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November 010 EPL, 9(01037001 doi:10.109/095-5075/9/37001 www.epljournal.org M.E.Zhitomirsky 1,,3(a andh.tsunetsugu 1 ServicedePhysiqueStatistique,MagnétismeetSupraconductivité,UMR-E9001CEA-INAC/UJF 17ruedesMartyrs,F-38054Grenoblecedex9,France InstituteforSolidStatePhysics,UniversityofTokyo-Kashiwanoha,5-1-5,Chiba77-8581Japan 3 Max-Planck-InstitutfürPhysikKomplexerSysteme-Nöthnitzerstr.38,D-01187Dresden,Germany received 0 October 010; accepted 5 October 010 published online 10 November 010 PACS 75.10.Jm Quantized spin models, including quantum spin frustration PACS 75.10.Kt Quantum spin liquids, valence bond phases and related phenomena PACS 75.10.Pq Spinchainmodels Abstract Competing ferro- and antiferromagnetic exchange interactions may lead to the formation of bound magnon pairs in the high-field phase of a frustrated quantum magnet. With decreasing field, magnon pairs undergo a Bose-condensation prior to the onset of a conventional one-magnon instability. We develop an analytical approach to study the zerotemperature properties of the magnon-pair condensate, which is a bosonic analog of the BCS superconductors. The representation of the condensate wave function in terms of the coherent bosonic states reveals the spin-nematic symmetry of the ground state and allows one to calculate various static properties. Sharp quasiparticle excitations are found in the nematic state with a small finite gap. We also predict the existence of a long-range ordered spin-nematic phase in the frustratedchainmateriallicuvo 4athighfields. editor s choice Copyright c EPLA, 010 Introduction. Frustrated spin systems are interesting in general, as their zero- and low-temperature properties are governed by quantum fluctuations. These strong fluctuations may inhibit the formation of longrange magnetic order, stabilizing instead a disordered spin liquid. Following Anderson[1], much of the interest inthepastdecadeshasbeenfocusedontheinvestigation of various quantum spin-liquid states[,3]. Another interesting possibility is the appearance of unconventional magnetic order characterized by partial breaking of the spin-rotational symmetry O(3. A specific example is provided by spin-nematic states, which are analogous to the ordered phases of needle-like molecules in liquid crystals. Spin-nematic phases have been discussed phenomenologically in[4,5], whereas identification of the relevant microscopic mechanism remains a challenging theoretical problem. It was suggested long time ago [6] that a sizable biquadratic exchange (S i S j in magnetic insulators with S 1 may stabilize a quadrupolar phase with vanishing sublattice magnetization S =0,but a nonzero second-ranktensor,e.g., S x S y = S z.inmostreal compounds the biquadratic exchange is, however, rather (a E-mail:mike.zhitomirsky@cea.fr small. Recently, the interest in the biquadratic mechanism for the spin-nematic order has been revived in connection withtheexperimentsoncoldatomicgases[7,8]andon thedisorderedmagneticmaterialniga S 4 [9 11]. In this letter we explore an alternative mechanism for the spin-nematic ordering based on competition between ferro- and antiferromagnetic interactions in magnetic insulators with an arbitrary value of the local spin includings =1/. Specifically, the mechanism operates in strong magnetic field and is based on the formation of bound magnon pairs in the fully polarized state[1 15]. This scenario has been studied numerically in a number of works on the so-called ferromagnetic J 1 -J chainmodel[16 0]anditsgeneralizationtotwo dimensions (D [1,]. Clear numerical evidence was found for the critical quadrupolar correlations below the saturation field for the 1D case. Note, however, that there is no true long-range order, nematic and otherwise, in one-dimensional quantum magnets at zero temperature. Therefore, a number of important questions on stability of the ordered nematic state and its excitation spectra arenotansweredbystudyingthepurely1dmodel. The purpose of this letter is to develop a simple analytical framework to treat the ground-state properties and low-energy magnetic excitations in the phase with 37001-p1
