We elucidate the reduction of the winding number (WN) caused by the onsite disorder in a higher WN next nearest neighbor XY model. When disorder becomes strong enough, Majorana edge modes become critically extended, beyond which they collapse into Anderson localized (AL) states in the bulk, resulting in a topological Anderson insulating state (TAI). We identify a resilience threshold Wt for every pair of Majorana fermions (MFs). In response to increasing disorder every pair of MFs collapse into AL bulk at their resilience threshold. For very strong disorder, all Majorana fermions collapse and a topologically trivial state is obtained. We show that the threshold values are deeply related to the localization length of Majorana fermions, which can be efficiently calculated by an appropriate modification of the transfer matrix method. At the topological transition point, localization length of the zero modes diverges and the system becomes scale invariant. The number of peaks in the localization length as the function of disorder strength determines the number of zero modes in the clean state before disorder is introduced. This finding elevates the transfer matrix method to the level of a tool for determination of the topological index of both clean and disordered systems.

Dirac electrons in clean graphene can mediate the interactions between two localized magnetic moments. The functional form of the RKKY interaction in pristine graphene is specified by two main features: (i) an atomic scale oscillatory part determined by a wave vector Q connecting the two valleys. Furthermore with doping another longer range oscillation appears which arise from the existence of an extended Fermi surface characterized by a single momentum scale kF . (ii) R α decay in large distances where the exponent α = −3 is a distinct feature of undoped Dirac sea (with a linear dispersion relation) in two dimensions. In this work, we investigate the effect of a few percent vacancies on the above properties. Depending on the doping level, if the chemical potential lies on the linear part of the density of states, the exponent α remains close to −3. Otherwise α reduces towards more negative values which means that the combined effect of vacancies and the randomness in their positions makes it harder for the carriers of the medium to mediate the magnetic interaction. Addition of a few percent of vacancies diminishes the atomic scale oscillations of the RKKY interaction signaling the destruction of two-valley structure of the parent graphene material. Surprisingly by allowing the chemical potential to vary, we find that the longer-range oscillations expected to arise from the existence of a kF scale in the vacant graphene are absent. This may indicate possible non-Fermi liquid behavior by "alloying" graphene with vacancies. The complete absence of oscillations in heavily vacant graphene can be considered an advantage for applications as a uniform sign of the exchange interaction is desirable for magnetic ordering.

As a low-energy effective theory on non-symmorphic lattices, we consider a generic triple point fermion Hamiltonian, which is parameterized by an angular parameter λ. We find strong λ dependence in both Drude and interband optical absorption of these systems. The deviation of the T 2 coefficient of the Drude weight from Dirac/Weyl fermions can be used as a quick way to optically distinguish the triple point degeneracies from the Dirac/Weyl degeneracies. At the particular λ = π/6 point, we find that the ‘helicity’ reversal optical transition matrix element is identically zero. Nevertheless, deviating from this point, the helicity reversal emerges as an absorption channel.

Topological index of a system specifies gross features of the system. However, in situations such as strong disorder where by level repulsion mechanism the spectral gap is closed, the topological indices are not welldefined. In this paper, we show that the localization length of zero modes determined from appropriate use of transfer matrix method reveals much more information than the topological index. The localization length can provide not only information about the topological index of the Hamiltonian itself, but it can also provide information about the topological indices of the "related" Hamiltonians. As a case study, we study a generalized XY model (2XY model) plus a generalized Dziyaloshinskii-Moriya-like (DM) interaction that after fermionization breaks the time-reversal invariance and is parameterized by φ. The parent Hamiltonian at φ = 0 which belongs to BDI class is indexed by integer winding number while the φ = 0 daughter Hamiltonian which belongs to class D is specified by a Z2 index ν = ±1. We show that the localization length in addition to determining the Z2 can count the number of Majorana zero modes left over at the boundary of the daughter Hamiltonianwhich are not protected by winding number anymore. Therefore the localization length outperforms the standard topological indices in two respects: (i) it is much faster and more accurate to calculate and (ii) it can count the winding number of the parent Hamiltonian by looking into the edges of the daughter Hamiltonian.