Entanglement entropy and negativity of disjoint intervals in CFT: some numerical extrapolations. (English) Zbl 1456.81106
Summary: The entanglement entropy and the logarithmic negativity can be computed in quantum field theory through a method based on the replica limit. Performing these analytic continuations in some cases is beyond our current knowledge, even for simple models. We employ a numerical method based on rational interpolations to extrapolate the entanglement entropy of two disjoint intervals for the conformal field theories given by the free compact boson and the Ising model. The case of three disjoint intervals is studied for the Ising model and the non compact free massless boson. For the latter model, the logarithmic negativity of two disjoint intervals has been also considered. Some of our findings have been checked against existing numerical results obtained from the corresponding lattice models.
MSC:
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
References:
| [1] | Amico L, Fazio R, Osterloh A and Vedral V 2008 Rev. Mod. Phys.80 517 · Zbl 1205.81009 · doi:10.1103/RevModPhys.80.517 |
| [2] | Eisert J, Cramer M and Plenio M B 2010 Rev. Mod. Phys.82 277 · Zbl 1205.81035 · doi:10.1103/RevModPhys.82.277 |
| [3] | Calabrese P, Cardy J and Doyon B 2009 J. Phys. A: Math. Theor.42 500301 · Zbl 1180.81014 · doi:10.1088/1751-8121/42/50/500301 |
| [4] | Callan C G and Wilczek F 1994 Phys. Lett. B 333 55 · doi:10.1016/0370-2693(94)91007-3 |
| [5] | Holzhey C, Larsen F and Wilczek F 1994 Nucl. Phys. B 424 443 · Zbl 0990.81564 · doi:10.1016/0550-3213(94)90402-2 |
| [6] | Calabrese P and Cardy J 2004 J. Stat. Mech. P06002 · Zbl 1082.82002 · doi:10.1088/1742-5468/2004/06/P06002 |
| [7] | Calabrese P and Cardy J 2009 J. Phys. A: Math. Theor.42 504005 · Zbl 1179.81026 · doi:10.1088/1751-8113/42/50/504005 |
| [8] | Caraglio M and Gliozzi F 2008 J. High Energy Phys. JHEP11(2008)076 |
| [9] | Furukawa S, Pasquier V and Shiraishi J 2009 Phys. Rev. Lett.102 170602 · doi:10.1103/PhysRevLett.102.170602 |
| [10] | Calabrese P, Cardy J and Tonni E 2009 J. Stat. Mech. P11001 · Zbl 1456.81360 · doi:10.1088/1742-5468/2009/11/P11001 |
| [11] | Calabrese P, Cardy J and Tonni E 2011 J. Stat. Mech. P01021 · Zbl 1456.81361 · doi:10.1088/1742-5468/2011/01/P01021 |
| [12] | Alba V, Tagliacozzo L and Calabrese P 2010 Phys. Rev. B 81 060411 · doi:10.1103/PhysRevB.81.060411 |
| [13] | Fagotti M and Calabrese P 2010 J. Stat. Mech. P04016 · doi:10.1088/1742-5468/2010/04/P04016 |
| [14] | Igloi F and Peschel I 2010 Europhys. Lett.89 40001 · doi:10.1209/0295-5075/89/40001 |
| [15] | Alba V, Tagliacozzo L and Calabrese P 2011 J. Stat. Mech. P06012 · Zbl 1456.81346 · doi:10.1088/1742-5468/2011/06/P06012 |
| [16] | Rajabpour M and Gliozzi F 2012 J. Stat. Mech. P02016 · Zbl 1456.81399 · doi:10.1088/1742-5468/2012/02/P02016 |
