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Uniform and staggered magnetizations induced by Dzyaloshinskii-Moriya interactions in isolated and coupled spin-1/2 dimers in a magnetic field

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We investigate the interplay of Dzyaloshinskii-Moriya interactions and an external field in spin 1/2 dimers. For isolated dimers and at low field, we derive simple expressions for the staggered and uniform magnetizations which show that the orientation of the uniform magnetization can deviate significantly from that of the external field. In fact, in the limit where the ${\bf D}$ vector of the Dzyaloshinskii-Moriya interaction is parallel to the external field, the uniform magnetization actually becomes {\it perpendicular} to the field. For larger fields, we show that the staggered magnetization of an isolated dimer has a maximum close to one-half the polarization, with a large maximal value of $0.35 g\mu_B$ in the limit of very small Dzyaloshinskii-Moriya interaction. We investigate the effect of inter-dimer coupling in the context of ladders with Density Matrix Renormalization Group (DMRG) calculations and show that, as long as the values of the Dzyaloshinskii-Moriya and of the exchange interaction are compatible with respect to the development of a staggered magnetization, the simple picture that emerges for isolated dimers is also valid for weakly coupled dimers with minor modifications. The results are compared with torque measurements on Cu$_{2}$(C$_{5}$H$_{12}$N$_{2}$)$_{2}$Cl$_{4}$. Comment: 8 pages, 9 figures
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Uniform and staggered magnetizations induced by Dzyaloshinskii-Moriya interactions in isolated
and coupled spin-1/2 dimers in a magnetic field
S. Miyahara,1J.-B. Fouet,2S. R. Manmana,3,4 R. M. Noack,4H. Mayaffre,5I. Sheikin,6C. Berthier,5,6 and F. Mila7
1Department of Physics and Mathematics, Aoyama Gakuin University, Sagamihara, Kanagawa 229-8558, Japan
2Institut Romand de Recherche Numérique en Physique des Matériaux (IRRMA), PPH-Ecublens, CH-1015 Lausanne, Switzerland
3Institut für Theoretische Physik III, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany
4Fachbereich Physik, Philipps Universität Marburg, D-35032 Marburg, Germany
5Laboratoire de Spectrométrie Physique, Université J. Fourier & UMR5588 CNRS, BP 87, 38402 Saint Martin d’Hères, France
6Grenoble High Magnetic Field Laboratory, CNRS, BP 166, F-38042 Grenoble Cedex 09, France
7Institute of Theoretical Physics, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Received 19 October 2006; revised manuscript received 1 February 2007; published 1 May 2007
We investigate the interplay of Dzyaloshinskii-Moriya interactions and an external field in spin-1/ 2 dimers.
For isolated dimers and at low field, we derive simple expressions for the staggered and uniform magnetiza-
tions which show that the orientation of the uniform magnetization can deviate significantly from that of the
external field. In fact, in the limit where the Dvector of the Dzyaloshinskii-Moriya interaction is parallel to the
external field, the uniform magnetization actually becomes perpendicular to the field. For larger fields, we
show that the staggered magnetization of an isolated dimer has a maximum close to one-half the polarization,
with a large maximal value of 0.35g
Bin the limit of very small Dzyaloshinskii-Moriya interaction. We
investigate the effect of interdimer coupling in the context of ladders with density-matrix renormalization-
group DMRGcalculations and show that, as long as the values of the Dzyaloshinskii-Moriya interaction and
of the exchange interaction are compatible with respect to the development of a staggered magnetization, the
simple picture that emerges for isolated dimers is also valid for weakly coupled dimers with minor modifica-
tions. The results are compared with torque measurements on Cu2C5H12N22Cl4.
