a

ets

ar sigmactively, inmperaturesvior of theshown to

e gapless,ted to thein the

urring in

Physics Letters A 346 (2005) 121–127

www.elsevier.com/locate/pl

Effects of disorder in two-dimensional quantum antiferromagn

C.M.S. Conceição, E.C. Marino∗

Instituto de Física, Universidade Federal do Rio de Janeiro, Cx.P. 68528, Rio de Janeiro, RJ 21941-972, Brazil

Received 9 June 2004; accepted 18 July 2005

Available online 27 July 2005

Communicated by A.R. Bishop

Abstract

We study the effects of disorder in two-dimensional quantum antiferromagnets on a square lattice, within the nonlinemodel approach, by using a random continuum distribution of spin stiffnesses or zero-temperature spin gaps, respethe renormalized classical and quantum disordered phases. The quenched staggered magnetic susceptibility at low teis evaluated in each case, showing that the phase structure of the clean system is preserved. The asymptotic behaquenched static and dynamic spin correlation functions is also obtained in the quantum disordered phase. Disorder isintroduce a change from exponential to power-law decay in these functions, indicating that the spin excitations becomin spite of the fact that the system is in a paramagnetic state. The scale determining the change in behavior is relavariance of the distribution function. A comparison is made with the dual behavior of skyrmion topological excitationsrenormalized classical phase. The effect found here is compared with the one produced by Griffiths singularities occdisordered systems. 2005 Elsevier B.V. All rights reserved.

PACS: 75.10.Jm; 75.10.Nr; 75.70.-i

Keywords: 2D antiferromagnets; Disorder

er-lot

ant

emsr-ingruc-cor-th-hentlyted-

1. Introduction

The physics of two-dimensional quantum antifromagnetic (AF) systems has been attracting aof interest, to a large extent due to the importrole played by them in the high-Tc superconducting

* Corresponding author.E-mail address: marino@if.ufrj.br(E.C. Marino).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2005.07.051

cuprates. The continuum description of these systin terms of a Nonlinear Sigma Model (NLSM), in paticular, has proved to be very convenient for extractrelevant physical information such as the phase stture, order parameters, magnetic susceptibilities,relation functions and critical exponents, among oers [1–5]. The presence of intrinsic disorder, on tother hand, is an important circumstance, frequeoccurring in real systems, which has been investigain several situations[6,7]. Special interest, in particu

.

122 C.M.S. Conceição, E.C. Marino / Physics Letters A 346 (2005) 121–127

ntin

-ings-

séel

is-ntges

r onci-asns.vesseti-her forver,, inneti-

as ancethed

ys-

ofeticonses

tiondis-

odou-se,

eof-therr

SM

seara-

t

ionm,

ic

mese-ion

zeroilethelessto

ndor-

gaptoara-tionion

av-

er-ere

idedreing

era-or-

lar, has been aroused in disordered systems presequantum phase transitions[8–10]. The clear experi-mental observation of a spin glass phase in highTc

cuprates, is another strong motivation for investigatthe introduction of disorder in 2D quantum AF sytems[11].

In a recent paper[12], the effect of disorder habeen investigated in the renormalized classical, Nordered, phase of the NLSM atT = 0. This was donethrough the introduction of a random continuum dtribution for the spin stiffness, which is the relevacontrol parameter in this phase. Quenched averawere then used to evaluate the effects of disordethe correlation functions of quantum topological extations (skyrmions). An interesting consequence wfound concerning the energy gap of these excitatioIn the Néel phase, without disorder, skyrmions haa finite energy gap, proportional to the spin stiffne(ρs ), which is the square of the staggered magnzation [13]. This fact usually enables one to use tskyrmion energy gap as a reliable order parametethe Néel phase. When disorder is introduced, howeone finds that the skyrmion energy gap may vanishspite of the fact that the quenched staggered magzation is different from zero[12]. This means that incertain situations one cannot use the skyrmion gapsuitable order parameter for AF order. The occurreof these situations is determined by the behavior ofdistribution function at the origin (zero coupling) antherefore is related to the amount of dilution in the stem[12].

