The phase transition of the two-level SP(4) model was investigated. By comparing the exact results with the approximate results of the static mean field theory, the importance of the dynamic effect in phase transitions of finite systems is exhibited.
One-dimensional non-equilibrium kinetic Ising models evolving under the competing effect of spin flips at zero temperature and nearest-neighbour spin exchanges exhibiting a parity-conserving (PC) phase transition on the level of kinks are now further investigated numerically, from the point of view of the underlying spin system. Critical exponents characterizing its statics and dynamics are reported. It is found that the influence of the PC transition on the critical exponents of the spins is strong and the origin of drastic changes as compared to the Glauber - Ising case can be traced back to the hyperscaling law stemming from directed percolation. The effect of an external magnetic field, leading to directed percolation-type behaviour on the level of kinks is also studied, mainly via the generalized mean-field approximation.
The extended Falicov-Kimball model is analyzed exactly in the ground state at half filling in the limit of large dimensions. In the model the on-site and the intersite density-density interactions between all particles are included. We determined the model's phase diagram and found a discontinuous transition between two different charge-ordered phases. Our analytical calculations show that the ground state of the system is insulating for any nonzero values of the interaction couplings. We also show that the dynamical mean-field theory and the static broken-symmetry Hartree-Fock mean-field approximation give the same results for the model at zero temperature. In addition, we prove using analytical expressions that at infinitesimally small, but finite, temperatures the system can be metallic.
Methods which include the effect of mean-field fluctuations on the nuclear partition function and level density of hot nuclei are tested by applying them to a many-body system for which exact solutions are available, the Lipkin model. It is found that for finite systems fluctuation corrections to the mean field partition function and level density can be substantial, and especially so near a shape phase transition where these fluctuations smooth out discontinuities in the mean-field quantities. The model calculations suggest that only time-independent fluctuations are important at high temperatures or at high excitation energies. Moreover, even when fluctuations are restricted to a small set of constraints, a significant improvement over the mean-field calculations can result. This is thus a practical approximation since it requires only knowledge of the deformation free energy surfaces at finite temperature.