The critical equation of state of three-dimensional XY systems

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J a n 2000IFUP-TH/2000-02

The critical equation of state of three-dimensional XY systems

Massimo Campostrini 1,Andrea Pelissetto 2,Paolo Rossi 1,Ettore Vicari 11Dipartimento di Fisica dell’Universit`a di Pisa and I.N.F.N.,I-56126Pisa,Italy 2Dipartimento di Fisica dell’Universit`a di Roma I and I.N.F.N.,I-00185Roma,Italy e-mail:campo@mailbox.difi.unipi.it ,rossi@mailbox.difi.unipi.it ,Andrea.Pelissetto@roma1.infn.it ,vicari@mailbox.difi.unipi.it (February 1,2008)Abstract We address the problem of determining the critical equation of state of three-dimensional XY systems.For this purpose we ?rst consider the small-?eld expansion of the e?ective potential (Helmholtz free energy)in the high-temperature phase.We compute the ?rst few nontrivial zero-momentum n -point renormalized couplings,which parametrize such expansion,by analyz-ing the high-temperature expansion of an improved lattice Hamiltonian with suppressed leading scaling corrections.These results are then used to construct parametric representations of the critical equation of state which are valid in the whole critical regime,satisfy the correct analytic properties (Gri?th’s analyticity),and take into account the Goldstone singularities at the coexistence curve.A systematic approxi-mation scheme is introduced,which is limited essentially by the number of known terms in the small-?eld expansion of the e?ective potential.From our approximate representations of the equation of state,we derive estimates of universal ratios of amplitudes.For the speci?c-heat amplitude ra-tio we obtain A +/A ?=1.055(3),to be compared with the best experimental estimate A +/A ?=1.054(1).Keywords:Critical Phenomena,three-dimensional XY systems,λ-transition of 4He,Critical equation of state,E?ective potential,High-Temperature Expansion.

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I.INTRODUCTION

In the theory of critical phenomena,continuous phase transitions can be classi?ed into universality classes determined only by a few basic properties characterizing the system,such as the space dimensionality,the range of interaction,the number of components and the symmetry of the order parameter.The renormalization-group theory predicts that,within a given universality class,the critical exponents and the scaling functions are the same for all systems.Here we consider the XY universality class,which is characterized by a two-component order parameter and e?ective short-range interactions.The lattice spin model described by the Hamiltonian

H L=?J s i· s j+ i h i· s i,(1)

where s i is a two-component spin satisfying s i· s i=1,is one of the systems belonging to the XY universality class.It may be viewed as a magnetic system with easy-plane anisotropy, in which the magnetization plays the role of order parameter and the spins are coupled to an external magnetic?eld h.

The super?uid transition of4He,occurring along theλ-line Tλ(P)(where P is the pres-sure),belongs to the three-dimensional XY universality class.Its order parameter is related to the complex quantum amplitude of helium atoms.It provides an exceptional opportunity for an experimental test of the renormalization-group predictions,thanks to the weakness of the singularity in the compressibility of the?uid and to the purity of the sample.Moreover, experiments in a microgravity environment lead to a reduction of the gravity-induced broad-ening of the transition.Recently a Space Shuttle experiment[1]performed a very precise measurement of the heat capacity of liquid helium to within2nK from theλ-transition obtaining an extremely accurate estimate of the exponentαand of the ratio A+/A?of the speci?c-heat amplitudes:

α=?0.01285(38),A+/A?=1.054(1).(2) These results represent a challenge for theorists because the accuracy of the test of the renormalization-group prediction is now limited by the precision of the theoretical calcula-tions.We mention the best available theoretical estimates forα:α=?0.0150(17)obtained using high-temperature expansion techniques[2],α=?0.0169(33)from Monte Carlo simu-lations using?nite-size scaling techniques[3],α=?0.011(4)from?eld theory[4].The close agreement with the experimental data clearly supports the standard renormalization-group description of theλ-transition[5].

