The Ising model [1] places a spin $s_i \in {+1, -1}$ on every site of a lattice. Neighboring spins interact through the Hamiltonian

$$ H = -J \sum_{\langle i,j \rangle} s_i s_j, $$

summed over nearest-neighbor bonds $\langle i,j \rangle$, with $J = 1$ favoring aligned neighbors (ferromagnetic coupling).

The system is sampled by single-spin-flip Metropolis Monte Carlo [2]. Each attempt picks a random site $i$, computes the energy change a flip would cost,

$$ \Delta E = 2 s_i \sum_{j \in \text{nn}(i)} s_j, $$

and accepts the flip with probability $\min(1, e^{-\beta \Delta E})$, where $\beta = 1/T$. One sweep (one call to step) makes as many such attempts as there are sites.

On the infinite 2D square lattice this model has an exact (Onsager) critical temperature $T_c = 2 / \ln(1 + \sqrt{2}) \approx 2.269$, separating an ordered ferromagnetic phase ($T < T_c$) from a disordered paramagnetic one ($T > T_c$).

This simulation runs on an $L \times L$ toroidal lattice: the edges wrap around, so interactions cross the boundary seamlessly.

P start/stop sampling · R randomize · C set T = Tc

L

T = 1.00 Tc (≈ 2.269)

Magnetization = 0.00 Energy = 0.00

Magnetization

Energy per site

References

  1. E. Ising, Beitrag zur Theorie des Ferromagnetismus, Zeitschrift für Physik 31, 253–258 (1925). doi:10.1007/BF02980577
  2. N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, Equation of State Calculations by Fast Computing Machines, The Journal of Chemical Physics 21, 1087–1092 (1953). doi:10.1063/1.1699114
  3. L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Physical Review 65, 117–149 (1944). doi:10.1103/PhysRev.65.117
  4. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, (Clarendon Press, 1971).
  5. H. E. Stanley, Scaling, universality, and renormalization: Three pillars of modern critical phenomena, Reviews of Modern Physics 71, S358–S366 (1999). doi:10.1103/RevModPhys.71.S358
  6. S. R. A. Salinas, Introduction to Statistical Physics, (Springer, 2001). doi:10.1007/978-1-4757-3508-6
  7. M. E. J. Newman, G. T. Barkema, Monte Carlo Methods in Statistical Physics, (Oxford University Press, 1999). doi:10.1093/oso/9780198517962.001.0001
  8. D. P. Landau, K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, (Cambridge University Press, 2014). doi:10.1017/CBO9781139696463
  9. E. Venites Filho, R. da Silva, J. R. Drugowich de Felício, A Spectral Investigation of Criticality and Crossover Effects in Two and Three Dimensions: Short Timescales with Small Systems in Minute Random Matrices, Entropy (2024). doi:10.3390/e26050395
  10. R. da Silva, H. C. M. Fernandes, E. Venites Filho, S. D. Prado, J. R. Drugowich de Felício, Mean-Field Criticality Explained by Random Matrices Theory, Brazilian Journal of Physics (2023). doi:10.1007/s13538-023-01295-9