Abstract
We study the entanglement entropies of an interval for the massless compact boson either on the half line or on a finite segment, when either Dirichlet or Neumann boundary conditions are imposed. In these boundary conformal field theory models, the method of the branch point twist fields is employed to obtain analytic expressions for the two-point functions of twist operators. In the decompactification regime, these analytic predictions in the continuum are compared with the lattice numerical results in massless harmonic chains for the corresponding entanglement entropies, finding good agreement. The application of these analytic results in the context of quantum quenches is also discussed.
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P. Calabrese and J. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
V. Eisler and I. Peschel, Reduced density matrices and entanglement entropy in free lattice models, J. Phys. A 42 (2009) 504003 [arXiv:0906.1663] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
J. Eisert, M. Cramer and M.B. Plenio, Area laws for the entanglement entropy - a review, Rev. Mod. Phys. 82 (2010) 277 [arXiv:0808.3773] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic Entanglement Entropy, vol. 931, Springer (2017) [https://doi.org/10.1007/978-3-319-52573-0] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
H.-Q. Zhou, T. Barthel, J.O. Fjærestad and U. Schollwöck, Entanglement and boundary critical phenomena, Phys. Rev. A 74 (2006) 050305 [cond-mat/0511732].
I. Affleck, N. Laflorencie and E.S. Sørensen, Entanglement entropy in quantum impurity systems and systems with boundaries, J. Phys. A 42 (2009) 504009 [arXiv:0906.1809].
C. Berthiere and S.N. Solodukhin, Boundary effects in entanglement entropy, Nucl. Phys. B 910 (2016) 823 [arXiv:1604.07571] [INSPIRE].
L. Taddia, J.C. Xavier, F.C. Alcaraz and G. Sierra, Entanglement Entropies in Conformal Systems with Boundaries, arXiv:1302.6222 [https://doi.org/10.1103/PhysRevB.88.075112].
I. Affleck, Conformal field theory approach to the Kondo effect, Acta Phys. Polon. B 26 (1995) 1869 [cond-mat/9512099] [INSPIRE].
N. Laflorencie, E.S. Sørensen, M.-S. Chang and I. Affleck, Boundary effects in the critical scaling of entanglement entropy in 1D systems, Phys. Rev. Lett. 96 (2006) 100603 [cond-mat/0512475] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008 [hep-th/0011156] [INSPIRE].
T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].
M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].
M. Nozaki, T. Takayanagi and T. Ugajin, Central Charges for BCFTs and Holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].
G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, JHEP 09 (2020) 002 [arXiv:1905.08255] [INSPIRE].
A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063 [arXiv:1905.08762] [INSPIRE].
A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, The page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149 [arXiv:1908.10996] [INSPIRE].
A. Almheiri et al., Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013 [arXiv:1911.12333] [INSPIRE].
G. Penington, S.H. Shenker, D. Stanford and Z. Yang, Replica wormholes and the black hole interior, JHEP 03 (2022) 205 [arXiv:1911.11977] [INSPIRE].
B. Estienne and J.-M. Stéphan, Entanglement spectroscopy of chiral edge modes in the Quantum Hall effect, Phys. Rev. B 101 (2020) 115136 [arXiv:1911.10125] [INSPIRE].
B. Estienne, B. Oblak and J.-M. Stéphan, Ergodic Edge Modes in the 4D Quantum Hall Effect, SciPost Phys. 11 (2021) 016 [arXiv:2104.01860] [INSPIRE].
V. Crépel, N. Claussen, B. Estienne and N. Regnault, Model states for a class of chiral topological order interfaces, Nature Commun. 10 (2019) 1861 [arXiv:1806.06858] [INSPIRE].
V. Crépel, N. Claussen, N. Regnault and B. Estienne, Microscopic study of the Halperin–Laughlin interface through matrix product states, Nature Commun. 10 (2019) 1860 [arXiv:1904.11023] [INSPIRE].
M.B. Plenio and S. Virmani, An introduction to entanglement measures, Quant. Inf. Comput. 7 (2007) 001 [quant-ph/0504163] [INSPIRE].
S. Hollands and K. Sanders, Entanglement measures and their properties in quantum field theory, arXiv:1702.04924 [INSPIRE].
