An Analytic Equation of State for Ising-like Models

Abstract

Using an Environmentally Friendly Renormalization we derive, from an underlying field theory representation, a formal expression for the equation of state, $y=f(x)$, that exhibits all desired asymptotic and analyticity properties in the three limits $x\to 0$, $x\to \infty$ and $x\to -1$. The only necessary inputs are the Wilson functions ${\gamma }_{\lambda }$, ${\gamma }_{\varphi }$ and ${\gamma }_{{\varphi }^2}$, associated with a renormalization of the transverse vertex functions. These Wilson functions exhibit a crossover between the Wilson-Fisher fixed point and the fixed point that controls the coexistence curve. Restricting to the case N=1, we derive a one-loop equation of state for $2<d<4$ naturally parameterized by a ratio of non-linear scaling fields. For $d=3$ we show that a non-parameterized analytic form can be deduced. Various asymptotic amplitudes are calculated directly from the equation of state in all three asymptotic limits of interest and comparison made with known results. By positing a scaling form for the equation of state inspired by the one-loop result, but adjusted to fit the known values of the critical exponents, we obtain better agreement with known asymptotic amplitudes.

10 pages, 2 figures

References in corpus (8)

• Critical Phenomena and Renormalization-Group Theory
• 3D Ising Model:The Scaling Equation of State
• Precise determination of critical exponents and equation of state by field theory methods
• Critical equation of state from the average action
• Equation of state and Goldstone-mode effects of the three-dimensional O(2) model
• Environmentally Friendly Renormalization
• The critical equation of state of three-dimensional XY systems
• Monte Carlo computation of the effective potential for the three dimensional Ising system

Cited by in corpus (5)

• The free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model
• Evidence for divergent baryon-number susceptibility in QCD matter
• Effective potential of the three-dimensional Ising model: the pseudo-$\epsilon$ expansion study
• An ab initio Calculation of the Universal Equation of State for the O(N) Model
• Two-Loop Crossover Scaling Functions of the O(N) Model