Boundary Time Crystals

In this work we introduce boundary time crystals. Here continuous time-translation symmetry breaking occurs only in a macroscopic fraction of a many-body quantum system. After introducing their definition and properties, we analyze in detail a solvable model where an accurate scaling analysis can be performed. The existence of the boundary time crystals is intimately connected to the emergence of a time-periodic steady state in the thermodynamic limit of a many-body open quantum system. We also discuss connections to quantum synchronization.

Figure 1

(Upper panel) A sketch of a boundary time crystal. The system is composed by a bulk ($B$) and a boundary ($b$) interacting trough an interaction term $\widehat{V}$. The Hamiltonian of the system is time independent. After tracing out the d.o.f. of the bulk, the dynamics of the boundary is described by the reduced density matrix ${\widehat{\rho }}_b$. In the time-crystal phase, the behavior of collective variables will show persistent oscillations in the thermodynamic limit. (Lower panel) The boundary magnetization of the BTC model discussed in the Letter is shown as a function of time, for different boundary sizes. In the asymptotic condition, spontaneous symmetry breaking appears in persistent oscillations when $N_b\to \infty$.

Figure 2

The eigenvalues $\lambda$ of the Liouvillian are shown in the weak dissipative case (${\omega }_0/\kappa =1.5$, right panel) and in the strong dissipative one for (${\omega }_0/\kappa =0.5$, left panel), in a system with $N_b=36$ spins. The insets show an enlargement over the eigenvalues with the largest real part. The eigenvalues are plotted in units of $\kappa$.

Figure 3

(Left) Finite size scaling for the real part of the Liouvillian eigenvalues in the BTC phase. The index $j$ labels the eigenvalues. The Liouvillian eigenvalues ${\lambda }_j$ are ordered as a function of their real part ($|\mathrm{Re}({\lambda }_j)|\le |\mathrm{Re}({\lambda }_{j+1})|$, and $j=0$ has zero real part). In the ${\omega }_0/\kappa >1$ phase they scale to zero as a power law of the inverse system size. (Right) The imaginary parts of the eigenvalues show a band structure, with a fundamental frequency separation ${\mathrm{\Gamma }}_{{\omega }_0/\kappa }$. For fixed excitation thresholds (we only select ${\lambda }_j$ such that $\nu =j^2/N_b\le \epsilon$) the width of the bands remains finite in the thermodynamic limit (here we choose $\nu <0.025$). The widths of the bands tend to decrease as we constrain to lower excitation thresholds. The eigenvalues are plotted in units of $\kappa$.

Figure 4

In the right panel we plot the decay rate of the oscillations of the magnetization $\eta$ for distinct system sizes. In the same plot we compare $\eta$ with the eigenvalues with the greatest real part—thus smallest absolute values for the real part—of the Liouvillian. In the left panel we plot the Fourier transform of the average magnetization (see lower panel in Fig. 1), highlighting the oscillation frequencies of the dynamics. The peaks are associated with the band separations in the imaginary part discussed in Fig. 3. The inset of the left panel is the solution in the thermodynamic limit where the oscillations persist indefinitely.