Local Dephasing Sustains Continuous Time Crystals in a Spin-1/2 Ensemble

The paper in detail

The model is \(H=\tfrac{h_z}{2}\sum_i\sigma_i^z-\tfrac{J}{N}\sum_{i<j}\sigma_i^x\sigma_j^x\), with each spin subject to decay \(\gamma\mathcal D[\sigma_i^-]\) and pure dephasing \(\gamma_\phi\mathcal D[\sigma_i^z]\), so \(\Gamma_1=\gamma\) and \(\Gamma_2=\gamma/2+2\gamma_\phi\). In homogeneous mean field the Bloch vector obeys nonlinear Bloch equations. At \(J=h_z\approx105\,\Gamma_1\) and from the same initial state, the magnetization relaxes when \(\gamma_\phi=0\) and oscillates indefinitely when \(\gamma_\phi=1.5\,\Gamma_1\). Two symmetry-related fixed points \(F_\pm\) exist when \(2h_zJ>h_z^2+\Gamma_2^2\), and linearization shows they are always stable. The limit cycle therefore does not come from a Hopf bifurcation but coexists with them, and the outcome depends on the basin of the initial state.

The mechanism comes from the shifted energy \(Q_N=2H/N+h_zI\). Because it commutes with \(H\), only the dissipators change it, and \(\mathcal L^\dagger Q_N=-\Gamma_1Q_N+4J\gamma_\phi U_N\), where \(U_N\) is the transverse pair correlation. Ferromagnetically aligned transverse spins lower the interaction energy, and dephasing destroys that alignment, so it raises the energy. In mean field \(\dot Q=-\Gamma_1Q+4J\gamma_\phi m_x^2\). Eliminating \(m_y\) and \(m_z\) leaves a damped nonlinear oscillator for \(m_x\) with an extra force \(-2JQm_x\). Without dephasing \(Q\) decays to zero, and an energy argument over one period forces \(\dot m_x\equiv0\), so no periodic orbit exists.

Charging alone does not explain motion, because charging and loss can also balance at a fixed point. What matters is that \(Q\) follows \(m_x^2\) through a memory kernel \(e^{-\Gamma_1s}\), so each harmonic lags by \(\arctan(\nu/\Gamma_1)\). The lag makes the orbit enclose an area in the \((m_x^2,Q)\) plane, and the feedback work around this loop equals \(\langle\dot Q^2\rangle/(4\gamma_\phi)>0\). It exactly balances the damping \(2\Gamma_2\oint\dot m_x^2\,dt\). An instantaneous response would retrace a line and do no work.

Beyond mean field, the Liouvillian is diagonalized in Dicke blocks for \(N=80\)–\(300\). The odd and even oscillatory branches become longer lived (\(\kappa\propto N^{-0.323}\), \(N^{-0.458}\)), their frequencies approach \(\Omega\) and \(2\Omega\), and the per-spin oscillatory weights in stationary correlations stay of comparable size. Projecting the identity onto an eigenmode gives \(q_k/u_k=4J\gamma_\phi/(\Gamma_1+\lambda_k)\), the quantum version of the mean-field lag. Semiclassical trajectories at \(N=10^5\) show oscillatory episodes interrupted by stays near \(F_\pm\), about 13% of the time, followed by reactivation. Simulated photon counting at efficiency 0.3 resolves the \(2\Omega\) peak from \(N=10^5\) upward.

Transferable methods

1. Exact energy balance from an observable that commutes with the Hamiltonian

   An observable that commutes with \(H\) changes only through the dissipators, so its Heisenberg equation is exact at any \(N\) and often closes on a few operators. Here it separates energy loss from a dephasing-driven source and constrains every Liouvillian mode.

   With \(Q_N=h_z(M_z+I)-JU_N\) and \([H,Q_N]=0\), \(\mathcal L^\dagger Q_N=\mathcal L^\dagger_{\rm diss}Q_N\). Local decay gives \(\mathcal L^\dagger_{\rm diss}(M_z+I)=-\Gamma_1(M_z+I)\), and each product \(\sigma_i^x\sigma_j^x\) of distinct spins decays at \(2\Gamma_2\), so \(\mathcal L^\dagger_{\rm diss}U_N=-2\Gamma_2U_N\). Hence \(\mathcal L^\dagger Q_N=-\Gamma_1Q_N+(2\Gamma_2-\Gamma_1)JU_N\), and \(2\Gamma_2-\Gamma_1=4\gamma_\phi\).

