Properties of the spin liquid phase in the vicinity of the Néel - Spin-Spiral Lifshitz transition in frustrated magnets
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Properties of the spin liquid phase in the vicinity of the N´eel - Spin-Spiral Lifshitztransition in frustrated magnets
Yaroslav A. Kharkov, Jaan Oitmaa, and Oleg P. Sushkov School of Physics, University of New South Wales, Sydney 2052, Australia
Three decades ago Ioffe and Larkin pointed out a generic mechanism for the formation of agapped spin liquid . In the case when a classical two-dimensional (2D) frustrated Heisenberg magnetundergoes a Lifshitz transition between a collinear N´eel phase and a spin spiral phase, quantumeffects usually lead to the development of a spin-liquid phase sandwiched between the N´eel and spinspiral phases. In the present work, using field theory techniques, we study properties of this universalspin liquid phase. We examine the phase diagram near the Lifshitz point and calculate the positionsof critical points, excitation spectra, and spin-spin correlations functions. We argue that the spinliquid in the vicinity of 2D Lifshitz point (LP) is similar to the gapped Haldane phase in integer-spin1D chains. We also consider a specific example of a frustrated system with the spiral-N´eel LP, the J − J antiferromagnet on the square lattice that manifests the spin liquid behavior. We presentnumerical series expansion calculations for this model and compare results of the calculations withpredictions of the developed field theory. PACS numbers: 75.10.Jm, 75.10.Kt, 75.50.Ee, 42.50.Lc
I. INTRODUCTION
Quantum spin liquids (SL) are “quantum disordered”ground states of spin systems, in which zero-point fluctu-ations are so strong that they prevent conventional mag-netic long-range order. The main avenues towards real-izing SL phases in magnetic systems are frustration andquantum phase transitions. A particularly interestingexample of SL is realized by tuning a frustrated mag-netic system close to a Lifshitz point (LP) that sepa-rates collinear and spiral states. In the vicinity of theLifshitz transition the quantum fluctuations are stronglyenhanced, resulting in a plethora of novel intermediatequantum phases .A general argument in favour of a universal gappedSL phase near LP in two-dimensional frustrated Heisen-berg antiferromagnets (AF) was first proposed by Ioffeand Larkin . They showed that in the proximity of theLP quantum fluctuations destroy long-range spin corre-lations and create a region in the phase diagram witha finite magnetic correlation length. Subsequent studiesfound evidence for SL phases in various two-dimensionalsystems near the LP, including Heisenberg models onsquare and honeycomb lattices with second and thirdnearest neighbor antiferromagnetic couplings . How-ever, the universality of the SL phase near LP, its ubiqui-tous properties, and the relation of the general argumentto specific Heisenberg models has not previously beenaddressed.In the present paper we revisit the Ioffe-Larkin scenarioand consider a field theory for a quantum Lifshitz transi-tion between collinear and spiral phases in D = 2+1. Dis-regarding microscopic properties of specific lattice mod-els we focus on the generic infrared physics at the LP.We develop a field-theoretic description of the O (3) Lif-shitz point based on the extended nonlinear sigma model.The nonlinear sigma model provides a unifying theoret- ical framework that allows us to analyze the phase di-agram, calculate positions of critical points, excitationspectra, and static spin-spin correlations functions. Wedemonstrate universal scalings of observables (gaps, po-sition of critical points, etc) in terms of the dimensionlessSL gap at the LP, δ , and show that the correlation lengthin the SL phase scales as ξ ∼ / √ δ . We also argue thatthe LP spin liquid has a similarity to the gapped Haldanephase in integer-spin 1D chains. However, for the 2DSL there is no significant difference between the integerand half-integer spin cases.A particular example of a system that has a N´eel-spiral LP and hence manifests the spin liquid behavioris the frustrated antiferromagnetic J − J − J Heisen-berg model on the square lattice with the second andthird nearest neighbour couplings as well as it’s simpli-fied version, the J − J model. We perform numericalseries expansion calculations for the J − J model andcompare results of the calculations with predictions ofthe developed field theory.The structure of the paper is as follows. In Sec. II weintroduce the effective field theory describing the N´eelto Spin Spiral Lifshitz point. Section III addresses thequantum LP, quantum fluctuations, and the criterion forquantum ‘melting’. Next, in Sec. IV we calculate thespin-wave gap and positions of critical points. SectionV addresses the static spin-spin correlator in the spinliquid phase. In Sec. VI we describe our numerical seriescalculations for the J − J model with spin S = 1 / S = 1 and compare results of these calculations withpredictions of the field theory. Finally our conclusionsare presented in Sec. VII. II. EFFECTIVE FIELD THEORY
