Strong to Weak Symmetry Breaking in Free Fermion Systems

Different correlators have been put forward to probe symmetry breaking patterns that can only occur when going from one mixed state to another; in particular, the Reyni-2 correlator has been proposed to capture an intrinsically mixed state phase transition dubbed strong-to-weak symmetry breaking (SW-SSB). In this work, we calculate the Renyi-2 correlator to explore the potential breaking of strong fermionic parity in two different contexts: (i) translationally invariant (TI) fermionic gaussian states and (ii) free fermion hybrid circuits. In case (i), we exactly calculate the Renyi-2 correlator and find that for any TI fermionic gaussian state, the strong symmetry cannot be broken. In case (ii), we calculate the Renyi-2 correlator using an effective field theory proposed to capture the universal behavior of such circuits and show that the purification transition observed in this system can be thought of as SW-SSB. Additionally, we put exact bounds on the ratio of the Renyi-2 correlator with another proposed SW-SSB measure, the fidelity correlator, for arbitrary fermionic gaussian states. Specifically, we show that this ratio is bounded by order one constants. This shows that fermionic gaussian state SW-SSB in the Reyni-2 sense is also present in the fidelity sense, and vice versa.