In the present Letter, we analyze the effectsdisorder both in the Néel and quantum paramagnphases of the NLSM. We study these effects firstlythe staggered magnetic susceptibility in both phaand then on the static and dynamic spin correlafunctions in the quantum paramagnetic (quantumordered) phase. We assume that disorder can be meled by a continuum distribution of the exchange cpling, J of the original spin system. In the Néel phathe control parameter is the spin stiffnessρs , which isproportional toJ . It is natural, therefore, to describdisorder in the continuum by a random distributionρs in that phase[12]. The relevant control parameter in the quantum paramagnetic phase, on the ohand, is the spin gap∆s . In order to model disordein this phase, let us observe thatρs and ∆s are ex-pressed in terms of the coupling constant of the NL

g

-

(g0) as

(1)ρs = 1

g0− 1

gc

, ∆s = 8π

(1

gc

− 1

g0

),

wheregc is the critical coupling for the quantum phatransition separating the Néel and the quantum pmagnetic phases atT = 0 [2]. From(1), we concludethat corresponding to a random distribution ofρs , weshall have an analogous distribution for 1/g0. Now us-ing the second part of(1), we see that this implies thathere will be a similar distribution for the spin gap∆s .In summary, corresponding to a random distributof the exchange coupling in the original spin systewe should have a similar distribution forρs , in theNéel phase, and for∆s in the quantum paramagnetphase.

A further justification for this way of introducingdisorder in the quantum paramagnetic phase cofrom the order-disorder duality relation existing btween the spin excitations and the quantum skyrmtopological excitations[13] occurring in the system. Inthe ordered Néel phase, skyrmions possess a nonenergy gap, proportional to the spin stiffness, whspin-waves (magnons) are gapless. Conversely, inquantum paramagnetic phase the former are gapand the latter acquire a nonzero gap. Accordingthe duality relation existing between skyrmions aspin excitations, the spin gap, in the quantum disdered phase, is related by duality to the skyrmion(spin stiffness)[13]. Consequently, the natural waydescribe the presence of disorder in the quantum pmagnetic phase, is to introduce a random distribufor the spin gap corresponding the similar distributof spin stiffnesses in the Néel phase.

2. The quenched average

The basic methodology for obtaining quenchederages in disordered systems is the replica method[6].This is unavoidable, for instance, for obtaining thmodynamic potentials such as the free energy. Thare special cases, however, where it can be avo[6]. Here we will explore one situation where theis an alternative to the replica method for computquenched averages[12].

Consider the quantum average of a certain optor, performed either in the Néel or quantum dis

C.M.S. Conceição, E.C. Marino / Physics Letters A 346 (2005) 121–127 123

ith-hethe

lhaveings

alethege

ra-sgo-e

ge.forereeal-

nc-

ustt

pa-t of

-

-re

,

te-uat-l tothenge

mwith

pintri-hat

thetrolr-edis

ularspinatl-

-

isn-

-t

dered phases of the NLSM. In the clean system (wout intrinsic disorder) this must be a function of trelevant control parameter in that phase, namely,quantum average of an operatorO is of the form:〈O〉q = F(αs) (αs = ρs,∆s , respectively, in the Néeor quantum disordered phases). Suppose now wea inhomogeneous configuration of exchange couplin the original spin system,J = Jij . Assuming thatthis configuration is slowly varying on a distance sccomparable to the lattice spacing, we will have incontinuum NLSM, accordingly, the quantum avera〈O〉q = F(αs(�r)), whereαs(�r) is the slowly varyingcontinuum configuration of the relevant control pameter that corresponds toJij (either the spin stiffnesor the spin gap). This is the key property we areing to use: for slowly varying configurations of thcontrol parameter, we may exchangeαs for αs(�r) inthe expression resulting from the quantum averaThis procedure would not be valid, for instance,the type of disorder occurring in spin glasses, whtheJij -configurations vary quite rapidly from positiv(antiferromagnetic) to negative (ferromagnetic) vues, from site to site.

