Reply to Comment on "Quantum Phase Transition of Randomly-Diluted Heisenberg Antiferromagnet on a Square Lattice"

1 Pages·0.06 MB·English

by  S. Todo

Reply to Comment on “Quantum Phase Transition of Randomly-Diluted Heisenberg
Antiferromagnet on a Square Lattice”
Synge Todo1∗, Hajime Takayama1, and Naoki Kawashima2
1Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
2Department of Physics, Tokyo Metropolitan University, Tokyo 192-0397, Japan
(February 1, 2008)
1
0
0
2 In Ref. [1], we have studied the quantum phase tran-
sition of the two-dimensionaldiluted Heisenberg antifer- 0.05
n
romagnet by using the quantum Monte Carlo method.
a
J Thetwomainconclusionsinthearticlearei)thecritical 0.03
1 concentration of magnetic sites is equal to the classical 32m / cl
3 percolation threshold (pcl = 0.592746), and ii) the value 0.02 slope = 0.53

of the critical exponents, such as β and Ψ, significantly
]
n dependsonthespinsizeS. In[2],Sandvikhaspresented
n theresultofhisquantumMonteCarlosimulationforthe 0.01
0.02 0.03 0.05 0.07 0.1 0.2
- same system only in the case of S = 1, in which the 1 / L
s 2 FIG. 1. Log-log plot of cluster magnetization (Ref. [2]).
di temperature dependence of the static structure factor is The solid line is obtained by least-squares fittingfor L≥28.
studied quite precisely. Based on his data, he claimed
.
t
a that the data in Ref. [1] should be seriously affected by
m the finiteness of temperature in our simulation. In addi- icantly differs from those for S ≥1. Therefore, we think
- tion, he calculatedthe cluster magnetizationmcl(L) just our second conclusionon the non-universalbehavior can
d atp=pcl,andclaimedthatitcouldremainfiniteevenin not be excluded.
n thethermodynamiclimitinthesamewayastheclassical It was also suggested [2], based on the calculation of
o
(S =∞)case,althoughquitelargefinite-sizecorrections the magnetization of the largest connected cluster for
c
[ to its asymptotic behavior should be considered in the each sample, that all the critical exponents, including

quantum case. As we will discuss below, however, his Ψandβ,areequaltotheclassicalvalues. Thedatawere
2 data do not conflict with our data nor with our conclu- plotted against x ≡ 1/LD/2, and appeared to suggest a
v
6 sions. In particular, the data presented in Fig. 1(b) in finite value in the thermodynamic limit. From our point
2 [2] can be explained from our view point assuming non- of view, however, his data can be interpreted as follows.
2 magnetic ground states. Thequantitym2cl(L)definedinRef.[2]shouldbeasymp-
1 As for his first claim on the effect of the temperature, totically scaled as
0
we agree that there may be a small underestimation for
01 thezero-temperaturevaluesofthestaticstructurefactor. m2cl(L)∼Ss(L,0,pcl)/L2D−d, (1)
/ However,evenifwereplaceourdatapointinFig.3in[1]
t where S (L,T,p) is the static structure factor (Eq. (4)
a for L = 48 by what we can expect from Fig. 1(a) in [2], s
m in Ref. [1])and LD is the averagenumber ofspins in the
it wouldresult in only a minor changein the estimate of
largestconnectedclustersinasystemoflinearsizeL. Ac-
- Ψ. Therefore, the error caused by this underestimation,
d ifany,doesnotseemtobelargeenoughtoinvalidateour cordingtoourresults[1],ontheotherhand,Ss(L,0,pcl)
n diverges as L2D−d−α with α = 0.52 for S = 1, and con-
o conclusions. sequently m2(L) should be scaled as L−0.52.2 In Fig. 1,
c Inaddition,inRef.[1],wehavepresentedtheresultof cl
v: another finite-size scaling analysis (Fig. 4), in which the wbyeSshanowdvaiklo[2g]-.loOgnpelosteeosftthhaetctlhuestdeartmafaogrnLeti≥za2t8iocnledaartlya
Xi data at higher temperatures are also used together with scalesasL−0.53,whichisfullyconsistentwithourresults
the one at the lowest temperature in the simulation. In
r mentioned above.
this analysis, we have explicitly taken into account the
a TheauthorsaregratefultoA.W.Sandvikforallowing
finite-temperatureeffectonthestaticstructurefactorby
us to use his numerical data.
introducing the dynamical exponent z. For S = 1, the
value of the exponent Ψ coincides with that obtained by
theformeranalysis. Onthe otherhand,forS = 1,itbe- ∗ Present address: Theoretical Physics, ETH Zu¨rich, 8093
2
Zu¨rich, Switzerland. E-mail: [email protected]
comesslightlylarger(about9%)thantheotherestimate
[1] K. Kato, S. Todo, K. Harada, N. Kawashima, S.
based on Fig. 3 in [1]. This small difference may be due
Miyashita, and H. Takayama, Phys. Rev. Lett. 84, 4204
tothefinite-temperatureeffectpointedoutin[2],though
(2000).
we have not commented on it in [1]. However, even the
[2] A. W.Sandvik,preprint (cond-mat/0010433).
estimate based on the finite-temperature analysis signif-
1