M. E. Zhitomirsky and H. Tsunetsugu Energy magnon J 1 J 1 magnon x y J 3 Hc H s1 H s Field spin cone spin nematic Fig. 1: (Color online Energy-field diagram for a frustrated quantum magnet close to the saturation field. Dot-dashed lines show lowest one- and two-magnon states. Solid lines represent theground-stateenergyfortheone-magnon(spin-coneand the two-magnon(spin-nematic condensate. a long-range spin-nematic order. Our description of the condensate of bound magnon pairs resembles in many aspects the BCS theory for the condensate of bound electron pairs in superconductors. In addition, we predict thataspin-nematicphasemustexistathighfieldsinthe frustratedchainmateriallicuvo 4 [3 8]. The phenomenon of magnon pair condensation has a close relationship to the old problem of particle vs. pair-superfluidity in an attractive Bose gas [9]. A common outcome is the density collapse prior to the pair-condensation transition[30]. Unrestricted growth of the local magnon density in spin-(1/ antiferromagnets is cured by their hard-core repulsion. In addition, reduced dimensionality of a spin subsystem found in many real materials helps to stabilize bound pairs and creates favorable conditions for their condensation. Experiments on LiCuVO 4 andotherrelatedcompoundsinhighmagnetic fields may, therefore, lead to the first observation of such an exotic off-diagonal long-range order in solid-state systems. In order to demonstrate the occurrence of magnon pair condensation at high magnetic fields we start with a general quantum Heisenberg antiferromagnet on an N-site lattice Ĥ= 1 J(rS i S j H S i, z (1 i,r i where r=r j r i. In the following S=1/issetfor definiteness. In strong magnetic fields the Zeeman energy dominates over the exchange interactions and stabilizes thefullypolarizedstate 0 =....Thisstateisthe vacuum for single spin-flips or magnons with the excitation energy ε q =H+ 1 1 q = 1 e iqr i S i 0 ( N i J(r[e iqr 1]=H+ 1 (J q J 0, (3 r Fig.:(ColoronlineTwo-dimensionalarrayofcopperionsin LiCuVO 4withprincipalexchangecouplings. where J q = r J(reiqr. In ordinary antiferromagnets spin-flipsrepeleachother.then,oncethebandgapin ε q vanishesatacertainq,j Q min{j q },anantiferromagnet undergoes a second-order transition into a canted spin structure at the saturation field H s1 = 1 (J 0 J Q, (4 as illustrated in fig. 1. The antiferromagnetic state below H s1 can be regarded as a Bose-condensate of single magnons[31,3]. Two-magnon bound states. The conventional scenario for an antiferromagnetic transition in a strong magneticfieldmaychangeifsomeoftheexchangebonds are ferromagnetic. In this case two spin-flips occupying the samebondwithj(r<0lowertheirinteractionenergy andmayformaboundpair[1,13].totreattheboundstate problem we follow the standard approach[33 35] and define a general two-magnon state = 1 f ij S i S j 0, (5 i,j withf ij =f ji beingthemagnonpairwavefunction.separating the center-of-mass motion f ij =e ik(ri+rj/ f k (r and calculating the matrix elements of Ĥ for states(5, we obtain the Bethe-Salpeter equation ( ε ε k/+q ε k/ q fk (q= 1 N ( Jp+q +J p q J k/+q J k/ q fk (p, (6 p where f k (q is the Fourier transform of f k (randε measures the magnon pair energy relative to the energy of the ferromagnetically polarized state. The above equation extends the previous theories[1,13,15] to an arbitrary geometry of exchange interactions and with a trivial replacementofε q italsoremainsvalidforanarbitrary valueofspins. While the subsequent theoretical consideration is entirely general, we introduce now for illustration a specificspinmodelshowninfig.,whichisrelatedtothe 37001-p
