Abstract We argue for an exponential bound characterizing the chaotic properties of modular Hamiltonian flow of QFT subsystems. In holographic theories, maximal modular chaos is reflected in the local Poincare symmetry about a Ryu-Takayanagi surface. Generators of null deformations of the bulk extremal surface map to modular scrambling modes — positive CFT operators saturating the bound — and their algebra probes the bulk Riemann curvature, clarifying the modular Berry curvature proposal of arXiv:1903.04493.
Abstract We argue for an exponential bound characterizing the chaotic properties of modular Hamiltonian flow of QFT subsystems. In holographic theories, maximal modular chaos is reflected in the local Poincare symmetry about a Ryu-Takayanagi surface. Generators of null deformations of the bulk extremal surface map to modular scrambling modes — positive CFT operators saturating the bound — and their algebra probes the bulk Riemann curvature, clarifying the modular Berry curvature proposal of arXiv:1903.04493.
Abstract We consider a Hayden & Preskill like setup for both maximally chaotic and sub-maximally chaotic quantum field theories. We act on the vacuum with an operator in a Rindler like wedge R and transfer a small subregion I of R to the other wedge. The chaotic scrambling dynamics of the QFT Rindler time evolution reveals the information in the other wedge. The holographic dual of this process involves a particle excitation falling into the bulk and crossing into the entanglement wedge of the complement to r = R\I. With the goal of studying the locality of the emergent holographic theory we compute various quantum information measures on the boundary that tell us when the particle has entered this entanglement wedge. In a maximally chaotic theory, these measures indicate a sharp transition where the particle enters the wedge exactly when the insertion is null separated from the quantum extremal surface for r. For sub-maximally chaotic theories, we find a smoothed crossover at a delayed time given in terms of the smaller Lyapunov exponent and dependent on the time-smearing scale of the probe excitation. The information quantities that we consider include the full vacuum modular energy R\I as well as the fidelity between the state with the particle and the state without. Along the way, we find a new explicit formula for the modular Hamiltonian of two intervals in an arbitrary 1+1 dimensional CFT to leading order in the small cross ratio limit. We also give an explicit calculation of the Regge limit of the modular flowed chaos correlator and find examples which do not saturate the modular chaos bound. Finally, we discuss the extent to which our results reveal properties of the target of the probe excitation as a “stringy quantum extremal surface” or simply quantify the probe itself thus giving a new approach to studying the notion of longitudinal string spreading.
Abstract We discover the connection between the Berry curvature and the Riemann curvature tensor in any kinematic space of minimal surfaces anchored on spherical entangling surfaces. This new holographic principle establishes the Riemann geometry in kinematic space of arbitrary dimensions from the holonomy of modular Hamiltonian, which in the higher dimensions is specified by a pair of time‐like separated points as in CFT1 and CFT2. The Berry curvature that we constructed also shares the same property of the Riemann curvature for all geometry: internal symmetry; skew symmetry; first Bianchi identity. We derive the algebra of the modular Hamiltonian and its deformation, the latter of which can provide the maximal modular chaos to the modular scrambling modes. The algebra also dictates the parallel transport, which leads to the Berry curvature exactly matching to the Riemann curvature tensor. Finally, we compare CFT1 to higher dimensional CFTs and show the difference from the OPE block.
In this Dissertation, different aspects of holographic entanglement entropy are explored. In quantum information theory, entanglement is a useful resource needed to perform quantum operations. Entanglement entropy quantifies this resource and even if computable in quantum field theories, it is quite hard to calculate explicitly. However, in the context of gauge/gravity duality, (holographic) entanglement entropy was proposed by Ryu and Takayanagi (RT) to be captured by the area of a minimal surface in Anti DeSitter (AdS) space, which is extremely simple to compute. We begin by using the holographic dictionary to derive the RT formula. This is done by first considering smooth bulk geometries which are dual to n copies of the boundary geometry glued in a particular way. The action of these geometries computes the Renyi entropies S_n for any integer n. We give a prescription to analytically continue the action to non-integer n and argue that, as n goes to 1,S_1, the entanglement entropy, is given by the area of a minimal surface, reproducing the RT conjecture. That is, we reduced the RT proposal to the equality between bulk and boundary partition functions, a standard entry in the dictionary. We then generalize the previous formalism to account for corrections due to bulk quantum fields (1/N corrections in the boundary). This allows us to derive a new formula: boundary entanglement entropy is given by the area of the minimal surface plus bulk entanglement entropy. This extends RT beyond the planar limit and we present several predictions of the proposal. Our exploration of holographic entanglement entropy continues by considering different states: by comparing the quantum corrected entropy for nearby states we obtain the modular hamiltonian operator in bulk perturbation theory. From this expression of the modular hamiltonian, we derive that relative entropy is bulk relative entropy and that the modular flow is the bulk modular flow.
Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.
A bstract The local two-dimensional Poincaré algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincaré algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincaré algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic. Interacting quantum systems in the thermodynamic limit and large N theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.
Abstract Recent work showed holographic error correcting codes to have simple universal features at O(1/G). In particular, states of fixed Ryu-Takayanagi (RT) area in such codes are associated with flat entanglement spectra indicating maximal entanglement between appropriate subspaces. We extend such results to one-loop order (O(1) corrections) by controlling both higher-derivative corrections to the bulk effective action and dynamical quantum fluctuations below the cutoff. This result clarifies the relation between the bulk path integral and the quantum code, and implies that i) simple tensor network models of holography continue to match the behavior of holographic CFTs beyond leading order in G, ii) the relation between bulk and boundary modular Hamiltonians derived by Jafferis, Lewkowycz, Maldacena, and Suh holds as an operator equation on the code subspace and not just in code-subspace expectation values, and iii) the code subspace is invariant under an appropriate notion of modular flow. A final corollary requires interesting cancelations to occur in the bulk renormalization-group flow of holographic quantum codes. Intermediate technical results include showing the Lewkowycz-Maldacena computation of RT entropy to take the form of a Hamilton-Jacobi variation of the action with respect to boundary conditions, corresponding results for higher-derivative actions, and generalizations to allow RT surfaces with finite conical angles.