Exploring Lee-Yang and Fisher zeros in the 2D Ising model through multipoint Padé approximants

We present a numerical calculation of the Lee-Yang and Fisher zeros of the 2D Ising model using multipoint Padé approximants. We perform simulations for the 2D Ising model with ferromagnetic couplings both in the absence and in the presence of a magnetic field using a cluster spin-flip algorithm. We show that it is possible to extract genuine signature of Lee-Yang and Fisher zeros of the theory through the poles of magnetization and specific heat, using the multipoint Padé method. We extract the poles of magnetization using Padé approximants and compare their scaling with known results. We verify the circle theorem associated to the well known behavior of Lee-Yang zeros. We present our finite volume scaling analysis of the zeros done at $T=T_c$ for a few lattice sizes, extracting to a good precision the (combination of) critical exponents $\beta \delta$. The computation at the critical temperature is performed after the latter has been determined via the study of Fisher zeros, thus extracting both ${\beta }_c$ and the critical exponent $\nu$. Results already exist for extracting the critical exponents for the Ising model in two and three dimensions making use of Fisher and Lee-Yang zeros. In this work, multipoint Padé is shown to be competitive with this respect and thus a powerful tool to study phase transitions.

Figure 1

Top: average energy (scaled by $1/L$ instead of $1/L^2$ for better visualization) as a function of $\beta$ from simulations. Bottom: specific heat capacity per lattice site calculated from the configurations generated.

Figure 2

Top: average magnetization per site as a function of $H$ from simulations. Bottom: susceptibility per site calculated from the configurations generated.

Figure 3

Average magnetization as a function of the external magnetic field, rational function approximation vs data. Top: $L=15$, 20; bottom: $L=10$, 30 (plotted separately for sake of clarity). The rational approximation shown has the order $[m,n]=[25,25]$.

Figure 4

Derivative of the rational function obtained in Fig. 3 for $L=10$, 15 plotted against the susceptibility data. Note that this is not an interpolation and no data on the susceptibility was used in the construction of the rational function. Since this is a derivative of an $[m,n]=[25,25]$ rational function, the order is $[m,n]=[24,25]$.

Figure 5

Zeros of the numerator (black pentagons) and of the denominator (red crosses) of the rational approximant $R_n^m(H)$ for the magnetization on $L=15$ (top) and $L=30$ (bottom), with $[m,n]=[25,25]$. The pale blue circles are the points used as input for the Padé. Notice that the closest singularity to the real axis gets closer to the real $H$ axis as $L$ gets larger, with real parts being consistent with $\mathrm{Re}(H_0)=0$.

Figure 6

Genuine poles extracted from Fig. 5 along with their error bars, shown for two lattice sizes (top: $L=15$, bottom: $L=30$). We want to highlight that these poles follow the circle theorem. They lie on the imaginary $H$ axis (within errors) and the closest pole moves closer to the real $H$ axis as the lattice size increases. Moreover, the number of poles increases.

Figure 7

Stability of closest pole for $L=10$. The green cloud represents closest poles extracted from varying the Taylor coefficients with noise drawn from a Gaussian distribution. The singularity structure shown, red crosses and black pentagons, is for the central values of the data at $L=10$. This is the result of drawing the coefficients $\sim 700$ times using a rational function of order $[m,n]=[25,25]$.

Figure 8

Stability of closest pole for $L=15$. The dark blue cloud of points represents the closest pole extracted by varying the order of the input Taylor coefficients using only the central values. The orders of the rational approximation used to obtain the blue cloud have been varied between [5, 5] to [25, 25].

Figure 9

Top: the scaling in $1/L$ of $\mathrm{Im}({\beta }_0)$, i.e., the imaginary part of the Fisher zero, detected as the closest singularity of the cumulant to the real axis. The correct critical exponent $\nu =1$ is reproduced with fairly good accuracy. Bottom: once $\nu$ has been extracted from the data, one can fit the value of the critical inverse temperature ${\beta }_c$ given by the intercept $A$ marked by a green star, which is reconstructed to 1% accuracy. The red star marks the intercept of $\mathrm{Im}{\beta }_0$ as $L\to \infty$ which is recovered to be zero.

Figure 10

Finite size scaling of $\mathrm{Im}(H_0)$. To guide the eye, we plot data versus $L^{1/8-2}$, where the correct critical exponents $\beta$ and $\nu$ are taken. The value obtained from fits is $\beta /\nu -d=-\beta \delta =-1.881(93)$, as shown in Table 1, which also gives $\beta =0.119(93)$. The dash-dotted red line indicates that $\mathrm{Re}[H_0]$ is consistent with zero for each lattice size.