discontinuities of critical amplitude for specific heat

Volume 45A, number 1 PHYSICS LETTERS 27 August 1973 DISCONTINUITIES OF CRITICAL AMPLITUDE FOR SPECIFIC HEAT R. ABE and S. HIKAMI Department of Pure andApplied Sciences, University of Tokyo, Komaba, Meguro-Ku, Tokyo, Japan Received 2 July 1973 On the basis of 1/n expansion, the critical amplitude for specific heat is shown to be discontinuous at space dimensions d = 2 + (2/rn), (m = 2, 3, ... ). This property is confirmed up to the order of 1/n 3. In previous papers [1], the present authors dis- and compare only (in e )/n terms, one would have cussed an anomalous behavior of logarithmic term re- = ~ 7~ ~ This is the previous lated to specific heat exponent a. It was shown that result [1]. However, eq. (1) suggests that the quantity: the coefficient of logarithmic term of the order of {&~ +.~ 7l(_B)mó(d,dm)} appears as a common 1/n has discontinuities at dimensions d = dm = 2 +(2/m), multiplicative factor in the higher order terms. If we (m = 2,3,...), in the case of short-range interaction, neglect the 1/n2 term of exponent a, eq. (1) is con- We interpreted this anomaly as discontinuities of a at sistent with the following form: d = drn. However, there remains another possibility that the anomaly can be interpreted as discontinuities E = + B 6(d~dm)] ~ ~O —al/n (2) of critical amplitude. The purpose of this letter is to clarify the situation by considering the leading logarithmic terms up to the order of 1/n3. (_B)rn 1 We have derived the expression for the in Z(Z: 2(1—d) ~(d,dm)e ~° + partition function) up to the order of i/n3. On the with basis of this expression, we have calculated the leading logarithmic terms up to the order of(ln e)3/n3, a 1 = &~ =~ ‘y~(~ —d). e being defined by e = Kc — K. It turns out that the energy in a reduced form is expressed as From eq. (2) we can see that the result [2—4] without 1 ~ anomalous term is restored, so that a1 is continuous E~~e~00 —-{&i +.}7i(_B)rn~(d,drn) } e °ln at d = dm. However, the critical amplitude: [1 + (_B)m6(d, dm )/2 (1 — d)] does not joint continu- 1-—~--- {&~ +~ ~ (_B)m~(d,drn)}&i e 1~°(ln e)2 ously to its value at n = °° in the case of d = drn, and 2n2 therefore the critical amplitude shows discontinuities 1 {&i ÷~ ~ _BY~~id,dm)} &? e’~0(ln e)3 at d = dm when it is regarded as a function of d. 3!n3 The above situation may be stated in the following (1) way. For the unperturbed system (n = oo), E is expressed asE = e’~0. In the presence of perturbation, where however, this term is split off into two parts as shown = 7~ (1 —d), in eq. (2) at d = drn. In this sence, the anomalous case is supposed to correspond to the degenerate unper- ( By~ =4 {mr [(1/2)+(1/m)]lm turbed state. — r’(I/2)r(1/rn) j for d 2+2/rn, It is also expected that the anomalous behavior of = 0 for d*2+2/m. a for the long-range interaction [1,5] is interpreted in a similar way. Details of this article will be publish- If one assumes a simple power law in z = 2_an_al “i ed elsewhere. :11

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Volume45A,number1 PHYSICSLETTERS 27 August 1973

DISCONTINUITIES OF CRITICAL AMPLITUDE FOR SPECIFIC HEAT

R. ABE andS. HIKAMIDepartmentof PureandAppliedSciences,Universityof Tokyo,Komaba,Meguro-Ku,Tokyo,Japan

Received2 July 1973

Onthebasisof 1/n expansion,the criticalamplitudefor specificheatis shownto bediscontinuousat spacedimen-sionsd= 2 + (2/rn), (m = 2, 3, ... ). This propertyis confirmedup to theorderof 1/n3.

