Excitations and quantum fluctuations in site-diluted two-dimensional antiferromagnets

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  • Excitations and quantum fluctuations in site-diluted two-dimensional antiferromagnets

    Eduardo R. Mucciolo,1 A. H. Castro Neto,2 and Claudio Chamon21Department of Physics, University of Central Florida, Orlando, Florida 32816, USA;

    Department of Physics, Duke University, Durham, North Carolina 27708, USA;Departamento de Fsica, Pontifcia Universidade Catlica do Rio de Janeiro, Caixa Postal 37801, 22452-970 Rio de Janeiro, Brazil

    2Department of Physics, Boston University, Boston, Massachusetts 02215, USA(Received 3 February 2004; published 24 June 2004)

    We study the effect of site dilution and quantum fluctuations in an antiferromagnetic spin system on a squarelattice within the linear spin-wave approximation. By performing numerical diagonalization in real space andfinite-size scaling, we characterize the nature of the low-energy spin excitations for different dilution fractionsup to the classical percolation threshold. We find nontrivial signatures of fractonlike excitations at high fre-quencies. Our simulations also confirm the existence of an upper bound for the amount of quantum fluctuationsin the ground state of the system, leading to the persistence of long-range order up to the percolation threshold.This result is in agreement with recent neutron-scattering experimental data and quantum Monte Carlo nu-merical calculations. We also show that the absence of a quantum critical point below the classical percolationthreshold holds for a large class of systems whose Hamiltonians can be mapped onto a system of couplednoninteracting massless bosons.

    DOI: 10.1103/PhysRevB.69.214424 PACS number(s): 75.10.Jm, 75.50.Ee, 75.30.Ds, 75.40.Mg

    I. INTRODUCTION

    The problem of the interplay of quantum fluctuationsand disorder in low dimensional systems is of fundamentalimportance in modern condensed matter physics. It is rel-evant for the understanding of the metal-insulator transi-tion in metal-oxide-semiconductor field-effect transistors(MOSFETS),1 impurity effects ind-wave superconductors,2

    non-Fermi-liquid behavior in U and Ce intermetallics,3 andthe persistence of long-range order(LRO) in two-dimensional (2D) spin-1/2 quantum antiferromagnets(AFM),4 among others.

    The 2D square lattice with nearest-neighbor hopping un-dergoes a classical percolation transition upon random dilu-tion. For the case of site dilution, the transition occurs at thehole concentrationxc

  • present and discuss the numerical methods used to generatethe random dilution, the identification of the largest con-nected cluster, and the diagonalization of the eigenvalueproblem. The numerical results for the spin-wave excitationspectrum and the quantum fluctuation corrections to theAFM ground state are presented and discussed in Sec. IV. InSec. V we argue that the absence of quantum phase transi-tions in site dilute systems occurs for spin Hamiltonians thatcan be mapped onto a system of coupled harmonic oscilla-tors. Finally, in Sec. VI, we draw our conclusions and pointto future directions.

    II. DILUTE HEISENBERG ANTIFERROMAGNET IN THESPIN-WAVE APPROXIMATION

    We begin by reviewing the well-known connection be-tween magnetic and bosonic systems. In particular, we areinterested in spin Hamiltonians of the form

    H = oki,jl,a

    hih jJaSiaSj

    a, s1d

    whereSia is thea=x,y,z component of a spinSat sitei, Ja is

    the nearest-neighbor exchange constant, andhi =0s1d if thesite is empty(occupied). The empty sites are randomly dis-tributed over the whole sample with uniform probability.Calling N the total number of sites, we define the fraction ofoccupied sites as

    p =1

    Noi=1

    N

    hi . s2d

    The dilution fraction is then defined asx=1p.The isotropic AFM Heisenberg model corresponds toJa

    =J.0 for a=x,y,z. When LRO is present(say, along thezdirection), the spin operators can be written in terms ofbosonic operators using the Holstein-Primakoff method.14

    Since the square lattice is bipartite, it can be divided up intotwo square sublattices,A=L+ andB=L. Thus,

    Si z = S aiai , s3d

    Si+ = f2S ai

    aig1/2ai , s4d

    Si = ai

    f2S aiaig1/2, s5d

    with i PA, andSj z = S+ bj

    bj , s6d

    Sj+ = bj

    f2S bjbjg1/2, s7d

    Sj = f2S bj

    bjg1/2bj , s8d

    with j PB. The bosonic operators obey the usual commuta-tion relations, namely,

    fai,ai8 g = dii8 and fbj,bj8

    g = d j j 8, s9d

    with all other commutators equal to zero.For the large spin casesS@1d or when the number of spin

    waves is small(ni =kai ail, nj =kbj bj

    l !S), we can expand

    the square root in Eqs.(3) and (6) in powers ofni and nj,keeping only linear terms. That allows us to write the ap-proximate bilinear bosonic Hamiltonian

    H < JS2oki j l

    hih j + JSoki j l

    hih jsaiai + bj

    bj+ aibj + aibj

    d.

    s10d

    The first term on the right-hand side of Eq.(10) representsthe classical ground-state energy of the AFM. Using thebosonic commutation relations, we can rearrange the Hamil-tonian in the following form:

    H < JSsS+ 1doki j l

    hih j + HSW, s11d

    where, now, the first term on the right-hand side representsthe ground-state energy in the absence of quantum fluctua-tions, while the latter is described by the second term,

    HSW =JS

    2 oki j l hih jsaiai + aiai

    + bjbj + bjbj

    + aibj + bjai + aibj

    + bjai

    d. s12d

    Hereafter we will drop the constant ground-state energy termand will only study the eigenstates ofHSW. The spin-waveHamiltonian contains bilinear crossed terms which, sepa-rately, do not conserve particle number. At this point, it isworth simplifying the notation by dropping the distinctionbetween bosonic operators living on different sublattices andreordering the summation over sites,

    HSW =JS

    2 oi,j=1N

    fKijsaiaj + aiaj

    d+ Di jsaiaj + aiaj

    dg, s13d

    where both indexes in the sum run over all sites in the lattice.The matricesK andD are defined as

    Kij = di j hiokil l

    hl s14d

    (the sum run over all nearest-neighbor sites toi) and

    Di j = Hhih j for i, j nearest neighbors0, otherwise. s15dNotice that both matricesK andD are real and symmetric.

