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

Chiral chains with two valleys and disorder of finite correlation length