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Spatially Anisotropic Heisenberg Kagome Antiferromagnet

Andreas P. Schnyder,1 Oleg A. Starykh,2 and Leon Balents11Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA

2Department of Physics, University of Utah, Salt Lake City, Utah 84112, USA(Dated: July 2, 2008)

We study the quasi-one-dimensional limit of the spin-1/2 quantum Heisenberg antiferromagnet on the kagomelattice. The lattice is divided into antiferromagnetic spin-chains (exchangeJ) that are weakly coupled via inter-mediate “dangling” spins (exchangeJ ′). Using one-dimensional bosonization, renormalization group methods,and current algebra techniques the ground state is determined in the limitJ ′

≪ J . We find that the danglingspins and chain spins form a spiral withO(1) andO(J ′/J) static moments, respectively, atop of which the chainspins exhibit a smallerO[(J ′/J)2] antiferromagnetically ordered component along the axis perpendicular to thespiral plane.

I. INTRODUCTION

The nearest-neighbor Heisenberg antiferromagnet on thekagome lattice, a two-dimensional network of corner-sharingtriangles, is one of the most geometrically frustrated mag-nets. Frustration suppresses the magnetic ordering tendencyand leads to an extensive classical ground state degeneracy.Order-by-disorder effects lift the degeneracies in the classicalsystem and are believed to select the coplanar

√3×

√3 pattern

as the ground state.1,2,3,4

The spin-1/2 quantum kagome antiferromagnet is muchless understood. Exact diagonalization5,6,7,8,9,10,11and seriesexpansion studies12,13indicate the absence of long-range mag-netic order, but the precise nature of the ground state remainsmysterious. For small systems, numerical simulations reveala large number of singlet states below a small (and possi-bly vanishing) spin gap.8,9,14 This observation has lead to thespeculation that the ground state of the kagome lattice mightbe a gapless critical spin liquid15,16,17or a particular type ofvalence-bond crystal that exhibits many low energy singletstates.18,19,20

Recently, two new candidate materials for an ideal spin-1/2 kagome antiferromagnet have attracted considerable at-tention. First, the mineral ZnCu3(OH)6Cl2, also known asHerbertsmithite, realizes structurally undistorted, magneti-cally isolated kagome layers with Cu2+ moments on the lat-tice sites.21,22,23,24,25,26,27Neither magnetic ordering nor spinfreezing has been observed for this material down to thelowest currently achievable temperature of 50 mK, whichis well below the energy scale of the antiferromagneticinteraction.22,23,24 While Herbertsmithite is structurally per-fect, Dzyaloshinskii-Moriya interactions and a small num-ber of impurities might complicate the experimental study ofthe ideal quantum kagome system.28 Secondly, there is Vol-borthite Cu3V2O7(OH)2·2H2O, a spin-1/2 (Cu2+) antiferro-magnet, whose magnetic sublattice consists of well-separatedkagome-like planes.29,30,31,32This material has a monoclinicdistortion, which deforms the equilateral kagome trianglesinto isosceles triangles leading to a difference between twoof the nearest-neighbor exchange constants (J ′) and the thirdone (J). Similar to Herbertsmithite the spins do not orderdown to 1.8 K, an energy scale fifty times smaller than theexchange coupling strength in Volborthite.29 However, at very

low temperatures evidence for a spin freezing transition hasbeen reported.32 The anisotropy ratio of the exchange cou-plings,α = J/J ′, could not be determined experimentally sofar, but the differing side lengths of the kagome triangles seemto favorα > 1. A recent comparison of exact diagonalizationcalculations with thermodynamic measurements suggests thatthe spatial anisotropy of the exchange couplings is small, andthat additional interactions beyond the nearest-neighborcou-plings might be present in the mineral Volborthite.33

In this article, motivated by the renewed interest in thekagome systems and the recent experiments on Volborthite,we investigate the spatially anisotropic version of the quan-tum kagome antiferromagnet. We shall focus on the quasi-one-dimensional limit,J ′ ≪ J , where the model consistsof quantum-critical spin-1/2 chains weakly coupled togethervia intermediate “dangling” spins (see Fig. 1). In this situ-ation the competition between quantum fluctuations and thestrong geometric frustration of the kagome lattice is particu-larly keen. The anisotropic quantum kagome antiferromag-net has been studied previously by a variety of techniques,all of which employ perturbation theories in some “artificial”small parameter. Examples include large-N expansions of theSp(N) symmetric generalization of the model,34,35 a block-spin perturbation approach to the trimerized kagome lattice,35

and semiclassical calculations in the limit of large spin.2,35,36

Our approach is complementary to these studies in that it of-fers a fairly controlled analysis of the quasi-one-dimensionallimit using powerful field-theoretical methods37,38,39that haveoriginally been developed for the investigation of quantumcritical systems in one dimension. Recently, the quasi-one-dimensional version of the kagome antiferromagnet in astrong magnetic field, assuming that the intermediate spinsarefully polarized, has been studied using similar techniques.40

The present article treats the case of zero external field.

Our approach rests on the assumption that in the quasi-one-dimensional limit,J ′ ≪ J , the intermediate spins order at atemperature scaleTs much higher than the ordering temper-atureTch of the weakly coupled chains. This is justified aposteriori by our finding that the effective interaction amongthe interstitial spins, which sets the temperature scaleTs, is oforder(J ′)2/J , whereas the most relevant effective interactionamong the weakly coupled spin-1/2 chains, which determinesTch, is of order(J ′)4/J3. Consequently, we can divide the

2

k, j

k, j

k, j

"

JJJ!!!

J

J'

2i 2i+1 2i+2

2y

2y+1

2y-1

J'

FIG. 1: (Color online) Spatially anisotropic kagome lattice withnearest-neighbor exchangeJ (blue) among chain spins (S) andJ ′

(black) among chain spins and interstitial spins (s).

theoretical analysis into three separate stages. First, wede-rive the effective interaction among the intermediate danglingspins using perturbative renormalization group (RG) transfor-mations in the time direction. These considerations are com-plemented by numerical estimates of the induced short dis-tance couplings among the dangling spins. Second, we an-alyze the ground state of the resulting, spatially anisotropictriangular lattice of dangling spins as a function of first andfurther-neighbor interactions. We find that in the ground statethe interstitial spins form a rotating spiral with a small (andpossibly vanishing) wavevector parallel to the chain direction.Third we determine the most relevant interchain interactionsusing a symmetry analysis and RG considerations. Besidesthe effective interchain interaction, the spiral magneticfieldproduced by the intermediate spins induces another perturba-tion to the system of decoupled Heisenberg chains. Finally,we analyze these perturbations to the fixed point of the inde-pendent (decoupled) spin-1/2 chains with the help of operatorproduct expansions (OPEs).

The ultimate result of our analysis is that all spins order,with the non-coplanar configuration shown in Fig. 2, in whichthe interstitial and chain spins predominantly form coplanarspirals with a wavevector(q, 0), but with a reducedO(J ′/J)static moment on the chains. The chain spins are weaklycanted out of the plane, with theO[(J ′/J)2] normal compo-nents forming an antiferromagnetically ordered pattern. Theprecise value ofq cannot be reliably determined from ouranalysis, but we expectq ≪ 1 and q = 0 is a distinctpossibility. In theq = 0 case, the state becomes coplanar.This ordered state differs from those found in two other re-cent studies35,36 using other methods, but there are similari-ties. These are discussed in Sec. VII.

The remainder of the article is organized as follows. Sec. IIdescribes the lattice Hamiltonian, its symmetries, and itslow-energy field theory description. In Sec. III we analyze thelow-energy interactions using symmetry considerations and aperturbative renormalization treatment. In Sec. IV we use nu-merical methods to estimate the interaction strength amongthe interstitial spins and derive an effective model for thedan-gling spins on the triangular lattice. Sec. V deals with theground state analysis of the triangular lattice. The perturba-

FIG. 2: (Color online) Three-dimensional view of the spin orderingpattern. The dangling spins form a coplanar spiral with anO(1)local moment. Projection of the chain spins into the spiral planealso forms a spiral, but with static moment ofO[J ′/J ]. The out ofplane components of the chain spins are non-zero but even smaller –O[(J ′/J)2] – and ordered antiferromagnetically along and betweenthe chains.

tive analysis of weakly coupled chains in a spiral magneticfield with a small wavevector,q ≪ 1, is presented in Sec. VIand our conclusions and a discussion of the relation to otherresults are given in Sec. VII. Some technical aspects of the RGcalculations are relegated to appendix A. For completeness,we study in appendix B the ordering of the weakly coupledchains in the presence of a spiral field with a large wavevec-tor.

II. MODEL DEFINITION AND LOW-ENERGYHAMILTONIAN

The Hamiltonian of the Heisenberg antiferromagnet on theanisotropic kagome lattice (see Fig. 1) is given byH =H0 + V , whereH0 describes the decoupled set of chains withnearest-neighbor antiferromagnetic Heisenberg interactions

H0 = J∑

i,y

Si,y · Si+1,y, (1a)

andV the interaction among the chains and the intermediatespins

V = J ′∑

i,y

s2i± 1

2,2y∓ 1

2

·(

S2i,2y + S2i±1,2y

+S2i,2y∓1 + S2i±1,2y∓1

)

. (1b)

For the interstitial spins we shall use the symbols, whereasthe chain spins are denoted byS. The anisotropic kagomelattice has rotation, translation and reflection symmetries. Thetranslational subgroup is generated by the translationsT1 andT2, which move the lattice by two units along the horizontal,and one of the diagonal axes, respectively. The rotational sub-group consists ofπ rotations about the lattice sites and aboutthe centers of the hexagons. We can distinguish two types

3

of reflection symmetries: reflectionsR1 about a vertical linethrough a midpoint of a chain bond (link parity), and reflec-tionsR2 about a horizontal line passing through the interme-diate dangling spins (see Fig. 1).