In this paper we address the problem of determining the critical equation of state char-acterizing the XY universality class.The critical equation of state relates the thermody-namical quantities in the neighborhood of the critical temperature,in both phases.It is usually written in the form(see e.g.Ref.[8])

H= MMδ?1f(x),x∝tM?1/β,(3) where f(x)is a universal scaling function(normalized in such a way that f(?1)=0and f(0)=1).The universal ratios of amplitudes involving quantities de?ned at zero-momentum

(i.e.integrated in the volume),such as speci?c heat,magnetic susceptibility,etc...,can be obtained from the scaling function f(x).

It should be noted that,for theλ-transition in4He,Eq.(3)is not directly related to the conventional equation of state that relates temperature and pressure.Moreover,in this case, the?eld H does not correspond to an experimentally accessible external?eld,so that the function appearing in Eq.(3)cannot be determined directly in experiments.The physically interesting quantities are universal amplitude ratios of quantities formally de?ned at zero external?eld.

As our starting point for the determination of the critical equation of state,we com-pute the?rst few nontrivial coe?cients of the small-?eld expansion of the e?ective poten-tial(Helmholtz free energy)in the high-temperature phase.For this purpose,we analyze the high-temperature expansion of an improved lattice Hamiltonian with suppressed lead-ing scaling corrections[9,10].If the leading non-analytic scaling corrections are no longer present,one expects a faster convergence,and therefore an improved high-temperature ex-pansion(IHT)whose analysis leads to more precise and reliable estimates.We consider a simple cubic lattice and theφ4Hamiltonian

H=?β x,y φx· φy+ x φ2x+λ( φ2x?1)2 ,(4)

where x,y labels a lattice link,and φx is a real two-component vector de?ned on lattice sites.The value ofλat which the leading corrections vanish has been determined by Monte Carlo simulations using?nite-size techniques[3],obtainingλ?=2.10(6).In Ref.[2]we have already considered the high-temperature expansion(to20th order)of the improved φ4Hamiltonian(4)for the determination of the critical exponents,achieving a substantial improvement with respect to previous theoretical estimates.IHT expansions have also been considered for Ising-like systems[10],obtaining accurate determinations of the critical ex-ponents,of the small-?eld expansion of the e?ective potential,and of the small-momentum behavior of the two-point function.

We use the small-?eld expansion of the e?ective potential in the high-temperature phase to determine approximate representations of the equation of state that are valid in the whole critical regime.To reach the coexistence curve(t<0)from the high-temperature phase(t>0),an analytic continuation in the complex t-plane[8,11]is required.For this purpose we use parametric representations[12–14],which implement in a rather simple way the known analytic properties of the equation of state(Gri?th’s analyticity).This approach was successfully applied to the Ising model,for which one can construct a systematic ap-proximation scheme based on polynomial parametric representations[11]and on a global stationarity condition[10].This leads to an accurate determination of the critical equation of state and of the universal ratios of amplitudes that can be extracted from it[11,4,10].XY systems,in which the phase transition is related to the breaking of the continuous symme-try O(2),present a new important feature with respect to Ising-like systems:the Goldstone singularities at the coexistence curve.General arguments predict that at the coexistence curve(t<0and H→0)the transverse and longitudinal magnetic susceptibilities behave respectively as

χT=M

?H～H d/2?2.(5)

In our analysis we will consider polynomial parametric representations that have the correct singular behavior at the coexistence curve.

The most important result of the present paper,from the point of view of comparison with experiments,is the speci?c-heat amplitude ratio;our?nal estimate

A+/A?=1.055(3)(6) is perfectly consistent with the experimental estimate(2),although not as precise.

The paper is organized as follows.In Sec.II we study the small-?eld expansion of the e?ective potential(Helmholtz free energy).We present estimates of the?rst few nontriv-ial coe?cients of such expansion obtained by analyzing the IHT series.The results are then compared with other theoretical estimates.In Sec.III,using as input parameters the critical exponents and the known coe?cients of the small-?eld expansion of the e?ective potential,we construct approximate representations of the critical equation of state.We obtain new estimates for many universal amplitude ratios.These results are then compared with experimental and other theoretical estimates.

