Lee-Yang theory, high cumulants, and large-deviation statistics of the magnetization in the Ising model

We investigate the Ising model in one, two, and three dimensions using a cumulant method that allows us to determine the Lee-Yang zeros from the magnetization fluctuations in small lattices. By doing so with increasing system size, we are able to determine the convergence point of the Lee-Yang zeros in the thermodynamic limit and thereby predict the occurrence of a phase transition. The cumulant method is attractive from an experimental point of view since it uses fluctuations of measurable quantities, such as the magnetization in a spin lattice, and it can be applied to a variety of equilibrium and nonequilibrium problems. We show that the Lee-Yang zeros encode important information about the rare fluctuations of the magnetization. Specifically, by using a simple ansatz for the free energy, we express the large-deviation function of the magnetization in terms of Lee-Yang zeros. This result may hold for many systems that exhibit a first-order phase transition.

Figure 1

The Ising model, Lee-Yang zeros, and the large-deviation statistics of the magnetization. (a, b) The two-dimensional Ising model on a torus with $N=L^d$ spins for $d=2$ and $L=100$. Red (blue) spins point up (down). The temperature is above the critical temperature, $\beta =0.9{\beta }_c$, in the left panel and below it, $\beta =1.1{\beta }_c$, in the right panel. No magnetic field is applied, $h=0$. (c) Lee-Yang zeros in the complex magnetic-field plane extracted from the high magnetization cumulants in Ising lattices of linear size $L=5,...,9$, here at the inverse temperature $\beta =0.9{\beta }_c$. The red points show the convergence points in the thermodynamic limit. (d) Large-deviation statistics of the magnetization per site, $m=M/N$, for large lattices, $N=L^2\gg 1$. The red curve is numerically exact, while the blue curve is obtained by inserting the convergence points from the left panel into the ansatz (58) with $m_0\simeq -1$.

Figure 2

High cumulants of the magnetization and extracted Lee-Yang zeros for the Ising chain. (a) High magnetization cumulants as a function of the system size $L$ at the inverse temperature $\beta \mathcal{J}=0.6$ and zero magnetic field. The odd cumulants vanish. The lines are guides to the eye. (b) Imaginary part of the Lee-Yang zeros extracted from the cumulants at two different temperatures. The extracted Lee-Yang zeros are shown with filled blue circles, while exact results are indicated with open red circles. The lines are guides to the eye. (c) Determination of the convergence points in the thermodynamic limit.

Figure 3

Lee-Yang zeros for the Ising square lattice in two dimensions. (a) Imaginary part of the Lee-Yang zeros, shown with blue points, extracted from the magnetization cumulants with increasing system size for three different temperatures, where ${\beta }_c=ln(1+\sqrt{2})/2\mathcal{J}\simeq 0.44/\mathcal{J}$ is the critical inverse temperature. For comparison, we show with red circles numerically exact results for the zeros. (b) Extraction of the ratio of critical exponents, $\mathcal{B}/\nu$, based on Eq. (37). For the Ising lattice, it is known that $\mathcal{B}=1/8$ and $\nu =1$. (c) Extrapolation of the Lee-Yang zeros in the thermodynamic limit above, $\beta =0.8{\beta }_c$, and below, $\beta =1.2{\beta }_c$, the critical temperature. At low temperatures, the Lee-Yang zeros reach the real axis, signaling a phase transition.

Figure 4

Lee-Yang zeros for the Ising lattice in three dimensions. (a) Imaginary part of the Lee-Yang zeros extracted from the magnetization cumulants at three different temperatures, where ${\beta }_c\simeq 0.22165/J$ is the critical inverse temperature. The cumulants were obtained by averaging over five sets of Monte Carlo simulations, each with $5\times {10}^5$ measurements. The red bands indicate the standard error over the five sets. (b) Extraction of the ratio of critical exponents, $\mathcal{B}/\nu$, based on Eq. (37). The extracted value is close to the best known value of $\mathcal{B}/\nu \simeq 0.5181$, with the uncertainty given by the standard error over the five sets. (c) Extrapolation of the Lee-Yang zeros in the thermodynamic limit above and at the critical temperature.

Figure 5

Lee-Yang zeros for the Ising lattice in three dimensions. (a) Imaginary part of the Lee-Yang zeros extracted from the magnetization cumulants at three different temperatures, where ${\beta }_c\simeq 0.22165/J$ is the critical inverse temperature. The cumulants were obtained by averaging over 15 sets of Monte Carlo simulations, each with $2\times {10}^7$ measurements. The red bands indicate the standard error over the 15 sets. (b) Extraction of the ratio of critical exponents, $\mathcal{B}/\nu$, based on Eq. (37). The extracted value is close to the best known value of $\mathcal{B}/\nu \simeq 0.5181$, with the uncertainty given by the standard error over the 15 sets. (c) Extrapolation of the Lee-Yang zeros in the thermodynamic limit above and at the critical temperature.

Figure 6

Large-deviation statistics of the magnetization for the Ising chain. (a) Large-deviation statistics at a finite temperature in the absence of a magnetific field. The solid line corresponds to Eq. (61), while the dashed line is an ellipse whose upper part is given by Eq. (58) with the Lee-Yang zeros inserted and $m_0=1$. (b) Large-deviation statistics as in the left panel, but with an applied magnetic field, which tilts the ellipse. (c) At very low temperatures, the ellipse collapses to a nearly straight line.

Figure 7

Large-deviation statistics for the two-dimensional Ising lattice. Numerically exact results are shown in red, while the blue lines are the approximation (58) with the convergence points of the Lee-Yang zeros inserted, having used $m_0=-\left(1+\mathrm{Im}[h_c/J]\right)$, which provides a good approximation. For $\beta =0.8{\beta }_c$, the convergence point is $\mathrm{Im}[h_c/J]=0.0865\left(9\right)$ and $m_0\approx -1.0865$, while for $\beta =0.9{\beta }_c$, we have $\mathrm{Im}[h_c/J]=0.0209\left(5\right)$ and $m_0\approx -1.0209$. In the last two columns, we have $\mathrm{Im}[h_c/J]=0$.

Figure 8

Large-deviation statistics for the three-dimensional Ising lattice. Results based on Monte Carlo simulations are shown in red, while the blue lines are the approximation (58) with the convergence points of the Lee-Yang zeros inserted. The width of the red markers indicate the bin size. In the first three columns, we have used the linear size $L=8$, and in the fourth one $L=10$. For the first two columns, the Lee-Yang zeros converge to the complex points $\mathrm{Im}[h_c/J]=0.0876\left(4\right)$ and $\mathrm{Im}[h_c/J]=0.0275\left(4\right)$, respectively, and we have used $m_0=-0.94$ and $m_0=-0.84$ for the fitting. In the last two columns, the Lee-Yang zero converge to $h_c=0$. For the Monte Carlo simulations, we have used $2\times {10}^7$ measurements for each panel.