Latticeφ4Theory of Finite-Size Effects Above the Upper Critical Dimension

X. S. Chen and V. Dohm
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X. S. Chen: Institut für Theoretische Physik, Technische Hochschule Aachen, D-52056 Aachen, Germany;
V. Dohm: Institut für Theoretische Physik, Technische Hochschule Aachen, D-52056 Aachen, Germany

International Journal of Modern Physics C (IJMPC), 1998, vol. 09, issue 07, 1073-1105

Abstract: We present a perturbative calculation of finite-size effects nearTcof theφ4lattice model in ad-dimensional cubic geometry of sizeLwith periodic boundary conditions ford>4. The structural differences between theφ4lattice theory and theφ4field theory found previously in the spherical limit are shown to exist also for a finite number of components of the order parameter. The two-variable finite-size scaling functions of the field theory are nonuniversal whereas those of the lattice theory are independent of the nonuniversal model parameters. One-loop results for finite-size scaling functions are derived. Their structure disagrees with the single-variable scaling form of the lowest-mode approximation for any finiteξ/Lwhere ξ is the bulk correlation length. AtTc, the large-Lbehavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close toTcof the lattice model, such asTmax(L)of the maximum of the susceptibility χ, are found to scale asymptotically asTc-Tmax(L) ~L-d/2, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predictχmax~Ld/2asymptotically. On a quantitative level, the asymptotic amplitudes of this large-Lbehavior close toTchave not been observed in previous MC simulations atd=5because of nonnegligible finite-size terms~L(4-d)/2caused by the inhomogeneous modes. These terms identify the possible origin of a significant discrepancy between the lowest-mode approximation and previous MC data. MC data of larger systems would be desirable for testing the magnitude of theL(4-d)/2andL4-dterms predicted by our theory.

Keywords: Finite-Size Effects; φ4Theory; Upper Critical Dimension; Five-Dimensional Ising Model; 05.70.Jk; 64.60.-i; 75.40.Mg (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1142/S012918319800100X

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