We start with the following O (3) symmetric La-grangian describing a transition from the N´eel to a spiralphase in two dimensional antiferromagnets: L = χ ⊥ ∂ t n µ ) − n µ K ( ∂ i ) n µ , ( n µ ) = 1 . (1)Here χ ⊥ is the transverse magnetic susceptibility, n µ is aunit length vector with N = 3 components correspondingto the staggered magnetization, ∂ i are the spatial gradi-ents. The general form of the “elastic energy” operator K ( ∂ i ) in inversion symmetric systems reads K ( ∂ i ) = − ρ ( ∂ i ) + b ∂ x + ∂ y ) + b ∂ x ∂ y + O ( ∂ i ) , (2)where we assume that the n -field is sufficiently smooth.The spin stiffness ρ is the tuning parameter that drivesthe system across the Lifshitz transition. The spin stiff-ness is positive in the N´eel phase, negative in the spiralphase and vanishes at the Lifshitz point. The b -termscontaining higher order spatial derivatives are necessaryfor stabilization of spiral order at negative ρ , and we willassume that b , >
0. While the kinematic form of theLagrangian (1) is dictated by global symmetries of thesystem, a formal derivation starting from a frustratedHeisenberg model can be found e.g. in Ref. Note thatin Lagrangian (1) we do not take into account topologi-cal terms. We will discuss their possible role later in thetext.The Lagrangian (1) is relevant to a number of mod-els and systems mentioned in the Introduction. Herewe would like to mention another example motivatedby rare-earth manganite materials (Tb,La,Dy)MnO (seeRef. ). These materials have a layered structure withthe individual ferromagnetic layers coupled antiferromag-netically. Due to the antiferromagnetic interlayer cou-pling the dynamics of the system is described by thesecond-order time derivative as in usual antiferromag-nets in agreement with Eq. (1). Within each planethere are ferromagnetic nearest neighbour and antifer-romagnetic second nearest neighbour Heisenberg interac-tions leading to an inplane frustration. These compoundscould be tuned to the N´eel-Spin-Spiral LP by performingchemical substitution. Of course real materials are three-dimensional and contain many planes, however thin filmscan manifest some physics considered here.In the AF phase of (1), ρ >
0, the rotational symme-try is spontaneously broken and the N´eel vector has anonzero expectation value, e.g. is directed along the z axis h n i = e z . In the spin spiral phase, with ρ <
0, thereis an incommensurate ordering n ( r ) = e cos( Qr ) + e sin( Qr ) , (3)where e , are orthogonal unit vectors and Q is thepitch of the spiral. For b ≤ b the spiral wave vec-tor is directed along x or y : Q = ( ± Q, , (0 , ± Q ),where Q = | ρ | /b . In the opposite case b > b thewave vector is directed along the main diagonals: Q = √ ( ± Q, ± Q ) , √ ( ± Q, ∓ Q ), where Q = 2 | ρ | / ( b + b ).The relation between the coefficients b and b depends Lifshitz point b)c)
Classical Lifshitz point a) Spin-Spiral Neel
Figure 1: Schematic phase diagram in the vicinity of the Lif-shitz transition between collinear antiferromagnetic and spi-ral states: a) classical Lifshitz transition, b) quantum phasediagram; strong quantum fluctuations in the vicinity of theLifshitz point result in the intermediate spin liquid phase. c)Excitation energy ω q in the spin liquid phase below and aboveLP. on the specific choice of the lattice model. In the“isotropic” case, b = b , the system has additionalrotational degeneracy in the momentum space due tothe arbitrary orientation of wave vector Q . The addi-tional degeneracy can destabilize spiral states and resultin quantum spin liquid states that have been predictedfor 3D antiferromagnets. In the present paper we willstay away from this special critical point. The classicalphase diagram is shown schematically in Fig.1a.We would like to make a comment regarding La-grangian (1). Parameters of any field theory dependon the momentum and energy scales that is describedby renormalization group procedure. We assume thatparameters in (1),(2) are fixed at the ultraviolet cutoffΛ ≈
1, where unity corresponds to the inverse latticespacing. Quantum fluctuations at scales larger than Λbut smaller than the boundary of magnetic Brillouin zonelead to a renormalization of the parameters ρ → ρ ren , b , → b ren , , . . . . Therefore, the values of the parametersin (1),(2) can be different from those naively derived us-ing spin wave theory. As was pointed out by Ioffe andLarkin this renormalization is especially relevant for thespin stiffness. The correction to the spin stiffness arisesdue to the the b -terms in (2). The easiest way to un-derstand the correction is to consider the N´eel phaseand decompose the order parameter into two transversecomponents and a longitudinal component n = ( π , n z ) , n z = p − π ≈ − π / . (4)Hence the following contribution from the b -term arises ∂ n z ∂ n z ∼ b ( ∂ π )( ∂ π ) . (5)The field π has fluctuations with momenta smaller thanΛ, π < , and fluctuations with momenta larger than Λ, π > , π = π < + π > . Substitution in (5) and averagingover high energy fluctuations gives b ( ∂ π )( ∂ π ) → b ( ∂ π < ) h ( ∂ π > ) i = δρ Λ ( ∂ π < ) . (6)Note, when averaging ( ∂ π ) × ( ∂ π ) each multipliermust