Journal of Holography Applications in Physics
Photonic Exceptional Points in Holography and QCD3
Document Type : Regular article
Author
International Centre for Theoretical Physics Asia-Pacific, University of Chinese Academy of Sciences, 100190 Beijing, China
Abstract
In this work, based on an analogy with holographic confining geometries and using complexified fields, we build a holographic toy model of third order photonic exceptional points (EPs) of ternary coupled microrings with gain and loss, which makes an open, non-Hermitian quantum system. In our model, we discuss the Ferrell-Glover-Tinkham sum rule for various combinations of gain and loss systems, and numerically find the behavior of spectra which matches with the experiments. We also discuss the inhomogeneous case of a holographic lattice for three-site photonic EPs. Additionally, we numerically find the behavior of phase rigidity and the Petermann factor around EPs versus various parameters of the model. We also discuss the connections between recent developments in complexified, time-dependent entanglement entropy and EPs, and then, we connect EPs and the $\theta$-vacuum of QCD through topological structures, partition functions, and winding numbers, and find a second-order EP in a perturbed $\theta$-vacuum model. Finally, we examined a controlled non-Hermitian deformation of $\theta$-vacuum toy model, by using the Lindblad formalism and Liouvillian eigenvalues.
Keywords
Main Subjects
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[1] M. Ghodrati, ”Schwinger effect and entanglement entropy in confining geometries”, Phys. Rev. D 92(6), 065015 (2015). DOI: 10.1103/PhysRevD.92.065015.
[2] M. Ghodrati, ”Hyperscaling violating solution in coupled dilaton-squared curvature gravity”, Phys. Rev. D 90(4), 044055 (2014). DOI: 10.1103/PhysRevD.90.044055.
[3] M. Ghodrati, ”Beyond ads space-times, new holographic correspondences and applications”, other thesis (2016). arXiv:1609.04168.
[4] M. Ghodrati, ”Complexity growth in massive gravity theories, the effects of chirality, and more”, Phys. Rev. D 96(10), 106020 (2017). DOI: 10.1103/PhysRevD.96.106020.
[5] M. Ghodrati, ”Complexity growth rate during phase transitions”, Phys. Rev. D 98(10), 106011 (2018). DOI: 10.1103/PhysRevD.98.106011.
[6] M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang, and Y.-T. Zhou, ”The connection between holographic entanglement and complexity of purification”, JHEP 09, 009 (2019). DOI: 10.1007/JHEP092019009.
[7] Y.-T. Zhou, M. Ghodrati, X.-M. Kuang, and J.-P. Wu, ”Evolutions of entanglement and complexity after a thermal quench in massive gravity theory”, Phys. Rev. D 100(6), 066003 (2019). DOI: 10.1103/PhysRevD.100.066003.
[8] M. Ghodrati, ”Complexity and emergence of warped ads3 space-time from chiral liouville action”, JHEP 02, 052 (2020). DOI: 10.1007/JHEP022020052.
[9] M. Ghodrati, ”Entanglement wedge reconstruction and correlation measures in mixed states: Modular flows versus quantum recovery channels”, Phys. Rev. D 104(4), 046004 (2021). DOI: 10.1103/PhysRevD.104.046004.
[10] M. Ghodrati, ”Correlations of mixed systems in confining backgrounds”, Eur. Phys. J. C 82(6), 531 (2022). DOI: 10.1140/epjc/s10052-022-10481-z.
[11] M. Ghodrati, ”Critical distance and crofton form in confining geometries”, J. Korean Phys. Soc. 81(2), 77 (2022). DOI: 10.1007/s40042-022-00523-w.
[12] M. Ghodrati, ”Encoded information of mixed correlations: The views from one dimension higher”, JHEP 08, 059 (2023). DOI: 10.1007/JHEP082023059.
[13] M. Ghodrati, ”String scattering amplitudes and mutual information in confining backgrounds: The partonic behavior”, Phys. Rev. D 112(4), 046011 (2025). DOI: 10.1103/PhysRevD.112.046011.
[14] M. Ghodrati and D. Gregoris, ”On the curvature invariants of the massive banadosteitelboim-zanelli black holes and their holographic pictures”, Int. J. Mod. Phys. A 37(34), 2250202 (2022). DOI: 10.1142/S0217751X22502025.
[15] M. Ghodrati and A. Naseh, ”Phase transitions in bergshoeff-hohm-townsend massive gravity”, Class. Quant. Grav. 34(7), 075009 (2017). DOI: 10.1088/1361-6382/aa634f.
