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The Information Geometry of the One-Dimensional Potts Model | | B.P. Dolan
; D.A. Johnston
; R. Kenna
; | | Date: |
6 Jul 2002 | | Journal: | J.Phys. A35 (2002) 9025-9036 | | Subject: | Statistical Mechanics | cond-mat.stat-mech hep-lat | | Abstract: | In various statistical-mechanical models the introduction of a metric onto the space of parameters (e.g. the temperature variable, $eta$, 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 ${cal R} = 1 + cosh (h) / sqrt{sinh^2 (h) + exp (- 4 eta)}$. 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 ${cal R} = A(q,eta,h) + B (q,eta,h)/sqrt{eta(q,eta,h)}$, where $eta(q,eta,h)$ is the Potts analogue of $sinh^2 (h) + exp (- 4 eta)$. 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. | | Source: | arXiv, cond-mat/0207180 | | Services: | Forum | Review | PDF | Favorites | |
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