Abstract
The entanglement entropy corresponding to a smooth region in general three-dimensional CFTs contains a constant universal term, −F ⊂ SEE. For a disk region, F|disk ≡ F0 coincides with the free energy on 𝕊3 and provides an RG-monotone for general theories. As opposed to the analogous quantity in four dimensions, the value of F generally depends in a complicated (and non-local) way on the geometry of the region and the theory under consideration. For small geometric deformations of the disk in general CFTs as well as for arbitrary regions in holographic theories, it has been argued that F is precisely minimized by disks. Here, we argue that F is globally minimized by disks with respect to arbitrary regions and for general theories. The proof makes use of the strong subadditivity of entanglement entropy and the geometric fact that one can always place an osculating circle within a given smooth entangling region. For topologically non-trivial entangling regions with nB boundaries, the general bound can be improved to F ≥ nBF0. In addition, we provide accurate approximations to F valid for general CFTs in the case of elliptic regions for arbitrary values of the eccentricity which we check against lattice calculations for free fields. We also evaluate F numerically for more general shapes in the so-called “Extensive Mutual Information model”, verifying the general bound.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].
D.V. Fursaev, Entanglement Rényi entropies in conformal field theories and holography, JHEP 05 (2012) 080 [arXiv:1201.1702] [INSPIRE].
B.R. Safdi, Exact and numerical results on entanglement entropy in (5 + 1)-dimensional CFT, JHEP 12 (2012) 005 [arXiv:1206.5025] [INSPIRE].
R.-X. Miao, Universal terms of entanglement entropy for 6d CFTs, JHEP 10 (2015) 049 [arXiv:1503.05538] [INSPIRE].
P. Bueno, H. Casini and W. Witczak-Krempa, Generalizing the entanglement entropy of singular regions in conformal field theories, JHEP 08 (2019) 069 [arXiv:1904.11495] [INSPIRE].
J.S. Dowker, Entanglement entropy for odd spheres, arXiv:1012.1548 [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
M. Mezei, Entanglement entropy across a deformed sphere, Phys. Rev. D 91 (2015) 045038 [arXiv:1411.7011] [INSPIRE].
N. Bobev, P. Bueno and Y. Vreys, Comments on squashed-sphere partition functions, JHEP 07 (2017) 093 [arXiv:1705.00292] [INSPIRE].
S. Fischetti and T. Wiseman, On universality of holographic results for (2 + 1)-dimensional CFTs on curved spacetimes, JHEP 12 (2017) 133 [arXiv:1707.03825] [INSPIRE].
S. Fischetti, L. Wallis and T. Wiseman, What spatial geometries do (2 + 1)-dimensional quantum field theory vacua prefer?, Phys. Rev. Lett. 120 (2018) 261601 [arXiv:1803.04414] [INSPIRE].
S. Fischetti, L. Wallis and T. Wiseman, Does the round sphere maximize the free energy of (2 + 1)-dimensional QFTs?, JHEP 10 (2020) 078 [arXiv:2003.09428] [INSPIRE].
K. Cheamsawat, S. Fischetti, L. Wallis and T. Wiseman, A surprising similarity between holographic CFTs and a free fermion in (2 + 1) dimensions, JHEP 05 (2021) 246 [arXiv:2012.14437] [INSPIRE].
H. Casini and M. Huerta, Entanglement and alpha entropies for a massive scalar field in two dimensions, J. Stat. Mech. 0512 (2005) P12012 [cond-mat/0511014] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
F.C. Marques and A. Neves, The Willmore conjecture, Jahresbericht Deutschen Math. Verein. 116 (2014) 201 [arXiv:1409.7664].
T. Willmore, Riemannian geometry, Oxford science publications, Clarendon Press, Oxford, U.K. (1996).
P. Djondjorov et al., Willmore energy and Willmore conjecture, Chapman and Hall/CRC, (2017).
S. Alexakis and R. Mazzeo, Renormalized area and properly embedded minimal surfaces in hyperbolic 3-manifolds, arXiv:0802.2250.
A.F. Astaneh, G. Gibbons and S.N. Solodukhin, What surface maximizes entanglement entropy?, Phys. Rev. D 90 (2014) 085021 [arXiv:1407.4719] [INSPIRE].
P. Fonda, D. Seminara and E. Tonni, On shape dependence of holographic entanglement entropy in AdS4/CFT3, JHEP 12 (2015) 037 [arXiv:1510.03664] [INSPIRE].
E. Perlmutter, M. Rangamani and M. Rota, Central charges and the sign of entanglement in 4D conformal field theories, Phys. Rev. Lett. 115 (2015) 171601 [arXiv:1506.01679] [INSPIRE].
H.B. Lawson, Complete minimal surfaces in S3, Ann. Math. 92 (1970) 335.
R. Kusner, Comparison surfaces for the Willmore problem, Pacific J. Math. 138 (1989) 317.
T. Faulkner, R.G. Leigh and O. Parrikar, Shape dependence of entanglement entropy in conformal field theories, JHEP 04 (2016) 088 [arXiv:1511.05179] [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
H. Liu and A.A. Tseytlin, D = 4 super Yang-Mills, D = 5 gauged supergravity, and D = 4 conformal supergravity, Nucl. Phys. B 533 (1998) 88 [hep-th/9804083] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
P. Bueno, R.C. Myers and W. Witczak-Krempa, Universality of corner entanglement in conformal field theories, Phys. Rev. Lett. 115 (2015) 021602 [arXiv:1505.04804] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
P. Fonda, L. Giomi, A. Salvio and E. Tonni, On shape dependence of holographic mutual information in AdS4, JHEP 02 (2015) 005 [arXiv:1411.3608] [INSPIRE].