    A. Bogoliubov transformation

    It is possible to diagonalize the spin-wave Hamiltonianthrough an operator transformation of the Bogoliubov type,

    ai = on

    suin an + vin and, s16d

    ai = o

    n

    svin* an + uin

    * and, s17d

    or, in matrix notation,

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  • S aa*D = SU V

    V* U*DS a

    a*D , s18d

    where the column vectorsa and a contain the operatorsaiandan, respectively, while theN3N matricesU andV con-tain the coefficientshuinj and hvinj, respectively, withi ,n=1, . . . ,N. Assuming that the new operators also obey thecanonical commutation relations,

    fan,am g = dnm and fan,amg = fan

    ,am g = 0, s19d

    one arrives at the following constraints for the transforma-tion coefficients:

    on=1

    N

    suin* ujn vinv jn

    * d = di j s20d

    and

    on=1

    N

    suin v jn vin ujnd = 0. s21d

    In matrix notation,

    U U V V = IN s22d

    and

    U VT V UT = 0, s23d

    whereIN is theN3N unit matrix. These relations can be putinto a more compact form by defining the matrices

    T = SU VV* U*

    D s24dand

    S = SIN 00 IN

    D . s25dThus, Eqs.(22) and (23) become one,

    T S T = S. s26d

    SinceS2= I2N, we find, after a simple algebra, that

    TS T = S. s27d

    As a result we have two additional(though not independent)sets of orthogonality equations,

    oi=1

    N

    suin* uim vin vim

    * d = dnm s28d

    and

    oi=1

    N

    suin* vim vin uim

    * d = 0. s29d

    B. Non-Hermitian eigenvalue problem (Ref. 15)

    The transformation defined by Eq.(18) allows us to diag-onalize the spin-wave Hamiltonian in terms of the new

    bosonic operators. For that purpose, we chooseT such that

    TSK DD K

    DT = SV+ 00 V

    D , s30dwhereV are diagonal matrices containing the eigenfrequen-cies: fVgnn=vn

    sd for n=1, . . . ,N. The eigenvalue problemdefined by Eq.(30) can be further simplified. Recalling Eq.(27), we have that

    SK DD K

    D T = S T SSV+ 00 V

    D . s31dIn fact, it is not difficult to prove that eigenfrequency matri-ces obey the relationV+=V

    * =V, with V being a diagonalmatrix with real entries only, as one physically expects. As aresult,

    SK DD K

    D SU VV* U*

    D = S U V V* U*

    D SV 00 V

    D . s32dWe can break up this 2N32N matrix equation into twocoupledN3N matrix equations,

    K U + D V* = U V,

    D U + K V* = V* V, s33d

    or, alternatively, writing explicitly the matrix elements,

    oj

    fKij ujn + Di j v jn* g = vn uin, s34d

    oj

    fDi j ujn + Kij v jn* g = vn vin

    * , s35d

    for all n and i. Thus, for a given eigenstaten, we can definean eigenvalue matrix equation in the usual form, namely,

    S K D D K

    DSunvn

    * D = vnSunvn* D . s36d(Notice that eachun and vn is now a column vector withcomponents running through alli =1, . . . ,N lattice sites.) The2N32N matrix shown in Eq.(36) is clearly non-Hermitian,but its eigenvalues are all real. Notice also that ifvn is aneigenvalue with corresponding eigenvectorsun,vn

    *d, then vnis also an eigenvalue, but withsvn

    * ,und as the correspondingeigenvector. Thus, despite the fact that the non-Hermitianmatrix provides 2N eigenvalues (eigenfrequencies), weshould only keep thoseN that are positive and whose corre-sponding coefficientsuin andvin satisfy Eqs.(20), (21), (28),and (29).

    The non-Hermitian matrix in Eq.(36) contains only inte-ger elements: 0, 1, 2, or 4 in the diagonal(corresponding toKii , i.e., the number of nearest neighbors to sitei) and 0 or 1in the off-diagonal components(corresponding toDi j , i.e., 1when i and j are nearest neighbors and zero otherwise). It isstrongly sparse, although without any particularly simplepattern due to the presence of dilution disorder.

    It is easy to verify that there are at least two zero modes inEq. (36), i.e., two distinct nontrivial solutions with zero ei-genvalue:

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  • ui0sad = 1, vi0

    sad = 1, s37d

    for all i =1, . . . ,N, and

    ui0sbd = vi0

    sbd =H 1, i P A, 1, i P B. s38d

    (In order to prove that these are indeed eigenstates, noticethat o j=1

    N Di j =Kii .) These two zero modes do not obey theorthogonality relation of Eq.(28); they have zero hyperbolicnorm instead.

    C. Average magnetization per site

    The total staggered magnetization can be written in termsof the expectation value of the spin-wave number operator:

    Mzstagg= Ko

    iPA

    Si z ojPB

    Sj zL=NS oi=1

    N

    kaiail, s39d

    where we have assumed that the sublattices contain the samenumber of sites:NA=NB=N/2. As a result, the average stag-gered spin per site along thez direction can be written as

    mz =Mz

    stagg

    N= S dmz, s40d

    with

    dmz =1

    Noi=1

    N

    dmiz, s41d

    and

    dmiz = kai

    ail. s42d

    Notice that dmz describes the spin-wave correction to theaverage staggered magnetization(always a reduction).

    In order to expressdmz in terms of the coefficientsvin anduin, we use Eqs.(16) and (17) to first write the site magne-tization at zero temperature in terms of eigenmodes. Upontaking the ground-state expectation value, we have to recallthat the vacuum contains zero eigenmodes. Hence,

    kanaml = kanam

    l = kanaml = 0, s43d

    while

    kanam l = dnm. s44d

    As a result,

    dmiz = o

    n=2

    N

    uvinu2, s45d

    where the sum runs only through eigenmodes withpositivefrequency(the zero modes have been subtracted).

    D. Reduction to anNN non-Hermitian eigenvalue problem

    It is possible to rewrite the 2N32N eigenvalue problemas two coupled eigenproblems, each one of orderN3N in-stead. The non-Hermitian character of the matrices involved

    does not change. However, the amount of work for numericalcomputations decreases by a factor of 4[recall that diagonal-izing a N3N requiresOsN3d operations].

    We begin by summing and subtracting Eqs.(34) and(35),obtaining

    oj=1

    N

    sKij + Di jdsujn v jnd = vnsuin + vind s46d

    and

    oj=1

    N

    sKij Di jdsujn + v jnd = vnsuin vind. s47d

    Multiplying these equations byKD andK+D, we find thefollowing eigenvalue equations after a simple manipulation:

    oj=1

    N

    fsK DdsK + Ddgi jsujn v jnd = vn2suin vind s48d

    and

    oj=1

    N

    fsK + DdsK Ddgi jsujn + v jnd = vn2suin + vind. s49d

    Although these equations are in principle decoupled, for thepurpose of finding the local magnetization they are not so,since we are interested in finding mainlyv (and notu+v oruv alone). We will come back to this point later. Equations(48) and (49) can also be presented in more revealing form,namely(here we will drop indices to shorten the notation),

    sK2 D2 fD,Kgdfsd = l fsd, s50d

    wherefsd=uv andl=v2. At this point it is interesting tonotice that the nonzero commutator is the cause of non-Hermiticity in the eigenvalue problem. Had it been zero, theproblem would be become real and symmetric. In fact, it isnot difficult to show that

    fD,Kgi j = Di jsKii Kjjd. s51d

    Thus, it is only when all sites have the same number ornearest neighbors, i.e., when no dilution is present, that theproblem becomes real and symmetric(and can therefore besolved analytically by a Fourier transform). Dilution alwaysmakes the eigenproblem non-Hermitian, although with realeigenvalues[even if we did not know the origin of Eq.(50),it would be easy to prove that all eigenvaluesl0].

