Emergent statistical mechanics of entanglement in random unitary circuits

We map the dynamics of entanglement in random unitary circuits, with finite onsite Hilbert space dimension q, to an effective classical statistical mechanics, and develop general diagrammatic tools for calculations in random unitary circuits. We demonstrate explicitly the emergence of a "minimal membrane" governing entanglement growth, which in 1+1 dimensions is a directed random walk in spacetime (or a variant thereof). Using the replica trick to handle the logarithm in the definition of the nth Rényi entropy, Sn, we map the calculation of the entanglement after a quench to a problem of interacting random walks. A key role is played by effective classical spins (taking values in a permutation group) which distinguish between different ways of pairing spacetime histories in the replicated system. For the second Rényi entropy, S2, we are able to take the replica limit explicitly. This gives a mapping between entanglement growth and a directed polymer in a random medium at finite temperature (confirming Kardar-Parisi-Zhang scaling for entanglement growth in generic noisy systems). We find that the entanglement growth rate ("speed") vn depends on the Rényi index n, and we calculate v2 and v3 in an expansion in the inverse local Hilbert space dimension, 1/q. These rates are determined by the free energy of a random walk and of a bound state of two random walks, respectively, and include contributions of "energetic" and "entropic" origin. We give a combinatorial interpretation of the Page-like subleading corrections to the entanglement at late times and discuss the dynamics of the entanglement close to and after saturation. We briefly discuss the application of these insights to time-independent Hamiltonian dynamics.