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The bimetallic system Co/Cu(111) has been studied extensively both experimentally and theoretically. From the theoretical point of view, the studies of Co/Cu(111) system are mainly related to ab initio calculations in the framework of Density Functional Theory (DFT). DFT is commonly used method for calculations in condensed matter physics. Usually the results of DFT calculations for condensed matter systems are in a good agreement with the experimental data. Even better results are observed for metal systems [1,2]. The result of recent work [1] was totally unexpected. Based on ab initio calculations Zaki et al. [3] conclude that DFT qualitatively fails to predict the dimerized structural phase for a monatomic Co wire [4]. It was also demonstrated, that the dimerized Co wire exists at low temperature and shows a tip-bias dependency [4]. The ab initio calculations are performed to study the structure of atom-wide Co wire on a vicinal Cu(111) surface. We have found two ferromagnetic states of Co wires. In the first state the Co wire consists of dimers, while in the second state the distance between atoms in Co wire is equal. The low phase transition temperature is explained by small difference in energy between the states. Our results completely refute the conclusion made by Zaki et al. [3] that DFT qualitatively fails to predict the dimerized structural phase for a monatomic Co wire on a vicinal Cu(111) surface. In addition, it was found that the phase transition temperature depends drastically on the length of the wire and the tip bias. In the first instance, tip-bias dependency is related with deformation of wire and surface. Thanks This work was supported by Russian Foundation of Basic Researches (Project number 15- 32-20560). References [1] S.V. Kolesnikov, A.L. Klavsyuk, A.M. Saletsky Surface Science 612, 48 (2013). [2] V.S. Stepanyuk, A.N. Klavsyuk, L. Niebergall, and P. Bruno, Phys. Rev. B 72, 153407 (2005). [3] N. Zaki, H. Park, R.M Osgood, A.J. Millis, and C.A. Marianetti, Phys. Rev. B 89, 205427 (2014). [4] N. Zaki, C.A. Marianetti, D.P. Acharya, P. Zahl, P. Sutter, J. Okamoto, P.D. Johnson, A.J. Millis, and R.M. Osgood, Phys. Rev. B 87, 161406 (2013).