Critical exponents and equation of state of the three-dimensional Heisenberg universality class

TL;DR

This work delivers a high-precision characterization of the three-dimensional Heisenberg universality class by combining an improved lattice $\phi^4$ model with Monte Carlo finite-size scaling and high-temperature expansions. The authors report refined critical exponents $\gamma=1.3960(9)$, $\nu=0.7112(5)$, $\eta=0.0375(3)$, and derive $\alpha=-0.1336(15)$, $\beta=0.3689(3)$, $\delta=4.783(3)$, alongside a robust equation of state valid throughout the critical region via small-magnetization HT expansions and parametric representations. They compute universal amplitude ratios and demonstrate good agreement with experimental data, thereby strengthening the predictive power of universal scaling for $O(3)$ systems. The methodology—improved Hamiltonians combined with MC and HT analyses—offers a template for precision studies of critical behavior in other universality classes and experimental contexts.

Abstract

We improve the theoretical estimates of the critical exponents for the three-dimensional Heisenberg universality class. We find gamma=1.3960(9), nu=0.7112(5), eta=0.0375(5), alpha=-0.1336(15), beta=0.3689(3), and delta=4.783(3). We consider an improved lattice phi^4 Hamiltonian with suppressed leading scaling corrections. Our results are obtained by combining Monte Carlo simulations based on finite-size scaling methods and high-temperature expansions. The critical exponents are computed from high-temperature expansions specialized to the phi^4 improved model. By the same technique we determine the coefficients of the small-magnetization expansion of the equation of state. This expansion is extended analytically by means of approximate parametric representations, obtaining the equation of state in the whole critical region. We also determine a number of universal amplitude ratios.

Figures (5)

• Figure 1: The scaling function $F(z)$ versus $z$.
• Figure 2: The scaling function $f(x)$ versus $x$.
• Figure 3: The scaling function $C(u)$, cf. Eqs. (\ref{['Cwb']}) and (\ref{['Cw']}).
• Figure 4: Plot of $m^2/B^2$ versus $C^+ h/m$.
• Figure 5: The scaling function $C(u)$ versus the experimental scaling variable ${u}_{\rm exp} \equiv t H^{-1/(\beta\delta)}$.