[16] M. Ghodrati, K. Hajian, and M. R. Setare, ”Revisiting conserved charges in higher curvature gravitational theories”, Eur. Phys. J. C 76(12), 701 (2016). DOI: 10.1140/epjc/s10052-016-4550-6.
[17] M. Ghodrati, ”Chaos, phase transitions and curvature invariants of (rotating, warped, massive) btz black holes”, AIP Conf. Proc. 2874(1), 020012 (2024). DOI: 10.1063/5.0216490.
[18] W. D. Heiss, ”The physics of exceptional points”, Journal of Physics A: Mathematical and Theoretical 45, 444016 (2012). DOI: 10.1088/1751-8113/45/44/444016.
[19] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, ”Exceptional topology of nonhermitian systems”, Rev. Mod. Phys. 93(1), 015005 (2021). DOI: 10.1103/RevModPhys.93.015005.
[20] A. Fatemiabhari and C. Nunez, ”From conformal to confining field theories using holography”, JHEP 03, 160 (2024). DOI: 10.1007/JHEP032024160.
[21] H. Geyer, W. Heiss, and F. Scholtz, ”The physical interpretation of non-hermitian hamiltonians and other observables”, Canadian Journal of Physics 86, 1195 (2008). DOI: 10.1139/p08-060.
[22] M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, ”Partonic behavior of string scattering amplitudes from holographic qcd models”, JHEP 05, 058 (2022). DOI: 10.1007/JHEP052022058.
[23] Z. Gong, M. Bello, D. Malz, and F. K. Kunst, ”Anomalous behaviors of quantum emitters in non-hermitian baths”, Phys. Rev. Lett. 129(22), 223601 (2022). DOI: 10.1103/PhysRevLett.129.223601.
[24] Z. Gong, M. Bello, D. Malz, and F. K. Kunst, ”Bound states and photon emission in non-hermitian nanophotonics”, Phys. Rev. A 106(5), 053517 (2022). DOI: 10.1103/PhysRevA.106.053517.
[25] M. Jahangiri, G.-M. Parsanasab, and L. Hajshahvaladi, ”Observation of anti-ptsymmetry and higher-order exceptional point pt-symmetry in ternary systems for singlemode operation”, Scientific Reports 15(1), 4823 (2025). DOI: 10.1038/s41598-025- 85623-w.
[26] M.-A. Miri and A. Alu, ”Exceptional points in optics and photonics”, Science 363, (2019). DOI: 10.1126/science.aar7709.
[27] N. Michel, W. Nazarewicz, J. Okołowicz, and M. Płoszajczak, ”Open problems in the theory of nuclear open quantum systems”, Journal of Physics G: Nuclear and Particle Physics 37, 064042 (2010). DOI: 10.1088/0954-3899/37/6/064042.
[28] W. D. Heiss, ”Chirality of wavefunctions for three coalescing levels”, Journal of Physics A: Mathematical and Theoretical 41, 244010 (2008). DOI: 10.1088/1751- 8113/41/24/244010.
[29] Z.-Y. Xian, D. Rodriguez Fernandez, Z. Chen, Y. Liu, and R. Meyer, ”Electric conductivity in non-hermitian holography”, SciPost Phys. 16(1), 004 (2024). DOI: 10.21468/SciPostPhys.16.1.004.
[30] W. D. Heiss and S. Radu, ”Quantum chaos, degeneracies, and exceptional points”, Phys. Rev. E 52, 4762 (1995). DOI: 10.1103/PhysRevE.52.4762.
[31] M. V. Berry and M. Tabor, ”Level clustering in the regular spectrum”, Proceedings of the Royal Society of London Series A 356, 375 (1977). DOI: 10.1098/rspa.1977.0140.
[32] M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, ”Measure for chaotic scattering amplitudes”, Phys. Rev. Lett. 129(26), 261601 (2022). DOI: 10.1103/PhysRevLett.129.261601.
[33] M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, ”Measuring chaos in string scattering processes”, Phys. Rev. D 108(6), 066006 (2023). DOI: 10.1103/PhysRevD.108.066006.
[34] N. Savic and M. Cubrovic, ”Weak chaos and mixed dynamics in the string s-matrix”, JHEP 03, 101 (2024). DOI: 10.1007/JHEP032024101.
[35] W. D. Heiss and A. L. Sannino, ”Transitional regions of finite fermi systems and quantum chaos”, Phys. Rev. A 43, 4159 (1991). DOI: 10.1103/PhysRevA.43.4159.
[36] M. Golubitsky and W. Langford, ”Pattern formation and bistability in flow between counterrotating cylinders”, Physica D: Nonlinear Phenomena 32(3), 362 (1988). DOI: 10.1016/0167-2789(88)90063-2.