Let us now describe the chains in the continuum limit,which is applicable as long asJ ≫ J ′. In this limit thelow energy properties of the antiferromagnetic spin-1/2 chainsare governed by the Wess-Zumino-Novikov-Witten (WZNW)SU(2)1 theory, and the chain spin operatorSi,y can be decom-posed into its uniformMy(x) and staggered spin magnetiza-tion Ny(x)

Si,y → a0 [My(x) + (−1)xNy(x)] , (2a)

wherex = ia0 anda0 denotes the lattice spacing. The uni-form magnetization can be written in terms of left- and right-movingSU(2) currents,My = Jy,R +Jy,L. Another impor-tant operator describing low energy properties of the spin-1/2chains is the staggered dimerizationεy(x), which is defined asthe continuum limit of the scalar product of two neighboringspins

Si,y · Si+1,y → (−1)xεy(x). (2b)

The scaling dimension of these continuum operators deter-mines the relevance of the operator in the RG sense withrespect to the Luttinger liquid fixed point of the decoupledchains. The uniform magnetizationMy has scaling dimen-sion1, whereas both the staggered spin magnetizationNy andthe staggered dimerizationεy have scaling dimension1/2.The microscopic lattice symmetries ofH , Eq. (1), leave animprint on the continuum description (2). That is, the actionof the space group symmetries on the continuum operators isgiven by

T1 : M → M , N → +N , ε → +ε, (3a)

R1 : M → M , N → −N , ε → +ε, (3b)

R2 ◦ T2 : M → M , N → −N , ε → −ε, (3c)

where, for brevity, we have suppressed the chain index. Othersymmetry operations on the continuum fields are either trivial,or can be rewritten as a product of the above transformations.

The three continuum fieldsM , N , andε form a closed op-erator algebra with respect to certain operator product expan-sions, which are widely used in the literature.37,38,39,41,42,43,44

For example, the right-movingSU(2) currentsJR satisfy thefollowing chiral OPEs

JaR(x, τ)Jb

R(0) =δab/(8π2)

(uτ − ix + a0στ )2+

iεabcJcR(0)/(2π)

uτ − ix + a0στ,

(4a)

with imaginary timeτ , στ = sgn τ , the short-distance cut-offa0, and spin velocityu = πJa0/2. Similar relations hold forthe left-moving spin currentsJL. The product ofJR andN

can be expanded as

JaR(x, τ)N b(0) =

iǫabcN c(0) − iδabǫ(0)

4π(uτ − ix + a0στ ). (4b)

The above equalities are understood to be valid only when in-serted into correlation functions and in the limit where thetwopoints(x, τ) and(0, 0) are close together. These current alge-bra relations will allow us to compute one-loop RG equationsby purely algebraic means (see Secs. III and VI).

Using relation (2a), we can derive a naive continuum limitof the interaction among the chains and the intermediate spins,Eq. (1b). First, we note that the intermediate spinss2i+ 1

2,2y− 1

2

couple symmetrically toS2i,2y andS2i+1,2y. Hence, the stag-gered spin magnetization enters only via its first derivative inthe continuum version of the interactionV , and we can set

S2i,2y + S2i+1,2y → 2a0M2y(2x) − a20

2∂xN2y(2x). (5)

In the above, and the following, we use the notation that thederivative in∂xN2y(2x) is with respect to the full argument,i.e. more explicitly

∂xN2y(2x) ≡ ∂XN2y(X)|X=2x . (6)

With this, the interaction Eq. (1b) readsV = V1 + V2, with

V1 = γ1

∑

x,y

s2x± 1

2,2y∓ 1

2

· [M2y(2x) + M2y∓1(2x)] ,

(7)

V2 = γ2

∑

x,y

(±)s2x± 1

2,2y∓ 1

2

· ∂x [N2y(2x) + N2y∓1(2x)] ,

where the bare coupling constants are given by,γ1 = 2J ′a0

and γ2 = −J ′a20/2, which follows from substituting (5)

into (1b). We have retained the next-to-leading interactionV2 as it will produce, in combination with the leading termV1, relevant interchain couplings with respect to the fixedpoint of the decoupled chains (see Sec. III). The contin-uum description of the interactionV , Eq. (7), is necessarilyinvariant under the symmetry transformations, Eq. (3). Inparticular, we note thatV does not contain a contributions2x+ 1

2,2y− 1

2

· N2y(2x) which is forbidden by the symme-tries of the spatially anisotropic kagome lattice (link parityR1 sendsN → −N ).

III. RENORMALIZATION GROUP TREATMENT

Here, we study the effective low-energy interactions amongthe dangling spins (H△), as well as between nearest neighborchains (Vch), using both a symmetry analysis and perturbativerenormalization group transformations. The goal is to under-stand which interactions amongst interstitial spins, amongstchains, or betwixt the two, are most relevant, and further, atwhat energy scales they become important in determining thelow energy physics. Technically, the Hamiltonian (1) with thechains treated in the continuum limit is formally rather simi-lar to a Kondo lattice. As the Hamiltonian retains some localcharacter due to the intermediate spins, in the framework ofthe RG approach, this problem develops only in time, and theenergy is the only variable that is being rescaled as the RGprogresses. Such an RG scheme is similar in spirit to the oneemployed in the context of a single impurity coupled to a Lut-tinger liquid (see for example Refs. 45,46,47).

4

A. Symmetry analysis

Before proceeding with the RG derivation of the low-energy effective interactions, we first write down the mostgeneral form ofVch andH△ that are allowed by the symme-tries of the spatially anisotropic kagome lattice. Such a gen-eral symmetry consideration will reveal all relevant symmetryallowed interactions that are expected to be generated throughRG transformations, when all nominally irrelevant terms aretaken into account. The space group symmetries needed forthis analysis have been discussed in Sec. II [see Eqs. (3)].

We begin with the interchain HamiltonianVch. In principle,the number of allowed interchain interaction terms is infinite.However, only a handful of terms will be important. Mostsignificant are those terms which are most relevant in the RGsense (with respect to the Luttinger liquid fixed point of thedecoupled chains). This amounts to two-chain interactionsinvolving no derivatives and only continuum fields that havesmall scaling dimension. Amongst these, we may further re-strict ourselves to nearest-neighbor chain interactions,as themagnitude of further-neighbor chain interactions is expectedto decrease with separation distance. The continuum oper-ators with the smallest scaling dimension are the staggereddimerizationεy and the staggered magnetizationNy. There-fore, we find

Vch =∑

y

∫

dx {γNNy · Ny+1 + γεεyεy+1} , (8)

where the value of the coupling constantsγN andγε will haveto be determined by microscopic calculations. Using Eqs. (3)it is straightforward to check that these are the only terms withlowest possible scaling dimension1 that satisfy the symmetryrequirements of the spatially anisotropic kagome lattice.

In addition to these most relevant interchain interactions,it is necessary to consider also a few less relevant terms, asthese will arise at lower order in the renormalization grouptreatment below, and are important forgeneratingthe morerelevant terms in Eq. (8) above. These are

V(1)ch =

∑

y

∫

dx {γ∂N∂xNy · ∂xNy+1 + γMMy · My+1} .

(9)Next, we turn to interactions among the interchain spins

H△. Considering only terms that arise atO[(J ′)2], we findthatH△ is given by Heisenberg interactions among danglingspins whosey-coordinate differ by at most one unit. The cou-pling constants of these Heisenberg interactions are restrictedby the symmetries of the lattice. With these conditions,H△

can be conveniently written in the form (see Fig. 3)

H△ =∑

x,y,r>0

J2r s2x± 1

2,2y∓ 1

2

· s2x± 1

2+2r,2y∓ 1

2

(10)

+∑

x,y,r

J|2r+1|s2x± 1

2,2y∓ 1

2

· s2x± 3

2+2r,2y± 1

2

.

Since the couplingsJr decrease in magnitude with increasing|r|, we can truncate the above sums overr after the first fewterms.

B. Renormalization group results

To determine the fluctuation-generated corrections to thelow-energy effective action we perform a perturbative RGanalysis of the interactionsV , Eq. (7), to one-loop order. Theperturbation theory is formulated by expanding the partitionfunctionZ =

∫

e−S0−R

dτV up to quadratic order in the cou-plings

Z ≃∫

e−S0

[

1 −∫

dτV +1

2T

∫

dτ1dτ2V (τ1)V (τ2)

]

.

(11)

Here,S0 denotes the fixed point action,T the time-orderingoperator, andτi the imaginary time. Implicit in Eq. (11) is aregularization, such that the integrals appearing in the expan-sion are taken only over the regions in which no two timesare closer than some short time cut-offα = a0/u. The RGproceeds by increasing this cut-off infinitesimally fromα tobα, whereb = edℓ and dℓ > 0 is the usual logarithmicchange of scale. To do this, all pairs of timesτ1, τ2 suchthat α < |τ1 − τ2| < bα must be fused using the operatorproduct expansion, and the integral overτ1 − τ2 in this rangecarried out. One thereby obtains a new partition function withrenormalized interactions and the increased cut-off. We thenperform an additional trivial rescaling of time and space coor-dinates for to restore the original cut-off:

Ny(x, τ) → b−1/2Ny(x/b, τ/b),

My(x, τ) → b−1My(x/b, τ/b), (12)

sx,y(τ) → sx,y(τ/b).

We note that the spatial rescaling is not required to restorethe cut-off but is natural and convenient for the continuumfields of the chains, which are described by a Lorentz invari-ant field theory. By contrast, we obviously cannot rescale thecoordinates of the discrete spin operators of the interstitialspins. This difference leads to the appearance of an explicitRG length scale in the effective couplings between the chainsand interstitial spins,V1 andV2, in the renormalized Hamilto-nian:

V1(ℓ) = γ1

∑

x,y

s2x± 1

2,2y∓ 1

2

· [M2y(2xℓ) + M2y∓1(2xℓ)] ,

V2(ℓ) = γ2

∑

x,y

(±)s2x± 1

2,2y∓ 1

2

· ∂x

[

N2y(2xℓ) (13)

+N2y∓1(2xℓ)]

,

wherexℓ = xe−ℓ.

