We use finite--size scaling of Lee--Yang partition function zeroes to study
the critical behaviour of the two dimensional step or sgn $O(2)$ model. We
present evidence that, like the closely related $XY$--model, this has a phase
transition from a disordered high temperature phase to a low temperature
massless phase where the model remains critical. The critical parameters
(including logarithmic corrections) are compatible with those of the
$XY$--model indicating that both models belong to the same universality class.Comment: 6 pages latex, 3 postscript figures, compressed and uuencoded
  (revised remarks on Lee_Yang theorem, version to appear in Phys Rev B

Irving, A. C.

Kenna, R.

English

arXiv

We use finite-size scaling of Lee-Yang partition function zeros to study the critical behavior of the two-dimensional step or sgn O(2) model. We present evidence that, like the closely related XY model, this has a phase transition from a disordered high-temperature phase to a low-temperature massless phase where the model remains critical. The critical parameters (including logarithmic corrections) are compatible with those of the XY model indicating that both models belong to the same universality class

Kenna, Ralph

Coventry University Pure Portal

CURVE is the Institutional Repository for Coventry University   Critical behavior of the two-dimensional step model  Irving, A. C. and Kenna, R.  Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Irving, A. C. and Kenna, R. (1996) Critical behavior of the two-dimensional step model. Physical Review B, volume 53 : 11568 http://dx.doi.org/10.1103/PhysRevB.53.11568  DOI 10.1103/PhysRevB.53.11568 ISSN 2469-9950 ESSN 2469-9969  Publisher: American Physical Society  Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.   This document is the author’s post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.        arXiv:hep-lat/9508033v2  1 Feb 1996Critical Behaviour of the Two DimensionalStep ModelA.C. Irving and R. Kenna1DAMTP, University of Liverpool L69 3BX, EnglandAugust 1995AbstractWe use finite–size scaling of Lee–Yang partition function zeroes to study the critical be-haviour of the two dimensional step or sgn O(2) model. We present evidence that, like theclosely related XY –model, this has a phase transition from a disordered high temperature phaseto a low temperature massless phase where the model remains critical. The critical parameters(including logarithmic corrections) are compatible with those of the XY –model indicating thatboth models belong to the same universality class.1Supported by EU Human Capital and Mobility Scheme Project No. CHBI–CT94–11251 IntroductionIn a recent paper [1], we demonstrated the power of finite–size scaling applied to Lee–Yang zeroes[2] in uncovering logarithmic corrections to scaling in the two–dimensional XY – or O(2) spin model.In this paper we apply the same techniques to the closely related ‘step model’ [3, 4], also known [5, 6]as the sgn O(2) model. The question of criticality of this model has, until now, been unresolveddespite several analyses based on high temperature series (see [6] for a review) and on numericalsimulation [7, 8]. The interest in the model arises from its possible membership of the XY –modeluniversality class which exhibits the Kosterlitz–Thouless [KT] phase transition [9]. Like the XY –model, the step model has a configuration space which is globally and continuously symmetric.Unlike the XY –model, however, the interaction function is discontinuous and the Mermin–Wagnertheorem [10] does not apply. Nonetheless, it is expected that if a phase transition exists in the stepmodel, it should not be to a phase with long range order [4, 8, 11].Sa´nchez–Velasco and Wills [8] presented evidence of critical behaviour starting at βc = 1/Tc =0.91 ± 0.04. This was based on finite–size scaling [FSS] of the spin susceptibility. Since the as-sociated critical index η(Tc) was significantly greater than that measured for the XY –model, itwas concluded that the step and XY models are not in the same universality class. In this paperwe present evidence that the step model is not critical at that temperature. However it is criticalat lower temperatures with a critical index η(Tc) compatible with the XY value. The accuracyafforded by the Lee–Yang zeroes study is a crucial part of the analysis.2 The step model and the XY –modelConsider the partition functionZ(β, h) =∑{~sx}e−βH+hnˆ·~M , (1)where the Boltzmann factor is β = 1/kT , nˆ is a unit vector defining the direction of the externalmagnetic field and h is a scalar parameter representing its strength. The summation is over allconfigurations open to the system and ~sx is a unit length two-component spin at each site x in thecubic lattice Λ ≡ Ld (d = 2). The magnetisation for a given configuration is~M =∑x∈Λ~sx .In the case of the XY –model, the interaction hamiltonian isHXY = −∑x∈Λd∑µ=1~sx · ~sx+µ ,while for the step model it isHstep = −∑x∈Λd∑µ=1sgn(~sx · ~sx+µ) .Thus the continuous (cosine) dependence of the interaction energy in the usual XY –model isreplaced by a discrete step function dependence.The leading infinite volume critical behaviour of the 2D XY –model is characterized by exponentialdivergences (essential singularities) in the thermodynamic functions [9]. In terms of the reduced1temperature t ≡ 1 − β/βc → 0+ the (leading) infinite volume scaling behaviour of the correlationlength and the zero-field magnetic susceptibility is (respectively) [9]ξ∞(t) ∼ eat−ν , (2)χ∞(t) ∼ ξ2−η∞ , (3)where ν = 1/2 and η = 1/4. The XY –model remains critical for all β > βc.For models obeying the Lee–Yang theorem [2], the partition function zeroes in