We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on R 1,d−1 . We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories. arXiv:1605.08072v1 [hep-th]

In this paper, we demonstrate the emergence of nonlinear gravitational equations directly from the physics of a broad class of conformal field theories. We consider CFT excited states defined by adding sources for scalar primary or stress tensor operators to the Euclidean path integral defining the vacuum state. For these states, we show that up to second order in the sources, the entanglement entropy for all ball-shaped regions can always be represented geometrically (via the Ryu-Takayanagi formula) by an asymptotically AdS geometry. We show that such a geometry necessarily satisfies Einstein's equations perturbatively up to second order, with a stress energy tensor arising from matter fields associated with the sourced primary operators. We make no assumptions about AdS/CFT duality, so our work serves as both a consistency check for the AdS/CFT correspondence and a direct demonstration that spacetime and gravitational physics can emerge from the description of entanglement in conformal field theories. arXiv:1705.03026v1 [hep-th] 8 May 2017 D Bulk-to-boundary propagators 48 E A simple example 49 1. The coordinate location of A(ε) is fixed.

We study the transport properties of topological insulators, encoding them in a generating functional of gauge and gravitational sources. Much of our focus is on the simple example of a free massive Dirac fermion, the so-called Chern insulator, especially in 2+1 dimensions. In such cases, when parity and time-reversal symmetry are broken, it is necessary to consider the gravitational sources to include a frame and an independent spin connection with torsion. In 2+1 dimensions, the simplest parity-odd response is the Hall viscosity. We compute the Hall viscosity of the Chern insulator using a careful regularization scheme, and find that although the Hall viscosity is generally divergent, the difference in Hall viscosities of distinct topological phases is well-defined and determined by the mass gap. Furthermore, on a 1+1-dimensional edge between topological phases, the jump in the Hall viscosity across the interface is encoded, through familiar anomaly inflow mechanisms, in the structure of anomalies. In particular, we find new torsional contributions to the covariant diffeomorphism anomaly in 1+1 dimensions. Including parity-even contributions, we find that the renormalized generating functionals of the two topological phases differ by a chiral gravity action with a negative cosmological constant. This (non-dynamical) chiral gravity action and the corresponding physics of the interface theory is reminiscent of well-known properties of dynamical holographic gravitational systems. Finally, we consider some properties of spectral flow of the edge theory driven by torsional dislocations.

We study universal features in the shape dependence of entanglement entropy in the vacuum state of a conformal field theory (CFT) on R 1,d−1 . We consider the entanglement entropy across a deformed planar or spherical entangling surface in terms of a perturbative expansion in the infinitesimal shape deformation. In particular, we focus on the second order term in this expansion, known as the entanglement density. This quantity is known to be non-positive by the strong-subadditivity property. We show from a purely field theory calculation that the non-local part of the entanglement density in any CFT is universal, and proportional to the coefficient C T appearing in the two-point function of stress tensors in that CFT. As applications of our result, we prove the conjectured universality of the corner term coefficient σ C T = π 2 24 in d = 3 CFTs, and the holographic Mezei formula for entanglement entropy across deformed spheres. arXiv:1511.05179v2 [hep-th]

We study the response of a class of topological systems to electromagnetic and gravitational sources, including torsion and curvature. By using the technology of anomaly polynomials, we derive the parity-odd response of a massive Dirac fermion in d = 2 + 1 and d = 4 + 1, which provides a simple model for a topological insulator. We discuss the covariant anomalies of the corresponding edge states, from a CallanHarvey anomaly-inflow, as well as a Hamiltonian spectral flow point of view. We also discuss the applicability of our results to other systems such as Weyl semi-metals. Finally, using dimensional reduction from d = 4 + 1, we derive the effective action for a d = 3 + 1 time-reversal invariant topological insulator in the presence of torsion and curvature, and discuss its various physical consequences.