Entanglement entropy at CFT junctions

We consider entanglement through permeable junctions of $N$ ($1+1$)-dimensional free boson and free fermion conformal field theories. In the folded picture we constrain the form of the general boundary state. We calculate replicated partition functions with interface operators inserted in the partially-folded picture, from which the entanglement entropy is calculated. The functional form of the universal and constant terms are the same as the $N=2$ case, depending only of the total transmission of the junction and the unit volume of the zero mode lattice. For $N>2$ we see a subleading divergent term which does not depend on the parameters of the junction. For $N=3$ we consider some specific geometries and discuss various limits.

Figure 1

Illustration of the parity transformation relating the interface between ${\mathrm{CFT}}_1$ and ${\mathrm{CFT}}_2$ to the tensor product ${\mathrm{CFT}}_1\otimes {\overline{\mathrm{CFT}}}_2$ with boundary. The folded picture is useful for characterizing classes of interfaces and some simple calculations (see [7]). For calculations such as ours the boundary states need to be unfolded once they are found.

Figure 2

On the right: A D1-brane wrapping the bosonic 2-torus continued into the compactification lattice so as to show the lattice intercept at $(k_1{R}_1,k_2{R}_2)$. On the left: A D1-brane wrapping the bosonic 2-torus (corresponding to the parameters $k_1=2$ and $k_2=3$) shown in the unit cell of the compactification lattice.

Figure 3

Illustrating the unfolded, folded, and partially folded pictures for a 3-junction. As before, the folded picture is used to characterize the boundary states. However, for the entanglement entropy calculations we will only unfold one CFT and work with interface operators in this partially folded picture.

Figure 4

The logarithmic map $z=\log w$ maps the $K$-sheeted Riemann surface, a single branch of which is shown on the left, to the geometry on the right. The circles on the left part of the figure correspond to an UV cutoff located $|w|=\epsilon$ and an IR cutoff located at $|w|=L$, with their image under the mapping forming the negative and positive real boundaries of the geometry on the right. This figure was adapted from [22].

Figure 5

A D2-brane wrapping the bosonic 3-torus continued into the compactification lattice so as to show the axis intercepts $q_i{R}_i{\widehat{\phi }}_i$; see Fig. 6 for the unit cell wrapping for a specific case. The polar and azimuthal angles that specify the rotation that takes the D2-brane in the ${\phi }^1{\phi }^2$-plane into this pictured D2-brane are also shown.

Figure 6

A D2-brane wrapping the bosonic 3-torus shown in the unit cell of the compactification lattice. The above corresponds to the parameters $q_1=3$, $q_2=2$, and $q_3=6$.