I. Peschel and J. Zhao, On single-copy entanglement, J. Stat. Mech. 2005 (2005) P11002 [quant-ph/0509002].
J. Eisert and M. Cramer, Single-copy entanglement in critical quantum spin chains, Phys. Rev. A 72 (2005) 042112 [quant-ph/0506250].
R. Orus, J.I. Latorre, J. Eisert and M. Cramer, Half the entanglement in critical systems is distillable from a single specimen, Phys. Rev. A 73 (2006) 060303 [quant-ph/0509023] [INSPIRE].
C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett. B 333 (1994) 55 [hep-th/9401072] [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].
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].
J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
J.L. Cardy, Effect of Boundary Conditions on the Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 275 (1986) 200 [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].
J.L. Cardy, Boundary conformal field theory, hep-th/0411189 [INSPIRE].
J.L. Cardy, O.A. Castro-Alvaredo and B. Doyon, Form factors of branch-point twist fields in quantum integrable models and entanglement entropy, J. Statist. Phys. 130 (2008) 129 [arXiv:0706.3384] [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].
A.B. Zamolodchikov, Conformal Scalar Field on the Hyperelliptic Curve and Critical Ashkin-teller Multipoint Correlation Functions, Nucl. Phys. B 285 (1987) 481 [INSPIRE].
M. Bershadsky and A. Radul, Conformal Field Theories with Additional Z(N) Symmetry, Int. J. Mod. Phys. A 2 (1987) 165 [INSPIRE].
V.G. Knizhnik, Analytic Fields on Riemann Surfaces. 2, Commun. Math. Phys. 112 (1987) 567 [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ’ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402 [hep-th/0312197] [INSPIRE].
H. Casini, I. Salazar Landea and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140 [arXiv:1607.00390] [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].
M. Caraglio and F. Gliozzi, Entanglement Entropy and Twist Fields, JHEP 11 (2008) 076 [arXiv:0808.4094] [INSPIRE].
H. Casini and M. Huerta, Reduced density matrix and internal dynamics for multicomponent regions, Class. Quant. Grav. 26 (2009) 185005 [arXiv:0903.5284] [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].
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].
M. Fagotti and P. Calabrese, Entanglement entropy of two disjoint blocks in XY chains, J. Stat. Mech. 1004 (2010) P04016 [arXiv:1003.1110] [INSPIRE].
A. Coser, L. Tagliacozzo and E. Tonni, On Rényi entropies of disjoint intervals in conformal field theory, J. Stat. Mech. 1401 (2014) P01008 [arXiv:1309.2189] [INSPIRE].
T. Hartman, Entanglement Entropy at Large Central Charge, arXiv:1303.6955 [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].
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].
C.A. Agon, M. Headrick, D.L. Jafferis and S. Kasko, Disk entanglement entropy for a Maxwell field, Phys. Rev. D 89 (2014) 025018 [arXiv:1310.4886] [INSPIRE].
C. De Nobili, A. Coser and E. Tonni, Entanglement entropy and negativity of disjoint intervals in CFT: Some numerical extrapolations, J. Stat. Mech. 1506 (2015) P06021 [arXiv:1501.04311] [INSPIRE].
B. Estienne, Y. Ikhlef and A. Rotaru, Second Rényi entropy and annulus partition function for one-dimensional quantum critical systems with boundaries, SciPost Phys. 12 (2022) 141 [arXiv:2112.01929] [INSPIRE].
M. Mintchev and E. Tonni, Modular Hamiltonians for the massless Dirac field in the presence of a boundary, JHEP 03 (2021) 204 [arXiv:2012.00703] [INSPIRE].
M. Fagotti and P. Calabrese, Universal parity effects in the entanglement entropy of XX chains with open boundary conditions, J. Stat. Mech. 1101 (2011) P01017 [arXiv:1010.5796] [INSPIRE].
F. Rottoli, S. Murciano, E. Tonni and P. Calabrese, Entanglement and negativity Hamiltonians for the massless Dirac field on the half line, J. Stat. Mech. 2301 (2023) 013103 [arXiv:2210.12109] [INSPIRE].
A. Bastianello, Rényi entanglement entropies for the compactified massless boson with open boundary conditions, JHEP 10 (2019) 141 [arXiv:1909.00806] [INSPIRE].