II.THE EFFECTIVE POTENTIAL IN THE HIGH-TEMPERATURE PHASE

A.Small-?eld expansion of the e?ective potential in the high-temperature phase

The e?ective potential(Helmholtz free energy)is related to the(Gibbs)free energy of the model.If M≡ φ is the magnetization and H the magnetic?eld,one de?nes

F(M)= M· H?1

(2j)!

a2j M2j.(8) This expansion can be rewritten in terms of a renormalized magnetization?

?F=1

(2j)!

g2j?2j(9)

where

?2=

ξ(t,H=0)2M(t,H)2

χ= x φα(0)φα(x) ,(11)

ξ=

1

√

g4

A(z),(13) where

A(z)=1

4!

z4+ j=31

g j?1

4

j≥3.(15)

One can show that z∝t?βM,and that the equation of state can be written in the form

H∝tβδ

?A(z)

N+2

χ4

r6=10?5(N+2)

χ24

,(19)

r8=280?280(N+2)

χ24

+

35(N+2)2

χ34

,

r10=15400?7700(N+2)

χ24

+

350(N+2)2

χ44

+1400(N+2)2

χ34?

35(N+2)3

χ44

.

The formulae relevant for the XY universality class are obtained setting N=2. Using theφ4lattice Hamiltonian(4),we have calculatedχand m2≡ x x2 φ(0)φ(x) to20th order,χ4to18th order,χ6to17th order,χ8to16th order,andχ10to15th order, for generic values ofλ.The IHT expansion,i.e.with suppressed leading scaling corrections, is achieved forλ=2.10(6)[3].In Table I we report the series of m2,χ,χ4,χ6,χ8and χ10forλ=https://www.doczj.com/doc/9210967483.html,ing Eqs.(18)and(19)one can obtain the HT series necessary for the determination of g4and r2j.We analyzed the series using the same procedure applied to the improved high-temperature expansions of Ising-like systems in Ref.[10].In order to estimate the?xed-point value of g4and r2j,we considered Pad′e,Dlog-Pad′e and?rst-order integral approximants of the series inβforλ=2.10,and evaluated them atβc.We refer to Ref.[10]for the details of the analysis.Our estimates are

g4=21.05(3+3),(20)

ˉg≡

5

TABLE I.Coe?cients of the high-temperature expansion of m2,χ,χ4,χ6,χ8,χ10.They have been obtained using the Hamiltonian(4)withλ=2.10.

i m2χχ4

00.41651792496748423277383?2.0727687239500861203205717.8224779282297038884825 18.19357083273717982769144?66.6409389277329689706620845.247991215093077905035 281.6954441314190452820070?1011.5010295209924286004317983.4651630934745766842 3577.734561420010600288632?10327.4974814820902607880248053.171125799025029980 43288.50395189429008814810?81557.44774033086145414012572480.03564061669203194 516096.6021109696857145564?536525.34584114957966023421718902.5365937343351130 670442.2092664445625747715?3075097.70592321924332973156737903.631432308532083 7282678.282078376456985396?15817337.6236306593687791998806344.593851737549288 81058442.95252211500336034?74543191.49651668032972855750852421.81974869997198 93744677.84665924694446919?326778866.93761039964032230428100948.8992362353297 1012635750.8707582942375013?1347823157.99877343602884149865516252.140441504474 1140959287.0765118834279901?5277067971.27466652475816694034037396.324618345020 12128268696.512235415964465?19750695382.32839078997883046464352917.44305602911 133********.322878356697650?71066114622.316375358117412757807364908.6407194560 141153973293.83499223649654?246972985495.95255738197651244942192011.6695801488 153337461787.55665279189948?832179913942.254869433972198319718730601.629272462 169454326701.58705373223814?2727546459564.59095483523

1726288539415.6469228911099

TABLE II.Estimates ofˉg≡5g4/(24π),r6,and r8.

