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Physical Review Letters

Three-Dimensional Ising Model Exponents and Scaled Equation of State from Lattice Renormalization Group

K. Subbarao

Published Dec 20, 1976• 28 citations
Publisher

Abstract

I correct the passage from eigenvalues to critical exponents in the recent Letter of Friedman on three-dimensional Ising exponents from the ground state of a two-dimensional model in a transverse field, and extend the approach to calculate the ground-state energy and thereby the scaled equation of state.
DOI
10.1103/physrevlett.37.1712

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The equivalence of the critical behavior of the ddd-dimensional Ising model at a finite temperature to that of a ddd- 1 dimensional system at zero temperature as a function of a transverse magnetic field is used to calculate the critical exponents for the three-dimensional model by applying the Niemeijer-Van Leeuwen renormalization-group method to the two-dimensional problem.
The leading correction-to-scaling exponent ω for the three-dimensional (3D) dilute Ising model is calculated in the framework of the field theoretic renormalization group approach. Both in the minimal subtraction scheme and in the massive field theory (resummed four-loop expansion) excellent agreement with recent Monte Carlo calculations (H. G. Ballesteros et al., Phys. Rev. B 58, 2740 (1998)) is achieved. The expression of ω as a series in a ε\sqrt \varepsilon ε​ expansion up to O(ɛ2) does not permit a reliable estimate for d=3.
An analysis is made of various methods of phenomenological renormalization based on finite-size scaling equations for inverse correlation lengths, the singular part of the free energy density, and their derivatives. The analysis is made using two-dimensional Ising and Potts lattices and the three-dimensional Ising model. Variants of equations for the phenomenological renormalization group are obtained which ensure more rapid convergence than the conventionally used Nightingale phenomenological renormalization scheme. An estimate is obtained for the critical finite-size scaling amplitude of the internal energy in the threedimensional Ising model. It is shown that the two-dimensional Ising and Potts models contain no finite-size corrections to the internal energy so that the positions of the critical points for these models can be determined exactly from solutions for strips of finite width. It is also found that for the two-dimensional Ising model the scaling finite-size equation for the derivative of the inverse correlation length with respect to temperature gives the exact value of the thermal critical exponent. PACS: 05.10.Cc, 64.60.Ak, 05.50.+q, 75.10.Hk 1 1
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