[37] K. Qu, S. Meuren, and N. J. Fisch, ”Creating pair plasmas with observable collective effects”, Plasma Physics and Controlled Fusion 65, 034007 (2023). DOI: 10.1088/1361- 6587/acb080.
[38] J. Wiersig, ”Petermann factors and phase rigidities near exceptional points”, Phys. Rev. Res. 5(3), 033042 (2023). DOI: 10.1103/PhysRevResearch.5.033042.
[39] W. D. Heiss, ”Repulsion of resonance states and exceptional points”, Phys. Rev. E 61, 929 (2000). DOI: 10.1103/PhysRevE.61.929.
[40] F. Klingl, N. Kaiser, and W. Weise, ”Effective lagrangian approach to vector mesons, their structure and decays”, Z. Phys. A 356(2), 193 (1996). DOI: 10.1007/s002180050167.
[41] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, ”Enhanced sensitivity at higher-order exceptional points”, Nature 548(7666), 187 (2017). DOI: 10.1038/nature23280.
[42] X.-L. Liu, C.-Y. Yue, J. Nian, and W. Zheng, ”Quantum-corrected holographic wilson loop correlators and confinement”, arXiv:2412.11107.
[43] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, ”Parity-time symmetric microring lasers”, Science 346(6212), 1258480 (2014). DOI: 10.1126/science.1258480.
[44] A. Einstein, B. Podolsky, and N. Rosen, ”Can quantum-mechanical description of physical reality be considered complete?”, Phys. Rev. 47, 777 (1935). DOI: 10.1103/PhysRev.47.777.
[45] N. Gisin and A. Go, ”Epr test with photons and kaons: Analogies”, American Journal of Physics 69, 264 (2001). DOI: 10.1119/1.1326080.
[46] STAR Collaboration, B. E. Aboona et al., ”Measuring spin correlation between quarks during qcd confinement”, Nature 650(8100), 65 (2026). DOI: 10.1038/s41586-025- 09920-0.
[47] X. Cao, M. Baggioli, H. Liu, and D. Li, ”Pion dynamics in a soft-wall ads-qcd model”, Journal of High Energy Physics 2022, 113 (2022). DOI: 10.1007/JHEP122022113.
[48] K. Debnath, Y. Zhang, and K. Molmer, ”Lasing in the superradiant crossover regime”, Phys. Rev. A 98, 063837 (2018). DOI: 10.1103/PhysRevA.98.063837.
[49] K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, ”Timelike entanglement entropy”, JHEP 05, 052 (2023). DOI: 10.1007/JHEP052023052.
[50] A. Milekhin, Z. Adamska, and J. Preskill, ”Observable and computable entanglement in time”, arXiv:2502.12240.
[51] J. Xu and W.-z. Guo, ”Imaginary part of timelike entanglement entropy”, JHEP 02, 094 (2025). DOI: 10.1007/JHEP022025094.
[52] P. Glorioso, X.-L. Qi, and Z. Yang, ”Space-time generalization of mutual information”, JHEP 05, 338 (2024). DOI: 10.1007/JHEP052024338.
[53] C. Nunez and D. Roychowdhury, ”Timelike entanglement entropy: A top-down approach”, Phys. Rev. D 112(2), 026030 (2025). DOI: 10.1103/PhysRevD.112.026030.
[54] W.-z. Guo and J. Xu, ”Duality of ryu-takayanagi surfaces inside and outside the horizon”, Phys. Rev. D 112(10), L101901 (2025). DOI: 10.1103/PhysRevD.112.L101901.
[55] K. Ikeda, ”Timelike quantum energy teleportation”, arXiv:2504.05353.
[56] K. Narayan and H. K. Saini, ”Notes on time entanglement and pseudo-entropy”, Eur. Phys. J. C 84(5), 499 (2024). DOI: 10.1140/epjc/s10052-024-12855-x.
[57] C. Nunez and D. Roychowdhury, ”Interpolating between spacelike and timelike entanglement via holography”, Phys. Rev. D 112(8), L081902 (2025). DOI: 10.1103/PhysRevD.112.L081902.
[58] J. Hales, U. Bajpai, T. Liu, D. R. Baykusheva, M. Li, M. Mitrano, and Y. Wang, ”Witnessing light-driven entanglement using time-resolved resonant inelastic x-ray scattering”, Nature Communications 14(1), 3512 (2023). DOI: 10.1038/s41467-023-38540-3.
