  Theoretical studies of quantum phase transitions have suggested critical
points with higher symmetries than those of the underlying Hamiltonian. Here we
demonstrate a surprising emergent symmetry of the coexistence state at a
strongly discontinuous phase transition between two ordered ground states. We
present a quantum Monte Carlo study of a two-dimensional $S=1/2$ quantum magnet
hosting the antiferromagnetic (AFM) and plaquette-singlet solid (PSS) states
recently detected in SrCu$_2$(BO$_3$)$_2$. We observe that the O(3) symmetric
AFM order and the Z$_2$ symmetric PSS order form an O(4) vector at the
transition. The control parameter $g$ (a coupling ratio) rotates the vector
between the AFM and PSS sectors and there are no energy barriers between the
two at the transition point $g_c$. This phenomenon may be observable in
SrCu$_2$(BO$_3$)$_2$.
