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Logarithmic Corrections to Scaling in the Two Dimensional $XY$--Model

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Liverpool Preprint:LTH 343Logarithmic Corrections to Scaling in the Two Dimensional XY –Model R.Kenna 1and A.C.Irving DAMTP,University of Liverpool L693BX,England December 1994Abstract By expressing thermodynamic functions in terms of the edge and density of Lee–Yang ze-roes,we relate the scaling behaviour of the speci?c heat to that of the zero ?eld magnetic susceptibility in the thermodynamic limit of the XY –model in two dimensions.Assuming that ?nite–size scaling holds,we show that the conventional Kosterlitz–Thouless scaling predictions for these thermodynamic functions are not mutually compatable unless they are modi?ed by multiplicative logarithmic corrections.We identify these logarithmic corrections analytically in the case of the speci?c heat and numerically in the case of the susceptibility.The techniques presented here are general and can be used to check the compatibility of scaling behaviour of odd and even thermodynamic functions in other models too.

The Kosterlitz–Thouless Scenario The partition function for the O(n)non–linearσ–model de?ned on a regular d–dimensional latticeΛwith periodic boundary conditions in the presence of an external magnetic?eld can be written as

1

Z L(β,h)=

,

βc

the scaling behaviour of the correlation length,susceptibility and the speci?c heat is given in[1]as

ξ∞(t)~e at?ν,(2)

,(3)

χ∞(t)~ξ2?η

C∞(t)~ξ?α∞+constant,(4) where for t→0+,ν=1/2,η=1/4and?α=?d=?2.The purpose of this work is to argue that the latter two scaling formulae are incompatible as they stand.To this end,a method is presented by which odd and even thermodynamic functions(the susceptibility and speci?c heat)can be related and expressed in terms of partition function https://www.doczj.com/doc/6014176340.html,ing certain reasonable assumptions regarding ?nite–size scaling,it is shown that there have to exist multiplicative logarithmic corrections2to (3)and(4).This method can be applied to any model.

Partition Function Zeroes Lee and Yang[2]showed the connection between critical behaviour and partition function zeroes.For a?nite system these are strictly complex(non–real).As L→∞one expects the zeroes to condense onto smooth curves.Zeroes in the plane of complex external magnetic?eld h are refered to as Lee–Yang zeroes.Lee and Yang further showed that for certain systems these zeroes are in fact restricted to the imaginary h axis(the Lee–Yang theorem)[2].In the symmetric phase,t>0,they lie away from the real h-axis,pinching it only as t→0+(in the

thermodynamic limit).For systems obeying the Lee–Yang theorem,such as the XY Model[3],this

pinching occurs at h=0prohibiting analytic continuation

from Re(

h)<0to Re(h)>0.This means that(for the?nite–size system or in the thermodynamic limit)the partition function is an entire function of h all of whose zeroes are purely imaginary.The zeroes in the complex fugacity plane(z≡exp(h))are distributed on the unit circle.Writing these zeroes as

z j(β)=e iθj(β),

the partition function is

Z L(β,h)=ρL(β,h) j z?e iθj(β) ,(5) whereρL is a non-vanishing function of h related to the spectral density(ignored in what follows since it contributes only to the regular part of the free energy in the thermodynamic limit).The free energy is then

f L(β,h)=L?d j ln(z?e iθj(β)).(6) De?ne the density of zeroes to be[4,5]

g L(β,θ)=L?d jδ(θ?θj(β))=?G L(β,θ)

2

(h+ln2)+ π0ln(cosh h?cosθ)dG L(β,θ),(9) where(8)has been used.

In the high temperature phase there exists a region aroundθ=0which is free from Lee–Yang zeroes[2].We de?ne the corresponding Yang–Lee edgeθYL(β)by

g(β,θ)=0for?θYL(β)<θ<θYL(β).

The integral in(9)can therefore be taken to begin atθYL(β).Salmhofer has proved the existence of a unique density of zeroes in the thermodynamic limit[5].In this limit,integrating(9)by parts and expanding the trigonometric functions gives for the singular part of the free energy[4,6,7]

f∞(β,h)=?2 πθYL(β)θ

θYL(β)

,(11)

Φ(x)being some function of x such thatΦ(|x|≤1)=0.From(10)and the fact that G(θYL,β)=0, one gets the speci?c heat

C∞(β)=?2f∞(β,h)

dβ2

dθ.(12)

The above formulae are quite general and hold for any model provided it obeys the Lee–Yang theorem.

To proceed further we insert the(model–speci?c)critical behaviour.Instead of(3)and(4),assume now the following modi?ed critical behaviour for the singular parts of the zero?eld susceptibility and the speci?c heat for t>~0.