IHT HT d=3g-exp.?-exp.

ˉg 1.396(4) 1.411(8), 1.406(8)[15] 1.403(3)[4] 1.425(24)[22,16]

1.415(11)[16] 1.40[20]

=z+1

?z

r2m.(26)

(2m?1)!

The function H(M,t)representing the external?eld in the critical equation of state(24) satis?es Gri?th’s analyticity:it is regular at M=0for t>0?xed and at t=0for M>0?xed.The?rst region corresponds to small z in Eq.(24),while the second is related to large z,where F(z)has an expansion of the form

F(z)=zδ n=0F∞n z?n/β.(27)

To reach the coexistence curve,i.e.t<0and H=0,one should perform an analytic continuation in the complex t-plane[8,11].The spontaneous magnetization is related to the complex zero z0of F(z).Therefore,the description of the coexistence curve is related to the behavior of F(z)in the neighbourhood of z0.

B.Goldstone singularities at the coexistence curve

The physics of the broken phase of N-vector models(including XY systems which cor-respond to N=2)is very di?erent from that of the Ising model,because of the presence of Goldstone modes at the coexistence curve.The singularity ofχL for t<0and H→0

is governed by the zero-temperature infrared-stable?xed point[23–25].This leads to the prediction

f(x)≈c f(1+x)2/(d?2)for x→?1,(28) where x∝tM?1/βand f(x)is the scaling function introduced in Eq.(3)(as usual,x=?1 corresponds to the coexistence curve).This behavior at the coexistence curve has been veri?ed in the framework of the large-N expansion to O(1/N)(i.e.next-to-leading order) [23,26].

The nature of the corrections to the behaviour(28)is less clear.Settingω=1+x and y=HM?δ,it has been conjectured thatωhas the form of a double expansion in powers of y and y(d?2)/2near the coexistence curve[27,28,25],i.e.for y→0

ω≡1+x=c1y+c2y1??/2+d1y2+d2y2??/2+d3y2??+ (29)

where?=4?d.This expansion has been derived essentially from an?-expansion analy-sis[29].Note that in three dimensions this conjecture predicts an expansion in powers of y1/2,or equivalently an expansion of f(x)in powers ofωforω→0.

The asymptotic expansion of the d-dimensional equation of state at the coexistence curve has been computed analytically in the framework of the large-N expansion[22],using the O(1/N)formulae reported in Ref.[23].It turns out that the expansion(29)does not strictly hold for values of the dimension d such that

2m

2

(f1(ω)+logωf2(ω))+O(N?2) ,(31)

N

where the functions f1(ω)and f2(ω)have a regular expansion in powers ofω.Moreover,

f2(ω)=O(ω2),(32) so that the logarithms a?ect the power expansion only at the next-next-to-leading order.A possible interpretation of the large-N analysis is that the expansion(31)holds for all values of N,so that Eq.(29)is not correct due to the presence of logarithms.The reason of their appearance is however unclear.Neverthless,it does not contradict the conjecture that the behavior near the coexistence curve is controlled by the zero-temperature infrared-stable Gaussian?xed point.In this case logarithms would not be unexpected,as they usually appear in the reduced-temperature asymptotic expansion around Gaussian?xed points(see e.g.Ref.[32]).

C.Parametric representations

In order to obtain a representation of the critical equation of state that is valid in the whole critical region,one may use parametric representations,which implement in a simple

way all scaling and analytic properties[12–14].One may parametrize M and t in terms of R andθaccording to

M=m0Rβm(θ),(33)

t=R(1?θ2),

H=h0Rβδh(θ),

where h0and m0are normalization constants.The variable R is nonnegative and measures the distance from the critical point in the(t,H)plane;it carries the power-law critical singularities.The variableθparametrizes the displacements along the line of constant R. The functions m(θ)and h(θ)are odd and regular atθ=0and atθ=1.The constants m0 and h0can be chosen so that m(θ)=θ+O(θ3)and h(θ)=θ+O(θ3).The smallest positive zero of h(θ),which should satisfyθ0>1,represents the coexistence curve,i.e.T