contain the high ( π > ) and the low ( π < ) energycomponents. The terms with one multiplier containingonly the high energy and another only the low energycomponents give rise to a total derivative contributionsto the Lagrangian and can be neglected. Equation (6)demonstrates a positive correction to the spin stiffness.Therefore quantum fluctuations always extend the N´eelphase compared to the prediction of spin-wave theorythat is indicated in Panel b of Fig. 1. The Lifshitz pointin the quantum case is shifted to the left compared to theLifshitz point in the classical case. In the quantum casethe Lifshitz point is “buried” in the spin liquid phase.Nevertheless, it is unambiguously defined as we discussin the following Sections. III. QUANTUM LIFSHITZ POINT: THE PHASEDIAGRAM AND THE SPIN LIQUID GAP
Quantum fluctuations destroy the classical N´eel tospin-spiral Lifshitz transition . Let us calculate localstaggered magnetization n z when approaching the LPfrom the N´eel phase. Representing the staggered magne-tization as h n z i ≈ − h π i , we obtain h π i ≈ ( N − X q Z idω (2 π ) 1 χ ⊥ ω − K ( q ) + i
0= ( N − Z d q (2 π ) /χ ⊥ ω q , (7)where ω q = χ − / ⊥ q ρq + b / q x + q y ) + b q x q y . In thevicinity of the LP, ρ →
0, the integral (7) is logarithmi-cally divergent, h π i ∝ ln (cid:16) Λ √ ρ (cid:17) , where Λ is the ultravio-let momentum cutoff. Hence at some critical value of thespin stiffness ρ = ρ cN the staggered magnetization h n z i vanishes, indicating a transition to the spin liquid phase.In the spin liquid phase, ρ < ρ cN , a gap ∆ must open toregularize the integral in Eq. (7) ω q → q ω q + ∆ = q ∆ + χ − ⊥ [ ρq + b / q x + q y ) + b q x q y ] . (8)Opening of the gap indicates an existence of a spin liq-uid phase at which the long range AF order is lost andthe order parameter correlations are exponentially de-caying. Importantly, this is a generic gapped spin liquidoriginating from long range fluctuations and is unrelated to a spin-dimer ordering. The SL gap is zero, ∆ = 0,at the critical point ρ cN and the gap increases when weproceed deeper into the spin liquid phase. The SL phasestretches across a finite window [ ρ cS , ρ cN ] in the vicinityof the LP, as depicted in Fig. 1b.The elementary spin excitations in the AF phase aretwo gapless Goldstone modes - transverse spin-waves anda massive longitudinal (’Higgs’) mode. Due to the unitlength constraint ( n = 1) the Higgs mode has a verylarge energy and can be disregarded. In the spiral phasethere are three Goldstone modes: a sliding mode and twoout of plane excitations. These three modes correspondto the three Euler angles defining the orientation of the( e , e , e ) triad, where e = [ e × e ]. .The excitation modes (8) in the SL phase are three-fold degenerate due to O (3) rotational invariance of themodel. Above the LP ( ρ >
0) the minimum of dispersionis located at q = 0, whereas below the LP ( ρ <
0) the dis-persion has four degenerate minima at the ’spiral’ wavevectors q = Q . The evolution of the dispersion acrossthe LP is schematically shown in Fig. 1c. The change ofthe shape of the dispersion indicates the Lifshitz point.The location of this critical point ρ cN can be found byimposing the condition h n z i →
0, which naively providesthe following criterion for the transverse spin fluctuations h π i c ≈
2. This critical value for h π i is largely overesti-mated and it is not consistent with the unit length con-straint. One can find a more accurate value of h π i c byaccounting for the next order terms in the Taylor seriesexpansion of n z = √ − π (see Appendix A), or alterna-tively by using the 1 /N expansion for O ( N ) theory. The1 /N expansion has been extensively applied to describequantum antiferromagnets. For the most relevant exam-ples see Refs. . In the 1 /N expansion approach welift the hard constraint n = 1 by introducing a Lagrangemultiplier L → L − λ ( n − . (9)After integrating out the n field in the new Lagrangian(9), we obtain an effective Lagrangian depending only onthe auxiliary field λ : L λ = N tr ln( − χ ⊥ ∂ tt − K ( q ) − λ ) + λ. (10)We can find the saddle point in the Lagrangian L λ bycalculating the variational derivative in (10) with respectto λ and regarding λ as a constant, λ = χ ⊥ ∆ : N X q Z idω (2 π ) 1 χ ⊥ ( ω − ∆ ) − K ( q ) = 1 . (11)The Lagrange multiplier in Eq. (11) has the mean-ing of the spin gap. Equation (11) determines the evo-lution of the gap ∆( ρ ) with the spin stiffness in theSL phase. Comparing Eq. (11) with Eq. (7) we con-clude that at the boundary between SL and AF phases h π i c = ( N − /N = 2 /
3. This criterion is quite naturalfor the O (3) symmetric quantum critical point separatingN´eel and SL states. Nevertheless, this criterion underes-timates h π i c . One can see this from the example of the S = 1 / h n z i = 2 h S z i = 1 − Z MBZ d q (2 π ) q − γ q − , (12)where γ q = (cos q x + cos q y ), and integration is per-formed over the magnetic Brillouin zone. In the limit q < χ ⊥ = 1 / J and ω q /J ≈ √ q , where J is the Heisen-berg AF coupling. Integration over q in (12) gives awell known result h n z i ≈ × .
305 which correspondsto h π i ≈ .
