Entanglement dynamics in short- and long-range harmonic oscillators

We study the time evolution of the entanglement entropy in the short- and long-range-coupled harmonic oscillators that have well-defined continuum limit field theories. We first introduce a method to calculate the entanglement evolution in generic coupled harmonic oscillators after quantum quench. Then we study the entanglement evolution after quantum quench in harmonic systems in which the couplings decay effectively as $1/r^{d+\alpha }$ with the distance $r$. After quenching the mass from a nonzero value to zero we calculate numerically the time evolution of von Neumann and Rényi entropies. We show that for $1<\alpha <2$ we have a linear growth of entanglement and then saturation independent of the initial state. For $0<\alpha <1$ depending on the initial state we can have logarithmic growth or just fluctuation of entanglement. We also calculate the mutual information dynamics of two separated individual harmonic oscillators. Our findings suggest that in our system there is no particular connection between having a linear growth of entanglement after quantum quench and having a maximum group velocity or generalized Lieb-Robinson bound.

Figure 1

Different configurations of systems and subsystems for entanglement dynamics with global quenches.

Figure 2

Quadratic growth in time for $S_A\left(t\right)$ in the region $t\ll t^{*}$ for the short-range harmonic oscillators with the configuration ${\mathfrak{g}}_1$. The total size of the system is $N=500$, and the subsystem size $l=60$. The solid red line corresponds to the $S_A\left(t\right)\equiv {\kappa }_2{t}^2$. Inset: The value of the scaling parameter ${\kappa }_2$ as a function of mass parameter $m_0$.

Figure 3

Top: (a) Entanglement entropy dynamics $S_A\left(t\right)$ for short-range harmonic oscillator with the configuration ${\mathfrak{g}}_1$ and different mass parameter $m_0$. The total size of the system is $N=400$, and the subsystem size $l=50$. The dashed line corresponds to the saturation time $t^{*}=l/2$. (b) The von Neumann entropy $S_A\left(t\right)$ in the $t<l/2$ limit obeys the $S_A\left(t\right)\sim {\kappa }_{1}t$, where ${\kappa }_1$ is a linear function of mass $m_0$. (c) In the $t>l/2$ limit $S_A\left(t\right)$ saturates to the $S_{\infty }$. The value of $S_{\infty }$ for the fixed value of $l=30$ is linear function of $m_0$. Bottom: (d) The von Neumann entropy $S_A\left(t\right)$ for short-range harmonic oscillators with the configuration ${\mathfrak{g}}_1$ and fixed value of the mass parameter $m_0=0.4$. The total size of the system is $N=400$, and the subsystem size $l\in \{20,30,40,50\}$. (e) In the $t>l/2$ limit, $S\left(t\right)$ saturates to the $S_{\infty }$. The value of $S_{\infty }$ is a linear function of $l$.

Figure 4

Prefactors $c_n$ and $c_n^{\prime }$ for dynamics of Rényi entropy for the short-range harmonic oscillator system with configuration ${\mathfrak{g}}_1$ and the system size $N=500$. The solid red line represents $\frac{1}{2}(1+1/n)$ [see Eq. (27)].

Figure 5

Top: (a) Entanglement entropy dynamics $S_A\left(t\right)$ for short-range harmonic oscillator with the configuration ${\mathfrak{g}}_2$ and different mass parameter $m_0$. The total size of the system is $N=500$, and the subsystem size $l=50$. The dashed line corresponds to the saturation time $t^{*}=l$. (b) $S_A\left(t\right)$ in the $t<l$ limit obeys $S\left(t\right)={\kappa }_{1}t$, where ${\kappa }_1$ is a linear function of mass $m_0$. (c) $S_A\left(t\right)$ in the $t>l$ limit saturates to $S_{\infty }$. The value of $S_{\infty }$ is a linear function of $m_0$. Bottom: (d) Entanglement entropy dynamics $S_A\left(t\right)$ for short-range harmonic oscillators with the configuration ${\mathfrak{g}}_2$ and fixed value of the mass parameter $m_0=0.05$. The total size of the system is $N=500$, and the subsystem size $l\in \{35,45,55,65\}$. (e) The value of $S_A\left(t\right)$ in the $t>l$ limit saturates to $S_{\infty }$, which is a linear function of $l$.

