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arXiv:2609.27830

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

Samuel L. Braunstein, Zhi-Wei Wang

Published Aug 19, 2026• 0 citations
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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 $M_{\max} = \frac{πR}{2}E + \frac{c}{6} - \frac{πc^{2}}{288\,RE} + O((RE)^{-2})$, and neither constant can be improved. The infimum is logarithmic, $M_{\min} \simeq -\frac{c}{12}[\ln(24πRE/c)-1]$, so the range is linear above and logarithmic below, the two meeting as $RE \to 0$; it is strictly wider above, by more than $\frac{2}{5}πRE$ 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, $ΔS_I \leq \frac{πR}{2}E + \frac{c}{6}$, recovering the original form of that inequality from the modular form that by itself imposes no constraint.

DOI
10.48550/arxiv.2609.27830

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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 RRR in a theory of central charge ccc, and states of mean energy at most EEE, the supremum of the modular energy is Mmax⁡=πR2E+c6−πc2288 RE+O((RE)−2)M_{\max} = \frac{πR}{2}E + \frac{c}{6} - \frac{πc^{2}}{288\,RE} + O((RE)^{-2})Mmax​=2πR​E+6c​−288REπc2​+O((RE)−2), and neither constant can be improved. The infimum is logarithmic, Mmin⁡≃−c12[ln⁡(24πRE/c)−1]M_{\min} \simeq -\frac{c}{12}[\ln(24πRE/c)-1]Mmin​≃−12c​[ln(24πRE/c)−1], so the range is linear above and logarithmic below, the two meeting as RE→0RE \to 0RE→0; it is strictly wider above, by more than 25πRE\frac{2}{5}πRE52​πRE 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/2πR/2πR/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/6c/6c/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/6nc/6nc/6 for nnn disjoint intervals of equal length. Composing with positivity of relative entropy gives a sharp Bekenstein bound, ΔSI≤πR2E+c6ΔS_I \leq \frac{πR}{2}E + \frac{c}{6}ΔSI​≤2πR​E+6c​, recovering the original form of that inequality from the modular form that by itself imposes no constraint.
We obtain new constraints for the modular energy of general states by using the monotonicity property of relative entropy. In some cases, modular energy can be related to the energy density of states and these constraints lead to interesting relations between energy and entropy. In particular, we derive new quantum energy inequalities that improve some previous bounds for the energy density of states in a conformal field theory. Additionally, the inequalities derived in this manner also lead us to conclude that the entropy of the state further restricts the possible amount of negative energy allowed by the theory.
We prove using invariance under the modular SSS- and STSTST-transformations that every unitary two-dimensional conformal field theory (CFT) of only even-spin operators (with no extended chiral algebra and with central charges c,c~>1c,\tilde{c}>1c,c~>1) contains a primary operator with dimension Δ1Δ_1Δ1​ satisfying 0<Δ1<(c+c~)/24+0.09280...0 < Δ_1 < (c+\tilde{c})/24 + 0.09280...0<Δ1​<(c+c~)/24+0.09280... After deriving both analytical and numerical bounds, we discuss how to extend our methods to bound higher conformal dimensions before deriving lower and upper bounds on the number of primary operators in a given energy range. Using the AdS3_33​/CFT2_22​ dictionary, the bound on Δ1Δ_1Δ1​ proves the lightest massive excitation in appropriate theories of 3D matter and gravity with cosmological constant Λ<0Λ< 0Λ<0 can be no heavier than 1/(8GN)+O(−Λ)1/(8G_N)+O(\sqrt{-Λ})1/(8GN​)+O(−Λ​); the bounds on the number operators are related via AdS/CFT to the entropy of states in the dual gravitational theory. In the flat-space approximation, the limiting mass is exactly that of the lightest BTZ black hole.
Abstract The modular Hamiltonian of reduced states, given essentially by the logarithm of the reduced density matrix, plays an important role within the AdS/CFT correspondence in view of its relation to quantum information. In particular, it is an essential ingredient for quantum information measures of distances between states, such as the relative entropy and the Fisher information metric. However, the modular Hamiltonian is known explicitly only for a few examples. For a family of states ρ λ that is parametrized by a scalar λ, the first order contribution in λ∼=λ−λ0 \overset{\sim }{\lambda } = \lambda -{\lambda}_0 λ∼=λ−λ0​ λ ∼ = λ − λ 0 of the modular Hamiltonian to the relative entropy between ρλ and a reference state ρλ 0 is completely determined by the entanglement entropy, via the first law of entanglement. For several examples, e.g. for ball-shaped regions in the ground state of CFTs, higher order contributions are known to vanish. In these cases the modular Hamiltonian contributes to the Fisher information metric in a trivial way. We investigate under which conditions the modular Hamiltonian provides a non-trivial contribution to the Fisher information metric, i.e. when the contribution of the modular Hamiltonian to the relative entropy is of higher order in λ∼ \overset{\sim }{\lambda } λ∼ λ ∼ . We consider one-parameter families of reduced states on two entangling regions that form an entanglement plateau, i.e. the entanglement entropies of the two regions saturate the Araki-Lieb inequality. We show that in general, at least one of the relative entropies of the two entangling regions is expected to involve λ∼ \overset{\sim }{\lambda } λ∼ λ ∼ contributions of higher order from the modular Hamiltonian. Furthermore, we consider the implications of this observation for prominent AdS/CFT examples that form entanglement plateaux in the large N limit.