Taking now a random distribution ofαs(�r), de-scribed by a functionalP [αs(�r)], we shall have thequenched average over disorder given by the futional integral

(2)⟨〈O〉q

⟩d

=∫

Dα(�r)P[α(�r)]〈O〉q

[α(�r)].

For this procedure to be valid, of course, we mchoose the distributionP [α(�r)] in such a way that ionly allows for slowly varying configurationsα(�r).

Observe now that the fluctuations of the controlrameter at different sites are usually independeneach other in quenched disordered systems[6]. Thisallows us to evaluate the functional integral in(2) inthe static ultralocal limit[14,15], in which

(3)⟨〈O〉q

⟩d

= 1

ΛΠ�r

αmax∫αmin

dα(�r)P(α(�r))〈O〉q

(α(�r)),

whereΛ ≡ (Π�r · 1) is a normalization factor corresponding, in the lattice, to a product over all links(ij)

[12]. Eq. (3) follows straightforwardly from the standard definition of the functional integration measuthrough the discretization of space[15]. Notice thatthe functional integral in(2) becomes, in this case

an infinite product of common integrals. Each ingral performs an independent average of the flucting variable on each site. This situation is identicathe one found in the cummulant expansion used inreplica method, where the integrals over the exchacoupling configurationsJij are common ones[6]. Inwhat follows, we will see how the fact that we perforindependent averages on each site is reconciledthe restriction that the random configurationsα(�r) areslowly varying.

We shall use, both for the spin stiffness and sgap, respectively in each phase, the following disbution: a Gaussian with a power law modulation thas been used in[12],

(4)P [α] ={

1N

αν−1e−(α−αs)2/2σ2

, α � 0,

0, α < 0,

whereν > 0 and

(5)N = σν(ν)D−ν

(−αs

σ

)e−α2

s /4σ2,

whereD−ν(x) is a parabolic cylinder function. In(4),α stands either for the spin stiffnessρ, in the orderedNéel phase (αs ≡ ρs ), or for the spin gap∆, in thequantum paramagnetic phase (αs ≡ ∆s ). Notice thatthe distribution vanishes for negative values ofargument. This is a natural choice since the conparametersρ and∆ are not defined in this case. Feromagnetic couplings, in particular, are not describby the distribution above and, therefore, frustrationabsent. Observe that a pure Gaussian is a particcase of(4) for ν = 1. This type of distribution habeen frequently used for describing disorder in ssystems (see[16], for instance). In order to ensure thonly slowly varying configurations of disorder are alowed, therefore guaranteeing the validity of(2) and(3), we always impose thatσ � αs . This implies thatwe will only have significant contributions from configurations satisfying, at each point�r

(6)αs − σ � α(�r) � αs + σ � αs.

This ensures that the relevant configurationsα(�r) arenecessarily slowly varying, thereby reconciling thcondition with the fact that we are making indepedent averages at each point.

Observe that for the distribution(4), whose parameters are constant, the integrals in(3) are independen

124 C.M.S. Conceição, E.C. Marino / Physics Letters A 346 (2005) 121–127

ex-

-es

e

cal

nd

illthatthe

is-e

ee.

onfor

orr

éel

e,usees-we

téel

a-eliza-

thetic

tumits

rgeorderin

us-ar-

sus-

of �r and the quenched average, therefore, may bepressed by a single integral, namely,

(7)⟨〈O〉q

⟩d

=∞∫

0

dα P (α)〈O〉q(α).

In the case of high-Tc cuprates, which are prototypes of 2D quantum antiferromagnets, typical valufor the parameters in(4) are [17,18]: ρs � 10−1 eV;∆s � 10−3 eV. We assumeσ � ρs , andσ � ∆s , re-spectively, in each phase, in order to ensure that thρ

and∆ configurations are slowly varying and(2) canbe safely used in the continuum limit. Hence, typivalues for the variance would beσ � 10−3 eV andσ � 10−5 eV, respectively, in the ordered Néel aquantum paramagnetic phases.

