Document)

PHYSICAL REVIEW B 83, 024413 (2011)

Effect of Dzyaloshinskii-Moriya interactions on the phase diagram and magneticexcitations of SrCu2(BO3)2

Judit Romhanyi,1,2 Keisuke Totsuka,3 and Karlo Penc1

1Research Institute for Solid State Physics and Optics, P.O. Box 49, H-1525 Budapest, Hungary2Department of Physics, Budapest University of Technology and Economics, Budafoki ut 8, H-1111 Budapest, Hungary

3Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwake-Cho, Kyoto 606-8502, Japan(Received 12 October 2010; published 26 January 2011)

The orthogonal dimer structure of the SrCu2(BO3)2 spin-1/2 magnet provides a realization of the Shastry–Sutherland model. Using a dimer-product variational wave function, we map out the phase diagram of theShastry–Sutherland model including anisotropies. Based on the variational solution, we construct a bond-waveapproach to obtain the excitation spectra as a function of the magnetic field. The characteristic features ofthe experimentally measured neutron and ESR spectra are reproduced, like the anisotropy-induced zero-fieldsplittings and the persistent gap at higher fields.

DOI: 10.1103/PhysRevB.83.024413 PACS number(s): 75.10.Jm, 75.10.Kt, 76.30.−v

I. INTRODUCTION

Spin gap systems, where the spin gap is of quantummechanical origin, are of interest to both theoretical andexperimental investigations. These systems have spin-disordered ground states that can be described as quantum spinliquids.1 A spin gap is known to open, for example, in S = 1/2spin systems that form lattices of coupled dimers,2 givingrise to many interesting phenomena. The Shastry–Sutherlandmodel3 provides a unique example of such two-dimensionalfrustrated networks of S = 1/2 dimers. The model includesnearest-neighbor (n.n.; J ) and next-nearest-neighbor (n.n.n.;J ′) antiferromagnetic interactions as shown in Fig. 1, with theHamiltonian

H = J∑n.n.

Si · Sj + J ′ ∑n.n.n.

Si · Sj . (1)

In the case of J ′ = 0 the model is reduced to a lattice ofindependent dimers, where in the ground state the S = 1/2spins of each dimer form a singlet, and the ground-state wavefunction is just the product of these independent dimer singlets(DSs). According to Shastry and Sutherland,3 the singlet-dimer product state is an exact eigenstate of the Hamiltonianeven for finite values of J ′, due to the particular geometry ofthe lattice.

An experimental realization of the Shastry–Sutherlandmodel is the quasi-two-dimensional antiferromagneticcompound4 SrCu2(BO3)2. This material has a tetragonalunit cell and is characterized by alternating layers ofCuBO3 molecules and Sr2+ ions; in the former, Cu2+ions occupying crystallographically equivalent sites carryspin S = 1/2 degrees of freedom and form a lattice oforthogonal dimers connected by triangular- shaped BO3

molecules.4,5 A schematic of the CuBO3 layer is shown inFig. 1.

Magnetic susceptibility measurements of this materialrevealed a peak at around 20 K and a sharp drop to 0 atdecreasing temperatures.4 Fitting the exponential curve that ischaracteristic for spin gap systems, Kageyama et al. estimated[4] the gap to be � ≈ 19 K, while from the NMR relaxationrate they obtained a gap of about 30 K. Magnetizationmeasurements4 clarified the presence of a gapped spin-singlet

ground state and a continuous transition to a gapless magneticstate at 20 T, which corresponds to a gap of 30 K, in goodagreement with the relaxation rate measurements. While earlymagnetization measurements in high fields revealed plateausonly at 1/4 and 1/8 of the saturated magnetization,4,6 refinedmeasurements have suggested7 more plateaus at 1/3 and othervalues of magnetization.

Miyahara and Ueda8 pointed out that SrCu2(BO3)2 canbe satisfyingly described by the Shastry–Sutherland model.They estimated the critical point where the singlet-dimerground state goes to the Neel state to be (J ′/J )c = 0.7by performing variational calculations and exact numericaldiagonalization [on the basis of series expansion9 and exactdiagonalization,10 it is now believed that the preceding twostates are mediated by a new plaquette-singlet phase and thata transition from the dimer phase to the plaquette singletoccurs at (J ′/J )c = 0.68]. Using the experimental findings inRef. 4, they estimated the nearest-neighbor coupling constantto be J = 100 K and the next-nearest-neighbor couplingJ ′ = 68 K. This yields J ′/J = 0.68, indicating SrCu2(BO3)2

to be close to the transition point (J ′/J )c = 0.7. Later thisestimate was updated to J = 7.3 meV with J ′/J = 0.63511 orJ = 6.16 meV with J ′/J = 0.603.12 Furthermore, Miyaharaand Ueda carried out perturbation theory in the dimer state upto the fourth order in J ′/J and found that the triplet excitationsare localized. Hopping of triplets is only possible throughclosed paths of dimer bonds, thus only from the sixth orderin perturbation. This property of triplet excitations is relatedto the formation of plateaus. At certain values of magnetizationthe excitations localize into a superlattice structure to minimizethe energy.8 Momoi and Totsuka13explained the appearance ofplateau states through a scenario of metal to Mott-insulatortransition where triplet excitations were treated as bosonsinteracting via various repulsive interactions arising fromhigher-order perturbation in J ′. Under dominating repulsiveinteraction, the triplet excitations localize and crystallize incommensurate patterns developing plateau states. In fact,NMR spectroscopy by Kodama et al. directly demonstrated14 asuperlattice structure at m/msat = 1/8. Recently new magne-tization plateaus have been found15 with the nonperturbativecontractor-renormalization method at 1/9, 1/6, and 2/9 ofsaturation, while an analysis16 using perturbative continuous

024413-11098-0121/2011/83(2)/024413(18) ©2011 American Physical Society

http://dx.doi.org/10.1103/PhysRevB.83.024413

JUDIT ROMHANYI, KEISUKE TOTSUKA, AND KARLO PENC PHYSICAL REVIEW B 83, 024413 (2011)

b4

a1 a2

a4 a3b3

b2b1

2

1

2

x

y

b

2

a

S4

2v

1

1

2

1

C

BA

A

FIG. 1. (Color online) (The Shastry–Sutherland lattice inSrCu2(BO3)2. Orthogonal dimers (thick lines) are denoted by A andB, while Cu ions on the dimer are numbered 1 and 2. Interdimer bonds(with Heisenberg exchange J ′) are shown by thin lines. We also showthe S4 and C2v symmetry point groups of the buckled CuBO3 layer.Open and filled circles indicate Cu ions that are below or above thelayer, respectively.

unitary transformation predicted, in addition to these, one moreplateau, at 2/15. It has also been argued that the inclusion ofthe spin-lattice effects determines the spin structure in theplateaus.17

In the past few years various experiments have beencarried out to examine spin excitations. While the originalShastry–Sutherland model is isotropic in spin space, its experi-mental realization SrCu2(BO3)2 exhibits anisotropic behavior;inelastic neutron scattering measurements,18 electron spinresonance,19 and Raman scattering20 indicated a splitting oftriplet excitations at the � point, which was explained to becaused by the effect of an interdimer Dzyaloshinskii-Moriya(DM) vector directed perpendicular to the copper plane.18

Later, another splitting was found21 at the q = (π,0) point,indicating the relevance of in-plane components of the DMinteraction. The ESR study by Nojiri et al.22 shows an antilevelcrossing at the critical magnetic field where the lowest-lyingtriplet excitation would cross the singlet level, which isconsistent with the persistent spin gap found in the specificheat23 and NMR measurements.24 These splittings and theantilevel crossing mean that states corresponding to differentmagnetizations (singlets and triplets) are mixing and Sz is nolonger a good quantum number. This mixing between singletand triplet states of a dimer can be explained by an intradimeranisotropy, for example, an intradimer DM vector. For moredetails on the Shastry–Sutherland model and SrCu2(BO3)2, werefer readers to the review articles Ref. 25 (theory) and Ref. 26(experiments).

The magnetization process in dimer systems such asTlCuCl3 is fairly well understood; the onset of magnetizationis triggered by Bose-Einstein condensation of gapped triplonsand the magnetic phase above the critical field is characterizedby broken XY symmetry perpendicular to the applied field(for a review see, e.g., Ref. 27). Dynamics at high fields is alsowell described within the preceding scenario.28 In contrast,the existence of DM interactions is known to substantiallymodify this picture, and new phases may even appear in thepresence of DM interactions. Moreover, the low kinetic energy

and relatively large (effective) interactions among dimers leadto various magnetic superstructures,14,29 which we cannotsimply neglect in considering the dynamics in high fields.Nevertheless, the global structure of the phase diagram andthe dynamics in the presence of DM interactions and magneticsuperstructures are only partially understood. The aim ofthis paper is to present a simple theoretical framework toinvestigate the ground-state phases and magnetic excitationsover them, with extension to cases with superstructures inmind. Specifically, by using the bond-wave approximation,we examine the excitation spectrum of SrCu2(BO3)2 at zeroand low magnetic fields below the plateaus and compare theresults with neutron scattering and ESR data.

The paper is structured as follows: In Sec. II we review thesymmetry group of SrCu2(BO3)2 and determine the allowedanisotropic terms in the Hamiltonian. In Sec. III we describethe variational approach we use to get the ground state andthe way we construct the excitation spectrum using the bondoperators. In Sec. IV we present the variational phase diagramand the spectrum for the zero-field case. In Sec. V we map outthe phase diagram in the presence of a field perpendicularto the basal plane. In Secs. VI and VII we describe theESR spectra in magnetic fields perpendicular and parallel tothe plane, respectively. Finally, we compare our results withneutron scattering experiments and ESR spectra in Sec. VIII.We conclude in Sec. IX.

