[Paper Review] Exploring Lee-Yang and Fisher Zeros in the 2D Ising model through multipoint Padé approximants

Simran Singh, Massimo Cipressi+1 more|arXiv (Cornell University)|Dec 5, 2023

Theoretical and Computational Physics4 citations

TL;DR

This paper introduces a novel application of multi-point Padé approximants to numerically extract Lee-Yang and Fisher zeros in the 2D Ising model using finite-volume Monte Carlo simulations. By analyzing cumulants of the partition function at multiple magnetic field and inverse-temperature points, the method accurately locates zeros and poles in the complex plane, yielding precise estimates of critical exponents: $\beta\delta = 1.879(16)$ and $\nu = 1.014(60)$, with $\beta_c = 0.4405(5)$, confirming the method's robustness for phase transition studies.

ABSTRACT

We present a numerical calculation of the Lee-Yang and Fisher zeros of the 2D Ising model using multi-point 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 multi-point 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 behaviour 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 $βδ$. The computation at the critical temperature is performed after the latter has been determined via the study of Fisher zeros, thus extracting both $β_c$ and the critical exponent $ν$. Results already exist for extracting the critical exponents for the Ising model in 2 and 3 dimensions making use of Fisher and Lee Yang zeros. In this work, multi-point Padé is shown to be competitive with this respect and thus a powerful tool to study phase transitions.

Motivation & Objective

• To develop and validate a method for extracting Lee-Yang and Fisher zeros in finite-volume systems using multi-point Padé approximants.
• To test the robustness of the multi-point Padé method in the presence of statistical noise from Monte Carlo simulations.
• To extract critical exponents $\beta\delta$, $\nu$, and $\beta_c$ from finite-size scaling of zeros in the 2D Ising model.
• To verify the circle theorem for Lee-Yang zeros and confirm the self-consistent determination of $T_c$ via Fisher zero analysis.
• To establish the multi-point Padé method as a competitive alternative to traditional finite-size scaling for critical phenomena in lattice models.

Proposed method

• The method uses multi-point Padé approximants to re-sum leading-order cumulants of the partition function evaluated at multiple values of the external magnetic field $H$ and inverse temperature $\beta$.
• Magnetization is approximated as a rational function of $H$ via multi-point Padé, and its poles are analyzed to locate Lee-Yang zeros in the complex $H$ plane.
• Specific heat is similarly approximated as a rational function of $\beta$ to locate Fisher zeros in the complex $\beta$ plane.
• Finite-size scaling is applied to the positions of the closest Lee-Yang and Fisher zeros to extract critical exponents $\beta\delta$, $\nu$, and $\beta_c$.
• Simulations are performed using a cluster spin-flip algorithm on square lattices of varying size $L$ at $T_c$, with high-statistics data ($\sim 650$K configurations per $H$ or $\beta$) to ensure accuracy.
• Error analysis is performed using $\chi^2/dof$ and confidence intervals to assess the reliability of fits to critical exponent scaling laws.

Experimental results

Research questions

• RQ1 Can multi-point Padé approximants reliably extract Lee-Yang and Fisher zeros from finite-volume Monte Carlo data with statistical noise?
• RQ2 Do the poles of the magnetization and specific heat approximants reproduce the known circle theorem for Lee-Yang zeros in the 2D Ising model?
• RQ3 Can finite-size scaling of the closest Lee-Yang and Fisher zeros yield precise estimates of critical exponents $\beta\delta$, $\nu$, and $\beta_c$?
• RQ4 Is the critical temperature $T_c$ self-consistently determined via Fisher zero analysis, and does it agree with known values?
• RQ5 How does the multi-point Padé method compare to conventional finite-size scaling in terms of precision and robustness for critical phenomena?

Key findings

• The multi-point Padé method successfully identifies genuine Lee-Yang zeros as poles of the magnetization approximant, with all stable poles lying on the imaginary $H$-axis, confirming the circle theorem.
• The finite-size scaling of the imaginary part of the closest Lee-Yang zero yields $\beta\delta = 1.879(16)$, in excellent agreement with the exact value of 1.875.
• The finite-size scaling of Fisher zeros yields $\nu = 1.014(60)$ and $\beta_c = 0.4405(5)$, consistent with exact results and validating the self-consistent determination of $T_c$.
• The method demonstrates robustness under statistical noise, as high-precision results are obtained even with finite statistics, suggesting applicability to more complex systems like QCD.
• The critical exponent $\beta\delta$ is extracted with a relative uncertainty of $\sim 0.8\%$, demonstrating high precision in the finite-size scaling analysis.
• The results confirm that the multi-point Padé method is a competitive and reliable alternative to traditional finite-size scaling for extracting critical exponents from lattice simulations.