  Effects of short-range electronic interactions in a three-dimensional
line-node semimetal that supports linearly dispersing quasiparticles around an
isolated loop in the Brillouin zone are discussed. Due to vanishing density of
states ($\varrho(E) \sim |E|$) various orderings in the bulk of the system,
such as the antiferromagnet and charge-density-wave, set in for sufficiently
strong onsite ($U$) and nearest-neighbor ($V$) repulsions, respectively. While
onset of these two orderings from the semimetallic phase takes place through
continuous quantum phase transitions, a first-order transition separates two
ordered phases. By contrast, topologically protected drumhead shaped surface
states can undergo charge or spin orderings, depending on relative strength of
$U$ and $V$, even when they are sufficiently weak. Such surface orderings as
well as weak long-range Coulomb interaction can be conducive to spontaneous
symmetry breaking in the bulk for weaker interactions. We numerically establish
such proximity effect driven spontaneous symmetry breaking in the bulk for
subcritical strength of interactions due to flat surface band and also discuss
possible superconducting orders in this system.

Mesoscale and Nanoscale Physics;Materials Science;Strongly Correlated Electrons;High Energy Physics - Theory