In previouspapers[1], thepresentauthorsdis- andcompareonly (in e )/n terms,one would havecussedananomalousbehaviorof logarithmic termre- = ~ 7~~ This is thepreviouslatedto specificheatexponenta. It wasshownthat result [1]. However,eq.(1) suggeststhat the quantity:the coefficientof logarithmic termof the orderof {&~+.~ 7l(_B)mó(d,dm)}appearsasa common1/n hasdiscontinuitiesat dimensionsd= dm=2 +(2/m), multiplicative factor in thehigherorderterms.If we(m = 2,3,...),in thecaseof short-rangeinteraction, neglectthe 1/n2term of exponenta,eq.(1) is con-We interpretedthis anomalyasdiscontinuitiesof aat sistentwith thefollowing form:d= drn. However,thereremainsanotherpossibilitythat the anomalycanbeinterpretedasdiscontinuities E= + B 6(d~dm)]~ ~O —al/n (2)of critical amplitude.Thepurposeof this letter is toclarify the situationby consideringtheleadingloga-rithmic termsup to the orderof 1/n3. (_B)rn 1

Wehavederivedthe expressionfor the in Z(Z: 2(1—d)~(d,dm)e ~° +

partition function)up to theorderof i/n3. On thewith

basisof this expression,we havecalculatedthelead-inglogarithmic termsup to theorderof(ln e)3/n3, a

1 = &~=~ ‘y~(~—d).e beingdefinedby e = Kc — K. It turnsoutthat theenergyin a reducedform is expressedas Fromeq.(2) wecanseethat the result [2—4]without

1 ~ anomaloustermis restored,so that a1 is continuousE~~e~00—-{&i +.}7i(_B)rn~(d,drn)} e °ln� at d = dm. However,the critical amplitude:

[1+ (_B)m6(d,dm)/2 (1 — d)] doesnotjoint continu-1-—~---{&~+~~ (_B)m~(d,drn)}&ie

1~°(lne)2 ously to its valueat n = °° in thecaseofd = drn, and2n2 thereforethecritical amplitude showsdiscontinuities

1 {&i ÷~~ _BY~~id,dm)}&? e’~0(lne)3 at d = dmwhen it is regardedasa function of d.

3!n3 The abovesituationmaybestatedin the following

(1) way. Forthe unperturbedsystem(n = oo), E is ex-pressedasE= e’~0. In the presenceof perturbation,

where however,this termis split off into two partsas shown

= 7~(1 —d), in eq.(2) at d = drn. In this sence,the anomalouscaseis supposedto correspondto thedegenerateunper-

( By~=4 {mr [(1/2)+(1/m)]lm turbedstate.— r’(I/2)r(1/rn) j ford 2+2/rn, It is alsoexpectedthat the anomalousbehaviorof

= 0 for d*2+2/m. afor the long-rangeinteraction[1,5] is interpretedin a similarway. Details of this articlewill be publish-

If oneassumesa simplepowerlawinz = �2_an_al “i edelsewhere.

:11

Page 2: Discontinuities of critical amplitude for specific heat

Volume45A, number1 PHYSICS LETTERS 27 AugUst 1973

It is ourpleasure-toexpressoursincerethanksto ReferencesProfessorsK.G. Wilson andF. Wegnerfor giving usvaluablecommentsandto ProfressorS. Ma for send- [1] R. AbeandS. Hikami, Phys.Lett.42A (1973)419,

ing us papersprior to publication. Frog. Theor.Phys.49 (1973)No. 6.[2] R. AbeandS.Hikami, Prog.Theor.Phys.49 (1973)442.3] S. Ma, Universityof California preprint.

[4] R.A. FerrellandD.J. Scalapino,Phys.Lett. 41A (1972)371.

[5] M. Suzuki,Frog.Theor.Phys.49 (1973)424.

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