The Information Geometry of the One-Dimensional Potts Model

Dolan, Brian P. and Johnston, D.A. and Kenna, R. (2002) The Information Geometry of the One-Dimensional Potts Model. Journal of Physics A: Mathematical and General, 35 (43). pp. 9025-9036.

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In various statistical-mechanical 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 one-dimensional 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 zero-temperature, zero-field ``critical point'' of the model. In this note we calculate ${\cal R}$ for the one-dimensional $q$-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 Lee-Yang edge.

Item Type: Article
Keywords: Renormalisation group, chaos
Academic Unit: Faculty of Science and 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

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