The self-duality transformation is applied to the Fisher zeroes near the
critical point in the thermodynamic limit in the q>4 state Potts model in two
dimensions. A requirement that the locus of the duals of the zeroes be
identical to the dual of the locus of zeroes (i) recovers the ratio of specific
heat to internal energy discontinuity at criticality and the relationships
between the discontinuities of higher cumulants and (ii) identifies duality
with complex conjugation. Conjecturing that all zeroes governing ferromagnetic
critical behaviour satisfy the latter requirement, the full locus of Fisher
zeroes is shown to be a circle. This locus, together with the density of zeroes
is shown to be sufficient to recover the singular form of all thermodynamic
functions in the thermodynamic limit.Comment: Contribution to Lattice 97, LaTeX, 3 pages, 0 figure

Anglès d'Auriac

Baxter

Bhattacharya

Chen

Fisher

Janke

Kaufman

Maillard

Martin

Matveev

Potts

Ralph Kenna

English

arXiv

Crossref

Singular behaviour of the Potts model in the thermodynamic limit

The self-duality transformation is applied to the Fisher zeroes near the critical point in the thermodynamic limit in the q &gt; 4 state Potts model in two dimensions. A requirement that the locus of the duals of the zeroes be identical to the dual of the locus of zeroes (i) recovers the ratio of specific heat to internal energy discontinuity at criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality with complex conjugation. Conjecturing that all zeroes governing ferromagnetic critical behaviour satisfy the latter requirement, the full locus of Fisher zeroes is shown to be a circle. This locus, together with the density of zeroes is shown to be sufficient to recover the singular from of all thermodynamic functions in the thermodynamic limit

