PHYSICAL REVIEW B 69, 184505 ~2004!

Quantum destruction of stiffness in diluted antiferromagnets and superconductors

N. Bray-Ali1 and J. E. Moore1,2

1Department of Physics, University of California, Berkeley, California 94720, USA2Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

~Received 29 December 2003; revised manuscript received 15 March 2004; published 20 May 2004!

The reduction of two-dimensional~2D! superconducting or antiferromagnetic order by random dilution isstudied as a model for the 2D diluted Heisenberg antiferromagnet La2Cu12p(Zn,Mg)pO4 and randomly inho-mogeneous 2D suerconductors. We show in simplified models that long-range order can persist at the perco-lation threshold despite the presence of disordered one-dimensional segments, contrary to the classical case.When long-range order persists to the percolation threshold, charging effects~in the superconductor! or frus-trating interactions~in the antiferromagnet! can dramatically modify the stiffness of the order. This quantumdestruction of stiffness is used to model neutron-scattering data on La2Cu12p(Zn,Mg)pO4. In a certain sim-plified model, there is a sharp stiffness transition between ‘‘stiff’’ and ‘‘floppy’’ ordered phases.

DOI: 10.1103/PhysRevB.69.184505 PACS number~s!: 75.10.Jm, 75.10.Nr, 75.40.Cx, 75.40.Mg

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I. INTRODUCTION

Randomly diluted superconductors and antiferromagninvolve a combination of classical percolation physics wthe quantum physics underlying superconductivity and aferromagnetism. Percolation is perhaps the simplest tration that can occur in a disordered system: random dilutof sites~or bonds! on a lattice induces a transition betweenphase with one infinite nearest neighbor connected clusteoccupied sites~bonds!, and a phase with only disconnectefinite clusters. Recent experiments on diluted twdimensional~2D! antiferromagnets and inhomogeneousperconductors require a theory of how quantum-mechaneffects modify the percolation transition in these systemThis question is also of practical importance for field-effedevices in which a thin film is tuned through the supercoducting transition.1

At the percolation thresholdpc the infinite connectedcluster is on the verge of being cut into disconnected finclusters ~Fig. 1!. For the connected cluster to have lonrange order~LRO!, the quasi-2D ‘‘blobs’’ must correlateacross quasi-1D ‘‘links.’’ This occurs easily in some othdiluted magnets2,3 and inhomogeneous superconductor4

since their degrees of freedom have LRO even in a 1D chThe two cases considered here, thes51/2 Heisenberg anti-ferromagnet and theO(N) quantum rotor, both have quantum degrees of freedom that order in 2D but not in 1D. Ththe question of whether LRO survives topc , when the infi-nite cluster is a fractal object of dimension91

48 '1.896,5 isunanswered.

The effect of quantum fluctuations is strongest onquasi-1D chains. For classical models on the percolacluster atT.0, the fact that these chains are disorde~have a finite correlation lengthz) implies that the clusterhas no LRO. Similar arguments have been made forquantum cases discussed here.2 The first part of this papeshows that the existence of arbitrarily long 1D segmentspc , and the fact that a spin or rotor in the middle of suchsegment has strong quantum fluctuations,does notpreventLRO for the cluster: the 2D blobs can order through disdered 1D links, in a manner that is impossible for classi

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models. These one-dimensional segments cause difficultyspin-wave calculations because, as seen below, the spinthe middle of such a segment fluctuate strongly. An adtional motivation for studying the effect of these 1D sements is that their effects are essentially unobservable inrent quantum Monte Carlo studies, as a typical realizatione.g., 105 spins will contain no 1D segments of length longthan 8. Eggertet al. connect 1D physics to diluted antiferromagnets differently.6

Randomly diluted quantum degrees of freedom appeatwo well-known nanoscale inhomogeneous materiaLa2Cu12p(Zn,Mg)pO4 is obtained by adding ‘‘static holes’~i.e., removing spins! at random in a quasi-2D antiferromagnet. For small hole densitiesp!pc , neutron-scatteringmeasurements7 agree well with quantum Monte Carlo7–9 andspin-wave10 calculations using thes5 1

