Bose-Einstein Condensation and the Lambda Transition for Interacting Lennard-Jones Helium-4

TL;DR

This work advances a phase-space, permutation-based view of the $\lambda$-transition by applying quantum loop Monte Carlo to interacting Lennard-Jones $^4$He. It formalizes a binary, entropy-driven condensation mechanism in which condensed bosons occupy multiple low-lying momentum states and compete with position permutation loops for available entropy, analyzed via a constrained free-energy minimization of $F(N_0|N,V,T)$. A key insight is that the heat capacity divergence on the near side arises from rapid growth of pure position loops, and that condensation and superfluidity re-emerge at the peak through mixed position chains linking condensed heads to uncondensed tails, i.e., a Lazarus-like transition. While acknowledging model limitations (e.g., neglected commutation function and LJ simplifications), the study provides a cohesive molecular-level mechanism linking occupation entropy, loop growth, and superfluid transport in the $\\lambda$-transition.

Abstract

An introduction to Bose-Einstein condensation and the $λ$-transition is given. Results of quantum loop Monte Carlo simulations are presented for interacting Lennard-Jones helium-4. The optimum condensation fraction is found by minimizing the constrained free energy. The results show that approaching the transition the growth of pure position permutation loops and the consequent divergence of the heat capacity are enabled by the suppression of condensation and consequently of superfluidity. Condensation and superfluidity emerge at the peak of the heat capacity due to mixed position permutation chains.

Figures (6)

• Figure 1: Most likely fraction of excluded Lennard-Jones $^4$He atoms, Eq. (\ref{['Eq:Free1']}). The circles are from homogeneous simulations and the triangles from simulations of a droplet. The dotted line and the lines connecting symbols are eye guides. Inset. The most likely fraction of Lennard-Jones $^4$He atoms in momentum states with $\overline N_{\bf a} > 1$.
• Figure 2: The specific heat capacity of Lennard-Jones $^4$He. The circles are from homogeneous simulations and the triangles from simulations of a droplet. Using Eq. (\ref{['Eq:Free1']}), the solid symbols have $\overline N_0$ atoms excluded from the position permutation loops, whereas the open symbols have all atoms included in the loops. The lines connecting symbols are eye guides. The error bars give the 95% confidence level. Inset. Magnification at higher temperatures.
• Figure 3: Most likely fraction of low-lying Lennard-Jones $^4$He atoms. Using Eq. (\ref{['Eq:Free2']}) for pure loops, the circles are from homogeneous simulations and the triangles are from simulations of a droplet. Using Eq. (\ref{['Eq:Free3']}) including pure loops and mixed chains, the plus symbols are from homogeneous simulations and the times symbols are from simulations of a droplet. The lines connecting symbols are eye guides.
• Figure 4: The specific heat capacity of Lennard-Jones $^4$He. The circles are from homogeneous simulations and the triangles from simulations of a droplet. Using Eq. (\ref{['Eq:Free2']}), the solid symbols have $\overline N_{0}'$ atoms excluded from the position permutation loops, whereas the open symbols have all atoms included in the loops. The lines connecting symbols are eye guides. The error bars give the 95% confidence level. Inset. Magnification at higher temperatures.
• Figure 5: Most likely fraction of condensed Lennard-Jones $^4$He atoms. The circles are from a homogeneous liquid (periodic boundary conditions) and the triangles are from simulations of a droplet. The empty symbols are for position loops and chains, and the filled symbols are for position loops only. The algebraic symbols are from Eq. (\ref{['Eq:Ftot3']}). The dotted line and the lines connecting the symbols are eye guides. Note that $\varepsilon_{\rm He}/k_{\rm B} = 10.22$ K.