Zero-modes in the random hopping model

If the number of lattice sites is odd, a quantum particle hopping on a bipartite lattice with random hopping between the two sublattices only is guaranteed to have an eigenstate at zero energy. We show that the localization length of this eigenstate depends strongly on the boundaries of the lattice, and can take values anywhere between the mean free path and infinity. The same dependence on boundary conditions is seen in the conductance of such a lattice if it is connected to electron reservoirs via narrow leads. For any nonzero energy, the dependence on boundary conditions is removed for sufficiently large system sizes.