  The study of low-dimensional quantum systems has proven to be a particularly
fertile field for discovering novel types of quantum matter. When studied
numerically, low-energy states of low-dimensional quantum systems are often
approximated via a tensor-network description. The tensor network's utility in
studying short range correlated states in 1D have been thoroughly investigated,
with numerous examples where the treatment is essentially exact. Yet, despite
the large number of works investigating these networks and their relations to
physical models, examples of exact correspondence between the ground state of a
quantum critical system and an appropriate scale-invariant tensor network have
eluded us so far. Here we show that the features of the quantum-critical
Motzkin model can be faithfully captured by an analytic tensor network that
exactly represents the ground state of the physical Hamiltonian. In particular,
our network offers a two-dimensional representation of this state by a
correspondence between walks and a type of tiling of a square lattice. We
discuss connections to renormalization and holography.