II. SYMMETRY CONSIDERATIONS AND THE MODELHAMILTONIAN

At high temperatures the space group of SrCu2(BO3)2

is a tetragonal I4/mcm.5,30 A structural distortion in theCuBO3 layers below Ts = 395 K shifts the two types oforthogonal dimer planes along the z axis in opposite directions,lowering the symmetry of the SrCu2(BO3)2 to I 42m at lowtemperatures, also with tetragonal symmetry.30 Restrictingourselves to the symmetries of the CuBO3 layer, above Ts

the two—orthogonal to each other—types of dimers lie inthe same plane and the wallpaper group p4g consists ofthe point group C2v = {E,C2(z),σxz,σyz} at the middle of thedimer bond and of the point group C4h in the middle of foursites belonging to different dimers. Lowering the temperature,while C2v remains a symmetry of the buckled CuBO3 layers,the loss of the σh reflection plane below Ts lowers the C4h

point group to S4 = {E,S4,C2(z),S34} (see Fig. 1).The unit cell

in both cases consists of two orthogonal dimers: dimer A,which is parallel to the x axis, and dimer B, parallel to they axis.

The symmetry of the lattice determines the possible termsin the Hamiltonian: the components of the g-tensor anisotropyand of the exchange interactions, including the components ofthe DM interactions; see Table I and, e.g., Ref. 24. We presentthe relevant terms shortly. Sites A1, A2, B1, and B2 correspondto sites 1, 2, 3, and 4 of Ref. 24, respectively.

The g tensor at site A1 takes the form

gA1 =

⎛⎜⎝

gx 0 −gs

0 gy 0

−gs 0 gz

⎞⎟⎠ , (2)

024413-2

EFFECT OF DZYALOSHINSKII-MORIYA INTERACTIONS . . . PHYSICAL REVIEW B 83, 024413 (2011)

TABLE I. Summary of symmetry analyses. If the g-tensoranisotropy is taken into account, the spin O(2) symmetry is lost.

High symmetry Low symmetry

Symmetry (unit cell) D4h D2d

DMIntradimer Forbidden D ‖ (ab) ∧ D ⊥ dimerInterdimer D′ ‖ c Arbitrary

Spinh ‖ z O(2) symmetry —h ‖ x — —

with gx = gy as required by the tetragonal symmetry. Whenthe field is in the z direction, the Zeeman term reads

Hh = −gzμBHz

(Sz

A1 + SzA2 + Sz

B1 + SzB2

)+ gsμBHz

(Sx

A1 − SxA2 + S

y

B1 − Sy

B2

), (3a)

while if the field is along the x axis, it readsHh = −gxμBHx

(Sx

A1 + SxA2

) − gyHx

(Sx

B1 + SxB2

)+ gsμBHx

(Sz

A1 − SzA2

). (3b)

For convenience, we choose hz = gzμBHz and introduce thescaled variable gs = gs/gz.

Next we consider the DM interactions HDM = HD + HD′ ,with the intradimer,

HD =∑n.n.

Dij · (Si × Sj ), (4)

and interdimer,

HD′ =∑n.n.n.

D′ij · (Si × Sj ), (5)

contributions. The summation is over the nearest-neighborand next-nearest-neighbor sites. Once we specify the DMvectors on a bond, the DM interactions on the remainingbonds of the unit cell follow from the symmetry of the cell,as shown in Fig. 2. To specify the sign of the DM interactionunambiguously, we have also denoted by the arrow the orderof the spins in the cross product: an arrow from site i to site j

in Fig. 2 means that we need to take Si × Sj .At temperatures above the structural transition,30 T > Ts =

395 K (high-temperature phase), the middle of a bond is aninversion center due to which there is no DM vector on thedimer bond, and only the interdimer D′ perpendicular to theCuBO3 plane is allowed:

D = 0, (6a)

D′ = (0,0,D′⊥) (6b)

(we denote the z component D′⊥). In the following, we call

this case the high-symmetry case.Below Ts (low-temperature phase), however, this inversion

symmetry is lost and the in-plane DM components are allowedas well. Correspondingly, D′ becomes an arbitrary vector andthe intradimer D is lying in the CuBO3 plane and perpendicularto the dimer, so that

DA = (0,D,0), (7a)

DB = (−D,0,0), (7b)

(b)

(a)

1

D

1

2

1

D'||nsD'⊥

||sD'

1 2

2

1

D

21

x

y

b

a

2

FIG. 2. (Color online) Components of symmetry allowed(a) intradimer and (b) interdimer Dzyaloshinskii-Moriya (DM)vectors shown in a unit cell. Arrows running from i to j onbonds indicate the ordering of the spin operators Si × Sj in the DMinteraction term.

with the site ordering convention as shown in Fig. 2. We referto this case as the the low-symmetry case.

A discussion of the different estimates for the strengthof the DM interactions and the g-tensor anisotropies forSrCu2(BO3)2 is presented in Sec. VIII.

III. THE VARIATIONAL APPROACH AND THEBOND-WAVE THEORY

A. Variational wave function

The ground state of the pure Shastry–Sutherland model,Eq. (1), can be written as a product of singlets |s〉 over the dimerbonds, � = ∏

dimers |s〉. In the presence of the DM interactionsand finite magnetic fields, we need to extend this wave functionto a variational one. Namely, we allow for a linear combinationof the singlet and triplet states on each dimer (we keep thedimer wave function entangled), while we retain the productform over the dimer bonds:

|�〉 =∏

A dimers

|ψA〉∏

B dimers

|ψB〉, (8)

where

|ψA〉 = us |s〉 +∑

α

uα|tα〉, (9a)

|ψB〉 = vs |s〉 +∑

α

vα|tα〉, (9b)

with |tα〉 being the three components of the triplets. Thiswave function can describe the phases that do not breakthe translational symmetry. Since we have two (i.e., A andB) dimers in the unit cell, the entire wave function |�〉 istranslationally invariant even when the wave functions of the

024413-3


two dimers are different. Certainly, this wave function cannotdescribe the plateaus except for the one at 1/2; for other valuesone would need to take a larger unit cell. The variationalparameters u and v are then determined by minimizing theenergy,

E = 〈�|H|�〉〈�|�〉 . (10)

The minimization is performed numerically, except for somesimple cases for which we could find analytical solutions.

B. Auxiliary boson formalism for the Hamiltonian

To find the excitation spectrum, we introduce, in the spiritof Sachdev and Bhatt,31 auxiliary bosons that create the singletand the triplet on each bond; the operator s† creates thesinglet state (|↑↓〉 − |↓↑〉)/√2, while the operators t

†x , t

†y ,

and t†z create the triplet states i(|↑↑〉 − |↓↓〉)/√2, (|↑↑〉 +

|↓↓〉)/√2, and −i(|↑↓〉 + |↓↑〉)/√2, respectively. This defi-nition is different from the one used in Ref. 31 by an additionalphase factor, −i, which ensures that the new bosons aretime-reversal invariant.32 Furthermore, for the four states oneach dimer to be faithfully represented, the number of bosonsper dimer is constrained:

s†s +∑

α=x,y,z

t†αtα = 1. (11)

The components of the spin operator at bond j are then givenas

Sαj,1 = i

2(t†α,j sj − s

†j tα,j ) − i

2εα,β,γ t

†β,j tγ,j , (12a)

Sαj,2 = − i

2(t†α,j sj − s

†j tα,j ) − i

2εα,β,γ t

†β,j tγ,j . (12b)

The intradimer part of the Heisenberg Hamiltonian H[Eq. (1)] in the bond representation reads

HJ = −3J

4

∑j

s†j sj + J

4

∑j

∑α=x,y,z

t†α,j tα,j , (13)

while the intradimer DM interaction reads

HD = D

2

∑j∈A

(t†y,j sj + s†j ty,j )

−D

2

∑j∈B

(t†x,j sj + s†j tx,j ) (14)

We can derive similar expressions for all the terms in theHamiltonian.

In the presence of a magnetic field along the z axis, it is moreconvenient to use the triplet bosons t

†1 , t

†0 = it

†z , and t

†1 , which

create |↑↑〉, (|↑↓〉 + |↓↑〉)/√2, and |↓↓〉, the eigenstates ofthe z component of the spin operator. The spin operators thenread

S+j,l = t

†1,j t0,j + t

†0,j t1,j√

2± s

†j t1,j − t

†1,j sj√

2, (15a)

S−j,l =

t†1,j

t0,j + t†0,j t1,j√

2∓

s†j t1,j − t

†1,j

sj√2

, (15b)

Szj,l =

t†1,j t1,j − t

†1,j

t1,j

2± s

†j t0,j + t

†0,j sj

2, (15c)

where the upper sign is for l = 1 spin and the lower sign forl = 2 spin in the dimer, as denoted in Fig. 1.

C. Bond-wave method

After rewriting the Hamiltonian in terms of the bondoperators using Eq. (12) or (15), we perform a bond-waveapproximation that is a natural extension of the usual spinwave theory. In the bond-wave approximation we extend thenumber of bosons per dimer from 1 to M , so that constraint(11) now reads

s†s +∑

α=x,y,z

t†αtα = M. (16)

The variational approach mentioned in Sec. III A is anal-ogous to finding the classical (S → ∞) ground state forspin models; M → ∞ is the classical solution where thequantum fluctuations between the dimers are neglected. Tosee this, it is convenient to rotate the “quantization axis.”Then the variational solution |ψA〉 in Eq. (9) can be writtenin terms of the “rotated” bosons as |ψA〉 = s

†A|0〉, where

s†A = uss

† + ∑α uαt†α . Similarly, we have |ψB〉 = s

†B|0〉, with

s†B defined using Eq. (9b). In the case of general M , we promote

these expressions to |ψA〉 = (s†A)M |0〉 and |ψB〉 = (s†B)M |0〉,which are direct analogues of the Bloch coherent states for thespin-M/2 system. In the classical-limit M → ∞, the coherentstate |ψA,B〉 may be thought of as the condensate of sA,B. Wealso rotate the remaining bosons into tα so that they obey theusual commutation relations and the local constraint Eq. (16).Accordingly, the expressions of the spin operators, Eq. (15),get modified.