We also impose the following relation, which wbe used later on, relating the size of the sample,we take asL � 1 mm to the scales associated tovariance and control parameters in each phase:

(8)

(L

hc

)σ

(αs

σ

) 1,

where c is the spin-wave velocity, the charactertic velocity of the system. The typical value for thordered phase ishc � 1 eV Å [17] and we assumhc � 10−2 eVÅ in the quantum paramagnetic phas

3. Renormalized classical phase

Let us investigate here the effects of disorderthe homogeneous, static staggered susceptibilityg0 < gc, or ρs > 0. Within the CP1 formulation of theNLSM, this is given by[5]

(9)χ(T ) = lim|�k|,ω→0

χ(ω, �k,T ) = T

4m2(T ),

whereχ(ω, �k,T ) is the vacuum polarization scalar fthe CP1 constraint field andm(T ) is the spin gap. FoT � ρs , we havem(T ) = T e−2πρs/T and

(10)χ(T ) = e4πρs/T

4T

T →0−→ ∞.

The spin gap vanishes atT = 0 and the susceptibilitydiverges, implying the occurrence of an ordered Nstate at zero temperature.

Sinceχ(T ) is proportional to a quantum averagnamely the spin–spin correlation function, we can(2) to obtain the quenched susceptibility in the prence of disorder. For slowly varying configurationshave[12]

(11)χ (T ) =∞∫

0

dρ P [ρ]χ(ρ,T ),

whereP [ρ] is given by(4). The conditionσ � ρs en-ables us to use the expression(10) in (11), obtaining,for T , σ � ρs

(12)

χ (T ) = 1

4T ν

[4πσ 2

ρs

+ T

]ν−1

exp

[8π2σ 2

T 2+ 4πρs

T

].

We see that we haveχ(T )T →0−→ ∞. This means tha

even after the inclusion of disorder the ordered Nstate persists forρs > 0. This result confirms, fromthe point of view of the susceptibility, the observtion made in[12] about the persistence of the Néphase, by looking directly at the quenched magnettion. Our results are also in agreement with[19], whereit is shown that the Néel order persists even afterinclusion of a certain amount of random ferromagnecouplings.

It has been shown, nevertheless, that the quanskyrmion quenched correlation function changesbehavior from exponential to power-law decay at ladistances as a consequence of the presence of disin this phase[12], despite the fact that the system isan ordered state.

4. Quantum paramagnetic phase

We now consider the effects of disorder on the sceptibility in the quantum paramagnetic phase, chacterized byg0 > gc, or ∆s > 0. χ(T ) is still given by(9) but the spin gap is now given, forT � ∆s , by [4]

(13)m(T ) = ∆s + 2T e−∆s/T ,

and no longer vanishes at zero temperature. Theceptibility is now given by

(14)χ(T ) = T

4∆2s

(1− 4

T

∆s

e−∆s/T

),

C.M.S. Conceição, E.C. Marino / Physics Letters A 346 (2005) 121–127 125

f

forg

n

-

ian

red,

atheoc-tav-s

-

be-an-gap

v-ion

onal

ndela-

and we see thatχ(T )T →0−→ 0, implying the absence o

an ordered state atT = 0 in this phase.We now introduce disorder in the system and,

slowly varying configurations will have, accordinto (2), the quenched susceptibility

(15)χ (T ) =∞∫

0

d∆P [∆]χ(∆,T ),

whereP [∆], the random distribution of∆, is given by(4) andχ(∆,T ), by (14). Observe that the conditioσ � ∆s allows us to introduce theT/∆s -expandedexpression(14) inside the integral in(15). After per-forming the∆ integration we get, forσ � T � ∆s ,

χ (T ) = T

4∆2s

− T 2

∆3s

exp

[−∆s

T

(1− σ 2

2T ∆s

)]

(16)×[1− (ν − 4)

σ 2

T ∆s

].