Kenna, Ralph

Coventry University Pure Portal

CURVE is the Institutional Repository for Coventry University   Singular behaviour of the Potts model in the thermodynamic limit  Kenna, R.   Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Kenna, R. (1998) Singular behaviour of the Potts model in the thermodynamic limit. Nuclear Physics B - Proceedings Supplements, volume 63 (1-3): 646-648 http://dx.doi.org/10.1016/S0920-5632(97)00859-1   DOI 10.1016/S0920-5632(97)00859-1 ISSN 0920-5632  Publisher: Elsevier  NOTICE: this is the author’s version of a work that was accepted for publication in Nuclear Physics B - Proceedings Supplements. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Nuclear Physics B - Proceedings Supplements, [63, 1-3, 1995] DOI: 10.1016/S0920-5632(97)00859-1 .   © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/  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/9712013v1  11 Dec 19971Singular Behaviour of the Potts Model in the Thermodynamic LimitRalph Kennaa∗aSchool of Mathematics,Trinity College Dublin,IrelandThe self–duality transformation is applied to the Fisher zeroes near the critical point in the thermodynamiclimit in the q > 4 state Potts model in two dimensions. A requirement that the locus of the duals of the zeroesbe identical to the dual of the locus of zeroes (i) recovers the ratio of specific heat to internal energy discontinuityat criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality withcomplex conjugation. Conjecturing that all zeroes governing ferromagnetic critical behaviour satisfy the latterrequirement, the full locus of Fisher zeroes is shown to be a circle. This locus, together with the density of zeroesis shown to be sufficient to recover the singular form of all thermodynamic functions in the thermodynamic limit.1. INTRODUCTIONThe question of the locus of Fisher zeroes inthe d = 2 Potts model is one which has recentlyreceived an increased amount of attention [1–6].In the two–state (Ising) case where the exact so-lution is available [7], the Fisher zeroes form twocircles in the complex u = exp (−β) plane in thethermodynamic limit [8].The partition function for the q–state Pottsmodel is ZL(β) =∑{σi}exp (β∑〈ij〉 δσiσj ) andin the thermodynamic limit is invariant underthe duality transformation u → D(u), whereD(u) = (1 − u)/(1 + (q − 1)u) [9]. The criticaltemperature at which the phase transition occursis invariant under duality and is uc = 1/(1+√q).Here the d = 2 q > 4 state model exhibits a firstorder phase transition [10].Based on similarities with the Ising case, Mar-tin and Maillard and Rammal [1] conjectured thatthe locus of Fisher zeroes in the d = 2, q–statePotts model be given by an extension of criticalduality to the complex plane, namely D(u) = u∗where u∗ is the complex conjugate of u. Thisidentification yields a circle with centre −1/(q−1)and radius√q/(q − 1). When q = 2 this recoversthe ferromagnetic Fisher circle of the Ising model[8]. In the Ising case, the partition function is∗Supported bt EU TMR Project No. ERBFMBI-CT96-1757actually a function of u2. There, the second (an-tiferromagnetic) Fisher circle comes from the mapβ → −β. Numerical investigations for small lat-tices [1] provided evidence that the Fisher zeroesindeed lie on the circle given by the identificationof duality with complex conjugation. Howeverthe numerics are highly sensitive to the bound-ary conditions used and the situation far fromcriticality remained unclear. Some progress wasrecently made in the non–critical region using lowtemperature expansions for 3 ≤ q ≤ 8 [4].Recently, and on the basis of numerical re-sults on small lattices with