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Journal ArticleDOI

The dilute quantum Heisenberg antiferromagnet

15 Apr 1991-Journal of Applied Physics (American Institute of Physics)-Vol. 69, Iss: 8, pp 5191-5193

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 077±010 in two dimensions The critical concentration for the appearance of long‐range order is indistinguishable from the percolation threshold
Topics: Heisenberg model (55%), Critical phenomena (55%), Percolation threshold (52%), Absolute zero (52%)

Summary (1 min read)

Jump to: [1 (9b a"] and [WpbJApl]

1 (9b a"

  • Finally, the authors consider the fourth-order staggered susceptibility.
  • These quantities have the asymptotic behavior.

WpbJApl

  • The aim is to determine pQ and the above critical exponents.
  • Critical exponents were obtained from the series by PadC analyses or more general methods as described previously.'.
  • Since the authors believe quantum randomness and classical randomness are in different universality classes, they assume the percolation fixed point to be unstable in the g direction.
  • For d=2 and 3 the authors used several of the above series including those for c2 and e to get estimates for the correlation length exponent v.

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University of Pennsylvania University of Pennsylvania
ScholarlyCommons ScholarlyCommons
Department of Physics Papers Department of Physics
4-15-1991
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
Follow this and additional works at: https://repository.upenn.edu/physics_papers
Part of the Quantum Physics Commons
Recommended Citation Recommended Citation
Wan, C., Harris, A., & Adler, J. (1991). The Dilute Quantum Heisenberg Antiferromagnet.
Journal of Applied
Physics,
69
(8), 5191-5193. http://dx.doi.org/10.1063/1.348123
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/436
For more information, please contact repository@pobox.upenn.edu.

The Dilute Quantum Heisenberg Antiferromagnet The Dilute Quantum Heisenberg Antiferromagnet
Abstract Abstract
Series expansions are used to treat the randomly diluted spin1/2 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 longrange 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 spin-i 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 long-range 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 zero-temperature
transition that occurs as a function of concentration
p
for
the spin-i 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 thatthe sum is
over pairs of nearest neighbors on a d-dimensional 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 bondis
unoccupied.
To highlight the differences between the spin-i 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 [ (SiSj)]~&Y~ i
iJ
(2)
where N is the total number of sites in the system, [ 1,
indicates an average over all configurations:(of the QS),
( ) denotes a zero-tetnperature 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-
wave-vector correlation function for a dilute ferromagnet.
Then all spins are parallel in the groundstates and C(p)
reduces to the mean-square size of connected clusters,
which is the percolation susceptibility. Thus for a classical
system the threshold concentration at which long-range
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 long-range 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$~ by writing
WT) -l&l -WT/]Apld). -c,= -. (4)
Also we may define the correlation. length 6 and-its asso-
ciated exponent Y by
. -
pkS c--$[ (si*Sj)]a&““‘/ x- [ (Si*Sj)l,&~ ., .: :(Sa)
.lip, - kv,
ki
(5b)
which we implemented for k= 1 and k=2. For such a
zero-temperature 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 zero-temperature quantum systems4 Likewise,
we write the singular part of the configurationally averaged
ground-state energy
E(p) as
.E(pj~+p[~-~, I
(7)
with3 (Y 4 2 - (d + z) Y, where z, the so-called dynamic ex-
ponent, is defined so thatthe characteristic energy scale is
of order 6 -:
The staggered susceptibility xs;- which is
defined to have-the 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
0021~8979/91/085191-03$03.00
@ 1991 American Institute of Physics
5191
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165.123.108.243 On: Thu, 09 Jul 2015 15:23:28

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 fourth-order staggered susceptibility. It is given
by more complicated expressions than Eqs. (8) and (9), so
we confine our attention to its zero-temperature 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 Lieb-Mattis 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
ground-state manifold can be simplified using the Wigner-
Eckart theorem. Then
[Xl,,= 2 p(r)x(r) = ; pbrXc(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. Fixed-point structure -of the
randomly dilute quantum antiferro-
magnet in the p-g 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 p-g 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
g-p plane we show in Fig. 1, that surrounds the Ntel or-
dered phase. The quantum random fixed point does not lie
in the p-g 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 high-dimensionality 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 two-point 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 previously3 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
4-d
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).
specific-heat 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. DMR-88-15469
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. DMR-88-19885.
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.
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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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