Long time dynamics following a quench in an integrable quantum spin chain: Local versus nonlocal operators and effective thermal behavior

We study the dynamics of the quantum Ising chain following a zero-temperature quench of the transverse field strength. Focusing on the behavior of two-point spin correlation functions, we show that the correlators of the order parameter display an effective asymptotic thermal behavior; i.e., they decay exponentially to zero with a phase coherence rate and a correlation length dictated by the equilibrium law with an effective temperature set by the energy of the initial state. On the contrary, the two-point correlation functions of the transverse magnetization or the density-of-kinks operator decay as a power law and do not exhibit thermal behavior. We argue that the different behavior is linked to the locality of the corresponding operator with respect to the quasiparticles of the model: nonlocal operators, such as the order parameter, behave thermally, while local ones do not. We study which features of the two-point correlators are a consequence of the integrability of the model by analyzing their robustness with respect to a sufficiently strong integrability-breaking term.

Figure 1

(Color online) (a) Phase coherence time and (b) correlation length extracted from the asymptotic decay of the two-point order parameter correlations as a function of the effective temperature. Data are for quenches ending in the ferromagnetic phase of the Ising chain at $\Gamma =0.5$ [see Eq. (3)]. Symbols refer to the case of a quench and correspond to different values of ${\Gamma }_0$ (empty symbols are for ${\Gamma }_0<\Gamma$, while filled ones are for ${\Gamma }_0>\Gamma$); continuous red curves denote the equilibrium values at finite temperatures, while dotted blue curves correspond to values obtained with a semiclassical analysis generalized to nonequilibrium cases.

Figure 2

With respect to the modes of the Hamiltonian at $t>0$, the boundary state $|\mathcal{B}⟩$ appears as a coherent superposition of an infinite number of pairs of particles with equal and opposite momentum.

Figure 3

(Color online) Absolute value of the on-site time-dependent correlation function ${\rho }_Q^{xx}\left(t,0\right)$ for quenches terminating in the ferromagnetic ($\Gamma =0.5$—left panel) and in the paramagnetic phase ($\Gamma =1.25$—right panel). The various curves are obtained for different values of ${\Gamma }_0$, as indicated in the graphs. The black curve corresponds to the equilibrium case (in the left panel, the horizontal dotted line indicates the asymptotic value ${⟨{\sigma }^x⟩}^2$, while the dashed one denotes the power-law envelope $\sim t^{-1}$). Colored curves stand for different quenches ${\Gamma }_0\ne \Gamma$; the straight dashed lines indicate the respective leading exponential decays. In the paramagnetic case, curves are rescaled by the zero-temperature $1/\sqrt{t}$ prefactor, analogous to the $K\left(t,0\right)$ term in Eq. (58).

Figure 4

(Color online) Phase coherence time as a function of the effective temperature in the off-critical region for different values of $\Gamma$ (main panel) and at criticality (inset). Symbols refer to ${\tau }_Q^{\phi }$ (here, as in Fig. 1, empty symbols stand for quenches starting with ${\Gamma }_0<\Gamma$; filled ones are for ${\Gamma }_0>\Gamma$), while continuous curves denote the equilibrium values at finite temperatures ${\tau }_T^{\phi }$, Eq. (55). For the system at criticality, we performed a finite-size scaling of ${\tau }_Q^{\phi }$ to study the asymptotic agreement with the equilibrium law ${\tau }_T^{\phi }=\frac{8}{\pi T}$ (black line). (data for $\Gamma =0.5$, 0.75, 1, 1.25 acknowledged from Fig. 3 of Ref. 31).

Figure 5

(Color online) Space-dependent correlation function ${\rho }_Q^{xx}\left(0,r\right)$ for quenches in the ferromagnetic phase (left panel, $\Gamma =0.5$) and in the paramagnetic phase (right panel, $\Gamma =1.25$). The various data sets are for different values of ${\Gamma }_0$, as explicitly indicated near each curve. On the right panel, dashed colored lines are exponential fits of the curves in the large-$r$ limit, while the dotted-dashed black curve represents $\mathcal{K}\left(r\right)$ at the corresponding $\Gamma$. In the insets we plot the correlation length ${\xi }_Q^{\phi }$ as a function of ${\Gamma }_0$, as extracted from the exponential decay in the space of correlations.

