Effect of solitons on the thermodynamic properties of a system with long-range interactions

Thermodynamic properties of a one-dimensional "Φ⁴" system, interacting via a pair potential which falls off exponentially with distance, is studied as a function of the range of this potential. From the classical equations of motion we find closed-form expressions for solitary wave solutions which reduce to Krumhansl and Schrieffer solutions in the short-range limit. The width and the energy (Eₛ) of the solitons are found to increase indefinitely as the range of the interaction increases. Transfer-integral techniques are used to evaluate the partition function by converting the functional integral into an equivalent nearest-neighbor problem. At low temperatures, the free energy contains a term proportional to exp(-EₛkBT) which signals the existence of solitons and ensures that there is no long-range order. However, when the range of interaction increases (i.e., Eₛ increases) this term approaches zero and the system undergoes a second-order phase transition in the infinite-range limit. The critical properties in this limit are found to be identical to a van der Waals model.