Rotating-state conjecture for the driven N-clock model

Consider the three-dimensional $N$-clock model with drift $d$. Let $\bar N$ and $\beta_N^{cr}$ be as above, and let $\langle\cdot\rangle_{\beta,d}^{k,T}$ denote the state after running the driven dynamics for time $T$ from the $k$th equilibrium phase.

Rotating-state conjecture. For every $N\geq\bar N$ and $\beta>\beta_N^{cr}$, there exists a critical drift $d_{cr}(\beta,N)>0$ such that, if $|d|<d_{cr}(\beta,N)$, then $\langle\cdot\rangle_{\beta,d}^{k,T}$ converges as $T\to\infty$ to a magnetized state $\langle\cdot\rangle_{\beta,d}^k$ close to $\langle\cdot\rangle_{\zeta_k,\beta}$ for small $d$. If $d>d_{cr}(\beta,N)$, there are periodic functions $m(T)>0$ and $\Phi(T)$, with period $\omega=\omega(\beta,N,d)$, and a phase shift $\phi_k=\phi_k(\beta,N,d)$ such that

$$\left|\left\langle\sigma_x\right\rangle_{\beta,d}^{k,T}-m(T)e^{i(\Phi(T)+\phi_k)}\right|\to0\quad\text{as }T\to\infty.$$

This predicts a transition from stationary magnetized states to rotating states under sufficiently strong drift. The rotating regime is one of the paper's proposed nonequilibrium analogues of the Goldstone-mode behavior of the symmetric model, and the claim remains open.