the magnetic fieldstrength (h) plane (the Lee–Yang zeroes) are all on the imaginary axis. In the high temperaturephase these zeroes remain away from the real axis, pinching it only as t→ 0+ (in the thermodynamiclimit). The zero closest to the real axis marks the edge of the distribution of zeroes and is knownas the Yang–Lee edge, zYL. The theorem has been proved only for certain models, the XY –modelincluded [12]. In [1] we used this fact to show the above leading critical behaviour for theXY –modeland the corresponding behaviour of the Yang–Lee edge, zYL are, in fact, modified by logarithmiccorrections:χ∞(t) ∼ ξ2−η∞ tr , (4)zYL(t) ∼ ξλ∞tp , (5)whereλ = −12(d+ 2− η) = −158(6)and the parameters r and p (= −r/2) are logarithmic correction indices [1]. The correspondingFSS behaviour for the susceptibility and the first zero z1 (= zY L) at t = 0 is [1]χL(0) ∼ L2−η(lnL)−rν , (7)z1(0) ∼ Lλ(lnL)r . (8)For the 2D XY –model the numerical value of r was found to be small but non-zero (−0.023±0.010)[1].The objects of the present analaysis were to establish1. if the scaling behaviour of the very precisely determined Yang–Lee edge would unequivocablydetermine whether the step model had a critical phase2. if so, whether the phase transition is in the same universality class as the XY –model.3 Method and resultsThe methods used are those described in [1]. A single cluster algorithm [13] is used to generate alarge number of measurements (100K for each lattice size L and temperature 1/β) of the energyH and the magnetisation ~M at zero external magnetic field. Such data were obtained for latticesizes L = 32, 48, 64, 128 and 256 covering the β range 0.86 to 1.40 with varying degrees of spacing.Multi–histogram techniques [14, 15] were used to combine data at different values of β and soobtain detailed β dependence. In the neighborhood of possible critical points we used sufficientlyfine spacing (typically 0.025) to ensure adequate overlap between histograms for a given size oflattice.The partition function for a complex magnetic field (h = hr + ihi, hr, hi ∈ R) can be written interms of a real and an imaginary part [1, 16, 17] asZ(β, hr + ihi) = ReZ(β, hr + ihi) + iImZ(β, hr + ihi) , (9)2whereReZ(β, hr + ihi) = Z(β, hr)〈cos (hiM)〉β,hr , (10)ImZ(β, hr + ihi) = Z(β, hr)〈sin (hiM)〉β,hr . (11)Here the subscripts indicate that the expectation values are taken at (inverse) temperature β andin a (real) external field hr.No specific proof of the Lee–Yang theorem exists for the step model. We can, however, use (10) and(11) to (numerically) determine the loci along which the real and imaginary parts of the partitionfunction separately vanish. These formulae concern expectation values of real quantities in a realexternal field and at no stage is a simulation involving a complex action involved. The Lee–Yangzeroes are then the points in the complex h–plane where the loci intersect [17]. We have determinedthese loci and thereby the Lee–Yang zeroes close to the real h–axis. We were able to determine thefirst 15 zeroes and found that they lie on the imaginary h–axis for all the lattices studied. Thus wehave numerical evidence that the step model obeys the Lee–Yang theorem. The following analysisapplies to the first zeroes only (the Yang–Lee edge) and we defer the study of higher zeroes to alater paper [18].The analysis began with a rough search for the leading critical behaviour predicted by (8). Anindependent test was also made using the (less accurate) susceptibility data and (7). Both methodsindicated critical behaviour setting in for β >∼ 1.2. In Fig. 1 we show a typical log–log plot ofz1 the Yang–Lee edge versus L. This is at a typical candidate value of the critical temperature(β = 1.22). The errors are considerably smaller than the symbols. For example, at β = 1.22 wefound z1 = 0.0024136(7) and 0.00017902(6) at L = 32 and 128 respectively. The slope of Fig. 1gives the effective leading index λeff ignoring corrections. At β = 1.22 this is λeff = −1.8761(2)where the chi-squared per degree of freedom (χ2/dof) for the linear fit shown is 0.85. In Fig. 2(a)we display the result of such fits as a function of β. The effective exponent λeff is just the slopeof the log–log linear fit which should obtain if critical behaviour is present (ignoring logarithmiccorrections). The corresponding χ2/dof for the linear fit is also shown. Acceptable values are onlyfound for β in excess of around 1.2. To quantify this statement we demandχ2/dof ≤ 2.0 (12)which means β >∼ 1.185. We note that the corresponding values of λeff (<∼ − 1.872(2)) includethat (−15/8 = −1.875) corresponding to the KT prediction.Fig. 2 is evidence for (i) the validity of FSS over a range of values β ≥ βc ≃ 1.185 and (ii) at βc theexponent λeff very close to the expected KT value (-15/8). Observation (i) means that, as in theXY case, the system remains critical for β ≥ βc and (ii) is evidence that λ is in fact -15/8 and thequestion of whether or not the step model belongs to the same universality class as the XY modelmust now be answered by determination of the correction exponent r.We therefore assume the behaviour (8) with λ = −15/8 at β = βc. The expected leading behaviouris removed and linear fits toln(z1L15/8)vs. ln lnL (13)performed. The results are shown in Fig. 3. Since the value λ = −15/8 (η = 1/4) is only expectedat βc these results can be used to identify the possible values of critical temperature and to test forthe presence of logarithmic corrections as in the XY –model [1].Applying the same criterion (12) as for the leading behaviour, we search for a range of βc valuesgiving an acceptable fit. We find1.195 ≤ βc ≤ 1.295 and correspondingly, 