Nonequilibrium Criticality at the Onset of Time-Crystalline Order

We explore the phase transitions at the onset of time-crystalline order in $\mathrm{O}(N)$ models driven out of equilibrium. The spontaneous breaking of time translation symmetry and its Goldstone mode are captured by an effective description with $\mathrm{O}(N)\times \mathrm{SO}(2)$ symmetry, where the emergent external SO(2) results from a transmutation of the internal symmetry of time translations. Using the renormalization group and the $\epsilon =4-d$ expansion in a leading two-loop analysis, we identify a new nonequilibrium universality class. Strikingly, it controls the long-distance physics no matter how small the microscopic breaking of equilibrium conditions is. The $\mathrm{O}(N=2)\times \mathrm{SO}(2)$ symmetry group is realized for magnon condensation in pumped yttrium iron garnet films and in exciton-polariton systems with a polarization degree of freedom.

Figure 1

Mean-field phase diagram for $r>0$. Tuning the damping to negative values stabilizes two time-crystal phases, depending on the sign of $u^{\prime }-u$. The symmetric and limit-cycle phases are separated by the transition lines $A$ and $A^{\prime }$, which are second-order transitions at mean field. The line $B$ separates the ordered phases and is a first-order transition, since the order parameter jumps from one configuration to the other. It is well described within mean-field approximation, unlike the other two transitions. Analysis of the latter is the main focus of this Letter.

Figure 2

Flow diagram at criticality ($\gamma =0$) projected on the $(g_d,{\kappa }_d,g_c)$ manifold, for $N=24$ and $\epsilon =0.1$. Similar flows are obtained for all $N>1$. In addition to the Gaussian fixed point, there are three equilibrium fixed points [57, 69, 70], labeled Wilson-Fisher (WF), chiral ($C_{+}$), and antichiral ($C_{-}$). They lie in the $g_c=0$ gray plane. $C_{+}$ is attractive at equilibrium, and the equilibrium black-dashed trajectory is attracted toward it. However, a small breaking of equilibrium grows at larger scale, and the gray solid trajectories go to the new nonequilibrium complex conjugated fixed points $N_{\pm }$.