DOI: 10.1103/PhysRevB.75.184402 PACS numbers: 75.10.Jm, 75.10.Pq, 75.40.Mg, 75.30.Kz
I. INTRODUCTION
In Mott insulators, the Heisenberg interaction JSi·Sjis in
most cases the dominant source of coupling between local
moments, and most theoretical investigations are based on
modeling in which only this type of interaction is included. It
has been known for a very long time, however, that other,
less symmetric, interactions are present. For instance, unless
there is an inversion center on a bond, spin-orbit coupling
induces an antisymmetric interaction of the form D·Si
Sj, which is known as the Dzyaloshinskii-Moriya DM
interaction.1,2Since it breaks the fundamental SU2symme-
try of the Heisenberg interactions, the DM interaction is at
the origin of many deviations from pure Heisenberg behav-
ior, such as canting3or small gaps.410 It is also known to
have a dramatic impact on the properties of antiferromagnets
in a magnetic field. Numerous experimental investigations of
quantum antiferromagnets currently in progress in large field
facilities call for a detailed understanding of this
problem.1115 Several issues have recently been the subject of
rather intensive research. For instance, the impact on triplon
Bose-Einstein condensation1618 of DM interactions has been
analyzed.19 The interplay of frustration and DM interactions
has also received significant attention.2023 The consequence
of the breaking of SU2symmetry on the excitation spec-
trum is also well understood thanks to the work of several
people including some of the present authors.410 It is by
now well established that a DM interaction can open a gap in
otherwise gapless regions. The scaling of this gap with the
magnitude of the DM interaction has been worked out for
several cases.6,8
Surprisingly, however, the other important consequence of
the breaking of the SU2symmetry on the ground-state
properties of weakly coupled dimers, namely the develop-
ment of a local magnetization, has not received much atten-
tion so far, although it is of immediate relevance to several
compounds. It was shown in the case of SrCu2BO32that a
DM interaction can lead to the development of a measurable
and in fact quite largestaggered magnetization,20 but a
simple picture of how the magnitude and the orientation of
the DM interaction with respect to the magnetic field influ-
ences these properties has not yet emerged. Besides, the fact
that a DM interaction can lead to the development of a trans-
verse uniform magnetization and its impact on torque mea-
surements of the magnetization have not been investigated in
detail. All these questions are central to the understanding of
several systems of current interest. In particular, recent NMR
results by Clémancey et al.24 have revealed the presence of a
staggered magnetization in the dimer compound
Cu2C5H12N22Cl4abbreviation: CuHpCl, and the inter-
pretation of these results requires a precise investigation of
the effect of DM interactions on weakly coupled dimer sys-
tems.
In this paper, our goal is threefold. First of all, we want to
put the theoretical results of Ref. 24 in a broader perspective,
investigating all aspects of the local magnetization, and not
simply the staggered magnetization that has been detected in
the NMR experiment reported in Ref. 24. We also want to
strengthen the case for the model proposed in Ref. 24 by
reporting torque measurements which can be very naturally
explained in the context of that model. Finally, a more gen-
eral goal is to come up with a simple picture of the develop-
PHYSICAL REVIEW B 75, 184402 2007
1098-0121/2007/7518/1844028©2007 The American Physical Society184402-1
ment of local magnetization in the presence of DM interac-
tions.
To this end, we first look at the case of an isolated dimer,
and derive simple expressions in the limits of weak and
strong magnetic field which we believe are very useful to get
a simple picture of subtle issues such as the effect of the
relative orientation of the magnetic field and the Dvector of
the DM interaction on the uniform magnetization. We then
turn to the case of coupled dimers and concentrate on a
simple ladder geometry. This choice is motivated partly by
the potential relevance of this geometry to actual compounds
such as CuHpCl, and by the possibility to obtain very ac-
curate results using the density-matrix renormalization-group
method DMRG兲共Refs. 2527in this quasi-one dimen-
sional geometry. We report in great detail on several quanti-
ties of direct experimental relevance such as the excitation
gap or the uniform magnetization. Finally, we report on dif-
ferent torque measurements on CuHpCl and discuss them
in the light of these results.
II. ISOLATED DIMER
The problem of an isolated dimer in a magnetic field in
the presence of a DM interaction is defined by the Hamil-
tonian
H=JS1·S2+D·S1S2g
BHS1
z+S2
z.1
The zaxis has been chosen to be that of the magnetic
field, and the yz plane as the plane defined by the magnetic
field and the Dvector. In actual systems, the direction of the
Dvector relative to the bond connecting the two sites is fixed
by the microscopic arrangement of atoms and orbitals, and it
is the orientation of the magnetic field that can be varied with
respect to the crystal, but the convention of having the mag-
netic field along the zaxis makes the discussion somewhat
simpler. The Dvector is written as D=0,Dsin
,Dcos
.
