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arXiv:2406.06121 · Journal of High Energy Physics

Hawking-Page and entanglement phase transition in 2d CFT on curved backgrounds

Akihiro Miyata, Masahiro Nozaki A, Kotaro Tamaoka, Masataka Watanabe

Published Aug 22, 2024• 12 citations
arXivHTMLPDF

Abstract

A bstract The thermodynamics and the entanglement properties of two-dimensional conformal field theories (2d CFTs) on curved backgrounds are studied. By means of conformal mapping we study the equivalent system on flat space governed by the deformed Hamiltonian, which is a spatial integral of the Hamiltonian density modulated by an enveloping function. Focusing on holographic CFTs, we observe Hawking-Page like phase transition for the thermal and the entanglement entropy as we vary the background metric. We also compute the mutual information to study the information theoretic correlation between parts of the curved spacetime. The gravity dual of 2d CFTs on curved background is also discussed.

DOI
10.1007/jhep08(2024)190

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•Nov 30, 2017•0 citations
A bstract The thermodynamics and the entanglement properties of two-dimensional conformal field theories (2d CFTs) on curved backgrounds are studied. By means of conformal mapping we study the equivalent system on flat space governed by the deformed Hamiltonian, which is a spatial integral of the Hamiltonian density modulated by an enveloping function. Focusing on holographic CFTs, we observe Hawking-Page like phase transition for the thermal and the entanglement entropy as we vary the background metric. We also compute the mutual information to study the information theoretic correlation between parts of the curved spacetime. The gravity dual of 2d CFTs on curved background is also discussed.
A bstract We will explore the dynamical property of non-equilibrium phenomena induced by two-dimensional holographic conformal field theory (2d holographic CFT) Hamiltonian on the curved spacetime by studying the time dependence of the entanglement entropy and mutual information. Here, holographic CFT is the CFT having the gravity dual. We will start from the boundary and thermofield double states, evolve the systems in Euclidean time with the Hamiltonian on the curved background, and then evolve them in real-time with the same Hamiltonian. We found that the early- and late-time entanglement structure depends on the curved background, while the entanglement growth does not, and is linear. Furthermore, in the gravity dual for the thermofield double state, this entanglement growth is due to the linear growth of the wormhole, while in the one for the boundary state, it is due to the in-falling of the end of the world brane to the black hole. We discussed the low temperature system can be regarded as the dynamical system induced by the multi-joining quenches. We also discussed the effective description of the high temperature system, called line tension picture.
A large part of the discussion on entanglement islands has explored the specific setup of 2d2d2d JT gravity with a flat heatbath coupled to a 2d2d2d CFT. In this paper, we consider a more general setup and treatment of islands in a d−d-d−dimensional AdS black hole background. The quantum fields modeling the Hawking radiation have a scale and are consistently inherited from a conformal parent theory; their symmetries are thus compatible with those of curved backgrounds. We demonstrate explicitly that the existence of islands is sensitive to the choice of CFT used to model the Hawking radiation. We compute the renormalised entanglement entropy of conformal fields on a negatively curved background in ddd dimensions at zero temperature as well as the thermal regulated entropy of an entangling region near the UV boundary. Using the latter quantity as the entropy of the Hawking radiation, we find that islands never emerge for d>2d>2d>2.
We study entanglement entropy (EE) in conformal field theories (CFTs) in Minkowski space with a planar boundary or with a planar defect of any codimension. In any such boundary CFT or defect CFT (DCFT), we consider the reduced density matrix and associated EE obtained by tracing over the degrees of freedom outside of a (hemi)sphere centered on the boundary or defect. Following Casini, Huerta, and Myers, we map the reduced density matrix to a thermal density matrix of the same theory on hyperbolic space. The EE maps to the thermal entropy of the theory on hyperbolic space. For boundary CFTs and DCFTs dual holographically to Einstein gravity theories, the thermal entropy is equivalent to the Bekenstein--Hawking entropy of a hyperbolic black brane. We show that the horizon of the hyperbolic black brane coincides with the minimal area surface used in Ryu and Takayanagi's conjecture for the holographic calculation of EE. We thus prove their conjecture in these cases. We use our results to compute the R\'enyi entropies and EE in DCFTs in which the defect corresponds to a probe brane in a holographic dual.
