Lee-Yang-zero ratio method in three-dimensional Ising model

TL;DR

This work applies the Lee-Yang-zero ratio (LYZR) method to the three-dimensional Ising model to locate the critical point with precision comparable to Binder cumulants, while simultaneously suppressing finite-size scaling violations and nonlinear temperature dependencies. By analyzing ratios of imaginary parts of Lee-Yang zeros and comparing with Binder cumulants, the authors extract universal LYZR values at the CP and demonstrate rapid convergence in system size. They also develop a single-LYZ/intersection approach when the second zero is hard to access, though with larger finite-size corrections. The study further clarifies the impact of irrelevant operators on FSS and provides guidelines for extrapolating CP quantities, establishing LYZR as a broadly applicable tool for CP searches in statistical and quantum field theories, including QCD-like systems.

Abstract

By performing Monte Carlo simulations of the three-dimensional Ising model, we apply the recently proposed Lee-Yang-zero ratio (LYZR) method to determine the location of the critical point in this model. We demonstrate that the LYZR method is as powerful as the conventional Binder-cumulant method in studying the critical point, while the LYZR method has the advantage of suppressing the violation of the finite-size scaling and non-linearity near the critical point. We also achieve a precise determination of the values of the LYZRs at the critical point, which are universal numbers. In addition, we propose an alternative method that uses only a single Lee-Yang zero and show that it is also useful for the search for the critical point.

Figures (8)

• Figure 1: Schmatic behavier of the LYZR $R_{nm}(t,L)$ for $n>m$, as functions of the reduced temperature $t$ for various system sizes $L$. The LYZRs for various $L$ intersect at the CP at $t=0$. The slope at the CP becomes steeper as $L$ increases. In the $L\rightarrow\infty$ limit, $R_{nm}(t,L)$ behaves as a step function shown by the red line, whose value is $(2n-1)/(2m-1)$ and unity for $t<0$ and $t>0$, respectively.
• Figure 2: Left: Imaginary parts of the first and second LYZs, ${\rm Im}\,h_{\rm LY}^{(n)}(T,L)$ with $n=1,2$, for various $L$. The vertical dashed line represents $T_\textrm{c}$ in Eq. \ref{['eq:Tc']}Ferrenberg:2018zst. Right: the same data with the vertical and horizontal axes rescaled according to the scaling relation Eq. \ref{['eq:tildeh_LY']} with $t=(T-T_{\rm c})/T_{\rm c}$.
• Figure 3: LYZRs $R_{n1}(T,L)$ for $n=2,3,4$ and the fourth-order Binder-cumulant $B_4(T,L)$ for various $L$. Statistical errors are indicated by the shaded bands. The circle markers with error bands show the fit results for various smallest system size $L_{\rm min}$ with $\Delta T \times 10^5 = 5$.
• Figure 4: LYZR and fourth-order Binder cumulant at $T=T_{\rm c}$, $R_{21}(T_{\rm c},L)$ and $B_4(T_{\rm c},L)$, as functions of $1/L$. The blue diamonds represent the results at $T=T_\textrm{c}$ evaluated in the intersection analysis with $L_{\rm min}=128$, while the red circles represent those at Eq. \ref{['eq:Tc']}. The horizontal solid lines are the values of $r_{21}$ and $b_{4}$ obtained by the intersection analysis at $L_{\rm min}=128$ together with the statistical errors depicted by the color bands. The horizontal black-dashed line in the lower panel denotes $b_4$ in Eq. \ref{['eq:B4atCP']}. The red solid/dashed lines are the fit result with Eq. \ref{['eq:largeV']} for the data at $L\ge96$, and the yellow symbols are the values of $r_{21}$ and $b_4$ obtained by the $L\to\infty$ extrapolations.
• Figure 5: Schematic picture of a geometric interpretation of Eq. \ref{['eq:C_f']}. The red band indicates the deviation of $T_{\rm c}$ from the true value caused by the linear approximation in Eq. \ref{['eq:Rlinear']}, which is proportional to $C_{R_{nm}}\delta T^2$.