78 in the equation h n z i ≈ − h π i . Theintegration in the corresponding long-wavelength approx-imation (7) with N = 3, χ ⊥ = 1 / J , ω q ≈ √ Jq and theultraviolet cutoff Λ = 1 gives a close value h π i ≈ . h π i c ≈ . (13)Equation (13) is an analogue of the Lindemann crite-rion for quantum melting of long range magnetic order in2D quantum magnets. Our approach implicitly violatesrotational invariance, but it allows us to calculate ap-proximately the positions of critical points and the valueof the spin liquid gap.The spin liquid gap ∆ is determined by Eqs. (7) and(8) from the condition h π i = h π i c ≈
1. At ρ > ρ <
0, the physical gap correspondsto the excitation energy at the “spiral” wave vector Q :∆ ph = min ω q = q ∆ + χ ⊥ K ( Q ), see Fig. 1c. This gapis closed at the spin-spiral-SL critical point. Therefore,the position of this critical point ρ cS is determined fromthe following two equations ( P q< Λ R idω (2 π ) 1 χ ⊥ ( ω − ∆ ) − K ( q )+ i = 1 , ∆ ph = ∆ + χ ⊥ K ( Q ) = 0 . (14)At ρ < ρ cS , the magnon Green’s function acquires a poleat imaginary frequency ω = ± i p | ∆ + K ( Q ) /χ ⊥ | . Thisis the indication of an instability of the SL phase towardscondensation of a static spiral with the wave vector Q .It is instructive to draw an analogy between the SLphysics at 2D Lifshitz point and the one-dimensionalHaldane spin chain. A condition similar to (11) deter-mines the value of the Haldane gap. Indeed, the integerspin S Heisenberg model in the continuous limit can bemapped to the O (3) relativistic nonlinear sigma modelin D = 1 + 1. The model parameters are the speedof the magnon, c = p ρ/χ ⊥ = 2 JS , and the transverse magnetic susceptibility, χ ⊥ = 1 / J ( J is the Heisenbergcoupling constant). Proceeding by analogy with (7) wefind the fluctuations of the spin in the Haldane model h π i c = 2 Λ Z dq π χ ⊥ p c q + ∆ ≈ πcχ ⊥ ln c Λ∆ , (15)As we already discussed, the ultraviolet cutoff is Λ ≈
1. The logarithmically divergent h π i in the Haldanemodel is analogous to the log-divergence in (7) at the LP.Numerical values of the Haldane gaps for S = 1 and S =2 are known from DMRG calculations: see e.g. Ref. ,∆ S =1 /J ≈ .
41, ∆ S =2 /J ≈ .
08. Taking these valuesof the gap Eq.(15) we obtain the following critical valuesof fluctuations, h π i c ≈ . S = 1) and h π i c ≈ . S = 2), which are smaller than (13). We believe thatthe difference is due to different dimensionality. WhileDMRG is more reliable it is interesting to note that therenormalization group analysis for the Haldane chaingives h π i c = 1.The differences in the values of h π i c is not crucialwhen making comparisons between 1D and 2D systems.However, it is well known that properties of the spinchains with half-integer and integer spins are very differ-ent. The gapped SL phase in 1D appears only in the inte-ger spin chains, while in contrast the excitations of half-integer spin chains are gapless spinons in agreement withthe Lieb-Shultz-Mattis theorem. We believe that the2D spin liquid in the vicinity of LP point is generic andindependent of the spin value. The Lieb-Shultz-Mattistheorem states that in systems with half-integer spin perunit lattice cell and full rotational SU (2) symmetry theexcitations are gapless or otherwise the ground state ofthe system is degenerate. The theorem was initially for-mulated for D = 1 + 1 systems and later generalized forhigher spatial dimensions . Technically in D = 1 + 1the dramatic difference between integer and half integerspin is due to the topological Berry phase term which isnot included in the Lagrangian (1). Topological effectsin D = 2 + 1 correspond to skyrmions or merons. In principle topological configurations become moreimportant when approaching the Lifshitz point. How-ever such topological solutions are unstable within themodel (1). Using scaling arguments one can see that dueto the fourth spatial derivative term in the Lagrangian(1) the energy of localized skyrmions at LP behaves as ∼ b , /R , where R is the skyrmion radius. Thereforeany localized skyrmions energetically prefer to have largesize R → ∞ and only contribute to the boundary terms.Although the topological solutions might play a role toreconcile with the Lieb-Shultz-Mattis theorem, these con-figurations are statistically irrelevant in the bulk. IV. POSITIONS OF N´EEL-SPIN LIQUID ANDSPIN-SPIRAL-SPIN LIQUID CRITICAL POINTS
In order to make our calculations more specific andhaving in mind comparison with the J − J model, inthis Section we set b = 0. It is convenient to intro-duce dimensionless spin stiffness and dimensionless gapparameters ¯ ρ = 2 ρb , δ = r χ ⊥ b ∆ . (16)At negative ρ the spiral wave vector is directed alongthe main diagonals Q = √ ( Q, ± Q ), Q = | ¯ ρ | . (17)As we already discussed in Section III the condition ofcriticality reads h π i c ≈ ≈ √ π ) √ χ ⊥ b Z d q q ¯ ρq + q x + q y + δ . (18)First, we determine the gap exactly at the LP, δ = δ ( ρ = 0). For δ ≪ δ = 1 . e − √ πζ √ χ ⊥ b . (19)The constant ζ in the exponent is given by the angu-lar part of the q -integral ζ = π K (cid:16) h − b b i(cid:17) , where K ( m ) = R π/ dφ √ − m sin φ is the complete elliptic in-tegral. In the specific case under consideration, b = 0, ζ = π K (1 / ≈ .