| [17] | Fagotti M 2012 Europhys. Lett.97 17007 · doi:10.1209/0295-5075/97/17007 |
| [18] | Calabrese P 2010 J. Stat. Mech. P09013 · Zbl 1456.81357 |
| [19] | Ryu S and Takayanagi T 2006 Phys. Rev. Lett.96 181602 · Zbl 1228.83110 · doi:10.1103/PhysRevLett.96.181602 |
| [20] | Ryu S and Takayanagi T 2006 J. High Energy Phys. JHEP08(2006)045 |
| [21] | Headrick M 2010 Phys. Rev. D 82 126010 · doi:10.1103/PhysRevD.82.126010 |
| [22] | Alcaraz F, Berganza M and Sierra G 2011 Phys. Rev. Lett.106 201601 · doi:10.1103/PhysRevLett.106.201601 |
| [23] | Berganza M, Alcaraz F and Sierra G 2012 J. Stat. Mech. P01016 · doi:10.1088/1742-5468/2012/01/P01016 |
| [24] | Taddia L, Xavier J, Alcaraz F and Sierra G 2013 Phys. Rev. B 88 075112 · doi:10.1103/PhysRevB.88.075112 |
| [25] | Essler F, Lauchli A and Calabrese P 2013 Phys. Rev. Lett.110 115701 · doi:10.1103/PhysRevLett.110.115701 |
| [26] | Calabrese P, Essler F and Lauchli A 2014 J. Stat. Mech. P09025 · Zbl 1456.82234 · doi:10.1088/1742-5468/2014/09/P09025 |
| [27] | Pálmai T 2014 Phys. Rev. B 90 161404 · doi:10.1103/PhysRevB.90.161404 |
| [28] | Gliozzi F and Tagliacozzo L 2010 J. Stat. Mech. P01002 · doi:10.1088/1742-5468/2010/01/P01002 |
| [29] | Coser A, Tagliacozzo L and Tonni E 2014 J. Stat. Mech. P01008 · Zbl 1456.82041 · doi:10.1088/1742-5468/2014/01/P01008 |
| [30] | Agón C M, Headrick M, Jafferis D L and Kasko S 2014 Phys. Rev. D 89 025018 · doi:10.1103/PhysRevD.89.025018 |
| [31] | Vidal G and Werner R F 2002 Phys. Rev. A 65 032314 · doi:10.1103/PhysRevA.65.032314 |
| [32] | Vidal G 2000 J. Mod. Opt.47 355 · doi:10.1080/09500340008244048 |
| [33] | Zyczkowski K, Horodecki P, Sanpera A and Lewenstein M 1998 Phys. Rev. A 58 883 · doi:10.1103/PhysRevA.58.883 |
| [34] | Peres A 1996 Phys. Rev. Lett.77 1413 · Zbl 0947.81003 · doi:10.1103/PhysRevLett.77.1413 |
| [35] | Eisert J 2006 arXiv:quant-ph/0610253 |
| [36] | Eisert J and Plenio M B 1999 J. Mod. Opt.46 145 · doi:10.1080/09500349908231260 |
| [37] | Plenio M 2006 Phys. Rev. Lett.95 090503 · doi:10.1103/PhysRevLett.95.090503 |
| [38] | Wichterich H, Molina-Vilaplana J and Bose S 2009 Phys. Rev. A 80 010304 · doi:10.1103/PhysRevA.80.010304 |
| [39] | Marcovitch S, Retzker A, Plenio M B and Reznik B 2009 Phys. Rev. A 80 012325 · doi:10.1103/PhysRevA.80.012325 |
| [40] | Calabrese P, Cardy J and Tonni E 2012 Phys. Rev. Lett.109 130502 · doi:10.1103/PhysRevLett.109.130502 |
| [41] | Calabrese P, Cardy J and Tonni E 2013 J. Stat. Mech. P02008 · Zbl 1456.81362 · doi:10.1088/1742-5468/2013/02/P02008 |
| [42] | Eisler V and Zimboras Z 2015 arXiv:1502.01369 |
| [43] | Coser A, Tonni E and Calabrese P 2015 arXiv:1503.09114 |
| [44] | Calabrese P, Cardy J and Tonni E 2015 J. Phys. A: Math. Theor.48 015006 · Zbl 1305.81122 · doi:10.1088/1751-8113/48/1/015006 |
