quantum percolation in two-dimensional antiferromagnets

PRL 94, 197204 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending20 MAY 2005

Quantum Percolation in Two-Dimensional Antiferromagnets

Rong Yu,1,2 Tommaso Roscilde,1 and Stephan Haas1

1Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089-0484, USA2Kavli Institute for Theoretical Physics, University of California, Santa Barbara, California 93106-4030, USA

(Received 1 October 2004; published 20 May 2005)

0031-9007=

The interplay of geometric randomness and strong quantum fluctuations is an exciting topic in quantummany-body physics, leading to the emergence of novel quantum phases in strongly correlated electronsystems. Recent investigations have focused on the case of homogeneous site and bond dilution in thequantum antiferromagnet on the square lattice, reporting a classical geometric percolation transitionbetween magnetic order and disorder. In this study we show how inhomogeneous bond dilution leads topercolative quantum phase transitions, which we have studied extensively by quantum Monte Carlosimulations. Quantum percolation introduces a new class of two-dimensional spin liquids, characterizedby an infinite percolating network with vanishing antiferromagnetic order parameter.

DOI: 10.1103/PhysRevLett.94.197204 PACS numbers: 75.10.Jm, 64.60.Ak, 75.10.Nr, 75.40.Cx

bb’

(b)(a)

FIG. 1 (color online). (a) A square lattice quantum antiferro-magnet composed of dimers (dark bonds, b) and ladders (light,vertical bonds, b0). (b) A segment of ladder in the percolatingcluster leads to local RVB states (ellipsis). The spins involved inthe RVB state are very weakly correlated to the rest of thecluster, causing an effective fragmentation of the percolatingcluster into nonpercolating subclusters.

Percolation occurs in a variety of contexts [1], rangingfrom blood vessel formation to clusters of atoms depositedon substrate surfaces. A fundamental question is whetherthe classical picture of permeating networks [2] applies aswell to strongly fluctuating quantum systems. In this re-spect, low-dimensional quantum magnets offer an idealplayground to probe the interplay between quantum fluc-tuations and geometric randomness both theoretically andexperimentally [3,4].

Two electronic spins coupled by an antiferromagneticHeisenberg interaction, H � JS1 � S2 (J > 0), combineinto a singlet state with purely quantum character. Suchsinglets are the fundamental building blocks of many novelphases in quantum antiferromagnets. For instance, theground state of a Heisenberg ladder is a coherent superpo-sition of singlet coverings involving mostly nearest-neighbor spins. This resonating valence bond (RVB) state[5] is realized in a number of antiferromagnetically corre-lated materials for which magnetometry and neutron scat-tering measurements indicate the absence of magneticorder down to the lowest accessible temperatures [6]. Incontrast, the S � 1=2 Heisenberg antiferromagnet on thesquare lattice has an antiferromagnetically long-range or-dered ground state [7]. However, as shown in Fig. 1(a), asquare lattice can be decomposed into dimers and laddersin such a way that there are no two adjacent dimers orladders. This suggests that, if lattice disorder can isolatesome of these substructures, it may lead to quantum-disordered ground states.

Recently, the effect of site dilution on the square-lattice S � 1=2 quantum Heisenberg antiferromagnethas been probed experimentally in the compoundLa2Cu1�p�Zn;Mg�p04, in which magnetic Cu2� ions arereplaced randomly by nonmagnetic Zn2� or Mg2� ions [3].The pure system (p � 1) displays long-range antiferro-magnetic order [7]. Upon doping with static nonmagneticimpurities, the order parameter is reduced until it vanishesat a critical doping concentration pc. Both experimental

05=94(19)=197204(4)$23.00 19720

data [3] and numerical simulations [3,4] indicate a mag-netic transition at the geometric percolation thresholdp� � 0:40 725 [2] beyond which the system breaks upinto finite clusters. An analogous result holds for the caseof bond dilution, where a fraction p of bonds is cut atrandom in the system: again the onset of disorder is ob-served at the percolation threshold pc � p� � 0:5 [4]. Inboth cases this implies that the magnetic transition iscontrolled by the underlying geometric transition.

