Localization by particle-hole symmetry breaking: a loop expansion

Localization by a broken particle-hole symmetry in a random system of non-interacting quantum particles is studied on a $d$--dimensional lattice. Our approach is based on a chiral symmetry argument and the corresponding invariant measure, where the latter is described by a Grassmann functional integral. Within a loop expansion we find for small loops diffusion in the case of particle-hole symmetry. Breaking the particle-hole symmetry results in the creation of random dimers, which suppress diffusion and lead to localization on the scale $\sqrt{D/|\mu|}$, where $D$ is the effective diffusion coefficient at particle-hole symmetry and $\mu$ is the parameter related to particle-hole symmetry breaking.