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Supplementary Information for “ Emergent Dirac fermions and broken symmetries in confined and deconfined phases of Z 2 gauge theories ”

To relate our work to more familiar condensed matter models, in this section we discuss the gauge invariant physical degrees of freedom of our model. In doing so, it is important to distinguish between Ising gauged charged and neutral operators. We first consider gauged operators, see Table. I. Here, similarly to the pure IGLT, we identify two bosonic degrees of freedom, the ‘m’ and ‘e’ particles and a fermion. The ‘m‘ particle (“vison”) corresponds to an Ising πflux excitation that is created by a non local string operator. The fermion is identified with the operator, cα, which unlike the pure IGLT also carries a U(1) charge and a SU(2) spin. This important difference, in the absence of pairing, allows us to define a Fermi surface and gives rise to the Deconfined Dirac phase. The ‘e‘ particle is then a bound state of the vison and fermion. We now discuss the gauge invariant operators, see Table.II, which are the physical degrees of freedom. This necessarily involves even powers of the fermion operator c, which can carry charge 2 eg. c↑c † ↓ or charge 0 eg. c †c which in turn can either transform as spin 1 or spin 0. In the Ising sector, gauge invariant operators correspond to energy density operators such the Ising magnetic flux or the Ising electric field.