[59] M. B. Hastings and T. Koma, ”Spectral gap and exponential decay of correlations”, Commun. Math. Phys. 265, 781 (2006). DOI: 10.1007/s00220-006-0030-4.
[60] B. Nachtergaele, Y. Ogata, and R. Sims, ”Propagation of correlations in quantum lattice systems”, Journal of Statistical Physics 124, 1 (2006). DOI: 10.1007/s10955-006-9143- 6.
[61] J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K.-Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H.-S. Lee, and P. Berini, ”Time-asymmetric loop around an exceptional point over the full optical communications band”, Nature 562(7725), 86 (2018). DOI: 10.1038/s41586-018-0523-2.
[62] H.-Z. Chen, T. Liu, H.-Y. Luan, R.-J. Liu, X.-Y. Wang, X.-F. Zhu, Y.-B. Li, Z.-M. Gu, S.-J. Liang, H. Gao, L. Lu, L. Ge, S. Zhang, J. Zhu, and R.-M. Ma, ”Revealing the missing dimension at an exceptional point”, Nature Physics 16(5), 571 (2020). DOI: 10.1038/s41567-020-0807-y.
[63] X. Dong, A. Lewkowycz, and M. Rangamani, ”Deriving covariant holographic entanglement”, JHEP 11, 028 (2016). DOI: 10.1007/JHEP112016028.
[64] W.-z. Guo, S. He, and Y.-X. Zhang, ”Relation between time- and spacelike entanglement entropy”, Phys. Rev. D 112(8), 086020 (2025). DOI: 10.1103/PhysRevD.112.086020.
[65] K.-W. Park, J. Kim, and K. Jeong, ”Double exceptional points generated by the strong imaginary coupling of a non-hermitian hamiltonian in an optical microcavity”, arXiv:2208.06860.
[66] A. Strohmaier and E. Witten, ”The timelike tube theorem in curved spacetime”, Commun. Math. Phys. 405(7), 153 (2024). DOI: 10.1007/s00220-024-05009-3.
[67] J. Maldacena, ”Real observers solving imaginary problems”, arXiv:2412.14014.
[68] Z. Yang, Y. Zhang, and W. Zheng, ”Comments on the de sitter double cone”, arXiv:2505.08647.
[69] Y. Chen, D. Stanford, H. Tang, and Z. Yang, ”On the phase of the de sitter density of states”, arXiv:2511.01400.
[70] J.-p. Zheng, J. Dukelsky, R. A. Molina, and A. M. Garcia-Garcia, ”Role of exceptional points in the dynamics of the lindblad sachdev-ye-kitaev model”, arXiv:2510.15793.
[71] X. Liu, J.-p. Zheng, and A. M. Garca-Garca, ”Inducing, and enhancing, many-body quantum chaos by continuous monitoring”, arXiv:2602.02750.
[72] F. Lenz, M. A. Shifman, and M. Thies, ”Quantum mechanics of the vacuum state in two-dimensional qcd with adjoint fermions”, Phys. Rev. D 51, 7060 (1995). DOI: 10.1103/PhysRevD.51.7060.
[73] J. Bersini, A. D’Alise, C. Gambardella, and F. Sannino, ”On the theta-angle physics of qcd under pressure: The strange and isospin phase diagram”, JHEP 07, 252 (2025). DOI: 10.1007/JHEP072025252.
[74] T. Vonk, F.-K. Guo, and U. G. Meissner, ”Aspects of the qcd theta-vacuum”, JHEP 06, 106 (2019). DOI: 10.1007/JHEP062019106.
[75] T. Yoshida, J. L. K. Konig, L. Rodland, E. J. Bergholtz, and M. Stalhammar, ”Winding topology of multifold exceptional points”, Phys. Rev. Res. 7(1), L012021 (2025). DOI: 10.1103/PhysRevResearch.7.L012021.
[76] R. Arouca, J. Cayao, and A. M. Black-Schaffer, ”Topological superconductivity enhanced by exceptional points”, Phys. Rev. B 108(6), L060506 (2023). DOI: 10.1103/PhysRevB.108.L060506.
[77] D. Arean, I. Iatrakis, M. Jarvinen, and E. Kiritsis, ”Cp-odd sector and theta dynamics in holographic qcd”, Phys. Rev. D 96(2), 026001 (2017). DOI: 10.1103/PhysRevD.96.026001.
[78] Y. Guan, J. M. Pawlowski, and M. Yamada, ”Theta-vacuum from functional renormalization”, Phys. Rev. D 113(2), 025003 (2026). DOI: 10.1103/PhysRevD.113.025003.
[79] D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, ”Theta, time reversal, and temperature”, JHEP 05, 091 (2017). DOI: 10.1007/JHEP052017091.