q ≤ 10, it has againbeen conjectured that for finite lattices with self–dual boundary conditions, and for other bound-ary conditions in the thermodynamic limit, thezeroes in the ferromagnetic regime are on theabove circle [5]. The conjecture of [5] was, in fact,proven for infinite q in [6]. This circle–conjectureis similar to another recent conjecture [3], namelythat the Fisher zeroes for the q–state Potts modelon a triangular lattice with pure three–site inter-action in the thermodynamic limit (which is alsoself–dual [10]) lie on a circle and a segment of thenegative real axis.All of the above conjectures regarding the locusof Fisher zeroes rely, at least in part, on numeri-cal approaches. In this paper, the problem is ad-dressed analytically and the origin of the circularlocus is clarified.22. Thermodynamic FunctionsFor finite L, the partition function is a poly-nomial in u and can be expressed in terms of itszeroes as ZL(β) ∝∏dVj=1 (u − uj(L)) [8]. The freeenergy is βf(β) = − lnZ(β)/V . The internal en-ergy is (up to a constant)e(β) =uVdV∑j=11u − uj(L). (1)The general nth cumulant is defined as γn(β) =(−)n+1∂n(βf)/∂βn. In conventional notation thespecific heat is c(β) = kBβ2γ2(β). With ∆γn ≡γn(β−) − γn(β+), the discontinuity in the nthcumulant at the critical temperature, the exactthermodynamic limit results include [10–13]∆c = kBβ2c∆e√q. (2)3. Partition Function ZeroesRecently, Lee [2] has derived a general theo-rem for first order phase transitions in which thezeroes can be expressed in terms of the discon-tinuities in the thermodynamic functions. For asystem with a temperature–driven phase transi-tion, Lee’s result is (in terms of t = 1 − β/βc)βcRetj(L) = A1I2j + A3I4j + A5I6j + . . . ,±βcImtj(L) = Ij + A2I3j + A4I5j + . . . , (3)where Ij = (2j − 1)π/(V ∆e) and . . . includesterms which vanish in the infinite volume limit.The first few coefficients An are [2,14] A1 =∆c/2kBβ2c ∆e, A2 = −2A21 + ∆γ3/3!∆e.3.1. The Locus of ZeroesFrom (3) the real part of the zeroes (in the ther-modynamic limit) can be expressed in terms oftheir imaginary parts as βcRet = L(βcImt) whereL(θ) = A1θ2+(−2A1A2+A3)θ4+ . . .. The zeroesare thus seen to lie on a curve. In the complexu upper half–plane the equation of this curve isγ(+)(θ) = uc exp (L(θ) + iθ).3.2. The Dual of the Locus and the Locusof the DualsApplying the duality transformation to γ(+)(θ)and expanding in θ gives ReD(γ(+)(θ))=uc[1 + (θ2/2q)(2√q − 2A1q − q) + . . .] andImD(γ(+)(θ))= −uc[θ − (θ3/6q)(6− 6q1/2 + q −12A1q1/2 + 6A1q) + . . .].Alternatively, applying the duality transforma-tion directly to the jth zero in the finite–size sys-tem givesβcRetDj (L) = AD1 I2j + AD3 I4j + AD5 I6j + . . .∓βcImtDj (L) = Ij + AD2 I3j + AD4 I5j + . . . (4)where terms vanishing in the thermodynamiclimit are included in . . . and the first few ADn areAD1 = q− 12 − A1 , (5)AD2 = −q−1 + 2q−12 A1 + A2 . (6)As in Sec.(3.1), (4) gives the locus of the dualof the upper half–plane zeroes in the thermody-namic limit to be γ(+)D(θ) = uc exp (LD(θ) − iθ),where LD(θ) = AD1 θ2 + (−2AD1 AD2 + AD3 )θ4 +. . .. The expansion of this locus of duals isReγ(+)D(θ) = uc[1+(θ2/2!)(−1+2AD1 )+ . . .] andImγ(+)D(θ) = −uc[θ + (θ3/3!)