The parametric representation satis?es the requirements of regularity of the equation of state.Singularities can appear only at the coexistence curve(due for example to the logarithms discussed in Sec.III B),i.e.forθ=θ0.Notice that the mapping(33)is not invertible when its Jacobian vanishes,which occurs when

Y(θ)≡(1?θ2)m′(θ)+2βθm(θ)=0.(34)

Thus the parametric representations based on the mapping(33)are acceptable only ifθ0<θl whereθl is the smallest positive zero of the function Y(θ).One may easily verify that the asymptotic behavior(28)is reproduced simply by requiring that

h(θ)≈(θ0?θ)2forθ→θ0.(35) The relation among the functions m(θ),h(θ)and F(z)is given by

z=ρm(θ) 1?θ2 ?β,(36)

F(z(θ))=ρ 1?θ2 ?βδh(θ),(37) whereρis a free parameter[11,10].Indeed,in the exact parametric equation the value ofρmay be chosen arbitrarily but,as we shall see,when adopting an approximation procedure the dependence onρis not eliminated.In our approximation scheme we will?xρto ensure the presence of the Goldstone singularities at the coexistence curve,i.e.the asymptotic behavior(35).Since z=ρθ+O(θ3),expanding m(θ)and h(θ)in(odd)powers ofθ,

m(θ)=θ+ n=1m2n+1θ2n+1,(38)

h(θ)=θ+ n=1h2n+1θ2n+1,

and using Eqs.(36)and(37),one can?nd the relations amongρ,m2n+1,h2n+1and the coe?cients F2n+1of the expansion of F(z).

One may also write the scaling function f(x)in terms of the parametric functions m(θ) and h(θ):

x=1?θ2

m(θ)

1/β

,(39)

f(x)= m(θ)h(1).

In App.A we report the de?nitions of some universal ratios of amplitudes that have been introduced in the literature,and the corresponding expressions in terms of the functions m(θ) and h(θ).

D.Approximate polynomial representations

In order to construct approximate parametric representations we consider polynomial approximations of m(θ)and h(θ).This kind of approximation turned out to be e?ective in the case of Ising-like sistems[11,10].The major di?erence with respect to the Ising case is the presence of the Goldstone singularities at the coexistence curve.In order to take them into account,at least in a simpli?ed form which neglects the logarithms found in Eq.(31), we require the function h(θ)to have a double zero atθ0as in Eq.(35).Polynomial schemes may in principle reconstruct also the logarithms,but of course only in the limit of an in?nite number of terms.

In order to check the accuracy of the results,it is useful to introduce two distinct schemes of approximation.In the?rst one,which we denote as(A),h(θ)is a polynomial of?fth order with a double zero atθ0,and m(θ)a polynomial of order(1+2n):

scheme(A):m(θ)=θ 1+n i=1c iθ2i ,(40)

h(θ)=θ 1?θ2/θ20 2.

In the second scheme,denoted by(B),we set

scheme(B):m(θ)=θ,(41)

h(θ)=θ 1?θ2/θ20 2 1+n i=1c iθ2i .(42)

Here h(θ)is a polynomial of order5+2n with a double zero atθ0.In both schemes the parameterρis?xed by the requirement(35),whileθ0and the n coe?cients c i are determined by matching the small-?eld expansion of F(z).This means that,for both schemes,in order to?x the n coe?cients c i we need to know n+1values of r2j,i.e.r6,...r6+2n.Note that for the scheme(B)

Y(θ)=1?θ2+2βθ2,(43) independently of n,so thatθl=(1?2β)?1.Concerning the scheme(A),we note that the analyticity of the thermodynamic quantities for|θ|<θ0requires the polynomial function Y(θ)not to have complex zeroes closer to origin thanθ0.

In App.B we present a more general discussion on the parametric representations.

