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Finite-size scaling analysis of the Φ4 field theory on the square lattice

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Abstract

Monte-Carlo calculations are performed for the model Hamiltonian ℋ = ∑i[(r/2)Φ 2(i)+(u/4)/gF4(i)]+∑<ij> (C/2)[Φ (i)−Φ(j)]2 for various values of the parametersr, u, C in the crossover region from the Ising limit (r→-∞,u+∞) to the displacive limit (r=0). The variableφ(i) is a scalar continuous spin variable which can lie in the range-∞<φ(i)<+∞, for each lattice site (i).φ(i) is a priori selected proportional to the single-site probability in our Monte Carlo algorithm. The critical line is obtained in very good agreement with other previous approaches. A decrease of apparent critical exponents, deduced from a finite-size scaling analysis, is attributed to a crossover toward mean-field values at the displacive limit. The relation of this model to the coarse-grained Landau-Ginzburg-Wilson free-energy functional of Ising models is discussed in detail, and, by matching local moments 〈Φ 2(i)〉, 〈Φ 4(i)〉 to corresponding averages of subsystem blocks of Ising systems with linear dimensionsl=5 tol=15, an explicit construction of this coarse-grained free energy is attempted; self-consistency checks applied to this matching procedure show qualitatively reasonable behavior, but quantitative difficulties remain, indicating that higher-order terms are needed for a quantitatively satisfactory description.

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Author notes

  1. A. Milchev (Alexander-von-Humboldt fellow)

    Present address: Institute for Physical Chemistry, Academy of Sciences, Sofia, Bulgaria

Authors and Affiliations

  1. Institut für Physik, Johannes-Gutenberg-Universität Mainz, Postfach 3980, D-6500, Mainz, Federal Republic of Germany

    A. Milchev (Alexander-von-Humboldt fellow), D. W. Heermann & K. Binder

Authors
  1. A. Milchev
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  2. D. W. Heermann
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  3. K. Binder
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Milchev, A., Heermann, D.W. & Binder, K. Finite-size scaling analysis of the Φ4 field theory on the square lattice. J Stat Phys 44, 749–784 (1986). https://doi.org/10.1007/BF01011906

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  • DOI: https://doi.org/10.1007/BF01011906

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