Topological Spin Excitations of Heisenberg Antiferromagnets in Two Dimensions

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  • VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999


    3504Topological Spin Excitations of Heisenberg Antiferromagnets in Two Dimensions

    Tai Kai NgDepartment of Physics, Hong Kong University of Science and Technology, Kowloon, Hong Kong

    (Received 14 December 1998)

    We discuss the construction and dynamics of vortexlike topological spin excitations in the Schwingerboson description of Heisenberg antiferromagnets in two dimensions. The topological spin excitationare Dirac fermions (with gap) when spin valueS is a half-integer. Experimental and theoreticalimplications of these excitations are being investigated. [S0031-9007(99)08982-6]

    PACS numbers: 71.10.Fd, 75.10.Jm, 75.40.Gb-










    Recently, strong interest has been shown in the studyHeisenberg antiferromagnets on a two-dimensional squlattice. Early theoretical analysis showed that the loenergy, long length scale behavior of the model shoube described by the renormalized classical regime of ts2 1 1dd O(3) nonlinear sigma model (NLsM) [1,2].The analysis was believed to be valid as long as the lenscale under investigation was much larger than lattispacing. More recently, in a series of experiments [3] aMonte Carlo simulations [46], it was discovered that thlength scale at which the renormalized classicals-modeldescription becomes valid is surprisingly long (L $ 200sites for spinS 1y2) [5,6]. A quantum Monte Carlostudy of the low energy spectrum of the model on finisize lattices also suggested that the spectrum in theS 1y2 case disagrees rather strongly with the predictioof NLsM even when the size of the systemN isnot too smallsN # 32d [7]. These anomalous findingssuggest that contrary to usual beliefs, there may existintermediate energy/ length scale where the behaviorthe system deviates strongly from NLsM.

    On the other hand, it has been shown by Read aChakraborty [8] that fermionic spin excitations exist atopological excitations in the short-ranged resonant vlence bond (RVB) phase of 2DS 1y2 Heisenbergmodel, and their analysis has been generalized [9] tocase (for arbitrary spinS) where the system is describedby a Schwinger-boson mean-field theory (SBMFT) [10These topological excitations are finite energy excitatioof 2D Heisenberg antiferromagnets that do not appearconventional NLsM description. However, the dynamicsof the topological spin excitations were not investigatedthose works [8,9].

    The purpose of this paper is to investigate, startinfrom SBMFT, the dynamics of the topological spin excitations constructed in Ref. [9] under some very geneassumptions. In particular, we shall show that the extence of these topological excitations is consistent with trecently observed anomalous behaviors in the Heisenbmodel. We shall also point out, in the case ofS 1y2 Heisenberg antiferromagnet, an interesting connectbetween the topological spin excitations and the spin e0031-9007y99y82(17)y3504(4)$15.00ofarewldhe











    citations in the flux phase [11] of fermionic RVB meanfield theory.

    We begin by reviewing briefly the construction of topological spin excitations in SBMFT of the 2D Heisenbermodel. Following Ref. [9] we assume that the lowenergy physics of general spin-S 2D Heisenbergmodel is described by the Schwinger-boson mean-fiHamiltonian

    HMF 2JXi,j

    fDpsZi"Zj# 2 Zi#Zj"d

    1 H.c. 2 jDj2g 1Xis

    lsZisZis 2 2Sd ,


    whereD sDpd andl are mean-field parameters determineby the mean-field equationsD ksZi"Zj# 2 Zi#Zj"dl andkZi"Zi"l 1 kZi#Zi#l 2S. Notice that in two dimensions,long-ranged antiferromagnetic order exists at zero teperature corresponding to bose condensationkZl fi 0 inSBMFT. We shall consider finite temperatureT fi 0 andkZl 0 in the following. (The effect of Bose condensation will be addressed at the end of this paper.) Notithat SBMFT offers a fairly accurate description of the loenergy physics of a Heisenberg model at two dimensio[10], thus justifying the starting point of our theory.

    To look for topological excitations in SBMFT, we notice that the structure of the mean-field theory resembvery much the Bardeen-Cooper-Schrieffer (BCS) theofor superconductivity, except that the spin pairs of bosoreplace the electron (fermion) Cooper pairs in BCS theoThe resemblance of the two theories leads us to studyvor-tex excitations in SBMFT, since vortices are stable toplogical excitations in BCS theory at two dimensions. IBCS theory, a vortex located at$r 0 is a solution ofthe BCS mean-field equation, where the order parameDBCSs$rd has a formDBCSs$rd fsrdeiu in a polar co-ordinate, wherefsrd is real and positive. To minimizeenergy, a magnetic flux ofp-flux quanta is trapped in thevortex core. The vortex solution in SBMFT has the samstructure, except that the BCS order parameterDBCS isreplaced by the Schwinger-boson order parameterDij and 1999 The American Physical Society

  • VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999








    -undisexn-the vector potential$A does not represent the physical magnetic field, but is a fictitious gauge field arising from phasfluctuations of order parameterDij [9,12]. The existenceand stability of the vortex solution was demonstratedRef. [9] in an effective Ginsburg-Landau theory. Thesvortex solutions are bosonic,S 0, topological excita-tions in SBMFT [9].

