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  Critical behavior in spherical and hyperbolic spaces

Benedetti, D. (2015). Critical behavior in spherical and hyperbolic spaces. Journal of Statistical Mechanics: Theory and Experiment, 2015(1): P01002. doi:10.1088/1742-5468/2015/01/P01002.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0018-C26F-4 Version Permalink: http://hdl.handle.net/11858/00-001M-0000-0024-8D85-6
Genre: Journal Article

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1403.6712.pdf (Preprint), 346KB
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 Creators:
Benedetti, Dario1, Author              
Affiliations:
1Microscopic Quantum Structure & Dynamics of Spacetime, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, escidoc:67201              

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Free keywords: Condensed Matter, Statistical Mechanics, cond-mat.stat-mech,General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th
 Abstract: We study the effects of curved background geometries on the critical behavior of scalar field theory. In particular we concentrate on two maximally symmetric spaces: $d$-dimensional spheres and hyperboloids. In the first part of the paper, by applying the Ginzburg criterion, we find that for large correlation length the Gaussian approximation is valid on the hyperboloid for any dimension $d\geq 2$, while it is not trustable on the sphere for any dimension. This is understood in terms of various notions of effective dimension, such as the spectral and Hausdorff dimension. In the second part of the paper, we apply functional renormalization group methods to develop a different perspective on such phenomena, and to deduce them from a renormalization group analysis. By making use of the local potential approximation, we discuss the consequences of having a fixed scale in the renormalization group equations. In particular, we show that in the case of spheres there is no true phase transition, as symmetry restoration always occurs at large scales. In the case of hyperboloids, the phase transition is still present, but as the only true fixed point is the Gaussian one, mean field exponents are valid also in dimensions lower than four.

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 Dates: 2014-03-262015
 Publication Status: Published in print
 Pages: 24 pages, 3 figures
 Publishing info: -
 Table of Contents: -
 Rev. Method: -
 Identifiers: arXiv: 1403.6712
DOI: 10.1088/1742-5468/2015/01/P01002
 Degree: -

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Title: Journal of Statistical Mechanics: Theory and Experiment
Source Genre: Journal
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Publ. Info: Bristol, England : Institute of Physics Publishing
Pages: - Volume / Issue: 2015 (1) Sequence Number: P01002 Start / End Page: - Identifier: ISSN: 1742-5468
CoNE: http://pubman.mpdl.mpg.de/cone/journals/resource/111076098244006