Aharonov-Bohm effect on entanglement entropy in conformal field theory

We consider the Aharonov-Bohm effect on entanglement entropy for one interval in ($1+1$) dimensional conformal field theory on a one dimensional ring. The magnetic field is confined inside the ring, i.e., there is a Wilson loop on the ring. The Aharonov-Bohm phase factor which is proportional to the Wilson loop is represented as insertion of twist operators. We compute exactly the Rényi entropy from a four point function of twist operators in a free charged scalar field.

Figure 1

One dimensional ring studied in this paper. The circumference of the ring is $L$ and the subsystem $A$ is one interval whose length is $l$. The space coordinate $x$ has the periodicity $x\sim x+L$. The magnetic field is confined inside the ring and induces the Aharonov-Bohm phase on the ring.

Figure 2

The Euclidean path integral for $\mathrm{Tr}[{\rho }_A^n]$ in $w$ and $z$ coordinates. In $z$-plane, the boundary condition (7) can be expressed by inserting twist operators ${\sigma }_{\nu }$ and ${\sigma }_{1-\nu }$ at $z=0$ and $z=\infty$.

Figure 3

The Rényi entropy $S_A^{(n=2)}$ as a function of $\nu$. From top to bottom: $l/L=1/3,1/4,1/6$. $S_A^{(n=2)}$ diverges when $\nu \to 0$ and $\nu \to 1$. $S_A^{(n=2)}$ becomes a minimum value for $\nu =1/2$.

Figure 4

The Rényi entropy $S_A^{(n=2)}$ as a function of $l/L$. From top to bottom: $\nu =1/10,1/5,1/2$.

Figure 5

The two closed loops ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ we consider as a basis of the loops in the complex plane.