Finite-size scaling in the theory above the upper critical dimension |
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Authors: | XS Chen V Dohm |
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Institution: | (1) Institut für Theoretische Physik, Technische Hochschule Aachen, 52056 Aachen, Germany, DE;(2) Institute of Particle Physics, Hua-Zhong Normal University, Wuhan 430079, P.R. China, CN |
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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 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 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.
Received: 20 October 1997 / Accepted: 5 March 1998 |
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Keywords: | PACS 05 70 Jk Critical point phenomena[:AND:] 64 60 i General studies of phase transitions - 75 40 Mg Numerical simulation studies |
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