  The Mermin-Wagner theorem states that spontaneous continuous symmetry
breaking is prohibited in systems with short-range interactions at spatial
dimension $D\le 2$. For long-range interactions with a power-law form
($1/r^{\alpha}$), the theorem further forbids ferromagnetic or
antiferromagnetic order at finite temperature when $\alpha\ge 2D$. However, the
situation for $\alpha \in (2,4)$ at $D=2$ is not covered by the theorem. To
address this, we conduct large-scale quantum Monte Carlo simulations and field
theoretical analysis. Our findings show spontaneous breaking of $SU(2)$
symmetry in the ferromagnetic Heisenberg model with $1/r^{\alpha}$-form
long-range interactions at $D=2$. We determine critical exponents through
finite-size analysis for $\alpha<3$ (above the upper critical dimension with
Gaussian fixed point) and $3\le\alpha<4$ (below the upper critical dimension
with non-Gaussian fixed point). These results reveal new critical behaviors in
2D long-range Heisenberg models, encouraging further experimental studies of
quantum materials with long-range interactions beyond the Mermin-Wagner
theorem's scope.
