Continuum Limits and Exact Finite-Size-Scaling Functions for One-Dimensional $O(N)$-Invariant Spin Models

Abstract

We solve exactly the general one-dimensional $O(N)$-invariant spin model taking values in the sphere $S^{N-1}$, with nearest-neighbor interactions, in finite volume with periodic boundary conditions, by an expansion in hyperspherical harmonics. The possible continuum limits are discussed for a general one-parameter family of interactions, and an infinite number of universality classes is found. For these classes we compute the finite-size-scaling functions and the leading corrections to finite-size scaling. A special two-parameter family of interactions (which includes the mixed isovector/isotensor model) is also treated, and no additional universality classes appear. In the appendices we give new formulae for the Clebsch-Gordan coefficients and 6--$j$ symbols of the $O(N)$ group, and some new generalizations of the Poisson summation formula; these may be of independent interest.

541038 bytes uuencoded gzip'ed (expands to 1301207 bytes Postscript); 88 pages including all figures

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Cited by in corpus (16)

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• Corrections to Finite-Size Scaling in the Lattice N-Vector Model for Infinite N
• Pathologies of the large-N limit for RP^{N-1}, CP^{N-1}, QP^{N-1} and mixed isovector/isotensor sigma-models
• Modeling Finite-Volume Effects and Chiral Symmetry Breaking in Two-Flavor QCD Thermodynamics
• Effective potential in three-dimensional O(N) models
• Critical structure factors of bilinear fields in O(N)-vector models
• On the spectrum of the Faddeev-Popov operator in topological background fields
• Finite-Size Scaling behavior in the O(4)-Model
• Finite-size effects in canonical and grand-canonical quantum Monte Carlo simulations for fermions
• Shape dependence of the finite-size scaling limit in a strongly anisotropic $O(\infty )$ model
• Exact duality transformations for sigma models and gauge theories
• More on coupling coefficients for the most degenerate representations of SO(n)
• New Universality Classes in One--Dimensional $O(N)$--Invariant Spin--Models with an $n$--Parametric Action
• Comparison among HB-inspired algorithms for continuous-spin systems and gauge fields
• Gauged O(n) spin models in one dimension