The model is illustrated in Fig. 1in the case where both the
field and the Dare perpendicular to the dimer, but all results
are valid whatever the actual orientation of the dimer see the
discussion of symmetry below.
The isolated dimer problem is, of course, very simple.
The Hilbert space is of dimension 4, and it will prove con-
venient to work in the basis,
s=1
2共兩↑↓↓↑典兲,
t1=↑↑,
t0=1
2共兩↑↓+↓↑典兲,
t−1=↓↓.2
In order to come up with an intuitive picture of how local
magnetizations develop, we will first discuss the problem
from the point of view of symmetry. We will then derive
useful expressions for small Din weak field and close to
saturation, and we will present plots of some representative
results obtained by exact diagonalizations in the last para-
graph of that section.
A. Symmetry analysis
Without DM interaction, the Hamiltonian is invariant un-
der the real-space permutation of sites 1 and 2 denoted P12 in
the following. Without a magnetic field, the Hamiltonian is
invariant under all rotations in spin space, and under time
reversal T, which changes the sign of all components of spin
operators. All these operations commute with each other. In a
magnetic field, the SU2symmetry of the Heisenberg model
is reduced to a U1symmetry corresponding to a rotation in
spin space around the field direction Rz
, and time reversal
is not a symmetry any more.
As soon as a DM interaction is introduced, the Hamil-
tonian is no longer invariant with respect to the permutation
P12. If the magnetic field is parallel to the Dvector, a rota-
tion in spin space around their common direction is still a
symmetry, but in the general case, all elementary symmetries
are lost. However, the operation that exchanges sites 1 and 2
and simultaneously changes the sign of the xcomponent of
the spin operators the component perpendicular to both the
field and the Dvectoris a symmetry operation, as can be
easily checked directly in Eq. 1. This operation can be de-
scribed as the composition of a rotation by
around the
direction perpendicular to both the field and the Dvector
Rx
, the time-reversal operation T, and the permutation
P12. As a consequence, the expectation values of local spin
operators in any eigenstate of the Hamiltonian satisfy the
relations
S1
x=−S2
x,
S1
y=S2
y,
S1
z=S2
z.3
These relations imply that the staggered magnetization per
site, defined as ms=共具S1S2典兲/2, is perpendicular to the
plane defined by the magnetic field and the Dvector, while
the uniform magnetization per site defined by mu=共具S1
+S2典兲/2 must lie in that plane.
If the Dvector is parallel to the field, the U1rotational
symmetry in spin spaceis still present, which can be easily
checked since Sz
tot=S1
z+S2
zcommutes with S1
xS2
yS1
yS2
x. The
states t−1and t1are still eigenstates with energies
J/4±g
BH, but sand t0get coupled. The staggered mag-
netization is identically zero, while the uniform magnetiza-
FIG. 1. Color onlinePictorial representation of the model of
Eq. 1of a dimer with DM interaction in a magnetic field.
MIYAHARA et al. PHYSICAL REVIEW B 75, 184402 2007
184402-2
tion jumps abruptly from 0 to 2g
Bz
ˆz
ˆis the direction of the
applied magnetic fieldat a critical field Hclarger than its
D=0 value J/g
B.
B. Low-field limit
In the limit D/J1 and below the saturation field Hc
=J/g
B, the ground-state wave function, up to second order
in D/J, reads
0=
1− D2
4J2
sDsin
22Jg
BHt1+iDcos
2Jt0
Dsin
22J+g
BHt−1.4
In the low-field limit, first-order perturbation theory in H
can be used to derive simple expressions for the expectation
value of the various spin operators:
S1
x=−S2
x=g
BHD sin
2J2,
S1
y=S2
y=−g
BHD2cos
sin
4J3,
S1
z=S2
z=g
BHD2sin2
4J3.5
These expressions lead to compact and suggestive expres-
sions for the uniform and staggered magnetizations:
mu=g
B
4J3DHD,
ms=g
B
2J2DH.6
As required by symmetry, the staggered magnetization is
perpendicular to both the field and the Dvector. As far as the
uniform magnetization is concerned, symmetry only requires
that it lies in the plane of the magnetic field and of the D
vector, but in the low-field limit, Eq. 6shows that it is
perpendicular to the Dvector. So the uniform magnetization
is in general not parallel to the magnetic field, as it would be
in a system with SU2symmetry, and it can in fact deviate
strongly: In the limit where the Dvector becomes parallel to
the field, the uniform magnetization becomes perpendicular
to the magnetic field, a rather anomalous behavior that
should have important consequences for torque measure-
ments of the magnetization.
