Abstract
We study conformal twist field four-point functions on a ℤN orbifold. We examine in detail the case N = 3 and analyze theories obtained by replicated N-times a minimal model with central charge c < 1. A fastly convergent expansion of the twist field correlation function in terms of sphere conformal blocks with central charge Nc is obtained by exploiting covering map techniques. We discuss extensive applications of the formalism to the entanglement of two disjoint intervals in CFT, in particular we propose a conformal block expansion for the partially transposed reduced density matrix. Finally, we refine the bounds on the structure constants of unitary CFTs determined previously by the genus two modular bootstrap.
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A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
S. Ribault, Conformal field theory on the plane, arXiv:1406.4290 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville field theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE].
G. Delfino and J. Viti, On three-point connectivity in two-dimensional percolation, J. Phys. A 44 (2011) 032001 [arXiv:1009.1314] [INSPIRE].
M. Picco, S. Ribault and R. Santachiara, On four-point connectivities in the critical 2d Potts model, SciPost Phys. 7 (2019) 044 [arXiv:1906.02566] [INSPIRE].
Y. He, J.L. Jacobsen and H. Saleur, Geometrical four-point functions in the two-dimensional critical Q-state Potts model: The interchiral conformal bootstrap, JHEP 12 (2020) 019 [arXiv:2005.07258] [INSPIRE].
A.W.W. Ludwig, Critical Behavior of the Two-dimensional Random Q State Potts Model by Expansion in (Q − 2), Nucl. Phys. B 285 (1987) 97 [INSPIRE].
J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge University Press (1996) [INSPIRE].
V. Dotsenko, J.L. Jacobsen, M.-A. Lewis and M. Picco, Coupled Potts models: Self-duality and fixed point structure, Nucl. Phys. B 546 (1999) 505 [cond-mat/9812227] [INSPIRE].
Z. Komargodski and D. Simmons-Duffin, The Random-Bond Ising Model in 2.01 and 3 Dimensions, J. Phys. A 50 (2017) 154001 [arXiv:1603.04444] [INSPIRE].
G. Delfino, Particles, conformal invariance and criticality in pure and disordered systems, Eur. Phys. J. B 94 (2021) 65 [Erratum ibid. 94 (2021) 87] [arXiv:2010.12275] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
M. Caraglio and F. Gliozzi, Entanglement Entropy and Twist Fields, JHEP 11 (2008) 076 [arXiv:0808.4094] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory, J. Stat. Mech. 0911 (2009) P11001 [arXiv:0905.2069] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement entropy of two disjoint intervals in conformal field theory II, J. Stat. Mech. 1101 (2011) P01021 [arXiv:1011.5482] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in quantum field theory, Phys. Rev. Lett. 109 (2012) 130502 [arXiv:1206.3092] [INSPIRE].
P. Calabrese, J. Cardy and E. Tonni, Entanglement negativity in extended systems: A field theoretical approach, J. Stat. Mech. 1302 (2013) P02008 [arXiv:1210.5359] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for MN/SN orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
L.J. Dixon, D. Friedan, E.J. Martinec and S.H. Shenker, The Conformal Field Theory of Orbifolds, Nucl. Phys. B 282 (1987) 13 [INSPIRE].
V.G. Knizhnik, Analytic Fields on Riemann Surfaces. 2, Commun. Math. Phys. 112 (1987) 567 [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
A.B. Zamolodchikov, Conformal Scalar Field on the Hyperelliptic Curve and Critical Ashkin-teller Multipoint Correlation Functions, Nucl. Phys. B 285 (1987) 481 [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
J. Cardy, A. Maloney and H. Maxfield, A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance, JHEP 10 (2017) 136 [arXiv:1705.05855] [INSPIRE].
M. Cho, S. Collier and X. Yin, Genus Two Modular Bootstrap, JHEP 04 (2019) 022 [arXiv:1705.05865] [INSPIRE].
C.A. Keller, G. Mathys and I.G. Zadeh, Bootstrapping Chiral CFTs at Genus Two, Adv. Theor. Math. Phys. 22 (2018) 1447 [arXiv:1705.05862] [INSPIRE].
