Quantum time crystals with programmable disorder in higher dimensions

We present fresh evidence for the presence of discrete quantum time crystals in two spatial dimensions. Discrete time crystals are intricate quantum systems that break discrete time translation symmetry in driven quantum many-body systems undergoing nonequilibrium dynamics. They are stabilized by many-body localization arising from disorder. We directly target the thermodynamic limit using instances of infinite tensor network states, and we implement disorder in a translationally invariant setting by introducing auxiliary systems at each site. We discuss how such disorder can be realized in programmable quantum simulators: This gives rise to the interesting situation in which a classical tensor network simulation can contribute to devising a blueprint of a quantum simulator featuring prethermal time crystalline dynamics, one that will ultimately have to be built in order to explore the stability of this phase of matter for long times.

Figure 1

Effect of imperfect flip ($\epsilon =0.5$) on a single uncoupled spin. One can see the emergence of a beating pattern along with the associated spikes leading to the formation of a knot at around Floquet cycle 32.

Figure 2

Equal space correlator evaluated at different times for coupled spins without disorder starting from the Néel state (left) and the polarized state (right) for the perfect (top) and imperfect flips (bottom). Time crystalline behavior cannot survive for the Néel state due to thermalization, while there is no dynamics for the initial polarized state. Also shown are simulations with $D=5$ for stroboscopic times. Insets: growth of local Rényi entropies $(\alpha =1/2,1)$ for the blue and red curve, respectively.

Figure 3

Equal space correlator evaluated at different times for coupled spins with strong disorder of two levels for different initial states and $\epsilon$. Also shown are the simulations with $D=5$ for stroboscopic times. Insets: growth of local Rényi entropies $(\alpha =1/2,1)$ for the blue and red curve, respectively.

Figure 4

Same as Fig. 3, but with five levels of disorder. The time crystal has stabilized due to the increasing number of disorder levels, thereby surviving much longer times for both of the initial states.

Figure 5

Stability of the time crystal with respect to different disorder strength $h$ and perturbation to the perfect flip $\epsilon$. The results are plotted only for stroboscopic times.

Figure 6

Implementing disorder in a translationally invariant setting. We choose our initial global state to be a tensor product of the physical state (represented by gray tensors) and the auxiliary state (represented by green tensors). $L$, $U$, $R$, and $Do$ correspond to the different links we need to update for a particular site.

Figure 7

Errors corresponding to Fig. 2 (left panels). Top: $\delta \ge 0$ is the absolute value of the difference of the expectation values of the correlators corresponding to simulations with $D=4$ and 5. Bottom: growth of local Rényi entropies for one-site reduced density matrix for the full time evolution.

Figure 8

Errors corresponding to Fig. 3. Top: $\delta$ is the absolute value of the difference of the expectation values of the correlators corresponding to simulations with $D=4$ and 5. Bottom: growth of local Rényi entropies for one-site reduced density matrix for the full time evolution. $\left(a\right)$, $\left(c\right)$, and $\left(d\right)$ refer to the different subplots of Fig. 3 in Sec. 3.

Figure 9

Errors corresponding to Fig. 4. Top: $\delta$ is the absolute value of the difference of the expectation values of the correlators corresponding to simulations with $D=4$ and 5. Bottom: growth of local Rényi entropies for one-site reduced density matrix for the full time evolution. $\left(a\right)$, $\left(c\right)$, and $\left(d\right)$ refer to the different subplots of Fig. 4 in Sec. 3.