Abstract
In this paper, we investigate the mixed-state entanglement in a model of p-wave superconductivity phase transition using holographic methods. We calculate several entanglement measures, including holographic entanglement entropy (HEE), mutual information (MI), and entanglement wedge cross-section (EWCS). Our results show that these measures display critical behavior at the phase transition points, with the EWCS exhibiting opposite temperature behavior compared to the HEE. Furthermore, we explore the behavior of thermodynamics and holographic quantum information at the zeroth-order phase transition point and find that it is opposite to that observed in the first-order phase transition. Additionally, we find that the critical exponents of all entanglement measures are twice those of the condensate. Our findings also suggest that the EWCS is a more sensitive indicator of the critical behavior of phase transitions than the HEE. Lastly, we uncover a universal inequality in the growth rates of EWCS and MI near critical points in thermal phase transitions, such as p-wave and s-wave superconductivity, suggesting that MI captures more information than EWCS when a phase transition first occurs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. Eisert, Entanglement in quantum information theory, Ph.D. Thesis, University of Potsdam, Potsdam, Germany (2006) [quant-ph/0610253].
A. Osterloh, L. Amico, G. Falci and R. Fazio, Scaling of Entanglement close to a Quantum Phase Transitions, Nature 416 (2002) 608 [arXiv:0202029].
L. Amico, R. Fazio, A. Osterloh and V. Vedral, Entanglement in many-body systems, Rev. Mod. Phys. 80 (2008) 517 [quant-ph/0703044] [INSPIRE].
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
G. Vidal and R.F. Werner, Computable measure of entanglement, Phys. Rev. A 65 (2002) 032314 [quant-ph/0102117] [INSPIRE].
M.B. Plenio, Logarithmic Negativity: A Full Entanglement Monotone That is not Convex, Phys. Rev. Lett. 95 (2005) 090503 [quant-ph/0505071] [INSPIRE].
R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81 (2009) 865 [quant-ph/0702225] [INSPIRE].
G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C 930308 (1993) 284 [gr-qc/9310026] [INSPIRE].
L. Susskind, The World as a hologram, J. Math. Phys. 36 (1995) 6377 [hep-th/9409089] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
R.-G. Cai, S. He, L. Li and Y.-L. Zhang, Holographic Entanglement Entropy on P-wave Superconductor Phase Transition, JHEP 07 (2012) 027 [arXiv:1204.5962] [INSPIRE].
Y. Peng and Q. Pan, Holographic entanglement entropy in general holographic superconductor models, JHEP 06 (2014) 011 [arXiv:1404.1659] [INSPIRE].
Y. Peng, Holographic entanglement entropy in superconductor phase transition with dark matter sector, Phys. Lett. B 750 (2015) 420 [arXiv:1507.07399] [INSPIRE].
X.-X. Zeng, H. Zhang and L.-F. Li, Phase transition of holographic entanglement entropy in massive gravity, Phys. Lett. B 756 (2016) 170 [arXiv:1511.00383] [INSPIRE].
T. Takayanagi and K. Umemoto, Entanglement of purification through holographic duality, Nature Phys. 14 (2018) 573 [arXiv:1708.09393] [INSPIRE].
K. Umemoto and Y. Zhou, Entanglement of Purification for Multipartite States and its Holographic Dual, JHEP 10 (2018) 152 [arXiv:1805.02625] [INSPIRE].
S. Dutta and T. Faulkner, A canonical purification for the entanglement wedge cross-section, JHEP 03 (2021) 178 [arXiv:1905.00577] [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].
N. Jokela and A. Pönni, Notes on entanglement wedge cross sections, JHEP 07 (2019) 087 [arXiv:1904.09582] [INSPIRE].
K. Babaei Velni, M.R. Mohammadi Mozaffar and M.H. Vahidinia, Some Aspects of Entanglement Wedge Cross-Section, JHEP 05 (2019) 200 [arXiv:1903.08490] [INSPIRE].
M.J. Vasli, M.R. Mohammadi Mozaffar, K. Babaei Velni and M. Sahraei, Holographic study of reflected entropy in anisotropic theories, Phys. Rev. D 107 (2023) 026012 [arXiv:2207.14169] [INSPIRE].
H.A. Camargo, P. Nandy, Q. Wen and H. Zhong, Balanced partial entanglement and mixed state correlations, SciPost Phys. 12 (2022) 137 [arXiv:2201.13362] [INSPIRE].
P. Liu, Y. Ling, C. Niu and J.-P. Wu, Entanglement of Purification in Holographic Systems, JHEP 09 (2019) 071 [arXiv:1902.02243] [INSPIRE].
Y.-f. Huang, Z.-j. Shi, C. Niu, C.-y. Zhang and P. Liu, Mixed State Entanglement for Holographic Axion Model, Eur. Phys. J. C 80 (2020) 426 [arXiv:1911.10977] [INSPIRE].