With this, the derivation of the RG equations is straightfor-ward, and follows closely the methods used in Refs. 45,46,47.Details of the calculations can be found in appendix A. Tak-ing into accountV = V1 + V2 to one loop, we obtain RGequations for both the interchain couplings and the couplings

5

J 4

J 3

k, j

k, j

k, j

"

b

a c

2x 2x+1 2x+2

k, j

k, j

k, j

"

2y

2y+1

2y-1

i k []

i j

kj

[]

⟨ ⟩

e

d

J 1

i j⟨ ⟩'

i k []'

J 1

J 2

FIG. 3: (Color online) Exchange paths of the triangular latticeformed by the intermediate spinss of the spatially anisotropickagome lattice. Solid black lines denote the nearest-neighbor ferro-magnetic exchangeJ1 (a - b and b - c bonds), dotted black lines thenearest-neighbor antiferromagnetic exchangeJ2 (a - c bond). Thefurther neighbor interactionsJ3 (a - d bond) andJ4 (a - e bond) forthe spinsa are illustrated with red arrows. The chain bonds of theanisotropic kagome lattice are indicated in light blue (cf.Fig. 1).

between the interstitial spins. The former are

dγ1

dℓ= +

1

πuγ21 ,

dγ2

dℓ= −1

2γ2 +

1

πuγ1γ2,

dγ∂N

dℓ= −γ∂N − γ2

2

u,

dγM

dℓ= −γ2

1 +γ2

M

2πu, (14)

dγN

dℓ= γN +

γ∂NγM

8πα2u,

dγε

dℓ= γε −

3γ∂NγM

16πα2u.

Note that the strongly relevant interactionsγN , γǫ arenotgen-erated at one loop fromγ1, γ2. Instead, the former are gener-ated from the sub-dominantγ∂N , γM terms. This is the reasonthe latter terms needed to be included in our treatment.

The RG equations for the interstitial spin couplings arefunctional, insofar as they describe the flow of the full set ofinteractionsJr. We find

dJ2r

dℓ= J2r + IM (re−ℓ)

γ21

u+ IN (re−ℓ)

γ22

u, (15a)

where

IM (r) =∑

σ′=±1

α

4π2

1

[α + σ′i2r]2 ,

IN (r) = 8CNα(α2 − 8r2)

(α2 + 4r2)5/2, (15b)

whereCN is the amplitude of the〈NaNa〉 correlator. Anal-ogous expressions can be derived for the interactionsJ|2r+1|,i.e., the second term inH△, Eq. (10).

In passing, we note that the form of the RG flows inEqs. (14) is highly constrained by the symmetry of the full(spatially anisotropic) kagome lattice, and this is necessary toavoid the generation of strongly relevant interactions at thisorder. By contrast, for the kagomestrip of extension one inthe y-direction, there is no translational symmetryT2, and

consequently the symmetry (3c) is absent. This allows for theappearance of a term in the renormalized Hamiltonian propor-tional to

∫

dxεy(x, τ), which is indeed generated proportion-ally to γ1γ2 in that case (compare with Azariaet al., Ref. 48,and see Ref. 49). This term is strongly relevant and generatesa gapped dimerized state in that model.

C. Implications

The RG flows in Eqs. (14,15) give considerable insight intothe emergent energy scales in the problem. Though interac-tions between the chains and between the interstitial spinsareboth generated from the bare interactionsγ1, γ2, their charac-ters are distinctly different. Crucially, we see from Eqs. (14)that amongst the chains, only marginal and irrelevant interac-tions –γ∂N , γM – are generated at this order. Since the barevaluesγ1(ℓ = 0) ∼ γ2(ℓ = 0) are both ofO[J ′], this includesall effective generated interactions up toO[(J ′)2]. A simplescaling argument shows that the low energy excitations of thechains can be modified from those of decoupled chains onlyon energy scalesparametrically smallerthan(J ′)2.

To understand this, let us consider the solutions to Eqs. (14)in somewhat more detail. The most important observation isthat the strongly relevant interactions,γN , γε [third line ofEqs. (14)], once they are generated, grow rapidly and expo-nentially with ℓ. They are therefore dominated by the ef-fects of their “source” terms (proportional toγ∂NγM ) atsmallℓ, which are most amplified under the growth. The sourceterms at small scale are themselves of order(J ′)2/J , as de-scribed above. Therefore we expect that the relevant interac-tions will be, at least until they renormalize to large values, oforder(J ′)4/J3. Indeed, the ultimate effect of all this short-scale renormalization is completely equivalent to includingfluctuation-generated “bare” interactions of this same order.This procedure has been described in considerable detail inprior publications, in which the formal manipulations closelyparallel those used here. See Refs. 37,39 for further informa-tion. Therefore we will in Sec. VI simply take the relevantcouplings to have the appropriate values:

γN (ℓ ∼ 1) = +2(J ′)4

4π4J3, γε(ℓ ∼ 1) = −3(J ′)4

4π4J3. (16)

By contrast, the interactionsJr generated in Eqs. (15) atO[(J ′)2] amongst the dangling spins will clearly modify theirenergetics at this same order, since they act within an oth-erwise completely degenerate manifold. More formally, thisfollows directly from the RG flows, Eq. (15), which shows thatthe interstitial interactions will grow toO(1) values (actuallyto any fixed value) by a scaleJeℓ ∼ J(γ2

1 , γ22) ∼ (J ′)2/J .

Therefore we see that, as claimed in the introduction, the or-dering (as we shall see, this is the result of these interactions)of the interstitial spins occurs at an energy scale at which thechains are unaffected by their couplings to the interstitial spinsand each other.

Having understood the hierarchy of energy scales resultingfrom the RG, we can proceed to try to understand in more de-tail the ordering of the interstitial spins. In principle, one may

6

do so by integrating the flow equations for theJr couplings,Eqs. (15), fromℓ = 0 to the ordering scale,ℓ = 2 ln(J/J ′).This is equivalent to performing the time integrals not overthe infinitesimal shell in the vicinity of the running cut-off,but instead from microscopic timesα up to times of orderα(J/J ′)2. Inspection of expressions in Eq. (15b) shows thatthese integrals in fact are dominated by times of order the mi-croscopic time, and indeed the upper limit of the time integra-tion can be extended to infinity without significant modifica-tion of the result.

The dominance of short times has physical significance:it implies that the generated interstitial spin interactions arein fact controlled by high energy physics of the Heisenbergchains, i.e., correlations of the chain spins on the scale ofafew lattice spacings. These short distance correlations arenotprecisely captured by the continuum bosonization/currental-gebra approach. Therefore, if an accurate determination ofthese dangling spin interactions is desired, we must abandon(for the moment) the field theory methodology and return to adirect study of the lattice model. Conversely, because the gen-erated interactions are dominated by short distance physics,we expect that no serious divergences will arise in a micro-scopic calculation. In the following section we show how per-turbative and numerical methods may be used to perform thenecessary calculations and obtain the fluctuation generated in-teractions amongst the interstitial spins. Using these preciseresults we determine the interstitial spin ordering in Sec.V.

Following this, in Sec. VI we will return to the lower energyphysics resulting from the fluctuation generated interactionsγ∂N , γM in Eqs. (14), which will ultimately lead to orderingamongst the chain spins as well.

IV. NUMERICAL ESTIMATES OF INTERACTIONSAMONG DANGLING SPINS

The dynamical structure factor of the spin-1/2 antiferro-magnetic Heisenberg chain,S(q, ω), can be related to the cou-pling constants of the effective interactions among the dan-gling spins,Jr, by means of second-order degenerate pertur-bation theory in the interactionV , Eq. (1b). Using availablenumerical data ofS(q, ω), both in the two-spinon approxima-tion and for finite-size systems, this gives an estimate for thesign and relative strength of the interactionsJr.

To formulate the perturbation theory inV we denote theground state of the unperturbed HamiltonianH0, Eq. (1a), by|0〉 and introduce the ground-state projection operatorP =|0〉〈0|. With Q = 1 − P , any given state|Ψ〉 of chain (S)and interchain (s) spins can be written as|Ψ〉 = |Ψ0〉+ |ΨQ〉,where|Ψ0〉 = P |Ψ〉 and|ΨQ〉 = Q|Ψ〉. We note that|Ψ0〉describes the state with the chain spins in the ground state,while the state of thes-spins remains arbitrary, i.e.,|Ψ0〉 =|0〉|{s}〉. Following along the lines of Refs. 40,50, we obtainan eigenvalue equation for|Ψ0〉

(

H0 + PV1

1 − RQVRV)

|Ψ0〉 = E|Ψ0〉, (17)

with the resolventR = (E − H0)−1. Eq. (17) is highly non-

linear in the energyE, since the left-hand side of this equationdepends onE through the resolventR. Performing a perturba-tion expansion on the operator(1 − RQV )−1 gives at secondorder inJ ′

(

H0 + PV R0V)

|Ψ0〉 = E|Ψ0〉, (18)

whereR0 = (E0 − H0)−1 is a function of the ground state

energyE0 of the decoupled chains. Multiplying Eq. (18)from the left by〈0| and inserting the perturbationV , Eq. (1b),yields, after some algebra, a Schrodinger equation for thein-terchain spins alone

H△ |{s}〉 = (E − E0) |{s}〉 , (19a)

with the effective interaction among interstitial spins

H△ = 4(J ′)2∑

x,y,r>0

A(2r)[

s2x± 1

2,2y∓ 1

2

· s2x± 1

2+2r,2y∓ 1

2

]

+2(J ′)2∑

x,y,r

A(|2r + 1|)[

s2x± 1

2,2y∓ 1

2

· s2x± 3

2+2r,2y± 1

2

]

.