Another remarkable feature of these results is that the
staggered magnetization is first order in D, while the uniform
magnetization is second order. Thus at low field the response
is dominated by the staggered magnetization, as already ob-
served in SrCuBO32.
Finally, let us emphasize that, as implied by Eq. 6as
well as by the symmetry arguments developed in Sec. II A,
the uniform and staggered magnetizations have universal ex-
pressions in terms of the magnetic field and of the Dvector,
which are valid regardless of the orientation of the dimer
with respect to them.
C. Critical field
At the critical field Hc=J/g
B, one has to turn to degen-
erate perturbation theory since, for D=0, sand t1are de-
generate. When the Dvector is not parallel to the field, these
states get coupled by an off-diagonal term Dsin
.Inthe
limit D0, the ground-state wave function is then simply
given by
0=共兩st1典兲/2, and the staggered magnetization
per site is equal to 2/4g
B0.35g
B, independently of
the angle
. Such a large value in the limit D0 might come
as a surprise since, when D=0, the staggered magnetization
should be identically zero. But in fact, when Dis small, the
staggered magnetization becomes peaked around Hc,inan
interval of width of the order of Dsin
, which shrinks to
zero in the limit where D0. So, for HHc, the staggered
magnetization indeed goes to zero when D0. In addition
at the critical field H=Hc, the staggered magnetization is ill
defined in the case D=0 since the ground state becomes
degenerate.
When DJ, the uniform magnetization at this field is
equal to g
B, which corresponds to half the polarization
value. When the angle between Dand His not
/2, a small
uniform component develops along ydue to the coupling of
swith t0. This transverse with respect to the fielduni-
form magnetization is given by mu
y=−2/4cos
D/Jg
B
−0.35 cos
D/Jg
B. In contrast to the small-field result,
it is now linear in D, but remains much smaller than the
staggered magnetization, which is of order 1.
D. Exact results
To get an idea of the accuracy of the expressions obtained
at low field and close to the saturation field, we have plotted
in Fig. 2the exact value of ms
x,mu
y, and mu
zfor a representa-
tive case D/J=0.04 and
=
/4. The small-field expression
is quantitatively accurate up to H0.25J/g
B, and the width
of the peak of the staggered magnetization and the maximal
value of mu
yare indeed of order D.
III. COUPLED DIMERS (LADDER)
A. Model
In this section, our goal is to check to which extent the
properties of a system of weakly coupled dimers resemble
those of isolated dimers. In particular, the transition between
zero magnetization and polarization takes place through an
extended region of magnetic field of the order of the inter-
dimer coupling, and we would like to know how the system
behaves within and outside this region. We will attack this
problem numerically, and in order to perform simulations on
large systems, we have chosen to work in a ladder geometry
and to use the DMRG. The model is defined by the Hamil-
tonian
UNIFORM AND STAGGERED MAGNETIZATIONS INDUCEDPHYSICAL REVIEW B 75, 184402 2007
184402-3
H=J
i
Si,1 ·Si,2 +
i
−1iD·Si,1 Si,2
+J
i
Si,1 ·Si+1,1 +Si,2 ·Si+1,2g
BH
i
Si,1
z+Si,2
z.
7
As for the isolated dimer, the Dvector is assumed to lie in
the yz plane, i.e., D=0,Dsin
,Dcos
. Our choice of an
alternating Dvector from one rung to the other see Fig. 3is
motivated by symmetry considerations. Indeed, in a canoni-
cal ladder, the middle of each rung is an inversion center, and
the DM interaction vanishes by symmetry. A simple way to
allow for the DM interaction to become finite without modi-
fying the symmetry of the exchange couplings is to assume
that some buckling is present along the ladder, as sketched in
Fig. 4. In that case, the only mirror plane that contains a
bond is the xz plane, and a DM interaction with a Dvector
parallel to yis allowed by symmetry. But, in this geometry,
the presence of a C2axis see Fig. 4implies that the D
vector alternates from one rung to the other. The buckling
realized in CuHpCl is slightly more subtle successive
rungs are connected by an inversion symmetry in the middle
of a plaquette, but this symmetry also implies alternating D
vectors. Note, however, that other ways of breaking the in-
version symmetry of the rungs can lead to other arrange-
ments of Dvectors.
