  We study a classical model of fully-packed loops on the square lattice, which
interact through attractive loop segment interactions between opposite sides of
plaquettes. This study is motivated by effective models of interacting quantum
matter arising in frustrated magnets or Rydberg atom arrays, for which loop
degrees of freedom appear at low energy. Through a combination of Monte Carlo
simulations and an effective height field theory, we find that the critical
point known to occur at infinite temperature gives rise to a high-temperature
critical phase with floating exponents. At lower temperature, the system
transitions via a Kosterlitz-Thouless phase transition to a nematic phase where
lattice rotation symmetry is broken. We discuss consequences for the phase
diagram of the quantum loop model on the same lattice.