| [45] | Coser A, Tonni E and Calabrese P 2014 J. Stat. Mech. P12017 · Zbl 1456.81055 · doi:10.1088/1742-5468/2014/12/P12017 |
| [46] | Eisler V and Zimboras Z 2014 New J. Phys.16 123020 · doi:10.1088/1367-2630/16/12/123020 |
| [47] | Hoogeveen M and Doyon B 2014 arXiv:1412.7568 |
| [48] | Wen X, Chang P-Y and Ryu S 2015 arXiv:1501.00568 |
| [49] | Sørensen E S, Chang M-S, Laflorencie N and Affleck I 2007 J. Stat. Mech. P08003 · Zbl 1456.82189 |
| [50] | Bayat A, Sodano P and Bose S 2010 Phys. Rev. B 81 064429 · doi:10.1103/PhysRevB.81.064429 |
| [51] | Sodano P, Bayat A and Bose S 2010 Phys. Rev. B 81 100412 · doi:10.1103/PhysRevB.81.100412 |
| [52] | Bayat A, Bose S, Sodano P and Johannesson H 2012 Phys. Rev. Lett.109 066403 · doi:10.1103/PhysRevLett.109.066403 |
| [53] | Alba V 2013 J. Stat. Mech. P05013 · Zbl 1456.82969 · doi:10.1088/1742-5468/2013/05/P05013 |
| [54] | Calabrese P, Tagliacozzo L and Tonni E 2013 J. Stat. Mech. P05002 · Zbl 1456.82099 · doi:10.1088/1742-5468/2013/05/P05002 |
| [55] | Press W, Teukolsky S, Vetterling W and Flannery B 2007 Numerical Recipes: the Art of Scientific Computing (Cambridge: Cambridge University Press) |
| [56] | Cardy J, Castro-Alvaredo O and Doyon B 2008 J. Stat. Phys.130 129 · Zbl 1134.81043 · doi:10.1007/s10955-007-9422-x |
| [57] | Dixon L J, Friedan D, Martinec E J and Shenker S H 1987 Nucl. Phys. B 282 13 · doi:10.1016/0550-3213(87)90676-6 |
| [58] | Zamolodchikov Al B 1987 Nucl. Phys. B 285 481 · doi:10.1016/0550-3213(87)90350-6 |
| [59] | Alvarez-Gaumé L, Moore G W and Vafa C 1986 Commun. Math. Phys.106 1 · Zbl 0605.58049 · doi:10.1007/BF01210925 |
| [60] | Verlinde E and Verlinde H 1987 Nucl. Phys. B 288 357 · doi:10.1016/0550-3213(87)90219-7 |
| [61] | Knizhnik V G 1987 Commun. Math. Phys.112 567 · Zbl 0656.58044 · doi:10.1007/BF01225373 |
| [62] | Bershadsky M and Radul A 1987 Int. J. Mod. Phys. A 02 165 · Zbl 1165.81373 · doi:10.1142/S0217751X87000053 |
| [63] | Dijkgraaf R, Verlinde E P and Verlinde H L 1988 Commun. Math. Phys.115 649 · Zbl 0649.32019 · doi:10.1007/BF01224132 |
| [64] | Fay J 1973 Theta functions on Riemann surfaces(Lecture Notes in Mathematics vol 352) (Berlin: Springer) · Zbl 0281.30013 · doi:10.1007/BFb0060090 |
| [65] | Mumford D 1991 Tata lectures on Theta III(Progress in Mathematics vol 97) (Boston: Birkhäuser) · Zbl 0744.14033 · doi:10.1007/978-0-8176-4579-3 |
| [66] | Igusa J 1972 Theta Functions (Berlin: Springer) · Zbl 0251.14016 · doi:10.1007/978-3-642-65315-5 |
| [67] | Giamarchi T 2004 Quantum Physics in One Dimension (New York: Oxford University Press) · Zbl 1075.81001 |
| [68] | Tagliacozzo L, Evenbly G and Vidal G 2009 Phys. Rev. B 80 235127 · doi:10.1103/PhysRevB.80.235127 |
| [69] | Peschel I and Chung M C 1999 J. Phys. A: Math. Gen.32 8419 · Zbl 0955.82005 · doi:10.1088/0305-4470/32/48/305 |