To consider the small “transverse” fluctuations around theclassical solution, we solve the constraint explicitly for s andtreat ts as the Holstein-Primakoff bosons. Using the formalexpansions (valid to order shown)

s† = s =√

M −∑

α

t†αtα

≈√

M − 1√M

∑α

t†αtα + · · · , (17)

we perform a 1/M expansion in the spin operators andsubsequently in the Hamiltonian. Then the Hamilton operatorcan be written as

H = M2H(0) + M3/2H(1) + MH(2) + · · · , (18)

where H(0) = E0 is just the variational energy and H(1) is acollection of the terms that are linear in

tk = (tx,A,k,ty,A,k,tz,A,k,tx,B,k,ty,B,k,tz,B,k) (19)

and in the similarly defined t†k. As expected from the variationalnature, it turns out that H(1) is identically equal to 0 for thevariational solution. The quadratic part H(2) is of the form

H(2) = 1

2

∑k∈BZ

(t †k

t−k

)T (M N

N∗ M

)(tk

t †−k

). (20)

024413-4


The H(2) can be diagonalized by the method described inAppendix A, and we find three excitations (one for each tαboson) per dimer.

IV. BOND-WAVE SPECTRUM IN A ZERO FIELD

We start with a discussion of the zero-field excitationspectra in the low-symmetry (finite D) case. Early neutronscattering results33 indicated that the spectrum consists ofan essentially dispersionless (localized) single-triplet branchand other multitriplet ones. This is the consequence of theorthogonal dimer structure, and triplets get dispersion inthe sixth order of the perturbation expansion8,25 in J ′/J .Later, higher-resolution neutron scattering experiments21 re-vealed that the first triplet excitation actually splits intothree sub-bands with well-defined dispersions. The splittingindicates the presence of anisotropies. In the following,we calculate these spectra starting from the bond-wavetheory.

The variational wave function that minimizes the energy,Eq. (10), in a zero magnetic field takes the following form:

|ψA〉 ∝ |s〉 + w√2

(|t1〉 + |t1〉) = |s〉 + w|ty〉, (21a)

|ψB〉 ∝ |s〉 + w√2

(i|t1〉 − i|t1〉) = |s〉 − w|tx〉, (21b)

with

w = − D

J + √J 2 + D2

= − D

2J+ O(D3/J 3). (22)

The corresponding energy is given by

EZ1[D2d ] = −J

2−

√D2 + J 2. (23)

This wave function is time reversal invariant and it does notbreak any of the symmetries of the D2d , the plane group of theHamiltonian. We denote this phase Z1[D2d ].

To get the excitation spectrum following the recipe outlinedin Sec. III C, we rotate the states on each bond of type A as⎛

⎜⎜⎜⎜⎝s†A

t†x,A

t†y,A

t†z,A

⎞⎟⎟⎟⎟⎠ =

⎛⎜⎜⎜⎝

1√1+w2 0 w√

1+w2 0

0 1 0 0

− w√1+w2 0 1√

1+w2 0

0 0 0 1

⎞⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

s†A

t†x,A

t†y,A

t†z,A

⎞⎟⎟⎟⎟⎠ , (24)

with an analogous rotation on bonds B, so that the variationalwave functions in Eqs. (21) are given as |ψA〉 = s

†A|0〉 and

|ψB〉 = s†B|0〉. Next, we condense the s

†A and s

†B singlets. The

expression of the bond-wave Hamiltonian is complicated foran arbitrary point in the Brillouin zone, except at the � point,where it assumes the following form:

H(2) = �

2(t†z,A tz,A + t

†z,B tz,B + tz,A t

†z,A + tz,B t

†z,B) + 1

2

⎛⎜⎜⎜⎜⎝

t†x,B

t†y,A

ty,A

tx,B

⎞⎟⎟⎟⎟⎠

T ⎛⎜⎜⎜⎜⎝

√J 2 + D2 −2D′

⊥ −2D′⊥ 0

−2D′⊥

√J 2 + D2 0 −2D′

⊥−2D′

⊥ 0√

J 2 + D2 −2D′⊥

0 −2D′⊥ −2D′

⊥√

J 2 + D2

⎞⎟⎟⎟⎟⎠

×

⎛⎜⎜⎜⎜⎝

tx,B

ty,A

t†y,A

t†x,B

⎞⎟⎟⎟⎟⎠ + 1

2

⎛⎜⎜⎜⎜⎝

t†y,B

t†x,A

tx,A

ty,B

⎞⎟⎟⎟⎟⎠

T ⎛⎜⎜⎜⎝

� 0

� 0

0 �

0 �

⎞⎟⎟⎟⎠

⎛⎜⎜⎜⎜⎝

ty,B

tx,A

t†x,A

t†y,B

⎞⎟⎟⎟⎟⎠ , (25)

with

= 2(D′⊥ + wD′

|| + J ′w2)

1 + w2, (26)

� = J + √J 2 + D2

2. (27)

For simplicity, we have introduced the quantity

D′|| = D′

||,ns − D′||,s (28)

for the in-plane components of the interdimer DM interaction,as only this combination enters the variational ground-state

energy of the translationally invariant dimer-product wavefunction, Eq. (8), and excitations. Actually, the matrices inthe Hamiltonian, Eq. (25), can be reduced to 2 × 2 ones byusing the symmetries of the S4 point group [see Eq. (B3)].

Following Appendix A, Sec. I, we diagonalize the Hamil-tonian, Eq. (25), and get the excitation energies:

ω1,2 = �, (29a)

ω±3 =

√�(� ± 2 ), (29b)

ω±4 =

√J 2 + D2 ± 4D′

⊥√

J 2 + D2. (29c)

We note that ω±3 ≈ J ± 2D′

⊥ and ω±4 ≈ J ± 2D′

⊥ for smallvalues of D′

⊥/J , thus the pairs of excitations ωa3 and ωa

4(a = ±) are essentially indistinguishable. Furthermore, thesplitting between the two branches ω−

n and ω+n (n = 3,4) at

024413-5


the � point is 4D′⊥ + O(D′

⊥2/J ), which is used to estimate

the value of D′⊥ in Sec. VIII A.

A similar calculation at the q = (π,π ) point gives

ω1 = J√J 2 + D2

, (30)

ω2 = J

2+ J 2

2√

J 2 + D2, (31)

and splittings that are quadratic in DM interactions:

ω2 − ω1 = J

2− J 2

2√

J 2 + D2≈ D2

4J 2. (32)

Let us note that for larger values of |D′⊥| the dispersion

becomes comparable to the gap, and new phases appear. Thebranches ω±

3 become gapless when√

J 2 + D2 = ∓4(D′⊥ +

wD′|| + J ′w2), while ω±

4 = 0 for 4D′⊥ = ∓√

J 2 + D2.Assuming that D′

‖ is absent and keeping only the leading termin D/J , we get that the phase Z1[D2d ] is stable for

−J

4− D2

8J< D′

⊥ <J

4− D2

8J 2(2J ′ − J ). (33)

in the zero field. Beyond these boundaries, Z2 twofolddegenerate phases with the symmetry group C2v (when ω−

3 →0 for D′

⊥ > 0) or S4 (when ω+4 → 0 for D′

⊥ < 0) are realized(see also Fig. 6). We discuss these Z2 phases in more detail ina separate publication.

After a first-order expansion with respect to D/J , D′⊥/J ,

D′||,s/J , and D′

||,ns/J , it is possible to solve the eigenvalueproblem analytically. In this case we get three twofolddegenerate branches: a dispersionless one with eigenvalue J

and two branches with

ωq =√

J 2 ± J�q ≈ J ± 12�q, (34)

where

�q =[(

J ′DJ

− 2D′‖,s

)2

(1 − cos qa cos qb)

+ 16D′2⊥ cos2 qa

2cos2 qb

2

]1/2

. (35)

The dispersion of the quasi-triplet excitations is shown inFig. 3.

The dispersion in a zero field was also considered by Chenget al. in Ref. 34, where they used a different approach; bysuitable rotation of the spin operators, they removed the D

and arrived at an effective Hamiltonian, where they carriedout a first-order perturbation expansion to get the dispersionsof the effective triplets. Even though they considered a unitcell that has C4 symmetry, our dispersion agrees with theirresult, up to ambiguity in the sign in front of the D′

||,s inEq. (35). Furthermore, they extended their analysis by exactdiagonalization calculations of the spectra.

V. PHASE DIAGRAM IN A FIELD PARALLEL TOTHE z AXIS

In this section, we consider the variational ground-statephase diagram in the presence of an external field along the

0.8

0.9

1

1.1

1.2

(0,0) (π,π) (π,0) (0,0)

ω/J

(qa,qb)

FIG. 3. (Color online) Dispersions of quasi-triplet excitations ina zero magnetic field: ω−

3,4 (bottom), ω1,2 (middle), and ω+3,4 (top). We

chose D = D′⊥ = 0.1J and J ′ = 0.6J .

z axis. The full Hamiltonian is now invariant under the mag-netic group S4 + �σxz×S4, which is isomorphic to D2d . Forclarity of the argument, we investigate the high-symmetry case(where D = 0 and gs = 0) and the low-symmetry case [whichis realized in the low-temperature phase of SrCu2(BO3)2]separately.

A. High-symmetry case

When the space group is I4/mcm (which is relevant in thehigh-temperature phase T > Ts) and the symmetry group ofthe two–dimer unit cell is D4h, the intradimer DM interactionD is absent and only the interdimer D′

⊥ DM interactionis finite. With this type of anisotropy the z component ofthe spin is a conserved quantity, and this greatly simplifiesthe form of the variational ground states and of the bond-waveHamiltonian.

Numerically minimizing the variational energy〈�|H |�〉/〈�|�〉 in the presence of a magnetic fieldalong the z direction, we have found three gapped phases(see Fig. 4): the DS, the one-half magnetization plateau,and the fully polarized phase. Furthermore, there are fourgapless phases associated with the symmetry breaking ofthe continuous O(2) symmetry: the Neel, the O(2)[C4], theO(2)[S4], and the O(2) × Z2 phase. In these O(2) phases, theO(2) rotational symmetry in the xy plane perpendicular tothe field hz is spontaneously broken and they are theconsequence of the Sz being a good quantum number. Herewe consider the different phases and their excitations in moredetail.

1. Dimer-singlet phase

As mentioned earlier, the exact ground state of the SU(2)symmetric Shastry–Sutherland model is the product of singletson dimers: |ψA〉 = |ψB〉 = |s〉 for 0 � J ′ � 0.68J , as shownin Ref. 9. In the variational approach this ground states turnsout to be stable for finite values of D′

⊥ and magnetic fieldsh < hc, where the critical field is given by

hc =√

J 2 − 4|D′⊥|J . (36)

024413-6


DS

0

O(2)xZ 2

O(2)xZ

O(2)[C ]

O(2)[S

m=1/2 plat.

