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Using path integral formalism, we show that the Abrikosov-Tkachenko vortexlattice may equivalently be understood as an array of Josephson junctions. TheTkachenko modes are found to be basically equivalent to the low energyexcitations (Goldstone modes) of an ordered state. The calculated frequenciesare in very good agreement with recent experimental data. Calculations of thefluctuations of the relative displacements of the vortices show that vortexmelting is a result of quantum fluctuations around the ordered state due to thelow energy excitations (Tkachenko modes)and occurs when the ratio of thekinectic energy to the potential energy of the vortex lattice is 0.001.Comment: revised paper 11 pages with 2 figures, all in Pdf forma

Tkachenko modes and quantum melting of Josephson junction types of vortex array in rotating Bose–Einstein condensates