    Let us callMsd=K2D2 fD ,Kg. An important feature ofthese matrices is that the left eigenvectorfsd of Msd is theright eigenvector ofMs7d. Thus, if we use an algorithm thatis capable of finding both the right and left eigenvectors of anon-Hermitian matrix, we only need to solve the problem forMs+d, for instance. In this case, we may say that the 2N32N problem has really been reduced toN3N.

    In terms of the eigenvectorsfsd, the orthogonality rela-tions of Eqs.(28) and (29) now read

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  • oi=1

    N

    ffins+d*fim

    sd + finsd*fim

    s+dg = 2 dnm s52d

    and

    oi=1

    N

    ffinsd*fim

    s+d fins+d*fim

    sdg = 0, s53d

    respectively. Moreover, using these relations and the defini-tion of fsd, it is straightforward to show that the average sitemagnetization can be written as

    dmz =1

    4Non=2N

    oi=1

    N

    fufins+du2 + ufin

    sdu2g 1

    2. s54d

    One issue that appears when diagonalizing the problemthrough solving Eq.(50) is that each eigenstateln may havean eigenvector corresponding to any linear combination oftheun andvn vectors, and not just thatunvn (that is becauseeachln corresponds to at least two eigenfrequencies, namelyv, with vn=ln). Provided that there are no other degen-eracies, one can sort out which combination is generated bynoticing the following. Suppose that

    fs+d = c+su vd and fsd = csu + vd, s55d

    then, it is easy to see that the normalization conditions forboth fsd andu,v imply c+c=1. We can then use Eqs.(46)and (47) to find that

    sK Ddfs+d =v

    c+2f

    sd s56d

    and

    sK + Ddfsd = c+2 v fs+d. s57d

    These equations provide a way of determining the coefficientc+ (and thus the actual mixing of degenerate eigenvectors).For instance, for a given eigenstate,

    c+2 =

    1l

    oi,j=1

    N

    fis+dsK + Ddi jf j

    sd

    oi=1

    fis+dfi

    sd. s58d

    Once thec+ coefficient has been determined, it is straightfor-ward to determine theun andvn vectors corresponding to agiven (positive), nondegenerate eigenfrequencyvn in aunique way. If additional degeneracy occurs, then one needsto introduce more coefficients(and consider combinations ofall degenerate eigenvectors) in Eq. (55).

    III. NUMERICAL SOLUTION OF THE NON-HERMITIANEIGENPROBLEM

    The numerical solution of the eigenproblem representedby Eq. (50) requires the full diagonalization of at least onereal nonsymmetric matrix. However, before that, we need togenerate the random dilution on a square lattice and set theappropriate boundary conditions. Another important point is

    that we can simplify the diagonalization by breaking the ma-trix into diagonal blocks, each one related to a single discon-nected cluster. The diagonalizations can then be carried outseparately on each block(for each disconnected cluster).Thus, the first task is to reorganize the matricesK and Dfollowing a hierarchy of disconnected cluster sizes. That in-volves only searching and sorting sites on the lattice(withoutany arithmetic or algebraic manipulation). Moreover, sincewe are interested only in what happens within the largestconnected cluster(the only one relevant in the thermody-namic limit and below the percolation threshold),7 we canconcentrate our numerical effort into the diagonalization ofthe matrix block corresponding to that cluster alone.

    The first step is to create a square lattice of sizeN=L3L (L being the lateral size of the lattice) with periodicboundary conditions in both directions. In order to haveNA=NB, we chooseL to be an even number. We fix the numberof holes as the integer part ofs1pdN and randomly distrib-ute them over the lattice with uniform probability.

    The second step is to identify all connected clusters thatexist in the lattice for a given realization of dilution. Sincemost lattices we work with are quite dense(relatively fewholes), we begin by finding all sites that belong to the clusterwhose sites are nearest to one of the corners of the lattice.Once all sites in that cluster are found, they are subtractedfrom the lattice and the search begins again for another clus-ter. The process stops when all sites have been visited andthe whole lattice is empty. The process of identifying sitesfor a particular cluster is the following. Starting from a fixedsite i (the cluster seed), we check whether its four neighborsare occupied or empty. The occupied ones get the same tagnumber as the first site visited. Then we move on to the nextside, i +1, and repeat the procedure. We continue until wereach theNth site.

    Along with identifying all sites belonging to each cluster,we also count then. That allows us to identify immediatelythe largest connected cluster in the lattice, whose number ofsites we callNc. We set a conversion table where the sequen-tial number identifying a site in the largest cluster is associ-ated to its coordinate in the original lattice. That allows us tolater retrace the components of the eigenvectors of this clus-ter to their locations in theL3L lattice.

    The process of identifying clusters is carried out underhard wall boundary conditions. In fact, it is only after thelargest cluster is found that we force periodic boundary con-ditions. This is the third step. For that purpose, we sweep thebottom and top rows, as well as the right and left columns ofthe square lattice, and check whether these sites are neigh-bors of sites belonging to the largest cluster once periodicboundary conditions are assumed. It turns out that it is easierand faster to do that than to search and classify clusters di-rectly from a lattice with periodic boundary conditions.

    The fourth step consists of storing the information neces-sary to assemble the non-Hermitian matrices of the typeshown in Eq.(36) for the largest cluster only. The algorithmis quite fast and allows one to generate and find the largestconnected cluster in lattices as large asL=100 in less than asecond.

    The information generated in the process of identifyingthe largest cluster and its structure is fed into a second rou-

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  • tine. There, one assembles both the non-Hermitian matricesof Eqs.(36) and (50). We have checked that the solution ofboth the 2Nc32Nc and Nc3Nc problems provide identicalsolutions up to several digits for a particular realization ofthe dilution problem at various lattice sizes and dilution frac-tions. However, only theNc3Nc problem was used to gen-erate the data presented here.

    It is important to point out that there exists an alternativeformulation of the problem defined by Eq.(13), using gen-eralized position and momentum operators(see AppendixA). In this formulation one can derive a sequence transfor-mation that permits the calculation of eigenvalues and eigen-vectors of the system Hamiltonian through the diagonaliza-tion of Hermitian matrices alone. However, from thecomputational point of view there is no substantial advantageof this approach with respect to the non-Hermitian one.

