Non-perturbative aspects of entanglement structures in $T\bar{T}$-deformed CFTs

TL;DR

This work investigates the non-perturbative entanglement structure of TT̄-deformed CFTs, focusing on how the TT̄-induced length-scale ε_{TT̄} ∝ √μ dynamically regulates UV entanglement. Utilizing a string-theoretic formulation of TT̄ and a replica-trick framework, the authors derive a general integral representation for deformed replica partition functions as a weighted integral over undeformed CFT results with a dynamical cut-off, governed by a kernel K(L,ℓ,μ) computed from the on-shell target-space action. A detailed semi-classical analysis reveals a perturbative saddle aligning the dynamical cut-off with the bare entanglement cut-off and uncovers a non-perturbative saddle that ties the dynamical cut-off to the TT̄ length-scale, contributing exponentially small corrections to Renyi and von Neumann entropies. For the semi-infinite case this non-perturbative sector provides a mechanism by which ε_{TT̄} may eventually override the bare cut-off, while in the finite-interval case the interpretation is more subtle due to boundary effects; the results suggest rich non-perturbative physics and potential phase transitions, with broader implications for holography and resurgence in TT̄-deformed systems.

Abstract

Turning on the $T\bar{T}$-deformation in a two-dimensional CFT provides a unique window to study explicitly how non-local features arise in the UV as a result of the deformation. A sharp signature is the dynamical emergence of an effective length-scale $\propto \sqrtμ$ that separates the local and non-local regimes of the deformed theory, effectively serving as a UV cut-off for computing observables in the local regime. In this paper, we study this phenomenon through the entanglement structures of the deformed theory. We focus on computing the Renyi entropies of single-interval sub-regions in the deformed vacuum states. We pay particular attention to the interplay between the bare entanglement cut-off inherited from the CFT computation and the effects from the $T\bar{T}$ deformations. Applying the general replica trick to the string theory formulation of $T\bar{T}$-deformed CFTs, we derive an explicit representation of the deformed replica partition function as a weighted integral of the CFT results evaluated at a dynamical cut-off, which is integrated over. We computed in detail the kernel functions of the integral representation, and performed the saddle-point analysis in the semi-classical limit of small $μ$. We found that in addition to the perturbative saddle-point which identifies the dynamical cut-off with the bare entanglement cut-off, there exists another non-perturbative saddle-point that identifies the dynamical cut-off with the $T\bar{T}$ length-scale $\propto \sqrtμ$, but whose contribution is exponentially small. We discuss how these non-perturbative effects can shed lights on the mechanism through which the $T\bar{T}$ length-scale may eventually replace the bare counter-part and become the effective entanglement cut-off.

Figures (4)

• Figure 1: A graphic illustration for the classical solution used to compute the kernel function $K(L,\ell,\mu)$. For better visualization, the conical singularities are shown as defects instead of excesses.
• Figure 2: Schematic plot of the action $\lambda^{-1}A_0(\ell) + A_1(\ell)$, i.e. the exponent of the integrand in \ref{['eq:Saddle Point Analysis']}. The perturbative saddle is marked with blue wedges while the non-perturbative saddle is marked with red wedges. The colored curves correspond to increasing values of $\lambda$ from top to bottom. As $\lambda$ increases, the two saddles approach each other and merge into a single saddle when $c\lambda \sim \epsilon^2$.
• Figure 3: The full solution of $\theta(\sigma)$ with $a=1,\; b=2$ and $\epsilon=0.01$, extrapolates from $\theta(\sigma)=2\pi \Theta(\sigma-\pi)$ for small $\ell$ to $\theta(\sigma)=\sigma$ for large $\ell$.
• Figure 4: A schematic figure contrasting the saddle points between the half-line and finite-interval cases.