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On the nature of the order parameter in quantum antiferromagnets
Joseph Reger
To cite this version:
Joseph Reger. On the nature of the order parameter in quantum antiferromagnets. Journal de
Physique II, EDP Sciences, 1991, 1 (3), pp.259-263. �10.1051/jp2:1991166�. �jpa-00247514�
Classification
Physics
AbsDnctg05.30 05.70F 64.60C 67.40D 75,10J
Sho« Comlnonication
On the nature of the order parameter in quantum
antiferromagnets
Joseph
D.Reger
Institut fur
Physik,
UniversithtMainz,
D-65W Mainz,Germany (Received
14Janumy
lwl,accepted 17Janumy
lwl)
Abstract. The first moments of the magnetization mz of the d
= 2, S
=
j
quantumHeisenberg
antiferromagnet
with nearestneighbor
interactions on the square lattice arecomputed
for T= 0 vta
path integral
Monte Carlo Simulations. The results for the ratio R=
(mf / (ml )~
show a cross-overat a system size of N m 100 to the classical behavtor, hmN-m R
=
9/5.
This supports the arguments that thelong
range order in theground
state is reachedby
a spontaneous symmetrybreaking,
just as in the classical case.1. Intmduction.
It seems that an
agreement
has been reached in the literature on the existence oflong
range orderin the
ground
state of the mo-dimensional nearestneighbor
S =) hotropic ant#erromagnetic quantum Heisenberg
model on the squarelattice,
e,g.[1-8],
andprobably
on allregular
unfrus-trated lattices in mo dimensions [9]. The
question
has along history
and the interest in it has been revivedrecently by
thediscovery
of the newhigh-Tc
materials.More
precisely,
the consensus is that thequantum fluctuations, althrough they
reduce thelong
range order in the
ground
state, are notstrong enough
tocompletely destroy
iL There is no con- senthowever,
on theprecise
value of themagnetization
at T = 0(although
a clearmajonty
favors about 60ib of the classicalvalue),
or on the mechanismby
which thelong
range order is built up.Some
reports
ofcomputer
simulationsquote
rather low numericalvalues,
but thisdiscrepancy
stems from a numerical factor which is omitted with the
argument
that the usual(classical)
sym-metry breaking
scenario is notapplicable
here [3]. Thisis_because
the orderparameter
vector will have fluctuations in itslength
forquantum
mechanical reasons, as it has beenargued subsequently [10].
In this case one would not bepermitted
toanticipate
in finitesystems
the effects of spon- taneoussymmetry breaking
in the usual way [3]. In a differentstudy
it has been evenexplicitely argued
that thelong
range order is reached withoutsymmetry breaking [4].
Symmetry breaking
occurs, of course, in thethermodynamic bmit only. Hence,
the correct definition of the orderparameter
in finitesystems
simulates the effect ofspontaneous
symmetry260 JOURNALDEPHYSIQUEII N°3
breaking by applying
an infinitesimalsymmetry breaking field, taking
thethermodynamic
limit in the presenceof
thefield
andletting
thb field tend to zero as the laststep. unfortunately,
thisprocedure
bimpractical
for mostsystems
because therange
of sizes that can be studied is too small. One arguesinstead,
that forlarge systems
without external fields the dbtribution of the or- derparameter
vector will bestrongly peaked
at aparticular
value of themagnitude,
but otherwiserotationally
invarianL Theapplication
of an externalfield, prodded
it isinfinitesimal,
willmerely
fix the direction of the order
parameter
vector. Therefore themagnitude
of the orderparameter
is the correct measure of order in finitesystems.
It has been
pointed
outrecently [10]
that thisargument
isonly
valid ifimplidt assumptions
about the lack of fluctuations in the
magnitude
of the orderparameter
vector are fulfilled. The authors of[10]
argue thatthey
are not fulfilled forquantum anfiferromagnets.
