Charge-4e/6e superconductivity and chiral metal from 3D chiral superconductor

Unconventional superconductivity (SC) characterized by multi-fermion orderings has attracted substantial attention. However, previous studies have largely focused on 2D systems or 3D systems with effective 2D symmetries. Here, we investigate the vestigial phases arising from thermal fluctuations of chiral SC in 3D systems governed by the cubic $O_h$ point group. By constructing low-energy effective Hamiltonians via Ginzburg-Landau analysis and conducting Monte Carlo simulations, we systematically investigate the phase fluctuations of chiral orders within the $E_g$ and $T_{2g}/T_{1u}$ irreducible representations (IRRPs). We identify a phase diagram topology different from 2D counterparts, where the multi-phase intersection manifests as a tetracritical point rather than the triple point typically found in 2D systems. We elucidate the evolution of these phases under thermal fluctuations. Our findings reveal that for both $E_g$ and $T_{2g}/T_{1u}$ IRRPs, the primary chiral orders could melt into a chiral metallic phase across specific parameter regimes. Moreover, for the $E_g$ IRRP, phase fluctuation could also induce a charge-$4e$ phase under certain regime, while for the $T_{2g}$ and $T_{1u}$ IRRPs, it leads to a higher-order charge-$6e$ SC state. Our work paves the way for exploring exotic vestigial orders driven by non-trivial 3D crystalline symmetries.