0.009 ≥ r ≥ −0.034 . (14)3The range of acceptable r values includes that found [1] for the XY –model (−0.023 ± 0.010) andthat corresponding to no logarithmic corrections (r = 0). As in our previous work [1], this rangeexcludes the prediction r = −1/16 coming from an approximate renormalisation group treatmentof the XY –model [9]. Thus we conclude that the present data are compatible with the stepmodel being in the same universality class as the XY –model. We do not, however, exclude otherpossibilities.The susceptibility data are consistent with the above analysis. If one assumes the KT value ofη(βc) = 1/4, one can construct a so-called Roomany–Wyld beta function approximant [19] fromthe finite–size data and use its zero to locate βc [1]. These approximants, based on pairs of latticesize L,L′, converge very rapidly [19]. We estimate βc = 1.22 ± 0.02. This last analysis of courseneglects possible logarithmic corrections to scaling.We have also studied the specific heat. As for the XY –model, the step model data show a broadpeak with no obvious relationship to the position of the leading critical point. The finite–sizedependence is not dramatic and is likely to be of little value in further elucidating the criticialbehaviour. A related question is to what extent one can make use of the Fisher zeroes [16, 20],i.e. zeroes of the partition function in the complex β plane at zero external magnetic field h. Forboth this and the XY –model, these are much harder to locate than the Lee–Yang zeroes and areconsequently less accurately determined.4 ConclusionsThe use of finite–size scaling applied to Lee–Yang zeroes has allowed us to present detailed evidenceof critical behaviour in the two-dimensional step (sgn O(2) spin) model. The data are consistentwith this model being in the same universality class as the XY – ( O(2) spin) model. That is,it undergoes a Kosterlitz–Thouless type transition with susceptibility index η(βc) = 1/4 and wedetermine that βc lies in the range 1.195 ≤ βc ≤ 1.295. With the available statistics, we found thelogarithmic correction exponent to lie in the range −0.034 ≤ r ≤ 0.009. This should be comparedwith our measurement for the XY correction exponent [1], −0.033 ≤ r ≤ −0.013 with which it iscompatable. The step model results are however also compatable with no logarithmic corrections(r = 0 corresponds to βc = 1.21 and χ2/dof = 0.92).The Mermin–Wagner theorem [10] does not apply directly to the step model because of the dis-continuous nature of the interaction hamiltonian. However, it has long been believed [4, 8] thatif a phase transition does exist, it will not involve a phase with long range order. The evidencepresented here supports this view.This raises a question as to the nature of the mechanism driving the phase transition in thestep model. The KT phase transition of the XY –model is understood to be driven by the bind-ing/unbinding of topological solutions (vortices). However, the energetics of vortex formation arevery different in the step model [11, 6, 8]. Since vortices with effectively zero excitation energy canbe created at all non-zero temperatures, the usual KT argument does not naturally lead one toexpect such a phase transition in the step model.If this is indeed the case, some other driving mechanism must be responsible for any phase transition.It would then be remarkable if — as the evidence presented here indicates – both models belongto the one universality class.References[1] R. Kenna and A. C. Irving, Phys. Lett. B 351, 273 (1995).4[2] C. N. Yang and T. D. Lee, Phys. Rev. 87 404 (1952); ibid. 410.[3] A. J. Guttmann, G. S. Joyce and C. J. Thompson, Phys. Lett. A 38, 297 (1972).[4] A. J. Guttmann and G. S. Joyce, J. Phys. C 6, 2691 (1973).[5] I-H. Lee and R. E. Shrock, Phys. Rev. B 36, 3712 (1987).[6] I-H. Lee and R. E. Shrock, J. Phys. A 21, 2111 (1988).[7] A. Nymeyer and A. C. Irving, J. Phys. A 19, 1745 (1986).[8] E. Sa´nchez-Velasco and P. Wills, Phys. Rev. B 37, 406 (1988).[9] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); J. M. Kosterlitz J. Phys.C 7, 1046 (1974).[10] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 1133 (1966).[11] A. J. Guttmann and A. Nymeyer, J. Phys. A 11 1131 (1978).[12] F. Dunlop and C.M. Newman, Commun. Math. Phys. 44 223 (1975).[13] U. Wolff, Phys. Rev. Lett. 62 361 (1989).[14] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61 2635 (1988); Computers inPhysics, Sep/Oct 1989.[15] K. Kajantie, L. Ka¨rkka¨inen and K. Rummukainen Nucl. Phys. B 357 693 (1991).[16] R. Kenna and C. B. Lang, Phys. Lett. B 264 396 (1991); Nucl. Phys. B (Proc. Suppl.)30 697 (1993); Nucl. Phys. B 393 461 (1993); Err. ibid. B 411 (1994) 340.[17] R. Kenna and C. B. Lang, Phys. Rev. E 49 5012 (1994).[18] R. Kenna and A. C. Irving, Liverpool Preprint LTH 364.[19] M.P. Nightingale, Physica A 83 (1976) 561; H. Roomany and H.W. Wyld, Phys. Rev.D 21 (1980) 3341.[20] M.E. Fisher, in Critical Phenomena, Proc. 51th Enrico Fermi Summer School, Varena,ed. M.S. Green (Academic Press, NY, 1972).53 4 5 6Ln(L)-10-9-8-7-6Ln(Z) 1Figure 1: Scaling behaviour of the Yang–Lee edge at a typical critical β value (β = 1.22).6-1.890-1.885-1.880-1.875-1.870-1.865λ1.15 1.20 1.25 1.30 1.35 1.40β0246810χ2           (a)(b)/dfeffFigure 2: Test of expected leading critical behaviour: (a) the effective exponent λeff (slope of astraight line fit) and (b) the corresponding χ2/dof versus β.7-0.06-0.04-0.020.000.020.04r1.15 1.20 1.25 1.30 1.35 1.40β0510152025χ2      (a)(b)/dfFigure 3: Logarithmic corrections: (a) the logarithmic correction exponent r to the Yang–Lee edgeis shown as a function of the assumed critical coupling βc and (b) the corresponding χ2/dof.8