(19b)

Here,A(r) is the linear combination

A(r) = GM (r − 1) + 2GM (r) + GM (r + 1) (20)

of the ground state expectation values

GM (r) = 〈0|Sa2x,y

1

E0 − H0Sa

2x+r,y|0〉. (21)

In deriving equation (19) we made use of the fact that theexpectation value (21) is independent of both thex- andy-coordinate, due to translational invariance. The linear combi-nation in Eq. (20) arises, because every interstitial spins iscoupled to pairs of chain spinsS. From inspection of Eq. (19)we find the following expressions for the exchange couplingsJr, which have been defined in Eq. (10),

J2r = 4(J ′)2A(2r),

J2r+1 = 2(J ′)2A(2r + 1), (22)

with r > 0. The factor2 difference between the first and thesecond line in Eq. (22) accounts for the fact that interstitialspinss with the samey-coordinate,y − 1

2 , say, are connectedvia both, they andy − 1 chains.

Finally, by inserting a resolution of identity in Eq. (21),

GM (r) =∑

n6=0

〈0|Sa2x,y|n〉

1

E0 − En〈n|Sa

2x+r,y|0〉,

with |n〉 the eigenstates ofH0, we realize thatGM (r),Eq. (21), is just a spectral representation of the zero-frequencyMatsubara spin Green’s function

GM (r) = −∫ ∞

0

dτ〈TτSa(2x + r, τ)Sa(2x, 0)〉. (23)

7

The zero frequency Green’s functionGM (r) in turn isconnected to the dynamical structure factorS(q, ω) via aKramers-Kronig transform

GM (r) =2

π

∫ ∞

0

dω′

∫ +π

0

dq S(q, ω′)cos(qr)

ω′. (24)

Inserting the above equation into the definition ofA(r),Eg. (20), gives

A(r) =8

π

∫ ∞

0

dω′

∫ π

0

dq cos2q

2S(q, ω′)

cos(qr)

ω′.(25)

Having related the interstitial exchange interactionsJr

to the dynamical structure factor, we evaluate numericallyS(q, w) in the subsequent sections to obtain precise estimatesfor the first few interstitial couplingsJ1 · · · J4.

A. Two-spinon dynamical structure factor

First we compute the couplingsJr using the two-spinonapproximation forS(q, ω). The two-spinon contribution tothe dynamical structure factor can be explicitly written as(seeRefs. 51,52)

S2(q, ω) =1

2π

e−I(ρ(q,ω))

√

ω2U (q) − ω2

Θ[ωUq − ω]Θ[ωL

q − ω], (26)

with the fundamental integral

I(ρ) =

∫ ∞

0

dtet

t

cosh(2t) cos(4ρt) − 1

cosh t sinh(2t), (27)

and with the lower continuum boundaryωLq = π

2 sin q andthe upper continuum boundaryωU

q = π sin q2 . The auxiliary

variableρ is a function ofq andω

ρ(q, ω) =4

πacosh

√

ω2U (q) − ω2

L(q)

ω2 − ω2L(q)

. (28)

InsertingS2(q, ω) into formula (25), we obtainA(r) in thetwo-spinon approximation

A2(r) =4

π2

∫ π

0

dq cos2q

2cos(qr) (29)

×∫ ωU

q

ωLq

dω′

ω′

e−I(ρ(q,ω′))

√

ω2U (q) − ω′2

.

The expression (29) can be evaluated numerically in an effi-cient way, provided one splits off the singular part from theintegral (27) (see Ref. 51). In doing so, we obtain for the ratiobetween the couplingsJ2 andJ1 the following result withinthe two-spinon approximation

J2

J1≃ 2A2(2)

A2(1)≃ −0.7446, (30a)

and find thatJ1 is ferromagnetic, whileJ2 is antiferromag-netic. Similarly, the magnitude of further neighbor interac-tions is estimated to be

J3

J1≃ +0.1909,

J4

J1≃ −0.2312. (30b)

with J3 < 0 andJ4 > 0.The two-spinon intensity accounts for about73% of the to-

tal structure factor intensity.51 The remaining part is carried bystates with a higher number of spinons, and it is believed thatthe four-spinon intensity together with the two-spinon inten-sity cover about98% of the spectral weight.52 In principle, itwould be possible to evaluate the ratios (30) within the four-spinon approximation, however the involved numerical inte-grals are rather expensive to compute. Instead, we shall usefinite size results for the structure factor to obtain a secondestimate for the magnitured of the couplingsJr.

B. Finite size results

We computeA(r) using numerical data from the ABACUSdatabase53 for the dynamical structure factor in a finite systemof N = 500 sites. This numerical data was obtained using amethod based on the Bethe ansatz framework, which involvesa summation over so-called determinant representations forform factors of spin operators on the lattice.54 In these com-putations the momentum delta functions are smoothed out byincluding a finite broadeningη ∼ 1/N . Using the numericaldata with a smearingη = 0.01 we obtain for the ratioJ2/J1

J2

J1≃ 2AN (2)

AN (1)≃ −0.7013, (31a)

with J1 < 0 andJ2 > 0, in accordance with Eq. (30a). Theestimated magnitude of further neighbor interactions is

J3

J1≃ +0.2349,

J4

J1≃ −0.1453, (31b)

whereJ3 is ferromagnetic andJ4 antiferromagnetic. As an-ticipated from the RG treatment in Sec. III B, we observe thatthe magnitude of the coupling strengths decreases with spinseparation distance.

Comparing Eqs. (30) to Eqs. (31), we infer that the valuesof J2/J1 agree well (within 10%), whereas the agreement offurther neighbor interactions is worse. Since the two-spinonapproximation misses 27% of the structure factor intensity,we suspect that the true values of the coupling strengths arecloser to the finite size results (31) than to the two-spinon es-timates. In the next section, we are going to work out the in-terstitial spin ordering using the results (30,31) and neglectingany longer range interactions withJr>4.

V. SPATIALLY ANISOTROPIC TRIANGULAR LATTICE

As the discussions in the previous sections have shown,the ordering of the dangling spinss, which occurs at an en-ergy scaleO[(J ′)2], is independent of the chain ordering. At

8

the energy scaleO[(J ′)2] the spinss, which form a spatiallyanisotropic triangular lattice55,56,57,58, are described by the ef-fective HamiltonianH△, Eq. (10), with antiferromagneticinteractions along the horizontal bonds (J2) and ferromag-netic interactions along the diagonal bonds (J1), see Fig. 3.Neglecting third-nearest neighbor and longer ranged interac-tions, we can truncate the sums overr in Eq. (10) after thefirst two terms and obtain

H△ ≈ J1

∑

[ij]

si · sj + J2

∑

〈ij〉

si · sj ,

+J3

∑

[ij]′

si · sj + J4

∑

〈ij〉′

si · sj , (32)

where[ij] and [ij]′ denote the diagonal bonds and〈ij〉 and

〈ij〉′ the horizontal bonds connecting first and second-nearestneighbors, respectively (see Fig. 3). The values of the cou-pling strengthJ1 · · · J4 are given by Eqs. (30,31).

This is a non-trivial spin-1/2 quantum model. However,it is also one which has been heavily studied, at least in thenearest-neighbor limit. In this case, over the vast majority ofthe phase diagram, the quantum ground state agrees with theclassical one. This is particularly true when the couplingsaresuch that the classical ground state is “close” to ferromagnetic(indeed the fully polarized ferromagnetic state is of course anexact eigenstate, as usual). We therefore expect that a classicalanalysis is reliable, and pursue it below.

The classical phase diagram of Hamiltonian (32) is foundby replacing the spin operators by classical, coplanar spiralvectors,

si = x cos(q · ri) + y sin(q · ri), (33)

and minimizing the energy, which amounts to minimizing theFourier transform of the exchange coupling59,60

J(q) =∑

i,j

Ji,j cos [q · (ri − rj)] , (34)

whereri denotes the position of sitei in real space, andx andy are two orthogonal unit vectors. In our case, Eq. (34) gives

J(q) = 2J1 cos qx cos qy + J2 cos(2qx) (35)

+2J3 cos(3qx) cos qy + J4 cos(4qx).

with J1, J3 < 0 andJ2, J4 > 0.For simplicity, let us first analyze the minima of Eq. (35)

for the caseJ3 = J4 = 0. This givesq = (qx, 0) with

qx =

{

0, J2 < |J1|/2,arccos[−J1/(2J2)], J2 < |J1|/2.

(36)

That is, forJ2 < |J1|/2 the classical ground state is fer-romagnetic, whereas forJ2 > |J1|/2 the ground state is aspiral state rotating along thex-direction with the wave vec-tor q = (qx, 0). [To study how quantum fluctuations alterthe classical ground state in the caseJ3 = J4 = 0 we haveperformed a linear spin wave analysis (not shown). The mag-netization decreases smoothly from1/2 to zero asJ2/|J1| isincreased, with a kink atJ2/|J1| = 1/2.]

In the case of non-zero further neighbor interactions,J3 6=0 andJ4 6= 0, there is no simple explicit expression describ-ing the global minima ofJ(q), Eq. (35). Numerically we findthat the minimum ofJ(q) with the coupling values given byEq. (31) (the finite size results) occurs atq = 0, i.e., the ferro-magnetic state is the ground state. However, the coupling pa-rameters of Eq. (31) are rather close to the boundary of a spiralphase in the coupling parameter space{Jr | r = 1, · · · , 4}.In particular, the coupling parameters (30) (the two-spinon re-sult) yield a spiral state withq ≃ 2π(0.08, 0). Therefore, weshall consider in what follows a cycloidal spiral ground statewith wavevectorq = (qx, 0), with qx small, which includesas a limiting case the ferromagnetic state (qx = 0), that is,

〈s2x± 1

2,2y∓ 1

2

〉 = (37)

s0

(

x cos[

2qxx ± qx

2

]

+ y sin[

2qxx ± qx

2

])

.