quasi-1dhelicalantiferromagnetlicuvo 4.Thismaterial consists of planar arrays of spin-(1/ copper chains with a ferromagnetic nearest-neighbor exchange J 1 <0and an antiferromagnetic second-neighbor coupling J >0. Chains are linked by diagonal bonds, whereas interplanar interactions are an order of magnitude smaller. Neutron scattering measurements provide the following estimate fortheexchangeparametersinlicuvo 4 :J 1 = 1.6meV, J =3.8meV,andJ 3 = 0.4meV [4]. The importance ofquantumeffectsinthismaterialisrevealedbyasmall value of ordered moments 0.3μ B in zero magnetic field[3].thepurely1dj 1 -J modelhasbeenstudied intensively in the past[1,16 0] though no results exist for a realistic planar model. To solve the integral equation (6 we expand f k (q into lattice harmonics and obtain a finite algebraic system. Bound states exist for 0.8π k y π with the minimum of ε (katk=(π,π. The binding energy defined by ε (K=ε Q E B is found numerically to be E B 0.030J. In the absence of bound states the two-magnon continuum has a gap that is twice larger thanthelowestone-magnonenergyε Q =H H s1.the two gaps vanish, therefore, in the same magnetic field. When bound states are present, condensation of magnon pairs starts in a higher magnetic field: H s =H s1 + 1 E B, (7 seefig.1.thecondensationfieldforlicuvo 4 iscalculated to be H s =47.1T(g=, whereas the single magnon branchsoftensath s1 =46.5T.TherelationH s >H s1 holdsuptoj 3 0.6meV.Hence,theconclusionabout the magnon pair condensation in LiCuVO 4 is rather robust and should not be affected by a possible uncertainty in the experimental coupling constants[4]. We finish our analysis of the linear problem by presenting the wave function of the lowest-energy bound pairs in the momentum representation: f K (q= λj cosq y J (1 cosq y +4J 3 sinq x sinq y +δε, (8 whereanumericalconstantδε=0.130j isrelatedtothe binding energy andλ is the normalization factor. In real spacethefunctione ikr/ f K (rhastheoddparityunder thereflectiony yandvanishes,therefore,atr=0. Coherent condensate of magnon pairs. Below H s the bound magnon pairs acquire negative energy and start to condense. It is convenient at this point to transform from spin-(1/ operators to the Holstein- Primakoff bosons: Si= z 1 a i a i, S i =a i 1 a i a i, (9 expanding subsequently square-roots to the first order ina i a i. The many-body state with a macroscopic number of thelowest-energypairsbelowh s canbeexpressedasthe coherent boson state of the pair creation operator: =e N / exp 1 f ij a i a j 0. (10 i,j Heref ij =e ik(ri+rj/ f K (risthewavefunctionofthe lowestenergypairsand isthecomplexamplitudeof the condensate. The state (10 is a bosonic equivalent ofthebcspairingwavefunctionforfermionsandmust be regarded as a variational ansatz, which can be further improved by taking into account pair-pair correlations. We use the ground-state wave function(10 to compute simple boson averages: a q =0, a K/+q a K/ q = f K (q 1 f K (q, n K/+q = a K/+q a K/+q = f K (q 1 f K (q, as well as a more complicated four-boson correlator: a p/+q a p/ q a p/+q a p/ q = δ p,k (1+n K/+q +n K/+q f K (qf K (q 1 4 f K (qf K (q + 4 (δ q,q +δ q, q (1+n p/+q +n p/ q ( fk p K +q fk 1 4 fk ( p K ( p K +q f K q ( p K q (11. (1 From these one can derive various spin correlators. In particular, the absence of the single-magnon condensate a q =0translatesinto S x,y i =0, (13 which signifies a lack of a usual antiferromagnetic order parameter. The transverse and longitudinal spin correlationsaregivenintheleadingorderby S i S+ j l f ilf lj, δs z iδs z j f ij (14 with δsi z=sz i Sz i. In accordance with the behavior ofthebound-statewavefunctionf K (rthetwocorrelatorsdecayexponentiallyas r i r j.thetransverse magneticstructurefactors (qhasnobraggpeaksand exhibits diffuse liquid-like spin correlations with a characteristic shape in momentum space determined by the bound-state wave function: S q S q [f K ( K +q +f K ( ] K q. (15 ThelongitudinalresponseS zz (qbeingformallyofthe sameorderin appearstobemuchweakerthans (q inthevicinityofthesaturationfieldh s. 37001-p3
M. E. Zhitomirsky and H. Tsunetsugu The general definition of the spin nematic order parameter in the O(-symmetric case is Q αβ ij =1 Sα is β j +Sβ i Sα j 1 δ αβ S i S j, (16 where i,jbelong to a nearest-neighbor bond and α,β= x,y. The quadrupolar tensor Q αβ ij acquires a nonzero expectation value in the presence of the pair condensate: Q xx ij +iq xy ij =1 S+ i S+ j f ij. (17 The phase of the condensate amplitude determines the orientation of the spin-nematic director in the(x,y-plane. The director forms a periodic structure in the real space determined by the momentum K. Using explicit expressions for the spin correlators, one can calculate the ground-state energy. Below, we shall use for this purpose a complementary approach, which