Gaussian dominance and phase transitions in systems with continuous symmetry

Within the Fröhlich strategy of the phase transitions theory in systems with continuous symmetry, the existence of nonunique state in the nonideal Bose gas for sufficiently small temperatures is proved. We use the technique of majorizing estimates for the correlation expectations and the holomorphic representation of the functional integral method. The main role in the approach is played by the condition of the Gaussian domination by Fr̈ohlich–Simon–Spencer which we extend to the continuous case under consideration. Equation for the critical temperature and an upper bound for the energy of elementary excitations is derived.