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We consider longitudinal nonlinear atomic vibrations in uniformly strainedcarbon chains with the cumulene structure ($=C=C=)_{n}$. With the aid of abinitio simulations, based on the density functional theory, we have revealedthe phenomenon of the $\pi$-mode softening in a certain range of its amplitudefor the strain above the critical value $\eta_{c}\approx 11\,{\%}$.Condensation of this soft mode induces the structural transformation of thecarbon chain with doubling of its unit cell. This is the Peierls phasetransition in the strained cumulene, which was previously revealed in [NanoLett. 14, 4224 (2014)]. The Peierls transition leads to appearance of theenergy gap in the electron spectrum of the strained carbyne, and this materialtransforms from the conducting state to semiconducting or insulating states.The authors of the above paper emphasize that such phenomenon can be used forconstruction of various nanodevices. The $\pi$-mode softening occurs becausethe old equilibrium positions (EQPs), around which carbon atoms vibrate atsmall strains, lose their stability and these atoms begin to vibrate in the newpotential wells located near old EQPs. We study the stability of the new EQPs,as well as stability of vibrations in their vicinity. In previous paper[Physica D 203, 121(2005)], we proved that only three symmetry-determinedRosenberg nonlinear normal modes can exist in monoatomic chains with arbitraryinterparticle interactions. They are the above-discussed $\pi$-mode and twoother modes, which we call $\sigma$-mode and $\tau$-mode. These modescorrespond to the multiplication of the unit cell of the vibrational state bytwo, three or four times compared to that of the equilibrium state. We studyproperties of these modes in the chain model with arbitrary pair potential ofinterparticle interactions.

Nonlinear atomic vibrations and structural phase transitions in strained carbon chains