  Projective measurements in random quantum circuits lead to a rich breadth of
entanglement phases and extend the realm of non-unitary quantum dynamics. Here
we explore the connection between measurement-only quantum circuits in one
spatial dimension and the statistical mechanics of loop models in two
dimensions. While Gaussian Majorana circuits admit a microscopic mapping to
loop models, for non-Gaussian, i.e., generic Clifford, circuits a corresponding
mapping may emerge only on a coarse grained scale. We then focus on a
fundamental symmetry of loop models: the orientability of world lines. We
discuss how orientability enters in the measurement framework, acting as a
separatrix for the universal long-wavelength behavior in a circuit. When
orientability is broken, the circuit falls into the universality class of
closely packed loops with crossings (CPLC) and features a Goldstone phase with
a peculiar, universal $\log^2(L)$-scaling of the entanglement entropy. In turn,
when orientability is preserved, the long-wavelength behavior of the circuit
mimics that of (coupled) two-dimensional Potts models. We demonstrate the
strength of the loop model approach by numerically simulating a variety of
measurement-only Clifford circuits. Upon varying the set of measured operators,
a rich circuit dynamics is observed, ranging from CPLC to the $1$-state Potts
model (percolation), the $2$-state Potts model (Ising) and coupled Potts models
(BKT) universality class. Loop models thus provide a handle to access a large
class of measurement-only circuits and yield a blueprint on how to realize
desired entanglement phases by measurement.