2 site-diluted Heisen-berg antiferromagnet~DHAF! model

H5J(̂i j &

pipjsi•sj , ~1!

where the sites form a square lattice, andpi ,pj equal 0 withprobabilityp and 1 with probability 12p. @Vajk et al. arguethat other interactions besides those in Eq.~1! may not benegligible close to threshold.11 We consider some such inteactions below.#

The second type of material, inhomogeneous higtemperature superconductors like Bi2Sr2CaCu2O81d~BSCCO!, comes from doping mobile holes into a quasi-2

FIG. 1. The backbone of the incipient infinite cluster~the back-bone is the portion that carries current from one end of the samto another! showing 2D ‘‘blobs’’ and 1D ‘‘links.’’

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N. BRAY-ALI AND J. E. MOORE PHYSICAL REVIEW B69, 184505 ~2004!

antiferromagnet. Scanning tunneling spectroscopyBSCCO surfaces show grains of size'3 nm that are eithersuperconducting or insulating.12 The grains are larger thathe coherence length'1 nm, so it is reasonable to assumthat they contain Cooper pairs, and that collectively thresemble a bond-diluted Josephson-junction~JJ! model

HJJ5(i

EC~ni2n̄!22EJ(̂i j &

pi j cos~u i2u j !. ~2!

Here the bond variablespi j have the same distribution aspiin Eq. ~1!. The above is not expected to be as accuratmodel of the microscopics as the diluted Heisenberg mo~1! is for La2Cu12p(Zn,Mg)pO4, since the microscopic origin of the disorder in BSCCO is unknown.

The model~2! is in the universality class of theO(2)rotor model, and forECÞ0, charging effects in the grainprevent LRO in a 1D chain.13 The local chargeni does notcommute with the superconducting phasef i : charging tendsto fix the localnumber, which is conjugate to the localphase.In the zero-charging-energy limit,4 HJJ orders in 1D atT50. In the Heisenberg spin-half case, and for small nonzEC /EJ in the Josephson-junction case, the 1D model is ccal ~has power-law correlations!; for largeEC /EJ or Heisen-berg spin one, the 1D model is short ranged.

One of our main conclusions for both the Heisenbergtiferromagnet andO(N) rotor is that order atpc is alloweddespite the existence of 1D links. We find that possibledered states atpc must have extremely low stiffness relativto the undiluted 2D case. Superconducting phases with ssuperfluid density have previously been proposed, e.g.,‘‘gossamer superconductor.’’14 Our picture differs in that thelow superfluid density results from randomness on scalarger than the coherence length,12,15 rather than from a uni-form theory.

We first calculate correlations in a toy model~3! to showhow quantum disordered or criticalT50 1D systems such athe spin-half chain are fundamentally different from classidisorderedT.0 1D systems. Even though both may be dordered in 1D, the quantum systems can order throughordered 1D regions while similar classical systems canThis statement underlies the possible existence of ordepc . We next solve a rather simplified limit of Eq.~2! thatshows a stiffness transition between two ordered phase‘‘stiff’’ phase and a ‘‘floppy’’ phase. In a less simplified bustill approximate model, we show that the stiff phase obthe classical scaling law for stiffness. Finally, we compaexperiment and simulations of Eq.~1! to this renormalizedclassical theory.

II. RESULTS FOR HEISENBERG AND O„N… MODELS

A. Order across a disordered link

First consider a chain of spins with uniform couplingJ,plus one spin at each end attached by a couplingJ8:

H15 (i 51

N21

Jsi•si 111J8~s0•s11sN•sN11!. ~3!

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cally, J@J8.0. We have in mind a single link together witits two neighboring blobs from Fig. 1. The weakJ8 exchangerepresents an effective coupling between the two boundsites of a blob. Thus, the limitJ@J8 is supposed to capturthe approximation that blob spins fluctuate slowly~on alower energy-scale! compared to those in the link. For versmall J8, the state of theN link spins is nearly undisturbedby it, and in particular,s1 andsN are only weakly correlatedwith each other.6 However, the two spins at the endss0 andsN11 can be made to form a perfect singlet with each othfor N even: limJ8/J→01s0•sN11523/4.

