Correspondence between entanglement growth and probability distribution of quasiparticles

We study the excess of (Renyi) entanglement entropy in various free field theories for the locally excited states defined by acting with local operators on the ground state. It is defined by subtracting the entropy for the ground state from the one for the excited state. Here the spacetime dimension is greater than or equal to 4. We find a correspondence between entanglement and a probability. The probability with which a quasiparticle exists in a subregion gives the excess of the entropy. We also propose a toy model which reproduces the excess in the replica method. In this model, a quasiparticle created by a local operator propagates freely and its probability distribution gives the excess.

Figure 1

The total Space divided into two subspaces $A$ and $B$ at $t=0$.

Figure 2

Operator insertion points before taking the analytic continuation in (a) $(\tau ,x^1)$ and (b) $(r,\theta )$.

Figure 3

Sketch of (a) $n$-sheeted Riemann surface, (b) $n$-sheeted Riemann surface with operator insertions.

Figure 4

A quasiparticle picture. At $t=-t$, quasiparticles appear at the point where $\varphi$ is located. (a) shows that all of them are included in $B$ in $l\ge t>0$. Therefore entanglement between them can not contribute to $\mathrm{\Delta }{S}_A^{(n)}$. (b) shows that their entanglement can contribute to $\mathrm{\Delta }{S}_A^{(n)}$ in $t>l>0$ because some of them are included in $A$. In the late time, it can be interpreted in terms of entanglement between two quasiparticles.

Figure 5

The ratios of diagrams. The diagram constructed of $G^{(n)}({\theta }_1-{\theta }_2)$ is called $D_1^{(n)}$. The diagram constructed of $G^{(n)}({\theta }_1-{\theta }_2+2\pi )$ and $G^{(n)}({\theta }_1-{\theta }_2-2(n-1)\pi )$ is $D_2^{(n)}$. The one constructed of $G^{(1)}({\theta }_1-{\theta }_2)$ is $D_1^{(1)}$. (a) is the ratio of $D_1^{(n)}$ to $(D_1^{(1)}{)}^n$. (b) is the ratio of $D_2^{(n)}$ to $(D_1^{(1)}{)}^n$.

Figure 6

Free particle propagation.

Figure 7

The plot of $S$. The parallel axis is $t$. The vertical axis is $S$. $(l,L)=(10,30)$.

Figure 8

A schematic explanation of the subsystems $A$, $B$, and $C$. $A$ is an infinite strip. $B$ is a finite strip.

Figure 9

The Plot of ${\mathcal{I}}_{A,B}$. The horizontal axis is $t$. The vertical axis is ${\mathcal{I}}_{A,B}$. $(l_A,l_B,L_B)=(5,10,15)$.

Figure 10

The Plot of ${\mathcal{I}}_{A,B}$. The horizontal axis is $t$. The vertical axis is ${\mathcal{I}}_{A,B}$. $(l_A,l_B,L_B)=(10,5,15)$.

Figure 11

A schematic explanation of the subsystems $A$, $B$, and $C$. $A$ and $B$ are finite strips.

Figure 12

The plot of ${\mathcal{I}}_{A,B}$. The horizontal axis is $t$. The vertical axis is ${\mathcal{I}}_{A,B}$. $(l_A,l_B,L_A,L_B)=(5,10,15,20)$.

Figure 13

The plot of ${\mathcal{I}}_{A,B}$. The horizontal axis is $t$. The vertical axis is ${\mathcal{I}}_{A,B}$. $(l_A,l_B,L_A,L_B)=(5,10,20,15)$.