A. Bastianello, J. Dubail and J.-M. Stéphan, Entanglement entropies of inhomogeneous Luttinger liquids, J. Phys. A 53 (2020) 155001 [arXiv:1910.09967] [INSPIRE].
H.J. Schnitzer and K. Tsokos, Partition Functions and Fermi-bose Equivalence for Simple Laced Groups on Compact Riemann Surfaces, Nucl. Phys. B 291 (1987) 429 [INSPIRE].
L. Alvarez-Gaume, G.W. Moore and C. Vafa, Theta Functions, Modular Invariance and Strings, Commun. Math. Phys. 106 (1986) 1 [INSPIRE].
L. Alvarez-Gaume et al., Bosonization on Higher Genus Riemann Surfaces, Commun. Math. Phys. 112 (1987) 503 [INSPIRE].
P.H. Ginsparg and C. Vafa, Toroidal Compactification of Nonsupersymmetric Heterotic Strings, Nucl. Phys. B 289 (1987) 414 [INSPIRE].
S. Hamidi and C. Vafa, Interactions on Orbifolds, Nucl. Phys. B 279 (1987) 465 [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, C = 1 Conformal Field Theories on Riemann Surfaces, Commun. Math. Phys. 115 (1988) 649 [INSPIRE].
D. Bernard, Z(2) Twisted Fields and Bosonization on Riemann Surfaces, Nucl. Phys. B 302 (1988) 251 [INSPIRE].
J.J. Atick, L.J. Dixon, P.A. Griffin and D. Nemeschansky, Multiloop Twist Field Correlation Functions for Z(N) Orbifolds, Nucl. Phys. B 298 (1988) 1 [INSPIRE].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
P. Calabrese and J. Cardy, Entanglement and correlation functions following a local quench: a conformal field theory approach, J. Stat. Mech. 0710 (2007) P10004 [arXiv:0708.3750] [INSPIRE].
O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive 1+1-dimensional quantum field theories, J. Phys. A 42 (2009) 504006 [arXiv:0906.2946] [INSPIRE].
B. Estienne, Y. Ikhlef and A. Morin-Duchesne, Finite-size corrections in critical symmetry-resolved entanglement, SciPost Phys. 10 (2021) 054 [arXiv:2010.10515] [INSPIRE].
B.Q. Jin and V.E. Korepin, Quantum Spin Chain, Toeplitz Determinants and the Fisher—Hartwig Conjecture, J. Statist. Phys. 116 (2004) 79 [INSPIRE].
R. Blumenhagen and E. Plauschinn, Introduction to conformal field theory: with applications to String theory, Springer, Berlin Heidelberg (2009) [https://doi.org/10.1007/978-3-642-00450-6] [INSPIRE].
E. Alvarez, J.L.F. Barbon and J. Borlaf, T duality for open strings, Nucl. Phys. B 479 (1996) 218 [hep-th/9603089] [INSPIRE].
C. Schweigert, J. Fuchs and J. Walcher, Conformal field theory, boundary conditions and applications to string theory, in the proceedings of the Eotvos Summer School in Physics: Nonperturbative QFT Methods and Their Applications, Budapest, Hungary, August 19–21 (2000) [https://doi.org/10.1142/9789812799968_0002] [hep-th/0011109] [INSPIRE].
H.M. Farkas and I. Kra, Riemann surfaces, Springer (1992).
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].
M. Oshikawa and I. Affleck, Boundary conformal field theory approach to the critical two-dimensional Ising model with a defect line, Nucl. Phys. B 495 (1997) 533 [cond-mat/9612187] [INSPIRE].
C. Restuccia, Limit theories and continuous orbifolds, Ph.D. thesis, Humboldt University, Berlin, Germany (2013) [arXiv:1310.6857] [INSPIRE].
I. Runkel and G.M.T. Watts, A nonrational CFT with c = 1 as a limit of minimal models, JHEP 09 (2001) 006 [hep-th/0107118] [INSPIRE].
A. Recknagel and V. Schomerus, Boundary Conformal Field Theory and the Worldsheet Approach to D-Branes, Cambridge University Press (2013) [https://doi.org/10.1017/CBO9780511806476] [INSPIRE].
I. Peschel, Calculation of reduced density matrices from correlation functions, J. Phys. A 36 (2003) L205 [cond-mat/0212631] [INSPIRE].