We derive sharp lower bounds on the energy cost of Page-like entanglement entropy excursions in two-dimensional conformal field theory. For states satisfying the primary quantum null energy condition (QNEC), the positive null energy E+E_+E+​ obeys a sharp lower bound with two branches separated by the threshold ΔS=(c/6)ln⁡2ΔS=(c/6)\ln 2ΔS=(c/6)ln2. We also derive a sharp lower bound on the net energy EnetE_{\rm net}Enet​, with a coefficient larger by a factor of two than the value quoted by Abdolrahimi and Page [Phys. Rev. D 92, 083005 (2015)] and, in the QNEC-saturating sector, we obtain a sharp lower bound on the negative energy E−E_-E−​. The constants are sharp as infima over smooth conformal vacua. We determine the entropy profiles that minimize E+E_+E+​ for general QNEC states and the one that minimizes E−E_-E−​ in the saturating sector. Finally, for an excursion containing an infinite sequence of small sub-excursions within a finite duration, we derive a necessary condition for the total positive energy to remain finite, which is stronger than requiring the entropy to have finite total variation.
In this article, we extend our study on a new class of modular Hamiltonians on an interval attached to the origin on the semi-infinite line, introduced in a recent work dedicated to scalar fields. Here, we shift our attention to fermions and similarly to the scalar case, we investigate the modular Hamiltonians of theories which are obtained through dimensional reduction, this time, of a free massless Dirac field in ddd dimensions. By following the same methodology, we perform dimensional reduction on both the physical and modular Hamiltonians. This process enables us to establish a correspondence: we identify the modular Hamiltonian in an interval connected to the origin to the one obtained from the reduction of the modular Hamiltonian pertaining to the conformal parent theory on a sphere. Intriguingly, although the resulting one-dimensional theories lack conformal symmetry due to the presence of a term proportional to 1/r1/r1/r, the corresponding modular Hamiltonians are local functions in the energy density. This phenomenon mirrors the well known behaviour observed in the conformal case and indicates the existence of a residual symmetry, characterized by a subset of the original conformal group. Furthermore, through an analysis of the spectrum of the modular Hamiltonians, we derive an analytic expression for the associated entanglement entropy. Our findings also enable us to successfully recover the conformal anomaly coefficient from the universal piece of the entropy in even dimensions, as well as the universal constant FFF term in d=3d=3d=3, by extending the radial regularization scheme originally introduced by Srednicki to perform the sum over angular modes.
A bstract In this article, we extend our study on a new class of modular Hamiltonians on an interval attached to the origin on the semi-infinite line, introduced in a recent work dedicated to scalar fields. Here, we shift our attention to fermions and similarly to the scalar case, we investigate the modular Hamiltonians of theories which are obtained through dimensional reduction, this time, of a free massless Dirac field in d dimensions. By following the same methodology, we perform dimensional reduction on both the physical and modular Hamiltonians. This process enables us to establish a correspondence: we identify the modular Hamiltonian in an interval connected to the origin to the one obtained from the reduction of the modular Hamiltonian pertaining to the conformal parent theory on a sphere. Intriguingly, although the resulting one-dimensional theories lack conformal symmetry due to the presence of a term proportional to 1/ r , the corresponding modular Hamiltonians are local functions in the energy density. This phenomenon mirrors the well known behaviour observed in the conformal case and indicates the existence of a residual symmetry, characterized by a subset of the original conformal group. Furthermore, through an analysis of the spectrum of the modular Hamiltonians, we derive an analytic expression for the associated entanglement entropy. Our findings also enable us to successfully recover the conformal anomaly coefficient from the universal piece of the entropy in even dimensions, as well as the universal constant F term in d = 3, by extending the radial regularization scheme originally introduced by Srednicki to perform the sum over angular modes.