Fluctuation-dissipation relations and critical quenches in the transverse field Ising chain

Dynamic correlation and response functions of classical and quantum systems in thermal equilibrium are connected by fluctuation-dissipation theorems, which allow an alternative definition of their (unique) temperature. Motivated by this fundamental property, we revisit the issue of thermalization of closed many-body quantum systems long after a sudden quench, focusing on the nonequilibrium dynamics of the Ising chain in a critical transverse field. We show the emergence of distinct observable-dependent effective temperatures, which rule out Gibbs thermalization in a strict sense but might still have a thermodynamic meaning.

Figure 1

Comparison between the effective temperature defined via the frequency-domain FDR applied to ${\sigma }_i^z$ (yellow solid line) and $M$ (blue dash-dotted line), and the energy definition (black dashed horizontal line) for ${\Gamma }_0=0.3$. The inset highlights the logarithmic divergence of $T_{\mathrm{eff}}^z\left(\omega \right)$ for $\omega \to 0$.

Figure 2

Decay of the order-parameter correlation $C_{+}^x$ (black line) and the linear response $R^x$ (yellow line) for ${\Gamma }_0=0.3$. Upper inset: zoom into the long-$t$ decay that demonstrates the exponential relaxation with the characteristic time $\tau$ defined in the text (dashed green line). Lower inset: $(e^{t/\tau }{C}_{+}^x/A_C-1)/a_C$ and $(e^{t/\tau }{R}^x/A_R-1)/a_r$ vs $t$; the green dashed line is the $t^{-1/2}$ envelope of the damped oscillations, in agreement with Eqs. (11) and (12).

Figure 3

${\Gamma }_0$-dependence of the order parameter effective temperature $T_{\mathrm{eff}}^x$ compared to $T_{\mathrm{eff}}^E$ defined from the energy [see Eq. (3)] (dashed line). The solid lines, from bottom to top, indicate the values determined on the basis of the classical limit of the FDR in the time domain, of the limit $\omega \to 0$ of the frequency-domain FDR, of Eq. (13), and of the limit $\omega \to 0$ of the frequency-domain FDR but for spins separated by a distance $r=10$. The dependence on ${\Gamma }_0$ can be read from the one of $\tau$ and Eq. (13).