Relaxation of Antiferromagnetic Order in Spin-$1/2$ Chains Following a Quantum Quench

We study the unitary time evolution of antiferromagnetic order in anisotropic Heisenberg chains that are initially prepared in a pure quantum state far from equilibrium. Our analysis indicates that the antiferromagnetic order imprinted in the initial state vanishes exponentially. Depending on the anisotropy parameter, oscillatory or nonoscillatory relaxation dynamics is observed. Furthermore, the corresponding relaxation time exhibits a minimum at the critical point, in contrast to the usual notion of critical slowing down, from which a maximum is expected.

Figure 1

Dynamics of the staggered magnetization $m_s(t)$ in (a) the $XXZ$ chain and (b) the $XZ$ chain. Symbols correspond to numerical results, and lines represent analytical results or fits by corresponding laws (see text). For $\Delta =0$ the typical behavior of the error is illustrated by comparing the numerical iTEBD result with 2400 retained states to the exact curve: The absolute deviation from the exact curve is less than ${10}^{-6}$ for $t<t_{\mathrm{runaway}}$. For $\Delta \ne 0$ data beyond $t_{\mathrm{runaway}}$ is omitted.

Figure 2

Absolute value of the staggered magnetization in the $XXZ$ model. Symbols represent numerical results, solid curves correspond to fits by the exponential law (3), and straight lines point out the exponential decay.

Figure 3

(a) Focus on the $XXZ$ chain close to the critical point $\Delta =1$. (b) Comparison of the $XXZ$ chain (symbols) and the $XZ$ chain (dashed lines) for strong anisotropies; solid lines correspond to an exponential fit. The dynamics of the staggered magnetization of the $XXZ$ and $XZ$ chains converge towards each other in the large-$\Delta$ limit.

Figure 4

Relaxation time $\tau$ and oscillation period $T=\frac{2\pi }{\omega }$ as a function of anisotropy in the $XXZ$ and $XZ$ models. Logarithmic or algebraic laws are emphasized by solid lines. In the region close to the critical point of the $XXZ$ model (indicated by the question mark), it becomes impossible to extract a relaxation time from the numerical results.