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 Mmax = πR/2E + c/6 − πc2/(288 RE) + O((RE)−2), and neither constant can be improved. The infimum is logarithmic, Mmin ≃ −c/12[ln(24πRE/c)−1], so the range is linear above and logarithmic below, the two meeting as RE → 0; it is strictly wider above, by more than 2/5πRE 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 πR/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, ΔSI ≤ πR/2E + c/6, recovering the original form of that inequality from the modular form that by itself imposes no constraint.
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BibTeX
@misc{braunstein2026modular,
title = {{The modular energy range of an interval in chiral conformal field theory}},
author = {Samuel L. Braunstein and Zhi-Wei Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27830},
eprint = {2609.27830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27830},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The modular energy range of an interval in chiral conformal field theory
AU - Samuel L. Braunstein
AU - Zhi-Wei Wang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27830
UR - https://arxiv.org/abs/2609.27830
AB - 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 = πR/2E + c/6 − πc^2/(288 RE) + O((RE)^−2), and neither constant can be improved. The infimum is logarithmic, M_min ≃ −c/12[ln(24πRE/c)−1], so the range is linear above and logarithmic below, the two meeting as RE → 0; it is strictly wider above, by more than 2/5πRE 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 πR/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, ΔS_I ≤ πR/2E + c/6, recovering the original form of that inequality from the modular form that by itself imposes no constraint.
ER -