Exactly solvable model with an unusual phase transition: the rolling transition of a pinned Gaussian chain

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We consider a Gaussian polymer chain in an external potential field of a Heaviside form. The chain is constrained by one of its ends to the point where the potential changes. For this model, the exact partition function is available. In the appropriate thermodynamic limit the chain ‘rolls’ from one half-space to the other upon changing the sign of the external potential by way of a rather special phase transition. The derivative of the free energy is discontinuous indicating a first-order character. Its complex zero distribution is consistent with literature predictions for this. However, from the exact analytical equation for the Landau function it is found that there are no metastable states associated with the transition. Moreover, the derivative of the energy is discontinuous pointing to a second order classification. Finally, there are no singularities with respect of derivatives of the entropy. All this is consistent with the hypothesis that the model features a (multi) critical point at the condition that the potential changes sign.