Interference effects in low-dimensional disordered metals are universal and determined by the general symmetry and topology properties of the system. Full symmetry classification of disordered systems includes three families of symmetry classes: conventional (Wigner-Dyson), chiral, and superconducting (Bogoliubov-de Gennes). We consider two-dimensional (2D) models of chiral symmetry implying that the Hamiltonian can be arranged in the form of a block off-diagonal matrix. A standard realization of such a system is provided by a bipartite lattice with random hopping. In contrast to conventional Wigner-Dyson classes, chiral systems exhibit very unusual localization properties. A remarkable feature of the chiral metal is the exact absence of localization corrections to all orders in the perturbation theory. At the same time, the density of states is strongly modified by the quantum interference effects and diverges at the center of the band. As was shown in Ref. , localization effects do emerge in chiral models when the theory is treated non-perturbatively. Specifically, the localization is controlled by topological vortex-like excitations of the sigma model, in similarity with the Berezinskii-Kosterlitz-Thouless phase transition.
An important realization of a 2D chiral model is given by disordered graphene provided the disorder is predominantly due to vacancies or chemical adsorbents, such as hydrogen, attached to individual graphene atoms. Bipartite lattices with randomly located vacancies constitute a very peculiar realization of chiral metals. We have shown that vacancies crucially modify interference effects close to the center of the band (Dirac point in case of graphene) leading to enhanced DOS and reduced localization length as compared to other realizations of chiral systems. These modifications are intimately related to zero modes arising in bipartite systems with unequal number of sites in the two sublattices. We have developed the non-linear sigma model formalism for chiral systems with vacancies and demonstrated how the zero modes affect localization phenomena.
On the quasiclassical level, disorder effects can be taken into account with the help of self-consistent T-matrix approximation. It amounts to solving a pair of self-consistency equations for the two components of the self energy: ∑A,B = –nA,BgA,B(0). Here nA,B are the average concentrations of vacancies in two sublattices and g(0) is the Green function at coincident points. The resulting density of states is shown in Fig. 1. Different curves correspond to different values of the sublattice imbalance nA – nB. The density of states develops a hard gap when the vacancies are imbalanced and a delta peak at zero energy appears due to finite concentration of zero modes.
Beyond quasiclassics, one has to take into account local fluctuations of the impurity densities as well as quantum interference effects. Both effects can be captured within the suitable non-linear sigma model. We have developed such a description and identified an additional terms in the sigma model action that appears only due to vacancies. In total, the sigma model action involves four terms with the coupling constants σ (dimensionless conductivity; for graphene, it is of order unity), c (Gade coupling), E (electron energy), and n (total average concentration of vacancies). Classical and quantum fluctuations are accounted for by perturbative renormalization of the sigma model. The conductivity σ and the concentration n stay constant under renormalization. The other two couplings flow as dc/d ln L = 1 and d ln E/d ln L = (c + 4πnL2)/σ2. The renormalization flow stops at the critical length scale Lc ~ (σ/E)1/2.
In the absence of vacancies, n = 0, the correlation length very weakly diverges with lowering energy, Lc ~ exp(σ | ln E |1/2) and the resulting density of states is ρ ~ 1/ELc2. With vacancies, renormalization of energy dramatically accelerates leading to a shorter correlation length Lc ~ σn–1/2 |ln E|1/2 and a stronger singularity in the density of states ρ ~ σ2/E √n Lc3 ~ n/σ E |ln E|3/2. These findings are supported by a large-scale numerical simulations  of disordered graphene. As shown in Fig. 2, the energy scaling of the density of states and correlation length are very well fitted by our analytical results.
To summarize, we have demonstrated that randomly distributed vacancies in a two-dimensional chiral metal strongly modify its critical properties at zero energy. They reduce the correlation length and increase the density of states as compared to the conventional chiral metals. Technically, this modification occurs due to an additional term in the sigma-model action. This term is responsible for local fluctuations of sublattice imbalance and emergent zero modes.