18. The numerical prefactor A = 1 . δ ≪ δ ≤ . − . ρ cN we evaluate the integral in (18) at δ ≪ ¯ ρ ≪ π Z d q q ¯ ρq + q x + q y + δ ≈ ζ (cid:18) . ¯ ρ (cid:19) − δ ¯ ρ (20)The condition δ = 0 gives the position of the N´eel-SLcritical point ¯ ρ cN :¯ ρ cN ≈ . e − √ πζ √ χ ⊥ b ≈ . δ . (21)According to (20) in the vicinity of the N´eel-SL criticalpoint, ¯ ρ < ¯ ρ cN , the gap grows linearly as δ ≈ . ρ cN − ¯ ρ ), that corresponds to a mean-field prediction.The spin stiffness ρ cN at the transition point from theN´eel phase to the spin liquid phase is small but still finite.Therefore, we believe that the transition belongs to thestandard O (3) universality class, the same as that in thebilayer quantum antiferromagnet, see e.g. Ref. The − − − = 0.02= 0.04= 0.06= 0.20= 0.70 Figure 2: Dimensionless spin liquid gap versus spin stiffnessfor different values of δ . correct critical index for O (3) transition is ν ≈ .
7, whichimplies δ ∝ (¯ ρ cN − ¯ ρ ) ν .On the side of negative spin stiffness, ¯ ρ cS < ¯ ρ <
0, thedimensionless physical gap reads α = r χ ⊥ b ∆ ph = p δ − ¯ ρ / . (22)The condition α = 0 determines the position of thespin-spiral to SL critical point ρ cS . Calculating the inte-gral in (18) at α ≪ Q ≪ π Z d q q Q / − Q q + q x + q y + α ≈ ζ ln (cid:18) . Q (cid:19) − αQ . (23)The condition α = 0 gives the position of the criticalpoint ¯ ρ cS : ¯ ρ cS = − Q ≈ − δ . (24)The gap in the vicinity of this critical point is α =0 . ρ − ¯ ρ cS ). This is a mean-field result and we believethat the transition at ρ cS does not belong to a standarduniversality class.The dimensionless gap found by numerical solution ofEq. (18) for different values of δ in the entire SL region ρ cS < ρ < ρ cN is presented in Fig. 2. From this figurewe conclude that asymptotic solutions given by Eqs. (21)and (24) become valid only at sufficiently small values of δ (i.e large values of S): Eq. (21) is valid at δ . . δ . . ρ cS and ρ cN evident from Fig.2 is due to stronger quantum fluctuations in the spiral( ρ <
0) region compared to the ρ > ρ cS is to approachthe spiral-SL critical point from the spiral phase and findthe condition when quantum fluctuations melt the spi-ral. The fluctuations of spiral consist of the out-of-plane h ( r , t ) and in-plane modes φ ( r , t ), can be parametrizedin the form ~n = ( p − h cos( Q · r + φ ) , p − h sin( Q · r + φ ) , h ) . (25)The total quantum fluctuation orthogonal to the spinalignment in the spiral state reads h π i = h φ i + h h i , (26) h φ i = 1(4 π ) √ χ ⊥ b Z d q q Q q + q x + q y , h h i = 1(4 π ) √ χ ⊥ b Z d q q Q / − Q q + q x + q y . The denominators in the integrals for h φ i and h h i in(26) represent the dispersions for the Nambu-Goldstoneexcitations: the sliding mode and the out of plane mode,see details in Appendix B. Evaluating the integrals withlogarithmic accuracy, we obtain h π i ≈ π ) √ √ χ ⊥ b ζ ln (cid:18) . Q (cid:19) . (27)Now, applying the same criterion for the critical point, h π i c ≈
1, we find the critical ¯ ρ cS ¯ ρ cS ≈ − . e − √ πζ √ χ ⊥ b ≈ − δ . (28)The prefactor in (28) is significantly smaller then theprefactor in Eq.(24). This emphasizes the fact that ourcalculation is only approximate. Pragmatically this un-certainty is not very significant. We already pointed outthat Eq. (24) is valid only for extremely small gaps, δ . .
02. At larger values of δ the position of thecritical point ρ cS is different from (24), see Fig. 2. Nu-merical evaluation of (26) combined with the criticalitycondition (13) gives the following locations of the criticalpoints ρ cS : ¯ ρ cS /δ = − . δ = 0 .
06; ¯ ρ cS /δ = − . δ = 0 .
2; ¯ ρ cS /δ = − . δ = 0 .
7. Comparing thesevalues with positions of the critical point that follow fromFig.2 we conclude that, for the practically interesting case δ (cid:19) .