Another motivation to work with alternating Dvectors is
to keep the perturbation caused by the interdimer coupling as
small as possible. In that respect, this choice is natural. In-
deed, as we have seen in the previous section, the presence
of a Dvector on a rung induces a staggered magnetization. If
the Dvectors of neighboring rungs iand i+1 are equal, the
moments Si,1and Si+1,1will also be equal, which is in
conflict with antiferromagnetic interrung exchange interac-
tions. If, on the contrary, the Dvectors are opposite on
neighboring rungs, the local moments will adopt configura-
tions that are compatible with the exchange.
Note that a ladder structure is always consistent with DM
interactions along the legs of the ladder. In the context of
weakly coupled dimers, these interactions, which should be a
small fraction of the exchange, are expected to be extremely
small, and we have checked that the effect of a small inter-
rung DM interaction is indeed negligible on the results dis-
cussed below.
B. Symmetry analysis
While symmetry considerations related to the microscopic
origin of the DM interaction will usually force it to lie in a
FIG. 3. Color onlineLadder with staggered DM
interaction.
FIG. 2. Color onlineField dependence of the uniform and
staggered magnetizations per site mu
y,mu
z, and ms
xof the isolated
dimer model for D/J=0.04 and
=
/4. The dashed lines are the
analytical results derived in the limit D/J1. Note the difference
of scale for positive and negative magnetizations.
FIG. 4. Color onlineStruc-
ture of a buckled ladder. In such a
ladder, a staggered DM interaction
in the ydirection is allowed by
symmetry.
MIYAHARA et al. PHYSICAL REVIEW B 75, 184402 2007
184402-4
specific direction with respect to the lattice, as for a single
dimer, the discussion of symmetry is simpler if real space
and spin space are separated. For the general case where the
magnetic field and the Dvectors are not parallel, the Hamil-
tonian has two types of symmetries acting only in real space:
ithe inversion centers around the middles of the plaquettes
formed by two consecutive rungs; iiall even translations
along the ladder direction. It also has a symmetry that acts in
both real space and spin space the global version of the
symmetry identified for a dimer, namely iiithe operation
that exchanges sites i,1and i,2or all dimers and simul-
taneously changes the sign of the xcomponent of all spin
operators. As long as these symmetries are not broken, the
following relations between the expectation values of local
spin operators on two neighboring rungs are expected to be
satisfied:
Si,1
x=−Si,2
x=−Si+1,1
x=Si+1,2
x,
Si,1
y=Si,2
y=Si+1,1
y=Si+1,2
y,
Si,1
z=Si,2
z=Si+1,1
z=Si+1,2
z,8
where xis the direction perpendicular to both the field and
the Dvector. As for the single dimer case, these relations are
valid regardless of the actual orientation of the lattice with
respect to the magnetic field and Dvector. They only require
the DM interaction to alternate from one rung to the next. We
thus define the staggered and uniform magnetizations per site
as
ms=1/N
i
−1i共具Si,1Si,2典兲,
mu=1/N
i
共具Si,1+Si,2典兲,9
where Nis the total number of sites, and with the convention
that the angle
is positive for ieven. As in the isolated dimer
case, the staggered magnetization msis along the xaxis,
while the uniform magnetization mulies in the yz plane.
C. Uniform and staggered magnetizations
Let us now turn to the discussion of the numerical results
we have obtained for the model of Eq. 7. We are interested
in the regime DJ
J. For D=0, the model is a simple
ladder in a field, and the properties are well understood.
There is, of course, no staggered magnetization because of
the U1symmetry, and the uniform magnetization is parallel
to the field for the same reason. It vanishes below a critical
field Hc1, takes off with a square-root singularity, and reaches
saturation with another square-root singularity at a second
critical field Hc2. The difference Hc2Hc1scales with J
.