We analyze how excitations affect the entanglement entropy for an arbitrary entangling interval in a 2d conformal field theory (CFT) using the holographic entanglement entropy techniques as well as direct CFT computations. We introduce the excitation entanglement entropy ${\mathrm{\ensuremath{\Delta}}}_{h}S$, the difference between the entanglement entropy generic excitations and their arbitrary conformal descendants denoted through hhh. The excitation entanglement entropy, unlike the entanglement entropy, is a finite quantity (independent of the cutoff), and hence a good physical observable. We show that the excitation entanglement entropy for any given interval is uniquely specified by a local second order differential equation sourced by the one point function of the energy momentum tensor computed in the excited background state, and two boundary and smoothness conditions. We analyze low and high temperature behavior of the excitation entanglement entropy and show that ${\mathrm{\ensuremath{\Delta}}}_{h}S$ grows as a function of temperature. We prove an ``integrated positivity'' for the excitation entanglement entropy, that although ${\mathrm{\ensuremath{\Delta}}}_{h}S$ can be positive or negative, its average value is always positive. We also discuss the mutual and multipartite information and (strong) subadditivity inequality in the presence of generic excitations and their conformal descendants.
Abstract In this note, we consider entanglement and Renyi entropies for spatial subsystems of a boundary conformal field theory (BCFT) or of a CFT in a state constructed using a Euclidean BCFT path integral. Holographic calculations suggest that these entropies undergo phase transitions as a function of time or parameters describing the subsystem; these arise from a change in topology of the RT surface. In recent applications to black hole physics, such transitions have been seen to govern whether or not the bulk entanglement wedge of a (B)CFT region includes a portion of the black hole interior and have played a crucial role in understanding the semiclassical origin of the Page curve for evaporating black holes. In this paper, we reproduce these holographic results via direct (B)CFT calculations. Using the replica method, the entropies are related to correlation functions of twist operators in a Euclidean BCFT. These correlations functions can be expanded in various channels involving intermediate bulk or boundary operators. Under certain sparseness conditions on the spectrum and OPE coefficients of bulk and boundary operators, we show that the twist correlators are dominated by the vacuum block in a single channel, with the relevant channel depending on the position of the twists. These transitions between channels lead to the holographically observed phase transitions in entropies.
In this thesis we study the time evolution of Rényi and entanglement entropies of thermal states in Conformal Field Theory (CFT). These quantities are usually hard to compute but Ryu-Takayanagi (RT) and Hubeny-Rangamani-Takayanagi (HRT) proposals allow us to find the same quantities using calculations in general relativity. We will introduce main concepts of holography, quantum information and conformal field theory that will be used to derive the results of this thesis. In the first part of the thesis, we explicitly compute entanglement entropy of the rotating BTZ black hole by directly applying HRT proposal and finding lengths of spacelike geodesics. Rényi entropy of thermal state perturbed by a local quantum quench is computed by mapping correlators on two glued cylinders to the plane for field theory containing a single free boson and for 2d CFTs in the large c limit. We consider Thermofield Double State (TFD) which is an entangled state in direct product of two 2D CFTs. It is conjectured to be holographically equivalent to the eternal BTZ black hole. TFD state is perturbed by a local quench in one CFT and mutual information between two intervals in two CFTs is computed. We find when mutual information vanishes and interpret this as scrambling time, i.e. time scale required for the system to thermalize. This field theory result is modelled with a massive free falling particle in the BTZ black hole. We have computed the back-reaction of the particle on the metric of BTZ and used RT proposal to find holographic entanglement entropy. Finally, we generalize this calculation to the case of rotating BTZ with inner and outer horizons. It is dual to the CFT with different temperatures for left and right moving modes. We calculate mutual information and scrambling time and find exact agreement between results in the gravity and those in the CFT.