(−1 + 6AD1 ) + . . .].Even when the finite–L system does not haveduality–preserving boundary conditions, takingthe thermodynamic limit restores self–duality.There the dual of the locus of zeroes and the locusof duals must be identical,D(γ(+))≡ γ(+)D . (7)Up to O(θ2) this is trivial. To O(θ3) and (sep-arately at) O(θ4) they are identical if A1 =1/(2√q). This is the result (2). The iden-tity (7) at O(θ5) and (separately at) O(θ6) givesA3 = A2/√q − q−3/2(q − 3)/24, or [11–13]∆γ4 =6√q∆γ3 +q − 6q3/2∆e . (8)Higher order results are obtainable using a com-puter algebra system such as Maple [14].4. The Full Locus and the Singular Partsof the Thermodynamic FunctionsPutting the above equations ((2) and (8)) into(5), (6) (and their higher order equivalents) yieldsADj = Aj (this has been verified up to j = 8).3Therefore (at least up to θ10) the dual of the locusof zeroes is the complex conjugate of the originallocus of zeroes. We now assume that this is thecase for all θ. Then, the full feromagnetic locusof zeroes (that part of the full locus which in-tersects the real temperature axis at the physicalferromagnetic critical point) is found by identify-ing [1] D(γ(+)(θ))= γ(+)∗(θ), where γ(+)∗rep-resents the complex conjugate of γ(+). The fullferromagnetic locus is then [1]γ(θ) =1q − 1(−1 + √qeiθ). (9)The density of zeroes is given by [2]2πg(θ) =(1 +1(q − 1)γ(θ))[1+∞∑n=2∆γn(n − 1)! (ln ((√q + 1)γ(θ)))(n−1)]. (10)The internal energy is (from (1) or [15])e = cnst. + u∫ 2π0g(θ)u − γ(θ)dθ . (11)Therefore, from (9), (10) and (11), the internalenergy is e(β < βc) = e0 ande(β > βc) = e0−∆e+∞∑n=2∆γn(βc − β)(n−1)(n − 1)! , (12)with e0 a constant (one expects that when sepa-rate Fisher loci which don’t cross the positive realtemperature axis are accounted for, e0 becomestemperature dependent). At βc the internal en-ergy discontinuity e(β = β−c ) − e(β = β+c ) = ∆eis recovered. Appropriate differentiation recov-ers the discontinuities in specific heat and highercumulants.5. ConclusionsThe requirement that the dual of the locus ofzeroes be identical to the locus of the duals ofzeroes (i) recovers the ratio of specific heat to in-ternal energy discontinuity at criticality and re-lations between the discontinuities of higher cu-mulants and (ii) identifies duality with complexconjugation.Conjecturing that all zeroes governing ferro-magnetic critical behaviour satisfy (ii) gives thatthis locus is the circle (9). This puts the conjec-turs of [1,3,5] on an analytic footing. The locus(9), together with the density of zeroes (10) is suf-ficient to recover the singular parts of all thermo-dynamic functions in the thermodynamic limit.It is to be expected that the regular parts comefrom separate loci of zeroes which don’t cross thepositive real temperature axis.The author thanks A. Irving, W. Janke and J.Sexton.REFERENCES1. P.P. Martin, Nucl. Phys. B 225 (1983) 497;J.M. Maillard and R. Rammal, J. Phys. A 16(1983) 353; P.P. Martin and J.M. Maillard, J.Phys. A 19 (1986) L547; ibid. 20 (1987) L601.2. K.-C. Lee Phys. Rev. Lett. 73 (1994) 2801.3. J.C. Anglès d’Auriac, J. M. Maillard, G. Rol-let and F.Y. Wu Physica A 206 (1994) 441.4. V. Matveev and R. Shrock, Phys. Rev. E 54(1996) 6174.5. C.-N. Chen, C.-K. Hu and F.Y. Wu, Phys.Rev. Lett., 76 (1996) 169.6. F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Mail-lard, C.-K. Hu and C.-N. Chen, Phys. Rev.Lett., 76 (1996) 173.7. B. Kaufman, Phys. Rev. 76 (1949) 1232.8. M.E. Fisher, in Lectures in Theoretical Phys-ics, Vol. VIIC, ed. W.E. Brittin, (Gordon andBreach, New York: 1968) 1.9. R.B. Potts, Proc. Camb. Phil. Soc. 48 (1952)106.10. R.J. Baxter, J. Phys. C 6 (1973) L445; J. Stat.Phys. 28 (1982) 1; R.J. Baxter, H.N.V. Tem-perley and S. Ashley Proc. Roy. Soc. London,Ser. A 358 (1978) 535.11. F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235.12. T. Bhattacharya, R. Lacaze and A. Morel,Nucl. Phys. B 435 (1995) 526; J. Phys. IFrance 7 (1997) 81.13. W. Janke and S. Kappler, Phys. Lett. A 197(1995) 227; J. Phys. I France 7 (1997) 663.14. R. Kenna cond-mat/9707039.15. R. Abe, Prog. Theor. Phys. 37, (1967) 1070;ibid. 38 322.