Figure 6

(Color online) Distance $r^{\ast }$ at which the first maximum in the oscillations for the spatial correlations crossing the critical point from the disordered phase appears as a function of $\Gamma <1$ and for fixed ${\Gamma }_0>1$. Such value is a measure of the period of oscillations. In the inset we show ${\rho }_Q^{xx}\left(r\right)$ for a paramagnetic-to-ferromagnetic quench. The oscillations are superimposed to an exponential decay behaving thermally, which is depicted with a straight red line.

Figure 7

(Color online) Correlation length as a function of the effective temperature in the off-critical region for different values of $\Gamma$ (main panel) and at criticality (inset). Symbols refer to the quenched situation ${\xi }_Q^{\phi }$; continuous curves denote the equilibrium values at finite temperatures ${\xi }_T^{\phi }$, Eq. (56), while dotted lines are obtained with a semiclassical analysis for the system out of equilibrium, Eq. (65). In the inset we show a finite-size scaling of data at criticality; the straight black curve indicates a $T_{\text{eff}}^{-1}$ behavior, and it is plotted as a guideline.

Figure 8

(Color online) Phase coherence times extracted by means of the semiclassical argument; data are shown for two values of $\Gamma$ in the ferromagnetic phase. The continuous red curves indicate ${\tilde{\tau }}_T^{\phi }$ which is computed using the effective quasiparticle distribution $n_k$, while the dotted blue ones are for ${\tilde{\tau }}_Q^{\phi }$ and are computed according to the occupation factor $f_k$. We also show the phase coherence times ${\tau }_Q^{\phi }$ evaluated numerically from the nonequilibrium situations (symbols—same data of Fig. 4). In the inset we plot $n_k$ and $f_k$ for a particular value of $\left({\Gamma }_0=0.75,\Gamma =0.5\right)$, corresponding to the point in the main panel that is indicated with an arrow.

Figure 9

(Color online) Asymptotic value reached by the transverse-field correlation functions, which is the square of the magnetization $⟨{\sigma }^z⟩$. The magnetization for a quenched dynamics (symbols) corresponding to a given effective temperature, as defined in Eq. (61), is compared to the corresponding value at equilibrium at the same temperature (continuous lines). Different colors and symbols stand for the various values of $\Gamma$, as declared in the legend: $\Gamma =0.5$ (black circles), 0.75 (red squares), 1 (green diamonds), and 1.25 (blue triangles). Inset: ${⟨{\sigma }^z⟩}_Q$ reached asymptotically after a quench from a given ${\Gamma }_0$ to $\Gamma =0$. Notice that the corresponding equilibrium value ${⟨{\sigma }^z⟩}_T$ for $\Gamma =0$ is rigorously zero at any temperature.

Figure 10

(Color online) Asymptotic value of the density of kinks as a function of the effective temperature $T_{\text{eff}}$. Different colors stand for various values of $\Gamma$, as shown in the inset. Symbols denote the density of kinks ${⟨{\rho }^{\mathcal{N}}⟩}_Q$ after a quench; to each value of ${\Gamma }_0$ corresponds a different initial state, therefore a different effective temperature $T_{\text{eff}}$. Straight lines denote the finite-temperature equilibrium values ${⟨{\rho }^{\mathcal{N}}⟩}_{T=T_{\text{eff}}}$.

Figure 11

(Color online) Fourier transform of the equal-time order-parameter correlation function for the Ising model in presence of a non-integrable perturbation $J_2$. Symbols connected by lines denote $n_Q^{xx}\left(k\right)$ after a quench from ${\Gamma }_0=5$ to $\Gamma =0.4$, according to the diagonal ensemble prediction. Dashed/dotted lines stand for the corresponding expectation values in the canonical ensemble $n_{T_{\text{eff}}}^{xx}\left(k\right)$ at the corresponding effective temperature $T_{\text{eff}}$. Inset: absolute differences $\delta n^{xx}\left(k\right)$ between the diagonal and the canonical ensemble predictions. Data are for a chain with $L=12$ sites.