Disordered loops in the two - dimensional antiferromagnetic spin - fermion model

The spin-fermion model has long been used to describe the quantum-critical behavior of 2d electron systems near an antiferromagnetic (AFM) instability. Recently, the standard procedure to integrate out the fermions to obtain an effective action for spin waves has been questioned in the clean case. We show that in the presence of disorder, the single fermion loops display two crossover scales: upon lowering the energy, the singularities of the clean fermionic loops are first cut off, but below a second scale new singularities arise that lead again to marginal scaling. In addition, impurity lines between different fermion loops generate new relevant couplings which dominate at low energies. We outline a non-linear σ model formulation of the single-loop problem, which allows to control the higher singularities and provides an effective model in terms of low-energy diffusive as well as spin modes.