This wouldplace quantum ant#erromagnets
into a different class fromquantum ferromagnets
and classical mag, nets.The hsue raised has both theoretical and
practical importance. Firstly,
if thequantum
antifer-romagnet
is sodifferent,
it is difficult to understand theapparently
excellentagreement
withspin
wave
theory [2, 8, 11],
as well as themounting
evidence that thissystem's
behavior isessentially
classical
[12-14]. Secondly,
if the orderparameter
is not a classical vector and the conventionalsymmetry breaking
scenario is notapplicable,
it is very difficult tointerpret
the results of the Monte Carlosimulations,
whichsupply rotationally
invariant correlation functions of one com-ponent
of the orderparameter only [5, 9], typically
themagnetization
mz. The results of thesimulations and the
spin
wavetheory
are ingood agreement
with theexperimental
results [9].The purpose of the
present study
was to test the violation of the above mentionedassumptions
in the mo-dimensional nearest
neighbor
S=
isotropic antiferromagnetic quantum Heisenberg
model on the square lattice. We follow the
proposition
of[10]
andstudy
the ratio of moments of themagnetization
distributionj~4
~ ~ z
(~'))~
If the order
parameter
m =(m~,
my, mz is a classical vector of fixedlength
and lids anisotropic distribution,
then in 3 dimensions one obtains R =9/5 by performing
twosimple integrals
inspherical
coordinates.The
previous
results forR,
that showed a monotonic increase withincreasing system
size to a value of about 1.9 and led to the conclusion that the classical value of9/5
= 1 8 will not bereached,
were obtained from simulations on very smallsystems (N
<20).
Since this behavior wassurprising
and the issue raised veryimportan~
it wasclearly
of interest to check whether the above trend continues for muchlarger system
sizes.2. The method.
The simulations were carried out
using
the world-finepath integral
Monte Carlotechnique,
asdescribed in earlier work
[5, 9].
Thecomputations
werequite demanding,
since it is more difficult to determinehigher
moments of the orderparameter
withcomparable
statistical errors than lower moments.Fortunately, however,
one does not needvery high accuracy
to find the correct ~end of the data as a function ofsystem
size.Hence,
noattempt
has been made to obtain smaller stansncal errors in the final estimates than sufficient to establish theright
trend. Theinherent1ystemanc
errors of the method due to finitetemperature
and finite lYotter-dimension in the simulation werekept
under control in the same fashion as describedby [5, 9].
The simulations wereperformed
for lattice sizes of 2 x2,
4 x4,
8 x
8,10
x 10 and 12 x 12. Thetemperature
T usedupto
linearsystem
size L = 10 was T=
0.I,
a value that
guarantees temperature independent (T
=0)
results for thesesystem
sizes. For L = 12 the valuep
=
T-1
= 15 was used. The number of "time slices" were bemeen 100 and
200,
for L = 12 between 100 and 4tXl. Several different values were used toperform
aquadratic
fit,
from which theextrapolated
value for infinite lYotter-dimension was obtained. For thelargest system
size 500000MCS/spin
wereneeded,
out of which 200000MCS/spin
were discarded forequilibration.
5-7 MC runs wereaveraged
to estimate statistical errors.The simulations were carried out on a
multi-processor lYansputer
machinecontaining
88 pro-cessors
~T800j20).
Theexisting
code[~
wasparallelimd using geometrical decomposition
in thelYotter-dimension
(this
b the obviouschoice,
since it is an order ofmagnitude larger
than thespace-like dimensions).
In this scheme theprocessors
are connected into aring topology
to com-ply
withperiodic boundary
conditions and theadjacent processors update
thespin
variables inneighboring
blocks of "time slices". Onefeeds
rathersophisticated
communication between theprocessors
to ensure detailed balance.Despite
ofthis,
an overallparallel efficiency
of 80fb could be achieved.3. Results.
The results are shown in
figure
I. The ratio of moments(ml) / (m))~
isplotted against
IIN
Some of the data of
[10] (upto
N=
20)
are also included(open circles).
