Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks

TL;DR

This work reexamines the entanglement entropy of two disjoint intervals in 1+1D CFTs by applying a conformal-block expansion to the four-point function of twist fields and by iterating Zamolodchikov's recursion for each conformal block. By including multiple twist-field OPE channels and systematically truncating the Zamolodchikov recursion, the authors obtain analytic approximations for the Rényi entropies ${\cal F}_n(x)$ and, via analytic continuation, for the von Neumann entropy, demonstrated in detail for the Ising model and the compact boson. The results show that incorporating additional fusion channels and higher-order recursion terms significantly improves the accuracy relative to exact results, with complete agreement in the $n=2$ Ising case and good concordance with numerical data for other cases. The approach provides a practical, analytically tractable framework for disjoint-interval entanglement and suggests future extensions to entanglement negativity and other CFTs. Overall, the paper offers a controlled approximation scheme that bridges conformal blocks, twist-field fusion, and replica-limit entanglement measures in disjoint-interval configurations.

Abstract

We reconsider the computation of the entanglement entropy of two disjoint intervals in a (1+1) dimensional conformal field theory by conformal block expansion of the 4-point correlation function of twist fields. We show that accurate results may be obtained by taking into account several terms in the operator product expansion (OPE) of twist fields and by iterating the Zamolodchikov recursion formula for each conformal block. We perform a detailed analysis for the Ising conformal field theory and for the free compactified boson. Each term in the conformal block expansion can be easily analytically continued and so this approach also provides a good approximation for the von Neumann entropy.

Figures (4)

• Figure 1: The best approximation we derived for $\mathcal{F}_n (x)$ ($n=2,3, 6$) and for the entanglement entropy $F_{\rm vN} (x)$ in the Ising model, still at the 0-th order in the Zamolodchikov formula. The dots represent the exact functions. The red line is the curve derived with the approximation in GR. The green curve is our approximation (the fusion channels included in the OPE of twist fields are listed with $(k, l)$ denoting the inclusion of the fusion channel $[\sigma_1 \cdots \sigma_k \epsilon_1 \cdots \epsilon_l \mathbb{I}_{k+l+1} \cdots \mathbb{I}_n ]$ and all its permutations). The cyan curve is the expansion in power of $x$ derived in CCT.
• Figure 2: The best approximation we derived for $\mathcal{F}_n (x)$ ($n=2, 3, 6$) and the von Neumann entropy $F_{\rm vN} (x)$ in the Ising model, by including further terms in the Zamolodchikov formula. The dots represent the exact functions. The green curve is our approximation at the 0-th order. The orange curve is the approximation at the $2$-nd order. The fusion channels included in the OPE of twist fields are the same as in Figure \ref{['Fn_Ising_best']}.
• Figure 3: The function $\mathcal{F}_2 (x)$ for different values of the compactification radius ($\eta= 1/3, 1/2, 0.7$) and the function $\mathcal{F}_3(x)$ and $\mathcal{F}_4(x)$ with $\eta=1/2$ for a compactified boson. In all cases the truncation in the Zamolodchikov formula is at the 0-th order. Two different approximations in the OPE are considered: the fusion channels included are $(0, 0;0, 0) , (1, 0; 1, 0)$ for the red curves and $(0, 0;0, 0) , (1, 0; 1, 0), (1, 0; 2,0) , (0, 1; 0,1), (2, 0; 1, 0), (0, 2; 0, 1)$ for the green curves (with $(p, q; k, l )$ denoting the inclusion of the term in Eq. \ref{['cb-fusionchannel']}). The dots represent the exact functions.
• Figure 4: The continuous lines represent the approximation of the Von Neumann entropy $F_{\rm vN} (x)$ for a compactified boson in Eq. \ref{['FvN-CB-0th']}. The dots are the numerics of the XXZ chain in the gapless regime obtained via TTN techniques ATC-CB.