[80] X. Gong, W.-z. Guo, and J. Xu, ”Entanglement measures for causally connected subregions and holography”, arXiv:2508.05158.
[81] J. Liu, R. Meyer, and Z.-Y. Xian, ”Operator size growth in lindbladian syk”, JHEP 08, 092 (2024). DOI: 10.1007/JHEP082024092.
[2] M. Ghodrati, ”Hyperscaling violating solution in coupled dilaton-squared curvature gravity”, Phys. Rev. D 90(4), 044055 (2014). DOI: 10.1103/PhysRevD.90.044055.
[3] M. Ghodrati, ”Beyond ads space-times, new holographic correspondences and applications”, other thesis (2016). arXiv:1609.04168.
[4] M. Ghodrati, ”Complexity growth in massive gravity theories, the effects of chirality, and more”, Phys. Rev. D 96(10), 106020 (2017). DOI: 10.1103/PhysRevD.96.106020.
[5] M. Ghodrati, ”Complexity growth rate during phase transitions”, Phys. Rev. D 98(10), 106011 (2018). DOI: 10.1103/PhysRevD.98.106011.
[6] M. Ghodrati, X.-M. Kuang, B. Wang, C.-Y. Zhang, and Y.-T. Zhou, ”The connection between holographic entanglement and complexity of purification”, JHEP 09, 009 (2019). DOI: 10.1007/JHEP092019009.
[7] Y.-T. Zhou, M. Ghodrati, X.-M. Kuang, and J.-P. Wu, ”Evolutions of entanglement and complexity after a thermal quench in massive gravity theory”, Phys. Rev. D 100(6), 066003 (2019). DOI: 10.1103/PhysRevD.100.066003.
[8] M. Ghodrati, ”Complexity and emergence of warped ads3 space-time from chiral liouville action”, JHEP 02, 052 (2020). DOI: 10.1007/JHEP022020052.
[9] M. Ghodrati, ”Entanglement wedge reconstruction and correlation measures in mixed states: Modular flows versus quantum recovery channels”, Phys. Rev. D 104(4), 046004 (2021). DOI: 10.1103/PhysRevD.104.046004.
[10] M. Ghodrati, ”Correlations of mixed systems in confining backgrounds”, Eur. Phys. J. C 82(6), 531 (2022). DOI: 10.1140/epjc/s10052-022-10481-z.
[11] M. Ghodrati, ”Critical distance and crofton form in confining geometries”, J. Korean Phys. Soc. 81(2), 77 (2022). DOI: 10.1007/s40042-022-00523-w.
[12] M. Ghodrati, ”Encoded information of mixed correlations: The views from one dimension higher”, JHEP 08, 059 (2023). DOI: 10.1007/JHEP082023059.
[13] M. Ghodrati, ”String scattering amplitudes and mutual information in confining backgrounds: The partonic behavior”, Phys. Rev. D 112(4), 046011 (2025). DOI: 10.1103/PhysRevD.112.046011.
[14] M. Ghodrati and D. Gregoris, ”On the curvature invariants of the massive banadosteitelboim-zanelli black holes and their holographic pictures”, Int. J. Mod. Phys. A 37(34), 2250202 (2022). DOI: 10.1142/S0217751X22502025.
[15] M. Ghodrati and A. Naseh, ”Phase transitions in bergshoeff-hohm-townsend massive gravity”, Class. Quant. Grav. 34(7), 075009 (2017). DOI: 10.1088/1361-6382/aa634f.
[16] M. Ghodrati, K. Hajian, and M. R. Setare, ”Revisiting conserved charges in higher curvature gravitational theories”, Eur. Phys. J. C 76(12), 701 (2016). DOI: 10.1140/epjc/s10052-016-4550-6.
[17] M. Ghodrati, ”Chaos, phase transitions and curvature invariants of (rotating, warped, massive) btz black holes”, AIP Conf. Proc. 2874(1), 020012 (2024). DOI: 10.1063/5.0216490.
[18] W. D. Heiss, ”The physics of exceptional points”, Journal of Physics A: Mathematical and Theoretical 45, 444016 (2012). DOI: 10.1088/1751-8113/45/44/444016.
[19] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, ”Exceptional topology of nonhermitian systems”, Rev. Mod. Phys. 93(1), 015005 (2021). DOI: 10.1103/RevModPhys.93.015005.
[20] A. Fatemiabhari and C. Nunez, ”From conformal to confining field theories using holography”, JHEP 03, 160 (2024). DOI: 10.1007/JHEP032024160.