P. Liu and J.-P. Wu, Mixed state entanglement and thermal phase transitions, Phys. Rev. D 104 (2021) 046017 [arXiv:2009.01529] [INSPIRE].
C.-Y. Chen, W. Xiong, C. Niu, C.-Y. Zhang and P. Liu, Entanglement wedge minimum cross-section for holographic aether gravity, JHEP 08 (2022) 123 [arXiv:2109.03733] [INSPIRE].
Y.-Z. Li, C.-Y. Zhang and X.-M. Kuang, Entanglement wedge cross-section with Gauss-Bonnet corrections and thermal quench, Sci. China Phys. Mech. Astron. 64 (2021) 120413 [arXiv:2102.12171] [INSPIRE].
A.R. Chowdhury, A. Saha and S. Gangopadhyay, Entanglement wedge cross-section for noncommutative Yang-Mills theory, JHEP 02 (2022) 192 [arXiv:2106.04562] [INSPIRE].
M. Sahraei, M.J. Vasli, M.R.M. Mozaffar and K.B. Velni, Entanglement wedge cross section in holographic excited states, JHEP 08 (2021) 038 [arXiv:2105.12476] [INSPIRE].
A. Chowdhury Roy, A. Saha and S. Gangopadhyay, Mixed state information theoretic measures in boosted black brane, Annals Phys. 452 (2023) 169270 [arXiv:2204.08012] [INSPIRE].
S. Maulik, More on entanglement properties of \( Li{f}_4^{(2)} \) × S1 × S5 spacetime with string excitations, Eur. Phys. J. Plus 138 (2023) 288 [arXiv:2209.05207] [INSPIRE].
P. Jain and S. Mahapatra, Mixed state entanglement measures as probe for confinement, Phys. Rev. D 102 (2020) 126022 [arXiv:2010.07702] [INSPIRE].
P. Jain, S.S. Jena and S. Mahapatra, Holographic confining/deconfining gauge theories and entanglement measures with a magnetic field, arXiv:2209.15355 [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Holographic Superconductors, JHEP 12 (2008) 015 [arXiv:0810.1563] [INSPIRE].
G.T. Horowitz, Introduction to Holographic Superconductors, Lect. Notes Phys. 828 (2011) 313 [arXiv:1002.1722] [INSPIRE].
S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, Building a Holographic Superconductor, Phys. Rev. Lett. 101 (2008) 031601 [arXiv:0803.3295] [INSPIRE].
Y. Ling, P. Liu, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic Superconductor on Q-lattice, JHEP 02 (2015) 059 [arXiv:1410.6761] [INSPIRE].
M. Rogatko and K.I. Wysokinski, Holographic vortices in the presence of dark matter sector, JHEP 12 (2015) 041 [arXiv:1510.06137] [INSPIRE].
R.E. Arias and I.S. Landea, Backreacting p-wave Superconductors, JHEP 01 (2013) 157 [arXiv:1210.6823] [INSPIRE].
R.-G. Cai, L. Li, L.-F. Li and R.-Q. Yang, Introduction to Holographic Superconductor Models, Sci. China Phys. Mech. Astron. 58 (2015) 060401 [arXiv:1502.00437] [INSPIRE].
M. Rogatko and K.I. Wysokinski, P-wave holographic superconductor/insulator phase transitions affected by dark matter sector, JHEP 03 (2016) 215 [arXiv:1508.02869] [INSPIRE].
T. Albash and C.V. Johnson, Holographic Studies of Entanglement Entropy in Superconductors, JHEP 05 (2012) 079 [arXiv:1202.2605] [INSPIRE].
H.-S. Jeong, K.-Y. Kim and Y.-W. Sun, Holographic entanglement density for spontaneous symmetry breaking, JHEP 06 (2022) 078 [arXiv:2203.07612] [INSPIRE].
A.C. Neto, Charge density wave, superconductivity, and anomalous metallic behavior in 2d transition metal dichalcogenides, Phys. Rev. Lett. 86 (2001) 4382.
H. Wang et al., Deterministic Entanglement of Photons in Two Superconducting Microwave Resonators, Phys. Rev. Lett. 106 (2011) 060401 [arXiv:1011.2862].
R.-G. Cai, L. Li and L.-F. Li, A Holographic P-wave Superconductor Model, JHEP 01 (2014) 032 [arXiv:1309.4877] [INSPIRE].
R.-G. Cai, L. Li, L.-F. Li and R.-Q. Yang, Towards Complete Phase Diagrams of a Holographic P-wave Superconductor Model, JHEP 04 (2014) 016 [arXiv:1401.3974] [INSPIRE].
P. Liu, C. Niu, Z.-J. Shi and C.-Y. Zhang, Entanglement wedge minimum cross-section in holographic massive gravity theory, JHEP 08 (2021) 113 [arXiv:2104.08070] [INSPIRE].