0.3 0.2 0.1

24

4

]

0.3

0.2

0.1

0

0 0.2 0.4 0.6 1 1.2 1.4 1.6 0.8

mz

zh /J

0.3

D' /

J⊥

0.2

0.1

0

−0.1

−0.2

−0.3

0.5

0.4

(a)

(b)

FIG. 4. (Color online) (a) Phase diagram in the hz-D′⊥ plane for

D = 0 and J ′/J = 0.6. DS is the dimer-singlet phase that remainsa variational ground state even for finite values of D′

⊥. The spinconfigurations in the different phases are shown in Fig. 5. m = 1/2plat. denotes the half-magnetization plateau phase, with a singlet anda magnetized triplet in each unit cell. (b) Magnetization curves fordifferent values of |D′

⊥| as a function of the magnetic field.

The ground-state energy comes purely from the exchangewithin a dimer,

EDS = −3J

2; (37)

all the other bond energies are identically 0.

2. O(2) phases

Between the DS and the one-half magnetization plateauwe find O(2) symmetric phases (see Fig. 4) when we applythe field perpendicular to the plane. In these phases themagnetization increases continuously between 0 and 1/2 perdimer (or mz = 1 per unit cell). Since Sz is a conservedquantity, the Hamiltonian does not break the O(2) symmetry ofthe rotations around the z axis. This symmetry is spontaneouslybroken in the O(2) phase. From numerical minimization wefound that the wave function can be written as

|ψA〉 ∝ |s〉 + ueiϕ |t1〉 + de−iϕ |t1〉, (38a)

|ψB〉 ∝ |s〉 ± iueiϕ |t1〉 ∓ ide−iϕ |t1〉, (38b)

with the upper sign for D′⊥ > 0 and the lower sign for D′

⊥ < 0.This wave function is continuously connected to the DS phase,as setting u = 0 and d = 0, we get back to the product ofsinglets. In this phase the Sz expectation values for all spinsare equal (〈Sz〉 ∝ |u|2 − |d|2), and the spin components in thexy plane along the interdimer bonds are perpendicular to each

O(2)[S

O(2) Z×

1

]4

4

O(2)[C 2

] 2dZ [D ]

FIG. 5. (Color online) Schematic representation of the O(2)symmetric phases in the high-symmetry (D = 0) case with a magneticfield perpendicular to the CuBO3 layer (i.e., h ‖ z). We have plottedthe expectation values of the spin component in the xy plane. Sincein this case Sz is conserved, the spins can be arbitrarily rotated by aglobal O(2) rotation in the plane. Blue (open arrows) and red (filledarrows) represent inequivalent spins.

other in such a way that, going around on a void square, thespins also make a full turn (as shown in Fig. 5); in other words,the spin configurations are invariant with respect to either theC4 or the S4 rotation. It is the sense of the rotation that makesthe difference between the two different cases of the sign ofD′

⊥. To make a clear distinction, we use the symmetry groupsthat leave the variational ground state invariant to label thesetwo phases O(2)[C4] and O(2)[S4]. We found that they arerealized for positive and negative values of D′

⊥, respectively.The expectation value of the Hamiltonian with the wave

functions, Eqs. (38a) and (38b), reads

EO(2) = u2 + d2 − 3

u2 + d2 + 1

J

2+ 2(u2 − d2)2

(u2 + d2 + 1)2J ′

− 4(u − d)2

(u2 + d2 + 1)2|D′

⊥| − 2(u2 − d2)

u2 + d2 + 1hz, (39)

and the minimization gives a set of polynomial equations thatneeds to be solved numerically. Close to the phase boundaryto the DS phase given by Eq. (36), we can expand in δh =hz − hc. In the lowest order in δh,

u = − (J + hc)√

2hc

2√

4JJ ′h2c + J 4 − h4

c

√δh, (40a)

d = (J − hc)√

2hc

2√

4JJ ′h2c + J 4 − h4

c

√δh. (40b)

The magnetization below hc is 0, and above hc it grows as

mz = (J − 4|D′⊥|)δh

2J |D′⊥| − 4D′

⊥2 + J ′(J − 4|D′

⊥|) + O(δh2). (41)

In the absence of a magnetic field the moduli of theamplitudes u and d of the two triplet components become

024413-7


equal, and writing v/√

2 = u = −d, the wave function sim-plies to

|ψA〉 ∝ |s〉 + v√2

(eiϕ|t1〉 − e−iϕ |t1〉), (42a)

|ψB〉 ∝ |s〉 ± v√2i(eiϕ|t1〉 + e−iϕ |t1〉), (42b)

where, as we noted, the sign is determined by that of D′⊥ =

±|D′⊥|. The minimum of the energy, Eq. (39), in this case is

achieved for

v =√

4|D′⊥| − J

4|D′⊥| + J

, (43)

with

E = −J

2− 2|D′

⊥| − J 2

8|D′⊥| . (44)

From the preceding analysis it turns out that the O(2) phasesare realized for |D′

⊥| > J/4 in a zero field [this is consistentwith Eq. (36)].

3. The O(2) × Z2 phase

The O(2) phase(s) and the one-half magnetization plateauphase are connected via two continuous phase transitions. Theintermediate phase exhibits both the Z2 symmetry breakingof the plateau phase (the inequivalence of the z component ofthe magnetization on the A and the B dimers) and the O(2)symmetry breaking of the O(2) phase, as shown in Fig. 5.As we approach the boundary of the one-half magnetizationplateau, the component of the spins perpendicular to the fielddecreases, eventually vanishing at the phase boundary. Thoughwe do not break the translational symmetry, the fact that themagnetization along the field is not equal on the A and the Bdimers (a discrete symmetry is broken), and that at the sametime we break a continuous symmetry of the O(2) type, wemay call this phase a supersolid.35,36

B. Low-symmetry case

At low temperatures [specifically, T < Ts = 395 K inSrCu2(BO3)2], the symmetry of the two-dimer unit cell isD2d . As noted earlier, the lowering of the symmetry allowsfor a finite value of the intradimer DM interactions D. Fromnumerical minimization, we mapped out the phase diagram,and we found a ground state that does not break any ofthe symmetries of the Hamiltonian in the low-field regionof the experimentally relevant parameters (we call the phaseZ1[D2d ]) (see Fig. 6). Additionally, we found many twofolddegenerate Z2 phases that we will describe in more detail in aseparate paper.

The total Sz is not a good quantum number anymore andthe continuous symmetry of the O(2) phases gets reduced todiscrete symmetries, and in the absence of continuous sym-metry, all phases become gapped. This symmetry reduction isseen in the expectation values of the energy; the inclusion ofthe intradimer DM D and the staggered g tensor gs introducesanisotropy terms to Eq. (39):

EZ1[D2d ] = EO(2) +√

2Du + d

u2 + d2 + 1cos ϕ

− 2√

2gshz

u − d

u2 + d2 + 1cos ϕ, (45)

]

0.2

1[D2d]

[S4

Z

]Z2

0.1

0

Z

−0.2

2

−0.1

[C2v

0.4 0.6 1 1.2 1.4 1.6

0.3

0.8

hz/J

mz

D⊥' /

J

0.2

0.1

0

−0.1

−0.2

−0.3

0.5

0.4

0.3

0.2

0.1

0

0 0.2

(b)

(a)

FIG. 6. (Color online) (a) Phase diagram in the hz-D′⊥ plane for

J ′/J = 0.6 and D/J = 0.1 (low-symmetry case). In comparison tothe D = 0 case in Fig. 4, in a large region of the phase space the dimersinglet and the O(2)[S4] essentially merged to create the Z1[D2d ]phase, a small part of the O(2)[S4] phase became a twofold degenerateZ2[S4], and the O(2)[C4] merged with the m = 1/2 magnetizationplateau phase into the Z2[C2v] phase. (b) Magnetization curves for afew selected values of D′

⊥.

which determine the preferred direction of the xy components.Assuming D > 0 and positive u and v, the DM energy on thedimers is minimal when ϕ = π . The variational wave functionin the Z1[D2d ] is given by

|ψA〉 ∝ |s〉 − u|t1〉 − d|t1〉, (46a)

|ψB〉 ∝ |s〉 + iu|t1〉 − id|t1〉, (46b)

and the expectation values of spin components are shown inFig. 5. This phase is adiabatically connected to the DS productphase (when u → 0 and d → 0), and in general the intradimerDM interaction D mixes the triplet components to the singlet,as expected from Eq. (14). It is also a special case of the wavefunction of the O(2)[S4] phase [Eqs. (38a) and (38b) with thelower sign] with the phase locked to ϕ = 0. In other words,when D′ < 0 (the experimentally relevant case), turning on aninfinitesimal value of D removes the phase boundary betweenthe DS phase and the O(2)[S4] phase. For D′ > 0, the O(2)[C4]phase becomes frustrated with respect to D and will give riseto a Z2 symmetry breaking.

The conditions ∂EZ1[D2d ]/∂u = 0 and ∂EZ1[D2d ]/∂d = 0lead to a set of polynomial equations of high degree that onecan solve only numerically. However, we can search for thesolution of u and v as an expansion in D/J . For small values

024413-8


of D, both u and d are linearly proportional to D when hz < hc

[hc is defined in Eq. (36)], and we can expand the energy as

EZ1[D2d ] = −3J

2+ 2J (d2 + u2) + 4D′

⊥(u − d)2

− 2hz(u2 − d2) −

√2D(u + d)

+ 2√

2gshz(u − v), (47)

and in the lowest order in D and gs , the minimum is achievedwith

u = 1

2√

2

(D − 2gshz)(J + hz) + 4DD′⊥

J 2 + 4JD′z − h2

z

, (48a)

d = 1

2√

2

(D + 2gshz)(J − hz) + 4DD′⊥

J 2 + 4JD′z − h2

z

. (48b)

For hz = 0 we recover Eq. (22). To be more precise, the expan-sion is actually in D/(J 2 + 4JD′

z − h2z) = D/(h2

c − h2z), and

the denominator becomes 0 at the D = 0, gs = 0 boundarybetween the DS and the O(2)[S4] phase. The expression forthe magnetization of a dimer is then

mz = hz

J

h2c(

h2c − h2

z

)2 (D − 2gsJ )2 + O(D4), (49)

which grows quadratically with the anisotropy. We note thatJ ′ enters only in the next order in the expansion.