Critical behavior of the two-dimensional step model

We use finite--size scaling of Lee--Yang partition function zeroes to study the critical behaviour of the two dimensional step or sgn O(2) model. We present evidence that, like the closely related XY--model, this has a phase transition from a disordered high temperature phase to a low temperature massless phase where the model remains critical. The critical parameters (including logarithmic corrections) are compatible with those of the XY--model indicating that both models belong to the same universality class

Irving, A C

Kenna, R

CERN Document Server

Liverpool Preprint: LTH 355
Critical Behaviour of the Two Dimensional
Step Model
A.C. Irving and R. Kenna
1
DAMTP, University of Liverpool L69 3BX, England
August 1995
Abstract
We use nite{size scaling of Lee{Yang partition function zeroes to study the critical be-
haviour of the two dimensional step or sgn O(2) model. We present evidence that, like the
closely related XY {model, this has a phase transition from a disordered high temperature phase
to a low temperature massless phase where the model remains critical. The critical parameters
(including logarithmic corrections) are compatible with those of the XY {model indicating that
both models belong to the same universality class.
1
Supported by EU Human Capital and Mobility Scheme Project No. CHBI{CT94{1125
1 Introduction
In a recent paper [1], we demonstrated the power of nite{size scaling applied to Lee{Yang zeroes
[2] in uncovering logarithmic corrections to scaling in the two{dimensionalXY { or O(2) spin model.
In this paper we apply the same techniques to the closely related `step model' [3, 4], also known [5, 6]
as the sgn O(2) model. The question of criticality of this model has, until now, been unresolved
despite several analyses based on high temperature series (see [6] for a review) and on numerical
simulation [7, 8]. The interest in the model arises from its possible membership of the XY {model
universality class which exhibits the Kosterlitz{Thouless [KT] phase transition [9]. Like the XY {
model, the step model has a conguration space which is globally and continuously symmetric.
Unlike the XY {model, however, the interaction function is discontinuous and the Mermin{Wagner
theorem [10] does not apply. Nonetheless, it is expected that if a phase transition exists in the step
model, it should not be to a phase with long range order [4, 8, 11].
Sanchez{Velasco and Wills [8] presented evidence of critical behaviour starting at 
c
= 1=T
c
=
0:91  0:04. This was based on nite{size scaling [FSS] of the spin susceptibility. Since the as-
sociated critical index (T
c
) was signicantly greater than that measured for the XY {model, it
was concluded that the step and XY models are not in the same universality class. In this paper
we present evidence that the step model is not critical at that temperature. However it is critical
at lower temperatures with a critical index (T
c
) compatible with the XY value. The accuracy
aorded by the Lee{Yang zeroes study is a crucial part of the analysis.
2 The step model and the XY {model
Consider the partition function
Z(; h) =
X
f~s
x
g
e
 H+hn^
~
M
; (1)
where the Boltzmann factor is  = 1=kT , n^ is a unit vector dening the direction of the external
magnetic eld and h is a scalar parameter representing its strength. The summation is over all
congurations open to the system and ~s
x
is a unit length two-component spin at each site x in the
cubic lattice   L
d
(d = 2). The magnetisation for a given conguration is
~
M =
X
x2
~s
x
:
In the case of the XY {model, the interaction hamiltonian is
H
XY
=  
X
x2
d
X
=1
~s
x
 ~s
x+
;
while for the step model it is
H
step
=  
X
x2
d
X
=1
sgn(~s
x
 ~s
x+
) :
1
Thus the continuous (cosine) dependence of the interaction energy in the usual XY {model is
replaced by a discrete step function dependence.
The leading innite volume critical behaviour of the 2D XY {model is characterized by exponential
divergences (essential singularities) in the thermodynamic functions [9]. In terms of the reduced
temperature t  1  =
c
! 0
+
the (leading) innite volume scaling behaviour of the correlation
length and the zero-eld magnetic susceptibility is (respectively) [9]

1
(t)  e
at
 
; (2)

1
(t)  
2 
1
; (3)
where  = 1=2 and  = 1=4. The XY {model remains critical for all  > 
c
.
In [1] we showed that for the 2D XY {model this behaviour and the corresponding behaviour of
the Yang{Lee edge, z
YL
, is in fact modied by logarithmic corrections:

1
(t)  
2 
1
t
r
; (4)
z
YL
(t)  

1
t
p
; (5)
where
 =  
1
2
(d+ 2  ) =  
15
8
(6)
and the parameters r and p (=  r=2) are logarithmic correction indices [1]. The corresponding
FSS behaviour for the susceptibility and the rst zero z
1
(= z
Y L
) at t = 0 is [1]

L
(0)  L
2 
(lnL)
 
r

; (7)
z
1
(0)  L

(lnL)
r
: (8)
For the 2D XY {model the numerical value of r was found to be small but non-zero ( 0:0230:010)
[1].
The objects of the present analaysis were to establish
1. if the scaling behaviour of the very precisely determined Yang{Lee edge would unequivocably
determine whether the step model had a critical phase
2. if so, whether the phase transition is in the same universality class as the XY {model.
3 Method and results
The methods used are those described in [1]. A single cluster algorithm [12] is used to generate a
large number of measurements (100K for each lattice size L and temperature 1=) of the energy
H and the magnetisation
~
M at zero external magnetic eld. Such data were obtained for lattice
sizes L = 32, 48, 64, 128 and 256 covering the  range 0:86 to 1:40 with varying degrees of spacing.
Multi{histogram techniques [13, 14] were used to combine data at dierent values of  and so
obtain detailed  dependence. In the neighborhood of possible critical points we used suciently
2
ne spacing (typically 0.025) to ensure adequate overlap between histograms for a given size of
lattice.
The partition function for a complex magnetic eld (h = h
r
+ ih
i
, h
r
; h
i
2 R) can be written in
terms of a real and an imaginary part [1, 15, 16] as
Z(; h
r
+ ih
i
) = ReZ(; h
r
+ ih
i
) + iImZ(; h
r
+ ih
i
) ; (9)
where
ReZ(; h
r
+ ih
i
) = Z(; h
r
)hcos (h
i
M)i
;h
r
; (10)
ImZ(; h
r
+ ih
i
) = Z(; h
r
)hsin (h
i
M)i
;h
r
: (11)
Here the subscripts indicate that the expectation values are taken at (inverse) temperature  and
in a (real) external eld h
r
. If the Lee{Yang theorem [2] holds for this model then the zeroes are
on the imaginary h{axis where ImZ vanishes. Then the Lee{Yang zeroes are simply the zeroes in
h
i
of
ReZ(; ih
i
) / hcos (h
i
M)i
;0
: (12)
Thus the zeroes are easily found and at no stage is a simulation involving a complex action involved.
The analysis began with a rough search for the leading critical behaviour predicted by (8). An
independent test was also made using the (less accurate) susceptibility data and (7). Both methods
indicated critical behaviour setting in for 
>