A completely different picture of the effect of disordermight instead emerge when resonating valence bondsdominate the ground state of the system. In fact, the spinsinvolved in a local RVB/singlet state are weakly correlatedto the remainder of the system, so that, from the point ofview of magnetic correlations, we can roughly considerthese spins missing from the geometric cluster they belongto, as sketched in Fig. 1(b). This implies that the percolat-ing cluster is effectively fragmented by the formation ofRVB/singlet states, and therefore the magnetic transition is

4-1 2005 The American Physical Society


hindered with respect to the geometric transition, acquir-ing in turn the nature of a quantum phase transition.Nonetheless, according to this argument the magneticquantum phase transition appears as a renormalized perco-lative transition, with an effective geometry of the clustersdictated by quantum effects. This scenario is referred to asquantum percolation.

In this work we investigate the occurrence of quantumpercolation in the S � 1=2 quantum antiferromagnet onthe square lattice with inhomogeneous bond dilution, mod-eled by the Hamiltonian

H �X

b

JbS1;b � S2;b �X

b0Jb0S1;b0 � S2;b0 : (1)

Here S are S � 1=2 spin operators, the bonds b are dimerbonds in Fig. 1(b), and the bonds b0 are the interdimerbonds. Jb, Jb0 > 0 are random antiferromagnetic couplingsdrawn from a bimodal distribution taking the two values J(‘‘on’’) and 0 (‘‘off’’), with different probabilities of the onstate, p�Jb � J� � P and p�Jb0 � J� � P0. In the specialcase P � P0 the homogeneous bond dilution problem isrecovered: the percolation threshold coincides with a mag-netic order-disorder transition at the classical critical valuePc � P0

c � 0:5 [2]. In contrast, for P � P0 one encountersinhomogeneous bond dilution. In particular, at P � 1 thesystem is an array of randomly coupled dimers, and in theopposite limit, P0 � 1, the system is a set of randomlycoupled ladders.

We determine the geometric percolation threshold forthe inhomogeneous bond dilution problem by generalizinga very efficient algorithm recently developed for homoge-neous percolation [8]. The resulting phase boundary isshown with filled diamond symbols in Fig. 2. A continuous

0 0.2 0.4 0.6 0.8 1

P

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

P’ classical

quantum (S=1/2)

Finite clustersm = 0

Infinitecluster

Infinite clusterm ≠ 0

m = 0

θ

FIG. 2 (color online). Phase diagram of the inhomogeneouslybond-diluted S � 1=2 antiferromagnet on the square lattice. Thecolored area indicates the quantum-disordered region in whichthe system has developed an infinite percolating cluster, but themagnetization m of the system vanishes because of quantumfluctuations. The angle � parametrizes the critical curve.

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transition line, with constant critical exponents of 2Dpercolation [2], is found, connecting the percolationthreshold for randomly coupled dimers, P � �P;P0� ��1; 0:382� with the percolation point P � �1=2; 1=2� ofthe homogeneously diluted systems, and terminating atthe critical point for randomly coupled ladders, P ��0; 1�. In this limiting case an infinitesimal concentrationof dimer bonds is sufficient to create a percolating clusterin the system.

When classical spins (S � 1) are placed on each latticesite, long-range antiferromagnetic order follows triviallythe onset of geometric percolation. In contrast, this pic-ture changes drastically for S � 1=2 quantum spins.Approaching the two limits of randomly coupled dimers(P ! 1) and ladders (P0 ! 1), the formation of singletstates for the spins on the dimers (b bonds) or on theladders (b0 bonds) are favored statistically by geometry,such that quantum fluctuations are strongly enhanced. Thiscontrasts with the case of homogeneous bond percolation,P � P0, in which quantum effects do not change the clas-sical picture [4]. Therefore the crossover from homogene-ous to inhomogeneous percolation in the S � 1=2antiferromagnet is accompanied by a continuous enhance-ment of quantum fluctuations.

To determine the magnetic phase diagram in the quan-tum limit we use the stochastic series expansion quantumMonte Carlo (QMC) method based on the directed-loopalgorithm [9]. This technique enables us to access ultralowtemperatures on large L� L lattices whose size is wellinside the scaling range of the critical regime. For stronglyinhomogeneous systems, we have studied the T ! 0behavior by sitting at an inverse temperature J � 8L,at which the analysis of critical scaling leads to atemperature-independent estimate of the transition point,as shown in Fig. 3(a). For systems closer to the homoge-neous limit we found it necessary to cool the system toeven lower temperatures, which we can efficiently reach bya -doubling approach [4] . Finite temperature propertiesare studied on larger clusters with L � 200. Furthermore,simulations are also carried out at the percolation thresholdon percolating clusters of exact size Nc grown freely froman initial seed (see Ref. [4] for a similar analysis). For eachvalue of P and P0 we average over 100–300 differentrealizations of the bond disorder (disorder averages areindicated as h. . .i). The finite staggered magnetization��hm2i

p�

��3hS��;��i=L2

pof an antiferromagnetically or-

dered ground state is accompanied by a divergent spin-spincorrelation length along the x and y directions, �x and �y.The location of the magnetic transition is obtained from thebond concentration at which the T � 0 correlation lengthsdiverge linearly with lattice size, �x; �y � L.

