Nonadiabatic stationary behavior in a driven low-dimensional gapped system

We discuss the emergence of nonadiabatic behavior in the dynamics of the order parameter in a low-dimensional quantum many-body system subject to a linear ramp of one of its parameters. While performing a ramp within a gapped phase seems to be the most favorable situation for adiabaticity, we show that such a change leads eventually to the disruption of the order, no matter how slowly the ramp is performed. We show this in detail by studying the dynamics of the one-dimensional quantum Ising model subject to linear variation of the transverse magnetic field within the ferromagnetic phase, and then propose a general argument applicable to other systems.

Figure 1

Log-log plot of the correlation length $\xi$ as a function of the duration $\tau$ of the linear ramp for initial transverse field $g_0=0.3$ and different final values of $g_1$. ${\xi }_{\mathrm{sud}}$ is the value of the correlation length for a sudden quench from $g_0$ to $g_1$.

Figure 2

Correlation length $\xi$ as a function of the duration $\tau$ for $g_0=0.3$ and $g_1=0.6$. The numerical results (red circles) are compared with the perturbative expansion up to second and eighth order. The inset shows the same plot in log-log scale.

Figure 3

Log-log plot of the correlation length $\xi$ as a function of the duration $\tau$ of the linear ramp for initial transverse field $g_0=0.3$ and different final values of $g_1$. Numerical results are compared with the predictions of adiabatic perturbation theory at two different orders.