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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

共Received 19 October 2006; revised manuscript received 1 February 2007; published 1 May 2007兲

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 共DMRG兲calculations 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 Cu2共C5H12N2兲2Cl4.

DOI: 10.1103/PhysRevB.75.184402 PACS number共s兲: 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·共Si

⫻Sj兲, which is known as the Dzyaloshinskii-Moriya 共DM兲

interaction.1,2Since it breaks the fundamental SU共2兲symme-

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.4–10 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.11–15 Several issues have recently been the subject of

rather intensive research. For instance, the impact on triplon

Bose-Einstein condensation16–18 of DM interactions has been

analyzed.19 The interplay of frustration and DM interactions

has also received signiﬁcant attention.20–23 The consequence

of the breaking of SU共2兲symmetry on the excitation spec-

trum is also well understood thanks to the work of several

people including some of the present authors.4–10 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 SU共2兲symmetry 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 SrCu2共BO3兲2that a

DM interaction can lead to the development of a measurable

共and in fact quite large兲staggered 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

Cu2共C5H12N2兲2Cl4关abbreviation: Cu共Hp兲Cl兴, 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 共2007兲

1098-0121/2007/75共18兲/184402共8兲©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 Cu共Hp兲Cl, and by the possibility to obtain very ac-

curate results using the density-matrix renormalization-group

method 共DMRG兲共Refs. 25–27兲in 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 Cu共Hp兲Cl 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·共S1⫻S2兲−g

BH共S1

z+S2

z兲.共1兲

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=共0,Dsin

,Dcos

兲.

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 共see the

discussion of symmetry below兲.

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,

兩s典=1

冑2共兩↑↓典−兩↓↑典兲,

兩t1典=兩↑↑典,

兩t0典=1

冑2共兩↑↓典+兩↓↑典兲,

兩t−1典=兩↓↓典.共2兲

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 SU共2兲symmetry of the Heisenberg model

is reduced to a U共1兲symmetry corresponding to a rotation in

spin space around the ﬁeld direction Rz共

␣

兲, 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 共the component perpendicular to both the

ﬁeld and the Dvector兲is a symmetry operation, as can be

easily checked directly in Eq. 共1兲. 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共

兲, 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

具S1

x典=−具S2

x典,

具S1

y典=具S2

y典,

具S1

z典=具S2

z典.共3兲

These relations imply that the staggered magnetization per

site, deﬁned as ms=共具S1−S2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 U共1兲rotational

symmetry 共in spin space兲is still present, which can be easily

checked since Sz

tot=S1

z+S2

zcommutes with S1

xS2

y−S1

yS2

x. The

states 兩t−1典and 兩t1典are still eigenstates with energies

J/4±g

BH, but 兩s典and 兩t0典get coupled. The staggered mag-

netization is identically zero, while the uniform magnetiza-

FIG. 1. 共Color online兲Pictorial representation of the model of

Eq. 共1兲of a dimer with DM interaction in a magnetic ﬁeld.

MIYAHARA et al. PHYSICAL REVIEW B 75, 184402 共2007兲

184402-2

tion jumps abruptly from 0 to 2g

Bz

ˆ共z

ˆis the direction of the

applied magnetic ﬁeld兲at a critical ﬁeld Hclarger than its

D=0 value J/g

B.

B. Low-ﬁeld limit

In the limit D/JⰆ1 and below the saturation ﬁeld Hc

=J/g

B, the ground-state wave function, up to second order

in D/J, reads

兩

0典=

冉

1− D2

4J2

冊

兩s典−Dsin

2冑2共J−g

BH兲兩t1典+iDcos

2J兩t0典

−Dsin

2冑2共J+g

BH兲兩t−1典.共4兲

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:

具S1

x典=−具S2

x典=g

BHD sin

2J2,

具S1

y典=具S2

y典=−g

BHD2cos

sin

4J3,

具S1

z典=具S2

z典=g

BHD2sin2

4J3.共5兲

These expressions lead to compact and suggestive expres-

sions for the uniform and staggered magnetizations:

mu=g

B

4J3共D⫻H兲⫻D,

ms=g

B

2J2共D⫻H兲.共6兲

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. 共6兲shows 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 SU共2兲symmetry, 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 SrCu共BO3兲2.

Finally, let us emphasize that, as implied by Eq. 共6兲as

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, 兩s典and 兩t1典are de-

generate. When the Dvector is not parallel to the ﬁeld, these

states get coupled by an off-diagonal term Dsin

.Inthe

limit D→0, the ground-state wave function is then simply

given by

0=共兩s典−兩t1典兲/冑2, and the staggered magnetization

per site is equal to 共冑2/4兲g

B⯝0.35g

B, independently of

the angle

. Such a large value in the limit D→0 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 D→0. So, for H⫽Hc, the staggered

magnetization indeed goes to zero when D→0. 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 DⰆJ, 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

兩s典with 兩t0典. This transverse 共with respect to the ﬁeld兲uni-

form magnetization is given by mu

y=−共冑2/4兲cos

共D/J兲g

B

⯝−0.35 cos

共D/J兲g

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 共D/J=0.04 and

=

/4兲. The small-ﬁeld expression

is quantitatively accurate up to H⯝0.25J/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 INDUCED…PHYSICAL REVIEW B 75, 184402 共2007兲

184402-3

H=J兺

i

Si,1 ·Si,2 +兺

i

共−1兲iD·共Si,1 ⫻Si,2兲

+J

储

兺

i

共Si,1 ·Si+1,1 +Si,2 ·Si+1,2兲−g

BH兺

i

共Si,1

z+Si,2

z兲.