TABLE III.Results for the parameters and the universal amplitude ratios using the scheme (A),cf.Eq.(40),and the scheme(B),cf.Eq.(41).The label above each column indicates the scheme,the number of terms in the corresponding polynomial,and the input parameters employed, beside the critical exponentsαandη.Note that the quantities reported in the?rst four lines do not have a physical meaning,but are related to the particular parametric representation employed. Numbers marked with an asterisk are inputs,not predictions.

[(A)n=1;r6,r8][(B)n=1;r6,r8][(B)n=2;r6,r8,r10][(B)n=2;r6,r8,A+/A?]ρ 2.22(3) 2.07(2) 2.04(5) 2.01(4)

θ20 3.84(10) 2.97(10) 2.8(4) 2.5(2)

c1?0.024(7)0.28(3)0.10(6)0.15(5)

c2000.01(2)0.02(1)

0246810

z 0500

1000

1500

F (z ) (A) n =1

(B) n =1

|z 0|

FIG.1.The scaling function F (z )vs.z .The convergence radius of the small-z expansion is

expected to be |z 0|=R 1/24?2.8.

This agreement is not trivial since the small-z expansion has a ?nite convergence radius given by |z 0|=R 1/24?2.8.Therefore,the determination of F (z )on the whole positive real axis from its small-z expansion requires an analytic continuation,which turns out to be e?ectively performed by the approximate parametric representations we have considered.We recall that the large-z limit corresponds to the critical theory t =0,so that positive real values of z describe the high-temperature phase up to t =0.Instead,larger di?erences between the approximations given by the schemes (A)and (B)for n =1appear in the scaling function f (x ),especially in the region x <0corresponding to t <0(i.e.the region which is not described by real values of z ).Note that the apparent di?erences for x >0are essentially caused by the normalization of f (x ),which is performed at the coexistence curve x =?1and at the critical point x =0requiring f (?1)=0and f (0)=1.Although the large-x region corresponds to small z ,the di?erence between the two approximate schemes does not decrease in the large-x limit due to their slightly di?erent estimates of R χ(see Table III).Indeed,for large values of x ,f (x )has an expansion of the form

f (x )=x γ

n =0f ∞n x ?2nβ(45)

with f ∞0=R ?1χ.

We also considered the case n =2,using the estimate r 10=?13(7).In this case the scheme (A)was not particularly useful because it turned out to be very sensitive to r 10,whose estimate has a relatively large https://www.doczj.com/doc/9210967483.html,bining the consistency condition θ0<θl (which excludes values of r 10<～?10when using the central values of α,η,r 6and r 8)with the IHT estimate of r 10,we found a rather good result for A +/A ?,i.e.A +/A ?=1.053(4).

?1.0?0.50.00.5 1.0

x 0.00.5

1.0

1.5

2.0f (x ) (A) n =1 (B) n =1

FIG.2.The scaling function f (x )vs.x .

On the other hand,the results for the other universal amplitude ratios considered,such as R χ,R c ,R 4,etc...,although consistent,turned out to be much more imprecise than those obtained for n =1,for example R χ=1.2(3).This fact may be explained noting that,except for the small interval ?10<～r 10<～?9(this interval corresponds to the central values of the other input parameters),the function Y (θ),cf.Eq.(34),has zeroes in the the complex plane which are closer to the origin than θ0.Therefore,the parametric function g 2(θ)related to the magnetic susceptibility (see App.A 2),and higher-order derivatives of the free-energy,

have poles within the disk |θ|<θ0.On the other hand,in the case n =1,θ20was closer to

the origin than the zeroes of Y (θ)for the whole range of values of the input parameters.

