  Low dimensional fermionic quantum systems are exceptionally interesting
because they reveal distinctive physical phenomena, including among others,
topologically protected excitations, edge states, frustration, and
fractionalization. Two-dimensional $^3$He has indeed shown a remarkable variety
of phases including the unusual quantum spin liquid. Our aim was to lower the
dimension of the $^3$He system even more by confining it on a suspended carbon
nanotube. In our measurements the mechanical resonance of the nanotube with
adsorbed sub-monolayer of $^3$He was measured as a function of coverage and
temperature down to 10\;mK. At lowest temperatures and low coverages we have
observed a liquid-gas coexistence which transforms to the famous 1/3
commensurate solid phase at intermediate densities. However, at larger
monolayer densities we have observed a quantum phase transition from 1/3 solid
to a completely new, soft and mobile solid phase. We interpret this mobile
solid phase as a bosonic commensurate crystal consisting of helium dimers with
topologically protected zero-point vacancies which are delocalized at low
temperatures. We thus demonstrate that $^3$He on a nanotube merges both
fermionic and bosonic phenomena, with a quantum phase transition between
fermionic solid 1/3 phase and a newly observed bosonic dimer solid. The
mobility and softness of the bosonic dimer solid are conditioned by
topology-induced vacancies which become delocalized at low temperatures owing
to a large zero-point motion.
