Critical behaviour and Scaling functions for the three-dimensional O(6) spin model with external field

5 Pages·0.21 MB·English

by  Sven Holtmann

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3
0
0 CRITICAL BEHAVIOUR AND SCALING FUNCTIONS FOR
2 THE THREE-DIMENSIONAL O(6) SPIN MODEL

n WITH EXTERNAL FIELD∗
a
J

6
1 S.HOLTMANNAND T. SCHULZE

Fakult¨at fu¨r Physik, Universit¨at Bielefeld,
1
D-33615 Bielefeld, Germany.
v
E-mail: holtmann, [email protected]
6
1
0
1
We numericallyinvestigate the three-dimensional O(6) model on123 to 1203 lat-
0
tices. From Binder’s cumulant at vanishing magnetic field we obtain the critical
3
couplingJc=1.42865(5)andverifythisvaluewiththeχ2-method. Theuniversal
0
valueofBinder’scumulantatthispointisgr(Jc)=−1.94456(10). Atthecritical
/
t couplingwefindthecriticalexponentsν=0.818(5),β=0.425(2)andγ=1.604(6)
a
fromafinitesizescalinganalysis. Wealsodeterminethefinite-size-scalingfunction
l
- onthe criticallineandthe equation of state. Our O(6)-resultforthe equation of
p
stateiscomparedtotheIsing,O(2)andO(4)results.
e
h
:
v
1. Introduction
i
X
Our goal is the determination of universal quantities of the three-
r
a dimensionalO(6)-invariantnonlinearσ-model,especiallycriticalexponents
and scaling functions. These quantities shall eventually be compared to
thoseof2-flavourstaggeredQCDwithadjointfermions. Therelevantsym-
metry ofthis modelis SU(4),so itshould be in the sameuniversalityclass
as the O(6) spin model1. The Hamiltonian of our model is defined as
βH = −J X φ~x·φ~y − H~ · Xφ~x. (1)
<x,y> x
Here x and y are the nearest-neighbour sites on a three-dimensional hy-
percubic lattice, φ~ is a 6-component unit vector at site x and H~ is the
x
external magnetic field. The coupling constant J is considered as inverse
temperature,therefore J =1/T. AtH =|H~|=6 0 the orderparameter,the
∗ThisworkissupportedbytheDeutscheForschungsgemeinschaftunderGrantNo. FOR
339/1-2
1
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magnetisation M, is defined by
M = hφki, (2)
whereφk isthelongitudinal(paralleltoH~)componentofφ~. AtH =0this
quantity vanishes for all couplings on the lattice, but we can use
M = h|φ~|i (3)
as an approximate order parameter. We also measure the susceptibility χ
and Binder’s cumulant g , defined by
r
χ = V(hφ~2i − M2), (4)
h(φ~2)2i
g = −3. (5)
r hφ~2i2
We have simulated the model with a Wolff-Cluster-Algorithm, using a
ghostspin to emulate the external magnetic field. We used hypercubic lat-
ticeswithperiodicboundariesandlinearextensionsLbetween12and120.
NearthecriticalcouplingJ wehavemadeupto200000measurementsper
c
data point, elsewhere about 20000 measurements. Between the measure-
ments we have used up to 400 cluster updates in the broken phase and up
to 1500 updates in the symmetric phase.
2. Determination of the critical coupling
At H = 0 the Binder cumulant can be described by the finite-size-scaling
function
g =Q (tL1/ν;L−ω), (6)
r g
where t= T−Tc is the reduced temperature and ω is the leading irrelevant
Tc
exponent. Therefore g should be independent of L at the critical point
r
t = 0 apart from corrections due to irrelevant scaling fields. The left plot
in Fig. 1 shows our reweighteddata, with the dotted lines representing the
jackknife error corridors. Although he intersection points coincide within
the errors,there aresome small corrections. As the value ofω is unknown,
we used Binder’s approximation2
1 1 c
2
= + (7)
J J logb
ip c
to extrapolate the intersection points J (L,b) of two lattices with sizes L
ip
andL′ =bL. FittingtheresultsforthelatticesizesL=12,16,20,24,30,36
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-1.943 3
gr -1.7 gr ML/
2.0
-1.8
-1.944 -1.9 1.5 120
-2.0 J 96
1.3 1.4 1.5 72
1.0 Q 48
36
-1.945
24
J HL /
-1.946 0.5
1.4284 1.4286 1.4288 1.4290 5 10 50 100 500
Figure 1. The left plot shows the binder cumulant for the lattices with size L =
12,16,20,24,30,36,48,60,72. The large lattices have a steeper slope then the small lat-
tices. Therightplotshowsthefinite-size-scalingfunctionatthecriticalcoupling.
to a constant value we find
1
= 0.699960(14) ⇒ J = 1.42865(5). (8)
c
J
c
We havecheckedthis resultwith the χ2-method describedin Ref. 3. From
this value we derive the universal quantity g (J )=−1.94456(10).
r c
The result for J is comparable with the result J = 1.42895(6) of Butera
c c
and Comi4 using a high temperature expansion, but their errors seem to