Figure 6

Entanglement entropy dynamics $S_A\left(t\right)$ in the long-range harmonic oscillators with the configuration ${\mathfrak{g}}_1$ for different values of $\alpha$. The curves correspond to the linear change of $S_A\left(t\right)$ with time $t$ before the saturation time $t^{*}$ for $1<\alpha \le 2$. The total size of the system is $N=400$, the mass parameter $m_0=0.1$, and the subsystem size $l=50$.

Figure 7

Quadratic growth in time for $S_A\left(t\right)$ where $t\ll t^{*}$. The harmonic system is long range ($\alpha =1.5$) with the configuration ${\mathfrak{g}}_1$. The total size of the system is $N=500$, and the subsystem size $l=60$. The solid red line corresponds to $S_A\left(t\right)\equiv {\kappa }_2{t}^2$. Inset: The value of the scaling parameter ${\kappa }_2$ as a function of mass parameter $m_0$.

Figure 8

Top: (a) Entanglement entropy dynamics $S_A\left(t\right)$ for long-range harmonic oscillators ($\alpha =1.6$) with the configuration ${\mathfrak{g}}_1$ and different mass parameter $m_0$. The total size of the system is $N=500$, and the subsystem size $l=50$. (b) Before the saturation time $t<t^{*}$, entanglement entropy dynamics obeys $S\left(t\right)\sim {\kappa }_{1}t$, where ${\kappa }_1$ is a linear function of $m^{\alpha /2}$. Bottom: (c) The entanglement entropy dynamics $S_A\left(t\right)$ for long-range harmonic oscillator with the configuration ${\mathfrak{g}}_1$ and fixed value of mass parameter $m_0=0.3$. The total size of the system is $N=300$, and the subsystem size $l\in \{30,35,40,45,50\}$. (d) $S_A\left(t\right)$ in $t\gg t^{*}$ saturates to $S_{\infty }$, where it changes linearly with $m_{0}l$.

Figure 9

Numerical results for the coefficients $\mathcal{A}$ and $\mathcal{B}$ as a function of $\alpha$ [see Eq. (34)].

Figure 10

Scaling prefactors ${\mathcal{A}}_n$ and ${\mathcal{B}}_n$ vs $n$ for different $\alpha$'s ($\alpha =1.6,1.8$). The solid red line represents $\frac{1}{2}(1+1/n)$.

Figure 11

(a) The saturation time $t^{*}$ versus ${(l/2)}^{\alpha /2}$ for different values of $\alpha$. (b) Scaling parameter $\lambda$ with respect to $\alpha$.

Figure 12

Top: Entanglement entropy dynamics $S_A\left(t\right)$ in long-range harmonic oscillators ($\alpha <1$) with the configuration ${\mathfrak{g}}_1$. The total size of the system is $N=500$, the mass parameter $m_0=0.1$, and the subsystem size $l=50$. The red lines correspond to logarithmic change of $S_A\left(t\right)\sim \mathcal{P}(m_0,l,\alpha )lnt$ where the coefficient $\mathcal{P}(m_0,l,\alpha )$ is a function of $m_0$, $l$, and $\alpha$. Bottom: The coefficient $\mathcal{P}(m_0,l,\alpha )$ for different values of $m_0$ and $l$ is a linear function of ${\left(m_{0}l\right)}^{\alpha /2}$, i.e., $\mathcal{P}(m_0,l,\alpha )={\mathcal{V}}_1\left(\alpha \right){\left(m_{0}l\right)}^{\alpha /2}+{\mathcal{V}}_1\left(\alpha \right)$, where the coefficient ${\mathcal{V}}_1$ (${\mathcal{V}}_2$) as shown in the inset is an increasing (decreasing) function of $\alpha$.