In order to analyze theT → 0 limit, we must also consider the regionT � σ � ∆s of (15). We obtain in thiscase

χ (T ) = T

4∆2s

− (ν − 3)√2πσ

(T

∆s

)ν−1

(17)× e−∆2s /2σ2

[1+ (ν − 3)

T ∆s

σ 2

].

This expression is only valid forν > 3, which is thecondition for integrability in(15). This condition isnot needed in(16), because in that case the Gaussis strongly damped and the integrand in(15) is expo-nentially suppressed at the origin.

From (17) we see thatχ(T )T →0−→ 0, implying that

the inclusion of disorder, described by(4) does notchange the fact that the system is in a disordeground state, atT = 0, for ∆s > 0. Notice, howeverthat the subleading behavior of the susceptibilityT = 0 is modified by the presence of disorder. Tcritical exponent for the quantum phase transitioncurring for g → gc, at T → 0, nevertheless, is nochanged and we still have the susceptibility behing as(g − gc)

−2 at the quantum critical point. Thi

happens because for the caseν > 3, which we are considering here, the leading term in(17)for T → 0 is thefirst one.

5. Static and dynamic spin correlators inthe quantum paramagnetic phase

We are going to investigate here the asymptotichavior of the quenched spin correlators in the qutum paramagnetic phase, where we have a spingiven by(13). Let us firstly take the large-time behaior of the zero temperature spin correlation funct〈Sz(�0, τ )Sz(�0,0)〉. This is proportional to

χ(τ) =∫

dω

2πχ(ω, �0, T = 0)e−iωτ

(18)= −2π Ei(−2∆sτ)τ→∞−→ 2πe−2∆sτ ,

where we used the fact thatm(T = 0) = ∆s , accordingto (13). The static correlator〈Sz(�x,0)Sz(�0,0)〉 in theparamagnetic phase, on the other hand, is proportito [5]

χ(�x) =∫

d2k

(2π)2χ(ω = 0, �k,T = 0)ei�k·�x

(19)|�x|→∞−→ e−2∆s �x,

where again we used(13).We are interested in the leading large-time a

large-distance behavior of the quenched spin corrtion function, hence, Eq.(7) will give:

χ (r) =∞∫

0

d∆P [∆]χ(r,∆,T = 0)

(20)τ→∞−→

∞∫0

d∆P [∆]e−2∆r,

wherer stands either forτ or |�x|, respectively, in dy-namic and static cases. Performing the∆ integration,we get, forσ � ∆s ,

χ (r) = (ν)√2π

(σ

∆s

)ν−1

e−∆2s /2σ2

exp

[(σr − ∆s

2σ

)2]

(21)× D−ν

(2σr − ∆s

σ

).

126 C.M.S. Conceição, E.C. Marino / Physics Letters A 346 (2005) 121–127

iorsar-

dy-

r, as

spinhaseeate.gapelyen-

ioncalnghat

-

esv-ecita-that

esontde-

ar-ter,pro-

ngeor-re

wlytheas-t-e)

atice in

on

. There-the

nesthen

redase

inineticrder

stri-ase,

In order to obtain the large-distance (time) behavof (21), we use condition(8). Taking large times aτ � L/c and using the asymptotic behavior of the pabolic cylinder functions[20] we get

χ (r)τ→∞−→ (ν)

2ν√

2π

(∆1−ν

s

σ

)

(22)× e−∆2s /2σ2 1

rν

[1+ ν

∆s

2σ 2r

].

We see that the exponential decay of the cleannamic and static correlators(18) and(19) is modifiedto a power law decay as a consequence of disordesociated to a distribution of∆’s given by(4). Observethat this occurs even forν = 1, when the distributionis a pure Gaussian. We conclude that the quantumexcitations become gapless in the paramagnetic p(∆s > 0) at T = 0 for this type of disorder, in spitof the fact that the system is in a non-ordered stIn such sates, the spin excitations usually have aand therefore, the introduction of disorder completchanges the scenario in this case. As we have mtioned something similar happened with the skyrmtopological excitations in the renormalized classiregion thereby clearly exposing the duality existibetween skyrmions and spin excitations. Notice tthe large distance behavior of(22) is determined bythe parameterν, regulating the behavior of the distribution function at the origin.