K. Audenaert, J. Eisert, M.B. Plenio and R.F. Werner, Entanglement Properties of the Harmonic Chain, Phys. Rev. A 66 (2002) 042327 [quant-ph/0205025] [INSPIRE].
A. Botero and B. Reznik, Spatial structures and localization of vacuum entanglement in the linear harmonic chain, Phys. Rev. A 70 (2004) 052329 [quant-ph/0403233] [INSPIRE].
M.B. Plenio, J. Eisert, J. Dreissig and M. Cramer, Entropy, entanglement, and area: analytical results for harmonic lattice systems, Phys. Rev. Lett. 94 (2005) 060503 [quant-ph/0405142] [INSPIRE].
M. Cramer, J. Eisert, M.B. Plenio and J. Dreissig, An Entanglement-area law for general bosonic harmonic lattice systems, Phys. Rev. A 73 (2006) 012309 [quant-ph/0505092] [INSPIRE].
N. Schuch, J.I. Cirac and M.M. Wolf, Quantum States on Harmonic Lattices, Commun. Math. Phys. 267 (2006) 65.
S. Lievens, N.I. Stoilova and J. Van der Jeugt, Harmonic oscillator chains as Wigner Quantum Systems: Periodic and fixed wall boundary conditions in gl(1|n) solutions, J. Math. Phys. 49 (2008) 073502 [arXiv:0709.0180] [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].
C. Berthiere and W. Witczak-Krempa, Relating bulk to boundary entanglement, Phys. Rev. B 100 (2019) 235112 [arXiv:1907.11249] [INSPIRE].
P. Jain, S.M. Chandran and S. Shankaranarayanan, Log to log-log crossover of entanglement in (1 + 1)dimensional massive scalar field, Phys. Rev. D 103 (2021) 125008 [arXiv:2103.01772] [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, no. 3.616.7, Academic Press, 7th ed. (2007).
Y.K. Yazdi, Zero Modes and Entanglement Entropy, JHEP 04 (2017) 140 [arXiv:1608.04744] [INSPIRE].
V. Eisler and I. Peschel, Evolution of entanglement after a local quench, J. Stat. Mech. 2007 (2007) P06005 [cond-mat/0703379].
P. Calabrese and J. Cardy, Quantum quenches in 1 + 1 dimensional conformal field theories, J. Stat. Mech. 1606 (2016) 064003 [arXiv:1603.02889] [INSPIRE].
B. Estienne, Y. Ikhlef and A. Rotaru, Rényi entropies for one-dimensional quantum systems with mixed boundary conditions, arXiv:2301.02124 [INSPIRE].
O.A. Castro-Alvaredo and B. Doyon, Bi-partite entanglement entropy in massive QFT with a boundary: The Ising model, J. Statist. Phys. 134 (2009) 105 [arXiv:0810.0219] [INSPIRE].
J. Dubail, J.-M. Stéphan, J. Viti and P. Calabrese, Conformal field theory for inhomogeneous one-dimensional quantum systems: the example of non-interacting Fermi gases, SciPost Phys. 2 (2017) 002 [arXiv:1606.04401] [INSPIRE].
I. Peschel and V. Eisler, Exact results for the entanglement across defects in critical chains, J. Phys. A 45 (2012) 155301 [arXiv:1201.4104].
M. Gutperle and J.D. Miller, Entanglement entropy at CFT junctions, Phys. Rev. D 95 (2017) 106008 [arXiv:1701.08856] [INSPIRE].
M. Mintchev and E. Tonni, Modular Hamiltonians for the massless Dirac field in the presence of a defect, JHEP 03 (2021) 205 [arXiv:2012.01366] [INSPIRE].
A. Roy and H. Saleur, Entanglement Entropy in the Ising Model with Topological Defects, Phys. Rev. Lett. 128 (2022) 090603 [arXiv:2111.04534] [INSPIRE].
V. Fateev, A.B. Zamolodchikov and A.B. Zamolodchikov, Boundary Liouville field theory. 1. Boundary state and boundary two point function, hep-th/0001012 [INSPIRE].
A.B. Zamolodchikov and A.B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/0101152 [INSPIRE].
V. Eisler and I. Peschel, Free-fermion entanglement and spheroidal functions, J. Stat. Mech. 2013 (2013) P04028 [arXiv:1302.2239].
