Thermalization Dynamics Close to a Quantum Phase Transition

We investigate the dissipative dynamics of a quantum critical system in contact with a thermal bath, focusing on the response of the system to a sudden change of the bath temperature, in analogy to studies of aging. The specific example of the $XY$ model in a transverse magnetic field whose spins are locally coupled to a set of bosonic baths is considered. We analyze the spin-spin correlations and block correlations and identify some universal features in the out-of-equilibrium dynamics. Two distinct regimes, characterized by different time and length scales, emerge. The initial transient dynamics is characterized by the same critical exponents as those of the equilibrium quantum phase transition and resembles the dynamics of thermal phase transitions. At long times equilibrium is reached through the propagation along the chain of a thermal front in a manner similar to the classical Glauber dynamics.

Figure 1

Correlation function $C_{zz}$ for a quench from $T=\infty$ to $T=0.1$ at $h=1$; red thick line on the right at large $t$ is the thermal equilibrium $C_{zz}(R)$. During the initial transient ($t\ll \alpha$), $C_{zz}\propto t^2$ for all $R$, as marked by the fits on the left.

Figure 2

Initial transient: snapshots of $C_{zz}$ (left) and ${\partial }_L\mathcal{I}$ (right) at a fixed $t/\alpha \ll 1$ after a quench from $T=\infty$ to $T=0.1$ and for $h=0.8$, 0.9, 0.95, 0.975, 1 (from bottom to top). The spikes relative to $C_{zz}$ in the left panel mark the distances ${\xi }_t(h)$ at which $C_{zz}$ changes sign. Dashed lines are plotted for comparison. Inset: ${\xi }_t$ as a function of $|h-h_c|$; for ${\partial }_L\mathcal{I}$, ${\xi }_t$ is calculated as the maximum of ${\partial }_L^2\mathcal{I}$ which marks the crossover between $L^{-1}$ and $L^{-4}$ scaling.

Figure 3

Thermalization of the spin-spin correlation function $C_{zz}$ after a quench from $T=\infty$ to $T=0.1$ at $h=1$. Left: snapshots of $C_{zz}(R)$ at $t/\alpha =10$, 20, 30, 40 from top to bottom; thick red line is the (exponential) thermal equilibrium $C_{zz}^{\mathrm{th}}$. At a given time after the quench $C_{zz}$ is thermalized up to a distance $R_{\mathrm{th}}$ that increases with time. Right: corresponding time dependence of $R_{\mathrm{th}}$; the linear fit gives $v_{\mathrm{th}}\simeq 0.32$.

Figure 4

Thermal front velocity $v_{\mathrm{th}}$ as a function of $1/T$. From top to bottom $h=1$, 0.9, 1.2 (so that $\Delta =0$, 0.1, 0.2). Lines are the fit $Ta(1+b\frac{\Delta }{T})e^{-\Delta /T}$ with $a=3.3$, $b=0.9$ for the specific case $\gamma =1$, $s=1$.