CURVE is the Institutional Repository for Coventry University   Singular behaviour of the Potts model in the thermodynamic limit  Kenna, R.   Author post-print (accepted) deposited in CURVE April 2016  Original citation & hyperlink:  Kenna, R. (1998) Singular behaviour of the Potts model in the thermodynamic limit. Nuclear Physics B - Proceedings Supplements, volume 63 (1-3): 646-648 http://dx.doi.org/10.1016/S0920-5632(97)00859-1   DOI 10.1016/S0920-5632(97)00859-1 ISSN 0920-5632  Publisher: Elsevier  NOTICE: this is the author’s version of a work that was accepted for publication in Nuclear Physics B - Proceedings Supplements. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Nuclear Physics B - Proceedings Supplements, [63, 1-3, 1995] DOI: 10.1016/S0920-5632(97)00859-1 .   © 2015, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International http://creativecommons.org/licenses/by-nc-nd/4.0/  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/9712013v1  11 Dec 19971Singular Behaviour of the Potts Model in the Thermodynamic LimitRalph Kennaa∗aSchool of Mathematics,Trinity College Dublin,IrelandThe self–duality transformation is applied to the Fisher zeroes near the critical point in the thermodynamiclimit in the q > 4 state Potts model in two dimensions. A requirement that the locus of the duals of the zeroesbe identical to the dual of the locus of zeroes (i) recovers the ratio of specific heat to internal energy discontinuityat criticality and the relationships between the discontinuities of higher cumulants and (ii) identifies duality withcomplex conjugation. Conjecturing that all zeroes governing ferromagnetic critical behaviour satisfy the latterrequirement, the full locus of Fisher zeroes is shown to be a circle. This locus, together with the density of zeroesis shown to be sufficient to recover the singular form of all thermodynamic functions in the thermodynamic limit.1. INTRODUCTIONThe question of the locus of Fisher zeroes inthe d = 2 Potts model is one which has recentlyreceived an increased amount of attention [1–6].In the two–state (Ising) case where the exact so-lution is available [7], the Fisher zeroes form twocircles in the complex u = exp (−β) plane in thethermodynamic limit [8].The partition function for the q–state Pottsmodel is ZL(β) =∑{σi}exp (β∑〈ij〉 δσiσj ) andin the thermodynamic limit is invariant underthe duality transformation u → D(u), whereD(u) = (1 − u)/(1 + (q − 1)u) [9]. The criticaltemperature at which the phase transition occursis invariant under duality and is uc = 1/(1+√q).Here the d = 2 q > 4 state model exhibits a firstorder phase transition [10].Based on similarities with the Ising case, Mar-tin and Maillard and Rammal [1] conjectured thatthe locus of Fisher zeroes in the d = 2, q–statePotts model be given by an extension of criticalduality to the complex plane, namely D(u) = u∗where u∗ is the complex conjugate of u. Thisidentification yields a circle with centre−1/(q−1)and radius√q/(q− 1). When q = 2 this recoversthe ferromagnetic Fisher circle of the Ising model[8]. In the Ising case, the partition function is∗Supported bt EU TMR Project No. ERBFMBI-CT96-1757actually a function of u2. There, the second (an-tiferromagnetic) Fisher circle comes from the mapβ → −β. Numerical investigations for small lat-tices [1] provided evidence that the Fisher zeroesindeed lie on the circle given by the identificationof duality with complex conjugation. Howeverthe numerics are highly sensitive to the bound-ary conditions used and the situation far fromcriticality remained unclear. Some progress wasrecently made in the non–critical region using lowtemperature expansions for 3 ≤ q ≤ 8 [4].Recently, and on the basis of numerical re-sults