VI. BOND-WAVE SPECTRUM FOR THE FIELD PARALLELTO THE z AXIS

In this section, we calculate the bond-wave excitationspectrum in the presence of an external magnetic field h ‖ z

and examine qualitatively the effect of intra- and interdimerDM interactions.

A. High-symmetry case

First, we consider the case where D = 0. In the DS productstate we condense the singlet states on both the A and the Bbonds, and there is no need to rotate the bosons (t = t). Theenergy is then (up to a constant shift)

H = EDS +∑q∈BZ

H2(q), (50)

where

H2(q) = J [t†0,A(q)t0,A(q) + t†0,B(q)t0,B(q)] +

⎛⎜⎜⎜⎝

t†1,B (q)t†1,A(q)t1,A

(−q)

t1,B(−q)

⎞⎟⎟⎟⎠

T

×

⎛⎜⎝

J − hz −2iD′⊥γ1 −2iD′

⊥γ1 0i2D′

⊥γ1 J − hz 0 i2D′⊥γ1

i2D′⊥γ1 0 J + hz i2D′

⊥γ1

0 −2iD′⊥γ1 −2iD′

⊥γ1 J + hz

⎞⎟⎠

×

⎛⎜⎜⎜⎜⎝

t1,B(q)

t1,A(q)

t†1,A

(−q)

t†1,B

(−q)

⎞⎟⎟⎟⎟⎠ (51)

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω/J

hz/J

To-1

Te-1

Te0

To0

To1

Te1D = 0

D′⊥ = −0.1 J

(b)

0

0.5

mz

(a)

FIG. 7. (Color online) Bond-wave spectrum at the � point whenwe keep the singlets and the Sz = 1 triplets only. The magnetic fieldis parallel to the z axis, J ′ = 0.6J , and D′

⊥ = 0.1J . The transitioninto the O(2) phase happens at hz ≈ 0.77J , and that into the plateauphase, at hz ≈ 1.1J . For 0.77 � hz/J � 1.03, one of the excitationsbecomes identically 0; this is the Goldstone mode of the O(2)[S4]phase. The Goldstone mode of the O(2) × Z2 phase is 0 for 1.03 �hz/J � 1.1. The two-dimer variational solution is unstable in the grayregion; the dispersion goes to 0 at some wave vector away from the �

point at hz = 0.96 and 1.08. The filled area above the dispersion lineshows the strength of the spin structure factor Sxx + Syy . The dashedline is the approximation from Ref. 37.

and

γ1 = cosqa

2cos

qb

2. (52)

The Hamiltonian matrix is of the form of Eq. (A6) and can bediagonalized following the procedure outlined in Appendix A,Sec. II. The operators in the momentum space are defined bythe t†(k) = N

−1/2�

∑j eik·rj t

†j , where rj is the position of the

j th spin.Actually, we can introduce the combinations

t†1,±(q) = 1√

2[t†1,A(q) ∓ it

†1,B(q)], (53a)

t†1,±(q) = 1√

2[t†1,A

(q) ± it†1,B

(q)], (53b)

together with the corresponding annihilation operators, so thatthe original 4 × 4 matrix in Eq. (51) decomposes into two2 × 2 problems, with the Hamiltonians

H(2)± (q) = (J − hz ± D′

⊥)t†1,±(q)t1,±(q)

± 2D′⊥γ1[t†1,±(q)t†1,±(q) + t1,±(q)t1,±(q)]

+ (J + hz ± D′⊥)t†1,±(q)t1,±(q). (54)

The bond-wave spectrum consists of six modes: twofolddegenerate nondispersing excitations with ω(q) = J (denotedT

e,o0 in Fig. 7) and four dispersing modes,

ω+,± =√

J 2 + 4JD′⊥ cos

qa

2cos

qb

2± hz, (55)

024413-9


that come from H(2)+ (q) [solid (blue) lines denoted T e

±1 inFig. 7] and

ω−,± =√

J 2 − 4JD′⊥ cos

qa

2cos

qb

2± hz (56)

from H(2)− (q) that we call the T o

±1 modes [shown by solid (red)lines in Fig. 7]. The dispersions have a finite gap in the DSproduct state. Let us also mention that for hz = 0 and D = 0,we recover Eq. (35).

From the foregoing equations, the gap closes at q = 0when the magnetic field reaches hc defined by Eq. (36), andwe enter into the O(2) phases. Unfortunately, the explicitform of the variational wave function and the bond-waveHamiltonian in the O(2) phases is too complicated, thus herewe discuss the numerical solution only. We just mention thatthe combinations t

†1,±(q) and t

†1,±(q) introduced in Eqs. (53)

are decoupling the bond-wave Hamiltonian in the O(2)case as well. When D′

⊥ > 0, the closing of the gap leadsto a Goldstone mode—the consequence of the continuoussymmetry breaking—that appears as a continuation of theω−,− mode, and the condensation of a linear combination ofthe t

†1,−(q = 0) and t

†1,−(q = 0) bosons results in the O(2)[C4]

phase described by the wave functions with the upper sign,Eq. (38) (see also Fig. 4). For D′

⊥ < 0, in contrast, theGoldstone mode evolves from the ω+,− mode, leading tothe O(2)[S4] phase (Fig. 4). As shown in Fig. 7, the lowestgapped mode for q = 0 in the DS phase remains gapless,while the O(2) symmetry is broken, and this is the case untilthe half-magnetization plateau.

Also, from the q-dependent excitation spectrum, we learnthat the spectrum may become gapless not only at q = 0,but also at some other q values in the Brillouin zone, thusindicating helical instability of the O(2) phases. In Fig. 7 weshow the boundary of this instability (the hatched region) thatwe obtained from the numerical calculations of the spectra.

The strength of the magnetic probe response is determinedby the structure factor Sαα(q,ω). In particular, Sxx(q = 0,ω)and Syy(q=0,ω) determines the strength of the ESR lines inthe first approximation, when the static magnetic field is alongthe z axes. The structure factor is given by

Sαα(q,ω) ∝∑∣∣〈f |Sα

q |0〉∣∣2δ(ω − Ef + E0), (57)

where |0〉 is the ground state (in our case, the |�〉 variationalwave function), f denotes the excited states, and E0 and Ef

are the energies of the respective states.As the first step, it is instructive to look at the ω-

integrated (static) structure factor, Sαα(q) = ∫dωSαα(q,ω),

which is actually the sum of the (positive) matrix elementsand is equal to 〈�|Sα

−qSαq |�〉. In the (pure) DS ground state,

limq→0 Sαq |0〉 → 0, so we expect to see no response in ESR

experiments, unless there are anisotropies that mix the tripletcomponents with the singlet.

In the O(2) phase (discussed in Sec. V A 2), the staticstructure factor is

Sαα(q = 0) = u2 + d2

1 + u2 + v2(58)

≈(J 2 + h2

c

)hc

4JJ ′h2c + J 4 − h4

c

δh (59)

for α = x, y, and z. Examining the individual matrix elements,it turns out that the matrix elements for Sxx and Syy areall vanishing except for the T e

0 line. In contrast, the matrixelements for Szz are nonzero for the T e

1 and T e−1 lines. Since

the ESR line width is proportional to Sxx and Syy when thefield is along the z direction, we expect a strong signal for theT e

0 line.

B. Low-symmetry case

First, let us discuss the case D′⊥ = 0 shown in Fig. 8. At low

fields, the spectrum looks like the usual one-triplet excitationZeeman split by the magnetic field: we see three branchesthat are twofold degenerate, as we have two dimers in a unitcell. Without any kind of anisotropies, these excitations wouldcorrespond to pure one-triplet excitations. However, in thelow-symmetry case the intradimer DM coupling D mixes thesinglets with these excitations. From the zero-field equations,(29), we get that the splitting is of the order of D2 for smallvalues of D/J , which is much smaller than the linear splittingcaused by D′

⊥. In contrast, the effect of D is much morepronounced at higher fields, where the gap becomes small andsinglet-triplet mixing is enhanced. Instead of the Goldstonemode, the anisotropy induces a “level repulsion,” and we geta finite gap that is roughly proportional to

√D/J , which

is consistent with the usual form of the anisotropy gap. Wenote that the “level repulsion” happens to only one of thetwo almost-degenerate branches that come down with theapplied field, and it depends crucially on the symmetry ofthis state as noted by Miyahara and Mila in Ref. 37, wherethey considered the dispersion of a single triplet bond movingin the singlet background by the standard perturbation theory.As we increase the field, the gap closes for the T o

1 level atthe phase boundary of the Z2 phase with C2v symmetry (seeFig. 6).

For finite inter- and intradimer DM interactions, we observeboth the zero-field splitting and the antilevel crossing around

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω/J

hz/J

To-1,Te

-1

Te0

To0

Te1

To1

D = 0.1 JD′⊥ = 0

(b)

0

0.5

mz

(a)

FIG. 8. (Color online) Excitation spectrum in a magnetic fieldparallel to the z axis when D′

⊥ = 0, J ′/J = 0.6. The instabilitiestoward helical states are at hz/J = 0.9421 and 1.1002, while thek = 0 instability into the Z2 phase is at 0.9515.

