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We study infinite time blow-up phenomenon for the half-harmonic map flow\begin{equation}\label{e:main00} \left\{\begin{array}{ll}  u_t = -(-\Delta)^{\frac{1}{2}}u +\left(\frac{1}{2\pi}\int_{\mathbb{R}}\frac{|u(x)-u(s)|^2}{|x-s|^2}ds\right)u\quad\text{in }\mathbb{R}\times (0, \infty),  u(\cdot, 0) = u_0\quad\text{ in }\mathbb{R},  \end{array} \right. \end{equation} with a function $u:\mathbb{R}\times [0,\infty)\to \mathbb{S}^1$. Let $q_1,\cdots, q_k$ be distinct points in$\mathbb{R}$, there exist an initial datum $u_0$ and smooth functions$\xi_j(t)\to q_j$, $0<\mu_j(t)\to 0$, as $t\to +\infty$, $j = 1, \cdots, k$,such that the solution $u_q$ of Problem (\ref{e:main00}) has the form\begin{equation*} u_q =\omega_\infty +\sum_{j= 1}^k \left(\omega(\frac{x-\xi_j(t)}{\mu_j(t)} )-\omega_\infty \right)+\theta(x, t),\end{equation*} where $\omega$ is the canonical least energy half-harmonic map,$\omega_\infty=\begin{pmatrix}  1 \end{pmatrix} $, $\theta(x, t)\to 0$ as $t\to +\infty$, uniformly away fromthe points $q_j$. In addition, the parameter functions $\mu_j(t)$ decay to $0$exponentially.

Infinite time blow-up for half-harmonic map flow from $\mathbb{R}$ into $\mathbb{S}^1$