To understand this result, note that atJ850, the fourstates of the end spins, which can be classified into a sinand triplet, are degenerate. Perturbation theory inJ8 pre-serves rotational symmetry and, forJ8!J, just splits thesinglet and triplet. The two spins at the ends correlate pfectly with each other, even though the internal spinsweakly correlated.

Diagonalization, using thick-restarted Lanczos,16 up to N520 confirms the second-order perturbation theory arment. At small enoughJ8/J there is an arbitrarily strongsinglet correlation

^s0sN11&'23

41 f 1~N!S J8

J D 2

, ~4!

but with a vanishing energy splitting between singlet atriplet:

Etriplet2Esinglet' f 2~N!JS J8

J D 2

. ~5!

Here f 1 and f 2 are dimensionless functions of the chalength that incorporate the matrix elements and energynominators of the unperturbed chain. Note thatf 1(N) growsslowly with N. In fact, from exact diagonalization up toN520, we find limN→` f 1(N)/N→0. In terms of our physicalpicture, this suggests blobs can order across a disorderdprovided that their internal fluctuations are slow enough.

B. Critical Josephson-junction array

At pc , the Josephson junction array~2!, in a certain ap-proximation, also exhibits long-range order. Consider~fornow! neglecting all fluctuations within 2D blobs. The scalinproperties of this model can be found exactly; later, fluctutions within the blobs will be partially restored. In this approximation, the phases of the blobs,$f i%,i 51, . . . ,N,commute withH, so choose a basis of simultaneous eigestates of Eq.~2! andf i . In this basis, the energy

E'2(i

Jicos~f i2f i 11!, ~6!

depends on exchange constantsJi.0 chosen from a distri-bution P(J) which we calculate below using propertiesthe 1D links@see Eq.~9!#. Regardless of the distribution, w

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QUANTUM DESTRUCTION OF STIFFNESS IN DILUTED . . . PHYSICAL REVIEW B69, 184505 ~2004!

have long-range order, sinceJi.0 andf i are classical. Wecalculate the stiffness of this order below using theJ→0behavior ofP(J).

Consider a single 1D link ofL sites bounded by two superconducting blobs of phasef1 andf2. After the quantum-classical mapping, we obtain a 2DXY model with fixedboundary conditions.13 The lowest energy is obtained iff15f2, but the energy cost to create a phase differencepends on the couplingK5EC /EJ in the 2D XY model.Cardy discusses this problem: The stiffness per site goesfinite limit in the algebraic~ALRO! phase of the 2DXYmodel but falls off exponentially withL in the short-rangedphase~SRO!.17 For small phase difference between bounariesDf5f12f2,

E~Df!;H k1~Df!2L21 in ALRO phase

k2~Df!2e2L/z in SRO phase.~7!

The exchange strength isJ(L)52E(Df)/(Df)2.Now consider the total energy costE(R,Df)5Jeff(R)

3(Df)2/2 to create a small phase differenceDf betweentwo faraway pointsA and B separated by distanceR. Here,Jeff(R) is what we mean by ‘‘stiffness.’’ Two geometric properties of the percolation cluster enter. The total number oflinks betweenA and B goes asN;R3/4 and the fraction oflinks P(L) of lengthL falls off exponentially withL: P(L);e2aL.3,18

The fraction of linksP(J) with exchange strengths Jjust a sum ofd functions:

P~J!5 (L51

N;R3/4

P~L !d@J2J~L !#. ~8!

The peaks become closely spaced asJ→0, so we can pass toan integral overL. Using Eq.~7! for J(L), we find the fol-lowing asJ→0:

P~J→0!5H k1 /J2e2k1a/J in ALRO phase

~J/k2!az21 in SRO phase.~9!

For the classicalXY chain with a random distribution ocouplingsP(J), Straley showed that there is a stiffness trasition depending on the exponentP(J)}Ja as J→0.19 Fora>0, there is a nonzero mean stiffness, andJeff(R);R23/4. For a,0, there is a qualitatively weaker stiffneswith exponent depending continuously ona:

Jeff~R!;R23/4(11a). ~10!

We refer to the phase with nonzero mean stiffness as ‘‘stiand the phase with continuously variable stiffness exponas ‘‘floppy.’’

