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We study the spin-pseudospin Hamiltonian of the Ising Model in TransverseField (IMTF) for pseudospins, coupled to the XY-spins on a triangular lattice.This model appears from analyses of the quarter-filled ladder compound$NaV_2O_5$, and pseudospins represent its charge degrees of freedom. In themolecular-field approximation we find that the model possesses two phases:charge-disordered without spin gap; and a low-temperature phase containing boththe anti-ferroelectric (zigzag) charge order and spin dimerization (spin gap).The phase transition is of the second kind, and the calculated physicalquantities are as those one expects from the Landau theory. One of particularfeatures of the phase diagram is that the inter-ladder spin-pseudospincoupling, responsible for the spin gap generation, also destroys the IMTFquantum critical point, resulting in the exponential behavior of $T_c$ in theregion of Ising's coupling where the IMTF is always disordered. We concludethat our mean-field results give a qualitatively correct description of thephase transition in $NaV_2O_5$, while a more sophisticated analysis iswarranted in order to take into account the thermal fluctuations and, probably,the proximity of the IMTF quantum critical point.Comment: 17 pages, 2 figure

Simultaneous charge ordering and spin dimerization in quasi-two-dimensional quarter-filled ladders