Limit-cycle phase in driven-dissipative spin systems

We explore the phase diagram of interacting spin-$1/2$ systems in the presence of anisotropic interactions, spontaneous decay, and driving. We find a rich phase diagram featuring a limit-cycle phase in which the magnetization oscillates in time. We analyze the spatiotemporal fluctuations of this limit-cycle phase based on a Gaussian-Floquet analysis. Spatial fluctuations destroy long-range limit-cycle ordering for dimension $d\le 2$, as a time-dependent generalization of the Mermin-Wagner theorem. This result can be interpreted in terms of a spatiotemporal Goldstone mode corresponding to phase fluctuations of the limit cycle. We also demonstrate that the limit-cycle phase exhibits an asymmetric power spectrum measurable in fluorescence experiments.

Figure 1

(a) Mean-field phase diagram of the driven-dissipative spin system for $(J_y,J_z)/\gamma =(6,2)$, highlighting the limit cycle and its vicinity. The nonequilibrium spin system has different phases: uniform (${\text{U}}_1$), bistable (${\text{U}}_1/{\text{U}}_2$), staggered ($\text{S}$), and LC phases. LC/${\text{U}}_2$ denotes phase coexistence. In the LC phase, the spin system undergoes self-sustaining collective oscillations in time. The solid and dashed lines represent continuous and discontinuous phase transitions, respectively. (b) An example trajectory of the time-dependent LC at $(\Omega ,J_x,J_y,J_z)/\gamma =(1,-7,6,2)$ that breaks both the time-translational and sublattice symmetry.

Figure 2

Zoomed-out phase diagram showing different phases for $(J_y,J_z)/\gamma =(6,2)$.

Figure 3

(a) Phase transitions of the LC phase as a function $J_x$ at $(\Omega ,J_y,J_z)/\gamma =(0.6,6,2)$. The sublattice spin order parameter $S^x$ develops bistability, Hopf bifurcation, and sublattice order. The LC phase has a continuous and discontinuous phase transition to the ${\text{U}}_1$ and S phase, respectively. Vertical bars represent the amplitudes of the sublattice spin oscillations. Circles and solid squares denote the two sublattice stationary states. The LC amplitude shows a scaling of ${(\delta J_x/\gamma )}^{1/2}$ near the continuous LC-to-${\text{U}}_1$ transition. (b) LC period $T$ as we vary $\Omega$ for fixed $(J_x,J_y,J_z)/\gamma =(-6.37,6,2)$. The divergence of period scales as $T\sim {(\delta \Omega /\gamma )}^{-1/2}$ as we approach the tricritical point at $(J_{x,c},{\Omega }_c)/\gamma \approx (-6.37,0.37)$.

Figure 4

(a)–(c) Dynamical spin correlations when $(\Omega ,J_x,J_y,J_z)/\gamma =(0.6,-6.09,6,2)$. (a) A typical plot of the sublattice correlation function as a function of momentum and time in the long-wavelength limit. (b) When $k=0$, the correlation diverges linearly with time, while it saturates to finite values for $k\ne 0$. The oscillations have the same period as the LC. (c) The stroboscopic correlation in the long time limit shows a $1/k^2$ dependence, reflecting the gapless excitation of the broken time-translational symmetry of the LC phase. (d) Correlation length $\xi$ scales as $\xi \sim {(\delta J_x/\gamma )}^{-1/2}$ near the continuous LC-${\text{U}}_1$ phase transition.

Figure 5

The first and second largest Floquet exponents ${\mu }_1$ and ${\mu }_2$ for the correlation functions $C(k,t)$ in the LC phase at $(\Omega ,J_y,J_z)/\gamma =(0.6,6,2)$. ${\mu }_1\propto -k^2$ and ${\mu }_2\sim -(k^2+k_0^2)$ in the small $k$ limit. The correlation length $k_0^{-1}$ diverges as $J_x/\gamma$ approaches the critical value.

Figure 6

Power spectra as a function of detuning $\Delta$ with respect to the spin natural frequency in stationary and LC phases. (a) At $(\Omega ,J_x,J_y,J_z)/\gamma =(0.6,-6,6,2)$, the system is in the time-independent ${\text{U}}_1$ phase and the corresponding power spectrum has a standard Mollow triplet structure. (b) The sublattice power spectrum for the LC phase at $(\Omega ,J_x,J_y,J_z)/\gamma =(1,-7,6,2)$ and a particular time $t$. The asymmetry in $\Delta$ is caused by the complex-valued two-time correlation function in the LC phase. Both sublattice spectra have a time dependence and they do not cancel each other. (c) The total dynamical power spectrum $S_A(t,\omega )+S_B(t,\omega )$ oscillates in time with the same periodicity of the LC.