R.-G. Cai, Y.-P. Hu, Q.-Y. Pan and Y.-L. Zhang, Thermodynamics of Black Holes in Massive Gravity, Phys. Rev. D 91 (2015) 024032 [arXiv:1409.2369] [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].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
Y. Ling, P. Liu, C. Niu, J.-P. Wu and Z.-Y. Xian, Holographic Entanglement Entropy Close to Quantum Phase Transitions, JHEP 04 (2016) 114 [arXiv:1502.03661] [INSPIRE].
Y. Ling, P. Liu and J.-P. Wu, Characterization of Quantum Phase Transition using Holographic Entanglement Entropy, Phys. Rev. D 93 (2016) 126004 [arXiv:1604.04857] [INSPIRE].
M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge University Press, Cambridge, U.K. (2010), https://doi.org/10.1017/CBO9780511976667.
P. Hayden, M. Headrick and A. Maloney, Holographic Mutual Information is Monogamous, Phys. Rev. D 87 (2013) 046003 [arXiv:1107.2940] [INSPIRE].
P.H. Nguyen, An equal area law for holographic entanglement entropy of the AdS-RN black hole, JHEP 12 (2015) 139 [arXiv:1508.01955] [INSPIRE].
R.-G. Cai, S. He, L. Li and Y.-L. Zhang, Holographic Entanglement Entropy in Insulator/Superconductor Transition, JHEP 07 (2012) 088 [arXiv:1203.6620] [INSPIRE].
H.-B. Zeng, X. Gao, Y. Jiang and H.-S. Zong, Analytical Computation of Critical Exponents in Several Holographic Superconductors, JHEP 05 (2011) 002 [arXiv:1012.5564] [INSPIRE].
Q. Pan, J. Jing, B. Wang and S. Chen, Analytical study on holographic superconductors with backreactions, JHEP 06 (2012) 087 [arXiv:1205.3543] [INSPIRE].
M. Ammon, J. Erdmenger, V. Grass, P. Kerner and A. O’Bannon, On Holographic p-wave Superfluids with Back-reaction, Phys. Lett. B 686 (2010) 192 [arXiv:0912.3515] [INSPIRE].
P. Li and Y. Ling, Entanglement wedge cross section inequalities in AdS/BCFT, Phys. Rev. D 106 (2022) 086021 [arXiv:2206.13417] [INSPIRE].
N. Bao and I.F. Halpern, Holographic Inequalities and Entanglement of Purification, JHEP 03 (2018) 006 [arXiv:1710.07643] [INSPIRE].
A. Garg, M. Randeria and N. Trivedi, Strong correlations make high-temperature superconductors robust against disorder, Nat. Phys. 4 (2008) 762.
H.-H. Wen, Developments and Perspectives of Iron-based High-Temperature Superconductors, Adv. Mat. 20 (2008) 3764.
A. Donos and J.P. Gauntlett, Holographic Q-lattices, JHEP 04 (2014) 040 [arXiv:1311.3292] [INSPIRE].
K. Landsteiner, Y. Liu and Y.-W. Sun, Quantum phase transition between a topological and a trivial semimetal from holography, Phys. Rev. Lett. 116 (2016) 081602 [arXiv:1511.05505] [INSPIRE].
Y. Ling, P. Liu, J.-P. Wu and Z. Zhou, Holographic Metal-Insulator Transition in Higher Derivative Gravity, Phys. Lett. B 766 (2017) 41 [arXiv:1606.07866] [INSPIRE].
M. Baggioli, B. Padhi, P.W. Phillips and C. Setty, Conjecture on the Butterfly Velocity across a Quantum Phase Transition, JHEP 07 (2018) 049 [arXiv:1805.01470] [INSPIRE].
M. Baggioli and D. Giataganas, Detecting Topological Quantum Phase Transitions via the c-Function, Phys. Rev. D 103 (2021) 026009 [arXiv:2007.07273] [INSPIRE].
Acknowledgments
We are very grateful to the referee for the comments on this paper. Peng Liu would like to thank Yun-Ha Zha for her kind encouragement during this work. Zhe Yang appreciates Feng-Ying Deng’s support and warm words of encouragement during this work. We are also very grateful to Chong-Ye Chen, Mu-Jing Li, and Wei Xiong for their helpful discussion and suggestions. This work is supported by the Natural Science Foundation of China under Grant No. 11905083, 12005077 and 11805083, as well as the Science and Technology Planning Project of Guangzhou (202201010655) and Guangdong Basic and Applied Basic Research Foundation (2021A1515012374).
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: 2301.13574
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
Yang, Z., Cheng, FJ., Niu, C. et al. The mixed-state entanglement in holographic p-wave superconductor model. J. High Energ. Phys. 2023, 110 (2023). https://doi.org/10.1007/JHEP04(2023)110
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2023)110