Heres0 . 1/2 is the local static moment (staggered magneti-zation) of the spiral state.

VI. COUPLED HEISENBERG CHAINS IN A SPIRALMAGNETIC FIELD

We are now in a position to address the spin-ordering pat-tern of the chains. We assume that the interstitial spins haveordered into the spiral state given by Eq. (37), and focus on thelower temperature scale of order(J ′)4/J3, at which the chainspins become affected by relevant interchain interactions. Theunperturbed system of antiferromagnetic Heisenberg chains isgiven by [note that we letx → x − 1/2 compared to Sec. V]

H0 = J∑

x,y

Sx− 1

2,y · Sx+ 1

2,y. (38)

The perturbations are: (i) coupling between chain and inter-chain spins, described by (40), (ii) marginal “backscattering”term, which is already present for a single Heisenberg chain,and (iii) the fluctuation generated interchain interactions (8).

Since the interchain spins form a two-dimensional orderedspiral state, described in the preceding section, the main effectof their interaction with the chain spinsS is described by thespiral magnetic field vector〈sx〉 given by [note that we letqx → q compared to Sec. V]

〈sx〉 = s0[x cos(qx) + y sin(qx)], (39)

with q small. The spiral magnetic field introduces the follow-ing perturbations to this system of decoupled spins

HS = J ′∑

x,y

{

S2x−1/2,y · 〈s2x−1 + s2x〉 (40)

+S2x+1/2,y · 〈s2x + s2x+1〉}

=∑

x,y

hx+1/2 · Sx+1/2,y, (41)

with

hx = 2h0 cos(q/2)[x cos(qx) + y sin(qx)], (42)

9

and the field strengthh0 = s0J′.

The simplest effect of the spiral field, as for any field, isto induce a corresponding spiral magnetization in linear re-sponse. This implies that, generically, there will be static com-ponents of the chain spins in the spiral (x−y) plane. Since thespiral field is proportional toJ ′, and the chain susceptibilityis generally proportional to1/J , these static components areof orderJ ′/J .

This simple linear response, however, is not the completestory. A much more detailed analysis is needed to resolve themore subtle effects of the spiral field beyond linear response,in conjunction with the inter-chain couplings described inSec. III. The nature of this more complete analysis dependscrucially upon the magnitude ofq. Whenq is small, the spinsrespond in a way which is similar to the response to a uni-form magnetic field. A systematic approach is then possible,in which the Hamiltonian is transformed into the slowly ro-tating frame in which the external field is uniform. This istractable because for smallq the non-Heisenberg interactionsinduced by the change of frame are weak. Because of the ex-pected smallness ofq, we focus on this case in the remainderof this section.

In the opposite limit of a “large” wavevector, which isincommensurate with the dominant fluctuations of the 1dHeisenberg chain, i.e.q, |π − q| ∼ O(1), the field weaklycouples to the spin chain. Indeed, the leadingO(h) effects av-erage out over space, and instead only sub-dominant terms areinduced atO(h2). In this limit these weakerO(h2) terms arecrucial in determining the final state of the system. Thoughthis limit is actually conceptually simpler than the oppositeone, it is technically challenging, because the effects of spiralfield are determined by very short-wavelength properties ofthe Heisenberg spins. As a result, we are not unambiguouslyable to resolve the ground state in this limit. However, theambiguity is small: we show in Appendix B that the system atzero temperature is in one of onlytwo possible phases. Oneof these is thesamenon-coplanar state which we find in thesmallq limit (the other is a coplanar state). This supports thenotion that, at least up to some criticalO(1) value ofq (andpossibly for allq), the ground state evolves smoothly from thesmallq limit.

A. Transformation to rotated frame

From here on, we assumeq ≪ 1. It is advantageous torotate the chain spins towards the direction of the spiral mag-netic field

Sx,y −→ RxSx,y,

Rx =

0 − sin(qx) cos(qx)0 cos(qx) sin(qx)−1 0 0

. (43)

The rotationRx amounts to−π/2 rotation about they-axisfollowed by a rotation about thez-axis with angleqx. We find

that under rotation (43) the magnetic field term becomes

HS = −2h0 cos(q/2)∑

x,y

Szx+1/2,y. (44)

H0 transforms into

H0 = J∑

x,y

(

Sxx− 1

2,ySx

x+ 1

2,y + cos q

[

Syx−1/2,ySy

x+1/2,y

+Szx−1/2,yS

zx+1/2,y

]

+ sin q[

Syx−1/2,ySz

x+1/2,y

−Szx−1/2,yS

yx+1/2,y

]

)

. (45)

In the limit of smallq, which we focus on, this Hamiltonian isconveniently split into Heisenberg oneH0 with the modifiedexchange constantJ = J cos q,

H0 = J∑

x,y

Sx− 1

2,y · Sx+ 1

2,y, (46)

an effective Ising anisotropyH1 along thex-axis,

H1 = J(1 − cos q)∑

x,y

Sxx− 1

2,yS

xx+ 1

2,y, (47)

and an effective Dzyaloshinskii-Moriya (DM) interactionH2

H2 = J sin q∑

x,y

[

Syx−1/2,yS

zx+1/2,y − Sz

x−1/2,ySyx+1/2,y.

]

(48)Thus,H0 = H0 + H1 + H2.

1. Low energy limit

It is appropriate now to take a low-energy limit, for whichwe use the non-Abelian spin current formulation. The zerothorder Heisenberg Hamiltonian in the continuum limit yieldsthe fixed point term plus a backscattering correction,H0 →H0+Hbs. The fixed point term, written in the Sugarawa form,is

H0 =2πu

3

∑

y

∫

dx [JL,y · JL,y + JR,y · JR,y ] , (49)

whereu = cos qu is the modified spinon velocity. To the orderwe work in this section, it is sufficient to takeu ≈ u = πJ/2.The backscattering correction is

Hbs =∑

y

∫

dx gbsJL,y · JR,y, (50)

wheregbs = cos qgbs < 0. Again we can approximategbs ≈gbs here.

The Ising anisotropy can also be expressed in terms of cur-rents. One must take care since it is a composite operator. Oneobtains

H1 = (1 − cos q)∑

y

∫

dx[2πu

3(Jx

L,yJxL,y + Jx

R,yJxR,y)

+gbsJxR,yJx

L,y

]

. (51)

10

The DM (see Ref. 61 for details) and external fieldHS termsadd up to

H2 + HS =∑

y

∫

dx[

d(JxR,y − Jx

L,y) − h(JzR,y + Jz

L,y)]

,

(52)whered = (3/π) sin qJ andh = 4h0 cos(q/2). We will con-sider both contributions in (52) on an equal footing. Formally,we considerq ≪ 1 andJ ′/J ≪ 1, but with qJ/J ′ arbitrary.In this limit we may approximateJ ≃ J , d ≃ 3qJ/π andh ≃ 2h0. Moreover,H1 can be dropped completely, sinceit represents next,q2, order anisotropy corrections to bothH0

and the marginal backscattering termHbs. (In principle, termsof orderq2 could be included by considering velocity shiftsand small anisotropy corrections to the backscattering cou-pling gbs. However, whenq is small enough, these higherorder corrections will not affect the outcome of the analysis inan essential way.)

Finally, the relevant interchain interactions read

H ′′ =∑

y

∫

dx {γNNy · Ny+1 + γεεyεy+1} . (53)

The analysis in Sec. III shows that coupling constantsγN,ε areof order(J ′)4/J3, and importantly,γN > 0.

B. Chiral SU(2) rotation

An unique feature of the WZNW field theory is its emer-gentchiral symmetry under independent SU(2) rotations forthe left and right moving sectors. We take advantage of this toremove the DM term in Eq. (52). Specifically, we rotate theright and left chiral spin currents about they axis byoppositeangles,+θ and−θ, respectively.

JR,y −→ RRJR,y , JL,y −→ RLJL,y, (54)

with

RR/L =

cos(θ) 0 ∓ sin(θ)0 1 0

± sin(θ) 0 cos(θ)

, (55)

whereθ = + atan(d/h). Under the rotations (54), the stag-gered magnetization and dimerization transform to

Nx,zy → Nx,z

y ,

Nyy → cos θ Ny

y + sin θ εy, (56)

εy → cos θ εy − sin θ Nyy .

Due to the chiralSU(2)R × SU(2)L symmetry , the low-energyH0 Hamiltonian is unaffected. The vector perturbation(52) simplifies toHz, where

Hz = −√

(h)2 + (d)2∑

y

∫

dx(JzL,y + Jz

R,y). (57)

Under the rotation the back-scattering term transforms into

Hbs =∑

x,y

{g1

2(J+

L,yJ−R,y + J−

L,yJ+R,y) (58)

+g2

2(J+

L,yJ+R,y + J−

L,yJ−R,y) + g4J

zL,yJ

zR,y

+g3

2(Jz

L,yJ+R,y + Jz

L,yJ−R,y − J+

L,yJzR,y − J−

L,yJzR,y)

}

,

where the couplingsgi can be expressed in terms of thebackscatteringgbs

g1 =gbs

2(1 + cos 2θ), g2 =

gbs

2(cos 2θ − 1),

(59)

g3 = gbs sin 2θ, g4 = gbs cos 2θ.

The interchain perturbationH ′′ is significantly affected aswell, and reads now

H ′′ =∑

x,y

{

cos2 θ(γεεyεy+1 + γNNyy Ny

y+1)

+ sin2 θ(γNεyεy+1 + γεNyy Ny

y+1) (60)

+(γN − γε) cos θ sin θ(εyNyy+1 + Ny

y εy+1)

+γNNxy Nx

y+1 + γNNzy Nz

y+1

}

.

