Statistical mechanics model for Clifford random tensor networks and monitored quantum circuits

We study entanglement transitions in Clifford (stabilizer) random tensor networks (RTNs) and monitored quantum circuits, by introducing an exact mapping onto a (replica) statistical mechanics model. For RTNs and monitored quantum circuits with random Haar unitary gates, entanglement properties can be computed using statistical mechanics models whose fundamental degrees of freedom ("spins") are permutations, because all operators commuting with the action of the unitaries on a tensor product Hilbert space are (linear combinations of) permutations of the tensor factors ("Schur-Weyl duality"). When the unitary gates are restricted to the smaller group of Clifford unitaries, the set of all operators commuting with this action, called the commutant, will be larger, and no longer form a group. We use the recent full characterization of this commutant by Gross et al. [Commun. Math. Phys. 385, 1325 (2021)] to construct statistical mechanics models for both Clifford RTNs and monitored quantum circuits, for on-site Hilbert-space dimensions which are powers of a prime number $p$. The elements of the commutant form the spin degrees of freedom of these statistical mechanics models, and we show that the Boltzmann weights are invariant under a symmetry group involving orthogonal matrices with entries in the finite number field $F_p$ ("Galois field") with $p$ elements. This implies that the symmetry group and consequently all universal properties of entanglement transitions in Clifford circuits and RTNs will, respectively, in general depend on and only on the prime $p$. We show that Clifford monitored circuits with on-site Hilbert-space dimension $d=p^M$ are described by percolation in the limits $d\to \infty$ at (a) $p=$ fixed but $M\to \infty$, and at (b) $M=1$ but $p\to \infty$. In the limit (a) we calculate the effective central charge, and in the limit (b) we derive the following universal minimal cut entanglement entropy $S_A=(\sqrt{3}/\pi )lnpln{L}_A$ for $d=p$ large at the transition. We verify those predictions numerically, and present extensive numerical results for critical exponents at the transition in monitored Clifford circuits for prime number on-site Hilbert-space dimension $d=p$ for a variety of different values of $p$, finding that projective and forced measurement schemes yield the same critical exponents and that they approach percolation values at large $p$. We clearly establish multifractal scaling of the purity, reflected in a continuous spectrum of critical exponents, while the typical exponent is the prefactor of the logarithm in the entanglement entropy. As a technical result, we generalize the notion of the Weingarten function, previously known for averages involving the Haar measure, to averages over the Clifford group.