  We demonstrate the nontrivial scaling behavior of Ising models defined on (i)
a donut-shaped surface and (ii) a curved surface with a constant negative
curvature. By performing Monte Carlo simulations, we find that the former model
has two distinct critical temperatures at which both the specific heat $C(T)$
and magnetic susceptibility $\chi(T)$ show sharp peaks.The critical exponents
associated with the two critical temperatures are evaluated by the finite-size
scaling analysis; the result reveals that the values of these exponents vary
depending on the temperature range under consideration. In the case of the
latter model, it is found that static and dynamic critical exponents deviate
from those of the Ising model on a flat plane; this is a direct consequence of
the constant negative curvature of the underlying surface.
