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This Hamiltonian is exactly the one obtained for fermionic systems in if we reverse the sign of operator L z. In, it was assumed that the number of the available fermionic states equals the total number of particles. Validity of this assumption was commented by D. Sun, A. Abanov, and V. L. Pokrovsky, e-print arXiv:0707.3630. Nevertheless, in the deduction of the Hamiltonian for our bosonic system, this assumption is not required. As will be shown later, a "renormalized" Landau-Zener formula derived from the Hamiltonian is able to account for the experimental findings in the nonadiabatic regime.
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