Description
I will discuss a model for fermions on a two-dimensional square lattice, minimally coupled to a Z2-valued dynamical gauge field, living on the bonds of the lattice. As observed numerically, this system displays a rich phase diagram, depending on the model parameters. In particular, at half-filling, the model at low temperature exhibits a semimetallic phase, in which the low-energy charge excitations are effectively described by 2+1 dimensional massless Dirac fermions. I will discuss the rigorous proof of this fact, building on Lieb’s seminal work about the solution of the pi-flux phase conjecture. In presence of a staggered mass term, the ground state of the gauge theory turns out to be four-fold degenerate, and separated by the rest of the spectrum by a gap. In particular, it supports anyonic excitations, equivalent to the ones of the toric code. The proofs are based on reflection positivity, chessboard estimates, fermionic cluster expansion and Hastings’ quasi-adiabatic flow. Based on collaborations with Sven Bachmann (UBC Vancouver) and Leonardo Goller (SISSA).