    To construct topologicalspin excitations we note, thatlike vortices in superconductors where electronic bounstates often exist inside the vortex core, boson boustates may exist inside vortices in SBMFT. In particulawe argued in Ref. [9] that, for a vortex centered atlattice site, a bound state of2S bosons must be formedat the vortex center because of the constraint that therealways2S bosons per site in the Heisenberg model. Iparticular, because of statistics transmutation associawith binding quantum particles to a flux tube ofp-fluxquantum in two dimensions [13], the resulting excitatiois a spin-S fermion when2S is odd [9].

    We shall now supply the mathematical details. Aimportant difference between SBMFT and BCS theorythat the mean-field Hamiltonian (1) breaks the translationsymmetry of the Heisenberg model by one lattice sitCorrespondingly, there exist two lattice sites per unit cein SBMFT and the fluctuations of the order parameterDijare described by two amplitude and two phase (uniforand staggered) fields in the continuum limit [9,12], i.e.,

    Di,i6n 12


    i 6



    1 q6n

    i 6



    3 eif

    Ri6ny2 $A?d $x1As6n si6ny2dg, (2)where q2n 2qn, As2n 2A

    sn are staggered com-

    ponents of the amplitude and phase fluctuations ofD,respectively.f and

    Rx $A ? d $x0 are the corresponding uni-form components. The effective Ginsburg-Landau actiofor the continuum field variables is derived in Ref. [9(see also Ref. [12]). We obtain to orderOsm0d [m isthe mass gap for spin-wave excitations in SBMFT whicis very small at low temperaturesm , Je2JSyT d], Seff Su 1 Ss, where

    Su ,Z



    a 2 4

    S 1


    f 1 f2


    S 1


    fsmu 2 Amd2



    Ss ,Z





    sqmd2 1



    1 icFmtqm


    where e2 , m, a , 4, and b and c are constants oforder O(1). Fmn mAsn 2 nA

    sm anduy2 is the uniform

    phase field for the Schwinger bosons. Notice that thgauge symmetry of the uniform gauge field$A is broken-e










    by kDijl f fi 0 in SBMFT. The existence of a stablevortex solution in SBMFT is tied with the$A field as in theusual superconductors.

    To derive vortex dynamics we concentrate atSu andintroduce vortex 3-current$jy in the phase field,mu !mu 1 mu

    0, where u0 is multivalued and jym emnlnlu

    0 fi 0. In the London limit where we treatfas constant, a duality transformation can be performwhere we can integrate out theu field to obtain aneffective actionSy for vortices [14],

    Sy Z




    4fs= 3 $ad2

    1 i $a ? s $jy 2 = 3 $Ad

    #1 Ss0dy ,


    where= 3 $a , s=u0 2 $Ad and corresponds to the transverse part of the supercurrent field in superconductoNotice that the amplitude fieldf goes to zero at the cen-ter of the vortex. As a result, there exists an additioncontribution to the vortex action,Ss0dy , h(energy neededto create vortex core,ey) 3 (length of vortex trajectoryin space-time)j. For N vortices,

    Ss0dy , eyNX


    Zdli ey



    s1 1


    d $xidt



    where $xistd represents the trajectory of theith vortexin Euclidean space-time, andcy is the limiting velocity.Notice that the suppression of thef field at the vortexcore also couples vortices to the staggered amplitude aphase fluctuations through the terms2 bf dq

    2m in Ss [9].

    We shall first considerSy in the following.The dynamics of a single vortex can be obtained b

    minimizing Ss0dy at real time. We find thatSs0dy describes

    relativistic particles with energyE gey, where g


    1 2$y2

    c2yand $y is the vortex velocity. In the absence

    of the = 3 $A term, the particles have charge (vorticity)and interact with each other through an effective U(gauge field $a. For boson vortices the correspondinquantum field theory is a relativistic theory of scalaelectrodynamics with charged bosons (vortices) [14].the presence of a trapped magnetic flux inside vortex cos= 3 $A 2 $jy , 0d, the electric 3-current$jy is screenedand the bosons decoupled from the gauge field$a. Theresulting theory is a relativistic theory of bosons witshort-ranged interactions, as is the case of vorticesusual superconductors [14].