C. Absorption of the field hz

The benefit of the chiral rotation is that within Abelianbosonization41,62,63 we can now absorb the magnetic field

hz :=

√

(h)2 + (d)2 by the usual shift64,65

ϕs,y → ϕs,y +1√2π

hz

ux, (61)

for all y. Note thathz = J ′√

(2s0)2 + (3qJ/πJ ′)2 is O(J ′)in the scaling limit considered here [see the discussion afterEq. (50)].

This transforms the currents in the following way

J±L,y → J±

L,ye±i hz

ux, J±

R,y → J±R,ye∓i hz

ux, (62)

JzL,y → Jz

L,y +hz

4πu, Jz

R,y → JzR,y +

hz

4πu. (63)

The latter transformation explicitly embodies the linear re-sponse of the chain magnetizationMz

y = JzR,y + Jz

L,y to thefield.

The scaling dimension1/2 fields transform according to

Nzy → cos

(

hz

ux

)

Nzy + sin

(

hz

ux

)

ǫy, (64)

ǫy → cos

(

hz

ux

)

ǫy − sin

(

hz

ux

)

Nzy , (65)

andNxy , Ny

y remain unchanged. Making this shift renders sev-eral terms in Eqs. (58,60) oscillatory, at scalesx > u/hz.

11

D. Renormalization group equations

Now we consider the effect of the various couplings. Be-cause of the explicit oscillatory factors introduced by theshiftin Eq. (61), we must consider two separate regimes of the flow.First, on scales shorter than the period of these oscillations,the oscillations themselves can be neglected, and we shouldconsider all the couplings in Eqs. (58,60). On longer scales,the oscillatory couplings may be dropped entirely. The readermay be familiar with a similar treatment of the effects of afield on a Heisenberg chain by Affleck and Oshikawa.64

1. Short scale flows

Consider first the short scale flows, i.e. the regime whenhz

u eℓa0 . 1. This means

0 ≤ ℓ . ℓ∗ = lnu

hza0∼ ln(J/J ′). (66)

We neglect completely the effect of the oscillatory factorsin-duced inHbs andH ′′. Although the form in Eq. (58) is com-plicated, the flows remain simple. This is because Eqs. (58,59)are obtain from the chiralSU(2) rotation which is a symmetryof the fixed point Hamiltonian. Thus, since the fieldhz has noeffect in this energy range, the flows remain fully SU(2) sym-metric. They are simply

dgbs

dℓ=

[gbs(ℓ)]2

2πu, (67)

and

dγN

dℓ= γN − 1

4πugbsγN , (68)

dγε

dℓ= γε +

3

4πugbsγε. (69)

One can check this simple result by directly calculating theflow equations forg1 · · · g4, and showing that the forms inEqs. (59) are preserved by these equations. This is the resultof the simple argument above.

We note that it is sufficient to work only to linear order inthe relevant couplingsγN , γε, since their initial values are oforder(J ′)4, and therefore remain small over this range ofℓ(they increase only by a factoreℓ ∼ J/J ′). To this order, theydo not feed back into the flow ofgbs. The usual solution toEq. (67) obtains:

gbs(ℓ) =gbs

(1 − gbs

2πuℓ). (70)

Sincegbs < 0, it becomes small under renormalization, andspecifically of orderu/ℓ for ℓ ≫ 1. Inserting this into theremaining equations and solving gives

γN (ℓ) =

(

1 − gbsℓ

2πu

)1/2

eℓ γN (0), (71a)

γε(ℓ) =

(

1 − gbsℓ

2πu

)−3/2

eℓ γε(0). (71b)

Evaluating this atℓ = ℓ∗ and using Eqs. (16) for the initialconditions gives

γN (ℓ∗) ∼ 1

2π4

( |gbs| ln(J/J ′)

2πu

)1/2(J ′)3

J2, (72a)

γε(ℓ∗) ∼ − 3

4π4

( |gbs| ln(J/J ′)

2πu

)−3/2(J ′)3

J2. (72b)

Note that these couplings indeed remain small at this scale.Furthermore, the staggered magnetization couplingγN is al-ready parametrically enhanced over the dimerization couplingγε, by a factor ofln2(J/J ′).

2. Long scale flows

Now we consider the renormalization on scales longer thanthe period of the oscillations induced by the field shift. Herethe SU(2) symmetry is truly broken, and the RG deviatesfrom the simple one above. We drop all oscillating terms (thisincludesg1, g3 and several of the terms inH ′′), so that theremaining perturbations to the bare HamiltonianH0 are

Hbs =∑

x,y

{

g2

2(J+

L,yJ+R,y + J−

L,yJ−R,y) + g4J

zL,yJ

zR,y

}

,

H ′′ =∑

x,y

{

γNyNyy Ny

y+1 + γNxNxy Nx

y+1 (73)

+γ+(Nzy Nz

y+1 + εyεy+1)}

,

where we have introduced the new coupling constantsg2, g4, γNx , γNy , γ+. They should be matched atℓ = ℓ∗ to thecouplings from the short-scale flows, defined in Eqs. (60,58),which implies

g2(ℓ∗) = −gbs(ℓ

∗) sin2 θ,

g4(ℓ∗) = gbs(ℓ

∗) cos 2θ,

γNx(ℓ∗) = γN (ℓ∗),

γNy(ℓ∗) = cos2 θγN (ℓ∗) + sin2 θγε(ℓ∗), (74)

γ+(ℓ∗) =[

cos2 θ γε(ℓ∗) + (1 + sin2 θ)γN (ℓ∗)

]

/2.

We now compute the RG equations for the perturbations (73)to the bare HamiltonianH0 using well established OPEs forthe non-Abelian spin currents (see for example Ref. 38). Aftersome lengthy calculations we find

dg2

dℓ= − g4g2

2πu,

dg4

dℓ= − g2

2

2πu,

dγNx

dℓ=

(

1 − 1

4πug4 +

1

2πug2

)

γNx ,

dγNy

dℓ=

(

1 − 1

4πug4 −

1

2πug2

)

γNy , (75)

dγ+

dℓ=

(

1 +1

4πug4

)

γ+.

It is important to understand how these equations lead to aninstability. The equations forg2, g4 are decoupled, and can

12

be solved separately. They have the standard form found inthe Kosterlitz-Thouless (KT) analysis.45 One recalls that thequantity

Y = g22 − g2

4 (76)

is a constant of the motion. The flows areunstableprovidedg4(ℓ

∗) < |g2(ℓ∗)|, which isalwayssatisfied except whenθ =

π/2 exactly, at which point it becomes an equality. That isfor θ ∈ [0, π/2[, the trajectories tend tog4 → −∞ andg2 →+∞. We fix Y by the initial conditions,

Y = |gbs(ℓ∗)|2

(

sin4 θ − cos2 2θ)

. (77)

Hence,Y is negative forθ ∈ [0, acos√

2/3] and positive forθ ∈ [acos

√

2/3, π/2]. Writing g22 = Y + g2

4 we can solvethe KT equations forg4. WhenY > 0, i.e., in the crossoverregime of the KT flow, we have

atan

(

g4(ℓ)√Y

)

= atan

(

gbs(ℓ∗) cos 2θ√

Y

)

−√

Y

2πu(ℓ − ℓ∗).

(78)

The couplingg4 clearly diverges when the right hand side ofthis equation reachesπ/2 plus an integer timesπ. The “time”ℓd of this divergence is

ℓd = ℓ∗ +2πu

|gbs(ℓ∗)|π/2 − atan

(

cos 2θΥ

)

Υ, (79)

where we defineΥ =√

Y /|gbs(ℓ∗)|. Usingℓ∗ ∼ ln(J/J ′)

andgbs(ℓ∗) ∼ 2πu/ℓ∗ we obtain

ℓd = ln(J/J ′)

[

1 +π/2 − atan

(

cos 2θΥ

)

Υ

]

. (80)

One can check thatℓd/ ln(J/J ′) increases monotonicallyfrom 4 whenθ = acos

√

2/3 to infinity asθ → π/2. Sim-ilarly, whenY < 0 (strong coupling regime)g4(ℓ) diverges atthe length scale

ℓd = ln(J/J ′)

[

1 + atanh

(

Υ

cos 2θ

)

(1/Υ)

]

, (81)

whereΥ =√−Y /|gbs(ℓ

∗)|. In this regimeℓd/ ln(J/J ′)

takes the value4 at θ = acos√

2/3, increases monotonicallywith decreasingθ, and diverges atθ = 0.

With the values atℓ = ℓ∗ given by Eqs. (72, 74) the relevantcouplingsγNx , γNy , andγ+ become of order one at the lengthscaleℓo ∼ 4 ln(J/J ′) (ℓo−ℓ∗ ∼ 3 ln(J/J ′)), which is alwayssmaller than the scaleℓd. The most relevant of these turns outto beγNx , as can be seen e.g. by examining the ratio

γNx(ℓ)

γNy(ℓ)=

γNx(ℓ∗)

γNy(ℓ∗)exp

[

+1

πu

∫ ℓ

ℓ∗dx g2(x)

]

, (82)

where (remember thatgbs < 0)

g2(ℓ∗) = |gbs(ℓ

∗)| sin2 θ =|gbs(ℓ

∗)|d2

d2 + h2. (83)

From this clearlyγNx becomes parametrically larger thanγNy

under renormalization (forq 6= 0). Similar analysis showsthat γNx is also enhanced relative toγ+. Thus, forq > 0,we find that the staggered magnetizationγNx = γNx dom-inates. Hence, the spins align antiferromagnetically alongthe x-direction, Sx

x, in the rotated (“comoving”) coordinateframe.

