Entanglement in a time-dependent coupled $\mathit{XY}$ spin chain in an external magnetic field

We consider an infinite one-dimensional anisotropic $\mathit{XY}$ spin chain with a nearest-neighbor time-dependent Heisenberg coupling $J(t)$ between the spins in presence of a time-dependent magnetic field $h(t)$. We discuss a general solution for the system and present an exact solution for particular choice of $J$ and $h$ of practical interest. We investigate the dynamics of entanglement for different degrees of anisotropy of the system and at both zero and finite temperatures. We find that the time evolution of entanglement in the system shows nonergodic and critical behavior at zero and finite temperatures and different degrees of anisotropy. The asymptotic behavior of entanglement at the infinite time limit at zero temperature and constant $J$ and $h$ depends only the parameter $\lambda =J/h$ rather than the individual values of $J$ and $h$ for all degrees of anisotropy but changes for nonzero temperature. Furthermore, the asymptotic behavior is very sensitive to the initial values of $J$ and $h$ and for particular choices we may create finite asymptotic entanglement regardless of the final values of $J$ and $h$. The persistence of quantum effects in the system as it evolves and as the temperature is raised is studied by monitoring the entanglement. We find that the quantum effects dominate within certain regions of the $\mathit{kT}$-$\lambda$ space that vary significantly depending on the degree of the anisotropy of the system. Particularly, the quantum effects in the Ising model case persist in the vicinity of both its critical phase transition point and zero temperature as it evolves in time. Moreover, the interplay between the different system parameters to tune and control the entanglement evolution is explored.

Figure 1

Dynamics of the nearest-neighbor concurrence $C(i,i+1)$, where $t$ is in units of $J_1^{-1}$, with $\gamma =1$ and (a) $h_0=h_1=1$ for various values of $J_0$ and $J_1$ at $\mathit{kT}=0$; (b) $J_0=J_1=1$ for various values of $h_0$ and $h_1$ at $\mathit{kT}=0$. (c) Dynamics of the magnetization per spin; (d) dynamics of the spin-spin correlation function in the $z$ direction for fixed $h=h_0=h_1=1$ for various values of $J_0$ and $J_1$ at $\mathit{kT}=0$ with $\gamma =1$.

Figure 2

$C(i,i+1)$ as a function of $\lambda$ for $h=h_0=h_1$ and $J=J_0=J_1$ at (a) $\mathit{kT}=0$ with any combination of $J$ and $h$; (b) $\mathit{kT}=1$ with $h_0=h_1=0.25,1,4$; (c) $\mathit{kT}=1$ with $J_0=J_1=0.25,1,4$; (d) $\mathit{kT}=3$ with $h_0=h_1=0.25,1,4$.

Figure 3

$C(i,i+1)$ as a function of ${\lambda }_1$ and $t$, in units of $J_1^{-1}$, at $\mathit{kT}=0$ with $\gamma =1$, $h_0=h_1=1$, and (a) $J_0=1$; (b) $J_0=5$. (c) The asymptotic behavior of $C(i,i+1)$ as a function of ${\lambda }_1$ with $\gamma =1$, $h_0=h_1=1$, and $J_0=0.5,1,2$ at $\mathit{kT}=0$; (d) dynamics of $C(i,i+1)$ with $\gamma =1$, $h_0=h_1=1$, $J_0=1,2$, and $J_1=0$ at $\mathit{kT}=0$.

Figure 4

The asymptotic behavior of $C(i,i+1)$ as a function of (a) $J_0$ and $J_1$ with $\gamma =1$ and $h_0=h_1=1$; (b) $h_0$ and $h_1$ at $\mathit{kT}=0$ with $\gamma =1$ and $J_0=J_1=2$; (c) $h_0$ and $J_0$ at $\mathit{kT}=0$ with $\gamma =1$ and $h_1=J_1=1$; (d) $h_1$ and $J_1$ at $\mathit{kT}=0$ with $\gamma =1$ and $h_0=J_0=1$, where $h_0$, $h_1$, and $J_0$ are in units of $J_1$.

Figure 5

The asymptotic behavior of $C(i,i+1)$ as a function of (a) $\lambda$ and $\mathit{kT}$, in units of $J_1$, with $\gamma =1$, $h_0=h_1=1$, and $J_0=J_1$; (b) ${\lambda }_1$ and $\mathit{kT}$ with $\gamma =1$, $h_0=h_1=1$, and $J_0=1$.

Figure 6

Dynamics of $C(i,i+2)$, where $t$ is in units of $J_1^{-1}$, with $\gamma =1$ and $h_0=h_1=1$ for various values of $J_0,J_1$ at $\mathit{kT}=0$ and $\mathit{kT}=0.1$.