For these reasons,for n =2,we present results only for the scheme (B).Combining the consistency condition θ0<θl ,which restricts the acceptable values of r 10(for example using the central estimates of the other input parameters it excludes values |r 10|>～12),with the IHT estimate r 10=?13(7),we arrive at the results reported in Table III (third column of data).The reported value has been obtained using r 10≈?9.The coe?cients c i ,reported in Table III,turn out to be relatively small in both schemes,and decrease rapidly,supporting our choice of the approximation schemes.The results of the various approximate parametric representations are in reasonable agreement.Their comparison is useful to get an idea of the systematic error due to the approximation schemes.There is a very good agreement for A +/A ?,which is the experimentally most important quantity.Our ?nal estimate is

A +/A ?=1.055(3).(46)

We mention that approximately one half of the error is due to the uncertainty on the critical

TABLE IV.Estimates of universal amplitude ratios obtained in di?erent approaches.The ?-expansion estimates of R c and Rχhave been obtained by setting?=1in the O(?2)series calculated in Refs.[34].

IHT–PR HT,LT d=3exp.?-exp.experiments A+/A? 1.055(3) 1.08[35] 1.056(4)[36] 1.029(13)[37] 1.054(1)[1]

1.058(4)[38]

1.067(3)[39]

1.088(7)[40]

R c0.12(1)0.123(3)[44]0.106

=0.12(1),(48)

B2

Rχ≡C+Bδ?1

=|z0|2=7.6(4),(50)

(C+)3

F∞0≡lim z→∞z?δF(z)=0.0303(3),(51)

0

F.Conclusions

Starting from the small-?eld expansion of the e?ective potential in the high-temperature phase,we have constructed approximate representations of the critical equation of state valid in the whole critical region.We have considered two approximation schemes based on polynomial representations that satisfy the general analytic properties of the equation of state(Gri?th’s analyticity)and take into account the Goldstone singularities at the coexistence curve.The coe?cients of the truncated polynomials are determined by matching the small-?eld expansion in the high-temperature phase,which has been studied by lattice high-temperature techniques.The schemes considered can be systematically improved by increasing the order of the polynomials.However,such possibility is limited by the number of known coe?cients r2j of the small-?eld expansion of the e?ective potential.We have shown that the knowledge of the?rst few r2j already leads to satisfactory results,for instance for the speci?c-heat amplitude ratio.Through the approximation schemes we have presented in this paper,the determination of the equation of state may be improved by a better determination of the coe?cients r2j,which may be achieved by extending the high-temperature expansion. We hope to return on this issue in the future.

Finally we mention that the approximation schemes which we have proposed can be applied to other N-vector models.Physically relevant values are N=3and N=4. The case N=3describes the critical phenomena in isotropic ferromagnets[45].The case N=4is interesting for high-energy physics:it should describe the critical behavior of ?nite-temperature QCD with two?avours of quarks at the chiral-symmetry restoring phase transition[47].

APPENDIX A:UNIVERSAL RATIOS OF AMPLITUDES

1.Notations

Universal ratios of amplitudes characterize the critical behavior of thermodynamic quan-tities that do not depend on the normalizations of the external(e.g.magnetic)?eld,order parameter(e.g.magnetization)and temperature.Amplitude ratios of zero-momentum quan-tities can be derived from the critical equation of state.We consider several amplitudes derived from the singular behavior of the speci?c heat

C H=A±|t|?α,(A1) the magnetic susceptibility in the high-temperature phase

1

χ=

C+4t?γ?2βδ,(A3)

3

the second-moment correlation length in the high-temperature phase

ξ=f+t?ν,(A4) and the spontaneous magnetization on the coexistence curve

M=B|t|β.(A5) Using the above normalizations for the amplitudes,the zero-momentum four-point coupling g4,cf.Eq.(18),can be written as

g4=?

