Abstract
We perform a comprehensive analysis of the symmetry-resolved (SR) entanglement entropy (EE) for one single interval in the ground state of a 1 + 1D conformal field theory (CFT), that is invariant under an arbitrary finite or compact Lie group, G. We utilize the boundary CFT approach to study the total EE, which enables us to find the universal leading order behavior of the SREE and its first correction, which explicitly depends on the irreducible representation under consideration and breaks the equipartition of entanglement. We present two distinct schemes to carry out these computations. The first relies on the evaluation of the charged moments of the reduced density matrix. This involves studying the action of the defect-line, that generates the symmetry, on the boundary states of the theory. This perspective also paves the way for discussing the infeasibility of studying symmetry resolution when an anomalous symmetry is present. The second scheme draws a parallel between the SREE and the partition function of an orbifold CFT. This approach allows for the direct computation of the SREE without the need to use charged moments. From this standpoint, the infeasibility of defining the symmetry-resolved EE for an anomalous symmetry arises from the obstruction to gauging. Finally, we derive the symmetry-resolved entanglement spectra for a CFT invariant under a finite symmetry group. We revisit a similar problem for CFT with compact Lie group, explicitly deriving an improved formula for U(1) resolved entanglement spectra. Using the Tauberian formalism, we can estimate the aforementioned EE spectra rigorously by proving an optimal lower and upper bound on the same. In the abelian case, we perform numerical checks on the bound and find perfect agreement.
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Acknowledgments
We thank Filiberto Ares, Pasquale Calabrese, Giuseppe Di Giulio, Michele Fossati, Kantaro Ohmori, Brandon Rayhaun, Shu-Heng Shao, Yuji Tachikawa, and Yijian Zou for useful discussions and comments on the draft. The work by YK, HO, and SP is supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under Award Number DE-SC0011632. In addition, YK is supported by the Brinson Prize Fellowship at Caltech. HO is supported in part by the Simons Investigator Award (MP-SIP-00005259), the World Premier International Research Center Initiative, MEXT, Japan, and JSPS Grants-in-Aid for Scientific Research 20K03965 and 23K03379. SP is supported in part by the Sherman Fairchild Postdoctoral Fellowship at Caltech. SM thanks support from Caltech Institute for Quantum Information and Matter and the Walter Burke Institute for Theoretical Physics at Caltech. This work was performed in part at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611.
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Kusuki, Y., Murciano, S., Ooguri, H. et al. Symmetry-resolved entanglement entropy, spectra & boundary conformal field theory. J. High Energ. Phys. 2023, 216 (2023). https://doi.org/10.1007/JHEP11(2023)216
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DOI: https://doi.org/10.1007/JHEP11(2023)216