15, both methods give close positions of the criti-cal point.As was mentioned in Sec. II in the presence of inplanerotational symmetry b = b (e.g. frustrated Heisen-berg model on the hexagonal lattice), quantum fluctua-tions become especially strong. In fact, when approach-ing the critical point ρ cS the integral R q √ ∆ + K ( q ) ∝ R q √ α +( q − Q ) is logarithmically divergent for α → q = Q . It implies that one has to keep higher orderterms O ( q i ) in the expansion (2) K ( q ) = ρq + b q + c ( q x + q y ) + d ( q x q y + q x q y ) (29)which break the symmetry with respect to spatial rota-tions in the { xy } plane and remove the degeneracy with respect to the choice of the direction of Q . After ac-counting for the higher order anisotropic terms ∝ O ( q i )the integral for h π i becomes convergent at | q | = Q andthe value ρ cS is well defined. V. SPIN-SPIN CORRELATION FUNCTION
Spin-spin correlations of a standard tool to analyzequantum critical properties of a magnetic system. In theSL phase the correlator provides an essential informationabout the properties of the ground state. The equal timetwo-point spin-spin correlation function reads C ( r ) = h n α ( r ) n α (0) i = 1 + 2[ R ( r ) − R (0)] + . . . , (30)where h π α ( r ) π β (0) i = δ αβ R ( r ) and indices α, β refer onlyto the x and y spin components. The two-point correlatoris normalized such that C (0) = h n α i = 1. In the SL phasethe correlation function should vanish at large distances C ( r → ∞ ) → R ( r → ∞ ) →
0. These conditionsare consistent with the “melting criterion” in Eq.(13) ifwe truncate the asymptotic expansion in Eq.(30) keepingonly the terms explicitly presented there.The h π ( r ) π (0) i correlation function in the SL phasereads R ( r ) = Z idωd q (2 π ) e i qr χ ⊥ ( ω − ∆ ) − K ( q ) + i . (31)Calculating (31) and substituting the result in Eq. (30),we obtain the two-point spin-spin correlation function C ( r ); the numerical results are plotted in Fig (3). Sim-ilar to the previous Section these plots correspond tothe case b = 0. Therefore, the correlator is somewhatanisotropic. There are two points to note, one is physi-cal and another is technical. (i) The correlation lengthscales as one over the square root of the gap, ξ ∝ / √ δ ,instead of the standard relation, ξ ∝ /δ . (ii) When in-tegrating in Eq.(31) we use the soft ultraviolet cutoff bymultiplying the integrand by e − q / (2Λ ) . The soft cutoffallows us to avoid nonphysical oscillations in R ( r ) due tothe Gibbs phenomenon. The Gibbs phenomenon resultsin spurious oscillations, which always exist for a sharpcutoff and are well known in Fourier analysis.The asymptotic behaviour of the correlation function R ( r → ∞ ) in the spin liquid phase at ρ = 0 can be analyt-ically obtained in the simplified isotropic approximation( b = b ): R ( r ) ∼ e − r q δ r cos r r δ − π ! , (32)Using Eq. (32) we deduce the spin-spin correlationlength ξ = q δ . In the case of negative spin-stiffness( ρ cS < ρ <
0) the correlation function R ( r ) becomesoscillating, see Fig. (3). In the vicinity of the critical .2 (cid:0) . (cid:1)(cid:2) .2 (cid:3) .4 (cid:4) .6 (cid:5) .81. (cid:6) = -1 (cid:7) . (cid:8) = (cid:9) . (cid:10) = 1.5 = -1 a)b)Figure 3: Static spin-spin correlation function C ( r ) in the spinliquid phase for positive and negative spin stiffness ( b = 0, δ ≈ . r is directed along a) the prin-cipal lattice axes ( x or y ), b) r is along the diagonal direction. point ρ cN the correlations decay as R ( r ) = 12 π √ χ ⊥ b I (cid:18) r √ ¯ ρ cN (cid:19) K (cid:18) r √ ¯ ρ cN (cid:19) ∼ r →∞ r . (33)Formula (33) is consistent with the well known ∝ /r decay of correlations of transverse spin components inthe N´eel phase (see e.g. Ref. ). We stress that the“isotropic approximation”, b = b , provides a qualita-tive and quantitative description of the correlation func-tion C ( r ) only away from the critical point ρ cS . In thevicinity of the point ρ cS the isotropic model (1) becomesunstable, see comments to Eq. (29).Now we would like to make a comparison between O (3)and O (2) quantum Lifshitz transitions. The O (2) versionof Lagrangian (1) describes the XY frustrated Heisenbergantiferromagnet in the continuous limit. The physics inthe O (2) model is quite different from the O (3) modeland the Ioffe-Larkin argument is inapplicable in this case.The O (2) Lagrangian can be mapped to the scalar Lif-shitz model described by a polar angle θ : n x + in y = e iθ .This model has an exact solution for the correlationfunction C ( r ) at the LP: C ( r ) decays algebraically atthe LP in contrast to the non-vanishing correlations at r → ∞ in long-range ordered N´eel or spin-spiral phase.Therefore we conclude that there exist a finite region inthe vicinity of the LP with algebraically decaying cor-relations. The region with algebraic spin correlations insome extent is analogous to the SL phase in the O (3) model addressed in the present paper. VI. J − J MODEL ON THE SQUARE LATTICE
In the present Section we compare the field theory pre-dictions with results of numerical calculations for the an-tiferromagnetic J − J Heisenberg model on the squarelattice. Frustrated J − J and J − J − J models havebeen discussed in numerous studies (see e.g. Refs. ):some references are also presented in the Introduction.In the classical limit both models exhibit the spin spi-ral state at a sufficiently large frustration. Quantumversions of the models show a magnetically disorderedstate at a sufficiently large frustration. Classically the J − J model at J /J = 1 / J − J modeldifferent from that considered in the present work. Onthe other hand if we set J = 0 and consider only the J frustration then classically there is a Lifshitz point witha transition to the spin-spiral at J = J /
