The rounding of the phase transition for disordered pinning with stretched exponential tails

TL;DR

This paper investigates how quenched disorder affects phase transitions in a disordered pinning model with stretched-exponential tail distributions, revealing a critical threshold at alpha=1/2 where the transition changes from first order to smooth.

Contribution

It demonstrates that the nature of the phase transition in a disordered pinning model with stretched-exponential tails depends on the tail parameter alpha, showing a transition from first order to smooth at alpha=1/2.

Findings

For alpha > 1/2, the transition remains first order.

For alpha ≤ 1/2, the free-energy curve is smoothed.

The rounding effect intensifies as alpha decreases.

Abstract

The presence of frozen-in or quenched disorder in a system can often modify the nature of its phase transition. A particular instance of this phenomenon is the so-called rounding effect: it has been shown in many cases that the free-energy curve of the disordered system at its critical point is smoother than that of the homogenous one. In particular some disordered systems do not allow first-order transitions. We study this phenomenon for the pinning of a renewal with stretched-exponential tails on a defect line (the distribution $K$ of the renewal increments satisfies $K(n) \sim c_K\exp(-n^{\alpha}),$ $\alpha\in (0,1)$) which has a first order transition when disorder is not present. We show that the critical behavior of the disordered system depends on the value of $\alpha$: when $\alpha>1/2$ the transition remains first order, whereas the free-energy diagram is smoothed for $\alpha\le 1/2$. Furthermore we show that the rounding effect is getting stronger when $\alpha$ diminishes.