Quench of a symmetry-broken ground state

We analyze the problem of how different ground states associated with the same set of Hamiltonian parameters evolve after a sudden quench. To realize our analysis we define a quantitative approach to the local distinguishability between different ground states of a magnetically ordered phase in terms of the trace distance between the reduced density matrices obtained by projecting two ground states in the same subset. Before the quench, regardless of the particular choice of subset, any system in a magnetically ordered phase is characterized by ground states that are locally distinguishable. On the other hand, after the quench, the maximum distinguishability shows an exponential decay in time. Hence, in the limit of very long times, all the information about the particular initial ground state is lost even if the systems are integrable. We prove our claims in the framework of the magnetically ordered phases that characterize both the $XY$ and the $N$-cluster Ising models. The fact that we find similar behavior in models within different classes of symmetry makes us confident about the generality of our results.

Figure 1

Behavior of the local distinguishability for two sudden quenches for the $XY$ model. In the upper panel we report our result for the case in which, with ${\gamma }_0={\gamma }_1=0.5$, the external field is quenched from $h_0=0.2$ to $h_1=0.8$. In the lower panel, always taking ${\gamma }_0={\gamma }_1=0.5$, we quench the external field from $h_0=0.4$ to $h_1=1.2$. In both cases, we show the behavior of the maximal distance $D_S\left(t\right)$ as a function of the time $t$ for $S$ made by one single spin (dotted black line), two neighbor spins (dot-dashed blue line), two next-neighbor spins (dashed red line), and three spins (solid green lines). Top-right inset: A zoom for very short times in which the transient is highlighted. Bottom-left inset: Behaviors, as a function of time $t$, of the absolute value of the magnetization along the $x \langle {\sigma }_i^x\rangle$ (solid black line) and along $y \langle {\sigma }_i^y\rangle$ (solid red line). In all our numerical evaluations we have used Planck units ($\hslash =1$) and we have assumed $J=1$. Therefore the time is dimensionless.

Figure 2

Time scale $\tau$, as a function of the final external field $h_1$, of the exponential decay of $D_S\left(t\right)$ for a quench in the $XY$ model that involves only the external field for several sets of initial parameters. Black circles represent the case in which ${\gamma }_0={\gamma }_1=0.8$ and the initial value of the external field is $h_0=0.2$; red squares show the case in which ${\gamma }_0={\gamma }_1=0.5$ and $h_0=0.5$; blue stars represent the case in which ${\gamma }_0={\gamma }_1=0.2$ and $h_0=0.8$. Having fixed the strength of the interaction terms of the Hamiltonian to 1 and having used Planck units, we have that both the time and the external field are dimensionless.

Figure 3

Behavior of the distinguishability for two sudden quenches for the $N$-cluster Ising models with $N=1$. In the upper panel we report our result for the case in which $\phi$ is sudden-quenched from ${\phi }_0=\frac{5}{16}\pi$ to ${\phi }_1=\frac{7}{16}\pi$, while in the lower panel, it is quenched from ${\phi }_0=\frac{3}{8}\pi$ to ${\phi }_1=\frac{1}{8}\pi$. In both panels we can see the behavior of the maximal distance $D_S\left(t\right)$ as a function of the time $t$ for $S$ made by one single spin, two neighbor spins, and two next-neighbor spins (dotted black line) and by three spins (solid red line). Inset: Zoom of the main inset for very short times in which the transient is highlighted. In these plots, having used Planck units ($\hslash =1$) and having assumed $J=1$, all the quantities are dimensionless.

Figure 4

Behavior of the absolute values of the differences $\mathrm{\Delta }$ between the values of the four broken-symmetry correlation functions obtained choosing $r_{\max }=r+\mathrm{\Delta }r$ and $r_{\max }=r$, as a function of the time for a fixed set of the Hamiltonian parameters $\gamma =0.8, h_0=0.2$, and $h_1=0.8$. We have arbitrarily chosen $\mathrm{\Delta }r=10$. Black circles (leftmost curve), $r=20$; red squares, $r=40$; blue stars, $r=60$; green upward triangles, $r=80$; orange downward triangles (rightmost curve), $r=100$. From the top left, in clockwise order, we have plotted the difference for the correlation functions $\langle {\sigma }_i^x\rangle , \langle {\sigma }_i^y\rangle , \langle {\sigma }_i^y{\sigma }_{i+1}^z\rangle$, and $\langle {\sigma }_i^x{\sigma }_{i+1}^z\rangle$. The lines at ${10}^{-9}$ indicate the border between computational noise and significant differences. Having fixed the strength of the interaction terms of the Hamiltonian to 1 and having used Planck units, we have that the time is dimensionless as well as the distance between the different expectation values.