Hence our model~6! has a transition whena5z21, thatis, when the typical length of a 1D link on the percolatiobackbonea21 is equal to the correlation length of the 1link z. There are estimates of the numbera for some stan-dard lattices.18 The stiffness transition isnot at the samevalue of EC /EJ as the KT transition where the 1D correltion length first becomes finite. The transition is caused

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competition between the 1D correlation lengthz and the per-colation physics that controlsa.

To improve Eq.~6!, let us allow static, spatial fluctuationin the 2D blobs. This is still approximate, since we neglequantum effects in the blobs. Again, the approximationlows us to find scaling properties exactly. Since the blobsstill classical, Eq.~6! still gives the energy, provided wemake the following changes. First, we add an extra phaseeach blob. That is,i 51, . . . ,2N21, where,N;R3/4 is thenumber of blobs. Second, we choose the exchange consbetween boundary sites of the same blob~i.e., Ji with i odd!from the classical distributionPC(J). This choice accountsfor the phase difference between the two boundary sitethe blob due to static, spatial fluctuations inside. Finally,continue to choose exchange constants between adjablobs~i.e., Ji with i even! from the link distributionP(J) inEq. ~9!.

Just as for Eq.~6!, the lowest energy for this model occuwhen the blob phases align. What is the stiffness of torder? Stiffness like conductance ‘‘adds in parallel:’’

Jeff~R!215Jblobs~R!211Jlinks~R!21, ~11!

where,Jblobs,Jlinks are the stiffness of blobs and links, respectively. For ALRO or large-z SRO links, blobs dominatethe stiffness. Since we choose blob exchange constantsthe classical distributionPC(J), they give an average stiffnessJC(R);R2t/n, wheret'1.31,20 is the classical stiffnessexponent, andn54/3. Thus, in the blob-dominated phasour model obeys the classical scaling form for stiffness. FSRO links withz!a21, the floppy links dominate. The stiffness scales with az-dependent exponent given by Eq.~10!,with a5az21. Thus, just as for Eq.~6!, this model has astiffness transition between two ordered phases as weEC /EJ . However, the transition now occurs whenJblobs5Jlinks , or equivalently, whena5t/z.

For what range ofEC /EJ , is Eq. ~6! or its variant abovean accurate model of the physical system Eq.~2!? For anyEC/EJÞ0, the blob phases$s i% arenot conserved quantumnumbers. Rather, we must add to Eq.~6! a charging termEC( i1/Ni(ni2n̄)2, where,ni is the number of superconducing pairs on the blobi, as in Eq.~2!, andNi is the number ofgrains in blobi @For the distribution ofNi , see Ref. 18#.Thus, Eq.~6! is an accurate model only if this charging teris irrelevant in the scaling limit. We suspect that it is irreevant for smallEC /EJ , but defer detailed treatment to a latwork.21

C. Critical O„NÌ2… rotor and Heisenberg antiferromagnet

Let us now consider theO(N) rotor models forN.2 andthe Heisenberg quantum antiferromagnet~1!. If all fluctua-tions in the blobs are negligible, then the ground-state eneis just that of a classicalO(N) chain with random distribu-tion of couplings: E'2( iJinW i•nW i 11, with ni

W anN-component unit vector. For the antiferromagnet,N53 andnW i is just the Ne´el moment of blobi. Our discussion imme-diately below Eq.~6! still applies. Regardless of the couplindistribution P(J) the ground state atpc must be long-range

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N. BRAY-ALI AND J. E. MOORE PHYSICAL REVIEW B69, 184505 ~2004!

ordered, sincenW i are classical andJi.0. Further, we canallow for static, spatial fluctuations in the blobs. Again, tground state is ordered. The stiffness, allowing for static bfluctuations, obeys classical finite-size scalingJeff(R);R2t/n for the antiferromagnet~1!, since its 1D links arealgebraically ordered withk15J/2p in Eq. ~7!. For theO(N) rotor with N.2, or the antiferromagnet with stronnext-nearest-neighbor interaction, there is a stiffness tration, since they have SRO in their 1D links.