[21] H. Geyer, W. Heiss, and F. Scholtz, ”The physical interpretation of non-hermitian hamiltonians and other observables”, Canadian Journal of Physics 86, 1195 (2008). DOI: 10.1139/p08-060.
[22] M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, ”Partonic behavior of string scattering amplitudes from holographic qcd models”, JHEP 05, 058 (2022). DOI: 10.1007/JHEP052022058.
[23] Z. Gong, M. Bello, D. Malz, and F. K. Kunst, ”Anomalous behaviors of quantum emitters in non-hermitian baths”, Phys. Rev. Lett. 129(22), 223601 (2022). DOI: 10.1103/PhysRevLett.129.223601.
[24] Z. Gong, M. Bello, D. Malz, and F. K. Kunst, ”Bound states and photon emission in non-hermitian nanophotonics”, Phys. Rev. A 106(5), 053517 (2022). DOI: 10.1103/PhysRevA.106.053517.
[25] M. Jahangiri, G.-M. Parsanasab, and L. Hajshahvaladi, ”Observation of anti-ptsymmetry and higher-order exceptional point pt-symmetry in ternary systems for singlemode operation”, Scientific Reports 15(1), 4823 (2025). DOI: 10.1038/s41598-025- 85623-w.
[26] M.-A. Miri and A. Alu, ”Exceptional points in optics and photonics”, Science 363, (2019). DOI: 10.1126/science.aar7709.
[27] N. Michel, W. Nazarewicz, J. Okołowicz, and M. Płoszajczak, ”Open problems in the theory of nuclear open quantum systems”, Journal of Physics G: Nuclear and Particle Physics 37, 064042 (2010). DOI: 10.1088/0954-3899/37/6/064042.
[28] W. D. Heiss, ”Chirality of wavefunctions for three coalescing levels”, Journal of Physics A: Mathematical and Theoretical 41, 244010 (2008). DOI: 10.1088/1751- 8113/41/24/244010.
[29] Z.-Y. Xian, D. Rodriguez Fernandez, Z. Chen, Y. Liu, and R. Meyer, ”Electric conductivity in non-hermitian holography”, SciPost Phys. 16(1), 004 (2024). DOI: 10.21468/SciPostPhys.16.1.004.
[30] W. D. Heiss and S. Radu, ”Quantum chaos, degeneracies, and exceptional points”, Phys. Rev. E 52, 4762 (1995). DOI: 10.1103/PhysRevE.52.4762.
[31] M. V. Berry and M. Tabor, ”Level clustering in the regular spectrum”, Proceedings of the Royal Society of London Series A 356, 375 (1977). DOI: 10.1098/rspa.1977.0140.
[32] M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, ”Measure for chaotic scattering amplitudes”, Phys. Rev. Lett. 129(26), 261601 (2022). DOI: 10.1103/PhysRevLett.129.261601.
[33] M. Bianchi, M. Firrotta, J. Sonnenschein, and D. Weissman, ”Measuring chaos in string scattering processes”, Phys. Rev. D 108(6), 066006 (2023). DOI: 10.1103/PhysRevD.108.066006.
[34] N. Savic and M. Cubrovic, ”Weak chaos and mixed dynamics in the string s-matrix”, JHEP 03, 101 (2024). DOI: 10.1007/JHEP032024101.
[35] W. D. Heiss and A. L. Sannino, ”Transitional regions of finite fermi systems and quantum chaos”, Phys. Rev. A 43, 4159 (1991). DOI: 10.1103/PhysRevA.43.4159.
[36] M. Golubitsky and W. Langford, ”Pattern formation and bistability in flow between counterrotating cylinders”, Physica D: Nonlinear Phenomena 32(3), 362 (1988). DOI: 10.1016/0167-2789(88)90063-2.
[37] K. Qu, S. Meuren, and N. J. Fisch, ”Creating pair plasmas with observable collective effects”, Plasma Physics and Controlled Fusion 65, 034007 (2023). DOI: 10.1088/1361- 6587/acb080.
[38] J. Wiersig, ”Petermann factors and phase rigidities near exceptional points”, Phys. Rev. Res. 5(3), 033042 (2023). DOI: 10.1103/PhysRevResearch.5.033042.
[39] W. D. Heiss, ”Repulsion of resonance states and exceptional points”, Phys. Rev. E 61, 929 (2000). DOI: 10.1103/PhysRevE.61.929.
[40] F. Klingl, N. Kaiser, and W. Weise, ”Effective lagrangian approach to vector mesons, their structure and decays”, Z. Phys. A 356(2), 193 (1996). DOI: 10.1007/s002180050167.
