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The European Physical Journal B
ISSN: 1434-6028 (printed version)
ISSN: 1434-6036 (electronic version)
Abstract Volume 5 Issue 3 (1998) pp 529-542
Finite-size scaling in the $\mathsf{\varphi^4}$ theory above the upper critical dimension
X.S. Chen (1)(2), V. Dohm (1) (a)
(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 Tc 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[:AND:] 64.60.i General studies of phase transitions - 75.40.Mg Numerical simulation studies
(a) email: vdohm@physik.rwth-aachen.de
Article in PDF format (455 KB)
Online publication: October 26, 1998
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