  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.
