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A Finite Size Scaling Study of Lattice Models in the three-dimensional Ising Universality Class

Martin Hasenbusch

TL;DR

The paper targets high-precision critical exponents in the 3D Ising universality class by simulating the spin-1/2 Ising and Blume-Capel models on a cubic lattice and employing finite-size scaling with improved observables to suppress leading corrections. A large-scale Monte Carlo campaign determines β_c and fixed-point couplings across several D values, identifies D^* ≈ 0.656, and extracts ν ≈ 0.63002 and η ≈ 0.03627 with ω ≈ 0.832, achieving improved control of systematic errors. The work demonstrates the power of model and observable improvement, cross-validates against high-temperature series and field-theoretic results, and provides robust benchmarks for theory and experiment. Overall, it underscores the practicality of improved finite-size scaling for precise critical phenomena studies in lattice models.

Abstract

We simulate the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter D on the simple cubic lattice. We perform a finite size scaling study of lattices of a linear size up to L=360 to obtain accurate estimates for critical exponents. We focus on values of D, where the amplitudes of leading corrections are small. Furthermore we employ improved observables that have a small amplitude of the leading correction. We obtain nu=0.63002(10), eta=0.03627(10) and omega=0.832(6). We compare our results with those obtained from previous Monte Carlo simulations and high temperature series expansions of lattice models, by using field theoretic methods and experiments.

A Finite Size Scaling Study of Lattice Models in the three-dimensional Ising Universality Class

TL;DR

The paper targets high-precision critical exponents in the 3D Ising universality class by simulating the spin-1/2 Ising and Blume-Capel models on a cubic lattice and employing finite-size scaling with improved observables to suppress leading corrections. A large-scale Monte Carlo campaign determines β_c and fixed-point couplings across several D values, identifies D^* ≈ 0.656, and extracts ν ≈ 0.63002 and η ≈ 0.03627 with ω ≈ 0.832, achieving improved control of systematic errors. The work demonstrates the power of model and observable improvement, cross-validates against high-temperature series and field-theoretic results, and provides robust benchmarks for theory and experiment. Overall, it underscores the practicality of improved finite-size scaling for precise critical phenomena studies in lattice models.

Abstract

We simulate the spin-1/2 Ising model and the Blume-Capel model at various values of the parameter D on the simple cubic lattice. We perform a finite size scaling study of lattices of a linear size up to L=360 to obtain accurate estimates for critical exponents. We focus on values of D, where the amplitudes of leading corrections are small. Furthermore we employ improved observables that have a small amplitude of the leading correction. We obtain nu=0.63002(10), eta=0.03627(10) and omega=0.832(6). We compare our results with those obtained from previous Monte Carlo simulations and high temperature series expansions of lattice models, by using field theoretic methods and experiments.

Paper Structure

This paper contains 12 sections, 54 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. The model
  3. The observables
  4. The simulation algorithm
  5. The simulations: CPU time and statistics
  6. $\beta_c$ and the fixed point values of phenomenological couplings
  7. The correction exponent $\omega$ and the improved model
  8. Improved Observables
  9. The exponent $\eta$
  10. The critical exponent $\nu$
  11. Summary and Conclusions
  12. Acknowledgements

Figures (5)

  • Figure 1: Results for the critical exponent $\eta$ obtained by fitting the standard and the improved magnetic susceptibility at $Z_a/Z_p=0.5425$ for the Ising model and the Blume-Capel model at $D=1.15$ using the ansatz (\ref{['chiback0']}). $L_{min}$ is the minimal lattice size that is taken into account. In the case of the improved magnetic susceptibility, the results obtained from the two different models fall nicely on top of each other. The dashed lines should only guide the eye.
  • Figure 2: Results for the critical exponent $\eta$ obtained by fitting the improved magnetic susceptibility at $Z_a/Z_p=0.5425$ and at $\xi_{2nd}/L=0.6431$ using the ansatz (\ref{['chisimple']}). Data for the Blume-Capel model at $D=0.641$ and $D=0.655$ are taken into account. $L_{min}$ is the minimal lattice size that is taken into account. The dashed lines should only guide the eye. For a discussion see the text.
  • Figure 3: Results for the critical exponent $\eta$ obtained by fitting the improved magnetic susceptibility at $Z_a/Z_p=0.5425$ and at $\xi_{2nd}/L=0.6431$ using the ansatz (\ref{['chiback']}). Data for the Blume-Capel model at $D=0.641$ and $D=0.655$ are taken into account. $L_{min}$ is the minimal lattice size that is taken into account. The dashed lines should only guide the eye. For a discussion see the text.
  • Figure 4: Results for the critical exponent $\nu$ obtained by fitting improved slopes of various phenomenological couplings at $Z_a/Z_p=0.5425$ as a function of $L_{min}^{-2}$, where $L_{min}$ is the minimal lattice size that is included into the fit. The dashed lines should only guide the eye. For a discussion see the text.
  • Figure 5: Results for the critical exponent $\nu$ obtained by fitting improved slopes of various phenomenological couplings at $Z_a/Z_p=0.5425$ with the ansatz (\ref{['nucorrection']}) as a function of $L_{min}$. In the upper part of the figure the correction exponent is fixed to $\epsilon=1.6$ and in the lower part it is fixed to $\epsilon=2$. The dashed lines should only guide the eye. For a discussion see the text.