3. The ordering pattern of chain spins

Let us now infer what this means in terms of the orig-inal spins, in the fixed coordinate frame. It is necessaryto trace back the transformations of the spin operators inEqs. (43,54,61). This is straightforward but tedious. We willnot give a general expression of the relation of the microscopicspins to the continuum operators after the final transformation,which is not illuminating. Instead, we give the result for theexpectation valueof the spin operators given that, as arguedabove, in the rotated variables the ordering is very simple:

〈Nay 〉 = M(−1)yδa,x, (84)

〈JaR〉 = 〈Ja

L〉 = 〈ǫ〉 = 0. (85)

HereM 6= 0 represents the spontaneous moment, and will be-come the staggered magnetization. The(−1)y factor obtainsbecauseγNx > 0.

Now we relate the spin operators as described above to thecontinuum fields:

〈Sxx+ 1

2,y〉 =

hz

2πucos θ cos qx, (86)

〈Sy

x+ 1

2,y〉 =

hz

2πucos θ sin qx, (87)

〈Szx+ 1

2,y〉 = −(−1)x+yM. (88)

Thus, indeed the dominatingγNx-term orders the chain spinsantiferromagnetically along thez-axis perpendicular to thex− y s-spiral plain. The non-zero components of the spins inthex− y plane are induced by the local field arising from theordered interstitial moments. Note that they are ofO(J ′/J)which is much larger thanM ∼ (J ′/J)2!. Thus the staticmoments on the chains are predominantly in the the plane ofthe spiral, with a smaller staggered component in the perpen-dicular (z) direction.

The case offerromagneticorder amongst the interchainspins is obtained by settingq = 0. This impliesd = 0 sothat g2 = 0, see (83). The chains are subjected to the uniformmagnetic fieldh only. This leads toθ = 0 and, as a result,symmetry in theNx − Ny plane: γNx/γNy = 1. The chainspinsS order nearly collinearly with a ferromagnetic compo-nent of orderO(J ′/J) and smaller antiferromagnetic compo-nent of orderO[(J ′/J)2] perpendicular to the ferromagneticmoments. Note that in this case, because the interstitial spinsand the predominant moment of the chains spin are ferromag-netically ordered and hence collinear, the full magnetic orderis actuallycoplanar. For instance, if the ferromagnetic mo-ments are aligned in thex direction, the antiferromagnetic

13

component of the chain spins will be aligned along some axise in they−z plane, and all the spins are contained in the planespanned byx ande.

VII. SUMMARY AND DISCUSSION

In this work we have analyzed the ground state phase ofthe quantum Heisenberg antiferromagnet on the kagome lat-tice with spatially anisotropic exchange. We have studied thisproblem in the quasi-1D limit, where the lattice is broken upinto antiferromagnetic spin-1/2 chains that are weakly inter-acting via intermediate “dangling” spinss (see Fig. 1). Thislimit lends itself to a perturbative RG analysis in the weak ex-change interactionJ ′ using bosonization and current algebratechniques. We find that there is a natural separation of en-ergy scales: the intermediate spins order at an energy scaleoforder(J ′)2/J , at which the chains are not influenced by theinteractions among themselves and with the interstitial spins.The low energy behavior of the chains, on the other hand,is only modified atO[(J ′)4], as geometric frustration pre-vents the generation of relevant interchain interactions at thelarger scale(J ′)2/J . We have used perturbative and numeri-cal methods to determine the effective interactionsJi amongthe interstitial spinss, that arise atO[(J ′)2]. It turns out thatthe spinss order in a coplanar cycloidal spiral with wavevec-tor q parallel to the chain direction. The magnitude of thewavevector is presumably rather small (if not vanishing). Itdepends sensitively on the strength of further neighbor inter-actions among the interstitial spins, which cannot be reliablydetermined from our approach. The ordered interstitial mo-ments induce a spiral order of the chain spins ofO(J ′/J).Besides this, the chain spins exhibit a small antiferromagneticcomponent ofO[(J ′/J)2] that points along the axis perpen-dicular to the spiral plane. Thisnon-coplanarground state ofthe spatially anisotropic kagome antiferromagnet is illustratedin Fig. 2.

It is interesting to compare to recent results for this lattice inthe spatially anisotropic limit. Wanget al.36 found a coplanarferrimagnetic chirality stripe order using a semiclassical anal-ysis (see also Ref. 35). In this state, the interstitial spins areferromagnetically ordered, and the chain spins are orderedinan antiferromagnetic fashion, nearly collinearly along anaxisperpendicular to the interstitial moments, but canted slightlyin that direction. The ordering of the interstitial spins isveryclose to our findings, i.e., it corresponds to the special caseq = 0, which as we have described cannot be excluded by ourcalculations. However, even in this case the ordering patternof the chain spins is quite different from that in Ref. 36, inso-far as we find that the chain spins have a predominantferro-magneticcomponent parallel to the interstitial spins (and onlya considerably smaller antiferromagnetic component perpen-dicular to the interstitial spins), while Wanget al. obtain apredominantlyantiferromagneticordering.

Yavorskiiet al. in Ref. 35, on the other hand, use a large-Nexpansion applied to theSp(N)-symmetric generalization ofthe model. In the limitJ ′ ≪ J they find that the chains arecompletely decoupled, and the interstitial spins show some

(short range) spin-spin correlation that is compatible with aspiral ordering pattern. While the mean field treatment of thislarge-N approach seems to miss the predominant spiral or-dering of the chain spins, the spiral ordering of the interstitialspins is in agreement with the findings of this work.

Numerical studies of the spatially anisotropic kagomemodel should be very helpful in establishing the range ofspatial anisotropy of exchange interaction where the non-coplanar ordered state found in this work represents theground state of the system. We hope that our work will in-spire further investigations of this interesting problem.

Acknowledgments

We thank J.-S. Caux for sharing his numerical data on thedynamical structure factor of the Heisenberg antiferromag-netic chain, and Ashvin Vishwanath and Fa Wang for conver-sations. This work has been supported in part by the NationalScience Foundation under Grant No. PHY05-51164. A. S.thanks the Swiss NSF for its financial support. L.B. was sup-ported by a David and Lucile Packard Foundation Fellowshipand the NSF through DMR04-57440.

APPENDIX A: RG CALCULATIONS

In this appendix we present a detailed derivation of the flowequations (14) and (15) using standard OPE relations for thecontinuum fields.

1. Derivation of Eqs. (14)

The first order terms in the RG equations (14) originatefrom the rescaling of the time and space coordinates and theredefinition of the fields [see Eqs. (12)]. As already discussedin the main text, the interaction termsV1 andV2, Eqs. (13), arelocal in space, which means that their scaling dimensions haveto be compared with1, the dimensionality of timeτ . Intersti-tial spins have scaling dimension0, whileM (N ) fields are ofscaling dimension1 (1/2). Consequently, the scaling dimen-sion ofV1 is1, while that ofV2 is3/2, from which follows thatthe RG equation for the couplingγ1 does not contain a linearterm, while that forγ2 starts with(1 − 3/2)γ2 = −γ2/2, inagreement with the first line of Eqs. (14).

The second and third lines of Eqs. (14) describe the flowof the fluctuation generated interchain couplingsVch, Eq. (8),andV

(1)ch , Eq. (9), which operate in two-dimensional space-

time. As a result, the scaling dimensions of the effective in-terchain interactions have to be compared with2. The firstorder terms in the second line of Eqs. (14) are then a directconsequence of the fact that the scaling dimension of theγ∂N -interaction is3, while that of theγM -interaction is2. Sim-ilarly, positive linear terms in the last line of Eqs. (14) fol-low from thestrong relevanceof the interchain couplingsVch,Eq. (8), which have scaling dimension1.

14

The second-order corrections to the flow equations (14)are derived from contracting terms in the perturbation ex-pansion (11). In order to obtain the one-loop corrections inEqs. (14), we need to consider contributions to the secondorder term in Eq. (11) that either yield a renormalization ofthe couplingsγi or generate newinterchaininteractions. Webegin by selecting from these contributions terms that con-tain a product of two intermediate spins from the same site,(2x + 1

2 , y + 12 ), say. Theselocal contributions read

1

2T

∫

dτ1dτ2 say+ 1

2

(τ1)sby+ 1

2

(τ2) (A1)

×(

γ21

[

May (τ1) + Ma

y+1(τ1)] [

M by(τ2) + M b

y+1(τ2)]

+2γ1γ2

[

May (τ1) + Ma

y+1(τ1)]

∂x

[

N by(τ2) + N b

y+1(τ2)]

+γ22∂x

[

Nay (τ1) + Na

y+1(τ1)]

∂x

[

N by(τ2) + N b

y+1(τ2)]

)

,

where we have suppressed thex-coordinate for brevity. Eventhough the interchain spinssy+ 1

2

have no dynamics of their

own at this level, they must be time-ordered as follows46

Tsay+ 1

2

(τ1)sby+ 1

2

(τ2)

= θτ1−τ2sa(τ1)s

b(τ2) + θτ2−τ1sb(τ2)s

a(τ1)

=δab

4+

i

2(θt − θ−t)ǫ

abcsc(τ), (A2)

whereθt is the step-function,τ = (τ1 + τ2)/2 is the center-of-mass time, andt = τ1 − τ2 is the relative time.

The off-diagonal term in Eq. (A2) (which is proportionalto ǫabc) is responsible for the renormalization of the Kondo-like couplingsγ1 and γ2. As an example we consider therenormalization ofγ1, which comes from the second line inEq. (A1). Separating slow and fast degrees of freedom we canapply the OPE (4) (and a similar expression for the left cur-rents) to the product of two spin currents at nearby points [i.e.,Ma

y (τ1)Mby(τ2) → iǫabdMd

y (τ)/(2πt)]. Combining this with(A2) leads to

− γ21

4πu

∫

dτ scy+ 1

2

(τ)M cy (τ)

∫

α<|t|<bα

dt

|t| . (A3)

The integral with|t| > bα does not contribute to the renor-malization. The one-loop correction to the flow equation forγ1 can now be read off (A3) as∝ dℓγ2

1/πu, which gives usthe first equation in (14).

