Finite-size scaling in the ϕ4 theory above the upper critical dimension

Issue		Eur. Phys. J. B Volume 5, Number 3, October_I 1998


Page(s)		529 - 542
DOI		http://dx.doi.org/10.1007/s100510050474

Contents | PDF_file_(435 Kb)

Eur. Phys. J. B 5, 529-542

Finite-size scaling in the $\mathsf{\varphi^4}$ theory above the upper critical dimension

X.S. Chen^1,2 - V. Dohm¹

¹ Institut für Theoretische Physik, Technische Hochschule Aachen, 52056 Aachen, Germany
² Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, P.R. China

Received: 20 October 1997 / Accepted: 5 March 1998

Abstract
We derive exact results for several thermodynamic quantities of the O(n) symmetric $\varphi^4$ field theory in the limit $n \rightarrow \infty$ in a finite d-dimensional hypercubic geometry with periodic boundary conditions. Corresponding results are derived for an O(n) symmetric $\varphi^4$ model on a finite d-dimensional lattice with a finite-range interaction. The leading finite-size effects near T_c of the field-theoretic model are compared with those of the lattice model. For 2 < d < 4, the finite-size scaling functions are verified to be universal. For d > 4, significant lattice effects are found. Finite-size scaling in its usual simple form does not hold for d > 4 but remains valid in a generalized form with two reference lengths. The finite-size scaling functions of the $\varphi^4$ field theory turn out to be nonuniversal whereas those of the $\varphi^4$ lattice model are independent of the nonuniversal model parameters. In particular, the field-theoretic model exhibits finite-size effects whose leading exponents differ from those of the lattice model. The widely accepted lowest-mode approach is shown to fail for both the field-theoretic and the lattice model above four dimensions.

PACS
05.70.Jk Critical point phenomena - 64.60.i General studies of phase transitions -
75.40.Mg Numerical simulation studies

Author for correspondance: vdohm@physik.rwth-aachen.de

Contents | PDF_file_(435 Kb)

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Finite-size scaling in the theory above the upper critical dimension

Finite-size scaling in the $\mathsf{\varphi^4}$ theory above the upper critical dimension