    To derive dynamics for the topological spin excitations we assume that, once the boson spins are boto the vortex core, their spatial degree of freedomquenched and the only modifications to the pure vortaction (4a) and (4b) are (1) the vortices now carry spin idicesm 2S, 2S 1 1, . . . , S, and there are2S 1 1 spin3505

  • VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999










    atcomponent of vortices, and (2) vortices become fermons when2S is odd. In particular, since the vortex action (4a) and (4b) is Lorentz invariant, the dynamicof the topological spin excitations must be describeds2 1 1dd-relativistic field theories of bosons whenS is aninteger and by s2 1 1dd-relativistic field theories offermions whenS is half-integer. To proceed further,we examine more carefully the symmetry constraintsSBMFT. The existence of two lattice sites per unit cell iSBMFT implies that there are also two quantum fieldscAsandcBs in the continuum theory, representing spin-s vor-tices centered atA- andB-sublattice sites, respectively [9].Under reflection or rotation bypy2 around the center of asquare plaquette, theA andB sublattices are interchangedand, correspondingly, alsocAs and c

    Bs . Notice that we

    have considered finite temperature wherekZl 0 in ourdiscussion and, correspondingly, parity (space-time refletion) symmetry is unbroken. In order to describe couplinof the bound boson spins at the vortex center to the fititious gauge fields$A and $As, the quantum fieldscAs andcBs must also carry charge and are complex.

    With these kinematics constraints, the quantum fietheories for the topological spin excitations can be dtermined [15]. In the case of integer spins where thtopological spin excitations are bosons, we construquantum fieldsc1s2ds c

    As 6 c

    Bs which are eigenstates

    of py2 rotation and reflection with eigenvalues61. Thedynamics of thec1s2ds fields are described separately brelativistic theories of complex scalar fields. In the caof half-integer spins where the spin excitations are fermons, the corresponding theory which respects parity istheory of Dirac fermions ins2 1 1dd [15]. In the fol-lowing we shall concentrate on the case of half-integspin systems.

    In this case, the topological spin excitations are Diraspinors described by four-component spinor fieldcs, with



    !, cAsBds sxd


    AsBd1s sxd

    cAsBd2s sxd

    !, (5)

    wherecAsBds s are two component fermion fields needeto describe positive and negative energy solutions of tDirac equation. In terms ofcs the effective Lagrangianwhich transforms correctly under parity is

    Leff Xs


    fcsgmsmcsd 2 smcsdgmcsg

    2 mcscs 2 csgmcsAm , (6)

    where gms are usual4 3 4 Dirac matrices ins2 11dd with m 0, 1, 2. $A is the uniform gauge field.Notice that the staggered gauge field$As decouples fromthe cs field in this level, as can be checked easifrom the transformation ofLeff under staggered gaugetransformationcAs ! cAseiu, andcBs ! cBse2iu. Noticealso that the spin degrees of freedoms appear asinternaldegrees of freedom for the Dirac fermions.

    It is interesting to compare the effective actionLeff withthe flux phase [11] of the fermionic mean-field theory o3506i--sby










    the Heisenberg model in the caseS 1y2. In both cases,pockets of Dirac fermions describe the fermionic spiexcitations at low energy. The number of species of Dirafermions are the same in both casesthere are four hapockets of Dirac fermions at aboutskx , kyd s6

    p2 , 6

    p2 d

    in the flux phase, and there are two full-pockets of Dirafermions coming from two independent combinations ocAs and c

    Bs in the present theory. The position of the

    Dirac-pockets in $k space cannot be determined withcertainty in our effective theory. Nevertheless, to makcomparison with the flux phase we shall assume ththe fermion pockets in our theory are centered arounskx , kyd s6

    p2 , 6

    p2 d. With this assumption the only

    difference betweenLeff and the flux phase is that thereis a gapey in the Dirac-fermion spectrum in our theory.In fact, in the limitey 0, it can be checked directly thatLeff describes the continuum theory for the flux phasThe strong similarity between our effective theory anflux phase leads us to speculate that the flux phase in fdescribes a new spin-disordered phase of 2D Heisenbantiferromagnets, where antiferromagnetism is destroyby driving the fermion mass gapey to zero. We shalldiscuss this scenario in a future publication.

