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4151991
The Dilute Quantum Heisenberg Antiferromagnet The Dilute Quantum Heisenberg Antiferromagnet
C. C. Wan
University of Pennsylvania
A. Brooks Harris
University of Pennsylvania
, harris@sas.upenn.edu
Joan Adler
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Recommended Citation Recommended Citation
Wan, C., Harris, A., & Adler, J. (1991). The Dilute Quantum Heisenberg Antiferromagnet.
Journal of Applied
Physics,
69
(8), 51915193. http://dx.doi.org/10.1063/1.348123
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The Dilute Quantum Heisenberg Antiferromagnet The Dilute Quantum Heisenberg Antiferromagnet
Abstract Abstract
Series expansions are used to treat the randomly diluted spin‐1/2 Heisenberg antiferromagnet at zero
temperature. A series is obtained at zero temperature in powers of the concentration for many correlation
functions and for the correlation length from which static, dynamic, and crossover exponents are
estimated. The correlation length exponent is found to be 0.77±0.10 in two dimensions. The critical
concentration for the appearance of long‐range order is indistinguishable from the percolation threshold.
Disciplines Disciplines
Physics  Quantum Physics
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/436
The dilute quantum Heisenberg antiferromagnet
C. C. Wan and A. B. Harris
:.
Department
of
Physics, University
of
Pennsylvania, Philadelphia, Pennsylvania 19104
Joan Adler@
Department
of
Physics, Technion, Haifa 32000, Israel
Series expansions are used to treat .the randomly diluted spini Heisenberg antiferromagnet at
zero temperature. A series is obtained at zero temperature in powers of the concentration
for many correlation functions and for the correlation length from which static, dynamic,
I
and crossover exponents are estimated. The correlation length exponent is found to be
0.77*0.10 in two dimensions. The critical concentration for the appearance of longrange E.
order is indistinguishable from the percolation threshold.
Series expansions have often been useful in determin
ing numerical values of critical exponents,. especially in
cases where existing theories have been unable to elucidate
the nature of the critical phenomena. Here we pursue a
program to obtain numerical data on the zerotemperature
transition that occurs as a function of concentration
p
for
the spini Heisenberg antiferromagnet in the presence of
quenched random dilution. The Hamiltonian for this sys
tem is
R’=J & Ei,Si*Sj,
(1)
where S is a quantum spin 4,
(i,j)
indicates that’the sum is
over pairs of nearest neighbors on a ddimensional hyper
cubic lattice, and eii is a random variable which assumes
the value unity with probability
p
if the bond (i,j) is oc
cupied and the value 0 with probability 1 
p
if the bond’is
unoccupied.
To highlight the differences between the spini model
and the classical model (viz., for the S’+.GC limit),Jet us
note the behavior of the staggered structure factor
C(p),
which is defined to be
’
C(p) =fV ’ 2 [ (Si’Sj)]~&“‘Y~ i
iJ
(2)
where N is the total number of sites in the system, [ 1,
indicates an average over all configurations:(of the Q’S),
( ) denotes a zerotetnperature average over all state(s) of
the ground manifold, and k is the wave vector associated
with the antiferromagnetic order. For a classccal system a
local “gauge” transformation in which one reverses the
sign of Si on one sublattice converts
C(p)’
into the zero
wavevector correlation function for a dilute ferromagnet.
Then all spins are parallel in the groundstates and C(p)’
reduces to the meansquare size of ‘connected clusters,
which is the percolation susceptibility. ’ Thus for a classical
system the threshold concentration at which longrange
order sets in coincides with the percolat& threshold. One
may define a critical exponent associated,with C(p) via

C@)IApl Y,
(3)
i,
‘)AIso at: School of Physics tid AStronbmy, Tel Aviv University, Tel
Aviv 69978, Israel.
.
where Ap =
pc

p
and the critical exponents reduce to
that of percolation. For the dilute quantum system Ap
=
pQ

p,
wherep, is the threshold concentration for the
appearance of longrange order
(pe
)
p,)
and y is not ex
pected to be identical to the percolation value.
We can similarly define other critical exponents. For
instance, if we consider sm+nonzero temperature,
T
then
we may introduce a crossover exponent’*2’$~ by writing
WT) l&l WT/]Apld). c,= . (4)
Also we may define the correlation. length 6 andits asso
ciated exponent Y by
. 
pkS c$[ (si*Sj)]a&““‘/ x [ (Si*Sj)l,&~” ., .: :(Sa)
.lip,  “kv,
ki
(5b)
which we implemented for k= 1 and k=2. For such a
zerotemperature transition one expects3
_. _
,’
,
r=(l rlh
(6)
where n is defined by I[(Si*Sj)]J  rb d 7: The differ2
ence between . Eq. (6) and the usual formula,
y = (2  7) Y, is due to the additional time dimension used
to describe zerotemperature quantum systems4 Likewise,
we write the singular part of the configurationally averaged
groundstate energy
E(p) as
.E(pj~+p[~~, I
(7)
with3 (Y 4 2  (d + z) Y, where z, the socalled dynamic ex
ponent, is defined so that’the characteristic energy scale is
of order 6 “:
The staggered susceptibility xs; which is
defined to havethe dimensions of (energy) L ‘,,consists of
two types of terms:’ I+’ 
i
xs=WT) + (NJ),
. (8)
where
L
I.