Note that the z-axis is IIN
in contrast to[10]
where I/N~
hused,
which hjustified
for the infinite range Lieb-Mattismodel
only.
Thepresent
choice enables one toplot
all the available data in asingle figure. Clearly,
the conclusion is not influenced
by
thin choice in any way.For small
system
sizes wereproduce
the results of[10],
where available. The ratio is mono-tonically increasing upto
N = M. Forlarger system
slurs it decreasesagain
andapproaches
the classical limit of 9/5. The error bars in the latterregion
are shown.200
90
'
~~
o ~
j~y~f
~
85l 80
1.75
~
i 70 o
1 65
60
0 00 0 02 0 04 0 06 0 08 0.10 0 12 0 14
1IN
Fig,
I. The ratio of moments R=
ml / (ml )~
of the magnetization of the nearestneighbor
S=
1/2
Heisenberg antiferromagnet
on the square lattice ve~sus1IN
Some of the results of[lo]
are also included(open circles).
A cross-ever to the classical value of9/5 is observed for N m 100 The error bars are shownin this region Only.
262 JOURNALDEPHYSIQUEII N°3
4. Discussion.
The non-mbnbtonic behavior of R h somewhat unusual and has to be contrasted with the results for the Lieb-Mattis model
[15],
which does not exhkit thisbehavior,
but rather converges mono-tonically
to the classical value[10].
The Lieb-Mattis modelis, however,
aninfinite
range Heisen-berg
model. It has beenitudied
indetail,
the exact solution h known[16].
One canexplicitly
examine its finite size behavior and find the
following [16,
17~: There bspontaneous symmetry
breaking
in thethermodynamic limi~
where at T = 0 all states with different values of the to- talspin
becomedegenerate.
The same scenario has been demonstrated for the infinite range XY-model[18].
The ratio R can beeasily computed analytically
[17~. Itapproaches
the classical valuemonotonically
frombelow,
inagreement
with the results of the simulations[10]. Moreover,
one can derive the dhtribution function of the
magnetization
P(mz
,
which b uniform since the
ground
state haslong
range order. Theon~/
finite size effect in P(mz
is the discreteness of theposskle
values of mz, but all values have the sameweight,
I.e. the distrkution hsharp
for all sizes.On the other hand it is obvious that in the short range model there will be an additional finite size effect: The
rounding
of theshape
of P(mz)
for smallersystem
sizes. This has also beenexplicitly
demonstrated insimulations,
seefigure
10 of[9].
There is apresumably complicated interplay
between the twoeffects,
which can lead to the cross-ever behavior found infigure
I. In thethermodynamic
limit P(m,
becomessharp
and the classical value of R is restored. It is clear that all states witharbitrarily large angular
momentum are contained in theground
state in this case, too(for
numerical evidence see[19]).
Theground
state of the model is asymmetry
brokenstate.
A detailed
understanding
of the non-monotonic behavior ofR, however,
is stillmissing
and isclearly
aninteresting question.
5. Conclusion.
In
conclusion,
the above results show that the nature of the orderparameter
inquantum
antifer-romagnets
h indeed classical in the sense that the fluctuations in itsmagnitude disappear
in thethermodynamic limit,
in other words the orderparameter
m is anhotropically
distributed classical vector. It is correct to think about thespontaneous symmetry breaking
in this model in the same fashion as in classicalspin systems:
in finitesystems
therotationally
invariantexpression
of theorder
parameter (magnitude)
is theappropriate
measure of order.Hence,
in thisrespect,
thequantum Heisenberg antiferromagnet belongs
to the same class of models as thequantum ferromagnetic Heisenberg model,
as well as the classicalcounterparts.
This result is in
agreement
with other studies which findtha~ although
this is aquantum system,
its behavior isessentially
classical[12-14].
Acknowledgements.
It is a
pleasure
to thank ARYoung
forhelpful
discussions and I. Peschel for a discussion andcorrespondence communicating
hiunpublished
results.References
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