024413-10


0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω/J

hz/J

Te-1

To-1

To0

Te0

Te1

To1D = 0.1 J

D′⊥ = 0.1 J

(b)

0

0.5m

z(a)

FIG. 9. (Color online) Excitation spectrum in a magnetic fieldparallel to the z axis in the low-symmetry case, for D′

⊥ > 0, D/J =0.1, and J ′/J = 0.6. Notation is the same as in Fig. 7.

the critical field. In Sec. IV, we presented a detailed calculationfor the zero-field dispersion and estimated this splitting in thefirst order of DM interactions to be 4D′

⊥. This is in excellentagreement with the findings of Cheng et al.34 The two low-lying modes in Figs. 9 and 10 curve differently in the O(2)phase; only one of them crosses the ground state, while theother is gapped. This can be explained by the fact that onlyone (the T e

1 ) low-lying excitation is coupled to D, and the gap isproportional to it.37 In the case of the bond-wave calculation,the gap opens as

√D as the effect of quantum fluctuations

(see Appendix B for more details). Flipping the sign of theinterdimer DM coupling D′

⊥ changes the lowest-lying mode(compare Figs. 9 and 10). The singlet-triplet mixing is differentaccording to the symmetry of the lowest-lying mode, and theantilevel crossing occurs only for D′

⊥ < 0.

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ω/J

hz/J

To-1

Te-1

Te0

To0

To1

Te1D = 0.1 J

D′⊥ = −0.1 J

(b)

0

0.5

mz

(a)

FIG. 10. (Color online) Excitation spectrum in a magnetic fieldparallel to the z axis for D′

⊥ < 0. J ′ = 0.6J .

Equation (58) for the static structure factor is valid alsofor finite D values, with the variational parameters u and v

in the wave function, Eq. (46), now being obtained by theminimization of Eq. (45). In the limit of small D/J , we canuse the values that are given by Eqs. (48), and at q = 0 we get,in lowest order in D/J,

Sxx = Syy = D2

4

(J + 4D′⊥)2 + h2

z(J 2 + 4JD′

⊥ − h2z

)2 (60)

for small field values (the apparent singularity at the criticalfield hz = hc = √

J 2 − 4|D′⊥|J is an artifact of the expan-

sion). Similarly to the D = 0 case, the weight of the spincorrelation function Sxx(q = 0,ω) in the Z1[D2d ] phase isconcentrated on the single T e

0 line. As we enter the Z2[C2v]phase, the Sxx(q = 0,ω) is split between the T e

0 and theT o

0 lines. The strength of the individual lines is shown inFigs. 8–10 by the filled region above the T e

0 and T o0 lines.

VII. PHASE DIAGRAM AND EXCITATION SPECTRUMFOR AN IN-PLANE MAGNETIC FIELD

In this section we consider the case when the magneticfield is applied in the CuBO3 plane, parallel to bond Aand perpendicular to bond B. This direction is denoted x

in Fig. 1 (we note that this direction is different from thecrystallographic a axis). This choice makes the two dimersinequivalent and breaks the rotational symmetry S4. Thisdirection of field lowers the D2v symmetry of the unitcell in the low-temperature phase to the magnetic group{E,σyz} + �C2(z) × {E,σyz}, which is isomorphic to C2v . Inthe following, we show the phase diagrams for finite values ofD and give a short discussion of the phases that appear. Wealso present the ESR spectrum at the end of this section.

A. Phase diagram

The numerically obtained phase diagram as a function of hx

and D′⊥ for a selected value of D is shown in Fig. 11. We denote

the ground state by Z1[C2v], which has the full C2v symmetryof the Hamiltonian and the variational wave function is of theform

|ψA〉 ∝ |s〉 − vy |ty〉 − iuz|tz〉, (61a)

|ψB〉 ∝ |s〉 + vx |tx〉, (61b)

with the energy expectation value

EZ1[C2v ] =−J(1 − v2

x

) + 2Dvx

2(1 + v2

x

) − J + 2hxuzvy + Duz

1 + u2z + v2

y

. (62)

The intradimer DM interaction D prefers states with dipoleexpectation values that are perpendicular to the vector D.As a consequence, the magnetic field induces the momentonly on dimers where the direction of D is perpendic-ular to the field. In our case, h||x, and only dimer Adevelops a finite magnetization: the spin components alongthe magnetic field, Sx

A,1 = SxA,2, as well as the components

perpendicular to the plane, SzA,1 = −Sz

A,2, become finite,as shown in Fig. 12. Increasing the magnetic field, theexpectation value of Sx

A increases smoothly up to mx = 1/2[see Fig. 13(a)].

024413-11


/J

/J

h

⊥

x

D'

(y)s2 [C ]

(x)sZ2[C ]

Z

(y)sZ2[C ]

]2vZ1[C

0.2

0.2

0

−0.2

0

−0.4

1.2 1.4 1 0.8 0.6 0.4

0.4

FIG. 11. (Color online) Phase diagram for the magnetic field inthe x direction. J ′/J = 0.6 and D/J = 0.1. The spin configurationsin the different phases are shown in Fig. 12.

The wave function |ψB〉, in contrast, is time reversalinvariant, where the expectation value of any spin componentis 0: 〈S1,2〉 = 0. However, it breaks the rotational symmetry, asthe vector chirality is finite: 〈S1 × S2〉 = −vx/(1 + v2

x). Thisis the so called p-type nematic state.38,39 The parameter vx

does not depend on the magnetic filed, and minimizing theenergy, Eq. (62), we find that vx = D/2J .

In the phase diagram, we found two additional phases, andthey are both twofold degenerate Z2 phases. For sufficiently

Z s

2v][C1Z

2 ]Z(y)[C s(x)

2 [C]

FIG. 12. (Color online) Schematic representation of the spinconfigurations in the phases Z1[C2v], Z2[C(x)

s ], and Z2[C(y)s ] that appear

when the field is along the x direction. The darker and lighter arrowsrepresent the two degenerate states.

=−0.1 JD=0.1 J

0.5

0.25

0

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 1.2

hx

ωm

(b)

x

(a)

zD'

FIG. 13. (Color online) Excitation spectrum in a magnetic fieldparallel to the x axis. J ′ = 0.6J .

large negative values of D′⊥, the phase Z2[C(y)

s ] is realized withthe wave function

|ψA〉 ∝ |s〉 − (vy ± iuy)|ty〉 − (iuz ∓ vz)|tz〉, (63a)

|ψB〉 ∝ |s〉 + (vx ± iux)|tx〉. (63b)

The magnetization on the A dimer consist of a uniform partSx

A,1 = SxA,2 and a staggered part S

y

A,1 = −Sy

A2and Sz

A,1 =−Sz

A,2 (as shown in Fig. 12). While in phase Z1[C2v] there wereno dipole components on dimer B, here the expectation valueof staggered magnetization mx

B,st is non-zero, SxB,1 = −Sx

B,2.The magnetization pattern is invariant under σyz.

At large enough positive D′⊥, we reach the phase Z2[C(x)

s ]where the ground state has the following form

|�〉A = |s〉 ± iux |tx〉 − vy |ty〉 − iuz|tz〉, (64a)

|�〉B = |s〉 + vx |tx〉 ± iuy |ty〉 ± vz|tz〉. (64b)

In this case, the spin expectation values are invariant under thereflexion �σxz, as shown in Fig. 12.

We note that in the limit of D → 0 the Z1[C2v] phase iscontinuously connected to the DS that is realized for

hx <

√J 2 − 16D′

⊥2 (65)

when D = 0. It is also continuously connected to one ofthe ground states of the twofold-degenerate m = 1/2 plateauphase (where the singlets are located on the B bonds).

B. ESR spectra

In the following we discuss the effect of the DM com-ponents on the ESR spectrum for a magnetic field parallelto the x axis. We remind the reader that in the absence ofanisotropy (i.e., DM interactions), the DS is the ground statefor low fields, hx < J , and that the excitations are the pure,Zeeman-split triplets with energies J − hx , J , and J + hx ,each of which is twofold degenerate, corresponding to the twodimers in the unit cell.

In Fig. 13, we show the calculated ESR spectrum for D =0.1J and D′

⊥ = −0.1J . In the absence of the field we observe

024413-12


the zero-field splitting 4D′⊥ that we discussed in Sec. IV. Now,

unlike the case of the field along the z direction, the spectrumconsists of three pairs of almost-degenerate levels (note thatin the absence of D, each pair is exactly degenerate in the DSphase), and only at higher fields near the phase transition dothe lines split. When D′

⊥ is large enough, the gap closes at theboundary to the Z2[C(y)

s ] phase. For D′⊥ = 0.1J the spectrum

looks essentially the same.For D′

⊥ = 0 (i.e., only D is present), the zero-field splittingdisappears, and as we approach the critical field the twofolddegenerate triplet branch splits. The different behaviors of thetwo low-lying excitations around the critical field are due totheir different singlet-triplet mixing. As discussed previouslywhen D is finite and D′

⊥ is 0, we are in phase Z1[C2v] (see thephase diagram, Fig. 11). Increasing the magnetic field from0, the value of the parameter uz in the ground-state wavefunction |ψA〉 [Eq. (61)] increases continuously, developing afinite magnetization mx at dimer A, while the magnetizationof dimer B remains 0.

VIII. COMPARISON TO EXPERIMENTAL SPECTRA

The ESR spectra have been considered previously byperturbation18 (for D = 0) and by exact diagonalizations.40 Inour approach, it is straightforward to take all the anisotropiesthat are relevant in experiments into account, and here weconsider the ESR spectra in a more realistic setting to test ourtheoretical framework.

Various attempts have been made to determine the valuesof the different terms in the Hamiltonian. For completeness,we write down the Hamiltonian in its full form (n.n., nearestneighbor; n.n.n., next-nearest neighbor, u.c., unit cell):

H = J∑n.n.

SiSj + J ′ ∑n.n.n.

SiSj +∑n.n.

Dij (Si × Sj )

+∑n.n.n.

D′ij · (Si × Sj ) −

∑u.c.

gzhz

(Sz

A1 + SzA2

+ SzB1 + Sz

B2

) +∑u.c.

gshz

(Sx

A1 − SxA2 + S

y

B1 − Sy

B2

).

(66)

Using the estimations of Cepas et al.18 for the value of theinterdimer DM component (D′

⊥/J = −0.02), Kodama et al.determined the intradimer DM component by fitting exactdiagonalization data and obtained24 D/J = 0.034. Fitting theneutron scattering data at zero field,21 Cheng et al. predicted34

D′⊥ = 0.18 meV and D′

||,s + J ′D/2J = 0.07 meV. Usingab initio LSDA + U calculation, Mazurenko et al. estimated41

D = 0.35 meV for intradimer and D′⊥ = 0.1 meV, D′

||,s =0.06 meV, and D′

||,ns = 0.04 meV for interdimer components.The values of the g-tensor anisotropies were estimated from

ESR and NMR measurements: gx = gy = 2.05 and gz = 2.28in Ref. 19, and gs = 0.023 from the tilt angle of the electricfield gradient in Ref. 24.