1:2. In Fig. 1 we show a typical log{log plot of
z
1
the Yang{Lee edge versus L. This is at a typical candidate value of the critical temperature
( = 1:22). The errors are considerably smaller than the symbols. For example, at  = 1:22 we
found z
1
= 0:0024136(7) and 0:00017902(6) at L = 32 and 128 respectively. The slope of Fig. 1
gives the eective leading index 
e
ignoring corrections. At  = 1:22 this is 
e
=  1:8761(2)
where the chi-squared per degree of freedom (
2
=dof) for the linear t shown is 0:85. In Fig. 2(a)
we display the result of such ts as a function of . The eective exponent 
e
is just the slope
of the log{log linear t which should obtain if critical behaviour is present (ignoring logarithmic
corrections). The corresponding 
2
=dof for the linear t is also shown. Acceptable values are only
found for  in excess of around 1.2. To quantify this statement we demand

2
=dof  2:0 (13)
which means 
c
>

1:185. We note that the corresponding values of 
e
(
<

  1:872(2)) include
that ( 15=8 =  1:875) corresponding to the KT prediction.
The evidence for critical behaviour is the existence of a range of acceptable chi-squared values for
a linear t. In view of the similarity of 
e
to the expected KT value, we now proceed to test the
further hypothesis that the step model is in fact in the same universality class as the XY {model.
We assume the behaviour (8) with a particular value of  (given by (6)). This should be valid at
 = 
c
. The expected leading behaviour is removed and linear ts to
ln

z
1
L
15=8

vs. ln lnL (14)
3
performed. The results are shown in Fig. 3. Since the value  =  15=8 ( = 1=4) is only expected
at 
c
these results can be used to identify the possible values of critical temperature and to test for
the presence of logarithmic corrections as in the XY {model [1].
Applying the same criterion (13) as for the leading behaviour, we search for a range of 
c
values
giving an acceptable t. We nd
1:195  
c
 1:295 and correspondingly, 0:009  r   0:034 : (15)
The range of acceptable r values includes that found [1] for the XY {model ( 0:023 0:010) and
that corresponding to no logarithmic corrections (r = 0). As in our previous work [1], this range
excludes the prediction r =  1=16 coming from an approximate renormalisation group treatment
of the XY {model [9]. Thus we conclude that the present data are compatible with the step
model being in the same universality class as the XY {model. We do not, however, exclude other
possibilities.
The susceptibility data are consistent with the above analysis. If one assumes the KT value of
(
c
) = 1=4, one can construct a so-called Roomany{Wyld beta function approximant [17] from
the nite{size data and use its zero to locate 
c
[1]. These approximants, based on pairs of lattice
size L; L
0
, converge very rapidly [17]. We estimate 
c
= 1:22 0:02. This last analysis of course
neglects possible logarithmic corrections to scaling.
We have also studied the specic heat. As for the XY {model, the step model data show a broad
peak with no obvious relationship to the position of the leading critical point. The nite{size
dependence is not dramatic and is likely to be of little value in further elucidating the criticial
behaviour. A related question is to what extent one can make use of the Fisher zeroes [15, 18],
i.e. zeroes of the partition function in the complex  plane at zero external magnetic eld h. For
both this and the XY {model, these are much harder to locate than the Lee{Yang zeroes and are
consequently less accurately determined.
4 Conclusions
The use of nite{size scaling applied to Lee{Yang zeroes has allowed us to present detailed evidence
of critical behaviour in the two-dimensional step (sgn O(2) spin) model. The data are consistent
with this model being in the same universality class as the XY { ( O(2) spin) model. That is,
it undergoes a Kosterlitz{Thouless type transition with susceptibility index (
c
) = 1=4 and we
determine that 
c
lies in the range 1:195  
c
 1:295. With the available statistics, we found the
logarithmic correction exponent to lie in the range  0:034  r  0:009. This should be compared
with our measurement for the XY correction exponent [1],  0:033  r   0:013 with which it is
compatable. The step model results are however also compatable with no logarithmic corrections
(r = 0 corresponds to 
c
= 1:21 and 
2
=dof = 0:92).
The Mermin{Wagner theorem [10] does not apply directly to the step model because of the dis-
continuous nature of the interaction hamiltonian. However, it has long been believed [4, 8] that
4
if a phase transition does exist, it will not involve a phase with long range order. The evidence
presented here supports this view.
This raises a question as to the nature of the mechanism driving the phase transition in the
step model. The KT phase transition of the XY {model is understood to be driven by the bind-
ing/unbinding of topological solutions (vortices). However, the energetics of vortex formation are
very dierent in the step model [11, 6, 8]. Since vortices with eectively zero excitation energy can
be created at all non-zero temperatures, the usual KT argument does not naturally lead one to
expect such a phase transition in the step model.
If this is indeed the case, some other driving mechanismmust be responsible for any phase transition.
It would then be remarkable if | as the evidence presented here indicates { both models belong
to the one universality class.
References
[1] R. Kenna and A. C. Irving, Phys. Lett. B 351, 273 (1995).
[2] C. N. Yang and T. D. Lee, Phys. Rev. 87 404 (1952); ibid. 410.
[3] A. J. Guttmann, G. S. Joyce and C. J. Thompson, Phys. Lett. A 38, 297 (1972).
[4] A. J. Guttmann and G. S. Joyce, J. Phys. C 6, 2691 (1973).
[5] I-H. Lee and R. E. Shrock, Phys. Rev. B 36, 3712 (1987).
[6] I-H. Lee and R. E. Shrock, J. Phys. A 21, 2111 (1988).
[7] A. Nymeyer and A. C. Irving, J. Phys. A 19, 1745 (1986).
[8] E. Sanchez-Velasco and P. Wills, Phys. Rev. B 37, 406 (1988).
[9] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); J. M. Kosterlitz J. Phys.
C 7, 1046 (1974).
[10] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 1133 (1966).
[11] A. J. Guttmann and A. Nymeyer, J. Phys. A 11 1131 (1978).
[12] U. Wol, Phys. Rev. Lett. 62 361 (1989).
[13] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61 2635 (1988); Computers in
Physics, Sep/Oct 1989.
[14] K. Kajantie, L. Karkkainen and K. Rummukainen Nucl. Phys. B 357 693 (1991).
[15] R. Kenna and C. B. Lang, Phys. Lett. B 264 396 (1991); Nucl. Phys. B (Proc. Suppl.)
30 697 (1993); Nucl. Phys. B 393 461 (1993); Err. ibid. B 411 (1994) 340.
5
[16] R. Kenna and C. B. Lang, Phys. Rev. E 49 5012 (1994).
[17] M.P. Nightingale, Physica A 83 (1976) 561; H. Roomany and H.W. Wyld, Phys. Rev.
D 21 (1980) 3341.
[18] M.E. Fisher, in Critical Phenomena, Proc. 51th Enrico Fermi Summer School, Varena,
ed. M.S. Green (Academic Press, NY, 1972).
6
Figure 1: Scaling behaviour of the Yang{Lee edge at a typical critical  value ( = 1:22).
7
02
4
6
8
10
1.15 1.2 1.25 1.3 1.35 1.4
ch
isq
/df
beta
(b)
-1.89
-1.885
-1.88
-1.875
-1.87
-1.865
lam
bd
a
(a)
Figure 2: Test of expected leading critical behaviour: (a) the eective exponent 
e
(slope of a
straight line t) and (b) the corresponding 
2
=dof versus .
8
05
10
15
20
25
1.15 1.2 1.25 1.3 1.35 1.4
ch
isq
/df
beta
(b)
-0.06
-0.04
-0.02
0
0.02
0.04
r
(a)
Figure 3: Logarithmic corrections: (a) the logarithmic correction exponent r to the Yang{Lee edge
is shown as a function of the assumed critical coupling 
c
and (b) the corresponding 
2
=dof.
9