M. Cho, S. Collier and X. Yin, Recursive Representations of Arbitrary Virasoro Conformal Blocks, JHEP 04 (2019) 018 [arXiv:1703.09805] [INSPIRE].
M.A. Rajabpour and F. Gliozzi, Entanglement Entropy of Two Disjoint Intervals from Fusion Algebra of Twist Fields, J. Stat. Mech. 1202 (2012) P02016 [arXiv:1112.1225] [INSPIRE].
P. Ruggiero, E. Tonni and P. Calabrese, Entanglement entropy of two disjoint intervals and the recursion formula for conformal blocks, J. Stat. Mech. 1811 (2018) 113101 [arXiv:1805.05975] [INSPIRE].
T. Dupic, B. Estienne and Y. Ikhlef, Entanglement entropies of minimal models from null-vectors, SciPost Phys. 4 (2018) 031 [arXiv:1709.09270] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088.
P. Calabrese, L. Tagliacozzo and E. Tonni, Entanglement negativity in the critical Ising chain, J. Stat. Mech. 1305 (2013) P05002 [arXiv:1302.1113] [INSPIRE].
V. Alba, Entanglement negativity and conformal field theory: a Monte Carlo study, J. Stat. Mech. 1305 (2013) P05013 [arXiv:1302.1110] [INSPIRE].
V. Eisler and Z. Zimboras, On the partial transpose of fermionic Gaussian states, New J. Phys. 17 (2015) 053048 [arXiv:1502.01369].
A. Coser, E. Tonni and P. Calabrese, Partial transpose of two disjoint blocks in XY spin chains, J. Stat. Mech. 1508 (2015) P08005 [arXiv:1503.09114] [INSPIRE].
A. Coser, E. Tonni and P. Calabrese, Towards the entanglement negativity of two disjoint intervals for a one dimensional free fermion, J. Stat. Mech. 1603 (2016) 033116 [arXiv:1508.00811] [INSPIRE].
A. Coser, E. Tonni and P. Calabrese, Spin structures and entanglement of two disjoint intervals in conformal field theories, J. Stat. Mech. 1605 (2016) 053109 [arXiv:1511.08328] [INSPIRE].
H. Shapourian, P. Ruggiero, S. Ryu and P. Calabrese, Twisted and untwisted negativity spectrum of free fermions, SciPost Phys. 7 (2019) 037 [arXiv:1906.04211] [INSPIRE].
T. Grava, A.P. Kels and E. Tonni, Entanglement of Two Disjoint Intervals in Conformal Field Theory and the 2D Coulomb Gas on a Lattice, Phys. Rev. Lett. 127 (2021) 141605 [arXiv:2104.06994] [INSPIRE].
M. Kulaxizi, A. Parnachev and G. Policastro, Conformal Blocks and Negativity at Large Central Charge, JHEP 09 (2014) 010 [arXiv:1407.0324] [INSPIRE].
J. Kudler-Flam and S. Ryu, Entanglement negativity and minimal entanglement wedge cross sections in holographic theories, Phys. Rev. D 99 (2019) 106014 [arXiv:1808.00446] [INSPIRE].
Y. Kusuki, J. Kudler-Flam and S. Ryu, Derivation of Holographic Negativity in AdS3/CFT2, Phys. Rev. Lett. 123 (2019) 131603 [arXiv:1907.07824] [INSPIRE].
B.A. Dubrovin, A.T. Fomenko and S.P. Novikov, Modern Geometry — Methods and Applications Part II. The Geometry and Topology of Manifolds, Springer (1985) [DOI].
R. Miranda, Algebraic Curves and Riemann Surfaces, American Mathematical Society (1995).
E.T. Whittaker and G.N. Watson, A Course in Modern Analysis, Cambridge University Press (1950).
A. Cappelli, C. Itzykson and J.B. Zuber, Modular Invariant Partition Functions in Two-Dimensions, Nucl. Phys. B 280 (1987) 445 [INSPIRE].
