Localized thermal states and negative energy

We construct localized states defined in a ball or the half-space of a conformal field theory (CFT) in Minkowski that are thermal with respect to the local modular flow. We compute their energy density at arbitrary temperature for a variety of CFTs, and find values for which it is negative and divergent at the boundary. Despite this singular behavior we show that the energy measured by an observer is consistent with the bounds present in the literature. For holographic CFTs these states are captured by hyperbolic black holes in anti-de Sitter, where the negative energy in field theory amounts to the well known negative mass of the black hole. As a byproduct, we show that the Casini-Huerta-Myers proof of the Ryu-Takayangi holographic entanglement formula for the vacuum reduced to a ball can be naturally extended to include half-space regions.