on small lattices with q ≤ 10, it has againbeen conjectured that for finite lattices with self–dual boundary conditions, and for other bound-ary conditions in the thermodynamic limit, thezeroes in the ferromagnetic regime are on theabove circle [5]. The conjecture of [5] was, in fact,proven for infinite q in [6]. This circle–conjectureis similar to another recent conjecture [3], namelythat the Fisher zeroes for the q–state Potts modelon a triangular lattice with pure three–site inter-action in the thermodynamic limit (which is alsoself–dual [10]) lie on a circle and a segment of thenegative real axis.All of the above conjectures regarding the locusof Fisher zeroes rely, at least in part, on numeri-cal approaches. In this paper, the problem is ad-dressed analytically and the origin of the circularlocus is clarified.22. Thermodynamic FunctionsFor finite L, the partition function is a poly-nomial in u and can be expressed in terms of itszeroes as ZL(β) ∝∏dVj=1 (u − uj(L)) [8]. The freeenergy is βf(β) = − lnZ(β)/V . The internal en-ergy is (up to a constant)e(β) =uVdV∑j=11u− uj(L) . (1)The general nth cumulant is defined as γn(β) =(−)n+1∂n(βf)/∂βn. In conventional notation thespecific heat is c(β) = kBβ2γ2(β). With ∆γn ≡γn(β−) − γn(β+), the discontinuity in the nthcumulant at the critical temperature, the exactthermodynamic limit results include [10–13]∆c = kBβ2c∆e√q. (2)3. Partition Function ZeroesRecently, Lee [2] has derived a general theo-rem for first order phase transitions in which thezeroes can be expressed in terms of the discon-tinuities in the thermodynamic functions. For asystem with a temperature–driven phase transi-tion, Lee’s result is (in terms of t = 1− β/βc)βcRetj(L) = A1I2j +A3I4j +A5I6j + . . . ,±βcImtj(L) = Ij +A2I3j +A4I5j + . . . , (3)where Ij = (2j − 1)pi/(V∆e) and . . . includesterms which vanish in the infinite volume limit.The first few coefficients An are [2,14] A1 =∆c/2kBβ2c∆e, A2 = −2A21 +∆γ3/3!∆e.3.1. The Locus of ZeroesFrom (3) the real part of the zeroes (in the ther-modynamic limit) can be expressed in terms oftheir imaginary parts as βcRet = L(βcImt) whereL(θ) = A1θ2+(−2A1A2+A3)θ4+ . . .. The zeroesare thus seen to lie on a curve. In the complexu upper half–plane the equation of this curve isγ(+)(θ) = uc exp (L(θ) + iθ).3.2. The Dual of the Locus and the Locusof the DualsApplying the duality transformation to γ(+)(θ)and expanding in θ gives ReD (γ(+)(θ)) =uc[1 + (θ2/2q)(2√q − 2A1q − q) + . . .] andImD (γ(+)(θ)) = −uc[θ− (θ3/6q)(6− 6q1/2 + q−12A1q1/2 + 6A1q) + . . .].Alternatively, applying the duality transforma-tion directly to the jth zero in the finite–size sys-tem givesβcRetDj (L) = AD1 I2j +AD3 I4j +AD5 I6j + . . .∓βcImtDj (L) = Ij +AD2 I3j +AD4 I5j + . . . (4)where terms vanishing in the thermodynamiclimit are included in . . . and the first few ADn areAD1 = q− 12 −A1 , (5)AD2 = −q−1 + 2q−12A1 +A2 . (6)As in Sec.(3.1), (4) gives the locus of the dualof the upper half–plane zeroes in the thermody-namic limit to be γ(+)D(θ) = uc exp (LD(θ) − iθ),where LD(θ) = AD1 θ2 + (−2AD1 AD2 + AD3 )θ4 +. . .. The expansion of this locus of duals isReγ(+)D(θ) = uc[1+(θ2/2!)(−1+2AD1 )+ . . .] andImγ(+)D(θ) = −uc[θ + (θ3/3!)