4, and the spin-stripe state has much higher energy than the spin-spiraland the N´eel states. Therefore the J − J model is agood testing ground for the generic theory of a “soft”Lifshitz transition developed in the present work. TheHamiltonian of the J − J model reads H = J X
4. As we already pointedout in Section II quantum fluctuations must shift the LPtowards larger values J /J > / χ ⊥ = 18 J . (35)The elasticity parameters of the Lagrangian can be foundin two ways. (i) The first way is a straightforward expan-sion of the classical elastic energy at small wave number q , that gives ρ = S ( J − J ) ,b = S (16 J − J )12 ,b = 0 . (36)(ii) An alternative way is to calculate the magnon dis-persion in the N´eel phase using the standard spin-wave J /J −0.75−0.70−0.65−0.60−0.55−0.50 E g s / J S=1/2 J /J −2.6−2.5−2.4−2.3−2.2−2.1−2.0−1.9−1.8−1.7 E g s / J S=1 a) b)Figure 4: J − J model ground state energy in the N´eeland in the Spin Spiral states for a) S = 1 / S = 1calculated by numerical series expansion method. theory. The dispersion reads : ω q = 4 SJ s(cid:18) − J J (1 − γ q ) (cid:19) − γ q , (37) γ q = 12 (cos q x + cos q y ) ,γ q = 12 (cos 2 q x + cos 2 q y ) . (38)Expanding ω q at small q and comparing the results withEq.(8) (at ∆ = 0) we find ρ = S ( J − J ) ,b = 4 J S " −
548 + 23 (cid:18) J J (cid:19) + (cid:18) J J (cid:19) ,b = 4 J S " −
18 + 2 (cid:18) J J (cid:19) . (39)Expressions for b and b in Eqs.(36) and (39) do notcoincide. At the LP, J = J /
4, both Eqs. give b = 0,however, values of b are different, Eq.(36) gives b =0 . S J while Eq.(39) gives b = 0 . S J . Of coursethe spin-wave theory value is more reliable.We have performed extensive series calculations bothin the N´eel phase and the spin-spiral phase. Unfortu-nately the series expansion method does not allow to as-sess properties of the spin liquid phase directly. However,it allows to estimate the range of parameters where thespin liquid exists which can be compared with predic-tions of the field theory. In the N´eel phase the seriesstarts from the simple Ising antiferomagnetic state. Inthe spiral phase the calculation is more tricky. We firstimpose a classical diagonal spiral with some wave vector Q and find the total energy of this state E ( Q ). This in-cludes the classical energy and the quantum correctionscalculated by means of series expansions. We performthis calculations for many values of Q and then find nu-merically the minimum of E ( Q ). Such procedure givesus the ground state energy E gs and the physical wavevector Q . The ground state energy E gs is plotted in Fig.4 versus J . The plot of the wave vector squared, Q ,versus J is presented in Fig. 5. From the field theorydescription we expect that near the LP the wave vector J /J Q S = 1S = 1/2
Figure 5: Spiral wave vector (squared) Q versus J . Dotsshow results of numerical series expansion. Blue (red) dotscorrespond S = 1 / S = 1). Dashed lines show fits of databy cubic polynomials, Q = a ( J − J LP ) + a ( J − J LP ) + a ( J − J LP ) . behaves as Q = 2 | ρ | b = 8 S b ( J − J LP ) . (40)Therefore, from Fig. 5 we determine positions of Lifshizpoints and, using Eq.(40) we find the values of the elasticconstant b at the LP: S = 1 / J LP ≈ . J , b /S ≈ . J ,S = 1 : J LP ≈ . J , b /S ≈ . J . (41)As expected, (see the very end of Section II), quantumfluctuations extend the N´eel phase in relation to the clas-sical LP J LP = 0 . J . Values of the elastic constant b are larger than that given by Eq.(36) and smaller thanthat given by Eq.(39).We have also calculated the magnon dispersion in theN´eel phase. The series expansion becomes erratic at J > . J and the errorbars in the calculations of ω q grow very quickly. The dispersion at J = 0 . J is shownin Fig. 6. We see that the shape of the dispersion is some-what different from the prediction of the spin-wave the-ory (37). On the the other hand the total bandwidth isconsistent with the spin-wave theory. The situation is dif-ferent in the case of a simple Heisenberg model ( J = 0),when the shape of magnon dispersion is consistent withthe spin-wave theory but the total bandwidth is about20% larger compared to the spin-wave theory value.We also compute the static on-site magnetization inthe N´eel and spiral phases. The magnetization vanishesat J cN and J cS critical points. We already pointed outthat the N´eel-SL transition at J cN belongs to the O (3)universality class. Therefore, we expect scaling h S z i ∝| J − J cN | β when approaching the critical point from theN´eel phase, here β = ( D − η ) ν/ ≈ ν/ ≈ .
35. Dueto this reason in Fig. 7 we show series expansion resultsfor the static on-site magnetization cubed. From here welocate the critical points. S = 1 / J cN ≈ . J , J cS ≈ . J ,S = 1 : J cN ≈ J cS ≈ . J . (42) (0, 0) (0, π) (π/2, π/2) (0, 0) q ω q / J Figure 6: Magnon dispersion ω q for J − J model on thesquare lattice in the N´eel phase at J /J = 0 .
2. Red circlescorrespond to the series expansion results, black line is thelinear spin-wave dispersion in Eq. (37). Figure 7: Average onsite magnetization cubed. Blue squares(red circles) show series expansion results for S = 1 / S = 1),solid lines are guides for the eye. Our result for the SL range ∆ J in the case S = 1 / , that suggest the SLphase at 0 . ≤ J /J ≤ .
