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The stabilizer group of honeycomb lattices and its application to deformed monolayers

Isospectral transformations of exactly solvable models constitute a fruitful method for obtaining new structures with prescribed properties. In this paper we study the stability group of the Dirac algebra in honeycomb lattices representing graphene or boron nitride. New crystalline arrays with conical (Dirac) points are obtained; in particular, a model for dichalcogenide monolayers is proposed and analyzed. In our studies we encounter unitary and non-unitary transformations. We show that the latter give rise to $\mbox{$\cal P\,$}\mbox{$\cal T\,$}$-symmetric Hamiltonians, in compliance with known results in the context of boosted Dirac equations. The results of the unitary part are applied to the description of invariant bandgaps and dispersion relations in materials such as MoS$_2$. A careful construction based on atomic orbitals is proposed and the resulting dispersion relation is compared with previous results obtained through DFT.