the limit of validity of our approach.
IV. PHASE DIAGRAM AND DISCUSSION
At finite temperature we expect the phase diagram to
look as presented in Fig. 5 (it is assumed that h ?= 0). The
higher transition temperature Tc1∼ J′3/J2corresponds
to the scale below which the translational symmetry is
broken (dimerization occurs), and is determined by the
energy scale inˆ Hefffor D = 0, Eq. (2). We expect Tc1
to have weak dependence on magnetic field when DM
interactions are present. At a lower scale Tc2 the Ising
Ty→ −Tysymmetry is spontaneously broken by Eq. (8).
For H = 0 we can estimate Tc2 ∼ J′˜D2(D/J′)2. At
fixed field this finite-temperature transition is in the 3D
Ising universality class, and the specific heat diverges as
C ∼ |T − Tc2(H)|−α, α ≈ 0.11.14We emphasize that
Fig. 5 shows only the low-field part of the phase diagram
(since Hc ∼ D ≪ J), while the physics at high fields
cannot be determined within the effective Hamiltonian
framework presented here.
<T > = 0
y
x,z
<T > = 0 partial dimer order
Tc
2
Hc
H
T
Tc


<T > = 0
x,z
<T > = 0
non−zero chirality
y
1
FIG. 5: Schematic phase diagram at non-zero temperature
in the presence of small magnetic field (H ≪ J) and DM
interactions.
In certain pyrochlores, such as the gadolinium tita-
nium oxides with S=7/2, field-driven phase transitions
have been observed,15although in this material the mag-
netic order is typically explained as originating from the
long-range dipolar interactions. For such large value of
the spin the DM mechanism for magnetic order, at least
the way it is developed in this work, should not be effec-
tive since our calculations were based on strong singlet
correlations in the ground state. At the moment it is hard
to point out a class of materials where the DM interac-
tions are definitely expected to be dominant with respect
to other anisotropies capable to produce ordering; some
possible examples are given in Ref. 13. In particular,
our results are specific to the case S = 1/2, while most