D. Off-critical Heisenberg antiferromagnet

The remainder of this paper discusses consequenceEq. ~1! of the picture that quantum fluctuations in the bloare negligible. As argued above, in this picture, the groustate atp5pc is long-range ordered and obeys classifinite-size scaling for stiffness. RecentT50 numerics atp5pc confirm these consequences.8 Below, we extend our‘‘renormalized classical’’ picture toT.0 andp,pc .

First, considerT.0, while still at p5pc , and definelength scalelc via Jeff(lc)5T. Heuristically, blobs that areseparated byR@lc should be independent, since it coslittle energy to misalign their Ne´el fields. Conversely, blobswith R!lc should align. In theT→0 limit, the ground-statestiffness calculated above,Jeff(R);R2t/n, still applies, giv-ing lc'a(J/T)n/t, wherea is the lattice spacing. A morecareful calculation confirms thatlc is indeed theT.0 cor-relation length atp5pc .21

Next, considerp,pc . On length scales that are smacompared to the percolation lengthj(p)'aup2pcu2n, thestructural properties of the largest connected cluster fop,pc are just those of the critical cluster.5 As the result, wehave correlation lengthl;a(J/T)n/t if ( J/T)n/t!j/a. Inother words, if at someT.0 the critical correlation lengthlc is very small, then the spins will not care if we reduce tdilution, since, structurally, the relevant local environmeremains critical.

What about low temperatures, such that (J/T)n/t@j/a?On length scales large compared toj, the structural proper-ties of the largest connected cluster are two dimension5

Further, the bonds in this ‘‘superlattice’’ look, structurallsimilar to the critical cluster in Fig. 1.

This can be exploited by doing the following one-stereal-space renormalization. First ‘‘integrate out’’ the bonreplacing them with their effective stiffnessJ̃5Jeff(j);up2pcu t. ~We again use the stiffness obtained by negleing quantum fluctuations in the blobs.! Next, rescale lengthsby j/a. The result is an undiluted 2D Heisenberg modwith small nearest-neighbor exchangeJ̃ and large latticespacing j. Plugging these into this model’s well-knowform for low-T correlation length,22 gives the resultl

nd

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dl

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l.

,,

t-

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;je2prs /T. Here, n54/3 and rs(p);up2pcux, with x't'1.31, the classical stiffness exponent. Results forT.0, p,pc are summarized in Fig. 2. Correlation length measuments on the DHAF from neutron scattering7 seem to agreewith our model. The data were fit to the following:10

l

a5

e

8

c/a

2prs

e2prs /T

11~4prs /T!2nT. ~12!

At low T, Eq. ~12! approximately follows our exponentiasince best fit to experiment gives our valuers(p)'up2pcux. At high-T, Eq. ~12! goes over to a power law, as iour model, but the exponentnT is hard to extract from theexperimental data~for an estimate ofnT from numerics, seeRef. 11!. Finally, the crossover between the two limits occuin Eq. ~12! when rs'T, which, agrees with our crossovecriterion (J/T)n/t'j/a.

III. CONCLUSIONS

We have shown that it is possible for long-range ordercoexist with arbitrarily strong quantum fluctuations on 1links: contrary to some previous claims, quantum systecan ‘‘order through disorder.’’ This supports the resultquantum Monte Carlo calculations, and suggests that suconducting or antiferromagnetic order can exist topc in 2Dbut has low stiffness because of both cluster geometryquantum fluctuations. This reducedrs results in a correlationlength much shorter forT.0 than without dilution. For di-luted La2CuO4, our analytic results explain and compare resonably well with existing experiment and numerics. Ocan hope that this quantum order through disorder alsopears in other problems of quantum ordering on randomometries.

ACKNOWLEDGMENTS

The authors acknowledge helpful conversations withGreven, D.-H. Lee, T. Senthil, O. Starykh, O. Vajk, andWu and support from DOE LDRD-366464, NSF DMR0238760, and NERSC.

FIG. 2. The three different stiff-phase forms for the correlatilength l near theT50, p5pc geometric critical point. Abovepc

the correlation length is just set by the percolation correlation len~cluster size! j. Below pc , rs vanishes asup2pcux, x'1.31.

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