    Since the solution of the non-Hermitian eigenvalue prob-lem is less standard than the Hermitian case, we provide adescription of the method in Appendix B.

    IV. RESULTS

    We have generated lattices with sizes ranging fromL=12 to 36 and dilution fractions going fromx=0 to xc. Thenumber of realizations for a given size and dilution fractionvaried between 500(nearly clean case) to 1000(at the clas-sical percolation threshold). The results of the numerical di-agonalizations are described below.

    A. Density of states

    The ensemble averaged density of eigenstates as a func-tion of frequency forL=36 is presented in Fig. 1 for severaldilution fractions and compared to the well-known result forthe clean case.16 For a small dilution, there is little departurefrom the clean case, although a small structure is alreadyvisible at aroundv /JS=3. As the dilution increases, a peakand an edge develop at around this frequency. Notice that the

    overall trend is a decrease in the number of high-frequencymodes, with the proportional increase in the number of low-frequency ones. Close to the percolation threshold, anotherstructure appears at aroundv /JS=2. Thus, we see that theeffect of dilution is to shift spectral weight from high to lowfrequencies in a nonuniform way. This tendency was alsoobserved in Ref. 9 for small dilution.

    Two additional very sharp peaks(not shown in Fig. 1)also exist in the density of states as the dilution increases.They occur at frequenciesv /JS=1,2 andcorrespond to con-figurations wherevin=0 for all sites in the cluster, whileuin=0 for all but two sites. Their typical spatial structures areshown in Fig. 2. Sincevin=0 for all sites, these states do notcontribute to the quantum corrections to the staggered mag-netization.

    For the clean case, it is simple to verify(based on theexact diagonalization of the problem) that the low frequencymodes provide the largest contribution todmz. For the dilutelattice, the same is true, as can be seen in Fig. 3, where wehave plotted

    dmzsvd =1

    N

    Koi=1

    N

    on=2

    N

    dmiz dsv vndL

    Kon=2

    N

    dsv vndL . s59dAs a result, we see that the transference of eigenmodes fromhigh to low frequencies is the mechanism by which quantumfluctuations are enhanced as the dilution increases.

    FIG. 1. Average density of states as a function of energy for fourdifferent site dilution fractions:x=0 (solid line), x=0.1 xc (dottedline), x=0.5 xc (dotted-dashed line), andx=xc (dashed line). Noticethe two structures just belowv /JS=2 and 3. The latter is alreadyvisible at x=0.1 xc, while the former becomes prominent only forx*0.4 xc. The data was obtained from 500 to 1000 realizations of a36336 lattice.

    FIG. 2. The two dangling structures that occur in thev /J=1 (a)and 2 (b) eigenstates. The filled circles indicateuin0, while theempty circles haveuin=0. All sites in these states havevin=0, thusthey do not contribute to the staggered magnetization.

    FIG. 3. Staggered magnetization per unit of magnetic site as afunction of energy,dmzsvd, under the same conditions of Fig. 1.

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  • B. Inverse participation ratio

    The nature of the eigenstates also changes as the dilutionincreases. The best way to characterize the nature of thestates is through the return probability, that is, the probabilitythat after some very long time a particle, moving in the per-colating lattice, will return to its originating point. The returnprobability can be expressed in terms of the inverse partici-pation ratio (IPR).17 Here, we use a definition of the IPRinvolving the eigenvector component related to the quantumfluctuation corrections to the magnetization, namely,

    Isvd =on=2

    N

    dsv vndIn

    on=2

    N

    dsv vnd

    , s60d

    where

    In =

    oi=1

    Nc

    vin4

    Soi=1

    Nc

    vin2 D2 . s61d

    In Fig. 4 we show the IPR as a function of energy forthree lattice sizes. According to its definition, the IPR forextended states decreases as the system size increases, whilefor localized states the IPR is insensitive to any size varia-tion. These trends are clearly visible in Fig. 4, namely, statesare mostly extended when dilution is small and tend to lo-calize as one gets closer to the percolation threshold. Forintermediate dilution[Fig. 4(c)], we see that the states closeto v /JS=3 are strongly localized while the remaining statesare quite extended. As expected, the low frequency statestend to remain extended up to strong dilution.

    C. Average magnetization

    In the thermodynamic limit, the magnitude of the stag-gered magnetization per unit of lattice sitemz can be writtenas

    mz = S NcNmDsS dmzd, s62dwhereNm is the total number of occupied sites in the latticesNm=pNd. While the first factor on the right-hand side of Eq.(62) is purely classical, the second factor is purely quantum,namely, it measures how quantum fluctuations reduce mag-netic order. Thus, sinceNc vanishes atxc, a quantum criticalpoint belowxc can only exist if, for somex

    * ,xc, we finddmz=S. If, on the contrary,dmz,Sat x=xc, then the order isonly lost at the percolation threshold and the transition isessentially classical. It is important to have in mind that thelinear spin-wave approximation is well defined only whendmi

    z!S, for all i =1, . . . ,N. Whendmiz

  • mental data, namely, a progressive decrease of the staggeredmagnetization up to the classical percolation threshold. At adilution fraction very close toxc, our simulations indicatethat the staggered magnetization should vanish. The inset inFig. 5 shows that the vanishing of the staggered magnetiza-tion occurs becausedmz goes to 1/2 very close to the clas-sical transition point. Thus, the same effect would not arisehad we usedS.1/2. The QMC simulations, on the otherhand, predictsdmz,1/2 at x=xc, thus indicating that thetransition is purely classical. The relatively small number ofexperimental points and the large error bars near the perco-lation threshold do not allow for an adequate distinction be-tween a classical and a quantum transition for LCMO.

    The discrepancy between our result and the QMC simu-lations for the staggered magnetization close toxc should beseen as an indication that, while qualitatively correct, ourapproach fails quantitatively when the order parameter mag-nitude is significantly reduced locally. This is expected if werecall the assumption used in the derivation of Eq.(10). Nev-ertheless, the spin-wave approximation, having access tolow-lying excited states and wave functions, allows us tounderstand in more detail, at least qualitatively, how the sup-pression of order due to quantum fluctuations takes placeupon dilution. This is not the case for the QMC simulations.In fact, it is surprising that our calculations seem to agreewith the experimental data better than the QMC. This can beunderstood by the fact that the experimental system maycontain extra oxygen atoms that introduce holes in the CuO2planes, as well as next-nearest neighboring interactions thatfrustrate the AFM state and also introduce larger quantumfluctuations that are captured by the overestimation producedin the linear spin-wave theory. In fact, it is known thatLa2CuO4, has a nonzero frustrating next-nearest neighborcoupling.4 That effectively decreases the spin per site to avalue smaller than 1/2, possibly bringing LCMO closer to aquantum critical point than the pureS=1/2 HeisenbergAFM.

