Universality of Rényi Entropy in Conformal Field Theory
Creators
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1.
Kyushu University
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2.
California Institute of Technology
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 SA(n) behaves as SA(n)=[f/(2πn)d−1][Area(∂A)/(d−2)εd−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.
Copyright and License
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.
Acknowledgement
The authors thank David Simmons-Duffin for helpful discussions and comments on the draft. This research was supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award No. DE-SC0011632 and by the Walter Burke Institute for Theoretical Physics at Caltech. H. O. is also supported in part by the Simons Investigator grant (MP-SIP-00005259) and by JSPS Grants-in-Aid for Scientific Research 23K03379. His work was performed in part at the Kavli Institute for the Physics and Mathematics of the Universe at the University of Tokyo, which is supported by the World Premier International Research Center Initiative, MEXT, Japan, at the Kavli Institute for Theoretical Physics (KITP) at the University of California, Santa Barbara, which is supported by grant NSF PHY-2309135, and at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611. Y. K. is also supported by the INAMORI Frontier Program at Kyushu University and JSPS KAKENHI Grant No. 23K20046. The authors thank Kyushu University Institute for Advanced Study and RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program. Discussions during the “Kyushu IAS-iTHEMS workshop: Nonperturbative methods in QFT” were useful in completing this work.
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Additional details
Funding
- United States Department of Energy
- Office of Science
- Office of High Energy Physics
- DE-SC0011632
- California Institute of Technology
- Walter Burke Institute for Theoretical Physics -
- Simons Foundation
- MP-SIP-00005259
- Japan Society for the Promotion of Science
- 23K03379
- Japan Society for the Promotion of Science
- 23K20046
- Ministry of Education, Culture, Sports, Science and Technology
- University of California, Santa Barbara
- National Science Foundation
- PHY-2309135
- National Science Foundation
- PHY-1607611
- Kyushu University
- World Premier International Research Center Initiative
Dates
- Available
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2025-08-05Published online