Zero Modes, Weyl Symmetries, and QBL Design

We present a set of new results related to general QBLs with an eye towards the search for non-interacting bosonic SPT physics. First, we describe the breakdown of Noether’s theorem in QBLs within the space of linear forms. We define ZMs of QBLs and, additionally, Weyl symmetry generators (SGs), which are generators of displacements in phase space that leave the overall dynamics invariant. Despite no obvious connection between these two operators in an open context, we prove a one-to-one canonical correspondence between them: To each zero mode, there must exists a canonically conjugate Weyl SG, and vice versa. We further generalize this result to accommodate approximate zero modes and generators of approximate symmetries. We proceed to discuss two novel design protocols for QBLs with certain desirable properties. The first takes, as an input, a quadratic fermionic Hamiltonian (QFH) with edge-localized, Hermitian, (possibly approximate) zero modes and provides, as an output, purely dissipative QBL (i.e., a QBL whose system Hamiltonian vanishes) possessing bosonic analogues of these (possibly approximate) zero modes. The second takes, as an input, a dynamically stable QBH and utilizes the duality of Chap. 4 to engineer a QBL whose unique steady state is given by the corresponding quasiparticle vacuum.