Critical behaviour of the two dimensional step model

CURVE is the Institutional Repository for Coventry University   Critical behavior of the two-dimensional step model  Irving, A. C. and Kenna, R.  Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Irving, A. C. and Kenna, R. (1996) Critical behavior of the two-dimensional step model. Physical Review B, volume 53 : 11568 http://dx.doi.org/10.1103/PhysRevB.53.11568  DOI 10.1103/PhysRevB.53.11568 ISSN 2469-9950 ESSN 2469-9969  Publisher: American Physical Society  Copyright © and Moral Rights are retained by the author(s) and/ or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This item cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder(s). The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders.   This document is the author’s post-print version, incorporating any revisions agreed during the peer-review process. Some differences between the published version and this version may remain and you are advised to consult the published version if you wish to cite from it.        arXiv:hep-lat/9508033v2  1 Feb 1996Critical Behaviour of the Two DimensionalStep ModelA.C. Irving and R. Kenna1DAMTP, University of Liverpool L69 3BX, EnglandAugust 1995AbstractWe use finite–size scaling of Lee–Yang partition function zeroes to study the critical be-haviour of the two dimensional step or sgn O(2) model. We present evidence that, like theclosely related XY –model, this has a phase transition from a disordered high temperature phaseto a low temperature massless phase where the model remains critical. The critical parameters(including logarithmic corrections) are compatible with those of the XY –model indicating thatboth models belong to the same universality class.1Supported by EU Human Capital and Mobility Scheme Project No. CHBI–CT94–11251 IntroductionIn a recent paper [1], we demonstrated the power of finite–size scaling applied to Lee–Yang zeroes[2] in uncovering logarithmic corrections to scaling in the two–dimensional XY – or O(2) spin model.In this paper we apply the same techniques to the closely related ‘step model’ [3, 4], also known [5, 6]as the sgn O(2) model. The question of criticality of this model has, until now, been unresolveddespite several analyses based on high temperature series (see [6] for a review) and on numericalsimulation [7, 8]. The interest in the model arises from its possible membership of the XY –modeluniversality class which exhibits the Kosterlitz–Thouless [KT] phase transition [9]. Like the XY –model, the step model has a configuration space which is globally and continuously symmetric.Unlike the XY –model, however, the interaction function is discontinuous and the Mermin–Wagnertheorem [10] does not apply. Nonetheless, it is expected that if a phase transition exists in the stepmodel, it should not be to a phase with long range order [4, 8, 11].Sánchez–Velasco and Wills [8] presented evidence of critical behaviour starting at βc = 1/Tc =0.91 ± 0.04. This was based on finite–size scaling [FSS] of the spin susceptibility. Since the as-sociated critical index η(Tc) was significantly greater than that measured for the XY –model, itwas concluded that the step and XY models are not in the same universality class. In this paperwe present evidence that the step model is not critical at that temperature. However it is criticalat lower temperatures with a critical index η(Tc) compatible with the XY value. The accuracyafforded by the Lee–Yang zeroes study is a crucial part of the analysis.2 The step model and the XY –modelConsider the partition functionZ(β, h) =∑{~sx}e−βH+hn̂·~M , (1)where the Boltzmann factor is β = 1/kT , n̂ is a unit vector defining the direction of the externalmagnetic field and h is a scalar parameter representing its strength. The summation is over allconfigurations open to the system and ~sx is a unit length two-component spin at each site x in thecubic lattice Λ ≡ Ld (d = 2). The magnetisation for a given configuration is~M =∑x∈Λ~sx .In the case of the XY –model, the interaction hamiltonian isHXY = −∑x∈Λd∑µ=1~sx · ~sx+µ ,while for the step model it isHstep = −∑x∈Λd∑µ=1sgn(~sx · ~sx+µ) .Thus the continuous (cosine) dependence of the interaction energy in the usual XY –model isreplaced by a discrete step function dependence.The leading infinite volume critical behaviour of the 2D XY –model is characterized by exponentialdivergences (essential singularities) in the thermodynamic functions [9]. In terms of the reduced1temperature t ≡ 1 − β/βc → 0+ the (leading) infinite volume scaling behaviour of the correlationlength and the zero-field magnetic susceptibility is (respectively) [9]ξ∞(t) ∼ eat−ν , (2)χ∞(t) ∼ ξ2−η∞ , (3)where ν = 1/2 and η = 1/4. The XY –model remains critical for all β > βc.For models obeying the Lee–Yang theorem [2], the partition function zeroes in the magnetic fieldstrength (h) plane (the Lee–Yang zeroes) are all on the imaginary axis. In the high temperaturephase these zeroes remain away from the real axis, pinching it only as t → 0+ (in the thermodynamiclimit). The zero closest to the real axis marks the edge of the distribution of zeroes and is knownas the Yang–Lee edge, zYL. The theorem has been proved only for certain models, the XY –modelincluded [12]. In [1] we used this fact to show the above leading critical behaviour for the XY –modeland the corresponding behaviour of the Yang–Lee edge, zYL are, in fact, modified by