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Multirange Ising model on the square lattice.

Charles S do Amaral1, Bernardo N B de Lima2, Ronald Dickman3

  • 1Departamento de Matemática, Centro Federal de Educação Tecnológica de Minas Gerais, Av. Amazonas 7675, Belo Horizonte, MG, Brazil.

Physical Review. E
|June 25, 2020
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Summary
This summary is machine-generated.

Adding longer-range interactions to the Ising model on a square lattice (Z^2) causes its critical temperature to approach that of higher-dimensional lattices (Z^4 and Z^6). This finding extends to percolation critical probabilities.

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Area of Science:

  • Statistical Mechanics
  • Condensed Matter Physics
  • Computational Physics

Background:

  • The Ising model is a fundamental model in statistical mechanics used to study magnetism and phase transitions.
  • Understanding the effect of interaction range on critical phenomena is crucial for theoretical and experimental physics.

Purpose of the Study:

  • To investigate how extending interaction ranges in the 2D Ising model influences its critical temperature.
  • To explore the convergence of critical temperatures towards higher-dimensional lattice models.

Main Methods:

  • Numerical simulations were employed to study the Ising model on a square lattice (Z^2).
  • Interactions were introduced between spins at distances 1 and m, and subsequently at distances 1, m, and u.
  • The critical temperature (T_c) was monitored as the interaction range parameter (m) increased.

Main Results:

  • The critical temperature T_c(m) monotonically converged to the critical temperature of the Ising model on Z^4 as m approached infinity.
  • For three interaction ranges (1, m, u), T_c(m,u) converged to the critical temperature of the Ising model on Z^6.
  • Analogous results were observed for percolation critical probabilities (p_c).

Conclusions:

  • Extending interaction ranges in lower-dimensional lattice models can effectively mimic the behavior of higher-dimensional systems.
  • This study provides insights into universality and dimensional crossover in statistical physics models.