K.A. Brakke, The surface evolver, Exper. Math. 1 (1992) 141.
H. Dorn, Holographic entanglement entropy for hollow cones and banana shaped regions, JHEP 06 (2016) 052 [arXiv:1602.06756] [INSPIRE].
D. Katsinis, I. Mitsoulas and G. Pastras, Geometric flow description of minimal surfaces, Phys. Rev. D 101 (2020) 086015 [arXiv:1910.06680] [INSPIRE].
F. Marques and A. Neves, The min-max theory and the Willmore conjecture, Ann. Math. 179 (2014) 683.
G. Anastasiou, J. Moreno, R. Olea and D. Rivera-Betancour, Shape dependence of renormalized holographic entanglement entropy, JHEP 09 (2020) 173 [arXiv:2002.06111] [INSPIRE].
G. Anastasiou, I.J. Araya, C. Arias and R. Olea, Einstein-AdS action, renormalized volume/area and holographic Rényi entropies, JHEP 08 (2018) 136 [arXiv:1806.10708] [INSPIRE].
G. Anastasiou, I.J. Araya, J. Moreno, R. Olea and D. Rivera-Betancour, Renormalized holographic entanglement entropy for quadratic curvature gravity, Phys. Rev. D 104 (2021) 086003 [arXiv:2102.11242] [INSPIRE].
M. Taylor and L. Too, Renormalized entanglement entropy and curvature invariants, JHEP 12 (2020) 050 [arXiv:2004.09568] [INSPIRE].
M. Babich and A. Bobenko, Willmore tori with umbilic lines and minimal surfaces in hyperbolic space, Duke Math. J. 72 (1993) 151.
A. Allais and M. Mezei, Some results on the shape dependence of entanglement and Rényi entropies, Phys. Rev. D 91 (2015) 046002 [arXiv:1407.7249] [INSPIRE].
P. Bueno, R.C. Myers and W. Witczak-Krempa, Universal corner entanglement from twist operators, JHEP 09 (2015) 091 [arXiv:1507.06997] [INSPIRE].
J. Helmes, L.E. Hayward Sierens, A. Chandran, W. Witczak-Krempa and R.G. Melko, Universal corner entanglement of Dirac fermions and gapless bosons from the continuum to the lattice, Phys. Rev. B 94 (2016) 125142 [arXiv:1606.03096] [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
M. Mariño, Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories, J. Phys. A 44 (2011) 463001 [arXiv:1104.0783] [INSPIRE].
C.A. Agón, P. Bueno and H. Casini, Is the EMI model a QFT? An inquiry on the space of allowed entropy functions, arXiv:2105.11464 [INSPIRE].
H. Casini, M. Huerta, R.C. Myers and A. Yale, Mutual information and the F-theorem, JHEP 10 (2015) 003 [arXiv:1506.06195] [INSPIRE].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [INSPIRE].
I.R. Klebanov, T. Nishioka, S.S. Pufu and B.R. Safdi, Is renormalized entanglement entropy stationary at RG fixed points?, JHEP 10 (2012) 058 [arXiv:1207.3360] [INSPIRE].
H. Casini, Entropy localization and extensivity in the semiclassical black hole evaporation, Phys. Rev. D 79 (2009) 024015 [arXiv:0712.0403] [INSPIRE].
H. Casini and M. Huerta, Remarks on the entanglement entropy for disconnected regions, JHEP 03 (2009) 048 [arXiv:0812.1773] [INSPIRE].
H. Casini, F.D. Mazzitelli and E. Testé, Area terms in entanglement entropy, Phys. Rev. D 91 (2015) 104035 [arXiv:1412.6522] [INSPIRE].
I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36 (2003) L205 [cond-mat/0212631].
H. Casini, C.D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 0507 (2005) P07007 [cond-mat/0505563] [INSPIRE].
W. Witczak-Krempa, L.E. Hayward Sierens and R.G. Melko, Cornering gapless quantum states via their torus entanglement, Phys. Rev. Lett. 118 (2017) 077202 [arXiv:1603.02684] [INSPIRE].
B. Estienne, J.-M. Stéphan and W. Witczak-Krempa, Cornering the universal shape of fluctuations, arXiv:2102.06223 [INSPIRE].
H. Casini and M. Huerta, Universal terms for the entanglement entropy in 2 + 1 dimensions, Nucl. Phys. B 764 (2007) 183 [hep-th/0606256] [INSPIRE].
T. Hirata and T. Takayanagi, AdS/CFT and strong subadditivity of entanglement entropy, JHEP 02 (2007) 042 [hep-th/0608213] [INSPIRE].
B. Swingle, Mutual information and the structure of entanglement in quantum field theory, arXiv:1010.4038 [INSPIRE].
H. Casini, E. Teste and G. Torroba, Modular Hamiltonians on the null plane and the Markov property of the vacuum state, J. Phys. A 50 (2017) 364001 [arXiv:1703.10656] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Markov property of the conformal field theory vacuum and the a theorem, Phys. Rev. Lett. 118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
Y. Nakaguchi and T. Nishioka, Entanglement entropy of annulus in three dimensions, JHEP 04 (2015) 072 [arXiv:1501.01293] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2107.12394
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Bueno, P., Casini, H., Andino, O.L. et al. Disks globally maximize the entanglement entropy in 2 + 1 dimensions. J. High Energ. Phys. 2021, 179 (2021). https://doi.org/10.1007/JHEP10(2021)179
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2021)179