be underestimated.
3. The critical exponents β, γ and ν
Forthedeterminationofthecriticalexponentsweusethe followingscaling
relations at t=H =0:
M = L−β/ν(a + a L−ω) (9)
0 3
χ = Lγ/ν(b + b L−ω) (10)
0 3
∂g
r = L1/ν(d + d L−ω) (11)
0 3
∂J
Fitting these relations to our data at lattices in the range L = 12−72
we get β/ν = 0.519(2) and γ/ν = 1.961(3) from M and χ. The error of
these quantities includes a variation of ω between 0.5 and 1.0. From the
derivativeofg weget1/ν =1.223(5). Hereweneglectthecorrectionterm,
r
becauseitiszerowithintheerrors. Fromtheseresultswecancalculatethe
criticalexponentsβ =0.425(2),γ =1.604(6)andν =0.818(5). Theyfulfil
the hyperscaling relations and are in complete agreement with the results
of Butera and Comi.4
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4. The equation of state
InthevicinityofT thecriticalbehaviourcanbedescribedbytheuniversal
c
equation of state. It can be written in the form
M = h1/δf (z′), z′ = t′/h1/βδ (12)
G
t′ and h are the normalized reduced temperature t′ = (T −T )/T and
c 0
magnetic field h=H/H with the normalizationconditions f (0)=1 and
0 G
f (z′) = (−z′)β as z′ → −∞. To get the normalization constants H and
G 0
T wehavetodeterminethecriticalamplitudesofthemagnetisationonthe
0
criticalisothermandonthecoexistenceline. Theinfinitevolumebehaviour
of M is given by
M(T ,H) = d H1/δ(1 + d1Hωνc) (13)
c c c
including the leading correction term. From hyperscaling relations we get
for the exponents δ = 4.780(22) and ν = 0.4031(24). With this ansatz,
c
using only the data of the largestlattice at each value of H, we get for the
critical amplitude d =0.642(1), which translates to H =d−δ =8.3(1).
c 0 c
In the broken phase the magnetisation is described by the ansatz
M(T <T ,H)=M(T,0)+c (T)H1/2+c (T)H. (14)
c 1 2
The H1/2 term is due to the Goldstone effect. As we have already seen
with the O(2)5 and O(4)6 spin model, this ansatz fits very well and allows
the extrapolation of data at nonzero H to H = 0. Fitting the results for
M(T,0)forseveralcouplings(J =1.45,1.47,1.5,1.55and1.6)totheansatz
M(T ≤T ,0)=B(T −T)β[1 + b (T −T)ων + b (T −T)] (15)
c c 1 c 2 c
we get B =1.22(1), which leads to T =B−1/β =0.63(1).
0
The left plot in Fig. 2 shows the scaling function f (z′). In the broken
G
phase (z′ < 0) the solid lines are reweighted data from different J-values.
There are visible corrections, so we use the ansatz
Mh−1/δ = f (z′) + hωνcf(1)(z′) + h2ωνcf(2)(z′) (16)
G G G
to obtain the universal value of f (z′) (dashed line). In the symmetric
G
phase the corrections are negligible. In the right plot of Fig. 2 we show a
comparisonoftheO(6)scalingfunctionstothoseofsomeotherO(N)mod-
els, which have been determined in Refs. 7 (Ising), 5 (O(2)) and 6 (O(4)).
As one can see, the functions get steeper with increasing N. These func-
tionscaneventuallybeusedtodeterminetheuniversalityclassofstaggered
2-flavour QCD both with fundamental or with adjoint fermions. But until
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2.0 5
2.5
M/h1/ O(1)
2.0 M/h1/ 1.5 O(2)
O(4)
1.5 O(6)
1.0
1.0
0.5
0.5
t’/h1/
t’/h1/
0 0
-10 -7.5 -5 -2.5 0 2.5 5 -5 -2.5 0 2.5 5
Figure2. TheequationofstateofO(6)(leftplot)andthecomparisonoftheequations
ofstateofO(1)(Ising),O(2),O(4)andO(6)(rightplot).
now QCD data has too low statistics to distinguish between these scaling
functions, and also the used lattice sizes are too small in QCD to be sure
that one has reached the infinite volume limit.
5. Finite-size-scaling function at Jc
Because of the small lattice sizes used in QCD, finite-size-scaling functions
are a better tool for the comparisonof spin models to QCD. At T and for
c
small H the different lattices should scale like
M(T ,H,L) = L−β/νQ (HLβδ/ν) (17)
c M
where Q is a universal scaling function. Our results are shown in the
M
right plot in Fig. 1. The solid line shows the asymptotic behaviour in the
limit L→∞
Q (z) = Q (z) = d z1/δ, (18)
M ∞ c
which is observable for z =HLβδ/ν>40.
∼
References
1. F. Karsch and M. Lu¨tgemeier, Nucl. Phys. B550, 449 (1999).
2. K.Binder, Phys. Rev. Lett. 47, 693 (1981).
3. J. Engels et al., Phys. Lett. B365, 219 (1996).
4. P.Butera and M. Comi, Phys. Rev. B58, 11552 (1998).
5. J.Engels,S.Holtmann,T.Mendes,T.Schulze,Phys.Lett. B492,219(2000).
6. J. Engels and T. Mendes, Nucl. Phys. B572, 289 (2000).
7. J. Engels, L. Fromme, M. Seniuch, Preprint cond-mat/0209492.