| [70] | Audenaert K, Eisert J, Plenio M B and Werner R F 2002 Phys. Rev. A 66 042327 · doi:10.1103/PhysRevA.66.042327 |
| [71] | Peschel I 2003 J. Phys. A: Math. Gen.36 L205 · Zbl 1049.82011 · doi:10.1088/0305-4470/36/14/101 |
| [72] | Botero A and Reznik B 2004 Phys. Rev. A 70 052329 · doi:10.1103/PhysRevA.70.052329 |
| [73] | Plenio M B, Eisert J, Dressig J and Cramer M 2005 Phys. Rev. Lett.94 060503 · doi:10.1103/PhysRevLett.94.060503 |
| [74] | Cramer M, Eisert J, Plenio M B and Dreissig J 2006 Phys. Rev. A 73 012309 · doi:10.1103/PhysRevA.73.012309 |
| [75] | Peschel I and Eisler V 2009 J. Phys. A: Math Theor.42 504003 · Zbl 1179.81032 · doi:10.1088/1751-8113/42/50/504003 |
| [76] | Calabrese P, Campostrini M, Essler F and Nienhius B 2010 Phys. Rev. L 104 095701 · doi:10.1103/PhysRevLett.104.095701 |
| [77] | Calabrese P and Cardy J 2010 J. Stat. Mech. P04023 · doi:10.1088/1742-5468/2010/04/P04023 |
| [78] | Calabrese P and Essler F 2010 J. Stat. Mech. P08029 · doi:10.1088/1742-5468/2010/08/P08029 |
| [79] | Chung C, Alba V, Bonnes L, Chen P and Lauchli A 2014 Phys. Rev. B 90 064401 · doi:10.1103/PhysRevB.90.064401 |
| [80] | Zaletel M, Bardarson J and Moore J 2011 Phys. Rev. Lett.107 020402 · doi:10.1103/PhysRevLett.107.020402 |
| [81] | Chandran A, Khemani V and Sondhi S L 2014 Phys. Rev. Lett.113 060501 · doi:10.1103/PhysRevLett.113.060501 |
| [82] | Cardy J 2013 J. Phys. A: Math. Theor.46 285402 · Zbl 1273.81192 · doi:10.1088/1751-8113/46/28/285402 |
| [83] | Casini H and Huerta M 2009 J. High Energy Phys. JHEP03(2009)048 |
| [84] | Shiba N 2012 J. High Energy Phys. JHEP07(2012)100 · Zbl 1397.83173 |
| [85] | Schnitzer H 2014 arXiv:1406.1161 |
| [86] | Hubeny V E and Rangamani M 2008 J. High Energy Phys. JHEP03(2008)006 |
| [87] | Tonni E 2011 J. High Energy Phys. JHEP05(2011)004 |
| [88] | Hayden P, Headrick M and Maloney A 2013 Phys. Rev. D 87 046003 · doi:10.1103/PhysRevD.87.046003 |
| [89] | Faulkner T 2013 arXiv:1303.7221 |
| [90] | Hartman T 2013 arXiv:1303.6955 |
| [91] | Faulkner T, Lewkowycz A and Maldacena J 2013 J. High Energy Phys. JHEP11(2013)074 · Zbl 1392.81021 |
| [92] | Fonda P, Giomi L, Salvio A and Tonni E 2015 J. High Energy Phys. JHEP02(2015)005 |
| [93] | Vidal G 2007 Phys. Rev. Lett.99 220405 · doi:10.1103/PhysRevLett.99.220405 |
| [94] | Vidal G arXiv:0912.1651 |
| [95] | Evenbly G and Vidal G 2011 J. Stat. Phys.145 891 · Zbl 1231.82021 · doi:10.1007/s10955-011-0237-4 |
| [96] | Singh S and Vidal G 2013 Phys. Rev. B 88 121108 · doi:10.1103/PhysRevB.88.121108 |
| [97] | Pfeifer R N C, Evenbly G and Vidal G 2009 Phys. Rev. A 79 040301 · doi:10.1103/PhysRevA.79.040301 |
| [98] | Evenbly G and Vidal G arXiv:1109.5334 |
| [99] | Swingle B arXiv:1010.4038 |
| [100] | Molina-Vilaplana J and Sodano P 2011 J. High Energy Phys. JHEP10(2011)011 011 · Zbl 1303.81174 |
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