  We examine dense self-gravitating stellar systems dominated by a central
potential, such as nuclear star clusters hosting a central supermassive black
hole. Different dynamical properties of these systems evolve on vastly
different timescales. In particular, the orbital-plane orientations are
typically driven into internal thermodynamic equilibrium by vector resonant
relaxation before the orbital eccentricities or semimajor axes relax. We show
that the statistical mechanics of such systems exhibit a striking resemblance
to liquid crystals, with analogous ordered-nematic and disordered-isotropic
phases. The ordered phase consists of bodies orbiting in a disk in both
directions, with the disk thickness depending on temperature, while the
disordered phase corresponds to a nearly isotropic distribution of the orbit
normals. We show that below a critical value of the total angular momentum, the
system undergoes a first-order phase transition between the ordered and
disordered phases. At the critical point the phase transition becomes
second-order while for higher angular momenta there is a smooth crossover. We
also find metastable equilibria containing two identical disks with mutual
inclinations between $90^{\circ}$ and $180^\circ$.