isanalogoustothebogoliubovmethodinthetheoryof superconductivity and yields the spectrum of quasiparticle excitations simultaneously with the static properties. Mean-field approach. The bosonic equivalent of the spin Hamiltonian is obtained upon substitution of(9 into(1. We restrict ourselves to quadratic and quartic termsina i sanddefinetwotypesofmean-fieldaverages for each exchange bond: ij = a i a j, n r = a i a j, (18 andthemagnondensityn= a i a i.theanomalouscorrelatorisfurtherfactorizedas ij =e ik(ri+rj/ r.both r and n r are even real functions of r with a proper choice of gauge. Performing the mean-field decoupling in the interaction term we obtain a quadratic form, which is then diagonalized with the canonical transformation. This yields the energy of one-magnon excitations ε K/+q =ǫ q r ǫ q = J(r ( 1 n n r A q B q,b q = r A q =H r sin 1 Krsinqr, J(r r cosqr, (19 ( 1 J(r n n r (1 cos 1 Krcosqr. In accordance with the exponential decay of spin correlations (14, the excitation spectrum acquires a gap in the presence of the magnon pair condensate. The above expressions are used to calculate bosonic averages and to obtain a closed form of the self-consistent equations: B q ǫ q cosqr, n r = q A q ǫ q cos ( 1 K+q r. r = q (0 InthelimitH H s onefinds r n r r,whilethe linearized equation for r transforms directly into the bound-state equation(6. ε(q [mev] 10 5 ε(q [mev] 1 0 0.3 [0 q 0] 0.6 H = 55T H = 50T H = 47T H = 45T H = 40T 0 0 0. 0.4 0.6 0.8 1 [0 q 0] Fig. 3:(Color online Dispersion of a single-magnon branch inlicuvo 4 inthefullypolarizedstate(h s 47Tandin the state with the magnon pair condensate. Field values for differentcurvesfromtoptobottomarelistedontheplot.the insetshowsthevicinityofthemagnongap. We have solved self-consistently the set of equations (19 and(0 and calculated the ground-state energy for thespinmodeloffig.assumingthesamesymmetryof bondvariables r andn r asinthemagnonpaircoherent state(10. With decreasing external field, the pairs overlap more appreciably and at a certain point give way to a conventional one-particle condensation. Comparing the ground-state energy of the pair condensate with the energy of a simple spin-cone structure we find the first-order transitionath c 44.5Tasillustratedschematicallyin fig.1.thespin-nematicstatehasthelowestenergyin afiniterangeoffieldsh c <H<H s,whichextendswell belowthecondensationfieldh s1 forsinglemagnons. The ground-state energy calculation allows to determine the slope of the magnetization curve M(HatH c < H<H s.inordinaryquasi-1dantiferromagnets,m(h deviatesfromastraightlineash H s resemblingthe square-rootsingularityofasinglequantumspinchain.our mean-field calculations for the high-field nematic phase in LiCuVO 4 yieldinsteadtheslopedm/dh 0.38M sat /J, whichamountstoonly54%oftheslopeoftheclassical magnetization curve. The quantum corrections beyond the mean-field approximation should somewhat modify this value.however,weexpectthemtobesmallforthesame reason as the suppression of critical thermal fluctuations in the BCS superconductors. Indeed, the size of the boundmagnonpairsisratherlargeextendingtoξ 10 interatomic spacing in the direction of chains. Already for small magnon densities each bound pair is surrounded by many neighboring magnon pairs, which enforces the meanfield behavior. Therefore, a distinct signature of the highfieldnematicphaseinlicuvo 4 willbeasharpchangein the slope of the magnetization curve. The dispersion of one-magnon excitations found together with the ground-state energy in the selfconsistent calculation is presented in fig. 3. In the fully polarizedstateath>h s 47T,variationoftheapplied field results in an overall shift of the magnon energy 37001-p4
