We calculate the amount of entanglement shared by two intervals in the ground state of a ($1+1$)-dimensional conformal field theory (CFT), quantified by an entanglement measure $\mathcal{E}$ based on the computable cross norm (CCNR) criterion. Unlike negativity or mutual information, we show that $\mathcal{E}$ has a universal expression even for two disjoint intervals, which depends only on the geometry, the central charge $c$, and the thermal partition function of the CFT. We prove this universal expression in the replica approach, where the Riemann surface for calculating $\mathcal{E}$ at each order $n$ is always a torus topologically. By analytic continuation, the result of $n=\frac{1}{2}$ gives the value of $\mathcal{E}$. Furthermore, the results of other values of $n$ also yield meaningful conclusions: The $n=1$ result gives a general formula for the two-interval purity, which enables us to calculate the Rényi-2 $N$-partite information for $N\le 4$ intervals; while the $n=\infty$ result bounds the correlation function of the two intervals. We verify our findings numerically in the spin-$1/2$ XXZ chain, whose ground state is described by the Luttinger liquid.

• Received 28 November 2022
• Accepted 9 March 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.131601