  We study the $q$-state clock models on heptagonal lattices assigned on a
negatively curved surface. We show that the system exhibits three classes of
equilibrium phases; in between ordered and disordered phases, an intermediate
phase characterized by a diverging susceptibility with no magnetic order is
observed at every $q \ge 2$. The persistence of the third phase for all $q$ is
in contrast with the disappearance of the counterpart phase in a planar system
for small $q$, which indicates the significance of nonvanishing surface-volume
ratio that is peculiar in the heptagonal lattice. Analytic arguments based on
Ginzburg-Landau theory and generalized Cayley trees make clear that the
two-stage transition in the present system is attributed to an energy gap of
spin-wave excitations and strong boundary-spin contributions. We further
demonstrate that boundary effects breaks the mean-field character in the bulk
region, which establishes the consistency with results of clock models on
boundary-free hyperbolic lattices.
