Prethermalization and persistent order in the absence of a thermal phase transition

We numerically study the dynamics after a parameter quench in the one-dimensional transverse-field Ising model with long-range interactions ($\propto 1/r^{\alpha }$ with distance $r$), for finite chains and also directly in the thermodynamic limit. In nonequilibrium, i.e., before the system settles into a thermal state, we find a long-lived regime that is characterized by a prethermal value of the magnetization, which in general differs from its thermal value. We find that the ferromagnetic phase is stabilized dynamically: as a function of the quench parameter, the prethermal magnetization shows a transition between a symmetry-broken and a symmetric phase, even for those values of $\alpha$ for which no finite-temperature transition occurs in equilibrium. The dynamical critical point is shifted with respect to the equilibrium one, and the shift is found to depend on $\alpha$ as well as on the quench parameters.

Figure 1

Schematic phase diagram of the long-range TFIM (1). The model exhibits an equilibrium quantum phase transition at a critical point $h_{\mathrm{c}}\left(\alpha \right)$ for all values of $\alpha$. A $\text{finite-}T$ phase transition occurs only for $\alpha <2$ (left), but not for $\alpha >2$ (right). Quenching from $h_{\text{i}}=0$ to $h_{\text{f}}=h$ and letting the system thermalize, equilibrium states on a line $T_{\text{f}}\left(h\right)$ (blue line in the left plot) are reached at long times. The phase-transition line is crossed at a field ${\tilde{h}}_{\text{c}}<h_{\text{c}}\left(\alpha \right)$, which results in a shift of the critical field with respect to the quantum critical point.

Figure 2

Time evolution of the order parameter $m$ as obtained from iMPS simulations for long-range exponent $\alpha =3$. (Data from finite-size $t\text{-DMRG}$ simulations look very similar.) For a strong quench from $h_{\text{i}}=0$ to $h_{\text{f}}=0.99$, the magnetization quickly decays towards zero (yellow line) and is well approximated by a power law (lower black line). For a small quench from $h_{\text{i}}=0$ to $h_{\text{f}}=0.28$, the magnetization shows an initial decay away from its initial value of 1 on a fast time scale (inset), and then saturates to a nonzero value for rather long times (blue line). Eventually, for the chosen parameter values and on a time scale not accessible in simulations, the system will thermalize to a state with zero magnetization.

Figure 3

Prethermal magnetization $\tilde{m}$ plotted as a function of the final quench parameter $h_{\text{f}}$. Both plots are for quenches starting from $h_{\text{i}}=0$, and for various system sizes as indicated in the legends. The existence of a magnetized phase for small $h_{\text{f}}$ and an unmagnetized phase for large $h_{\text{f}}$ is clearly visible for $\alpha =1.6$ (top) and $\alpha =3$ (bottom).

Figure 4

Prethermal magnetization $\tilde{m}$ plotted as a function of the final quench parameter $h_{\text{f}}$ for $\alpha =2.3$. Top: quenching from $h_{\text{i}}=0$; bottom: quenching from $h_{\text{i}}=0.2$. Both plots show qualitatively similar behavior. A slight dependence of the dynamical critical point on $h_{\text{i}}$, as expected for the thermal behavior in the long-time limit after the quench, might also be present in the prethermalized regime on intermediate time scales, but cannot be established beyond doubt.

Figure 5

Illustration of the fitting and extrapolation procedure for $\alpha =3, h_{\text{i}}=0, h_{\text{f}}=0.28$. The oscillating blue line shows the iMPS data. Dashed lines (yellow and green, almost on top of each other and hardly distinguishable on the scale of the plots) show the exponential fit functions $m_1$ and $m_2$, solid lines (orange, brown, and purple, again hardly distinguishable) show power law fits $m_3, m_4$, and $m_5$. For the three-parameter fits $m_2, m_4$, and $m_5$, the optimal values of the fit parameters come with large error bars, and those functions are therefore discarded. Among the two-parameter fits, $m_3$ (solid orange line) has a significantly smaller mean squared deviation from the simulation data than $m_1$ (dashed yellow), and is therefore used for determining the prethermal magnetization $\tilde{m}=m_3\left(10t_{\text{f}}\right)$.