   For a right eigenmode \(\mathcal LR_k=\lambda_kR_k\) with \(\operatorname{Tr}R_k=0\), projecting the identity gives \(\lambda_kq_k=-\Gamma_1q_k+4J\gamma_\phi u_k\), so \(q_k/u_k=4J\gamma_\phi/(\Gamma_1+\lambda_k)\) with \(q_k=\operatorname{Tr}(Q_NR_k)\) and \(u_k=\operatorname{Tr}(U_NR_k)\). As \(\kappa_k\to0\) this becomes the harmonic response \(4J\gamma_\phi/(\Gamma_1+i\nu)\).

   Editorial check: integrating the mean-field equations at the Fig. 1 parameters, \(\dot Q=-\Gamma_1Q+4J\gamma_\phi m_x^2\) holds to \(10^{-13}\) along the trajectory. With \(\gamma_\phi=0\) the state settles at \(m_z=-0.5\); with \(\gamma_\phi=1.5\), \(m_z\) keeps oscillating between \(-0.63\) and \(-0.38\) at late times.

   How to reuse it

   In an open many-body model, look for a conserved quantity shifted by a constant so that the dissipators map it into a small operator set. The resulting identity gives exact finite-\(N\) relations between eigenmode projections and between correlation functions, which also check numerics. It needs dissipators that do not generate a growing operator hierarchy.

2. Cycle work of a lagged feedback variable

   To decide whether a damped oscillator coupled to a relaxing variable can sustain a periodic orbit, multiply its equation by the velocity and integrate over a period. Conservative terms drop out, so damping must be paid by feedback work, and that work is nonzero only if the feedback lags.

   Eliminating \(m_y=-(\dot m_x+\Gamma_2m_x)/h_z\) and \(m_z=(Q+Jm_x^2)/h_z-1\) gives \[\ddot m_x+2\Gamma_2\dot m_x=-(h_z^2+\Gamma_2^2-2h_zJ)m_x-2J^2m_x^3-2JQm_x.\] Multiplying by \(\dot m_x\) and integrating over a period gives \(2\Gamma_2\oint\dot m_x^2\,dt=-J\oint Q\,d(m_x^2)\). If \(\gamma_\phi=0\), \(Q\to0\), the right side vanishes and \(m_x\) is constant.

   With \(u=m_x^2\) and \(\dot Q=-\Gamma_1Q+4J\gamma_\phi u\), integration by parts gives \[-J\langle Q\dot u\rangle=J\langle\dot Qu\rangle=\frac{\langle\dot Q(\dot Q+\Gamma_1Q)\rangle}{4\gamma_\phi}=\frac{\langle\dot Q^2\rangle}{4\gamma_\phi},\] since \(\langle Q\dot Q\rangle=0\) over a period. The work is strictly positive unless \(Q\) is constant. If \(Q\) were proportional to \(u\) at every instant, the loop would have zero area.

   Editorial check: on the \(\gamma_\phi=1.5\) orbit (period \(T\approx0.29\,T_1\)), averaging over 33 whole periods gives \(2\Gamma_2\langle\dot m_x^2\rangle\), \(-J\langle Q\dot u\rangle\) and \(\langle\dot Q^2\rangle/(4\gamma_\phi)\) all equal to 420.99. The charging–loss balance \(\Gamma_1\langle Q\rangle=4J\gamma_\phi\langle m_x^2\rangle=39.93\) holds as well.

   How to reuse it

   For any oscillator coupled to a slowly relaxing variable (temperature, population, cavity field), this energy method gives a necessary condition for limit cycles: a nonzero source and a finite relaxation time. The \(\langle\dot Q^2\rangle\) form fixes the sign without solving for the orbit. It does not prove that the cycle exists or say which initial states reach it.

Source: Section III.1 (Eqs. 7–8); Section III.2–III.3 (Eqs. 9–18); Section IV.2 (Eqs. 24–27). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. It turns a ubiquitous decoherence channel into a control parameter for autonomous temporal order. The identity gives a testable amplitude–phase relation between energy and magnetization in every oscillatory mode, which can be checked beyond all-to-all models. The oscillating state also takes part in metastable switching between two stationary states.

Limitations. The necessity proof holds in homogeneous mean field. At finite \(N\) the evidence is power-law fits over \(N\le300\), not a proof that decay rates vanish. The limit cycle coexists with two stable fixed points, so whether oscillations appear depends on the initial state. Fluctuations and photon-counting readout are treated with a semiclassical diffusion model rather than exact quantum trajectories, and no experiment is reported.

Description based on the full text of the linked arXiv version, read 2026-10-25; equations follow the paper’s notation.