Figure 13

The role of finite-size effects on the entanglement entropy dynamics $S_A\left(t\right)$ in long-range harmonic oscillators ($\alpha <1$) with the configuration ${\mathfrak{g}}_1$. The total size of the system is $N=500$; the mass parameter $m_0=0.1$. Top: $\alpha =0.4$. Bottom: $\alpha =0.6$.

Figure 14

Top: The time evolution of the von Neumann entropy $S_A\left(t\right)$ in the short-range harmonic oscillators with the total size $N=400$ when the initial state is a gapless power law with long-range scaling exponent ${\alpha }_0$. $S_A\left(t\right)$ starts from a nonzero value and grows linearly with time, then saturates. Middle: The time evolution of the von Neumann entropy $S_A\left(t\right)$ in the harmonic oscillators with weak long-range coupling ($\alpha =1.6$). Bottom: The same result for the harmonic oscillators with strong long-range couplings ($\alpha =0.6$).

Figure 15

Top: The numerical estimates of the prefactor $\mathcal{L}({\alpha }_0,\alpha )$ for different values $\alpha =2.0,1.6,1.2$ as a function of ${\alpha }_0$. The solid red lines correspond to $\mathcal{L}({\alpha }_0,\alpha )=\frac{1}{3}ln(\alpha /{\alpha }_0)$. Bottom: The prefactor ${\mathcal{J}}_n$ for the dynamics of the Rényi entropy after quench from gapless power-law initial state [see Eqs. (40) and (41)]. The solid red line corresponds to $\frac{1}{6}(1+1/n)$.

Figure 16

The relation between the saturation time $t^{*}$ and the subsystem size $l$ for different values $\alpha$. The solid red line represents $t^{*}=b\left(\alpha \right)l$. Inset: The prefactor $b\left(\alpha \right)$ as a function of $\alpha$ shows the linear behavior $b\left(\alpha \right)\simeq 0.4\alpha -0.3$.

Figure 17

Top: Entanglement entropy dynamics $S_A\left(t\right)$ for long-range harmonic oscillator Eq. (42) with an exponentially decaying correlation function. The total size of the system is $N=400$, the subsystem size $l=50$, and $\alpha =1.5$. Different lines correspond to different values of the mass parameter $m_0$. Bottom: Prefactor ${\mathcal{A}}_n$ for dynamics of the Rényi entropy for the long-range harmonic oscillator system ($\alpha =1.5$). The solid red line represents $\frac{1}{2}(1+1/n)$.

Figure 18

Entanglement entropy dynamics $S_A\left(t\right)$ for long-range harmonic oscillator Eq. (42) with strong coupling ($\alpha <1$) in which the initial state's correlation function decays exponentially. The total size of the system is $N=400$, the subsystem size $l=50$, and $\alpha =0.6$. Different lines corresponds to different values of the initial mass parameter $m_0$.

Figure 19

Top: The time evolution of the von Neumann entropy $S_A\left(t\right)$ in the short-range harmonic oscillators with the total size $N=500$ when the initial state is uncoupled. $S_A\left(t\right)$ starts from zero and grows linearly with time then saturates to the value $S_{\infty }$ which is a linear function of subsystem size $l$ (see inset). Bottom: The numerical estimates of the prefactors $c_n$ and $c_n^{\prime }$. The solid red line corresponds to $\frac{1}{2}(1+1/n)$.