    Next we consider the experimental consequencesthe topological spin excitations. We shall first consideS 1y2 as an example. In terms ofc, the spins carriedby the Dirac fermions in theS 1y2 case are describedby the operators

    $SAsBdsxd 12



    f: c1AsBdkm sxd $smm0cAsBdkm0 sxd :g ,


    where $SAsxd and $SBsxd are spin operators in theA andB sublattices, respectively, and$s are Pauli matrices,m, m0 6 12 . : O : O 2 kOlG are normal-ordered op-erators. The uniform and staggered spin operators canconstructed from$SAsBd, where $ms $nd $SA 1 s2d $SB de-scribes spin fluctuations around momenta$q s0, 0d and$q sp, pd. Notice that, because of the constraint thathere is one spin per site in the original Heisenberg modthe Dirac fermions together with the original boson spinmust satisfy the constraintX


    " Xk1,2

    : c1aks sxdcakssxd :

    !1 Zas sxdZ

    as sxd

    # 1 ,


    which introduces additional coupling between boson spiand topological spin excitations not included inLeff.

    Experimentally, the Dirac fermion spectrum can bobserved directly in a neutron scattering experimeat energy v . 2ey. Its contribution to the dynamicstructure factorSs $q, vd at $q , s0, 0d and $q , sp, pd canbe calculated directly using Eqs. (6) and (7). We find ththe contributions are similar toSs $q, vd calculated fromflux phase, except that a gap,2ey is found in the spectral

  • VOLUME 82, NUMBER 17 P H Y S I C A L R E V I E W L E T T E R S 26 APRIL 1999











    function. The existence of a Dirac-fermion spectrum alaffects spin correlation at lower temperatures. To see thwe consider the antiferromagnetic spin-correlation lengjsT d. In pure SBMFT,jsT d is given at low temperatureby [10]

    jsT d p





    !, (9)

    where nB is the density of bose-condensed spinsSBMFT at T 0. In the presence of the Dirac-fermionspectrum, the system gets more disordered atT fi 0because of the thermal effect associated with the exdegrees of freedom andjsT d decreases. The effectcan be estimated in a mean-field approximation usithe constraint (8) [16], where the average numberSchwinger bosons per siteknSBl is given by

    knSBl 1 2X


    k:c1aks caks:l .

    It is straightforward to show that, at low temperatureknSBl , 1 2 2 exps2

    eyT d, and the leading correction a

    low temperature tojsT d is obtained by replacingnB !nB 2 2 exps2

    eyT d in Eq. (9). The reduction injsT d

    continues untilT , ey, when the number of thermallyexcited fermion spin excitations becomes large, and tspin correlation becomes qualitatively different from thprediction of SBMFT or NLsM. For general spin valueS, ey , JS2 and the temperature at which the NLsMdescription becomes invalid occurs atT , JS2 which isdeep inside the quantum critical regime for large valueof S. For a small value ofS, where there is no clearseparation between energy scalesSJ and S2J, the wholequantum critical regime may be washed away by thexistence of topological spin excitations. Such a scenaindeed seemed to be observed in Monte Carlo simulatioof the S 1y2 2D Heisenberg antiferromagnet [46]where the quantum critical regime predicted by thNLsM description seems to be missing. Notice thfor integer spin systems the contribution to dynamicstructure factorSs $q, vd from topological spin excitationsis quite different from half-integer spin systems becauof different statistics. However the effect onjsT d shouldbe qualitatively similar.

    Finally we discuss the effects of gauge field$A. Inthe T ! 0 limit, where bose condensation of the bosospin takes placeskZl fi 0d, the gauge field$A becomesconfining [17] and the topological spin excitations arconfined in pairs by linear confining potential. At finitetemperatures the confining potential is effective uplength scale, correlation lengthjsT d, and the effect ofconfinement is expected to be strong at low temperatuwhen the system is at the renormalized classical regimAs a result, the topological spin excitation spectrum$q , s0, 0d is expected to be strongly modified from thfree-Dirac-fermion prediction (for half-integer spins) asoat,th















    low temperature. However, the behavior of the topologcal spin-excitation spectrum at$q , sp, pd is determinedby the short-distance behaviors of spin pairs and shounot be affected strongly by confinement. As a result, wexpect that our predictions forSs $q, vd at $q , sp, pd andjsT d remain qualitatively valid.

    In summary, using SBMFT, we show in this papethat stable topological spin excitations exist in Heisenbeantiferromagnets at two dimensions. The topologicaspin excitations are described by relativistic quantumfield theories of complex scalars when spin valueSis an integer, and they are Dirac fermions whenSis half-integer. We have discussed the theoretical anexperimental consequences of these excitations and hapointed out that the existence of these spin excitationmay be the reason for the anomalous results observin recent experiments [3] and Monte Carlo simulation[47]. In particular, these excitations can be observedirectly in neutron scattering experiments, and woulprovide a direct test of our theory.

    The author thanks Professor N. Nagaosa and ProfesP. A. Lee for many interesting discussions and also thanthe hospitality of the Aspen Center of Physics where paof this work was completed. This work is supported bHKRGC through Grant No. HKUST6143-97P.

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