The information geometry of the one-dimensional Potts model

B. P. Dolan, D. A. Johnston, R. Kenna

Research output: Contribution to journalArticle

Abstract

In various statistical-mechanical models the introduction of a metric into the space of parameters (e.g. the temperature variable, ß, and the external field variable, h, in the case of spin models) gives an alternative perspective on the phase structure. For the one-dimensional Ising model the scalar curvature, R, of this metric can be calculated explicitly in the thermodynamic limit and is found to be R = 1 + cosh(h)/vsinh 2(h) + exp(-4ß). This is positive definite and, for physical fields and temperatures, diverges only at the zero-temperature, zero-field 'critical point' of the model. In this paper we calculate R. for the one-dimensional q-state Potts model finding an expression of the form R = A(q, ß, h) + B(q, ß, h)/v(q, ß, h), where ?(q, ß, h) is the Potts analogue of sinh2(h) + exp(-4ß). This is no longer positive definite, but once again it diverges only at the critical point in the space of real parameters. We remark, however, that a naive analytic continuation to complex field reveals a further divergence in the Ising and Potts curvatures at the Lee-Yang edge.

Original languageEnglish
Pages (from-to)9025-9035
Number of pages11
JournalJournal of Physics A: Mathematical and General
Volume35
Issue number43
DOIs
Publication statusPublished - 1 Nov 2002

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geometry
critical point
curvature
Ising model
temperature
divergence
analogs
scalars
thermodynamics

Cite this

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title = "The information geometry of the one-dimensional Potts model",
abstract = "In various statistical-mechanical models the introduction of a metric into the space of parameters (e.g. the temperature variable, {\ss}, and the external field variable, h, in the case of spin models) gives an alternative perspective on the phase structure. For the one-dimensional Ising model the scalar curvature, R, of this metric can be calculated explicitly in the thermodynamic limit and is found to be R = 1 + cosh(h)/vsinh 2(h) + exp(-4{\ss}). This is positive definite and, for physical fields and temperatures, diverges only at the zero-temperature, zero-field 'critical point' of the model. In this paper we calculate R. for the one-dimensional q-state Potts model finding an expression of the form R = A(q, {\ss}, h) + B(q, {\ss}, h)/v(q, {\ss}, h), where ?(q, {\ss}, h) is the Potts analogue of sinh2(h) + exp(-4{\ss}). This is no longer positive definite, but once again it diverges only at the critical point in the space of real parameters. We remark, however, that a naive analytic continuation to complex field reveals a further divergence in the Ising and Potts curvatures at the Lee-Yang edge.",
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The information geometry of the one-dimensional Potts model. / Dolan, B. P.; Johnston, D. A.; Kenna, R.

In: Journal of Physics A: Mathematical and General, Vol. 35, No. 43, 01.11.2002, p. 9025-9035.

Research output: Contribution to journalArticle

TY - JOUR

T1 - The information geometry of the one-dimensional Potts model

AU - Dolan, B. P.

AU - Johnston, D. A.

AU - Kenna, R.

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