Exploring Lee-Yang and Fisher Zeros in the 2D Ising model through multipoint Pad\'e approximants

We present a numerical calculation of the Lee-Yang and Fisher zeros of the 2D Ising model using multi-point Pad\'{e} 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 Pad\'{e} method. We extract the poles of magnetization using Pad\'{e} 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 $\beta \delta$. 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 2 and 3 dimensions making use of Fisher and Lee Yang zeros. In this work, multi-point Pad\'{e} is shown to be competitive with this respect and thus a powerful tool to study phase transitions.