    D. Local fluctuations

    We now turn to the question of local fluctuations. We haveso far discussed the site-averaged demagnetizationdmz andused the criterion that it must be smaller thanS for the orderto persist. However, one could argue that some sites mayhave particularly large fluctuations; if these large fluctuationstake place exactly at weak links of the largest connectedcluster backbone, then they could be responsible for earlierdestruction of the long-range order.11 We have numerical evi-dence that this is unlikely, although the relatively small sizeof our lattices does not allow us to be conclusive. In Fig. 6we show an intensity plot of the local quantum fluctuationsdmi in the largest connected cluster, very close to the perco-lation threshold, for a typical realization of aL=32. Noticethat the largest fluctuations tend to appear only along dead-ends or dangling structures, and not in the links connectingblobs of the cluster backbone. The same trend is seen in allrealizations that we have inspected.

    E. Excited states

    Equation(13) represents a system ofN coupled harmonicoscillators withA=K+D andB=KD being the spring con-

    stant and the inverse mass tensors, respectively(for an alter-native description of the problem in terms of position andmomentum operators, see Appendix A). For the simplemodel of elastic vibrations in a lattice, whenqi =sai+ai

    d /2 has the meaning of a displacement of theith atomaround its equilibrium position, it is well known that theHamiltonian of such a system can be mapped onto a problemof diffusion in a disordered lattice.6 Analytical results for therelated diffusive problem, as well as numerical simulationswith large systems, allow one to understand several proper-ties of the diluted vibrational model, such as the density ofstates(DOS) and the dynamical structure factor. Perhaps oneof the most distinctive features is the existence of stronglylocalized excitations named fractons.20 For dilution fractionsx,xc, these excitations appear in the high-frequency portionof the spectrum, beyond a certain crossover scalevc, whilethe low-frequency part is dominated by acoustic phonon like,nearly extended, excitations. At exactlyx=xc, the systemsbecomes a fractal and the fractons take over the entirespectrum.6 Scaling considerations, as well as numericalsimulations, have shown that, for a square lattice, the DOSbehaves as

    rsvd ~ H v, v , vcv1/3, v . vc.

    s63d

    The crossover frequency depends critically on the dilution:vc~ sxcxdD, whereD

  • percolation threshold, indicating that quantum fluctuationsare likely not sufficiently enhanced by the dilution to destroythe existent long-range order.10 We will get back to this pointin Sec. V.

    In order to investigate the existence of such fractons in thedilute 2D Heisenberg AFM, as well as to clarify the nature ofits low-lying excitations, we have calculated the the dynami-cal structure factor,

    Ssq,vd =E dt eivt oi,j=1

    Nm

    eiqsRiR jd 3 kSi+s0dSj

    std + Sis0dSj

    +stdl.

    s65d

    By using Eqs.(3), (6), and (18), we can expressSsq ,vd interms of Fourier transformations of the site-dependent Bogo-liubov coefficientsuin andvin. We find

    Ssq,vd = 2S ovn0

    dsv vndfunAsqdun

    As qd+ vnAsqdvn

    As qd

    + unAsqdvn

    Bs qd+ vnAsqdun

    Bs qd + unBsqdvn

    As qd

    + vnBsqdun

    As qd + vnBsqdvn

    Bs qd+ unBsqdun

    Bs qd + g,

    s66d

    where the partial terms involvingunA,B are given by the Fou-

    rier transformation ofun, namely,

    unA,Bsqd = o

    iPA,Buin e

    iqr i , s67d

    and analogously forvnA,B. Notice that the sum over sites runs

    only over one of the sublattices,A or B, depending on theparticular term. Thus, only four two-dimensional Fouriertransformations are required in order to evaluateSsq ,vd.

    We have computed numerically the Fourier transforma-tions and calculated the average dynamical structure factorfor lattices of sizeL=32 at several dilution fractions. Onlysites within the largest connected cluster were taken into ac-count. Averages were performed over 50 realizations foreach case. The results are presented in the form of intensityplots in Fig. 7. To provide a better contrast, we have rescaledkSsq ,vdl by the function,fsvd=oq kSsq ,vdl. Only the dataalong two particular directions in momentum space areshown, namelyq=qx=qy and q=qx,qy=0, with 0qp(the lattice spacing is taken to be unit). For small dilution,the structure factor resembles closely that of the clean case,with some small broadening of the magnon branch due to theweak destruction of translation invariance. However, particu-larly along theqy=0 direction, one can already notice a smallhump at aroundv /JS=3, consistent with the peak-and-edgestructure seen in Fig. 1. This feature becomes more promi-nent with increasing dilution. For dilution fractions largerthan 0.6xc, another hump becomes visible at aroundv /JS=2, again consistent with the feature observed in Fig. 1 at thesame frequency. Close to the percolation threshold, there ex-ist three clear broad branches in the spectrum. While thedispersion of the high-frequency branch atv /JS.3 ishardly affected by the dilution, the opposite occurs with thelow-frequency one, atv /JS,2, where the slope(magnonvelocity) decreases with increasing dilution. We interpret the

    progressing breaking and bending of the magnon branch asthe system becomes more diluted to the appearance of frac-tons. At x=xc the excitation spectrum has little resemblanceto that ofx=0 and even the long wavelength part is stronglymodified. In between these two limits, there is a crossoverfrom a magnon-dominated to a fracton-dominated spectrum.A three-branch structure for the spectral function in the spin-wave approximation was also found in Ref.9 However, thepositioning of those branches and their relative intensitywere different from what we observed in our numerical so-lution. We believe the cause of this discrepancy is the limitedrange of applicability of the perturbative treatment, which isexpected to be accurate only in the weak dilution regime.

    The gradual appearance, broadening, and motion of thebranches are better represented in Fig. 8, where the rescaledaverage dynamical structure factor is shown as a function offrequency for a fixed momentumqx=0.4/p and qy=0. Thetwo-peak structure observed in our numerical data has someresemblance to the results of inelastic neutron scattering per-formed by Uemura and Birgeneau for the compoundMnxZn1xF2,

    13 whose magnetic properties are described bythe three-dimensional site-diluted random Heisenberg modelwith S=5/2.These authors observed two broad peaks whichwere associated with magnons(low-energy, extended) andfractons (high-energy, localized) excitations. The relativeamplitude, width, and dispersion relations of these excita-tions were measured and were found consistent with the the-oretical predictions by Orbach and Yu,21 and Aharony andco-workers.22 The scaling theory of the latter predicts that atwo-peak spectrum appears at momentaq,j1, wherej isthe percolation correlation length(i.e., the typical linear sizeof the disconnected clusters whenx,xc). The relativelysmall size of theL=32 lattice does not allow us to make asimilar quantitative comparison between these theories andour numerical results.

