On the symmetry of real-space renormalisation |
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| Institution: | 1. Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK;2. Theory Group, Blackett Laboratory, Imperial College, South Kensington, London SW7 2BZ, UK;1. Department of Biology, FFCLRP, University of Sao Paulo, Av. Bandeirantes, 3900, 14040-903, Ribeirao Preto, Sao Paulo, Brazil;2. CEFE UMR 5175, CNRS – Université de Montpellier – Université Paul-Valéry Montpellier – EPHE – IRD, 1919 Route de Mende, 34293, Montpellier, Cedex 5, France;3. INRA - UMR 1062 CBGP (INRA/IRD/CIRAD/supagro), 34988, Montferrier Sur Lez, France;4. Key Laboratory of Tropical Forest Ecology, Xishuangbanna Tropical Botanical Garden, Chinese Academy of Sciences, Menglun, Mengla, Yunnan, 666303, China;1. Department of Industrial and Information Engineering, Second University of Naples, Aversa (CE), Italy;2. Department of Mathematics and Physics, Second University of Naples, Caserta, Italy;3. ISTerre, IRD-CNRS-OSUG, University of Grenoble, Saint Martin d’Héres, France;1. University of Arizona, Department of Mathematics, Tucson, USA;2. Faculty of Physics, University of Warsaw, Poland;1. Department of Physics, Technical University of Košice, 042 00 Košice, Slovakia;2. Laboratory of Radiation Biology, Joint Institute for Nuclear Research, 141 980 Dubna, Russia |
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| Abstract: | A geometric structure, arising from the embedding into a Hilbert space of the parametrised probability measure for a given lattice model, is applied here to study the symmetry properties of real-space renormalisation group (RG) flow. In the projective state space this flow is shown to have two contributions: a gradient term, which generates a projective automorphism of the state space for each given length scale; and a correction term due to the scale change. We argue that this structure implies the absence of any symmetry of a geodesic type for the RG flow when restricted to the parameter space submanifold of the state space. This is demonstrated explicitly via a study of the one-dimensional Ising model in an external field. In this example we construct exact expressions for the beta functions associated with the flow induced by infinitesimal rescaling. These constitute a generating vector field for RG diffeomorphisms on the parameter space manifold, and we analyse the symmetry properties of this transformation. The results indicate an approximate conformal Killing symmetry near the critical point, but no generic symmetry of the RG flow globally on the parameter space. |
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