[41] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, ”Enhanced sensitivity at higher-order exceptional points”, Nature 548(7666), 187 (2017). DOI: 10.1038/nature23280.
[42] X.-L. Liu, C.-Y. Yue, J. Nian, and W. Zheng, ”Quantum-corrected holographic wilson loop correlators and confinement”, arXiv:2412.11107.
[43] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides, and M. Khajavikhan, ”Parity-time symmetric microring lasers”, Science 346(6212), 1258480 (2014). DOI: 10.1126/science.1258480.
[44] A. Einstein, B. Podolsky, and N. Rosen, ”Can quantum-mechanical description of physical reality be considered complete?”, Phys. Rev. 47, 777 (1935). DOI: 10.1103/PhysRev.47.777.
[45] N. Gisin and A. Go, ”Epr test with photons and kaons: Analogies”, American Journal of Physics 69, 264 (2001). DOI: 10.1119/1.1326080.
[46] STAR Collaboration, B. E. Aboona et al., ”Measuring spin correlation between quarks during qcd confinement”, Nature 650(8100), 65 (2026). DOI: 10.1038/s41586-025- 09920-0.
[47] X. Cao, M. Baggioli, H. Liu, and D. Li, ”Pion dynamics in a soft-wall ads-qcd model”, Journal of High Energy Physics 2022, 113 (2022). DOI: 10.1007/JHEP122022113.
[48] K. Debnath, Y. Zhang, and K. Molmer, ”Lasing in the superradiant crossover regime”, Phys. Rev. A 98, 063837 (2018). DOI: 10.1103/PhysRevA.98.063837.
[49] K. Doi, J. Harper, A. Mollabashi, T. Takayanagi, and Y. Taki, ”Timelike entanglement entropy”, JHEP 05, 052 (2023). DOI: 10.1007/JHEP052023052.
[50] A. Milekhin, Z. Adamska, and J. Preskill, ”Observable and computable entanglement in time”, arXiv:2502.12240.
[51] J. Xu and W.-z. Guo, ”Imaginary part of timelike entanglement entropy”, JHEP 02, 094 (2025). DOI: 10.1007/JHEP022025094.
[52] P. Glorioso, X.-L. Qi, and Z. Yang, ”Space-time generalization of mutual information”, JHEP 05, 338 (2024). DOI: 10.1007/JHEP052024338.
[53] C. Nunez and D. Roychowdhury, ”Timelike entanglement entropy: A top-down approach”, Phys. Rev. D 112(2), 026030 (2025). DOI: 10.1103/PhysRevD.112.026030.
[54] W.-z. Guo and J. Xu, ”Duality of ryu-takayanagi surfaces inside and outside the horizon”, Phys. Rev. D 112(10), L101901 (2025). DOI: 10.1103/PhysRevD.112.L101901.
[55] K. Ikeda, ”Timelike quantum energy teleportation”, arXiv:2504.05353.
[56] K. Narayan and H. K. Saini, ”Notes on time entanglement and pseudo-entropy”, Eur. Phys. J. C 84(5), 499 (2024). DOI: 10.1140/epjc/s10052-024-12855-x.
[57] C. Nunez and D. Roychowdhury, ”Interpolating between spacelike and timelike entanglement via holography”, Phys. Rev. D 112(8), L081902 (2025). DOI: 10.1103/PhysRevD.112.L081902.
[58] J. Hales, U. Bajpai, T. Liu, D. R. Baykusheva, M. Li, M. Mitrano, and Y. Wang, ”Witnessing light-driven entanglement using time-resolved resonant inelastic x-ray scattering”, Nature Communications 14(1), 3512 (2023). DOI: 10.1038/s41467-023-38540-3.
[59] M. B. Hastings and T. Koma, ”Spectral gap and exponential decay of correlations”, Commun. Math. Phys. 265, 781 (2006). DOI: 10.1007/s00220-006-0030-4.
[60] B. Nachtergaele, Y. Ogata, and R. Sims, ”Propagation of correlations in quantum lattice systems”, Journal of Statistical Physics 124, 1 (2006). DOI: 10.1007/s10955-006-9143- 6.
[61] J. W. Yoon, Y. Choi, C. Hahn, G. Kim, S. H. Song, K.-Y. Yang, J. Y. Lee, Y. Kim, C. S. Lee, J. K. Shin, H.-S. Lee, and P. Berini, ”Time-asymmetric loop around an exceptional point over the full optical communications band”, Nature 562(7725), 86 (2018). DOI: 10.1038/s41586-018-0523-2.
