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arXiv:hep-lat/9307018 · Journal of Physics A Mathematical and General

The Ising model on tetrahedron-like lattices: a finite-size analysis

Oscar Diego, J. González, Jesús Salas

Published May 7, 1994• 22 citations
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Abstract

We study, using dimer and Monte Carlo approaches, the critical properties and finite-size effects of the Ising model on honeycomb lattices folded on the tetrahedron. We show that the main critical exponents are not affected by the presence of conical singularities. The finite-size scaling of the position of the maxima of the specific heat does not match, however, with the scaling of the correlation length, and the thermodynamic limit is attained faster on the spherical surface than in corresponding lattices on the torus.

DOI
10.1088/0305-4470/27/9/013

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We study, using dimer and Monte Carlo approaches, the critical properties and finite-size effects of the Ising model on honeycomb lattices folded on the tetrahedron. We show that the main critical exponents are not affected by the presence of conical singularities. The finite-size scaling of the position of the maxima of the specific heat does not match, however, with the scaling of the correlation length, and the thermodynamic limit is attained faster on the spherical surface than in corresponding lattices on the torus.
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We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical
We perform a Monte Carlo analysis of the Ising model on many three-dimensional lattices. By means of finite-size scaling we obtain the critical points and determine the scaling dimensions. As expected, the critical exponents agree with the three-dimensional Ising universality class for all models. The irrelevant field, as revealed by the correction-to-scaling amplitudes, appears to be relatively large. Combining the Monte Carlo results for the hydrogen peroxide lattice with those for five other three-dimensional lattices, we obtain a set of data covering a wide range of the irrelevant temperature field. This is helpful in the determination of the parameters describing the corrections to scaling. As a consequence, new results are obtained for the universal parameters describing Ising criticality in three dimensions, with reduced error margins in comparison with earlier Monte Carlo analyses. The critical exponents describing the thermodynamic singularities are determined by the temperature renormalization exponent yt = 1.58693(9) and the magnetic renormalization exponent yh = 2.48178(5). The corrections to scaling are governed by the irrelevant exponent y1 = −0.821(5).
Thermal and magnetic effects in a system consisting of thin layers of coupled Ising spins with S=1/2 and S=1 are considered. The specific heat and the correlation length display maxima at two different temperatures. It is discussed in what sense these maxima can be interpreted as a finite-size rounding of a thermodynamic singularity associated with a phase transition. The connection with ordinary, extraordinary, and special surface phase transitions is made. In two dimensions (2D), the surface critical exponents are calculated from conformal invariance. The bulk and surface finite-size scaling of the order-parameter profiles at the transition points is discussed. In 2D, an exact scaling function for the profiles is suggested through conformal invariance arguments for the (extra)ordinary transition.