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Renormalisation Group Flow and Geodesics in the O(N) Model for Large N

Brian P. Dolan
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Abstract

A metric is introduced on the space of parameters (couplings) describing the large N limit of the O(N) model in Euclidean space. The geometry associated with this metric is analysed in the particular case of the infinite volume limit in 3 dimensions and it is shown that the Ricci curvature diverges at the ultra-violet (Gaussian) fixed point but is finite and tends to constant negative curvature at the infra-red (Wilson-Fisher) fixed point. The renormalisation group flow is examined in terms of geodesics of the metric. The critical line of cross-over from the Wilson-Fisher fixed point to the Gaussian fixed point is shown to be a geodesic but all other renormalisation group trajectories, which are repulsed from the Gaussian fixed point in the ultra-violet, are not geodesics. The geodesic flow is interpreted in terms of a maximisation principle for the relative entropy.

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Brian P. Dolan. 1997-10-31. Renormalisation Group Flow and Geodesics in the O(N) Model for Large N. https://doi.org/10.1016/s0550-3213(98)00457-x

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