Dynamical phase diagram of a quantum Ising chain with long-range interactions

We investigate the effect of short-range correlations on the dynamical phase diagram of quantum many-body systems with long-range interactions. Focusing on Ising spin chains with power-law decaying interactions and accounting for short-range correlations by a cluster mean field theory we show that short-range correlations are responsible for the emergence of a chaotic dynamical region. Analyzing the fine details of the phase diagram, we show that the resulting chaotic dynamics bears close analogies with that of a tossed coin.

Figure 1

The dynamical phase diagram for the long-range Ising model ($J=1$) obtained by CMFT with $\ell =5$ for a quench of the transverse field from $h_{\mathrm{in}}=0$ to $h$: The time-averaged longitudinal magnetization (up to a time $T=100$) vs $\alpha$ and $h$. Notice that, as soon as $\alpha >1$ (dashed line), the dynamical phase transition fans out in a critical region, where the asymptotic magnetization becomes very sensitive to the system parameters. The blue square denotes the region detailed in Fig. 2.

Figure 2

A portion of the phase diagram (blue square in Fig. 1) for $\ell =5$ with increasing the resolution of $\delta \alpha =\delta h=5\times {10}^{-3},1\times {10}^{-3},5\times {10}^{-4},1\times {10}^{-4}$. In the critical region the phases strongly intermingle giving rise to new structures whenever the resolution is increased. Bottom panels: Size of the maximum neighborhood $\varepsilon$ containing the point of the same phase evaluated for $\alpha =1.4$ (dotted line) as a function of the postquench transverse field $h$ normalized to the valued ${\varepsilon }_{\text{max}}={max}_h\varepsilon$. Independently of the resolution $\varepsilon /{\varepsilon }_{\text{max}}$ shrinks to zero for increasing $h$.

Figure 3

Dynamical phase diagram for the long-range transverse-field Ising model with $\alpha =1.2$ obtained by CMFT combined with TDVP. We show the order parameter as a function of the final $h$. At small final fields, the dynamical order parameter is close to the mean-field value (gray dashed line). Close to $h=1$ we observe that the final magnetization becomes very sensitive to the quench parameter and alternates between positive and negative values. At large transverse fields, the dynamical order parameter vanishes. The data are obtained with the bond dimension $D=64$. The region in red denotes the range of parameters for which convergence is attained at times longer than the one accessible by the MPS-TDVP.

Figure 4

Convergence of the time-dependent longitudinal magnetization $m^z\left(t\right)$ with the cluster size in the dynamical critical region at $\alpha =1.2$. Relatively large system sizes (around $\ell =200, D=64$) are necessary to observe convergence. The order parameter $m^z$ in Fig. 3 is obtained by averaging over the green region.