A. Cappelli, C. Itzykson and J.B. Zuber, The ADE Classification of Minimal and \( {A}_1^{(1)} \) Conformal Invariant Theories, Commun. Math. Phys. 113 (1987) 1 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer (1999).
N. Javerzat, R. Santachiara and O. Foda, Notes on the solutions of Zamolodchikov-type recursion relations in Virasoro minimal models, JHEP 08 (2018) 183 [arXiv:1806.02790] [INSPIRE].
R. Santachiara and A. Tanzini, Moore-Read Fractional Quantum Hall wavefunctions and SU(2) quiver gauge theories, Phys. Rev. D 82 (2010) 126006 [arXiv:1002.5017] [INSPIRE].
K.B. Alkalaev and V.A. Belavin, Conformal blocks of WN minimal models and AGT correspondence, JHEP 07 (2014) 024 [arXiv:1404.7094] [INSPIRE].
V. Belavin, O. Foda and R. Santachiara, AGT, N-Burge partitions and \( \mathcal{W} \)N minimal models, JHEP 10 (2015) 073 [arXiv:1507.03540] [INSPIRE].
R. Santachiara and J. Viti, Local logarithmic correlators as limits of Coulomb gas integrals, Nucl. Phys. B 882 (2014) 229 [arXiv:1311.2055] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
L. Borisov, M.B. Halpern and C. Schweigert, Systematic approach to cyclic orbifolds, Int. J. Mod. Phys. A 13 (1998) 125 [hep-th/9701061] [INSPIRE].
S. Furukawa, V. Pasquier and J. Shiraishi, Mutual Information and Compactification Radius in a c=1 Critical Phase in One Dimension, Phys. Rev. Lett. 102 (2009) 170602 [arXiv:0809.5113] [INSPIRE].
V. Alba, L. Tagliacozzo and P. Calabrese, Entanglement entropy of two disjoint blocks in critical Ising models, Phys. Rev. B 81 (2010) 060411 [arXiv:0910.0706] [INSPIRE].
V. Alba, L. Tagliacozzo and P. Calabrese, Entanglement entropy of two disjoint intervals in c = 1 theories, J. Stat. Mech. 1106 (2011) P06012 [arXiv:1103.3166] [INSPIRE].
G. Mussardo, Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford University Press (2020).
A. Feiguin et al., Interacting anyons in topological quantum liquids: The golden chain, Phys. Rev. Lett. 98 (2007) 160409 [cond-mat/0612341] [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
D. Poland, S. Rychkov and A. Vichi, The Conformal Bootstrap: Theory, Numerical Techniques, and Applications, Rev. Mod. Phys. 91 (2019) 015002 [arXiv:1805.04405] [INSPIRE].
S. Ribault and R. Santachiara, Liouville theory with a central charge less than one, JHEP 08 (2015) 109 [arXiv:1503.02067] [INSPIRE].
S H. Simon, E.H. Rezayi, N.R. Cooper and I. Berdnikov, Construction of a paired wave function for spinless electrons at filling fraction ν = 2/5, Phys. Rev. B 75 (2007) 075317.
E. Ardonne, J. Gukelberger, A.W.W. Ludwig, S. Trebst and M. Troyer, Microscopic models of interacting Yang-Lee anyons, New J. Phys. 13 (2011) 045006 [arXiv:1012.1080].
D. Bianchini, O.A. Castro-Alvaredo, B. Doyon, E. Levi and F. Ravanini, Entanglement Entropy of Non Unitary Conformal Field Theory, J. Phys. A 48 (2015) 04FT01 [arXiv:1405.2804] [INSPIRE].
V.S. Dotsenko and V.A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B 284 (1984) 312 [INSPIRE].
J. Teschner, Liouville theory revisited, Class. Quant. Grav. 18 (2001) R153 [hep-th/0104158] [INSPIRE].
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Ares, F., Santachiara, R. & Viti, J. Crossing-symmetric twist field correlators and entanglement negativity in minimal CFTs. J. High Energ. Phys. 2021, 175 (2021). https://doi.org/10.1007/JHEP10(2021)175
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DOI: https://doi.org/10.1007/JHEP10(2021)175