The transition line for the quantum antiferromagnet andfor the classical percolation problem is clearly different.This shows that the magnetic phase transition in the case ofinhomogeneous bond dilution turns into a quantum phase

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100

101

T |P’−P’c|−νz

10−1

100

ξ T

1/z

P’=0.50 0.52 0.53 0.54 0.56 0.57 0.58 0.60 0.62

y

x

y

x

(a)

0 0.5 1 1.5θ (rad)

1

1.5

2

zν

ν=1

ν=4/3

z=1.9

classicalquantum quantum

(b)

FIG. 4 (color online). (a) Scaling plot of �x (open symbols), �y(full symbols) for P � 1 (randomly coupled dimers). (b) Criticalexponents z and � for the magnetic transition in the inhomoge-neously bond-diluted S � 1=2 Heisenberg antiferromagnet. Theangle � parametrizes the critical line in Fig. 2. Open (full) sym-bols refer to estimates obtained through the scaling of �x��y�.The shaded areas correspond to the regions of parameters inwhich the transition is quantum. All lines are guides to the eye.

0 0.05 0.1

1/Nc

1/2

0.00

0.05

<m

c2 >

P=P’=0.5P=0.133, P’=0.7

0 2 4 6 8 10

n

0.44

0.46

0.48

0.50

0.52

0.54

Pc’

(n)

P=0.9375

(b)

P=0.9375P=0.9375

(a)

FIG. 3. (a) Estimates of the quantum critical point as the valueof the disorder showing the critical scaling �x; �y � L at atemperature T � J=nL, plotted as a function of n. For n � 8the estimate clearly becomes temperature independent.(b) Disorder-averaged square order parameter on the percolatingcluster only as a function of the size Nc of the percolating clusterfor the homogeneous percolation point (P � P0 � 0:5) and for aclassical inhomogeneous percolation point (P � 0:133, P0 �0:7). The temperature is T � 1=�8Nc�. The solid lines are cubicpolynomial fits.


transition for sufficiently strong inhomogeneities, leavingthe percolation transition line at two multicritical points. Inparticular, between the two transition lines there is anintermediate regime (colored area) in which geometricalpercolation has been reached, such that there is an infinitepercolating cluster [2]. Nonetheless, magnetic order isabsent, as shown by the scaling of the order parameter inFig. 3(b). Here the disorder-averaged squared magnetiza-tion mc of the percolating cluster only is shown as afunction of the cluster size. Unlike the case of the homo-geneous percolation point [4] P � P0 � 0:5, the percolat-ing cluster in the strongly inhomogeneous case shows avanishing magnetization at the onset of geometric perco-lation, so that the addition of extra bonds is required for themagnetic order to set in. The quantum-disordered regimeof the phase diagram corresponds to an infinite family oftwo-dimensional quantum spin liquids for which the anti-ferromagnetic Heisenberg interaction does not lead tospontaneous symmetry breaking in the thermodynamiclimit.

Moreover, the phase diagram in Fig. 2 shows remarkablereentrant behavior. Decreasing P�P0� from 1 to 0 with fixedP0�P� in the range 0:45 & P0 & 0:55 (0:2 & P & 0:32), thesystem experiences a magnetic transition from the spinliquid to the ordered antiferromagnetic state. This is a cleanexample of the order-by-disorder mechanism, in whichmagnetic order is induced by randomness due to the geo-metric destruction of local RVB/singlet states.

Unlike in other doped lattices [10], we do not observethe occurrence of residual magnetic order due to the long-range effective interactions [11] between spins not in-volved in a local RVB/singlet state. We ascribe this finding

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to a distinctive feature of bond dilution as opposed to otherforms of disorder: in fact, when the cut of a bond does notlead to the local enhancement of quantum fluctuations, italways leaves two nearby spins not forming a singlet withtheir neighbors, so that each unpaired spin immediatelyfinds its partner to build a longer-range singlet.

