  We introduce a disorder-free model of $S=1/2$ spins on the square lattice in
a constrained Hilbert space where two up-spins are not allowed simultaneously
on any two neighboring sites of the lattice. The interactions are given by
ring-exchange terms on elementary plaquettes that conserve both the total
magnetization as well as dipole moment. We show that this model provides a
tractable example of strong Hilbert space fragmentation in two dimensions with
typical initial states evading thermalization with respect to the full Hilbert
space. Given any product state, the system can be decomposed into disjoint
spatial regions made of edge and/or vertex sharing plaquettes that we dub as
``quantum drums''. These quantum drums come in many shapes and sizes and
specifying the plaquettes that belong to a drum fixes its spectrum. The spectra
of some small drums is calculated analytically. We study two bigger
quasi-one-dimensional drums numerically, dubbed ``wire'' and a ``junction of
two wires'' respectively. We find that these possess a chaotic spectrum but
also support distinct families of quantum many-body scars that cause periodic
revivals from different initial states. The wire is shown to be equivalent to
the one-dimensional PXP chain with open boundaries, a paradigmatic model for
quantum many-body scarring; while the junction of two wires represents a
distinct constrained model.

Statistical Mechanics