  The reduced density matrices (RDMs) of many-body quantum states form a convex
set. The boundary of low dimensional projections of this convex set may exhibit
nontrivial geometry such as ruled surfaces. In this paper, we study the
physical origins of these ruled surfaces for bosonic systems. The emergence of
ruled surfaces was recently proposed as signatures of symmetry-breaking phase.
We show that, apart from being signatures of symmetry-breaking, ruled surfaces
can also be the consequence of gapless quantum systems by demonstrating an
explicit example in terms of a two-mode Ising model. Our analysis was largely
simplified by the quantum de Finetti's theorem---in the limit of large system
size, these RDMs are the convex set of all the symmetric separable states. To
distinguish ruled surfaces originated from gapless systems from those caused by
symmetry-breaking, we propose to use the finite size scaling method for the
corresponding geometry. This method is then applied to the two-mode XY model,
successfully identifying a ruled surface as the consequence of gapless systems.
