Six-$\alpha$ cluster Bose-Einstein condensation and supersolid $^{12}$C($0_2^+)$+$^{12}$C($0_2^+)$ molecular structure in $^{24}$Mg

TL;DR

This paper demonstrates that low-spin six-alpha condensate states in magnesium-24 are well described by a superfluid alpha-cluster model, revealing a dual superfluid-crystalline nature indicative of a supersolid, and connecting to known molecular resonances.

Contribution

The study provides the first rigorous superfluid alpha-cluster model description of six-alpha condensate states in $^{24}$Mg, including the treatment of NG zero modes and the identification of supersolid characteristics.

Findings

Reproduces the roton rotational band with large moment of inertia.

Matches the molecular resonance at 32.5 MeV in $^{12}$C+$^{12}$C scattering.

Shows coexistence of superfluidity and crystallinity as a supersolid signature.

Abstract

We show for the first time that the low-spin ($J \le 4^+$) six-$\alpha$ condensate candidate states in $^{24}$Mg, recently reported by Fujikawa et al. [Phys. Lett. B 848, 138384 (2024)], are well described by the superfluid $\alpha$-cluster model (SCM). This is achieved by a rigorous treatment of the Nambu-Goldstone (NG) zero mode as the order parameter of condensation in the finite six-$\alpha$ system. We find that a roton rotational band with a large moment of inertia is built on the first excited NG $0^+$ state, analogous to the roton bands observed in three-, four-, and five-$\alpha$ condensates in $^{12}$C, $^{16}$O, and $^{20}$Ne, respectively. Remarkably, our calculated roton band reproduces the well-known molecular resonance with a $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) structure ($16^+$) observed at $E_{\rm c.m.} = 32.5$ MeV in inelastic $^{12}$C+$^{12}$C scattering. This result provides a unified description of both the low-spin six-$\alpha$ condensate states and the high-spin $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) molecular resonance. Analysis of the wave functions reveals a large overlap between the SCM states and a geometrical $^{12}$C($0_2^+$)+$^{12}$C($0_2^+$) configuration. This dual nature -the coexistence of superfluidity and crystallinity- identifies these states as a signature of a supersolid.