    While we do observe a two-peak structure at sufficientlylarge dilution, we also see a third peak, although quite weak.This additional peak may be characteristic of square lattices;in the body-centered-cubic MnxZn1xF2, there might exist adifferent multipeak structure. Moreover, th elarge broadeningand limited energy resolution of the neutron-scattering ex-periments may have made such features unobservable in thedata of Ref. 13. However, one aspect which seems differentbetween the experimental data and our numerical simulationsis the way the spectral weight is transferred between mag-nons and fractons as a function of dilution. While Uemuraand Birgeneau see an increase(decrease) of high-frequencyfractons(low-frequency magnons), we see the opposite: Thelow-frequency portion of the spectrum becomes more in-tense.

    The averaging procedure recovers, to some extent, thetranslation invariance broken by the dilution. This allowed usto identify the large momentum part of the branches which,otherwise, would not be visible. However, the low momen-tum part is clearly visible even when no averaging is per-formed on the data(not shown). This fact, together with thestrong dispersion of the lower energy branches, suggests thatlong wavelength propagating magnons are present for dilu-tion fractions below the percolation threshold, and lose out tofractons atx=xc. At T=0, these modes are responsible for the

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  • quantum fluctuations that bring down the long range order.Their spectral weight becomes more important as the high-frequency states become more localized with increasing di-lution. This feature is masked in Fig. 7 by the frequency-dependent rescaling, but is clear from Fig. 1. The high-frequency, high momentum modes are much less dispersivethan the low-frequency, low momentum ones. In fact, par-ticularly below v /JS=3, the magnon branch isq indepen-dent (i.e., broad in momentum) and thus strongly localized.

    The low-frequency modes are likely not fully ballistic(coherent), given the large broadening seen in the structurefactor, but should rather have a diffusive propagation at largescales. As to whether they remain two dimensional or gain a

    lower-dimensional character, our data is not conclusive.

    V. UPPER BOUND FOR QUANTUM FLUCTUATIONS INBOSONIC SYSTEMS

    The possibility of a classification of quantum random sys-tems into universality classes is an important theoreticalproblem with relevance to experiments. Random matrix clas-sification schemes introduced by Wigner limited the classesof problems initially to the orthogonal, unitary, and symplec-tic classes,23 but these were later extended to encompass theclasses of chiral models24 and models where the particlenumber is not a conserved quantity.25 While most of the ran-

    FIG. 7. Gray plots of the res-caled average dynamical structurefactor sliced along two distinct di-rections in momentum space:qx=qy=q [(a), (c), and (e)] and qx=q,qy=0 [(b), (d), and (f)]. Plotsat different dilution fractions donot necessarily have the same ar-bitrary scale for the gray intensity.The most prominent branches aremarked with dashed lines. Averag-ing for each plot was performedover 50 realizations of dilution.

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  • dom matrix problems are related to fermionic spectra, thereis renewed interest in the problem of bosonic random matrixtheory.26 Of particular importance is its application to theproblem of diluted quantum magnets since these systems, inthe limit of large spinS, can be approximately described, vialinear spin-wave theory, as a problem of noninteractingbosons.14

    One of the main differences between fermions and bosonsis that, in addition to the symmetries of the underlyingHamiltonian, one must ensure that the bosonic spectrum issemipositive definite; this stability condition is not an issuein fermionic systems. However, it is automatically satisfiedwhen the disorder is caused by site dilution.

    Let us concentrate our discussion on systems whoseHamiltonian can be mapped onto a set ofN coupled har-monic oscillators of the following kind:

    H =1

    2 oi,j=1N

    sqi Aij qj + pi Bij pjd. s68d

    Here, N is the total number of sites of the square lattice,while qi and pi represent generalized position and momen-tum operators at a lattice sitei, respectively, such thatfqi ,pjg= i di j , with i , j =1, . . . ,N. TheN3N matricesA andBare real, symmetric, and semipositive; in the most generalcase, they do not commute. The magnitude of quantum fluc-tuations are characterized by the mean-square deviation ofthe average value of these operators in the ground state:q2

    =oi kqi2l /N andp2=oi kpi

    2l /N. As mentioned in Sec. IV C, ifquantum fluctuations are unbounded at the percolating re-gime, then these mean values diverge and LRO is not pos-sible, implying that order has to be destroyed beforexc isreached; that is, a quantum critical point should exist beforethe percolation threshold. Here we show that this is not thecase and that LRO can persist up to and includingxc. As forthe spin Hamiltonians approximated by Eq.(68), the exactdisappearance of magnetic LRO may also depend on the spinvalue.

    The eigenfrequencies of the harmonic oscillators of Eq.(68) can be shown to be those of

    VL2 = AB or VR

    2 = BA. s69d

    Since, in general,fA,Bg is nonzero, the matricesVL2 andVR

    2

    are non-Hermitian and distinct. However, it is simple toshow that they have exactly the same real eigenvalues,vn

    2,for n=1, . . . ,N. This is also true for the matricesVC

    2

    =B1/2AB1/2 andVC2 =A1/2BA1/2. Therefore, the fluctuations of

    the position and momentum, averaged over all sites, can bewritten as27

    q2 =1

    2NtrsB VC1d

    b*2

    k, s70d

    p2 =1

    2NtrsA VC1d

    a*2

    k, s71d

    where a* and b* denote the maximum eigenvalues of thesemipositive definite matricesA andB, respectively, and weused that

    k =1

    Ntr VC

    1 =1

    Ntr VC

    1 =E0

    vmax

    dv rsvdv1, s72d

    sinceVC andVC share the same eigenvalues(vmax being thelargest), with spectral densityrsvd. Thus, the finiteness ofthe quantum fluctuations reduces to the problem of the con-vergence of the integral in Eq.(72) (quantum mechanics re-quires thatq2 p21/4). Notice that the quantum fluctuationcorrection to the order parameter per unit of lattice site canbe written as

    dmz = a q2 + b p2 + g, s73d

    wherea, b, andg are constants that depend on the particularmodel and order parameter under consideration.

    We now turn to applying these general results to specificbosonic models based on Heisenberg Hamiltonians.

