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The terms of topological and quantum stabilities of low-dimensionalcrystalline carbon lattices with multiple non-equivalent sublattices are coinedusing theoretical analysis, multilevel simulations, and available experimentalstructural data. It is demonstrated that complex low-dimensional lattices areprone to periodicity breakdown caused by structural deformations generated byPBC approach. To impose PBC limitations for low-dimensional lattices, theTopology Conservation Theorem (TCT) is formulated and proved. It is shown thatthe lack of perfect filling of 2D crystalline space units may cause formationof i) Structure waves; ii) Nanotubes or rolls; iii) Saddle structures; iv)Aperiodic atomic clusters; v) Stabilization of 2D lattices by aromaticresonance, correlation effects, or van-der-Waals interactions. The effect ofquantum instability of infinite structural waves is studied using quasiparticleapproach. It is found that both perfect finite-sized, or stabilized structuralwaves can exist and can be synthesized. It is shown that for low-dimensionallattices prone to breakdown of translation invariance (TI), complete activespace of normal coordinates cannot be reduced to a subspace of TI normalcoordinates. As a result, constrained TI subspace structural minimization mayartificially return a regular point at the potential energy surface as either aglobal/local minimum/maximum. It is proved that phonon dispersion cannot beused as solid and final proof of their stability. It is shown that ab initiomolecular dynamics (MD) PBC Nos\'e-Hoover thermostat algorithm constrains thelinear dimensions of the periodic slabs in MD box preventing their thermostatedequilibration. Based on rigorous TCT analysis, a flowchart algorithm forstructural analysis of low-dimensional crystals is proposed and proved to be apowerful tool for theoretical design of advanced complex nanomaterials.

Topological and Quantum Stability of Low-Dimensional Crystalline Lattices with Multiple Nonequivalent Sublattices