Microscopic derivation of effective field theory for hydrodynamics in U(1)-symmetric Brownian Ciruits

Late times dynamics of a symmetry-preserving Browinan circuit can be described by lowenergy states of effective frustration-free Hamiltonian in a doubled Hilbert space.

For U(1) circuit such states are magnon-like excitations on top of the groundstate manifold. It has been shown that such Brownian circuits exibit Strong-to-Weak symmetry

breaking, and it can be argued that the low-energy states can be identified with excitations of it’s goldstone bosons and the late time behaviour should be described by

the corresponding EFT.

In this work we derive a field theoretical description of a Super-Hamiltonian for a U(1) Brownian circuit. We find that simplest spin-1/2

and related free-fermion models can be described as two independent spin-1/2 chains coupled to a non-dynamical ”projector” field. We show that this system reduces to a Heisenberg

model at late times and describes diffusion of a ”fermionic pair” formed by fermions of ”top” and ”botom” layers of the doubled space. This model supports a ferromagnet -

like symmetry breaking, which we identify with strong-to-weak symmetry breaking. Using this field theoretical description, we investigate Brownian circuits with more general terms and emergence of KMS symmetry. We also discuss generalisation of our findings for models with larger N.

In addition, our work provides an alternative derivation of Non-Linear Sigma Model description of monitored dynamics of free fermions.