    A. O(2) model

    This model is realized, for example, in an array of Joseph-son junctions, where the variablesqi correspond to the lin-earized phase of the superconductor order parameter.10,11Thematrices in Eq.(68) take the simple formsAij =Kij Di j andBij =sU /Jddi j , whereDi j =1s0d when the nearest-neighboringsites i , j are both occupied(otherwise) and Kij =di j ol=1

    N Dil ,i.e., Kii counts the number of nearest neighbors of sitei.Here,J denotes the Josephson coupling andU is the islandcharging energy(usually,J@U). The same structure occursin the case of vibrations in a diluted lattice.6 For the O(2)model, the frequency eigenvalues are obtained fromV2

    =sU /JdA. The problem of determining the density of statesof the connectivity matrixA can be mapped onto a problemof diffusion in a disordered lattice.6 It can be shown that thedensity of states ofV goesrsvd~vd

    *1. Up to the percola-

    tion threshold,d* =d=2, while d* = d

  • B. Heisenberg antiferromagnetic model

    When we takeJx=Jy=Jz, the matrices take the formAij=Kij Di j andBij =Kij +Di j . It is useful to define a matrixL=LT that has the effect of changing the signs ofqi andpi forall sites i in one of the two sublattices. Thus,B=LAL, andVR

    2 =LALA. Notice thatG=LA, is not semipositive definite(in fact it is non-Hermitian), but has real eigenvalues withthe same magnitude as those ofV.

    In the O(2) model, becauseV2~A, one can directly relatethe energy eigenvalues ofV to those of the matrixA. In theAFM case, we need to obtain the density of states ofG=LA, which is not simply related to that ofA (sincefL ,Ag0 in general for diluted systems). This nontrivial relationbetween the eigenvalues ofV andA is generally present inbosonic problems. For example, a similar problem also ap-pears in the work of Gurarie and Chalker in the relationbetween their stiffness and frequency eigenvalues.26

    The problem of determining the density of states ofG fora random dilution problem is one of the interesting openquestions related to the important difference between randomfermionic and bosonic systems. In a fermionic problem thisquestion would be already answered by matching the sym-metries ofG to the Cartan classification table.25 However,one cannot substitute the matrixA by an arbitrary randommatrix with similar symmetries, that would violate the semi-positive definite constraint. In this work we do not attempt toanalytically resolve this problem; however, we find numeri-cal evidence that the density of states ofG2 follows that ofAat low energies.

    Our simulations show that the site-averaged fluctuationsdmi

    z are bounded, both below and at the percolation thresh-old. This means that order should exist up to and includingthe critical dilutionxc, as long asdmi is small compared tothe value of the spinS. Recalling Eq.(62), we conclude thatthere could be a minimum valueSmin for the spin, belowwhich there is a quantum phase transition for dilutionsx,xc.An effective local spin smaller than 1/2 can be realized in abilayer system with antiferromagnetic interlayer coupling.28

    C. XXZ model

    In this case,Jx=JyJz. We then haveAij =Kij Di j andBij =Kij gDi j , where g measures the anisotropy. Alterna-tively, we can writeB=s1+g /2dA+s1g /2dLAL, and soV2=s1+g /2dA2+s1g /2dLALA (notice that forg1 wehave the same problem as the AFM). The analysis is similarto the previous two cases. The amount of fluctuations is con-trolled by the anisotropy, and it can be shown to be boundedif the density of states follows that of the AFM case.

    VI. DISCUSSION AND CONCLUSIONS

    In this work we have studied the role played by site dilu-tion in enhancing quantum fluctuations in the ground state ofthe Heisenberg antiferromagnet in a square lattice. Using thelinear spin-wave approximation for this model, we have per-formed exact numerical diagonalizations for lattices up to36336, with dilution fractions going from the clean limit tothe classical percolation threshold. Our results indicate a pro-

    gressive, nonuniform shift of spectral weight in the spin-wave excitation spectrum from high and to low frequency asthe dilution increases, with the high-frequency part becom-ing more localized. The higher density of low-frequency,long wavelength excitations leads to strong quantum fluctua-tions and a decrease in the magnitude of the staggered mag-netization. For dilutions very close to the classical percola-tion threshold, we have found that quantum fluctuations aresufficiently strong to nearly match the clean-limit magnitudeof the magnetization whenS=1/2, but not forhigher spins.This is consistent with recent neutron-scattering experimentswith the S=1/2 dilute Heisenberg antiferromagnetLa2Cu1xsZn,MgdxO4, which show that long-range order dis-appears at around the classical percolation threshold. How-ever, quantum Monte Carlo simulations suggest that quan-tum fluctuations should remain small and that the destructionof long-range order is controlled only by the disappearanceof the infinite connected cluster. We understand this discrep-ancy between the quantum Monte Carlo results and the linearspin-wave theory near the classical percolation threshold asan indication that the latter has its validity limited, as themagnitude of the order parameter is too small.

    While perhaps not quantitatively accurate, our simulationsdo allow us to probe into nature of the low-lying excitedstates of the dilute antiferromagnet. We observe two clearhumps in the density of states at frequenciesv /JS=2 and 3.By calculating the ensemble-averaged dynamical structurefactor, we were able to associate the appearance of thesehumps with the breaking of the clean-limit magnon branchinto three distinct but broad branches. The new branches tendto be strongly localized(nondispersive) at high frequenciesand have a diffusive, rather than ballistic, nature at low fre-quencies. In the literature, the multipeak structure had beenassociated with the appearance of fractons in the excitationspectrum as the dilution increases. From our simulations, itseems that the picture is somewhat more complex. Besidesthe overall broadening, the position of the high-frequencybranch remains close to the corresponding portion of originalclean-limit magnon branch, while the magnon velocity(thedv /dq slope) in the low-frequency branch is continuouslyreduced with increasing dilution. Therefore, it appears thatthe fracton character also contaminates the low-frequencybranch. However, the lack of resolution due to the finite sizeof our lattices does not allow for a conclusive picture. We didnot attempt, however, to study fracton states which can pos-sibly occur above the percolation threshold.20,29

    It is important to remark that, in principle, due to Ander-son localization in two dimensions, we expect that in theinfinite system all excitations should in fact be localized forany finite dilution. In order to probe more carefully stronglocalization and the consequent exponential decay in realspace, we need not only much larger lattices, but also tocalculate two-point correlators, which goes beyond the appli-cability of linear spin-wave approximation. For that samereason, we were not able to evaluate quantities such as thespin stiffness, which involves matrix elements of higher thanbilinear operators.

    We also studied the question of local quantum fluctuationsas a way to destroy long-range order. For finite-size lattices,we found that the weak links do not show strong quantum

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  • renormalizations. That provides some indication that localquantum fluctuations may not be sufficient to change thedominance of the classical percolation picture.

    Using a more general analytical formulation, we have ar-gued that there exists an upper bound for the quantum fluc-tuations in any model with a classically ordered ground statewhose Hamiltonians can be mapped onto that of a system ofcoupled harmonic oscillators. The amount of quantum fluc-tuations depends directly on the low-energy behavior of thedensity of states of the associated bosonic problems. Ourexact diagonalization of the linear spin wave Hamiltonian ona percolating lattice led us to identify the value of the upperbound for one particular type of model and can readily beused to find similar values for any other bosonic model. Thiscould be used to study a large class of spin Hamiltonians thatcan be bosonized in the ordered phase.

