  We study effects of competing pairwise higher-order interactions (HOI) with
alternating signs and exponentially decreasing intensity on critical behavior
of the XY model. It is found that critical properties of such a generalized
model can be very different from the standard XY model and can strongly depend
on whether the number of HOI terms is odd or even. Inclusion of any odd number
of HOI terms results in two consecutive phase transitions to distinct
ferromagnetic quasi-long-range order phases. Even number of HOI terms leads to
two phase transitions only if the decay of the HOI intensities is relatively
slow. Then the high-temperature transition to the ferromagnetic phase is
followed by another transition to a peculiar competition-induced canted
ferromagnetic phase. In the limit of an infinite number of HOI terms only one
phase transition is confirmed, and under the conditions of fierce competition
between the even and odd terms the transition temperature can be suppressed
practically to zero.