Information on the excitation spectra are available fromESR,19,22 Raman,20,42 far-infrared (FIR),43,44 and neutronscattering18,21,33 measurements. We mainly compare ourspectra with the ESR, FIR, and neutron-scattering mea-surements. Our lines T o

±1 correspond to T0p(±) in the

FIR spectra in Ref. 43 and to O1 in the ESR spectra in Ref. 22,the lines T e

±1 correspond to T0m(±) in Ref. 43 and to O2 inRef. 22, and the lines T e

0 and T o0 are T0p,m(0) in Ref. 43.

A. Quantitative comparison to experiments at a zero field

In particular, high-resolution ESR measurements by Nojiriet al.22 show two triplet excitations, at 679 ± 2 and 764 ±2 GHz. The FIR measurements of Room et al.43 showthree triplet modes, at 22.72 ± 0.05 cm−1 (≈681 GHz),24.11 ± 0.05 cm−1 (≈723 GHz), and at 25.51 ± 0.05 cm−1

(≈765 GHz). The origin of the signal at 24.11 cm−1 is the�Sz = 0 triplet excitation, which is not seen in the zero-fieldESR spectra. From Eq. (35) we can deduce that the splittingbetween the �Sz = 1 and the �Sz = −1 triplet lines gives4D′

⊥ ≈ 85 GHz, that is, D′⊥ ≈ 21 GHz.

Furthermore, high-resolution inelastic neutron scatteringmeasurements in a zero field performed by Gaulin et al.21 haverevealed that the dispersion above the gap consists of threedistinct branches of triplet excitations. The splitting observedthere was fitted by Cheng et al.34 to yield a result that isidentical to our Eq. (35). From these dispersions, the splittingbetween the triplets at q = 0 is �(0,0) = 4D′

⊥ ≈ 0.4 meV (e.g.,≈95 GHz, close to the 85 GHz given previously), while atq = (π,0) it is �(π,0) = √

2(2D′||,s − DJ ′

J) = 0.2 meV.

We need to mention here that in our approach the dispersionof the triplets coming from the interdimer coupling J ′ ismissing altogether; this is why we have used, in estimatingD′

⊥, the “bare” value J of the single-triplet gap, insteadof using the renormalized value that is actually observed inESR measurements (the preceding value may be renormalizedif we go beyond the linear bond-wave approximation). Thenumerical diagonalization by Cheng et al.34 has shown thatthe effect of J ′ is only to modify the dispersions in such a waythat the splittings remain independent of J ′.

B. Quantitative comparison of the spectra ata finite magnetic field

In this subsection, we try to fit the ESR spectrum of Nojiriet al. in Ref. 22 by using the bond-wave method in additionto variational calculation. To obtain a quantitatively good fit,we need to include the DM interactions as well as the g-tensoranisotropies.

In the fitting, we use the values of the anisotropy constantsgz = 2.28 estimated in Ref. 19 and gs = 0.023 in Ref. 24. Avalue of the intradimer DM coupling D = 0.034J ≈ 60 GHzis obtained in Ref. 24, assuming that J = 85 K,8,11 anda similar value (D = 28.6 cm−1 = 54 GHz) in reported inRef. 43. The interdimer coupling constant is given by D′

⊥ =21 GHz, as already determined. For the reason describedpreviously, we choose J to be equal to 722 GHz, the valueof the experimentally observed gap.19 Furthermore, we findthat the spectrum is essentially independent of the value ofJ ′ inasmuch as we are in the Z1[D2d ] phase, so we havechosen J ′/J = 0.6 for internal consistency of the calculation.The calcualted bond-wave spectrum with the parametersmentioned previously is shown in Fig. 14.

We find a surprisingly good quantitative agreement with thehigh-field ESR of Nojiri et al.22 and the FIR measurements of

024413-13

JUDIT ROMHANYI, KEISUKE TOTSUKA, AND KARLO PENC PHYSICAL REVIEW B 83, 024413 (2011)ν

(GH

z)

0

200

400

600

800

1000

1200

0 5 10 15 20 25 30 35 40hz (T)

To-1

Te-1

Te0

To0

To1

Te1

(a)

(b)900

800

7000 10 20

Te0

FIG. 14. (Color online) (a) Qualitative comparison of the exci-tation spectrum with the h‖c ESR spectrum shown in Fig. 4(a) ofNojiri et al.22 (b) Diamonds are the far-infrared data from Ref. 43,and squares are the ESR data from Ref. 22.

Room et al.43 Our spectra reproduced not only the value ofthe high-field gap in the T e

1 excitation above 20T, but alsothe behavior of the T e

0 level, which follows nicely the main(i.e., highest-intensity) peak in the ESR spectrum [Fig. 14(b)],thus clearly identifying those lines as originating from tripletexcitations.

IX. SUMMARY AND CONCLUSIONS

We have studied the description of the magnetic propertiesof SrCu2(BO3)2 using the Shastry–Sutherland model extendedwith anisotropies. The possible form of the anisotropies, likethe DM interactions [Eqs. (4) and (5)] and the g-tensoranisotropy [Eq. (2)], follows from the structure and thesymmetry properties (space group) of the material.

We used a bond-factorized form of the variational wavefunction to study the effect of the anisotropies on theground-state properties in phases that are compatible with thecrystallographic unit cell comprising two orthogonal dimersin the presence of an external magnetic field. This includes theexperimentally relevant DS phase in low fields below the 1/8magnetization plateau and the 1/2 plateau.

We have found that the DS phase remains a good variationalground state in the so-called high-temperature phase, wherethe only anisotropy, which takes on a finite but small value,is the inter-dimer interaction D′

⊥ perpendicular to the CuBO3

planes, and the magnetic field is perpendicular to the plane.This phase is surrounded by gapless phases with an O(2)symmetry where, due to D′

⊥, the triplons can propagate witha well-defined dispersion.

In the less symmetrical, low-temperature structure ofSrCu2(BO3)2, the finite intradimer DM interaction D appears,and it gives rise to an admixture of triplet componentsand the singlet states in the variational wave function ofthe DS phase, without any symmetry breaking (we denotedthis phase Z1[D2v]). The finite D not only removes theO(2) degeneracy in the gapless phases, but also introducesfrustration, depending on the sign of D′

⊥; in the unfrustrated

case there is a crossover from the O(2) phase to Z1[D2d ]where there is a preferred direction in the O(2)-plane setby D, while in the frustrated case an Ising-like Z2 sym-metry breaking appears. This is a behavior similar to thatseen in ladders45 and square antiferromagnets46 with DMinteractions.

We have also studied the effect of anisotropies on excitationspectra. For that purpose, we used the “bond-wave” theory,which is based on the bosons representing the entangledstates of the dimers. In a zero field we recovered momentum-dependent splitting of the triplet states, in accordance withneutron-scattering experiments. Furthermore, we also recov-ered the experimentally measured ESR spectra for a physicallyreasonable set of parameters. In this respect, we note thefollowing in comparison to the usual perturbational approach,where a single triplet excitation propagates:37 (i) to describethe spectra in a field that is parallel to the CuBO3 plane, theinclusion of all triplet states is needed; and (ii) the finite-fieldgap in the perturbational approach is proportional to D,while in the bond-wave approach it is proportional to

√D,

thus a smaller value of D can already lead to observableeffects.

Regarding the ESR line intensities, we have found that inthe high-symmetry case, the weight in the spin-structure factorappears in the O(2) phases, is concentrated on the line that issplit from the Sz = 0 triplet excitation, and, loosely speaking,follows the “paramagnetic” ESR line. In the low-symmetrycase, the anisotropies make the DS-O(2) quantum critical pointa crossover, and the weight accordingly appears at the energiesof the order of the zero-field singlet-triplet gap. This largeweight is clearly observed in ESR spectra in the 15-25 T rangeat 700 GHz and above.

Finally, let us mention the main disadvantage of the bond-wave method. Namely, keeping only the quadratic terms inthe boson operators, the dispersion due to J ′ is not takeninto account, and similarly, the J ′ term does not decrease thesinglet-triplet gap from its bare value � = J (for that we needto go to higher orders in the 1/M expansion and to introduceterms with four bosons). In other words, the situation is, inthis respect, similar to the perturbational approaches that startfrom the decoupled dimers.

ACKNOWLEDGMENTS

We are pleased to thank D. Huvonen, G. Kriza, F. Mila,S. Miyahara, U. Nagel, T. Room, and M. Zhitomirsky forstimulating discussions and M. Hagiwara and S. Kimura forsharing their unpublished ESR results and discussions. Wealso thank the authors of Ref. 43 for sending us their experi-mental data. This work was supported under Grant-in-Aid forScientific Research (C) No. 20540375 from MEXT, Japan, bythe global COE program “The Next Generation of Physics,Spun from Universality and Emergence” of Kyoto University,and by Hungarian OTKA Grant Nos. K73455 and NN76727.The authors are also thankful for the hospitality of the YukawaInstitute in Kyoto, where this work was initiated; the NationalInstitute of Chemical Physics and Biophysics in Tallinn, wherewe learned about the far-infrared measurements; and the MPIPhysik Komplexer Systeme in Dresden, where this work wasfinalized.

024413-14


APPENDIX A: BOND-WAVE THEORY

1. General case

While it is a well-known procedure, for completenesswe present how to diagonalize a quadratic form of bosonicoperators of the form

K = 1

2

(a

a†

)T (B A

A† BT

)(a†

a

), (A1)

where B = B† is a hermitian d × d matrix and A = AT isa symmetric d × d matrix, so that the whole HamiltonianK is hermitian (however, it is not normal ordered). a†

denotes a vector of d bosons (a†1,a

†2, . . . ,a

†d ); similarly, a =

(a1,a2, . . . ,ad ). We also assume that the 2d × 2d matrix inEq. (A1) is positive definite; this ensures that all eigenvaluesassociated with creation operators are positive.