The renormalization ofγ2 is computed in a similar way,one only needs to realize that it comes from the third line inEq. (A1). FusingMa

y with ∂xN by via the following OPE (see

Ref. 38 for more details)

May (τ1)∂xN b

y(τ2) = limx′→x

May (x′, τ1)∂xN b

y(x, τ2)

=−δabεy(x, τ)

2π(ut + a0σt)2+

iǫabc∂xN cy(x, τ)

2π(ut + a0σt), (A4)

leads to the following one-loop correction of the flow equationfor γ2: δγ2 ∝ dℓγ1γ2/π. It is useful to note that rescaling of

space and time does not affect quadratic terms, as each of themis explicitly proportional to the RG stepdℓ, which comes fromthe shell integration of relative coordinates.

The second-order corrections to the flow equations for theinterchain couplingsγM andγ∂N follow from the diagonalterm in Eq. (A2) (which is proportional toδab). For example,applying relation (A2) to the second line of Eq. (A1) produces

γ21

4u

∫

dτMay (τ)Ma

y+1(τ)

∫

α<|t|<bα

dt.

To generate from this theγM term in Eq. (9) one needs tosum all local contributions like the one above using

∑

x ... =∫

dx/(2a0)... , as appropriate for the kagome geometry. Asa result one finds thatγM ∼ γ2

1dℓ. Similarly, the other inter-chain coupling,γ∂N in Eq. (9), can be derived starting fromthe last line in Eq. (A1).

Finally, we turn to the relevant interchain interactionsγN

andγε which are generated by fusing the marginal interactionγM and the irrelevant interactionγ∂N . Details of this pro-cedure have been described previously in Refs. 37,38. Herewe would only like to mention that the RG scheme that wehave adopted here (i.e., integrating the one-loopx-integralsover theentire space of relativex coordinates while restrict-ing the relative time integral to the shell,α < |t| < bα),which is different from the RG scheme of Refs. 37,38, doesnot modify the outcome of the calculation of Refs. 37,38 inany significant way. Namely, upon fusingMa

y (x1, τ1) with∂xN b

y(x2, τ2) on chainy (and, similarly, on chainy ± 1)one arrives at the following integral over the relative coor-dinatex = x1 − x2 and over relative timet = τ1 − τ2:I ∼

∫∞

−∞dx∫

α<|t|<bαdt(x2 + t2)−2 ∼ (b − 1)α−2. This

explains the structure of the quadratic terms in the last line ofEqs. (14).

Additionally, we note that the third line in Eq. (A1) in com-bination with the first term in Eq. (A4) results in a stronglyrelevant contribution (scaling dimension1/2)

−6γ1γ2

πa0u

∫

dτεy(x, τ). (A5)

In the two-dimensional version of the spatially anisotropickagome lattice this term cancels out, since the summationover local bow-tie crossings (that is, overx) brings in thefactor (−1)x [which originates from the staggering factor inEq. (2a)], resulting in

∫

dτdx(−1)xεy(x, τ) → 0. This ishow the symmetry (3c) manifests itself.

In contrast, for the kagomestrip of extension one in they-direction, the staggering factor(−1)x does not appear,since the bow-tie crossings are separated bytwo lattice spac-ings. As a consequence, the expression (A5) turns into∫

dτdx εy(x, τ), which implies spontaneous dimerization ofthe kagome strip, an ordering pattern that does not reducethe translational symmetry. This strongly relevant term wasmissed in previous analytical work48. Numerical studies, onthe other hand, did find dimerized ground states49.

15

2. Derivation of Eq. (15)

To derive the flow equation (15) we need to select fromthe second order term in Eq. (11) contributions that containproducts of two different intermediate spins with the samey-coordinate,y + 1

2 , say. Among these, the most importantcontributions are those, that involve products of uniform orstaggered magnetizations from the same chain

+2 × 1

2T∑

x1,x2

∫

dτ1dτ2 sa2x1+

1

2

(τ1)sb2x2+

1

2

(τ2)

×[

γ21Ma(2x1, τ1)M

b(2x2, τ2) (A6)

+γ22∂xNa(2x1, τ1)∂xN b(2x2, τ2)

]

,

where we have suppressed they-coordinate for brevity. Thefactor2 in the first line arises because there is an equal con-tribution from both they and the(y + 1)-chain. Since thespinss2x+ 1

2

at different sites commute and time-ordering ofthe continuum fieldsM andN is automatic, we can disregardthe operatorT in the above expression, provided we excludethe casex1 = x2, which was treated in the previous subsec-tion.

By splitting the integrals of Eq. (A6) into slow and fast de-grees, we can use the OPEs, Eqs. (4), to fuse the product oftwo continuum fields at nearby points. In this way, we derivefrom Eq. (A6) the one-loop renormalization to the first termof the interactionH△, Eq. (10)

∑

x1,x2

∫

dτsa2x1+ 1

2

sa2x2+

1

2

(

γ21I+

M + γ21I−M + γ2

2IN

)

,(A7a)

with the integrals

I±M =

∫

α<|t|<bα

dt1/(8π2)

[ut ± i(x1 − x2) + a0σt]2 , (A7b)

IN =

∫

α<|t|<bα

dt ∂x1∂x2

CN√

u2t2 + 4(x1 − x2)2,(A7c)

whereCN ≈ (2π)−3/2 is the amplitude of the〈NaNa〉 cor-relator. The integral over the infinitesimal interval[α, bα] inEq. (A7b) and Eq. (A7c) amounts to replacing t withαℓ. Thesubsequent rescaling turns the cut-offαℓ back into the mi-croscopic cut-offα, while x → xℓ = xe−ℓ. SimplifyingIM = I+

M + I−M and explicitly taking derivatives inIN leadsto equations (15). It is worth noting that the two contributionsIM andIN are of opposite sign, resulting in a fast decay ofinteractions between interchain spinss.

APPENDIX B: CHAIN ORDERING IN THE LIMIT OF ALARGE SPIRAL WAVEVECTOR ( q ∼ O[1])

In this appendix, we briefly discuss how the chain spinsmight order under the perturbation of a spiral magnetic fieldwith a large wavevectorq = (q, 0), whereq ∼ O[1]. Inthis case the transformation to a rotated frame as done in

Section VI A is not useful, as the generated interaction termscome with couplings of the order of the bare exchangeJ , see(47) and (48). Instead, we should remain in the original, non-rotated basis, which leaves the dominantO(J) interactionsin their simplest form. Moreover, the rapid oscillation of thespiral field, which is highly incommensurate with the “nat-ural” wavevectors0 andπ of correlations of the Heisenbergchains, ensures that its effects rapidly average out to leadingorder. However, toO(h2) we may expect non-oscillating in-teractions to be generated. In principle these can be obtainedby a perturbative analysis expanding in powers ofHS (40).

As usual, symmetry analysis is helpful in figuring out thetype of terms that can be expected from such a calculation.There are two important symmetries. First, the spiral fieldterm breaks both spin-rotational and translational symmetries,but is invariant under a translationx → x + 1 followed byasimultaneous spin rotation by the angleq about thez axis.The spiral state (39) is also invariant under spatial inversion(x → −x) followed by a change of sign for they- and z-components of the spins (Sy,z → −Sy,z). Taking into ac-count these constraints, the only possible marginal or relevantterms which may be generated in a single chain are

H(2)h =

∑

xy

∫

dx{

dz(JzyR − Jz

yL) + δgzJzyRJz

yL

}

. (B1)

We neglect here terms that are already present without the spi-ral field, and those that couple different chains as these cou-pling constants are necessarily smaller by at least one powerof J ′, since the latter cannot be generated without some bareinter-chain interactions.

A naive current algebra calculation using the continuum ap-proximation, Eq. (2a) for the spin operators, indeed producesprecisely these terms atO(h2). However, the resulting ex-plicit expressions are not reliable, as they are dominated byshort distances of orderq−1, in which the continuum limit isinappropriate. A lattice scale calculation, similar in spirit tothat in Sec. IV (yet more closely to that in Ref.40), is required.Unfortunately, it turns out that to do so requires detailed in-formation on the lattice scale properties of certainfour spincorrelation functions of an Heisenberg chain. This data is notavailable to our knowledge. Therefore, we must rely upon thesymmetry considerations alone, assuming no particular signsor magnitudes fordz andδgz above, apart from the fact thatboth are expected to be ofO(h2/J).

Fortunately, this does not result in significant ambiguity.This is largely because the nominally “relevant” DM correc-tion dz has trivial effects. Similar to the magnetic field in(61) it can be removed by a simple shift. Unlike the case inSec. III B, however, the shift does not affect the backscatteringHbs, which is written in terms of the fieldϕ, which is dual toθ.The only effect of the shift is to change the ordering momen-tum, if any, of theNx,y components. It does not determinethe nature of the ordering instability.

The anisotropy termδgz is important, as it tips the balanceof competition between different components ofN fields inthe interchain HamiltonianH ′′, Eq. (53). With the help of anOPE-based calculation similar to the one that led to (76) we

16

find

d

dℓln

γNz

γNx

=δgz

4πu. (B2)

This tells us that the type ofN -order is determined by thesign of the generatedδgz. Positiveδgz favorsNz compo-nents, leading to the non-coplanar ordering pattern found inSection VI D 3, in the small-q limit. Negative δgz, on theother hand, would preferNx,y components, without breakingthe symmetry between them. Such a state is clearlyco-planar,

and different from the one found in Section III B. Note thatthe DM termdz will affect the ordering wavevector of thisstate but not the non-coplanar one. Which of these two situ-ations obtains cannot be discriminated by our analysis, sincethe sign ofδgz is not determined. However, it is rather naturalto expect that asq is reduced fromO(1) values, one shouldobserve behavior consistent with the smallq analysis. Thiswould suggest thatδgz > 0 for a non-vanishing range ofqgreater than zero, and indeed it is possible that this is the casefor all q.

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