Effective Thermal Dynamics Following a Quantum Quench in a Spin Chain

We study the nonequilibrium dynamics of the quantum Ising model following an abrupt quench of the transverse field. We focus on the on-site autocorrelation function of the order parameter, and extract the phase-coherence time ${\tau }_Q^{\phi }$ from its asymptotic behavior. We show that the initial state determines ${\tau }_Q^{\phi }$ only through an effective temperature set by its energy and the final Hamiltonian. Moreover, we observe that the dependence of ${\tau }_Q^{\phi }$ on the effective temperature fairly agrees with that obtained in thermal equilibrium as a function of the equilibrium temperature.

Figure 1

Time dependence of $|{\rho }_Q^{xx}(t)|$ for a quench to a final ferromagnet $\Gamma =0.5$. Different curves, obtained numerically for a finite chain of $L=600$ sites, correspond to different initial ${\Gamma }_0$’s. Inset: $|{\rho }_Q^{xx}(t)|\sqrt{t}$ for $\Gamma =1.25$.

Figure 2

Phase-coherence time ${\tau }_Q^{\phi }$ as a function of the initial transverse field ${\Gamma }_0$. The different curves refer to a ferromagnetic ($\Gamma =0.5$, red circles), a critical ($\Gamma =1$, blue triangles) and a paramagnetic ($\Gamma =1.25$, green squares) quench dynamics. Inset: effective temperature $T_{\mathrm{eff}}$ vs ${\Gamma }_0$, as extracted from Eq. (3), for the same values of $\Gamma$.

Figure 3

Phase-coherence time ${\tau }_Q^{\phi }$ as a function of $T_{\mathrm{eff}}$. Different symbols are for various values of $\Gamma$. Empty symbols correspond to an initial field ${\Gamma }_0<\Gamma$, while filled ones are for ${\Gamma }_0>\Gamma$. Straight lines denote the finite-temperature equilibrium values of ${\tau }_{T=T_{\mathrm{eff}}}^{\phi }$. The inset shows the finite-size scaling of ${\tau }_Q^{\phi }$ at criticality. Data in the main panel are for $L=600$ (except for $\Gamma =1$ and $T_{\mathrm{eff}}<0.07$, where $L=1000$).

Figure 4

Phase-coherence time at $\Gamma =0.5$ computed according to the quasiparticle distribution $f_k$ (${\tilde{\tau }}_Q^{\phi }$, blue squares) and to the effective quasiparticle distribution $n_k(T_{\mathrm{eff}})$ (${\tilde{\tau }}_T^{\phi }$, red circles).