How to construct an example of a function inside $H^{1/2}(S^1)\setminus \mathcal{A}(S^1)$?

Question

In a lecture we proved that for all $s>\frac{1}{2}$, we have a continuous embedding $H^s(S^1)\hookrightarrow \mathcal{A}(S^1)$ where $S^1=\mathbb{R}/\mathbb{Z}$, i.e. we can embed the Sobolev space of regularity $s>\frac{1}{2}$ into the Wiener algebra on the circle. In the exercises we then saw that the logarithm provides an example of a function inside $H^s(S^1)$ for $s<\frac{1}{2}$ which isn't inside the Wiener algebra. But in the lecture it was also mentioned that $H^{1/2}(S^1)$ can't either be embedded into $\mathcal{A}(S^1)$. So I wondered: what is an example of a function inside $H^{1/2}(S^1)$ which is not in the Wiener algebra?