Topological and quantum stability of low-dimensional crystalline lattices with multiple nonequivalent sublattices* * Dedicated to the memory of our close friend, fellow colleague, former student (PA) and teacher (AK) Prof. A A Kuzubov (1974-2016).

The terms of topological and quantum stabilities of low-dimensional crystalline carbon lattices with multiple non-equivalent sublattices are coined using theoretical analysis, multilevel simulations, and available experimental structural data. It is demonstrated that complex low-dimensional lattices are prone to periodicity breakdown caused by structural deformations generated by linear periodic boundary conditions (PBC). To impose PBC mandatory limitations for complex low-dimensional lattices, the topology conservation theorem (TCT) is introduced, formulated and proved. It is shown that the lack of perfect filling of planar 2D crystalline space by structural units may cause the formation of (i) structure waves of either variable or constant wavelength; (ii) nanotubes or rolls; (iii) saddle structures; (iv) aperiodic ensembles of irregular asymmetric atomic clusters. In some cases the lattice can be stabilized by aromatic resonance, correlation effects, or van-der-Waals interactions. The effect of quantum instability and periodicity breakdown of infinite structural waves is studied using quasiparticle approach. It is found that both perfect finite-sized, or stabilized structural waves can exist and can be synthesized. It is shown that for low-dimensional lattices prone to breakdown of translation invariance (TI), complete active space of normal coordinates cannot be reduced to a subspace of TI normal coordinates. As a result, constrained TI subspace structural minimization may artificially return a regular point at the potential energy surface as either a global/local minimum/maximum. It is proved that for such lattices, phonon dispersion cannot be used as solid and final proof of either stability or metastability. It is shown that ab initio molecular dynamics (MD) PBC Nosé-Hoover thermostat algorithm constrains the linear dimensions of the periodic slabs in MD box preventing their thermostated equilibration. Based on rigorous TCT analysis, a flowchart algorithm for structural analysis of low-dimensional crystals is proposed and proved to be a powerful tool for theoretical design of advanced complex nanomaterials.