(−1 + 6AD1 ) + . . .].Even when the finite–L system does not haveduality–preserving boundary conditions, takingthe thermodynamic limit restores self–duality.There the dual of the locus of zeroes and the locusof duals must be identical,D(γ(+))≡ γ(+)D . (7)Up to O(θ2) this is trivial. To O(θ3) and (sep-arately at) O(θ4) they are identical if A1 =1/(2√q). This is the result (2). The iden-tity (7) at O(θ5) and (separately at) O(θ6) givesA3 = A2/√q − q−3/2(q − 3)/24, or [11–13]∆γ4 =6√q∆γ3 +q − 6q3/2∆e . (8)Higher order results are obtainable using a com-puter algebra system such as Maple [14].4. The Full Locus and the Singular Partsof the Thermodynamic FunctionsPutting the above equations ((2) and (8)) into(5), (6) (and their higher order equivalents) yieldsADj = Aj (this has been verified up to j = 8).3Therefore (at least up to θ10) the dual of the locusof zeroes is the complex conjugate of the originallocus of zeroes. We now assume that this is thecase for all θ. Then, the full feromagnetic locusof zeroes (that part of the full locus which in-tersects the real temperature axis at the physicalferromagnetic critical point) is found by identify-ing [1] D (γ(+)(θ)) = γ(+)∗(θ), where γ(+)∗ rep-resents the complex conjugate of γ(+). The fullferromagnetic locus is then [1]γ(θ) =1q − 1(−1 +√qeiθ) . (9)The density of zeroes is given by [2]2pig(θ) =(1 +1(q − 1)γ(θ))[1+∞∑n=2∆γn(n− 1)! (ln ((√q + 1)γ(θ)))(n−1)]. (10)The internal energy is (from (1) or [15])e = cnst.+ u∫ 2pi0g(θ)u− γ(θ)dθ . (11)Therefore, from (9), (10) and (11), the internalenergy is e(β < βc) = e0 ande(β > βc) = e0−∆e+∞∑n=2∆γn(βc − β)(n−1)(n− 1)! , (12)with e0 a constant (one expects that when sepa-rate Fisher loci which don’t cross the positive realtemperature axis are accounted for, e0 becomestemperature dependent). At βc the internal en-ergy discontinuity e(β = β−c ) − e(β = β+c ) = ∆eis recovered. Appropriate differentiation recov-ers the discontinuities in specific heat and highercumulants.5. ConclusionsThe requirement that the dual of the locus ofzeroes be identical to the locus of the duals ofzeroes (i) recovers the ratio of specific heat to in-ternal energy discontinuity at criticality and re-lations between the discontinuities of higher cu-mulants and (ii) identifies duality with complexconjugation.Conjecturing that all zeroes governing ferro-magnetic critical behaviour satisfy (ii) gives thatthis locus is the circle (9). This puts the conjec-turs of [1,3,5] on an analytic footing. The locus(9), together with the density of zeroes (10) is suf-ficient to recover the singular parts of all thermo-dynamic functions in the thermodynamic limit.It is to be expected that the regular parts comefrom separate loci of zeroes which don’t cross thepositive real temperature axis.The author thanks A. Irving, W. Janke and J.Sexton.REFERENCES1. P.P. Martin, Nucl. Phys. B 225 (1983) 497;J.M. Maillard and R. Rammal, J. Phys. A 16(1983) 353; P.P. Martin and J.M. Maillard, J.Phys. A 19 (1986) L547; ibid. 20 (1987) L601.2. K.-C. Lee Phys. Rev. Lett. 73 (1994) 2801.3. J.C. Angle`s d’Auriac, J. M. Maillard, G. Rol-let and F.Y. Wu Physica A 206 (1994) 441.4. V. Matveev and R. Shrock, Phys. Rev. E 54(1996) 6174.5. C.-N. Chen, C.-K. Hu and F.Y. Wu, Phys.Rev. Lett., 76 (1996) 169.6. F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Mail-lard, C.-K. Hu and C.-N. Chen, Phys. Rev.Lett., 76 (1996) 173.7. B. Kaufman, Phys. Rev. 76 (1949) 1232.8. M.E. Fisher, in Lectures in Theoretical Phys-ics, Vol. VIIC, ed. W.E. Brittin, (Gordon andBreach, New York: 1968) 1.9. R.B. Potts, Proc. Camb. Phil. Soc. 48 (1952)106.10. R.J. Baxter, J. Phys. C 6 (1973) L445; J. Stat.Phys. 28 (1982) 1; R.J. Baxter, H.N.V. Tem-perley and S. Ashley Proc. Roy. Soc. London,Ser. A 358 (1978) 535.11. F.Y. Wu, Rev. Mod. Phys. 54 (1982) 235.12. T. Bhattacharya, R. Lacaze and A. Morel,Nucl. Phys. B 435 (1995) 526; J. Phys. IFrance 7 (1997) 81.13. W. Janke and S. Kappler, Phys. Lett. A 197(1995) 227; J. Phys. I France 7 (1997) 663.14. R. Kenna cond-mat/9707039.15. R. Abe, Prog. Theor. Phys. 37, (1967) 1070;ibid. 38 322.

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Singular Behaviour of the Potts Model in the Thermodynamic Limit

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