    ACKNOWLEDGMENTS

    We thank I. Affleck, H. Baranger, J. Chalker, A. Cherny-shev, M. Greven, J. Moore, C. Mudry, A. Sandvik, T. Senthil,O. Sushkov, O. Vajk, A. Vishwanath, and M. Vojta for usefulconversations. We acknowledge financial support fromCNPq and PRONEX in Brazil(E.R.M.), and from the NSFthrough Grants No. DMR-0103003(E.R.M.), No. DMR-0343790(A.H.C.N.), and No. DMR-0305482(C.C.).

    APPENDIX A: FORMULATION IN TERMS OF COUPLEDHARMONIC OSCILLATOR

    The spin-wave Hamiltonian of Eq.(13) can be repre-sented in terms of position and momentum operators. In thislanguage, it becomes more transparent that the problem offinding the eigenvalues and eigenvectors of the Hamiltoniancan be solved by the diagonalization of two real symmetricmatrices, an alternative to the non-Hermitian eigenvalue for-mulation of Sec. II B.

    Let us perform the following operator transformation:

    qi =ai + ai

    2 and pi =ai ai

    i2, sA1d

    for all i =1, . . . ,N. Notice that the operatorsqi and pi obeythe canonical position-momentum commutation relations. Itis convenient to adopt a matrix formulation for the problem,namely,

    H =JS

    2sqTA q + pTB pd, sA2d

    wherex=hxiji=1. . .N and p=hpiji=1. . .N denote vectors of posi-tion and momentum operators, whilefAgi j =Kij +Di j , andfBgi j =Kij Di j . The quantum fluctuation correction to thesublattice magnetization can be written as

    dm=1

    2Ntrfkq qTl + kp pTlg

    1

    2. sA3d

    We can diagonalizeB through an orthogonal transformationU,

    UTBU = b sdiagonald sA4d

    and define new coordinates such that

    q = Uq8 and p = U p8. sA5d

    Defining A8=UTA U, we then have that

    H =JS

    2sq8TA8 q8 + p8Tb p8d sA6d

    and

    dm=1

    2Nftrskq8 q8Tld + trskp8 p8Tldg

    1

    2. sA7d

    It is not difficult to prove that all elements in the diagonalof b are positive except one,b1, which is zero. In order toeliminate this zero mode, we subtract the corresponding rowor line in all vectors and matrices, which amounts to a re-duction in the Hilbert space(or, alternatively, to setq08=0):hq8jN hq8jN1 and hp8jN hp8jN1. Also, fA8gN3N fA8gN13N1 andfbgN3N fbgN13N1. Notice that now allbk.0, k=2, . . .N. Thus, in this new space, we can performthe following rescaling:

    q9 = b1/2q8 and p9 = b1/2p8, sA8d

    which allows us to write

    H =JS

    2sq9TA9 q9 + p9Tp9d, sA9d

    with A9=b1/2A8b1/2 as well as

    dm=1

    2Nftrsbkq9q9Tld + trsb1kp9 p9Tld+ kp81

    2lg 1

    2,

    sA10d

    where the last term within the square brackets reflects theexistence of a zero mode. It is useful now to return to the sitebasis by carrying out the inverse rotation,

    q- = Uq9 and p- = U p9, sA11d

    such that

    H =JS

    2sq-TC q- + p-T p-d, sA12d

    where C=U A9 UT is the new connectivity matrix. If wedefine

    B1/2 = U b1/2 UT and B1/2 = U b1/2 UT sA13d

    within the subspaceN13N1 where no zero mode ispresent, we can write the connectivity matrix as

    C = B1/2AB1/2. sA14d

    Also, notice that the inverse rotation does not change theexpression of the sublattice magnetization,

    dm=1

    2Nftrsb1kq-q-Tld + trsbkp-p-Tld + kp80

    2lg 1

    2.

    sA15d

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  • We can now perform the last operation, namely, the di-agonalization of the connectivity matrix,

    qiv = Vq- and piv = V p-, sA16d

    yielding

    H =JS

    2sqiv Tc qiv + piv Tpivd, sA17d

    where

    VTC V = c sdiagonald, sA18d

    with all cn.0, n=1, . . .N1. We finally arrive to a system ofdecoupled harmonic oscillators. The quantum fluctuationpart of the magnetization becomes

    dm=1

    2Nftrsb1VTkqivqiv TlVd + trsb VTkpivpiv TlVd

    + kp802lg

    1

    2. sA19d

    However, we know that

    fkqivqiv Tlgnm= dnm kqniv 2l =

    1

    2fcgnm

    1/2 sA20d

    and

    fkpiv piv Tlgnm= dnmkpniv 2l =

    1

    2fcgnm

    1/2. sA21d

    Therefore,

    dm=1

    4Nftrsb1 VT c1/2 Vd + trsb VT c1/2 Vd + kp80

    2lg 1

    2.

    sA22d

    It is interesting to notice that, sinceA =O B O, we havethat

    C = V2, sA23d

    where

    V = B1/2 O B1/2. sA24d

    Thus, it is easy to see that ifC were a positive matrix, thenwe would be able to writeVTc1/2V ;C1/2=V. That wouldallow us to simplify the expression for the sublattice magne-tization a step further. However, the connectivity matrix isnot necessarily positive.

    APPENDIX B: NUMERICAL DIAGONALIZATION OF THENON-HERMITIAN MATRIX

    The diagonalization of the non-Hermitian matrices con-sists of five steps. First, the real(but asymmetric) matrixMs+d is reduced to an upper Hessenberg form through anorthogonal transformation, namely,A=QMs+dQT. This isdone by using theLAPACK subroutinesDGEHRD andDORGHR.Second, we use theLAPACK subroutineDHSEQR to performthe Schur factorization of the Hessenberg matrix:A=ZTZT.That allows us to obtain the eigenvalues and the Schur vec-tors, which are contained in the orthogonal matrixZ. Third,using anotherLAPACK subroutine,DTREVC, we extract fromZ both right and left eigenvectors ofMs+d. In the fourth stepwe renormalize all eigenvectorshfsdj such that they satisfyEq. (52) [the condition in Eq.(53) is automatically satisfied],sort the eigenvalueshlnj in ascending order, and extract thezero mode from the spectrum. For some realizations, thelowest eigenvalues next to the zero mode cannot be distin-guished from the zero mode itself and are therefore ne-glected. Finally, the correct linear combination offn

    sd thatprovides the correctun and vn for each positive eigenfre-quencyvn=ln is obtained according to the algorithm pre-sented in Sec. II D.

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