What are the critical exponents associated with thisquantum phase transition? The finite temperature quan-tum critical behavior of the correlation length �� T�1=z

determines the dynamical critical exponent z [12,13],whereas the critical exponent � governs the divergenceof the correlation length at the T � 0 critical point, ��jP � Pcj

��. Pc can be obtained as the value of disorder atwhich � exhibits a power-law behavior as a function of T,and it is in agreement with the estimates coming from theT � 0 analysis described above. From the scaling relation

��P;T� �1

T1=zF��TjP � Pcj

�z� (2)

the exponent � can be determined as the one realizingthe best collapse of T1=z��P;T� curves plotted versus

4-3


TjP � Pcj�z for different P values, as shown in Fig. 4(a).

The values extracted for � and z are shown in Fig. 4(b) as afunction of the angle � � tan�1��1� P0�=�1� P�� whichparametrizes the critical curve of Fig. 2. The � exponent isfound to satisfy the Harris criterion for disordered systems[14] � � 2=d � 1 within error bars. Moreover, we observethat z and � are almost constant over a large portion of thecritical line, taking the values z � 1:9 and � � 4=3, andthey both show a drop to z; � � 1 only near the edges of thecritical line in the strongly quantum regimes (P ! 1 andP0 ! 1). Remarkably, such a drop happens away from themulticritical points at which the percolation transition andthe magnetic transition depart from each other. This im-plies that, for moderate quantum fluctuations, the magneticquantum phase transition retains a percolative nature, infull agreement with the picture of quantum percolation, inwhich the main effect of quantum fluctuations is a spatialrenormalization of the percolating cluster. In particular, thecorrelation length exponent � is clearly consistent with theclassical value � � 4=3 for two-dimensional percolation[2]. The z exponent can be also related to the criticalexponents of the classical phase transition. At and aroundthe homogeneous percolation point P � P0 � 0:5, wherethe transition is classical, the exponent z is the scalingdimension of the finite-size gap of the percolating clusterdiverging in the thermodynamic limit, �� L�z. Since thenumber of sites Nc of this cluster diverges as Nc � LD withD � 91=48 [2], if the finite-size gap vanishes as �� N��

c

it is easy to see that z � �D. For clusters with the geometryof the chain or of the square lattice, � � 1 [5,15].Extending this result also to the fractal percolating clusterwe obtain z � D � 1:89 . . . , in agreement with the QMCresult.

The concept of quantum percolation has previously beeninvoked in the context of weakly interacting electronicsystems, such as in metal to insulator transitions of granu-lar metals [16] and in the integer quantum Hall effect ofsemiconductor heterostructures [17]. In these examples,quantum mechanics enters mostly through single-particlephenomena, such as localization effects of single-electronwave functions and quantum tunneling. In contrast, thequantum percolation discussed in the present Letter occursin a strongly correlated electron system. Its quantum as-pects reside in the formation of local many-body quantumstates, i.e., dimer singlet states or RVB states, and itultimately has the nature of a collective quantum phasetransition. The specific example of inhomogeneous bonddilution of the square-lattice Heisenberg antiferromagnetoffers a remarkable realization of such scenario, withcontinuous tuning of the quantum nature of the percolativetransition in the system. A continuous modulation of quan-tum effects at a percolation transition can also be achievedby progressively coupling two square-lattice layers whichare site diluted at exactly the same lattice sites [18]. In that

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case, nonetheless, the estimated critical exponents for themagnetic quantum phase transition differ substantiallyfrom those of classical percolation and are much closerto those of a quantum phase transition in a clean system,suggesting that quantum effects alter the percolative natureof the transition.

From the experimental point of view, if different non-magnetic ions mediate the intradimer interactions [b bondsof Fig. 1(a)] with respect to the interdimer ones [b0 bondsof Fig. 1(a)] in a square lattice antiferromagnet, selectivechemical substitution of these two types of ions offers away to realize inhomogeneous bond disorder. Extensionsof the picture given in this Letter would be necessary tomimic a realistic experimental situation. Nevertheless, theeffects leading to a quantum correction of the percolationthreshold should be robust, and they should persist in thepresence of more general models of inhomogeneous bonddisorder.

We acknowledge fruitful discussions with B. Normand,H. Saleur, and A. Sandvik. S. H. acknowledges hospitalityat KITP, Santa Barbara. This work is supported by the NSFthrough Grant No. DMR-0089882. Computational facili-ties have been generously provided by the HPCC-USCCenter.

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quantum percolation in two-dimensional antiferromagnets

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