Applying the ∂O/∂t = i[K,O] equation of motion of anoperator O to the bosonic operators a† and a, we get

∂

∂t

(a†

a

)= i

(B A

−A† −BT

) (a†

a

). (A2)

Our aim is to find a suitable linear combination of a and a†

operators that are energy eigenstates of the K. This is achievedby solving the(

B A

−A† −BT

)T (uj

vj

)= ωj

(uj

vJ

)(A3)

eigenvalue equation. Equation (A3) has a particular prop-erty: for each (uj ,vj ) eigenvector with eigenvalue ωj > 0,the (v∗

j ,u∗j ) is also an eigenvector with eigenvalue −ωj .

We associate eigenvectors with positive eigenvalues with thecreation operator and eigenvectors with negative eigenvalueswith the corresponding annihilation operator:

α†j = uj · a† + vj · a, (A4a)

αj = v∗j · a† + u∗

j · a, (A4b)

so that the [αj ,α†j ′ ] = δj,j ′ commutation relation is fulfilled.

With this choice, [K,α†j ] = ωjα

†j and [K,αj ] = −ωjαj hold,

and K takes the

K =d∑

j=1

ωj

(α†jαj + 1

2

)(A5)

diagonal form.

2. Restricted case

In certain cases the quadratic term is simpler and can bewritten as

K′ =(

a

b†

)T (B A

A† C

)(a†

b

), (A6)

where a† = (a†1,a

†2, . . . ,a

†d ′ ) and b = (b1,b2, . . . ,bd ′′ ), with

d = d ′ + d ′′, and the a and b bosons commute among eachother. This happens frequently for bosons in the momentumrepresentation, as the momentum conservation allows onlyterms of the type a

†kak, a

†−ka−k, a

†ka

†−k, and aka−k (i.e., terms

like a†ka−k and akak are missing), and we can associate ak and

a−k bosons with a and b bosons in Eq. (A6), respectively.B = B† and C = C† ensure that the whole Hamiltonianmatrix is hermitian; moreover, we assume it to be positivesemidefinite.

Using the equation-of-motion technique, we get the equiv-alent of Eq. (A3) for this case:(

B A

−A† −C

)T (uj

vj

)= ω′

j

(uj

vj

). (A7)

We define the creation operators and energies as

α†j = uj · a† + vj · b and ωj = ω′

j (A8)

when ω′j � 0 and

α†j = v∗

j · b† + u∗j · a and ωj = −ω′

j (A9)

when ω′j < 0, so that [αj ,α

†j ′ ] = δj,j ′ and the commutation

relations are satisfied. Eventually we arrive at the

K′ =d∑

j=1

ωj

(α†jαj + 1

2

)+ 1

2(TrB − TrC) (A10)

diagonal form.

APPENDIX B: KEEPING |s〉 AND |t1〉 ONLY

In the case of the field parallel to the z axis, it is a usualpractice to keep only the low-lying singlet and Sz = 1 triplet(the component aligned with the field) state of a bond. Herewe are interested in the behavior of the gap close to the criticalfield. For that reason, we restrict the discussion to the DS andthe O(2)[S4] phase for D = 0 and the Z1[D2d ] phase in thefinite D case.

As the first step, we define the rotated boson operators,

s†A(k) = cos

α

2s†A(k) + sin

α

2eiϕt

†1,A(k), (B1a)

t†A(k) = sin

α

2s†A(k) − cos

α

2eiϕt

†1,A(k), (B1b)

s†B(k) = cos

α

2s†B(k) − i sin

α

2eiϕt

†1,B(k), (B1c)

t†B(k) = sin

α

2s†B(k) − i cos

α

2eiϕt

†1,B(k), (B1d)

so that the variational wave function that comprises theaforementioned phases is given by |�〉A = s

†A|0〉 and |�〉B =

s†B |0〉, and we fix the phase ϕ = 0 for convenience [see

also Eq. (38) for comparison]. The expectation value of theHamiltonian is then given by

E0 = −J

(1

2+ cos α

)+ J ′

2(1 − cos α)2 + D′

⊥ sin2 α

+ 1√2D sin α − gzhz(1 − cos α) . (B2)

Here we introduce D = D − 2gshz, as in this section, D and gs

appear in this combination only. The minimization procedureinvolves solving a quartic polynomial equation that is tedious.Instead, we concentrate on the case when the anisotropy termD is small.

024413-15


We also need the bond-wave Hamiltonian. For that weintroduce

t†±(k) = 1√

2[t†A(k) ± t

†B(k)], (B3)

the symmetric and antisymmetric combinations of the rotatedtriplet operators that reduce the size of the matrices in theHamiltonian. Expanding in powers of M , we getH = E0M

2 +MH2 + . . ., where we omitted higher-order terms in 1/M . Thebond-wave Hamiltonian H(2) = H(2)

+ + H(2)− is given as

H(2)± =

∑k

(t†±(k)

t±(−k)

)T

×(

a ± (b + 2D′⊥)γ1 bγ1

bγ1 a ± (b + 2D′⊥)γ1

)

×(

t±(k)

t†±(−k)

), (B4)

where a can be conveniently expressed as

a = b − E0 − hz − J

2, (B5)

and

b = 12 (J ′ − 2D′

⊥) sin2 α, (B6)

while γ1 is defined in Eq. (52). The Bogoliubov transformationyields

ω± =√

(a ± 2D′⊥γ1)(a ± 2bγ1 ± 2D′

⊥γ1). (B7)

1. High-symmetry case

The minimal energy for D = 0 of Eq. (B2) is achieved for

cos αO(2) =

⎧⎪⎨⎪⎩

1 hz � hc1,hc1+hc2−2hz

hc2−hc1hc1 � hz � hc2,

−1 hc2 � hz.

(B8)

These solutions correspond to the DS, O(2)[S4], and fullypolarized phase, respectively. Note that the one-half magneti-zation plateau is missing; the form of the chosen wave functiondoes not allow for Z2 breaking.

hc1 = J − 2|D′⊥|, (B9a)

hc2 = J + 2J ′ + 2|D′⊥|. (B9b)

The variational energy of the unit cell is then

E0 =

⎧⎪⎪⎨⎪⎪⎩

− 3J2 hz � hc1.

− 3J2 − (hz−hc1)2

hc2−hc1hc1 � hz � hc2.

J2 + 2J ′ − 2gzhz hc2 � hz.

(B10)

It turns out that the boundary between the DS phase andthe O(2) phase is shifted at the expense of the O(2) phasecompared to the case where we keep all four states of a dimer[see Eq. (36)], and the boundaries overlap only in the limit ofsmall |D′

⊥| values, when the critical field is close to J .

Now let us turn to the excitation spectrum. In the DS phasea = J − hz and b = 0 in Eq. (B4), so that the H(2)

± matricesare actually diagonal,

H(2)± =

∑k

ω±(k)t†±(k)t±(k), (B11)

with excitation energies

ω±(k) = J − hz ± 2D′⊥ cos

qa

2cos

qb

2. (B12)

This is the same as the small D′⊥/J limit of the dispersions

given by Eqs. (55) and (56), when we kept all four bosons perdimer.

We can also write the bond-wave Hamiltonian in the

H(2) =∑

k

(t†A(k)

t†B(k)

)T (J − hz 2D′

⊥γ1

2D′⊥γ1 J − hz

)(tA(k)

tB(k)

)(B13)

form: here we recognize, up to phase factors, the upper leftcorner of the 4 × 4 matrix in Eq. (51).

In the O(2)[S4] phase, a = J − hc1 = −2D′⊥ and

b = (hz − hc1)(hc2 − hz)

(hc2 − hc1), (B14)

and from Eq. (B7) we get

ω±(k) = 2

√1 ∓ cos

a2

cosb2

×√

D′⊥

(D′

⊥ ∓ (D′⊥ + b) cos

a2

cosb2

). (B15)

0.01

0.02

0.84 0.82 0.8 0.78

1

0

0.9 0.8 0.7 0.6 0.5

0.15

0

0.05

0.1

0.2

0.25

0.3

0.35

z

ω/ D=0.1 J

D=0.01 J

D=0.001 J

J

D=0.001 J

=−0.1 JD'

(a)

(b)

/Jh

~

~

~

~

z

FIG. 15. (Color online) (a) The lowest-lying branch of theexcitation spectrum at k = 0 is shown when keeping two (thick lines)and two bosons (thin lines) per dimer for J ′ = 0.6J , D′

⊥ = 0.1J , anddifferent values of D. (b) The bond-wave spectrum has a dip at thehc1 = 0.8J critical field. The dotted line is the approximation fromRef. 37; the dashed line and the circle are the approximations givenby Eqs. (B17) and (B19), respectively.

024413-16


We note that ω+(k) → 0 as k → 0, thus it becomes theGoldstone mode associated with the continuous symmetrybreaking in the O(2) phase.

2. Low-symmetry case

In the presence of the D anisotropies, the ω+ Goldstonemode acquires a finite gap in the presence of anisotropies. Inthe case of small D we include the first-order correction in D

to the α given by Eq. (B8),

cos α =(

1 + D√2(hz − hc1)(hc2 − hz)

)cos αO(2), (B16)

and we end up with

ω+ = D1/2

[(hz − hc1)(hc2 − hz)

2

]1/4

(B17)

in the leading order in D. This approximation is shown by thedotted line in Fig. 15(b). It clearly fails as h → hc1, as in the

limit αO(2) → 0, Eq. (B16) is no longer valid. Instead, at thecritical field hc1 and in the α → 0 limit, the energy expression,Eq. (B2), simplifies considerably, in leading order

α = − 21/6D1/3

(J ′ − 2D′⊥)1/3

, (B18)

and for the gap we get

ωc1 =√

3D2/3(J ′ − 2D′⊥)1/3

22/3. (B19)

This approximation is shown by the circle in Fig. 15(b). We findthat on the boundary between the DS and the O(2) phase, thegap closes faster than D1/2, namely, with a power of 2/3. Sucha behavior at the quantum critical point has been discussed forquantum antiferromagnets in Refs. 47 and 48. We also notethat the perturbational

√(h − hc1)2 + D2 result in Ref. 37 does

not capture quantum fluctuation effects close to the criticalfield hc1.

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