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arXiv:1707.03791

Essential Singularities in Universal Scaling Functions at the Ising Coexistence Line

Jaron Kent-Dobias, James P. Sethna

Published Jul 12, 2017• 0 citations
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Abstract

Renormalization group ideas and results from critical droplet theory are used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at the abrupt transition. Analogous forms for the magnetization and susceptibility in two dimensions are fit to numeric data and show good agreement, especially when their nonsingular behavior is modified to match existing numeric results.

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Renormalization group ideas and results from critical droplet theory are used to construct a scaling ansatz for the imaginary component of the free energy of an Ising model in its metastable state close to the critical point. The analytic properties of the free energy are used to determine scaling functions for the free energy in the vicinity of the critical point and the abrupt transition line. These functions have essential singularities at the abrupt transition. Analogous forms for the magnetization and susceptibility in two dimensions are fit to numeric data and show good agreement, especially when their nonsingular behavior is modified to match existing numeric results.
The fermionic version of the two-dimensional Ising model in a uniform magnetic field is studied. Its approximate but rather accurate solution was obtained recently in a variational procedure based on the BCS ansatz for the ground state. Here, this solution is continued analytically into the complex H-plane in the vicinity of the origin. In the ferromagnetic phase an essential singularity in the free-energy is observed at the point H=0 of the type predicted by the droplet condensation theory. Simple phenomenological estimates for the radius of the critical droplet and for the rate of decay of metastable states are confirmed by microscopic calculations.
We apply a generalized transfer-matrix method to the d=2 Ising ferromagnet at H\ensuremath{\ne}0 and TTc{\mathit{T}}_{\mathit{c}}Tc​ to obtain complex constrained free energies. Although the method does not explicitly introduce droplets, the imaginary parts of those constrained free energies that correspond to the metastable state are accurately predicted by a field-theoretical droplet model. The free energy of a critical droplet is given by an equilibrium Wulff construction, and we find evidence for Goldstone modes on its surface. These results indicate that our approach provides a nonperturbative numerical continuation of the free energy around the essential singularity at H=0.
Differential renormalization-group equations for the one-particle-irreducible vertex functions are solved analytically to provide information about the scaling properties of the correlation function at small and moderate wave vectors everywhere in the critical region. For Ising-like spin systems, this extends previous work on the anomalous dimension crossover function to nonzero magnetization. The susceptibility-correlation-length relative-amplitude ratio is correctly given to $O(\ensuremath{\epsilon})$, $\ensuremath{\epsilon}=4\ensuremath{-}d$; moreover, a fully nonlinear crossover-amplitude function is obtained. For general nnn-component spins, the longitudinal part of the correlation function, ${\ensuremath{\Gamma}}_{2,L}$ is calculated to $O(\ensuremath{\epsilon})$ in crossover form for small and moderate kkk. The asymptotic singularity ${\ensuremath{\Gamma}}_{2,L}\ensuremath{\sim}{k}^{\ensuremath{\epsilon}}$ is automatically incorporated. The result is exact in the spherical limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$.
After a brief presentation of the two-dimensional Ising model, we study the analytic properties of the scaling function associated with the free energy in the critical domain T → Tc, H → 0. The analysis is based on numerical data obtained through the Truncated Free Fermion Space Approach. We determine the discontinuities across the Yang-Lee and Langer branch cuts. We confirm the standard analyticity assumptions and propose “extended analyticity”; roughly speaking, the latter states that the Yang-Lee branching point is the nearest singularity under Langer's branch cut. We support the extended analyticity by evaluating numerically the associated “extended dispersion relation”.
The conventional renormalization-group analysis of the tricritical behavior of a metamagnet is completed by a description of the first-order transition below the tricritical temperature, and the conditions for the discontinuity in the magnetization are explicitly verified in an Ising spin model. The singularity structure of the crossover scaling function is derived from relations between the scaling fields for the critical and tricritical fixed points. Numerical calculations illustrating various aspects of the theory are given for a square-lattice Ising model in a four-cell cluster approximation.
A droplet theory of Ising systems is constructed, valid near space dimension d=1d=1d=1, and based on a model incorporating only the configurational energy of surface tension. The droplet distribution function is calculated explicitly, with use of renormalization-group methods to control the droplet-shape fluctuation effects essential to its scaling form. Universal quantities are calculated; those (e.g., $\ensuremath{\beta}$) which reflect the droplet concentration have an essential singularity in $\ensuremath{\epsilon}=d\ensuremath{-}1$.