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キーワード:
Condensed Matter, Statistical Mechanics, cond-mat.stat-mech,General Relativity and Quantum Cosmology, gr-qc,High Energy Physics - Theory, hep-th
要旨:
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.