M. Mintchev, D. Pontello, A. Sartori and E. Tonni, Entanglement entropies of an interval in the free Schrödinger field theory at finite density, JHEP 07 (2022) 120 [arXiv:2201.04522] [INSPIRE].
M. Mintchev, D. Pontello and E. Tonni, Entanglement entropies of an interval in the free Schrödinger field theory on the half line, JHEP 09 (2022) 090 [arXiv:2206.06187] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensional space: Recursion representation of conformal block, Theor. Math. Phys. 73 (1987) 1088 [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].
J.-M. Stéphan and J. Dubail, Local quantum quenches in critical one-dimensional systems: entanglement, the Loschmidt echo, and light-cone effects, arXiv:1105.4846 [https://doi.org/10.1088/1742-5468/2011/08/P08019].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
A.M. Läuchli, Operator content of real-space entanglement spectra at conformal critical points, arXiv:1303.0741 [INSPIRE].
J. Cardy and E. Tonni, Entanglement hamiltonians in two-dimensional conformal field theory, J. Stat. Mech. 1612 (2016) 123103 [arXiv:1608.01283] [INSPIRE].
R. Arias, D. Blanco, H. Casini and M. Huerta, Local temperatures and local terms in modular Hamiltonians, Phys. Rev. D 95 (2017) 065005 [arXiv:1611.08517] [INSPIRE].
E. Tonni, J. Rodríguez-Laguna and G. Sierra, Entanglement hamiltonian and entanglement contour in inhomogeneous 1D critical systems, J. Stat. Mech. 1804 (2018) 043105 [arXiv:1712.03557] [INSPIRE].
V. Alba, P. Calabrese and E. Tonni, Entanglement spectrum degeneracy and the Cardy formula in 1+1 dimensional conformal field theories, J. Phys. A 51 (2018) 024001 [arXiv:1707.07532] [INSPIRE].
V. Eisler and I. Peschel, Analytical results for the entanglement Hamiltonian of a free-fermion chain, J. Phys. A 50 (2017) 284003 [arXiv:1703.08126] [INSPIRE].
V. Eisler and I. Peschel, Properties of the entanglement Hamiltonian for finite free-fermion chains, J. Stat. Mech. 1810 (2018) 104001 [arXiv:1805.00078] [INSPIRE].
R.E. Arias, H. Casini, M. Huerta and D. Pontello, Entropy and modular Hamiltonian for a free chiral scalar in two intervals, Phys. Rev. D 98 (2018) 125008 [arXiv:1809.00026] [INSPIRE].
J. Surace, L. Tagliacozzo and E. Tonni, Operator content of entanglement spectra in the transverse field Ising chain after global quenches, Phys. Rev. B 101 (2020) 241107(R) [arXiv:1909.07381] [INSPIRE].
G. Di Giulio and E. Tonni, On entanglement hamiltonians of an interval in massless harmonic chains, J. Stat. Mech. 2003 (2020) 033102 [arXiv:1911.07188] [INSPIRE].
G. Di Giulio, R. Arias and E. Tonni, Entanglement hamiltonians in 1D free lattice models after a global quantum quench, J. Stat. Mech. 1912 (2019) 123103 [arXiv:1905.01144] [INSPIRE].
V. Eisler, E. Tonni and I. Peschel, On the continuum limit of the entanglement Hamiltonian, J. Stat. Mech. 1907 (2019) 073101 [arXiv:1902.04474] [INSPIRE].
V. Eisler, G. Di Giulio, E. Tonni and I. Peschel, Entanglement Hamiltonians for non-critical quantum chains, J. Stat. Mech. 2010 (2020) 103102 [arXiv:2007.01804] [INSPIRE].
N. Javerzat and E. Tonni, On the continuum limit of the entanglement Hamiltonian of a sphere for the free massless scalar field, JHEP 02 (2022) 086 [arXiv:2111.05154] [INSPIRE].
V. Eisler, E. Tonni and I. Peschel, Local and non-local properties of the entanglement Hamiltonian for two disjoint intervals, J. Stat. Mech. 2208 (2022) 083101 [arXiv:2204.03966] [INSPIRE].