The effect just observed is similar to the onproduced by Griffiths singularities occurring in seeral disordered systems[21,22]. As a consequencof these, in some cases, the spectrum of spin extions also becomes gapless, in spite of the factthe system is in a disordered phase[10]. An impor-tant difference, however, is that Griffiths singularitiaffect dynamic correlators only, having no effectstatic correlation functions[21,22]. Nevertheless, ihas been observed recently that when disorder isscribed by unbounded distribution functions (in pticular Gaussian) similar to the one used in this Letthere are effects similar but stronger than the onesduced by Griffiths singularities[23]. The static spincorrelators, in particular are now affected and chatheir behavior to a power law inside a quantum disdered phase[23]. This is analogous to the effect we areporting here, in a different system.

-

6. The scale σ

Let us study here the meaning of the scaleσ , ap-pearing in our distribution function(4). As we ex-plained, we have always keptσ � ρs,∆s , in order toguarantee that the random configurations are slovarying. In the previous section, we investigatedlong-distance (time) regime of the spin correlators,suming that|�x| � L, the size of the system, which saisfies(8). We consider now a different distance (timscale characterized by (|�x| = cτ )

(23)

( |�x|hc

)σ � 1�

(∆s

σ

)

and investigate the behavior of the dynamic and stspin correlators in the quantum paramagnetic phasthe region characterized by(23). Imposing the condi-tion (23) in (21) and using the asymptotic expansifor the parabolic cylinder functions, we get, (h = c =1)

χ (r)rσ�1�∆s/σ−→ exp

[−2∆sr

(1− σ 2r

∆s

)]

(24)×[1+ (1− ν)

2σ 2r

∆s

],

wherer stands either forτ or |�x|. From(24) we cansee that for scales less thanhc/σ the spin correlatorspresent the same behavior as for the clean systemscalehc/σ , therefore, determines the size of thegion for the onset of the disordered regime, wherecorrelators have their behavior modified to the ofound in the previous section. Using the values ofparameters suitable for high-Tc cuprates, described iSection2, we would find a distance scale of∼103 Åfor the onset of disorder. This should be compawith the spin gap distance scale, which in that cwould be∼10 Å.

7. Concluding remarks

We have considered the inclusion of disordera two-dimensional quantum antiferromagnet, boththe renormalized classical and quantum paramagnphases, at low temperatures. The presence of disowas described by means of a continuum random dibution of the relevant control parameter in each ph

C.M.S. Conceição, E.C. Marino / Physics Letters A 346 (2005) 121–127 127

pinwly

ag-nsbe-nd

rtiesxci-neticpectap-al-

antme-utrip-ithesenot

q,.C.and

89)

94)

ess,

Sci-

1;

d.

niv.

62

nto

ory,

dge

5)

12.995)

nd

either the spin stiffness or the zero-temperature sgap. Quenched averages were evaluated, for slovarying configurations of disorder. Staggered mnetic susceptibilities and spin correlation functiohave been studied in the quenched regime. Thehavior of these indicate that the nature of the groustate is not modified but, nevertheless, the propeof the basic excitations are deeply changed. Spin etations, become gapless in the quantum paramagphase, thereby exhibiting a dual character with resto skyrmion quantum excitations, which become gless, in the presence of disorder, within the renormized classical region[12]. It would be interesting toinvestigate in more detail the properties of the relevexcitations in each phase as a function of the paraterν of the disorder distribution function. We point othat the present method is not suitable for the desction of the discrete type of disorder associated wdelta functions, in spite of their interest, because thare not slowly varying. For the same reason it canbe applied to spin glasses.

Acknowledgements

This work has been supported in part by CNPFAPERJ and PRONEX-66.2002/1998-9. C.M.Shas been supported by CAPES and FAPERJE.C.M. has been partially supported by CNPq.

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