A bstract In this work, we consider the atypical non-equilibrium state found in [1708.06328] which holographically represents a behind-the-horizon excitation in a black hole spacetime. The special feature of this state is that it looks like an equilibrium state when probed by a class of low-energy operators. First, we retrieve this property using the uniformization mapping in the limit of a large central charge, in the process we are able to derive rather than presume approximate thermal physics. Furthermore, in the large- c and high-energy limit, we realize these excitations as elements of the commutant algebra of a GNS-representation of the light operator algebra. Instead of analytically continuing a mixed heavy-light Euclidean correlator to a Lorentzian correlator, we identify the Euclidean correlator as a GNS-linear form and interpret the Lorentzian correlator as a vacuum expectation value of representatives of the light operator algebra on the GNS-vacuum.
A bstract In this work, we consider the atypical non-equilibrium state found in [1708.06328] which holographically represents a behind-the-horizon excitation in a black hole spacetime. The special feature of this state is that it looks like an equilibrium state when probed by a class of low-energy operators. First, we retrieve this property using the uniformization mapping in the limit of a large central charge, in the process we are able to derive rather than presume approximate thermal physics. Furthermore, in the large- c and high-energy limit, we realize these excitations as elements of the commutant algebra of a GNS-representation of the light operator algebra. Instead of analytically continuing a mixed heavy-light Euclidean correlator to a Lorentzian correlator, we identify the Euclidean correlator as a GNS-linear form and interpret the Lorentzian correlator as a vacuum expectation value of representatives of the light operator algebra on the GNS-vacuum.
We consider a class of non-equilibrium pure states, which are generally present in an isolated quantum statistical system. These are states of the form ∣Ψ⟩=e−2βHUe2βH∣Ψ0⟩, where U is a unitary made out of simple operators and ∣Ψ0⟩ is a typical equilibrium pure state with sharply peaked energy. We argue that in a system with a holographic dual these states have a natural interpretation as an AdS black hole with transient excitations behind the horizon. We explore the interpretation of these states as pure states undergoing a time-dependent spontaneous fluctuation out of equilibrium. While these states are atypical and the microscopic phases of the wavefunction are correlated with the matrix elements of simple operators, the states are partly disguised as equilibrium states due to cancellations between contributions from different coarse-grained energy bins. These cancellations are guaranteed by the KMS condition of the underlying equilibrium state ∣Ψ0⟩. However, in correlators which include the Hamiltonian H these cancellations are spoiled and the non-equilibrium nature of the state ∣Ψ⟩ can be detected. We discuss connections with the proposal that local observables behind the horizon are realized as state-dependent operators. The states studied in this paper may be useful for implementing an analogue of the "traversable wormhole" protocol for a 1-sided black hole, which could potentially allow us to extract the excitation from behind the horizon. We include some pedagogical background material.
A bstract Recently, it has been observed that the Hartle-Hawking correlators, a signature of smooth horizon, can emerge from certain heavy excited state correlators in the (manifestly non-smooth) BTZ stretched horizon background, in the limit when the stretched horizon approaches the real horizon. In this note, we develop a framework of quantizing the CFT modular Hamiltonian, that explains the necessity of introducing a stretched horizon and the emergence of thermal features in the AdS-Rindler and (planar) BTZ backgrounds. In more detail, we quantize vacuum modular Hamiltonian on a spatial segment of S 1 , which can be written as a particular linear combination of sl(2,ℝ) generators. Unlike radial quantization, (Euclidean) time circles emerge naturally here which can be contracted smoothly to the ‘fixed points’(end points of the interval) of this quantization thus providing a direct link to thermal physics. To define a Hilbert space with discrete normalizable states and to construct a Virasoro algebra with finite central extension, a natural regulator ( ϵ ) is needed around the fixed points. Eventually, in the dual description the fixed points correspond to the horizons of AdS-Rindler patch or (planar) BTZ and the cut-off being the stretched horizon. We construct a (Lorentzian) highest weight representation of that Virasoro algebra where vacuum can be identified with certain boundary states on the cut-off surface. We further demonstrate that two point function in a (vacuum) descendant state of the regulated Hilbert space will reproduce thermal answer in ϵ → 0 limit which is analogous to the recent observation of emergent thermality in (planar) BTZ stretched horizon background. We also argue the thermal entropy of this quantization coincides with entanglement entropy of the subregion. Conversely, the microcanonical entropy corresponding to high energy density of states exactly reproduce the BTZ entropy. Quite remarkably, all these dominant high lying microstates are defined only at finite ϵ in the regulated Hilbert space. We expect that all our observations can be generalized to BTZ in stretched horizon background where the boundary spatial coordinate is compactified.
