Universality, Lee-Yang Singularities, and Series Expansions

We introduce a new way of reconstructing the equation of state of a thermodynamic system near a second-order critical point from a finite set of Taylor coefficients computed away from the critical point. We focus on the Ising universality class (${\mathbb{Z}}_2$ symmetry) and show that, in the crossover region of the phase diagram, it is possible to efficiently extract the location of the nearest thermodynamic singularity, the Lee-Yang edge singularity, from which one can (i) determine the location of the critical point, (ii) constrain the nonuniversal parameters that maps the equation of state to that of the Ising model in the scaling regime, and (iii) numerically evaluate the equation of state in the vicinity of the critical point. This is done by using a combination of Padé resummation and conformal maps. We explicitly demonstrate these ideas in the celebrated Gross-Neveu model.

Figure 1

The phase diagram of the Gross-Neveu model assuming unbroken translational symmetry. Inset: the mapping between the Ising model parameters, $r$, $h$ and $T$, $\mu$ near the critical point given in Eq. (1).

Figure 2

The poles and zeros of Padé (red, yellow triangles, respectively) and conformal Padé (blue, green circles, respectively) approximants for two different temperatures compared with the exact locations of ${\mu }_{\mathrm{LY}}^2$.

Figure 3

The Lee-Yang singularity trajectory ${\mu }_{\mathrm{LY}}(T)$ reconstructed from conformal Padé with 20 and 10 terms. The vertical line denotes $T_c$.

Figure 4

The susceptibility as a function of $\mu$ for two different temperatures. Vertical lines denote $\mathrm{Re}{\mu }_{\mathrm{LY}}(T)$.