  Two dimensional systems with U(1) symmetry exhibit a peculiar phase, i.e.,
the Berezinskii-Kosterlitz-Thouless (BKT) phase. In particular situations, the
BKT phase exists as an intermediate temperature phase. There have been
scenarios for the phase transitions at the two endpoints of the intermediate
BKT phase, i.e., the phase transition at the low-temperature endpoint is a BKT
transition and that at the high-temperature endpoint is either a BKT transition
or a first-order transition. The present study gives a novel scenario, i.e., a
second-order transition with a new critical universality and a BKT transition.
We found that this new phase transition is realized in spin-crossover systems
on a triangular lattice with an antiferromagnetic short-range interaction. At
the low-temperature transition the elastic interaction plays as a ferromagnetic
infinite-range interaction and encourages the breaking of $Z_2$ symmetry
between high-spin rich and low-spin rich states.