Figure 7

Dynamics of the nearest-neighbor concurrence $C(i,i+1)$, where $t$ is in units of $J_1^{-1}$, with $\gamma =0.5$ and (a) $h_0=h_1=1$ for various values of $J_0$ and $J_1$ at $\mathit{kT}=0$; (b) $J_0=J_1=1$ for various values of $h_0$ and $h_1$ at $\mathit{kT}=0$. (c) Dynamics of the magnetization per spin and (d) the spin-spin correlation function in the $z$ direction for fixed $h=h_0=h_1=1$ for various values of $J_0$ and $J_1$ at $\mathit{kT}=0$ with $\gamma =0.5$.

Figure 8

The asymptotic behavior of $C(i,i+1)$ with $\gamma =0.5$ as a function of $\lambda$ when $h_0=h_1$ and $J_0=J_1$ at (a) $\mathit{kT}=0$ with any combination of constant $J$ and $h$; (b) $\mathit{kT}=1$ with $h_0=h_1=0.25,1,4$; (c) $\mathit{kT}=1$ with $J_0=J_1=0.25,1,4$; (d) $\mathit{kT}=3$ with $h_0=h_1=0.25,1,4$.

Figure 9

$C(i,i+1)$ as a function of ${\lambda }_1$ and $t$, in units of $J_1^{-1}$, at $\mathit{kT}=0$ with $\gamma =0.5$, $h_0=h_1=1$ and (a) $J_0=1$; (b) $J_0=5$. The asymptotic behavior of $C(i,i+1)$ as a function of (c) $J_0$ and $J_1$ at $\mathit{kT}=0$ with $\gamma =0.5$ and $h_0=h_1=1$; (d) $h_0$ and $h_1$ at $\mathit{kT}=0$ with $\gamma =0.5$ and $J_0=J_1=1$, where $h_0$, $h_1$, and $J_0$ are in units of $J_1$.

Figure 10

The asymptotic behavior of $C(i,i+1)$ as a function of (a) $\lambda$ and $\mathit{kT}$, in units of $J_1$, with $\gamma =0.5$, $h_0=h_1=1$, and $J_0=J_1$; (b) ${\lambda }_1$ and $\mathit{kT}$ with $\gamma =0.5$, $h_0=h_1=1$, and $J_0=1$.

Figure 11

Dynamics of $C(i,i+2)$, where $t$ is in units of $J_1^{-1}$, with $\gamma =0.5$ and $h_0=h_1=1$ for various values of $J_0,J_1$ at $\mathit{kT}=0$ and $\mathit{kT}=0.25$.

Figure 12

Dynamics of the nearest-neighbor concurrence $C(i,i+1)$, where $t$ is in units of $J_1^{-1}$, with $\gamma =0$ for various values of $J_0,J_1,h_0$, and $h_1$ at $\mathit{kT}=0$. (b) Dynamics of the magnetization per spin and the spin-spin correlation function in the $z$ direction for fixed $h=h_0=h_1=1$ for various values of $J_0$ and $J_1$ at $\mathit{kT}=0$ with $\gamma =0$.

Figure 13

The asymptotic behavior of $C(i,i+1)$ with $\gamma =0$ as a function of $\lambda$ when $h_0=h_1$ and $J_0=J_1$ at (a) $\mathit{kT}=0$ with any combination of constant $J$ and $h$; (b) $\mathit{kT}=1$ with $h_0=h_1=0.25,1,4$; (c) $\mathit{kT}=1$ with $J_0=J_1=0.25,1,4$; (d) $\mathit{kT}=3$ with $h_0=h_1=0.25,1,4$.

Figure 14

(a) $C(i,i+1)$ as a function of ${\lambda }_1$ and $t$, in units of $J_1^{-1}$, at $\mathit{kT}=0$ with $\gamma =0$, $h_0=h_1=1$ and $J_0=5$; (b) $C(i,i+1)$ as a function of ${\lambda }_0$ and $t$ at $\mathit{kT}=0$ with $\gamma =0$, $h_0=h_1=1$ and $J_1=5$.

Figure 15

The asymptotic behavior of $C(i,i+1)$ as a function of (a) ${\lambda }_0$ and $\mathit{kT}$, in units of $J_1$, with $\gamma =0$, $h_0=h_1=1$, and $J_1=1$; (b) $\mathit{kT}$ with $\gamma =0$, $h_0=h_1=1$, and $J_0=J_1=1$.

Figure 16

Dynamics of $C(i,i+2)$, where $t$ is in units of $J_1^{-1}$, with $\gamma =0$ and $h_0=h_1=1$ for various values of $J_0,J_1$ at $\mathit{kT}=0$ and $\mathit{kT}=0.1$.

Figure 17

(a) The asymptotic behavior of $C(i,i+1)$ as a function of ${\lambda }_1$ and $\gamma$ with $h_0=h_1=1$ and $J_0=1$ at $\mathit{kT}=0$. (b) The asymptotic behavior of $C(i,i+2)$ as a function of ${\lambda }_1$ and $\gamma$ with $h_0=h_1=1$ and $J_0=1$ at $\mathit{kT}=0$.