  We construct non-trivial weak solutions $\theta\in C_t^0C_x^{0-}$ to the
surface quasi-geostrophic (SQG) equations, which have compact support in time
and, thus, violate the conservation of the Hamiltonian. The result is sharp in
view of the fact that such a conservation law holds for all weak solutions in
the class $C_{t,x}^0 \subset L_{t,x}^3$ (Isett-Vicol, 2015) and resolves the
Onsager conjecture for SQG. The construction is achieved by means of a Nash
iteration together with the linear decoupling method recently introduced in
Giri-Radu (2023).
