Phase transitions in disordered systems: Exactly solvable model

Abstract

Study of correlation effects in an exactly solvable model twoelectron system J. Chem. Phys. 94, 517 (1991); 10.1063/1.460368Exact solvable threedimensional models of manybody systems Am.

Cubic interactions in coupled systems

“…Without the random fields, RG theory [2,3,12,13] predicts a fluctuation-induced first-order transition which is rigorously borne out [8] by the model. The effect of the random fields has not yet been considered in RG theory, but the model explicitly shows the second-order phase transition to be restored in the presence of one random field and that the ordered phase for space dimensionality d 4 is destroyed when both random fields are present [10]. The addition of a cubic anisotropy term into the free-energy functional of a pure coupled-parameter system changes its critical behaviour and this is the subject here.…”

“…Examples include systems with or without the influence of quenched disorder described by a single or multiple coupled order parameters, systems with cubic or dipole interactions, and systems with short-or long-range-correlation impurities either in the static or dynamic approach. Using exact models that partially take into account interactions of fluctuations [5][6][7][8][9][10][11], some of these systems were successfully treated theoretically. This provides not only an alternative approach for the study of complex-symmetry systems, but also a check on the validity of RG predictions which are very often based on various approximations such as the ε-expansion.…”

“…Therefore, either the anisotropic phase (17), or the isotropic one, equation (18), may occur under the appropriate conditions. The most interesting case is that when one uncoupled system, say i, satisfies the condition for a first-order transition, inequalities (10), and the other one, i , satisfies the conditions for a second-order transition, inequalities (8) or (9) with i → i . When such systems are coupled together, a phase transition will occur in system i, and, depending on the polarity of w, the transition may be a new first-order one, or a restored new second-order one into the anisotropic phase.…”

“…Explicitly, a positive coupling produces a new fluctuation-induced first-order phase transition into the anisotropic phase with a lower critical temperature than that of the zero-coupling case. This is obtained when systems i and i simultaneously satisfy inequalities (10), and ( 8) or (9) with i → i , respectively. The solution ϕ i+ = ϕ i− of equation ( 17) gives the critical temperature of the first-order transition into the anisotropic phase, which to order w is…”

“…This restoration is a transition into the anisotropic phase, with a critical temperature that decreases with increasing absolute value of the coupling strength w. This result is obtained by numerical analysis. The critical temperature of this transition is the solution of equation ( 17) when set equal to zero, ϕ 2 i± = 0, and when the interaction constants, in addition to w < 0, also satisfy inequalities (10), for system i, and ( 8) or (9) with i → i , for system i . This is a cumbersome expression and will not be provided here.…”