8. However, our predictions arereasonably close to the exact diagonalization results ,suggesting the gapped SL phase for 0 . ≤ J /J ≤ . J cS criticalpoint is smaller than the O (3) value, M ∝ ( J − J cS ) β , β ∼ . χ ⊥ and b . Hence, according to Eqs.(19) and(16) values of the gap at the LP are S = 1 / δ ≈ . , ∆ ≈ . J ,S = 1 : δ ≈ . , ∆ ≈ . J . (43)Formally the field-theoretical prediction (16) is derivedwithin logarithmic accuracy and valid at δ ≪
1, whilethese values, especially that at S = 1 /
2, are not small.Nevertheless, we believe that Eq.(43) gives a reasonable estimate of the gaps. Knowing the dimensionless gapsand using Fig. 2 we can deduce the window δ ¯ ρ occupiedby the spin liquid phase. Combining this with Eq.(40)we find the spin liquid window ∆ J = | J cS − J cN | thatfollows from the field theory,∆ J /J ≈ . , ( S = 1 / , ∆ J /J ≈ . , ( S = 1) . (44)These values while being slightly larger are in a rea-sonable agreement with the SL phase windows followingfrom series expansion data in Fig. 7.In conclusion of this Section we would like to commenton the anisotropic J − J model on square lattice. In this model J frustrates J only in one direction,say J connects only the third nearest neighbours inthe y -direction. This results in an anisotropic LP: thespin stiffness ρ y vanishes at some value of J while ρ x remains finite and positive. The wave vector of thespin spiral is always directed along the y-axis. In thiscase quantum fluctuations at the LP are described as h π i ∝ R d q √ q x + q y + ρ x q x . The integral is infrared conver-gent unlike that in the isotropic LP. Therefore generi-cally one cannot expect a spin liquid in this case. Thefluctuations are still enhanced and there must be a sup-pression of the on-site magnetization at the LP. This isexactly what series expansions for the anisotropic J − J model with S=1/2 indicate. It is likely that a similarscenario is valid for thin films of frustrated manganites(Tb,La,Dy)MnO tuned close to LP. VII. CONCLUSION
In this work, using field theory techniques, we havestudied properties of the universal spin liquid phase ina vicinity of an isotropic Lifshitz point in a system oflocalized frustrated spins. Our general analysis includesthe phase diagram, positions of critical points, excita-tion spectra, and spin-spin correlations functions. In thesemiclassical regime of large spin S the spin liquid phaseforms an exponentially narrow region in the vicinity ofthe Lifshitz point. The derivation of these results is ac-companied with a thorough discussion of the criterion forquantum melting of long range magnetic order in two di-mensions, an analogue of Lindemann criterion. We arguethe 2D Lifshitz point spin liquid is similar to the gappedHaldane phase in integer-spin 1D chains. In order tocheck our general field theory results, and in particular tocheck the quantum melting criterion, we have performednumerical series expansion calculations for the J − J model on square lattice. We demonstrate that results ofthese two different approaches are in a good agreement.0 VIII. ACKNOWLEDGMENTS
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Appendix A: The value of h π i c derived fromasymptotic Taylor expansion. After expanding n z = √ − π in a Taylor series andusing Wick’s theorem: h n z i = 1 − ∞ X k =1 h π i k (2 k − k − ( k − − h π i − h π i − h π i + . . . . (A1)The series (A1) is asymptotic and the coefficients atlarge k diverge. Since the series is asymptotic we truncateit when the coefficients in front of h π i k terms becomelarger then unity. Accounting for the leading terms in theexpansion up to h π i inclusive gives the critical value h π i c ≈ .
93 for h n z i = 0. Appendix B: Excitations in static spin-spiral phase
By considering fluctuations in the spin spiral state wefind the condition when quantum fluctuations melt thespiral. Here we derive the dispersions of in plane andout of plane fluctuations in the spin-spiral state. To bespecific let us assume that the spiral lies in { xy } plane: n = (cos Qr , sin Qr , . (B1)There are two different spin waves, the in-plane ϕ ( r , t ), n = (cos( Qr + φ ) , sin( Qr + φ ) , , (B2)and the out-of-plane h ( r , t ), n = ( p − h cos Qr , p − h sin Qr , h ) . (B3)Substituting parametrization (B2) and (B3) in the Euler-Lagrange equations of motion corresponding to the La-grangian (1) and linearising the equations with respectto φ and h we obtain the dispersion of the in-plane andout of plane modes. The derivation is straightforward,see e.g. Ref. . The dispersion of the in-plane mode is ω q = 1 χ ⊥ (cid:20) K ( Q ) −
12 ( K ( Q + q ) + K ( Q − q )) (cid:21) = b χ ⊥ (cid:2) Q q + q x + q y (cid:3) , (B4)and the dispersion of the out-of-plane mode isΩ q = 1 χ ⊥ [ K ( q ) − K ( Q )]= b χ ⊥ (cid:2) Q / − Q q + q x + q y (cid:3) . (B5)1The total quantum fluctuation orthogonal to the spinalignment in the spiral phase reads h π i = h φ i + h h i , (B6) h φ i = Z d q (2 π ) ω q , h h i = Z d q (2 π ) q . From the condition h π i = h π i c ≈ ρ cScS