logarithmiccorrections:χ∞(t) ∼ ξ2−η∞ tr , (4)zYL(t) ∼ ξλ∞tp , (5)whereλ = −12(d + 2 − η) = −158(6)and the parameters r and p (= −r/2) are logarithmic correction indices [1]. The correspondingFSS behaviour for the susceptibility and the first zero z1 (= zY L) at t = 0 is [1]χL(0) ∼ L2−η(ln L)−rν , (7)z1(0) ∼ Lλ(ln L)r . (8)For the 2D XY –model the numerical value of r was found to be small but non-zero (−0.023±0.010)[1].The objects of the present analaysis were to establish1. if the scaling behaviour of the very precisely determined Yang–Lee edge would unequivocablydetermine whether the step model had a critical phase2. if so, whether the phase transition is in the same universality class as the XY –model.3 Method and resultsThe methods used are those described in [1]. A single cluster algorithm [13] is used to generate alarge number of measurements (100K for each lattice size L and temperature 1/β) of the energyH and the magnetisation ~M at zero external magnetic field. Such data were obtained for latticesizes L = 32, 48, 64, 128 and 256 covering the β range 0.86 to 1.40 with varying degrees of spacing.Multi–histogram techniques [14, 15] were used to combine data at different values of β and soobtain detailed β dependence. In the neighborhood of possible critical points we used sufficientlyfine spacing (typically 0.025) to ensure adequate overlap between histograms for a given size oflattice.The partition function for a complex magnetic field (h = hr + ihi, hr, hi ∈ R) can be written interms of a real and an imaginary part [1, 16, 17] asZ(β, hr + ihi) = ReZ(β, hr + ihi) + iImZ(β, hr + ihi) , (9)2whereReZ(β, hr + ihi) = Z(β, hr)〈cos (hiM)〉β,hr , (10)ImZ(β, hr + ihi) = Z(β, hr)〈sin (hiM)〉β,hr . (11)Here the subscripts indicate that the expectation values are taken at (inverse) temperature β andin a (real) external field hr.No specific proof of the Lee–Yang theorem exists for the step model. We can, however, use (10) and(11) to (numerically) determine the loci along which the real and imaginary parts of the partitionfunction separately vanish. These formulae concern expectation values of real quantities in a realexternal field and at no stage is a simulation involving a complex action involved. The Lee–Yangzeroes are then the points in the complex h–plane where the loci intersect [17]. We have determinedthese loci and thereby the Lee–Yang zeroes close to the real h–axis. We were able to determine thefirst 15 zeroes and found that they lie on the imaginary h–axis for all the lattices studied. Thus wehave numerical evidence that the step model obeys the Lee–Yang theorem. The following analysisapplies to the first zeroes only (the Yang–Lee edge) and we defer the study of higher zeroes to alater paper [18].The analysis began with a rough search for the leading critical behaviour predicted by (8). Anindependent test was also made using the (less accurate) susceptibility data and (7). Both methodsindicated critical behaviour setting in for β >∼ 1.2. In Fig. 1 we show a typical log–log plot ofz1 the Yang–Lee edge versus L. This is at a typical candidate value of the critical temperature(β = 1.22). The errors are considerably smaller than the symbols. For example, at β = 1.22 wefound z1 = 0.0024136(7) and 0.00017902(6) at L = 32 and 128 respectively. The slope of Fig. 1gives the effective leading index λeff ignoring corrections. At β = 1.22 this is λeff = −1.8761(2)where the chi-squared per degree of freedom (χ2/dof) for the linear fit shown is 0.85. In Fig. 2(a)we display the result of such fits as a function of β. The effective exponent λeff is just the slopeof the log–log linear fit which should obtain if critical behaviour is present (ignoring logarithmiccorrections). The corresponding χ2/dof for the linear fit is also shown. Acceptable values are onlyfound for β in excess of around 1.2. To quantify this statement we demandχ2/dof ≤ 2.0 (12)which means β >∼ 1.185. We note that the corresponding values of λeff (<∼ − 1.872(2)) includethat (−15/8 = −1.875) corresponding to the KT prediction.Fig. 2 is evidence for (i) the validity of FSS over a range of values β ≥ βc ≃ 1.185 and (ii) at βc theexponent λeff very close to the expected KT value (-15/8). Observation (i) means that, as in theXY case, the system remains critical for β ≥ βc and (ii) is evidence that λ is in fact -15/8 and thequestion of whether or not the step model belongs to the same universality class as the XY modelmust now be answered by determination of the correction exponent r.We therefore assume the behaviour (8) with λ = −15/8 at β = βc. The expected leading behaviouris removed and linear fits toln(z1L15/8)vs. ln ln L (13)performed. The results are shown in Fig. 3. Since the value λ = −15/8 (η = 1/4) is only expectedat βc these results can be used to identify the possible values of critical temperature and to test forthe presence of logarithmic corrections as in the XY –model [1].Applying the same criterion (12) as for the leading behaviour, we search for a range of βc valuesgiving an acceptable fit. We find1.195 ≤ βc ≤ 1.295 and correspondingly, 0.009 ≥ r ≥ −0.034 . (14)3The range of acceptable r values includes that found [1] for the XY –model (−0.023 ± 0.010) andthat corresponding to no logarithmic corrections (r = 0). As in our previous work [1], this rangeexcludes the prediction r = −1/16 coming from an approximate renormalisation group treatmentof the XY –model [9]. Thus we conclude