Since, apart from this scaling, the properties depend very
little on J
, we quote results for a single value of J
, and
having in mind the compound CuHpCl,11 we have chosen
J
/J=0.2. For that ratio, the critical fields in the absence of
DM interactions are given by g
BHc1=0.82Jand g
BHc2
=1.40J.
For the model with DM interaction, we have performed
exact diagonalization ED,27 up to 20 sites 10 rungs, and
DMRG calculations on ladders with up to 80 rungs. The
results evolve smoothly with the size, and we only quote
DMRG results obtained for 80-rung clusters finite-size ef-
fects for the gap are discussed in the next section. Note that
in those calculations, Szis not a good quantum number. This
is well known to reduce greatly the maximal size available to
exact diagonalizations, but this also has an impact on the
number of states we were able to keep during the DMRG
runs. Here, we diagonalize by the Davidson methoda ma-
trix of size 4m2at each DMRG step. In a standard DMRG
run where Szis a good quantum number, the matrix of the
effective Hamiltonian in the variational basis is block-
diagonal, which can speed up the diagonalization by a factor
of 10 or more. The memory needed is also larger at fixed m
than for the standard DMRG. For those reasons, most of the
calculations were done with up to m=600 states kept during
five sweeps, and only up to N=80 sites. The discarded
weight was of the order of 10−10 when we targeted two states
to extract the gap, and of the order of 10−12 or less when we
targeted a single state to extract correlations. We also per-
formed a few runs with mup to 800 in order to confirm that
the numerical data were well converged.
The zcomponent of the magnetization is displayed in Fig.
5for several values of Dand
. It is reminiscent of that for
D=0; however, when D0, the magnetization develops as
soon as the magnetic field is switched on, only reaching satu-
ration asymptotically in the limit of infinite field. The square-
root singularities are removed. It was shown in Ref. 10 that,
at Hc1, the magnetization should depend on the magnitude of
the Dvector as mu
zDsin
2/5, in agreement with the
present results see the inset of Fig. 5.
When
/2 i.e., Dz0, a uniform magnetization
along the yaxis also develops, as in the isolated dimer case.
Figure 6shows the magnetization along the yaxis and the
angle
ubetween the uniform magnetization and the zaxis
as a function of the magnetic field for
=
/6. At low field,
FIG. 5. Color onlineExamples of the variation of the uniform
magnetization along the zaxis with the field. Inset: Plot of mu
zas a
function of Dsin
/J2/5 slightly below Hc1, which confirms the
scaling predicted in Ref. 8.
UNIFORM AND STAGGERED MAGNETIZATIONS INDUCEDPHYSICAL REVIEW B 75, 184402 2007
184402-5
the uniform magnetization is orthogonal to the DM vector,
again as for an isolated dimer. The magnetization along yis
maximal between the two critical fields. Its value in that
range is clearly much smaller than the component along the
field
ubecomes very small near Hc1, but this extra con-
tribution to the uniform magnetization will produce a torque
that should be detectable experimentally given the very high
sensitivity of torque measurements.
The staggered magnetization along xexhibits a kind of
plateau in the intermediate phase between Hc1and Hc2Fig.
7. Its magnitude inside the plateau is of the order of the
maximal value of the isolated dimer 0.35g
B, and it de-
pends relatively weakly on D. In contrast, the extent of the
tails outside this plateau region increases rapidly with D.
Remarkably, the magnetization per spin along xis larger than
along zup to Hc1and even slightly above. Note that the
staggered magnetization depends essentially on the value of
Dyand is very weakly affected by the value of Dz.
D. Gap
The effect of a SU 2breaking interaction on a ladder has
been studied in Ref. 10. It strongly depends on the nature of
the plateau phase. For the transition from the zero or full
polarization to the gapless phase, the effective-field theory is
expected to be the same as for the spin chain close to satu-
ration, and the gaps at Hc1and Hc2should open as
Dsin
4/5, as shown in Ref. 8. This prediction clearly
agrees with the results for
=
/2 shown in Fig. 8lower
right panel. Size effects are already very small for N=80
sites, as can be seen in Fig. 8lower left panel. Between the
two critical fields, the gap is expected to remain finite. The
closing of the gap in Ref. 10 was caused by a breaking of the
Z2symmetry which does not occur here as there is no m
=1/2 plateau when D/J=0. The effect of the zcomponent
of Dis expected to be very small. This is also confirmed by
our DMRG results not shown.