χ∞~ξ2?η

t r,C∞~ξ?αt q.(13) Similarly,assume that the leading scaling behaviour of the Yang–Lee edge in terms of the reduced temperature is

θYL(t)~ξλt p.(14) For t>~0the critical indices are independent of t[1].The scaling behaviour of C∞can be related to that ofχ∞andθYL through(11)and(12).These give

ξ(t)?αt q∝ξ(t)2?η+2λt2p+r?2?2ν.

Therefore,

λ=

1

2

(q?r)+1+ν.(16)

Finite–Size Scaling The?nite–size scaling(FSS)hypothesis,?rst formulated in1972by Fisher [8],is a relationship between the scaling behaviour of thermodynamic quantities in the in?nite volume limit and the size dependence of their?nite volume counterparts.The general statement of FSS,which is expected to hold in all dimensions including the upper critical one[9]is that if P L(t) is the value of some thermodynamic quantity P at reduced temperature t measured on a system of linear extent L,then

P L(0)

ξ∞(t)

,(17)

whereξL(t)is the correlation length of the?nite–size system.Luck has shown that,for the XY–model in two dimensions,ξL(0)is proportional to L[10].Fixing the scaling variable,one has

ξ∞(t)~x?1L,

and from(2)

t~(ln L)?1

ν,(18)

θ1(0)~Lλ(ln L)?p

For the?nite size system it is convenient to consider the zeroes in the complex h–plane.The singular part of the free energy corresponds to

f L(t,h)=L?d j ln(h?iθj(t))(20) whereθ1=θY L.The(reduced)magnetic susceptibility is therefore

χL(t)=?2f L

θ2j

.(21)

The lowest lying zeroes are expected to scale in the same way[11]so that

χL(t)∝?L?dθ1(t)?2.(22)

From(18)and(19)we have

L2?η(ln L)?rν,(23)

and conclude

?α=?d,q=?2(1+ν).(24)

Thus the scaling behaviour of the singular part of the speci?c heat indeed exhibits multiplicative logarithmic corrections.Accepting the KT predictionsν=1/2,η=1/4for t>~0,the leading critical exponent for the Yang–Lee edgeλand the correction exponents q and p are

λ=?15

2

.(25)

The above analytic considerations have yielded no information on the odd correction exponent r.The original renormalisation group analysis of Kosterlitz and Thouless[1],in fact,implicitly contained the prediction

r=?1/16(26)

as noted by[12].Subsequent analysis have concentrated on the r=0form of the scaling behaviour (18)and the veri?cation thatη(βc)=1/4.Allton and Hamer[13]have conjectured that the deviation of their determination ofηfrom1/4might be due to logarithmic corrections.

The study of the full scaling form and the numerical determination of r is the subject of what follows.

Numerical Procedures Numerical methods were used to test the scaling picture of the last sec-tion.Speci?cally,we analysed the FSS behaviour of the Yang–Lee edge so as to test the prediction (19),(25)

θ1(βc)~Lλ(ln L)r.(27) In particular,we sought to con?rmλ=15/8for the leading behaviour and to check if r=0.

An algorithm based on that of Wol?[14]was used to simulate the XY–model at zero magnetic ?eld(h=0)on square lattices of sizes L=32,64,128and256.The values ofβat which the simulations took place(βo)were evenly spaced in steps of0.02between1.00and1.20.At each

simulation point,50,000measurements were made from each of two separate starts.The second runs were used only to verify equilibration and to improve our understanding of the statistical errors.The latter were estimated using bootstrap as described below.The autocorrelation time of the susceptibility was also monitored.Histogram reweighting[15]was used to enable extrapolation betweenβo values.

The determination ofβc is inextricably bound up with the assumed critical scaling behaviour.For the present purposes,we adopted a two-stage strategy:we used estimates ofβc from our own and other analyses which had assumed no logarithmic corrections then tested the stability of these conclusions with respect both to uncertainties inβc and to the form of the scaling asumption.

One simple estimate was based on assumed conventional(r=0)scaling of the susceptibility(3).A generalised[16]Roomany-Wyld approximant[17]was used to give estimates of the corresponding renormalisation group beta function for each pair of lattice sizes related by factor of two scaling:

βRW L =(

2ln L)/ln2+ln(χ2L/χL)/ln2

[χ2LχL]

.(28)

A preliminary estimate,usingη=1/4and r=0,leads to toβc=1.11(1)which is at the lower end of the range of published estimates[12],[18],[19].These span1.11to1.13.The most recent high precision result is1.1197(5)[19].

