Determination of the conformal-field-theory central charge by the Wang-Landau algorithm

We present a simple method to estimate the central charge of the conformal field theory corresponding to a critical point of a two-dimensional lattice model from Monte Carlo simulations. The main idea is to use the Wang-Landau flat-histogram algorithm, which allows us to obtain the free energy of a lattice model on a torus as a function of torus radii. The central charge is calculated with good precision from a free-energy scaling at the critical point. We apply the method to the Ising, tricritical Ising (Blume-Capel), Potts, and site-diluted Ising models, and we also discuss an estimation of the conformal weights.

Figure 1

The free-energy density in the Ising model. The numerically obtained free energy for square lattices with sizes $N\times N$ together with the exact solution are shown. All shown curves visually overlap. The critical temperature is shown by the vertical line.

Figure 2

Thermal dependence of the free energy in the Ising model for square lattices with sizes $N\times N$ near the critical point. The critical temperature is shown by the vertical line.

Figure 3

Estimation of the minimal anomalous dimension, ${\mathrm{\Delta }}_{\text{min}}$, in the Ising model. The fitted dependence is given by Eq. (11).

Figure 4

A finite-size scaling of the free energy $f(M,N)$ as a function of $q^h=e^{-2\pi hM/N}$ (a) and $f\left(N\right)$ as a function of $N^{-2}$ (b) in the Ising model.

Figure 5

Our estimation of the central charge in the site-diluted Ising model as a function of the probability for a lattice site to be magnetic. The probability $p=1$ corresponds to the pure Ising model.

Figure 6

A finite-size scaling of the free energy $f(M,N)$ as a function of $q^h=e^{-2\pi hM/N}$ (a) and $f\left(N\right)$ as a function of $N^{-2}$ (b) in the tricritical Ising model.

Figure 7

A finite-size scaling of the free energy $f(M,N)$ as a function of $q^h=e^{-2\pi hM/N}$ (a) and $f(\infty ,N)$ as a function of $N^{-2}$ (b) in the three-state Potts model.

Figure 8

A finite-size scaling of the free energy $f(M,N)$ as a function of $q^h=e^{-2\pi hM/N}$ (a) and $f\left(N\right)$ as a function of $N^{-2}$ (b) in the four-state Potts model.