  Quantum information theory and strongly correlated electron systems share a
common theme of macroscopic quantum entanglement. In both topological error
correction codes and theories of quantum materials (spin liquid, heavy fermion
and high-$T_c$ systems) entanglement is implemented by means of an emergent
gauge symmetry. Inspired by these connections, we introduce a simple model for
fermions moving in the deconfined phase of a $\mathbb Z_2$ gauge theory, by
coupling Kitaev's toric code to mobile fermions. This permits us to exactly
solve the ground state of this system and map out its phase diagram. Reversing
the sign of the plaquette term in the toric code, permits us to tune the
groundstate between an orthogonal metal and an orthogonal semimetal, in which
gapless quasiparticles survive despite a gap in the spectrum of original
fermions. The small-to-large Fermi surface transition between these two states
occurs in a stepwise fashion with multiple intermediate phases. By using a
novel diagrammatic technique we are able to explore physics beyond the
integrable point, to examine various instabilities of the deconfined phase and
to derive the critical theory at the transition between deconfined and confined
phases. We outline how the fermionic toric code can be implemented as a quantum
circuit thus providing an important link between quantum materials and quantum
information theory.
