The critical exponents of the superfluid transition in He4

TL;DR

This work delivers highly precise estimates of the critical exponents for the three-dimensional XY universality class, benchmarked against the superfluid transition in $^4$He. By combining two improved lattice Hamiltonians (the $\phi^4$ and ddXY models) with extensive MC simulations and long HT expansions (up to 22nd order), the authors isolate and control leading and next-to-leading scaling corrections, achieving robust cross-validation between finite-size scaling and HT analyses. The final results include $\alpha=-0.0151(3)$, $\nu=0.6717(3)$, $\eta=0.0381(3)$, $\gamma=1.3178(2)$, $\delta=4.780(1)$, and $\beta=0.3486(1)$, with $\omega=0.785(20)$ and $\omega_2=1.8(2)$; they also compute universal amplitude ratios and highlight a residual tension with the experimental $\alpha$ in microgravity. The study provides a comprehensive cross-check among MC, HT, and IHT approaches, offering refined benchmarks for the XY universality class and informing ongoing experimental tests.

Abstract

We improve the theoretical estimates of the critical exponents for the three-dimensional XY universality class, which apply to the superfluid transition in He4 along the lambda-line of its phase diagram. We obtain the estimates alpha=-0.0151(3), nu=0.6717(1), eta=0.0381(2), gamma=1.3178(2), beta=0.3486(1), and delta=4.780(1). Our results are obtained by finite-size scaling analyses of high-statistics Monte Carlo simulations up to lattice size L=128 and resummations of 22nd-order high-temperature expansions of two improved models with suppressed leading scaling corrections. We note that our result for the specific-heat exponent alpha disagrees with the most recent experimental estimate alpha=-0.0127(3) at the superfluid transition of He4 in microgravity environment.

Figures (10)

• Figure 1: Summary of our results for the specific-heat exponent $\alpha$. Shorthands are explained in the text. The coloured region corresponds to our final estimate $\alpha=-0.0151(3)$.
• Figure 2: $\bar{U}_4$ (above) and $\bar{R}_\Upsilon$ (below) at fixed $R_Z=0.3202$ for various values of $\lambda$ ($\phi^4$ model), $D$ (ddXY model) and the standard XY model, vs. $L^{-\omega}$ with $\omega=0.785$.
• Figure 3: Determination of $\lambda^*$ and $D^*$: (a) results from fits of $\bar{U}_4$ at $R_Z=0.3202$ to $a+c L^{-\omega}$ with $\omega=0.785$; (b) results from fits of $\bar{U}_4$ at $R_\xi=0.5925$ to $a+c L^{-\omega}$ with $\omega=0.785$; (c) results from fits of $\bar{U}_4$ at $R_Z=0.3202$ to $a+c L^{-\omega}+e L^{-\omega_2}$ with $\omega=0.785$, $\omega_2=1.8$; (d) results from fits of $\bar{U}_4$ at $R_\xi=0.5925$ to $a+c L^{-\omega}+e L^{-\omega_2}$ with $\omega=0.785$, $\omega_2=1.8$. The dashed lines indicate our final estimates.
• Figure 4: Estimates of the improved quantity (\ref{['ttt']}) in the case of the Binder cumulant $U_4$ at $R_Z=0.3202$ and $R_\xi=0.5925$ (above), and of $R_\Upsilon$ at $R_Z=0.3202$ (below), vs. $L^{-2}$.
• Figure 5: Results for the exponent $\nu$ obtained by fits for several values of $L_{\rm min}$: (a) to $a L^{1/\nu}$ and (b) to $a L^{1/\nu}( 1 + e L^{-\omega_2})$ with $\omega_2=1.8$.