Entanglement and Rényi Entropy of Multiple Intervals in $T\overline{T}$-Deformed CFT and Holography

TL;DR

This work analyzes entanglement and Rényi entropies for multiple intervals in a 2d CFT deformed by $T\overline{T}$ at finite temperature, using both field-theoretic (replica/twist) and holographic methods with a finite cutoff. At first order in the deformation parameter $\mu$, entanglement entropy is additive across intervals in holographic CFTs, while Rényi entropy for two intervals is additive only when interval separations are large, due to cosmic-brane backreaction. The results, which include explicit single-interval and two-interval formulas and their holographic interpretations, provide a nontrivial consistency check of TTbar holography and reveal mixing effects in Rényi entropy when intervals are not well separated. The paper also maps out phase transitions between disconnected and connected minimal surfaces under varying deformation, temperature, and cutoff, highlighting how $\mu$ and finite cutoff modify the holographic entanglement structure.

Abstract

We study the entanglement entropy (EE) and the Rényi entropy (RE) of multiple intervals in two-dimensional $T\overline{T}$-deformed conformal field theory (CFT) at finite temperature by field theoretic and holographic methods. First, by the replica method with the twist operators, we construct the general formula of the RE and EE up to the first order of a deformation parameter. By using our general formula, we show that the EE of multiple intervals for a holographic CFT is just a summation of the single interval case even with the small deformation. This is a non-trivial consequence from the field theory perspective, though it may be expected by the Ryu-Takayanagi formula in holography. However, the deformed RE of the two intervals is a summation of the single interval case only if the separations between the intervals are big enough. It can be understood by the tension of the cosmic branes dual to the RE. We also study the holographic EE for single and two intervals with an arbitrary cut-off radius (dual to the $T\overline{T}$ deformation) at any temperature. We confirm our holographic results agree with the field theory results with a small deformation and high temperature limit, as expected. For two intervals, there are two configurations for EE: disconnected ($s$-channel) and connected ($t$-channel) ones. We investigate the phase transition between them as we change parameters: as the deformation or temperature increases the phase transition is suppressed and the disconnected phase is more favored.

Figures (9)

• Figure 1: The holographic entanglement entropy for two intervals $[x'_1, x'_2]\cup[x'_3, x'_4]$ at $u=u_c$: schematic pictures of minimal surfaces (red curves).
• Figure 2: Schematic picture of manifold $\mathcal{M}^{3}$.
• Figure 3: The mixing effect $\mathbb{M}$ in \ref{['mixing123']} of symmetric configuration, $\ell_{12} = \ell_{34}$, where $\ell_{12} := |x_2 - x_1|$ and $\ell_{34} := |x_4 - x_3|$. In order to satisfy the condition $\eta\to0$ (a) and $\eta\to1$ (b) only the ranges of $\ell_{23} \gg 1$ (a) or $\ell_{23} \ll 1$ with a fixed $\ell_{12}$(b) are valid.
• Figure 4: The mixing effect $\mathbb{M}$ in \ref{['mixing123']} vs $\eta$: $\ell_{12}$ = 0.5, 1, 5 (red, green, blue).
• Figure 5: Transition curves of symmetric case ($\bar{\ell}_{34}= \ell_{34}/\ell_{12} =1$), where $\bar{\ell}_{23} = \ell_{23}/\ell_{12}$, $\bar{u}_c = u_c \ell_{12}$ and $\bar{u}_h = u_h \ell_{12}$. All solid curves represent phase transition points satisfying $\mathcal{S}_{c} = 1$ (see \ref{['RatioEq']}) with the various temperature $\bar{u}_h$, and the black dashed curves represent approximate formulas \ref{['Asymp']}. The region above(below) the solid curves corresponds to the $s(t)$-channel.