Note that, because of the presence of a gap, the finite-size
effects are expected to be negligible if the cluster size is
larger than the correlation length. Since the correlation is
inversely proportional to the gap, the finite-size effects are
expected to be maximal at the two critical fields where the
gap is minimum. As shown in the lower left panel of Fig. 8,
these effects are already completely negligible for 40 sites,
and all the results presented in this paper can be considered
to accurately represent the thermodynamic limit.
E. Application to experiments
The existence of a field induced staggered magnetization
in dimer systems has been established by NMR in
SrCu2BO3Ref. 20and in CuHpCl.24 In this latter com-
pound, in which Hc1=7.5 T and Hc2= 13 T, the field depen-
dence of msat 50 mK between 5 and 15 T has been very
well reproduced by our calculation on a ladder for reasonable
parameter values: D/J=0.05, J
/J=0.2, J=13 K. The field-
induced transverse uniform magnetization mu, however, is an
order of magnitude smaller than ms, and very difficult to
detect in NMR measurements. On the contrary, magnetic
torque measurements are not sensitive to the transverse stag-
gered magnetization, but are ideally suited to detect the pres-
ence of mu, as explained in the following. Let us recall that
in the presence of an external magnetic field Happlied along
the zaxis, a crystal which is able to rotate around an axis iis
submitted to a torque which depends on the orientation of z
and iwith respect to the principal axis of the susceptibility
FIG. 6. Color onlineLower panel: ycomponent of the uniform
magnetization as a function of the field for
=
/6 and various
values of D/J. Significant values appear between Hc1and Hc2, and
far outside this interval as soon as D/Jis not too small. Upper
panel: Angle
ubetween the magnetic field and the uniform mag-
netization muas a function of the field for
=
/6 and several
values of D/J. Note that
ugoes to
/2 in the low-field limit, in
agreement with the prediction for an isolated dimer Eq. 6兲兴.
FIG. 7. Color onlineStaggered magnetization as a function of
the magnetic field for several values of D/Jand
. Large values are
achieved between Hc1and Hc2, and far outside this interval as soon
as D/Jis not too small. The value between Hc1and Hc2depends
relatively weakly on D/Jand
, and is of the same order as the
maximal value in the case of an isolated dimer 0.35g
B. In con-
trast, the value outside this interval depends very strongly on the
magnitude of Dsin
.
MIYAHARA et al. PHYSICAL REVIEW B 75, 184402 2007
184402-6
tensor
. The torque is equal to zero each time the magnetic
energy EM=−H2z
zpasses through an extremum when ro-
tating the crystal around i. Let us consider the simple geom-
etry where iis perpendicular both to the applied field and to
the easy axis of the susceptibility tensor, which we shall
assume to have an axial symmetry. In that case,
can be
simply expressed as
=
sin 2
H2,10
where
is the angle between zand the easy axis of
.Inthe
absence of Dzyaloshinskii-Moriya interaction, the anisotropy
of the magnetic susceptibility of a system of coupled dimers
of spins 1/2 is expected to come only from the macroscopic
resultant of the individual gtensors. Such an anisotropy is
both temperature and field independent. However, as shown
above, a staggered DM interaction along the rungs of a lad-
der can also induce a component of the magnetization with a
component mu
yperpendicular to the applied magnetic field,
which strongly varies with Hand disappears only well above
Hc2Fig. 6. Except for very peculiar orientations of D, this
component will exist even if the field is applied along the
easy axis, so that only mu
ywill contribute to the torque, which
should thus vanish for fields much larger than Hc2.