Determination of Lee–Yang Zeroes When the external?eld is complex(h=h r+ih i),the partition function(1)can be rewritten

Z L(β,h r+ih i)=Re Z L(β,h r+ih i)+i Im Z L(β,h r+ih i),(29) where

Re Z L(β,h r+ih i)=

1

N { σx}eβS+h r M sin(h i M),

=Z L(β,h r) sin(h i M) β,h

r

.(31) According to the Lee–Yang theorem the zeroes are on the imaginary h–axis(h r=0)where

sin(h i M) β,h

r

vanishes and so the Lee–Yang zeroes are simply the zeroes of

Re Z L(β,ih i)∝ cos(h i M) β,0.(32) Thus the Lee–Yang zeroes are easily found.Moreover,one can?nd them at anyβusing reweight-ing techniques[15].For the lowest lying zeroes atβ=1.11we foundθ1(L)=0.0023353(7), 0.0006350(2),0.00017278(5)and0.000047062(13)for L=32,64,128and256respectively.Er-rors were calculated using the bootstrap method where the data for eachβo are resampled100

times(with replacement)leading to100estimates forθ1,from which the variance and bias can be calculated.

In?gure1we plot the logarithm of the position of the?rst Lee–Yang zero against the logarithm of the lattice size L,atβ=1.11.In the absence of any corrections,the slope should give the leading power–law exponentλ.In fact the slope is-1.8776(2),the deviation from the KT value of ?15/8=?1.875being attributed to the presence of logarithmic corrections.To identify these,and the correction exponent r in(27),ln(θ1L15/8)is plotted against ln ln L in?gure2.A straight line is identi?ed.Its slope is?0.012(1)giving evidence for a non–zero value of r,albeit not in agreement with the RG predictions of?1/16=?0.0625from[1].

At this point,one must investigate the extent to which these conclusions regarding the existence of multiplicative logarithmic corrections depend upon the measurement ofβc.We have attempted to do this systematically by studying the above ln ln L?ts for r as a function of the assumed critical beta.Fits with a range of values of r(including zero)are possible but not all with acceptable χ2/degree of freedom.To be conservative,we take‘acceptable’to mean less than2.This corre-sponds to a minimum con?dence level of14%.In?gure3we show that acceptable values ofχ2 are possible only for1.110<~βc<~1.120.The r=0solution would correspond toβc≈1.105 andχ2/dof≈4.Similary,we?nd that a?t with r=?1/16would correspond toβc≈1.138and χ2/dof≈16.

In summmary,assumming the Kosteritz-Thouless valueη=1/4,we?nd non-zero logarithmic corrections to scaling and a corresponding estimate of the critical temperature:

r=?0.023±0.010,βc=1.115±.005.(33)

As a cross check,one may use the above values for r andηto?nd the Roomany-Wyld approximant (28)and estimateβc from its zeroes.We?ndβc=1.115±0.004in good agreement with the above. However,we emphasize the much higher precision available to an analysis based on Lee-Yang zeroes rather than on the spin susceptibility.

Conclusions Theoretical arguments concerning the consistency of the scaling behaviour of odd and even thermodynamic functions at a KT phase transition have been presented.The generally used scaling formulae have to be modi?ed by multiplicative corrections.These are identi?ed ana-lytically for the speci?c heat and numerically for the susceptibility.This numerical identi?cation comes via an analysis of Lee–Yang zeroes,the FSS of which is linked to that of the susceptibility.

We would like to thank https://www.doczj.com/doc/6014176340.html,cock for assistance with multihistogram reweighting techniques.

References

[1]J.Kosterlitz and D.Thouless,J.Phys.C6(1973)1181;J.Kosterlitz,J.Phys.C7

(1974)1046.

[2]C.N.Yang and T.D.Lee,Phys.Rev.87404(1952);ibid.410.

[3]F.Dunlop and C.M.Newman,Commun.Math.Phys.44(1975)223.

[4]R.Kenna and https://www.doczj.com/doc/6014176340.html,ng,Phys.Rev.E49(1994)5012.

[5]M.Salmhofer,UBC–Preprint(unpublished);Nucl.Phys.B(Proc.Suppl.)30,(1993),

81.

[6]R.Abe,Prog.Theor.Phys.37(1967)1070;Prog.Theor.Phys.38(1967)72;ibid.322;

ibid.568.

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39,(1968)349.

[8]M.E.Fisher,in Critical Phenomena,Proc.51th Enrico Fermi Summer School,Varena,

ed.M.S.Green(Academic Press,NY,1972).