A=
1
2
(S,(kPS,t
k)) ,
1
(9s)
a
cl”
.~.
.*
B
?=2 ‘2 (S,(k) (N 9) (2 
E#)  ‘S,
1
a
(,k)) , ”
1
(9b)
a”
5191 J.Appl.Phys.69(8),15 April 1991
0021~8979/91/08519103$03.00
@ 1991 American Institute of Physics
5191
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where a: labels Cartesian components, x is the unit oper
ator, and 9 is a projection operator that is unity only in
the ground manifold whose energy is Ea. Finally, we con
sider the fourthorder staggered susceptibility. It is given
by more complicated expressions than Eqs. (8) and (9), so
we confine our attention to its zerotemperature limit and
write
where h, is the staggered field. These quantities have the
asymptotic behavior
WpbJApl y=“, (114
d(p)ApI+‘=“++,
(lib)
The aim is to determine
pQ
and the above critical ex
ponents. For this purpose we constructed series expansions
of these quantities in powers ofp up to orderpi3. To obtain
such an expansion for any quantity X we must calculate X
for all possible isolated clusters I’ consisting of up to 13
bonds. In this task we are aided by the LiebMattis theo
rem6 which says that for any cluster the total spin Stat of
the true ground state is identical to the value of S, in the
NCel state. Also, when S,,#O, the sum over the degenerate
groundstate manifold can be simplified using the Wigner
Eckart theorem.’ Then
[Xl,,= 2 p(r)x(r) = ; p”b’r’Xc(r>,
r
(12)
where P( I’) is the occurrence probability of the cluster l?,
nb(I’) is the number of bonds in J?, and X,(F) is the
cumulant value of X (which is obtained recursively by
subtracting the contributions from cumulants of all subdi
agrams of r). For quantities that only depend on the to
pology (and not on the exact shape) of the cluster, the sum
over I? can be restricted to topologically inequivalent dia
grams, since we have the weak embedding constants for
general hypercubic lattice in d dimensions. (For the series
involving the correlation length this simplification could
not be made, and those results were confined to d=2 and
3.) Critical exponents were obtained from the series by
PadC analyses or more general methods as described pre
viously.’ For instance, for d=2 the the coefficients in the
series for the correlation correlation length, starting with
the constant term and going up to order p”, are 0.0000,
1 .oooo, 0.6667,
14.7718,
 28.9345, 396.6181,
 2507.5602,
21 322.9721,  163 559.2768,
1276 223.5022,  9 657 938.4516, and 71906 968.9306.
From the original series, we can obtain further series either
by dividing one series by another or by term by term divi
sion of two series. The latter procedure’ has the advantage
I
I
that it yields a series whose divergence occurs at the known
value p = 1 in contrast to the former series whose threshold
remains at the unknown value
pQ
In principle, the first question should be to determine
pp and in particular to see whetherpQ = pc. To answer this
5192 J. Appl. Phys., Vol. 69, No. 8,15 April 1991
FIG. 1. Fixedpoint structure of the
randomly dilute quantum antiferro
magnet in the pg plane, where g is
the coupling constant in the nonlinear
sigma model. At the dilute quantum
fixed point Q, a fixed distribution of
coupling of finite width (i.e., inequiv
alent to a pure system ) is maintained.
This point, although on the critical
surface, does not Iie in the pg plane.
question we show in Fig. 1 the phase diagram we expect for
this system. There we incorporate the result of Chakra
varty, Halperin, and Nelson” that there is a critical value
for the coupling constant in the pure system for the non
linear sigma model. Also it seems certain that the condi
tion for randomness to be relevant,3 viz., a: + v > 0, holds
so that the pure system fixed point is unstable to dilution.
Since we believe quantum randomness and classical ran
domness are in different universality classes, we assume the
percolation fixed point to be unstable in the g direction.
Therefore there is a critical surface, whose projection in the
gp plane we show in Fig. 1, that surrounds the Ntel or
dered phase. The quantum random fixed point does not lie
in the pg plane because it requires a nontrivial fixed dis
tribution of coupling constants. This picture indicates that
the critical concentration should in fact depend on l/S.
However, our series were not long enough and well enough
behaved to detect a nonzero value of
Sp =pQ  pc.
It might be possible to show that t3pfO by considering
the highdimensionality limit. Although we expect Ap to
decrease with increasing dimension,i’ the high
dimensionality limit is often quite simple for classical mod
els, because there loopless, or tree, clusters dominate. In
fact, for most classical models the twopoint correlation
function for high d is given in terms of chain diagrams. The
case of lattice animals is a notable exception in which loop
less diagrams with all numbers of free ends are important12
in high d. In the present case we find that in the high
dimensionality limit our series have important contribu
tions from all loopless diagrams. In fact, most of our series
alternate in sign, indicating the importance of an unphys
ical singularity at
small
negative
p.
