Monte Carlo Computation of the Finite-Size Scaling Function: an Alternative Approach

Chin‐Kun Hu, Jau-Ann Chen, Chai‐Yu Lin +1 more

Adrian Patrascioiu, Erhard Seiler

Jae-Kwon Kim, Adauto J. F. de Souza, David Paul Landau

Using single cluster flip Monte Carlo simulations we accurately determine new finite size scaling functions which are expressed only in terms of the variable x=${\ensuremath{\xi}}_{\mathit{L}}$/L, where ${\ensuremath{\xi}}_{\mathit{L}}$ is the correlation length in a finite system of size L. Data for the d=2 and d=3 Ising models, taken at different temperatures and for different size lattices, show excellent data collapse over the entire range of scaling variable for susceptibility and correlation length. From these finite size scaling functions we can estimate critical temperatures and exponents with rather high accuracy even though data are not obtained extremely close to the critical point. The bulk values of the renormalized four-point coupling constant are accurately measured and show strong evidence for hyperscaling. \textcopyright{} 1996 The American Physical Society.

Koo-Chul Lee

We extend the finite-size scaling theory into the complex temperature plane and present an alternative scenario for the second-order phase transition where the second-order derivatives of the partition function diverge at the transition point with the same exponent but different (discontinuous) amplitudes ${\mathit{A}}_{\ifmmode\pm\else\textpm\fi{}}$. The complex scaling partition function ${\mathit{Z}}_{\mathit{s}}$ for the d=3 Ising model on a simple-cubic lattice is calculated using microcanonical Monte Carlo technique and is used to verify the scenario. The method of complex ${\mathit{Z}}_{\mathit{s}}$ can be used as an alternative to the series method for the study of the second-order phase transition.

Francesco Parisen Toldin

In the finite-size scaling analysis of Monte Carlo data, instead of computing the observables at fixed Hamiltonian parameters, one may choose to keep a renormalization-group invariant quantity, also called phenomenological coupling, fixed at a given value. Within this scheme of finite-size scaling, we exploit the statistical covariance between the observables in a Monte Carlo simulation in order to reduce the statistical errors of the quantities involved in the computation of the critical exponents. This method is general and does not require additional computational time. This approach is demonstrated in the Ising model in two and three dimensions, where large gain factors in CPU time are obtained.