[9]R.Kenna and https://www.doczj.com/doc/6014176340.html,ng,Phys.Lett.B264(1991)396;Nucl.Phys.B(Proc.Suppl.)

30(1993)697;Nucl.Phys.B393(1993)461;Err.ibid.B411(1994)340.

[10]J.M.Luck,J.Phys.A15(1982)L169.

[11]C.Itzykson,R.B.Pearson,and J.B.Zuber,Nucl.Phys.B220[FS8](1983)415;

C.Itzykson and J.M.Luck,Progress in Physics,Critical Phenomena(1983Brasov

Conference)Birkh¨a user,Boston Inc.Vol11,45(1985).

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[13]C.Allton and C.Hamer,J.Phys.A21(1988)2417.

[14]U.Wol?,Phys.Rev.Lett.62(1989)361.

[15]K.Kajantie,L.K¨a rkk¨a inen and K.Rummukainen,Nucl.Phys.B357(1991)693.

[16]A.C.Irving and C.Hamer,Nucl.Phys.B230[FS10](1984)361;C.Hamer and A.C.

Irving,J.Phys.A17(1984)1649.

[17]M.P.Nightingale,Physica A83(1976)561;H.Roomany and H.W.Wyld,Phys.Rev.

D21(1980)3341.

[18]R.Gupta,J.De Lapp,G.Batrouni,G.C.Fox,C.F.Baillie and J.Apostolakis,Phys.

Rev.Lett.61(1988)1996;L.Biferale and R.Petronzio,Nucl.Phys.B328(1989)677;

C.F.Baillie and R.Gupta,Nucl.Phys.B(Proc.Suppl.)20(1991)669;A.Hulsebos,

J.smit and J.C.Vink,Nucl.Phys.B356(1991)775;R.G.Edwards,J.Goodman and

A.D.Sokal,Nucl.Phys.B354(1991)289.

[19]M.Hasenbusch,M.Marcu and K.Pinn,Physica A208(1994)124.

-10

-9

-8

-7

-6

3 3.5

4 4.5

5 5.56

L n (Z 1)

Ln (L)Figure 1:Leading FSS of Lee–Yang Zeroes at β=1.11.

0.4320.434

0.4360.4380.44

1.2 1.3 1.4 1.5 1.6 1.7

L n (Z 1) + (15/8)L n (L )

LnLn(L)Figure 2:Corrections to FSS of Lee–Yang Zeroes at β=1.11.

-0.06

-0.04

-0.020

0.02

r (a)

4

8

12161.1 1.12 1.14

C h i s q /d o f BETA (b)

Figure 3:(a)Correction Exponent r Measured over a Range of βand (b)corresponding χ2per Degree of Freedom (Goodness of Fit).

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男装、女装衣服尺码对照表
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cm)
cm)
西(腰裤围尺寸码) T
恤尺码
毛衣尺码 内裤尺码 统计比例
160 37(S) 44(S) 80(S) 72
28
S
S
S
0
165 38(M) 46(M) 84(M) 74,76 29
M
M
M
1
170 39(L) 48(L) 88(S) 78
30
L
L
L
2
175 40(XL) 50(XL) 92(M) 80
31
XL
XL
XL
3
180 41(2XL) 52(2XL) 96(L) 82
32
2XL
2XL
2XL
3
185 42(3XL) 54(3XL) 100(XL) 84,86 33
3XL
3XL
3XL
2
190 43(4XL) 56(4XL) 104(4XL) 88
34
4XL
4XL
4XL
1
195 44(5XL)
90
35
5XL
5XL
5XL
0
2、衬衫尺寸(除个别款尺寸,买前询问)
平铺尺寸 M
XL
胸围 97cm 99cm 101cm
肩宽 43cm 44cm 45cm
衣长 67cm 68cm 69cm
袖长 62cm 64cm 65cm
3、裤装尺码为: 26 代表腰围为:“尺” 28 代表腰围为:“尺” 30 代表腰围为:“尺” 32 代表腰围为:“尺” 34 代表腰围为:“尺” 38 代表腰围为:“尺” 42 代表腰围为:“尺” 50 代表腰围为:“尺” 54 代表腰围为:“尺”
27 代表腰围为:“尺” 29 代表腰围为:“尺” 31 代表腰围为:“尺” 33 代表腰围为:“尺” 36 代表腰围为:“尺” 40 代表腰围为:“尺” 44 代表腰围为:“尺” 52 代表腰围为:“尺”
4.女装尺码对照表
上装尺码
“女上装”尺码对照表(cm)
S
M
L
155/80A
160/84A
165/88A
XL 170/92A

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