Eur. Phys. J. B 5, 529-542 (1998)
https://doi.org/10.1007/s100510050474

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

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

Corresponding author: ^a vdohm@physik.rwth-aachen.de

Received: 20 October 1997
Accepted: 5 March 1998
Published online: 15 October 1998

Abstract

We derive exact results for several thermodynamic quantities of the O(n) symmetric field theory in the limit in a finite d-dimensional hypercubic geometry with periodic boundary conditions. Corresponding results are derived for an O(n) symmetric 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 field theory turn out to be nonuniversal whereas those of the 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.