The modular energy range of an interval in chiral conformal field theory

Abstract

We determine exactly the range of the modular Hamiltonian of an interval in a chiral conformal field theory over states of bounded energy. For an interval of length $R$ in a theory of central charge $c$, and states of mean energy at most $E$, the supremum of the modular energy is $M_{\max }=\frac{\pi R}{2}E+\frac{c}{6}-\frac{\pi c^2}{288RE}+O((RE{)}^{-2})$, and neither constant can be improved. The infimum is logarithmic, $M_{\min }\simeq -\frac{c}{12}[\ln (24\pi RE/c)-1]$, so the range is linear above and logarithmic below, the two meeting as $RE\to 0$; it is strictly wider above, by more than $\frac{2}{5}\pi RE$ at every energy, which reduces after a Stieltjes substitution to the positivity of a variance. Both bounds are attained, the one-parameter family of weights proving them being the Legendre structure of the answer. The two ingredients separate cleanly. The coefficient $\pi R/2$ is Möbius representation theory alone: the complementary weight is a translate of the special conformal generator, hence positive, so for the full modular Hamiltonian the bound needs no additive constant. The constant $c/6$ is precisely the price of truncating that weight at the endpoints, supplied by the quantum energy inequality of Fewster and Hollands; the saturating state places its negative energy exactly there. The constant is accordingly additive over entangling points, being $nc/6$ for $n$ disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, $\Delta S_I\le \frac{\pi R}{2}E+\frac{c}{6}$, recovering the original form of that inequality from the modular form that by itself imposes no constraint.

arXiv comment15 pages