Similar behavior was
found previously’3 for quantum percolation. For example,
in the limit of high dimension the series for
C(p)
becomes
a power series in the variable dp with,coefficients, starting
with the constant term, which are 0.2500, 0.5000,
 0.3333, 1.7299,  4.6647, 23.1059,  115.3064,
664.3666,  3935.6508, ii? 961.6815,  147 319.3240,
912 274.4025,  5 675 735.2061, and 35 459 841.9534. To
summarize: Quantum fluctuations, like lattice animals,
have interesting behavior in high dimension.
Finally, we discuss the numerical values we found for
the critical exponents. Since our analyses were not very
sensitive to the exact choice ofpQ, we set
pQ
=
pc
In most
cases the results given in Table I represent several indepen
dent methods of analysis and the error bars reflect the
degree of consistency between different methods. The series
for E(p) could not be analyzed, as is often the case for
Wan, Harris, and Adler
5192
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TABLE I. Critical exponents for quantum dilution.
d=2
d=3
d=8
V
0.77*0.1 0.575 *0.05
. . .
ZV
1.30*0.1 1.30*0.1 1.33 =ko.o3
4
1.6hO.3
..I
1.5hO.05
4d
0.25 f 0.2 0.25 kO.2 0.10~0.1
Y
0.70*0.1 0.35*0.1 0.19*0.01
“The entries for 4  EY are values obtained from the series A(p)/C(p).
specificheat series. For d=2 and 3 we used several of the
above series including those for c2 and e to get estimates
for the correlation length exponent v. The other exponents
could then be determined from the series of Eq. ( 11) . We
expected to find $=zv because zv is the exponent associ
ated with the time axis and #J is associated with the equiv
alent temperature axis. However, as can be seen from Ta
ble I, we did not quite get this result. Perhaps this is an
indication of the reliability of our results. The unusually
small values of y are a consequence of Eq. (6)) which is a
unique reflection of quantum critical phenomena. Perhaps
the most important result in Table I is that v is very defi
nitely much less than for percolation, where Y = $ in d=2.
The fact that v is different from its percolation value proves
that the quantum antiferromagnet at percolation is not de
scribed at the percolation fixed point as the classical system
is.112 Or in other words, at the percolation fixed point l/S
is relevant. Although v is not too different from the non
linear sigma model result,” (v,0.7), our value of z that is
greater than unity indicates that the nonlinear sigma model
fixed point is indeed unstable to randomness. So although
we have not shown that
Sp
is nonzero, our results are
5193
J. Appl. Phys., Vol. 69, No. 8,15 April 1991
otherwise consistent with Fig. 1. It is obvious that there is
much to be understood, although our results are the first to
indicate rough values for the critical exponents of this
quantum dilution fixed point.
This work was supported in part by the National Sci
ence Foundation (NSF) under Grant No. DMR8815469
and by a grant from the U.S.Israel Binational Science
Foundation. One of us (J. A.) acknowledges the support of
the Israel Academy of Science. We thank B. I. Halperin
and T. C. Lubensky for helpful discussions concerning
scaling relations. We thank A. Aharony for helpful sugges
tions. We thank R. Fisch for some helpful comments on
the location of po. The computations were done on a Star
dent Titan 3000 computer supported by the NSF MRL
program under Grant No. DMR8819885.
‘M. J. Stephen and G. S. Grest, Phys. Rev. Lett. 38, 567 (1977).
‘A. B. Harris and T. C. Lubensky, Phys. Rev. B 35, 6964 (1987).
ID. Boyanovsky and I. Cardy, Phys. Rev. 26, 154 (1982); D. Boy
anovsky and J. Cardy, Phys. Rev. 27, 6971 (1983).
4J. Hertz, Phys. Rev. B 14, 1165 (1976).
‘J. H. Van Vleck, Theory
of
Electric and Magnetic Susceptibilities (Ox
ford University Press, Oxford, 1927).
‘E. H. Lieb and D. C. Mattis, Ann. Phys. (NY) 16,407 (1961); E. H.
Lieb and D. C. Mattis, J. Math. Phys. 3. 749 (1962).
‘See M. E. Rose, Elementary Theory of Angular Momentum (Wiley,
New York, 1957), p. 85.
‘L. Klein, J. Adler, A. Aharony, A. B. Harris, and Y. Meir, Phys. Rev.
B 40, 4824 (1989).
“D. L. Hunter and G. A. Baker, Jr., Phys. Rev: B 7, 3346 (1973); Y.
Meir, J. Phys. A 20, L349 (1987).
‘OS Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. 39, 2344
(1989).
“R Raghavan and D. C. Mattis, Phys. Rev. B 23, 4791 (1981); Y.
Shapir, A. Aharony, and A. B. Harris, Phys. Rev. Lett. 49,486 (1982).
“A. B. Harris, Phys. Rev. B 26, 337 (1982).
I3 A. B. Harris, Phys. Rev. B 29, 2519 (1984), and Appendix A therein.
Wan, Harris, and Adler 5193
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