Destruction of string order after a quantum quench

We investigate the evolution of string order in a spin-1 chain following a quantum quench. After initializing the chain in the Affleck-Kennedy-Lieb-Tasaki state, we analyze in detail how string order evolves as a function of time at different length scales. The Hamiltonian after the quench is chosen either to preserve or to suddenly break the symmetry which ensures the presence of string order. Depending on which of these two situations arises, string order is either preserved or lost even at infinitesimal times in the thermodynamic limit. The fact that nonlocal order may be abruptly destroyed, what we call string-order melting, makes it qualitatively different from typical order parameters in the manner of Landau. This situation is thoroughly characterized by means of numerical simulations based on matrix product states algorithms and analytical studies based on a short-time expansion for several simplified models.

Figure 1

Results of the numerical simulations described in Sec. 3b. We simulate $L=60$ sites with OBC. (Upper panels) Expectation value of the string operator along the $y$ axis as a function of the string length $l$ and at fixed time after a quench using the Hamiltonian in Eq. (6) with $\mathrm{\Delta }=0.2$ and odd perturbation ${\sum }_n{\widehat{S}}_n^x$ for four different values of the coupling constant, here indicated with $b_x$. Each line corresponds to a different time, from $t=0$ (dark red) to $t=2.5J^{-1}$ (bright red) with time spacing $0.1J^{-1}$. For short lengths a steady region appears, whose size increases with time, and for longer lengths SO melting in the bulk is observed. (Lower panels) Expectation value of the string operator along the $y$ axis as a function of time at fixed string length, from $l=1$ (dark green) to $l=35$ (bright green). For $b_x=0.2$, the expectation value of the string operator becomes independent of time (a local stationary state is reached) at short lengths. For larger values of $b_x$, simulations are too short to reveal a steady region at short lengths. Thermalization of SO at short lengths to the corresponding canonical ensemble with effective temperature $\beta \simeq 1.5J^{-1}$ is reported in Fig. 2 for $b_x=0.2$.

Figure 2

Thermalization analysis of the data in Fig. 1. The system is quenched with Hamiltonian (6) by using $\mathrm{\Delta }=0.2$ and $b_x=0.2$ (see Fig. 1, leftmost panels). The thermal average of the expectation value of the string operator along the $x$ axis as a function of string length (blue thick line) with inverse temperature $\beta \simeq 1.5J^{-1}$ is overlapped to the expectation value of the string operator along the $x$ axis as a function of the string length, computed at fixed time $t=2.28J^{-1}$ (red dots). In the inset we show the same analysis for the expectation value of the string operator along the $y$ axis. In both cases, we numerically observe thermalization of SO at short lengths.

Figure 3

Numerical simulations for the $\epsilon$ and string-length dependence of the expectation value of the string operator. As in Fig. 1, the system is quenched with Hamiltonian (6). We use $L=60$ and $\mathrm{\Delta }=0.2$. (Left) Expectation value of the string operator along $y$ as a function of the coupling constant at time $t=1.6J^{-1}$ and string length $l=30$ to catch bulk value of the expectation value of the string operator, away from the thermalized region. The quadratic behavior predicted by Eq. (22) is confirmed by the quadratic fit (solid line). A similar behavior is seen for the expectation value of the string operator along $x$ and $z$ (insets). (Right) Expectation value of the string operator along $y$ in the bulk for the perturbation as in the left panel, at time $t=1.6J^{-1}$ and coupling constant $\epsilon =0.07$. SO along $y$ (red dots) linearly melts in $l$, as confirmed by the linear fit (red straight line). This is consistent with the fact that an exponential decay is observed away from the perturbative regime.

Figure 4

Quench of the pure AKLT model. Expectation value of the string operator $〈{\widehat{\mathcal{O}}}_l^{\left(\alpha \right)}〉\left(t\right)$ for the perturbation ${\widehat{G}}_x={\sum }_i{\widehat{S}}_i^y{\widehat{S}}_i^z+{\widehat{S}}_i^z{\widehat{S}}_i^y$ at $\epsilon =0.05$ and chain length $L=60$. (Top left) Data for $\alpha =x$ as a function of string length. No dependence on the string length arises as in Eq. (27). Curves are taken at fixed $t$, from $t=0$ (dark red uppermost curve) to $t=0.04J^{-1}$ (bright red lowermost curve), with time spacing $\mathrm{\Delta }t=0.004J^{-1}$. (Top right) Data for $\alpha =x$ as a function of time; curves for different $l$ are superimposed. A quadratic scaling is observed with space-independent curvature as in Eq. (27). (Bottom left) Data for $\alpha =y$ as a function of string length. A linear scaling is observed with time-dependent curvature as in Eq. (26). (Bottom right) Data for $\alpha =y$ as a function of time. A quadratic scaling is observed with space-dependent curvature as in Eq. (26). Curves are taken at fixed $l$, from $l=1$ (dark green uppermost curve) to $l=35$ (bright green lowermost curve), with length spacing $\mathrm{\Delta }l=1$. Color coding in bottom panels is the same as in upper panels.

Figure 5

Same simulation as in Fig. 4, but away from the perturbative regime. We display the long-time behavior of the expectation value of the string operator along the $x$ axis (top panels) and $y$ axis (bottom panels). Curves for string length dependence are taken from $t=0$ (dark red curve, left panels) to $t=4.48J^{-1}$ (bright red curve, left panels) by varying $\mathrm{\Delta }t=0.08J^{-1}$ for each curve. Curves for time dependence (right panels) are taken at fixed string length each, as in Fig. 4. Unfortunately, simulations are not able to reveal a thermalized region, due to the entanglement growth in time.

Figure 6

Quench of the pure AKLT model. Expectation value of the string operator along $x$ axis (red crosses), along the $y$ axis (blue circles), and along the $z$ axis (magenta dots, over the blue circles) as a function of the coupling constant $\epsilon$ for ${\widehat{G}}_x$ perturbation. In each case data are fitted by a parabola, confirming the quadratic scaling predicted in Eqs. (26) and (27). Data are taken at fixed time $t=0.04J^{-1}$ and string length $l=30$ with $L=60$. Numerical values of the curvature extracted by the quadratic fits are $-0.002839$ ($x$ axis), $-0.02816$ ($y$ axis), and $-0.02819$ ($z$ axis). These values can be compared to those predicted by Eqs. (26) and (27) which are $-0.002844$ ($x$ axis) and $-0.02821$ ($y$ and $z$ axes). Analytical predictions are confirmed.