FINITE-SIZE SCALING OF THE CORRELATION LENGTH IN ANISOTROPIC SYSTEMS

2007 ◽  
Vol 21 (23n24) ◽  
pp. 4212-4218 ◽  
Author(s):  
X. S. CHEN ◽  
H. Y. ZHANG
Keyword(s):  
Correlation Length ◽  
Periodic Boundary ◽  
Scaling Function ◽  
Finite Size ◽  
Periodic Boundary Conditions ◽  
Scaling Functions ◽  
Finite Size Scaling ◽  
Spatial Anisotropy ◽  
Size Scaling ◽  
Anisotropic Systems

The finite-size scaling functions of thermodynamic functions in anisotropic systems have been shown to be dependent on the spatial anisotropy [X.S. Chen and V. Dohm, Phys. Rev. E 70, 056136 (2004)]. Here we extend this study to the correlation length ξ‖ of the anisotropic O (n) symmetric φ4 model in an Ld−1 × ∞ cylindric geometry with periodic boundary conditions. We calculate the exact finite-size scaling function of correlation length ξ‖ for T ≥ Tc in 2 < d < 4 dimensions and in the limit n → ∞. The finite-size scaling function of ξ‖ is dependent on a normalized symmetric (d − 1) × (d − 1) matrix defined by the anisotropy matrix of anisotropic systems.

scholarly journals Continuum limits and exact finite-size-scaling functions for one-dimensionalO(N)-invariant spin models

1997 ◽  
Vol 86 (3-4) ◽  
pp. 581-673 ◽  
Author(s):  
Attilio Cucchieri ◽  
Tereza Mendes ◽  
Andrea Pelissetto ◽  
Alan D. Sokal
Keyword(s):  
Finite Size ◽  
Scaling Functions ◽  
Finite Size Scaling ◽  
Spin Models ◽  
Size Scaling

scholarly journals Lattice φ4 Theory of Finite-Size Effects Above the Upper Critical Dimension

1998 ◽  
Vol 09 (07) ◽  
pp. 1073-1105 ◽  
Author(s):  
X. S. Chen ◽  
V. Dohm
Keyword(s):  
Field Theory ◽  
Lattice Model ◽  
Size Effects ◽  
Lattice Theory ◽  
Finite Size ◽  
Model Parameters ◽  
Scaling Functions ◽  
Finite Size Scaling ◽  
Finite Size Effects ◽  
Size Scaling

We present a perturbative calculation of finite-size effects near Tc of the φ4 lattice model in a d-dimensional cubic geometry of size L with periodic boundary conditions for d>4. The structural differences between the φ4 lattice theory and the φ4 field 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 finite ξ/L where ξ is the bulk correlation length. At Tc, the large-L behavior becomes lowest-mode like for the lattice model but not for the field-theoretic model. Characteristic temperatures close to Tc of the lattice model, such as T max (L) of the maximum of the susceptibility χ, are found to scale asymptotically as Tc-T max (L) ~L-d/2, in agreement with previous Monte Carlo (MC) data for the five-dimensional Ising model. We also predict χ max ~Ld/2 asymptotically. On a quantitative level, the asymptotic amplitudes of this large-L behavior close to Tc have not been observed in previous MC simulations at d=5 because of nonnegligible finite-size terms ~L(4-d)/2 caused 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 the L(4-d)/2 and L4-d terms predicted by our theory.

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