Deformed LDPC codes with spontaneously broken non-invertible duality symmetries

Low-density parity check (LDPC) codes are a well known class of Pauli stabiliser Hamiltonians that furnish fixed-point realisations of nontrivial gapped phases such as symmetry breaking and topologically ordered (including fracton) phases. In this work, we propose symmetry-preserving deformations of these models, in the presence of a transverse field, and identify special points along the deformations with interesting features: (i) the special point is frustration-free, (ii) its ground states include a product state and the code space of the underlying code, and (iii) it remains gapped in the thermodynamic (infinite volume) limit. So the special point realises a first-order transition between (or the coexistence of) the trivial gapped phase and the nontrivial gapped phase associated with the code. In addition, if the original model has a non-invertible duality symmetry, then so does the deformed model. In this case, the duality symmetry is spontaneously broken at the special point, consistent with the associated anomaly. A key step in proving the gap is a coarse-graining/blocking procedure on the Tanner graph of the code that allows us to apply the martingale method successfully. Our model, therefore, provides the first application of the martingale method to a frustration-free model, that is not commuting projector, defined on an arbitrary Tanner graph. We also discuss several familiar examples on Euclidean spatial lattice. Of particular interest is the 2+1d transverse field Ising model: while there is no non-invertible duality symmetry in this case, our results, together with known numerical results, suggest the existence of a tricritical point in the phase diagram.