Figure 20

Top: The time evolution of the von Neumann entropy $S_A\left(t\right)$ in the long-range harmonic oscillators ($\alpha =1.5$) with the total size $N=500$ when the initial state is uncoupled. $S_A\left(t\right)$ starts from zero and grows linearly with time then saturates to the value $S_{\infty }$ which is a linear function of subsystem size $l$ (see inset). Bottom: The numerical estimates of the prefactors ${\mathcal{A}}_n$ and ${\mathcal{B}}_n$ for different values $\alpha =1.6,1.8$. The solid red lines correspond to $\frac{1}{2}(1+1/n)$.

Figure 21

The density plot of mutual information $I_{i,j}$ between two sites with the separation distance $r=|i-j|/2$ in short- and long-range harmonic oscillators. The total size and initial mass parameter are equal to $N=200$ and $m_0=0.3$, respectively. Top left, $\alpha =0.4$; top right, $\alpha =1.0$; bottom left, $\alpha =1.4$; bottom right, $\alpha =2.0$. Yellow (light gray) regions correspond to larger value of $I_{i,j}$ and black regions correspond to smaller value of $I_{i,j}$. Dashed lines correspond to $t_I^{*}=r^{\alpha /2}$. At time $t_I^{*}$, the mutual information $I_{i,j}$ suddenly changes from zero to nonzero value. For $t<t_I^{*}$ the mutual information $I_{i,j}$ is zero and for $t>t_I^{*}$ it remains nonzero and oscillates due to finite-size effects.

Figure 22

The numerical estimation for $t_I^{*}$ as a function of distance $r$. At time $t_I^{*}$, the mutual information $I_{i,j}$ of two particular points with distance $r=|i-j|/2$ suddenly changes from zero to nonzero value.

Figure 23

Numerical evaluation of $K^{-1/2}\left(r\right)$ vs $r$ for power-law initial state shows the scaling behavior $K^{-1/2}\left(r\right)\propto 1/r^{\beta }$. Inset: Scaling exponent $\beta$ as a function of $\alpha$. The solid red line represents $\beta =1+\alpha$.

Figure 24

Top: Entanglement entropy dynamics $S_A\left(t\right)$ in short-range harmonic oscillator with the configuration ${\mathfrak{g}}_2$ and different system size $N$ ($l=30$ and $m_0=0.05$). The same results for the configuration ${\mathfrak{g}}_1$ ($l=30$, $m_0=0.12$) shown in the inset. Bottom: $S_A\left(t\right)$ for long-range harmonic oscillator ($\alpha =1.5$) with different system size $N$. Notice that the results do not depend on $N$.

Figure 25

Top: Entanglement entropy dynamics $S_A\left(t\right)$ in short-range harmonic oscillator with the configuration ${\mathfrak{g}}_1$. The total size of the system is $N=500$, the mass parameter $m_0=0.12$, and the subsystem size $l=50$. Different lines corresponds to different values of the small mass parameter $m$. Middle: The same results for long-range harmonic oscillator with $\alpha =1.5$ with the configuration ${\mathfrak{g}}_1$, the system is $N=500$, the mass parameter $m_0=0.12$, and the subsystem size $l=50$. Bottom: Entanglement entropy dynamics $S_A\left(t\right)$ for long-range harmonic oscillators with strong coupling $\alpha =0.4$ and $\alpha =0.2$.

Figure 26

Top: The scaling prefactor $c_n$ for the entanglement dynamics in short-range harmonic oscillator before saturation regime [see Eq. (26)]. Different symbols correspond to different values of the small mass parameter $m$. Bottom: The scaling prefactor ${\mathcal{A}}_n$ for the entanglement dynamics in long-range harmonic oscillator ($\alpha =1.5$) before saturation regime [see Eq. (35)]. Different symbols correspond to different values of the small mass parameter $m$.

Figure 27

Top: The time evolution of the mutual information $I_{i,j}$ between two sites $i$ and $j$ in short-range harmonic oscillators with the total size $N=200$, mass parameter $m_0=0.3$, and different values of the small parameter $m$ and distance $\mid i-j\mid$. Bottom: The same results for long-range harmonic oscillator ($\alpha =1.5$).