Scaling behavior of quantum critical relaxation dynamics of a system in a heat bath

We study the scaling behavior of the relaxation dynamics to thermal equilibrium when a quantum system is near the quantum critical point. In particular, we investigate systems whose relaxation dynamics is described by a Lindblad master equation. We find that the universal scaling behavior not only appears in the equilibrium stage at the long-time limit but also manifests in the nonequilibrium relaxation process. While the critical behavior is dictated by the low-lying energy levels of the Hamiltonian, the dissipative part in the Lindblad equation also plays important roles in two aspects: First, the dissipative part makes the high-energy levels decay fast, after which the universal behavior controlled by the low-lying modes emerges. Second, the dissipation rate gives rise to a time scale that affects the scaling behavior. We confirm our theory by solving the Lindblad equation for the one-dimensional transverse-field Ising model.

Figure 1

Evolution of $M$ for $L=7$ in various cases. Initial magnetization is chosen as $M_0=1$. Inset: Fitting of curve for the pure dissipative case.

Figure 2

At zero temperature, the $M$ versus $t$ curves for various lattice sizes in (a) overlap well in (b) when $M$ and $t$ are rescaled with $L$. $A=0.1$ for different lattice sizes.

Figure 3

In the presence of symmetry-breaking field, the $M$ versus $t$ curves for various lattice sizes and fixed $T{L}^z\simeq 2.1$ and $h{L}^{\beta \delta /\nu }\simeq 0.573$ in (a) overlap well in (b) when $M$ and $t$ are rescaled with $L$. $A=0.1$ for different lattice sizes.

Figure 4

(a) The $M$ versus $t$ curves for various lattice sizes and fixed $T{L}^z\simeq 2.1, h{L}^{\beta \delta /\nu }\simeq 0.573$, and $c{L}^z\simeq 1.2$. (b) After rescaling, the outlines of these curves collapse well, but the high-frequency part cannot collapse.