that the present data are compatible with the stepmodel being in the same universality class as the XY –model. We do not, however, exclude otherpossibilities.The susceptibility data are consistent with the above analysis. If one assumes the KT value ofη(βc) = 1/4, one can construct a so-called Roomany–Wyld beta function approximant [19] fromthe finite–size data and use its zero to locate βc [1]. These approximants, based on pairs of latticesize L,L′, converge very rapidly [19]. We estimate βc = 1.22 ± 0.02. This last analysis of courseneglects possible logarithmic corrections to scaling.We have also studied the specific heat. As for the XY –model, the step model data show a broadpeak with no obvious relationship to the position of the leading critical point. The finite–sizedependence is not dramatic and is likely to be of little value in further elucidating the criticialbehaviour. A related question is to what extent one can make use of the Fisher zeroes [16, 20],i.e. zeroes of the partition function in the complex β plane at zero external magnetic field h. Forboth this and the XY –model, these are much harder to locate than the Lee–Yang zeroes and areconsequently less accurately determined.4 ConclusionsThe use of finite–size scaling applied to Lee–Yang zeroes has allowed us to present detailed evidenceof critical behaviour in the two-dimensional step (sgn O(2) spin) model. The data are consistentwith this model being in the same universality class as the XY – ( O(2) spin) model. That is,it undergoes a Kosterlitz–Thouless type transition with susceptibility index η(βc) = 1/4 and wedetermine that βc lies in the range 1.195 ≤ βc ≤ 1.295. With the available statistics, we found thelogarithmic correction exponent to lie in the range −0.034 ≤ r ≤ 0.009. This should be comparedwith our measurement for the XY correction exponent [1], −0.033 ≤ r ≤ −0.013 with which it iscompatable. The step model results are however also compatable with no logarithmic corrections(r = 0 corresponds to βc = 1.21 and χ2/dof = 0.92).The Mermin–Wagner theorem [10] does not apply directly to the step model because of the dis-continuous nature of the interaction hamiltonian. However, it has long been believed [4, 8] thatif a phase transition does exist, it will not involve a phase with long range order. The evidencepresented here supports this view.This raises a question as to the nature of the mechanism driving the phase transition in thestep model. The KT phase transition of the XY –model is understood to be driven by the bind-ing/unbinding of topological solutions (vortices). However, the energetics of vortex formation arevery different in the step model [11, 6, 8]. Since vortices with effectively zero excitation energy canbe created at all non-zero temperatures, the usual KT argument does not naturally lead one toexpect such a phase transition in the step model.If this is indeed the case, some other driving mechanism must be responsible for any phase transition.It would then be remarkable if — as the evidence presented here indicates – both models belongto the one universality class.References[1] R. Kenna and A. C. Irving, Phys. Lett. B 351, 273 (1995).4[2] C. N. Yang and T. D. Lee, Phys. Rev. 87 404 (1952); ibid. 410.[3] A. J. Guttmann, G. S. Joyce and C. J. Thompson, Phys. Lett. A 38, 297 (1972).[4] A. J. Guttmann and G. S. Joyce, J. Phys. C 6, 2691 (1973).[5] I-H. Lee and R. E. Shrock, Phys. Rev. B 36, 3712 (1987).[6] I-H. Lee and R. E. Shrock, J. Phys. A 21, 2111 (1988).[7] A. Nymeyer and A. C. Irving, J. Phys. A 19, 1745 (1986).[8] E. Sánchez-Velasco and P. Wills, Phys. Rev. B 37, 406 (1988).[9] J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973); J. M. Kosterlitz J. Phys.C 7, 1046 (1974).[10] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 1133 (1966).[11] A. J. Guttmann and A. Nymeyer, J. Phys. A 11 1131 (1978).[12] F. Dunlop and C.M. Newman, Commun. Math. Phys. 44 223 (1975).[13] U. Wolff, Phys. Rev. Lett. 62 361 (1989).[14] A.M. Ferrenberg and R.H. Swendsen, Phys. Rev. Lett. 61 2635 (1988); Computers inPhysics, Sep/Oct 1989.[15] K. Kajantie, L. Kärkkäinen and K. Rummukainen Nucl. Phys. B 357 693 (1991).[16] R. Kenna and C. B. Lang, Phys. Lett. B 264 396 (1991); Nucl. Phys. B (Proc. Suppl.)30 697 (1993); Nucl. Phys. B 393 461 (1993); Err. ibid. B 411 (1994) 340.[17] R. Kenna and C. B. Lang, Phys. Rev. E 49 5012 (1994).[18] R. Kenna and A. C. Irving, Liverpool Preprint LTH 364.[19] M.P. Nightingale, Physica A 83 (1976) 561; H. Roomany and H.W. Wyld, Phys. Rev.D 21 (1980) 3341.[20] M.E. Fisher, in Critical Phenomena, Proc. 51th Enrico Fermi Summer School, Varena,ed. M.S. Green (Academic Press, NY, 1972).53 4 5 6Ln(L)-10-9-8-7-6Ln(Z) 1Figure 1: Scaling behaviour of the Yang–Lee edge at a typical critical β value (β = 1.22).6-1.890-1.885-1.880-1.875-1.870-1.865λ1.15 1.20 1.25 1.30 1.35 1.40β0246810χ2           (a)(b)/dfeffFigure 2: Test of expected leading critical behaviour: (a) the effective exponent λeff (slope of astraight line fit) and (b) the corresponding χ2/dof versus β.7-0.06-0.04-0.020.000.020.04r1.15 1.20 1.25 1.30 1.35 1.40β0510152025χ2      (a)(b)/dfFigure 3: Logarithmic corrections: (a) the logarithmic correction exponent r to the Yang–Lee edgeis shown as a function of the assumed critical coupling βc and (b) the corresponding χ2/dof.8

Critical Behaviour of the Two Dimensional Step Model

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