[62] H.-Z. Chen, T. Liu, H.-Y. Luan, R.-J. Liu, X.-Y. Wang, X.-F. Zhu, Y.-B. Li, Z.-M. Gu, S.-J. Liang, H. Gao, L. Lu, L. Ge, S. Zhang, J. Zhu, and R.-M. Ma, ”Revealing the missing dimension at an exceptional point”, Nature Physics 16(5), 571 (2020). DOI: 10.1038/s41567-020-0807-y.
[63] X. Dong, A. Lewkowycz, and M. Rangamani, ”Deriving covariant holographic entanglement”, JHEP 11, 028 (2016). DOI: 10.1007/JHEP112016028.
[64] W.-z. Guo, S. He, and Y.-X. Zhang, ”Relation between time- and spacelike entanglement entropy”, Phys. Rev. D 112(8), 086020 (2025). DOI: 10.1103/PhysRevD.112.086020.
[65] K.-W. Park, J. Kim, and K. Jeong, ”Double exceptional points generated by the strong imaginary coupling of a non-hermitian hamiltonian in an optical microcavity”, arXiv:2208.06860.
[66] A. Strohmaier and E. Witten, ”The timelike tube theorem in curved spacetime”, Commun. Math. Phys. 405(7), 153 (2024). DOI: 10.1007/s00220-024-05009-3.
[67] J. Maldacena, ”Real observers solving imaginary problems”, arXiv:2412.14014.
[68] Z. Yang, Y. Zhang, and W. Zheng, ”Comments on the de sitter double cone”, arXiv:2505.08647.
[69] Y. Chen, D. Stanford, H. Tang, and Z. Yang, ”On the phase of the de sitter density of states”, arXiv:2511.01400.
[70] J.-p. Zheng, J. Dukelsky, R. A. Molina, and A. M. Garcia-Garcia, ”Role of exceptional points in the dynamics of the lindblad sachdev-ye-kitaev model”, arXiv:2510.15793.
[71] X. Liu, J.-p. Zheng, and A. M. Garca-Garca, ”Inducing, and enhancing, many-body quantum chaos by continuous monitoring”, arXiv:2602.02750.
[72] F. Lenz, M. A. Shifman, and M. Thies, ”Quantum mechanics of the vacuum state in two-dimensional qcd with adjoint fermions”, Phys. Rev. D 51, 7060 (1995). DOI: 10.1103/PhysRevD.51.7060.
[73] J. Bersini, A. D’Alise, C. Gambardella, and F. Sannino, ”On the theta-angle physics of qcd under pressure: The strange and isospin phase diagram”, JHEP 07, 252 (2025). DOI: 10.1007/JHEP072025252.
[74] T. Vonk, F.-K. Guo, and U. G. Meissner, ”Aspects of the qcd theta-vacuum”, JHEP 06, 106 (2019). DOI: 10.1007/JHEP062019106.
[75] T. Yoshida, J. L. K. Konig, L. Rodland, E. J. Bergholtz, and M. Stalhammar, ”Winding topology of multifold exceptional points”, Phys. Rev. Res. 7(1), L012021 (2025). DOI: 10.1103/PhysRevResearch.7.L012021.
[76] R. Arouca, J. Cayao, and A. M. Black-Schaffer, ”Topological superconductivity enhanced by exceptional points”, Phys. Rev. B 108(6), L060506 (2023). DOI: 10.1103/PhysRevB.108.L060506.
[77] D. Arean, I. Iatrakis, M. Jarvinen, and E. Kiritsis, ”Cp-odd sector and theta dynamics in holographic qcd”, Phys. Rev. D 96(2), 026001 (2017). DOI: 10.1103/PhysRevD.96.026001.
[78] Y. Guan, J. M. Pawlowski, and M. Yamada, ”Theta-vacuum from functional renormalization”, Phys. Rev. D 113(2), 025003 (2026). DOI: 10.1103/PhysRevD.113.025003.
[79] D. Gaiotto, A. Kapustin, Z. Komargodski, and N. Seiberg, ”Theta, time reversal, and temperature”, JHEP 05, 091 (2017). DOI: 10.1007/JHEP052017091.
[80] X. Gong, W.-z. Guo, and J. Xu, ”Entanglement measures for causally connected subregions and holography”, arXiv:2508.05158.
[81] J. Liu, R. Meyer, and Z.-Y. Xian, ”Operator size growth in lindbladian syk”, JHEP 08, 092 (2024). DOI: 10.1007/JHEP082024092.
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Volume 7, Issue 1
July 2026Pages 81-141
- Receive Date: 19 March 2026
- Revise Date: 14 May 2026
- Accept Date: 14 May 2026