C+4

m0Rγg2(θ),g2(θ)=

2βδθh(θ)+(1?θ2)h′(θ)

g(θ0)

,(A12)

R c≡αA+C+

(C+)3

=|z0|2=ρ2[m(θ0)]2 θ20?1 ?2β,(A14) Rχ≡C+Bδ?1

Using Eqs.(36)and(37)one can compute F(z)and obtain the small-z expansion coef-?cients of the e?ective potential r2j.The constant F∞0,which is related to the behavior of F(z)for z→∞,cf.Eq.(27),is given by

F∞0≡lim z→∞z?δF(z)=ρ1?δ[m(1)]?δh(1).(A16) Using the relations(39)concerning the scaling function f(x),one can easily obtain the constant c f,which is related to the behavior of f(x)for x→?1,cf.Eq.(28),

c f≡lim x→?1(1+x)?2f(x).(A17) We consider also the universal amplitude ratio R+ξ≡(A+)1/3f+which can be obtaine

d from th

e estimates o

f R4,R c and g4:

R+ξ≡(A+)1/3f+= R4R c

m≡ρm(θ)and h(θ),but it is easy to get convinced that one of the two functions can be chosen arbitrarily.For de?niteness,let’s take

m≡ρm(θ),which we may symbolically write in the form of a functional equation,

δ

m ρh(θ)

keeping z?xed.By expanding m(θ)according to Eq.(38)and treating the coe?cientsρand c n≡m2n+1as variational parameters,we may turn the above equation into a set of partial di?erential equations(keeping z?xed)

d

(1?θ2)βδ =0,(B2)

d

(1?θ2)βδ =0,(B3) which must be satis?ed exactly for all i by the function h(θ).Simple manipulations lead to the following explicit form:

Y(θ) h+ρ?h?θ+2βδθh ,(B4)

Y(θ)?h

?θ

+2βδθh ,(B5)

where Y(θ)is de?ned in Eq.(43).In turn,by expanding

h(θ,ρ,c i)=θ+

∞

n=1h2n+1(ρ,c i)θ2n+1,(B6)

and substituting into the above equations one obtains an in?nite set of linear di?erential recursive equations for the coe?cients h2n+1,which generalize the relations found in Ref.[10], where the case m(θ)=θwas analyzed.A typical approximation to the exact parametric equation of state amounts to a truncation of h to a polynomial form.We may in this case re?ne the approximation by reinterpreting the?rst recursion equations involving a coe?cient h2t+1which is forcefully set equal to zero as stationarity conditions,which force the parametersρand c i into the values minimizing the unwanted dependence of the truncated F(z)on the parameters themselves,i.e.on the choice of the function

θ20 p

,(B7)

which includes both the Ising model in three dimensions(p=1)and general O(N)symmetric models with Goldstone bosons in d dimensions(p=2/(d?2)).It is easy to recognize that the following relationship must then hold:

1

θ20

.(B8)

Global stationarity implies that the stability conditions may be extracted from the variation of any physical quantity.In particular we may concentrate on the universal zero of F(z), z0,noting that z0=z(θ0),cf.Eq.(36).As a consequence of the above results,the simplest models can all be described by the parametrization

z0

= (1?θ20)β(1+c1θ20).(B9)

6

Let us?rst consider the case c1=0.The minimization procedure leads to

6γ(γ?p)

ρ2=

.(B10)

(1?2β)γ

Setting p=1one immediately recognizes the linear parametric model representation for the Ising model[12–14].Unfortunately when p=2the solution is not physically satisfactory, because it givesθ20<0for all N≥2,and therefore the scheme is useless for models with Goldstone singularities.

Let us now include c1.Requiring z0to be stationary with respect to variations of both parameters,we obtain

6γ(γ?p+1)

ρ2=

,

(3?2β)γ

2(γ?p)+(2β?1)

c1=γ

√3?2β 32β β.(B12) In the Ising model the above solution reduces to

ρ2=2γ,(B13)

3

θ20=

(β+γ)?1.

3

Note that,substituting the physical values of the critical exponentsβandγfor the N=1 model,c1turns out to be a very small number(c1=0.04256)and the predicted numerical value of z0is2.8475,consistent within1%with the linear parametric model prediction[10]. It is fair to say that in the Ising case the above solution has a status which is comparable

伯努利方程推导

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伯努利方程的推导

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伯努利方程的推导及其实际应用

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能量方程(伯努利方程)实验

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