A. Roy, F. Pollmann and H. Saleur, Entanglement Hamiltonian of the 1+1-dimensional free, compactified boson conformal field theory, J. Stat. Mech. 2008 (2020) 083104 [arXiv:2004.14370] [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, Finite temperature entanglement negativity in conformal field theory, J. Phys. A 48 (2015) 015006 [arXiv:1408.3043] [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, Partial transpose of two disjoint blocks in XY spin chains, J. Stat. Mech. 1508 (2015) P08005 [arXiv:1503.09114] [INSPIRE].
V. Eisler and Z. Zimborás, Entanglement negativity in two-dimensional free lattice models, Phys. Rev. B 93 (2016) 115148 [arXiv:1511.08819] [INSPIRE].
C. De Nobili, A. Coser and E. Tonni, Entanglement negativity in a two dimensional harmonic lattice: Area law and corner contributions, J. Stat. Mech. 1608 (2016) 083102 [arXiv:1604.02609] [INSPIRE].
H. Shapourian, K. Shiozaki and S. Ryu, Partial time-reversal transformation and entanglement negativity in fermionic systems, Phys. Rev. B 95 (2017) 165101 [arXiv:1611.07536] [INSPIRE].
V. Eisler and Z. Zimborás, On the partial transpose of fermionic Gaussian states, New J. Phys. 17 (2015) 053048 [arXiv:1502.01369] [INSPIRE].
I.R. Klebanov, T. Nishioka, S.S. Pufu and B.R. Safdi, On Shape Dependence and RG Flow of Entanglement Entropy, JHEP 07 (2012) 001 [arXiv:1204.4160] [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].
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].
D. Seminara, J. Sisti and E. Tonni, Corner contributions to holographic entanglement entropy in AdS4/BCFT3, JHEP 11 (2017) 076 [arXiv:1708.05080] [INSPIRE].
D. Seminara, J. Sisti and E. Tonni, Holographic entanglement entropy in AdS4/BCFT3 and the Willmore functional, JHEP 08 (2018) 164 [arXiv:1805.11551] [INSPIRE].
P. Bueno, H. Casini, O.L. Andino and J. Moreno, Disks globally maximize the entanglement entropy in 2 + 1 dimensions, JHEP 10 (2021) 179 [arXiv:2107.12394] [INSPIRE].
I. Affleck, Edge magnetic field in thexxzspin- chain, J. Phys. A 31 (1998) 2761.
B. Nienhuis, Critical behavior of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Statist. Phys. 34 (1984) 731 [INSPIRE].
S. Eggert and I. Affleck, Magnetic impurities in half integer spin Heisenberg antiferromagnetic chains, Phys. Rev. B 46 (1992) 10866 [INSPIRE].
J. Sully, M. Van Raamsdonk and D. Wakeham, BCFT entanglement entropy at large central charge and the black hole interior, JHEP 03 (2021) 167 [arXiv:2004.13088] [INSPIRE].
B. Chen and J.-Q. Wu, Universal relation between thermal entropy and entanglement entropy in conformal field theories, Phys. Rev. D 91 (2015) 086012 [arXiv:1412.0761] [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].
B. Estienne, Y. Ikhlef and A. Rotaru, The operator algebra of cyclic orbifolds, J. Phys. A 56 (2023) 465403 [arXiv:2212.07678] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
Acknowledgments
We are grateful to Viktor Eisler, Paul Fendley, Mihail Mintchev, Giuseppe Mussardo, Ivan Kostov, Gregory Schehr and Barton Zwiebach for useful discussions. We thank in particular Alvise Bastianello for helpful correspondence. AR is grateful to SISSA for hospitality during part of this work. ET acknowledges the Galileo Galilei Institute (through the program Reconstructing the Gravitational Hologram with Quantum Information), the Institute Henri Poincaré, the Institut de Physique Théorique at Université Paris-Saclay, the Laboratoire de Physique Théorique et Hautes Energies at Sorbonne Université and the Center for Theoretical Physics at MIT for hospitality and financial support during part of this work.
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Estienne, B., Ikhlef, Y., Rotaru, A. et al. Entanglement entropies of an interval for the massless scalar field in the presence of a boundary. J. High Energ. Phys. 2024, 236 (2024). https://doi.org/10.1007/JHEP05(2024)236
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DOI: https://doi.org/10.1007/JHEP05(2024)236