Dolan, Brian P and Johnston, DA and Kenna, R
(2002)
The Information Geometry of the OneDimensional Potts Model.
Journal of Physics A: Mathematical and General, 35 (43).
pp. 90259036.
Abstract
In various statisticalmechanical models the introduction of a metric onto the space of parameters (e.g. the temperature variable, $\beta$, and the external field variable, $h$, in the case of spin models) gives an alternative perspective on the phase structure. For the onedimensional Ising model the scalar curvature, ${\cal R}$, of this metric can be calculated explicitly in the thermodynamic limit and is found to be ${\cal R} = 1 + \cosh (h) / \sqrt{\sinh^2 (h) + \exp ( 4 \beta)}$. This is positive definite and, for physical fields and temperatures, diverges only at the zerotemperature, zerofield ``critical point'' of the model.
In this note we calculate ${\cal R}$ for the onedimensional $q$state Potts model, finding an expression of the form ${\cal R} = A(q,\beta,h) + B (q,\beta,h)/\sqrt{\eta(q,\beta,h)}$, where $\eta(q,\beta,h)$ is the Potts analogue of $\sinh^2 (h) + \exp ( 4 \beta)$. 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 LeeYang edge.
Item Type: 
Article

Keywords: 
Renormalisation group, chaos 
Subjects: 
Science & Engineering > Experimental Physics 
Item ID: 
268 
Depositing User: 
Dr. Brian Dolan

Date Deposited: 
09 Nov 2005 
Journal or Publication Title: 
Journal of Physics A: Mathematical and General 
Publisher: 
Institute of Physics 
Refereed: 
Yes 
URI: 

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