accordingtoeq.(3.ath=h s magnonshaveasmall gap g =H s H s1 0.06meV.Thefielddependenceof ε k changesdrasticallyinthepresenceofthemagnonpair condensate.decreasingthefieldmodifiestheshapeofε k buttheexcitationgap g remainspracticallyunchanged, seetheinsetinfig.3.thelowestfieldh=40tusedin fig.3isbelowthetransitionfieldh c intothespin-cone magnetic structure. The corresponding dispersion curve ε k illustratesthatthespin-nematicstateremainslocally stableevenbelowh c.thisisincontrastwiththeprevious scenario suggested for an attractive Bose gas, for which the pair condensate was assumed to become unstable due to a softening of the single-particle branch[9,30]. The motion of the spin-nematic order parameter provides an additional gapless branch of collective excitations, which should yield a nonzero dynamical signal in the longitudinal channel. Transverse spin-spin correlations in the nematic phase are dominated by unpaired magnons(19. Finally,letusalsocommentonalow-fieldstateatH< H c.theconsideredscenarioofadirecttransitionbetween the spin-nematic state and the conventional canted antiferromagnetic phase applies most certainly to the D model[1],whichexhibitsamagneticorderingatq= (π,0inzerofield.inthecaseofweaklycoupledchains,the low-field phase might be more complicated than the simple spin-cone structure used above as an example. The NMR measurements[6,8] indicate that the intermediate-field phase,whichisobservedinlicuvo 4 above8t[5],has predominant longitudinal SDW-type correlations between local spins. This experimental finding agrees, in principle, with the numerical results for a single spin chain[17 19]. Understandingthefateofsucha1Dphaseinthepresence of interchain couplings as well as its relation to the transverse nematic order remains an open theoretical problem. To summarize, we have presented the analytical description for the Bose-condensate of bound magnon pairs in a frustrated quantum magnet in high magnetic fields. The theory applies to a number of real magnetic compounds with competing ferro- and antiferromagnetic interactions. After the first version of the present work has appeared, we learned about the experimental observation of a new phaseinlicuvo 4 inthefieldrange41 44T[36]. We are grateful to M. Enderle, B.Fåk, M.Hagiwara,and L.Svistov for stimulating discussions.partofthisworkhasbeenperformedwithinthe Advanced Study Group Program on Unconventional Magnetism in High Fields at the Max-Planck Institute for the Physics of Complex Systems. HT acknowledges support by Grants-in-Aid for Scientific Research (No. 17071011 and No. 1905003 and by the Next-Generation Supercomputing Project, Nanoscience Program, MEXT of Japan. REFERENCES [1] AndersonP.W.,Mater.Res.Bull.,8(1973153. [] Fradkin E., Field Theories of Condensed Matter Systems (Addison-Wesley, Reading 1991. [3] Misguich G. and Lhuillier C.,inFrustrated Spin Systems, edited by Diep H. T. (World Scientific, Singapore005. [4] AndreevA. F.andGrishchuk I. A., Sov. Phys. JETP, 60(198467. [5]Chandra P. and Coleman P., Phys. Rev. Lett., 66 (1991100. [6] BlumeM.andHsiehY.Y.,J.Appl.Phys.,40(1969 149. [7] StengerJ.etal.,Nature,396(1998345. [8] Demler E.andZhou F.,Phys.Rev.Lett.,88(00 163001. [9] NakatsujiS.etal.,Science,309(0051697. [10] Tsunetsugu H. andarikawa M., J.Phys.Soc.Jpn., 75(006083701. [11] LäuchliA.,MilaF.andPencK.,Phys.Rev.Lett.,97 (00608705. [1] ChubukovA.V.,Phys.Rev.B,44(19914693. [13] KuzianR.O.andDrechslerS.-L.,Phys.Rev.B,75 (00704401. [14] DmitrievD.V.andKrivnovV.Ya.,Phys.Rev.B,79 (00905441. [15] UedaH.T.andTotsukaK.,Phys.Rev.B,80(009 014417. [16] Heidrich-Meisner F., Honecker A. andvekua T., Phys.Rev.B,74(00600403(R. [17] VekuaT.etal.,Phys.Rev.B,76(00717440. [18] HikiharaT.etal.,Phys.Rev.B,78(008144404. [19] SudanJ.,LuscherA.andLäuchliA.M.,Phys.Rev. B,80(00914040(R. [0] Heidrich-Meisner F., McCulloch I. P. and KolezhukA.K.,Phys.Rev.B,80(009144417. [1] Shannon N., Momoi T. andsindzingre P., Phys. Rev. Lett.,96(0060713. [] Shindou R. and Momoi T., Phys. Rev. B, 80 (009 064410. [3] GibsonB.J.etal.,PhysicaB,350(004e53. [4] EnderleM.etal.,Europhys.Lett.,70(00537. [5] BanksM.G.etal.,J.Phys.:Condens.Matter,19(007 1457. [6] ButtgenN.etal.,Phys.Rev.B,76(007014440. [7] SchrettleF.etal.,Phys.Rev.B,77(008144101. [8] ButtgenN.etal.,Phys.Rev.B,81(01005403. [9] Valatin J. G. and Butler D., Nuovo Cimento, 10 (195837. [30] NozièresP.andSaintJamesD.,J.Phys.(Paris,43 (1981133. [31] Matsubara T. andmatsuda H., Prog.Theor.Phys., 16(1956569. [3] BatyevE. G.andBraginskii L. S., Sov. Phys. JETP, 60(1984781. [33] Wortis M., Phys. Rev., 13 (1963 85; 138 (1965 A116. [34] HanusJ.,Phys.Rev.Lett.,11(1963336. [35] Mattis D. C., The Theory of Magnetism I(Springer, Berlin 1988. [36] SvistovL.E.etal.,arXiv:1005.5668. 37001-p5