共7兲

As for the isolated dimer, the Dvector is assumed to lie in

the yz plane, i.e., D=共0,Dsin

,Dcos

兲. Our choice of an

alternating Dvector from one rung to the other 共see Fig. 3兲is

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 共see Fig. 4兲implies that the D

vector alternates from one rung to the other. The buckling

realized in Cu共Hp兲Cl is slightly more subtle 共successive

rungs are connected by an inversion symmetry in the middle

of a plaquette兲, 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 具Si,1典and 具Si+1,1典will 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. 共Color online兲Ladder with staggered DM

interaction.

FIG. 2. 共Color online兲Field 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/JⰆ1. Note the difference

of scale for positive and negative magnetizations.

FIG. 4. 共Color online兲Struc-

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 共2007兲

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:

共i兲the inversion centers around the middles of the plaquettes

formed by two consecutive rungs; 共ii兲all even translations

along the ladder direction. It also has a symmetry that acts in

both real space and spin space 共the global version of the

symmetry identiﬁed for a dimer兲, namely 共iii兲the operation

that exchanges sites 共i,1兲and 共i,2兲or 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:

具Si,1

x典=−具Si,2

x典=−具Si+1,1

x典=具Si+1,2

x典,

具Si,1

y典=具Si,2

y典=具Si+1,1

y典=具Si+1,2

y典,

具Si,1

z典=具Si,2

z典=具Si+1,1

z典=具Si+1,2

z典,共8兲

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=共1/N兲兺

i

共−1兲i共具Si,1典−具Si,2典兲,

mu=共1/N兲兺

i

共具Si,1典+具Si,2典兲,共9兲

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. 共7兲. We are interested

in the regime D⬍J

储

⬍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 U共1兲symmetry, 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 Hc2−Hc1scales with J

储

.

Since, apart from this scaling, the properties depend very

little on J

储

, we quote results for a single value of J

储

, and

having in mind the compound Cu共Hp兲Cl,11 we have chosen

J

储

/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 共ED兲,27 up to 20 sites 共10 rungs兲, 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 共ﬁnite-size ef-

fects for the gap are discussed in the next section兲. 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 共by the Davidson method兲a 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 D⫽0, 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

z⬀共Dsin

兲2/5, in agreement with the

present results 共see the inset of Fig. 5兲.

When

⫽

/2 共i.e., Dz⫽0兲, 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. 共Color online兲Examples of the variation of the uniform

magnetization along the zaxis with the ﬁeld. Inset: Plot of mu

zas a

function of 共Dsin

/J兲2/5 slightly below Hc1, which conﬁrms the

scaling predicted in Ref. 8.

UNIFORM AND STAGGERED MAGNETIZATIONS INDUCED…PHYSICAL REVIEW B 75, 184402 共2007兲

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 共

ubecomes very small near Hc1兲, 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 Hc2共Fig.

7兲. Its magnitude inside the plateau is of the order of the

maximal value of the isolated dimer 共0.35g

B兲, 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 共2兲breaking 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

共Dsin

兲4/5, as shown in Ref. 8. This prediction clearly

agrees with the results for

=

/2 shown in Fig. 8共lower

right panel兲. Size effects are already very small for N=80

sites, as can be seen in Fig. 8共lower left panel兲. Between the

two critical ﬁelds, the gap is expected to remain ﬁnite. 共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兲. The effect of the zcomponent

of Dis expected to be very small. This is also conﬁrmed by

our DMRG results 共not shown兲.

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

SrCu2共BO兲3共Ref. 20兲and in Cu共Hp兲Cl.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

储

/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. 共Color online兲Lower 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 关Eq. 共6兲兴.

FIG. 7. 共Color online兲Staggered 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 共0.35g

B兲. In con-

trast, the value outside this interval depends very strongly on the

magnitude of Dsin

.

MIYAHARA et al. PHYSICAL REVIEW B 75, 184402 共2007兲

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

=共

⬜−

储

兲sin 2

H2,共10兲

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

Hc2共Fig. 6兲. 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 共410 mK兲on the

same compound Cu共Hp兲Cl 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

Cu共Hp兲Cl 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

储

关100兴. 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

储

/J=0.2, Dy/J=0.05, and

Dz/J=0.086 共

=

/6兲is depicted on the same plot, with

scales adjusted to get the same value at Hc1. The values of Dy

and J

储

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. 共Color online兲Upper

panel: Field dependence of the ex-

citation gap 䉭for J

储

/J=0.2 and

several values of D/J:D/J=0.01

and

=

/2 共black circle兲,D/J

=0.02 and

=

/2 共red square兲,

D/J=0.04 and

=

/2 共green dia-

mond兲, and D/J=0.08 and

=

/2 共blue triangle兲for N=80

共DMRG兲. 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 共see text兲.

FIG. 9. 共Color online兲Transverse uniform magnetization mu

yfor

Dy/J=0.05, Dz/J=0.086, and J

储

/J=0.2 共blue circles兲and torque

divided by ﬁeld 关experimental curve obtained on Cu共Hp兲Cl兴as a

function of the ﬁeld 共red line兲. Inset: torque measurement raw data

共dashed-dotted black line兲and torque divided by ﬁeld 共red line兲.

UNIFORM AND STAGGERED MAGNETIZATIONS INDUCED…PHYSICAL REVIEW B 75, 184402 共2007兲

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 共still

not deﬁnitely identiﬁed兲should presumably be included. In

addition, there is a transition into a three-dimensional 共3D兲

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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