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arXiv:2503.24353 · Physical Review Letters

Universality of Rényi Entropy in Conformal Field Theory

Yuya Kusuki, Hirosi Ooguri, Sridip Pal

Published Jul 14, 2025• 3 citations
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Abstract

We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in d dimensions, the nth Rényi entropy S_{A}^{(n)} behaves as S_{A}^{(n)}=[f/(2πn)^{d-1}][Area(∂A)/(d-2)ε^{d-2}](1+O(n)) in the n→0 limit when the boundary of the entanglement domain A is spherical with the UV cutoff ε. The theory dependence is encapsulated in the cosmological constant f in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the modular Hamiltonian for the domain A. In two dimensions, we can use the hot spot idea, which describes the effective action in the high-temperature limit when the temperature is position-dependent, to derive more powerful formulas valid for arbitrary positive n. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the Cardy formula for boundary operators in higher dimensions.

DOI
10.1103/fsg7-bs7q

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We use the thermal effective theory to prove that, for the vacuum state in any conformal field theory in d dimensions, the nth Rényi entropy S_{A}^{(n)} behaves as S_{A}^{(n)}=[f/(2πn)^{d-1}]Area(∂A)/(d-2)ε^{d-2} in the n→0 limit when the boundary of the entanglement domain A is spherical with the UV cutoff ε. The theory dependence is encapsulated in the cosmological constant f in the thermal effective action. Using this result, we estimate the density of states for large eigenvalues of the modular Hamiltonian for the domain A. In two dimensions, we can use the hot spot idea, which describes the effective action in the high-temperature limit when the temperature is position-dependent, to derive more powerful formulas valid for arbitrary positive n. We discuss the difference between two and higher dimensions and clarify the applicability of the hot spot idea. We also use the thermal effective theory to derive an analog of the Cardy formula for boundary operators in higher dimensions.
We propose bounds relating thermal entropy to entanglement in unitary conformal field theories: f≤4π2af\le4π^2af≤4π2a in four dimensions and f≤2.99830949… Ff\le2.99830949\ldots\,Ff≤2.99830949…F in three dimensions. Here fff is the dimensionless coefficient of the cosmological constant term in the thermal effective action, aaa is the Euler anomaly coefficient, and FFF is the sphere free energy. Spherical Rényi entropy relates the four-dimensional bound to the conformal collider bounds. We formulate a stronger conjectural inequality involving the stress-tensor three-point function, saturated by every free theory, and establish it in controlled regimes of weakly coupled gauge theories, large-NNN vector models, and holographic CFTs. Finite-NNN numerical results provide additional tests. The bounds constrain the Cardy growth of the density of states at high energy.
We derive several new results for Rényi entropy, S n , across generic entangling surfaces. We establish a perturbative expansion of the Rényi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the n-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the n-dependence of the universal part of the ground state Rényi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Rényi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Rényi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Rényi entropy also fixes the n-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.
Two-dimensional conformal field theories with a large central charge and a small number of low-dimension operators are studied using the conformal block expansion. A universal formula is derived for the Renyi entropies of N disjoint intervals in the ground state, valid to all orders in a series expansion. This is possible because the full perturbative answer in this regime comes from the exchange of the stress tensor and other descendants of the vacuum state. Therefore, the Renyi entropy is related to the Virasoro vacuum block at large central charge. The entanglement entropy, computed from the Renyi entropy by an analytic continuation, decouples into a sum of single-interval entanglements. This field theory result agrees with the Ryu-Takayanagi formula for the holographic entanglement entropy of a 2d CFT, applied to any number of intervals, and thus can be interpreted as a microscopic calculation of the area of minimal surfaces in 3d gravity.
We derive several new results for Rényi entropy, S n , across generic entangling surfaces. We establish a perturbative expansion of the Rényi entropy, valid in generic quantum field theories, in deformations of a given density matrix. When applied to even-dimensional conformal field theories, these results lead to new constraints on the n-dependence, independent of any perturbative expansion. In 4d CFTs, we show that the n-dependence of the universal part of the ground state Rényi entropy for entangling surfaces with vanishing extrinsic curvature contribution is in fact fully determined by the Rényi entropy across a sphere in flat space. Using holography, we thus provide the first computations of Rényi entropy across non-spherical entangling surfaces in strongly coupled 4d CFTs. Furthermore, we address the possibility that in a wide class of 4d CFTs, the flat space spherical Rényi entropy also fixes the n-dependence of the extrinsic curvature contribution, and hence that of arbitrary entangling surfaces. Our results have intriguing implications for the structure of generic modular Hamiltonians.
We compute thermal corrections to Rényi entropies of d dimensional conformal field theories on spheres. Consider the nth Rényi entropy for a cap of opening angle 2θ on S d−1. From a Boltzmann sum decomposition and the operator-state correspondence, the leading correction is related to a certain two-point correlation function of the operator (not equal to the identity) with smallest scaling dimension. More specifically, via a conformal map, the correction can be expressed in terms of the two-point function on a certain conical space with opening angle 2πn. In the case of free conformal field theories, this two-point function can be computed explicitly using the method of images. We perform the computation for the conformally coupled scalar. From the n → 1 limit of our results, we extract the leading thermal correction to the entanglement entropy, reproducing results of arXiv:1407.1358.
We evaluate one-interval Rényi entropy and entanglement entropy for the excited states of two-dimensional conformal field theory (CFT) on a cylinder, and examine their differences from the ones for the thermal state. We assume the interval to be short so that we can use operator product expansion (OPE) of twist operators to calculate Rényi entropy in terms of sum of one-point functions of OPE blocks. We find that the entanglement entropy for highly excited state and thermal state behave the same way after appropriate identification of the conformal weight of the state with the temperature. However, there exists no such universal identification for the Rényi entropy in the short-interval expansion. Therefore, the highly excited state does not look thermal when comparing its Rényi entropy to the thermal state one. As the Rényi entropy captures the higher moments of the reduced density matrix but the entanglement entropy only the average, our results imply that the emergence of thermality depends on how refined we look into the entanglement structure of the underlying pure excited state.