In arXiv:2312.14108, we argued that a sliver at the black hole mass in the Hilbert space of a quantum field theory on an AdS black hole with a stretched horizon, has many desirable features of black hole microstates. A key observation is that the stretched horizon requires a finite Planck length, i.e., finite-N. Therefore it is best viewed as a ``quantum horizon" -- a proxy for the UV-complete bulk description, and not directly an element in the bulk EFT. It was shown in arXiv:2312.14108 that despite the manifest absence of the interior, the 2-point function in the sliver is indistinguishable from the smooth horizon correlator, up to the Page time. In this paper, instead of boundary correlators, we work directly in the bulk, and demonstrate the appearance of the bulk Hartle-Hawking correlator in the large-N limit. This is instructive because it shows that the analytic continuation across the horizon is emergent. It is a bulk transition to a Type III algebra and provides a structural distinction between black holes and weakly coupled thermal systems. We also identify a mechanism for universal code subspaces and interior tensor factors to appear via a quantum horizon version of thermal factorization. Our claims apply directly only within a ``Page window", so they are not in immediate tension with firewall arguments. During a Page window, typical heavy microstates probed by light single trace operators respond with effectively smooth horizons in low-point correlators. We work in 2+1 dimensions to be concrete, but expect our results to hold in all higher dimensions. We discuss some important differences between our approach and the conventional fuzzball program, and also argue that probe-notions (like infalling boundary conditions) must be distinguished from microstate-notions (like size of the Einstein-Rosen bridge) to make meaningful statements about post-Page smoothness.
We consider the entanglement entropy in 2d conformal field theory in a class of excited states produced by the insertion of a heavy local operator. These include both high-energy eigenstates of the Hamiltonian and time-dependent local quenches. We compute the universal contribution from the stress tensor to the single interval Renyi entropies and entanglement entropy, and conjecture that this dominates the answer in theories with a large central charge and a sparse spectrum of low-dimension operators. The resulting entanglement entropies agree precisely with holographic calculations in three-dimensional gravity. High-energy eigenstates are dual to microstates of the BTZ black hole, so the corresponding holographic calculation is a geodesic length in the black hole geometry; agreement between these two answers demonstrates that these individual microstates of holographic CFTs effectively thermalize at the level of the single-interval entanglement entropy. For local quenches, the dual geometry is a highly boosted black hole or conical defect. On the CFT side, the rise in entanglement entropy after a quench is directly related to the monodromy of a Virasoro conformal block.
We investigate the complex time evolution of a vacuum state with the insertion of a local primary operator in two-dimensional conformal field theories (2d CFTs). This complex time evolution can be considered as a composite process constructed from Lorentzian time evolution and a Euclidean evolution induced by a post-selected measurement. Our main finding is that in the spatially-compact system, this complex time evolution drives the state of the subsystems to those of the primary state with the same conformal dimensions of the inserted operator. Contrary to the compact system, the subsystems of the spatially non-compact system evolve to states that depend on the non-unitary process during a certain time regime. In holographic systems with a compact spatial direction, this process induced by a heavy local operator can correspond to the relaxation from a black hole with an inhomogeneous horizon to that with a uniform one, while in the ones with a non-compact spatial direction, it can correspond to the relaxation to that with a horizon depending on the non-unitary process.
A bstract We study quantum quench processes in (1+1)-dimensional conformal field theory (CFT) in which the initial thermal equilibrium (Gibbs) state is time-evolved by spatially inhomogeneous Hamiltonians, the so-called Möbius and sine-square-deformed (SSD) Hamiltonians. We found that, when the quench is induced by the SSD Hamiltonian, almost all the degrees of freedom are asymptotically gathered at a single point, resulting in a point-like excitation. This excitation, which we dub black hole-like excitation, carries as much information as the total thermal entropy. In contrast, other parts of the system approach the low-entropy (low-temperature) state at late times. For the quench by the Möbius Hamiltonian, we instead found an eternal periodic oscillation of physical quantities such as von Neumann entropy for subsystems. When the CFT admits a holographic dual description, the SSD quench induces a time-dependent, inhomogeneous deformation of the bulk black hole horizon, which, at late enough times, “touches” the boundary. Our quench setups can be used as a way to create low-temperature states, and, also, simulate the formation and evaporation processes of black holes.