The information geometry of the one-dimensional Potts model

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

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Abstract

In various statistical-mechanical models the introduction of a metric into the space of parameters (e.g. the temperature variable, β, 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, Script R, of this metric can be calculated explicitly in the thermodynamic limit and is found to be Script R = 1 + cosh(h)/√sinh2(h) + exp(−4β). 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 Script R for the one-dimensional q-state Potts model finding an expression of the form Script R = A(q, β, h) + B(q, β, h)/√η(q, β, h), where η(q, β, h) is the Potts analogue of sinh2(h) + exp(−4β). 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
Article number9025
JournalJournal of Physics A: Mathematical and General
Volume35
Issue number43
DOIs
Publication statusPublished - 15 Oct 2002

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Potts model
Information Geometry
Potts Model
One-dimensional Model
Diverge
Positive definite
Geometry
Critical point
geometry
Metric
critical point
Ising model
Analytic Continuation
curvature
Spin Models
Zero
Scalar Curvature
Thermodynamic Limit
Phase structure
Ising

Bibliographical note

The full text is also available from: http://de.arxiv.org/abs/cond-mat/0207180

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The information geometry of the one-dimensional Potts model. / Dolan, B. P.; Johnston, D. A.; Kenna, Ralph.

In: Journal of Physics A: Mathematical and General, Vol. 35, No. 43, 9025, 15.10.2002.

Research output: Contribution to journalArticle

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