Fisher zeros and singular behaviour of the two-dimensional Potts model in the thermodynamic limit

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Abstract

The duality transformation is applied to the Fisher zeros near the ferromagnetic critical point in the q>4 state two-dimensional Potts model. A requirement that the locus of the duals of the zeros be identical to the dual of the locus of zeros in the thermodynamic limit (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 for the zeros near the ferromagnetic critical point. The conjecture that all zeros governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeros to be a circle. This locus, together with the density of zeros is then shown to be sufficient to recover the singular part of the thermodynamic functions in the thermodynamic limit, their regular parts coming from separate loci of zeros not crossing the positive real temperature axis.
Original languageEnglish
Pages (from-to)9419
JournalJournal of Physics A: Mathematical and General
Volume31
Issue number47
DOIs
Publication statusPublished - 1998

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Potts model
Potts Model
Thermodynamic Limit
loci
two dimensional models
Thermodynamics
Locus
thermodynamics
Zero
critical point
discontinuity
Specific heat
requirements
roots of equations
Discontinuity
Critical point
Duality
internal energy
conjugation
Zero-crossing

Bibliographical note

The full text can also be found here: http://de.arxiv.org/abs/cond-mat/9707039
This is an author-created, un-copyedited version of an article published in Journal of Physics A: Mathematical and General. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0305-4470/31/47/004.

Cite this

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title = "Fisher zeros and singular behaviour of the two-dimensional Potts model in the thermodynamic limit",
abstract = "The duality transformation is applied to the Fisher zeros near the ferromagnetic critical point in the q>4 state two-dimensional Potts model. A requirement that the locus of the duals of the zeros be identical to the dual of the locus of zeros in the thermodynamic limit (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 for the zeros near the ferromagnetic critical point. The conjecture that all zeros governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeros to be a circle. This locus, together with the density of zeros is then shown to be sufficient to recover the singular part of the thermodynamic functions in the thermodynamic limit, their regular parts coming from separate loci of zeros not crossing the positive real temperature axis.",
author = "Ralph Kenna",
note = "The full text can also be found here: http://de.arxiv.org/abs/cond-mat/9707039 This is an author-created, un-copyedited version of an article published in Journal of Physics A: Mathematical and General. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0305-4470/31/47/004.",
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T1 - Fisher zeros and singular behaviour of the two-dimensional Potts model in the thermodynamic limit

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N1 - The full text can also be found here: http://de.arxiv.org/abs/cond-mat/9707039 This is an author-created, un-copyedited version of an article published in Journal of Physics A: Mathematical and General. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/0305-4470/31/47/004.

PY - 1998

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N2 - The duality transformation is applied to the Fisher zeros near the ferromagnetic critical point in the q>4 state two-dimensional Potts model. A requirement that the locus of the duals of the zeros be identical to the dual of the locus of zeros in the thermodynamic limit (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 for the zeros near the ferromagnetic critical point. The conjecture that all zeros governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeros to be a circle. This locus, together with the density of zeros is then shown to be sufficient to recover the singular part of the thermodynamic functions in the thermodynamic limit, their regular parts coming from separate loci of zeros not crossing the positive real temperature axis.

AB - The duality transformation is applied to the Fisher zeros near the ferromagnetic critical point in the q>4 state two-dimensional Potts model. A requirement that the locus of the duals of the zeros be identical to the dual of the locus of zeros in the thermodynamic limit (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 for the zeros near the ferromagnetic critical point. The conjecture that all zeros governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such Fisher zeros to be a circle. This locus, together with the density of zeros is then shown to be sufficient to recover the singular part of the thermodynamic functions in the thermodynamic limit, their regular parts coming from separate loci of zeros not crossing the positive real temperature axis.

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