Chiral chains with two valleys and disorder of finite correlation length
In one-dimensional disordered systems with a chiral symmetry it is well-known that electrons at energy $E = 0$ avoid localization and simultaneously exhibit a diverging density of states (DOS). For $N$ coupled chains with zero-correlation-length disorder, the diverging DOS remains for odd $N$, but a vanishing DOS is found for even $N$. We use a thin spinless graphene nanotube with disordered Semenoff mass and disordered Haldane coupling to construct $N = 2$ chiral chain models which at low energy have two linear band crossings at different momenta $\pm K$ (two valleys) and disorder with an arbitrary correlation length $ξ$ in units of lattice constant $a$. We find that the finite momentum $\pm K$ forces the disorder in one valley to depend on the disorder in the other valley, thus departing from known analytical results which assume having $N$ independent disorders (whatever their spatial correlation lengths). Our main numerical results show that for this inter-dependent mass disorder the DOS is also suppressed in the limit of strongly coupled valleys (lattice-white noise limit, $ξ/a = 0$) and exhibits a non-trivial crossover as the valleys decouple ($ξ/a\gtrsim 5$) into the DOS shapes of the $N = 1$ continuum model with finite correlation length $ξ$. We also show that changing the intra-unit-cell geometry of the disordered Haldane coupling can tune the amount of inter-valley scattering yet at lowest energies it produces the decoupled-valley behavior ($N = 1$) all the way down to lattice white noise.