A bstract Turning on the $$T\overline{T }$$ -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 ∝ $$\sqrt{\mu }$$ 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 $$T\overline{T }$$ deformations. Applying the general replica trick to the string theory formulation of $$T\overline{T }$$ -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 $$T\overline{T }$$ length-scale ∝ $$\sqrt{\mu }$$ , but whose contribution is exponentially small. We discuss how these non-perturbative effects can shed lights on the mechanism through which the $$T\overline{T }$$ length-scale may eventually replace the bare counter-part and become the effective entanglement cut-off.
Non-perturbative aspects of entanglement structures in TT-deformed CFTs
A bstract Turning on the TT -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 ∝ μ 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 TT deformations. Applying the general replica trick to the string theory formulation of TT -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 TT length-scale ∝ μ , but whose contribution is exponentially small. We discuss how these non-perturbative effects can shed lights on the mechanism through which the TT length-scale may eventually replace the bare counter-part and become the effective entanglement cut-off.
Non-perturbative aspects of entanglement structures in TTˉ-deformed CFTs
Turning on the TTˉ-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 ∝μ 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 TTˉ deformations. Applying the general replica trick to the string theory formulation of TTˉ-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 TTˉ length-scale ∝μ, but whose contribution is exponentially small. We discuss how these non-perturbative effects can shed lights on the mechanism through which the TTˉ length-scale may eventually replace the bare counter-part and become the effective entanglement cut-off.
Quantum entanglement measures from Hyperscaling violating geometries with finite radial cut off at general d, θ from the emergent global symmetry
The quantum entanglement measures for TT deformed field theory on boundary, deformation coefficient μ, with dual bulk geometry with finite radial cutoff ρc, for entangling region is single or disjoint intervals on the boundary, of length l is expected to give global description of these measures over the complete parameter-regime of (l,μ) or on 2D (l,ρc) plane, because it is solvable irrelevant deformation. Here, to find quantum-measures through RT prescription, from Hyperscaling violating bulk geometry with finite radial cut off, we found mathematically it is impossible, to obtain such global form, since the turning point ρ0(l,ρc), neither in its exact or in any perturbative form, is solvable globally, can describe these quantum measures at most locally over some specific regime in 2D (l,ρc) plane! However, to find such global form, we found, on application of RT formalism, a global symmetry structure, from the considered geometry emerges, over 2D parameter-space, irrespective of d, θ, which alongwith global b.c and other consistency conditions, fix ρ0(l,ρc) globally, exactly in $l >> ρ_c $ and ρc>>l regime and as some interpolating expressions, very close to the exact one in other regime. Some of the quantum entanglement measures with this ρ0(l,ρc), with our intuitively predicted behaviour for them in the deformed theory, derived and shown, behaving accordingly. The impact of this emergent symmetry on these quantum-measures is discussed, the possible space time origin of this symmetry is explored, although the later aspect is subjected to a proper and detailed study.
In this paper, we propose a novel expansion to compute the large interval limit of the R\'enyi entropy of 2D conformal field theory (CFT) at high temperature. Via the replica trick, the single interval R\'enyi entropy of 2D CFT at finite temperature could be read from the partition function on n-sheeted torus connected with each other along a branch cut. We calculate the partition function by inserting a complete basis across the branch cut. Because of the monodromy condition across the branch cut in the large interval limit, the basis of the states should be the ones in the twist sector. We study the twist sector of a general module of CFT and find that there is a one-to-one correspondence between the twist sector states and the normal sector states. As an application, we revisit the noncompact free scalar theory and discuss the large interval limit of the R\'enyi entropy of this theory by using our proposal. We find complete agreement in the leading and next-leading orders with direct expansion of the exact partition function. Moreover, we prove the relation (10) between thermal entropy and the entanglement entropy for a generic CFT with discrete spectrum.
Reflected entropy and timelike entanglement in TTˉ deformed CFT2s
We develop a covariant formalism to investigate the mixed state entanglement structure of time-dependent boosted subsystems in TTˉ deformed CFT2s through the reflected entropy. To this end we utilize the conformal perturbation theory to obtain the Rényi reflected entropy through the partition function on replica manifold. The correction to the reflected entropy to the first order in the deformation parameter μ in then obtained in the replica limit for finite temperature and finite sized systems. We verify our field theoretic computations by obtaining the dual EWCS in the corresponding bulk cut-off AdS3 geometries and find perfect agreement between the two.
A bstract Modular flows probe important aspects of the entanglement structures, especially those of QFTs, in a dynamical framework. Despite the expected non-local nature in the general cases, the majority of explicitly understood examples feature local space-time trajectories under modular flows. In this work, we study a particular class of non-local modular flows. They are associated with the relativistic vacuum state and sub-regions whose boundaries lie on a planar null-surface. They satisfy a remarkable algebraic property known as the half-sided modular inclusion, and as a result the modular Hamiltonians are exactly known in terms of the stress tensor operators. To be explicit, we focus on the simplest QFT of a massive or massless free scalar in 1 + 2 dimensions. We obtain explicit expressions for the generators. They can be separated into a sum of local and non-local terms showing certain universal pattern. The preservation of von Neumann algebra under modular flow works in a subtle way for the non-local terms. We derive a differential-integral equation for the finite modular flow, which can be analyzed in perturbation theory of small distance deviating from the entanglement boundary, and re-summation can be performed in appropriate limits. Comparison with the general expectation of modular flows in such limits are discussed.
We study the entanglement entropy (EE) and the R\'enyi entropy (RE) of multiple intervals in two-dimensional TT- deformed conformal field theory (CFT) at a 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 nontrivial 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 cutoff radius (dual to the TT 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.