In order to test this idea, magnetic torque measurements
have been performed at low temperature 410 mKon the
same compound CuHpCl in which the presence of field
induced staggered magnetization mswas observed by
NMR.24 Experiments were carried out in a resistive magnet
and
was measured from zero up to 23 T. A small crystal of
CuHpCl was glued on a beryllium bronze cantilever, the
displacements of which were measured capacitively. The
cantilever could be rotated in situ with respect to the applied
field H. The orientation of the crystal was adjusted so that
=0 at the highest values of H, as shown in the inset of
Fig. 9, which fully cancels the contribution due to the aniso-
tropy of the gtensor. This orientation indeed corresponds to
H
100. In spite of this, a large additional contribution
shows up between the two critical fields, and extends well
outside the intermediate region. For comparison, the calcu-
lated component of the uniform magnetization perpendicular
to the field of a ladder with J
/J=0.2, Dy/J=0.05, and
Dz/J=0.086
=
/6is depicted on the same plot, with
scales adjusted to get the same value at Hc1. The values of Dy
and J
are those used in Ref. 24 to fit the staggered magne-
tization, while the results depend very little on Dzup to an
overall scale factor. The two curves are in good qualitative
FIG. 8. Color onlineUpper
panel: Field dependence of the ex-
citation gap for J
/J=0.2 and
several values of D/J:D/J=0.01
and
=
/2 black circle,D/J
=0.02 and
=
/2 red square,
D/J=0.04 and
=
/2 green dia-
mond, and D/J=0.08 and
=
/2 blue trianglefor N=80
DMRG. Lower left panel: Scal-
ing of the excitation gap as a func-
tion of 1/Nfor D/J= 0.02 and
=
/2. Lower right panel: Scaling
of the gap as a function of D/Jfor
N=80 see text.
FIG. 9. Color onlineTransverse uniform magnetization mu
yfor
Dy/J=0.05, Dz/J=0.086, and J
/J=0.2 blue circlesand torque
divided by field experimental curve obtained on CuHpClas a
function of the field red line. Inset: torque measurement raw data
dashed-dotted black lineand torque divided by field red line.
UNIFORM AND STAGGERED MAGNETIZATIONS INDUCEDPHYSICAL REVIEW B 75, 184402 2007
184402-7
agreement, especially considering the fact that the only ad-
justable parameter is the overall scale factor. In order to go
beyond this qualitative agreement, it would be necessary to
consider several additional effects. First of all, inelastic
neutron-scattering data have challenged the description of
this system as a simple ladder,28 and further couplings still
not definitely identifiedshould presumably be included. In
addition, there is a transition into a three-dimensional 3D
ordered phase11 between Hc1and Hc2, and although the pre-
cise nature of the ordering is still unknown, it is very likely it
will affect the uniform magnetization. The onset of the 3D
ordering at Hc1is clearly visible on the experimental mea-
surements shown in Fig. 9. Obviously, at the present stage,
too little is known about these additional effects to be able to
take them into account, and this is left for future investiga-
tion.
IV. CONCLUSIONS
If spin-1/2 dimers are coupled in such a way that there is
no inversion center at the middle of the bond, very signifi-
cant modifications of the physics in a magnetic field have to
be expected. Indeed, unless it is forbidden by symmetry, a
DM interaction will always be present, and the analysis re-
ported in this paper shows that even a tiny DM interaction
can modify some aspects of the physics rather dramatically.
This is especially true for the staggered magnetization, which
immediately acquires large values in the intermediate phase
where the system gets polarized, and which can take on sig-
nificant values outside this phase for physically relevant val-
ues of the DM interaction. This is also true for the uniform
magnetization as soon as the Dvector of the DM interaction
and the field are neither parallel nor perpendicular. In that
case, a component of the uniform magnetization perpendicu-
lar to the magnetic field appears, which can induce a mea-
surable torque on the sample. This has been proven for an
isolated dimer and for a ladder with staggered DM interac-
tions, but these conclusions are expected to hold true for all
coupled-dimer systems as long as the Dvectors are arranged
in such a way that there is no competition with Heisenberg
exchange as far as the development of a staggered magneti-
zation is concerned. It is our hope that these results will help
understand some of the strange properties observed in
coupled-dimer systems.
ACKNOWLEDGMENTS
We acknowledge useful discussions with Karlo Penc and
Oleg Tchernyshyov. This work was supported by the Grant-
in-Aids for Scientific Research on Priority Areas “Invention
of Anomalous Quantum Materials” and for Aoyama Gakuin
University 21st COE Program from the Ministry of Educa-
tion, Culture, Sports, Science and Technology of Japan, by
the Swiss National Fund, by MaNEP, and by the SFB 382.
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184402-8
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