{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T22:39:34Z","timestamp":1775083174962,"version":"3.50.1"},"reference-count":74,"publisher":"Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften","license":[{"start":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T00:00:00Z","timestamp":1679529600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science and Technology Council","award":["112-2636-M-007-007"],"award-info":[{"award-number":["112-2636-M-007-007"]}]}],"content-domain":{"domain":["quantum-journal.org"],"crossmark-restriction":false},"short-container-title":["Quantum"],"abstract":"The fidelity susceptibility is a tool for studying quantum phase transitions in the Hermitian condensed matter systems. Recently, it has been generalized with the biorthogonal basis for the non-Hermitian quantum systems. From the general perturbation description with the constraint of parity-time (PT) symmetry, we show that the fidelity F<\/mml:mi><\/mml:mrow><\/mml:math> is always real for the PT-unbroken states. For the PT-broken states, the real part of the fidelity susceptibility R<\/mml:mi>e<\/mml:mi><\/mml:mrow>[<\/mml:mo>X<\/mml:mi><\/mml:mrow>F<\/mml:mi><\/mml:msub>]<\/mml:mo><\/mml:math> is corresponding to considering both the PT partner states, and the negative infinity is explored by the perturbation theory when the parameter approaches the exceptional point (EP). Moreover, at the second-order EP, we prove that the real part of the fidelity between PT-unbroken and PT-broken states is R<\/mml:mi>e<\/mml:mi><\/mml:mrow>F<\/mml:mi><\/mml:mrow>=<\/mml:mo>1<\/mml:mn>2<\/mml:mn><\/mml:mfrac><\/mml:math>. Based on these general properties, we study the two-legged non-Hermitian Su-Schrieffer-Heeger (SSH) model and the non-Hermitian XXZ spin chain. We find that for both interacting and non-interacting systems, the real part of fidelity susceptibility density goes to negative infinity when the parameter approaches the EP, and verifies it is a second-order EP by R<\/mml:mi>e<\/mml:mi><\/mml:mrow>F<\/mml:mi><\/mml:mrow>=<\/mml:mo>1<\/mml:mn>2<\/mml:mn><\/mml:mfrac><\/mml:math>.<\/jats:p>","DOI":"10.22331\/q-2023-03-23-960","type":"journal-article","created":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T14:33:52Z","timestamp":1679582032000},"page":"960","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":36,"title":["General properties of fidelity in non-Hermitian quantum systems with PT symmetry"],"prefix":"10.22331","volume":"7","author":[{"given":"Yi-Ting","family":"Tu","sequence":"first","affiliation":[{"name":"Department of Physics, University of Maryland, College Park, MD, USA"}]},{"given":"Iksu","family":"Jang","sequence":"additional","affiliation":[{"name":"Department of Physics, National Tsing Hua University, Hsinchu 300044, Taiwan"}]},{"given":"Po-Yao","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Physics, National Tsing Hua University, Hsinchu 300044, Taiwan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0380-1431","authenticated-orcid":false,"given":"Yu-Chin","family":"Tzeng","sequence":"additional","affiliation":[{"name":"Physics Division, National Center for Theoretical Sciences, Taipei 106319, Taiwan"},{"name":"Center for Theoretical and Computational Physics, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan"}]}],"member":"9598","published-online":{"date-parts":[[2023,3,23]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"W.-L. You, Y.-W. Li, and S.-J. Gu. ``Fidelity, dynamic structure factor, and susceptibility in critical phenomena''. Phys. Rev. E 76, 022101 (2007).","DOI":"10.1103\/PhysRevE.76.022101"},{"key":"1","doi-asserted-by":"publisher","unstructured":"S.-J. Gu. ``Fidelity approach to quantum phase transitions''. Int. J. Mod. Phys. B 24, 4371\u20134458 (2010).","DOI":"10.1142\/S0217979210056335"},{"key":"2","doi-asserted-by":"publisher","unstructured":"M.-F. Yang. ``Ground-state fidelity in one-dimensional gapless models''. Phys. Rev. B 76, 180403(R) (2007).","DOI":"10.1103\/PhysRevB.76.180403"},{"key":"3","doi-asserted-by":"publisher","unstructured":"John Ove Fj\u00e6restad. ``Ground state fidelity of Luttinger liquids: a wavefunctional approach''. Journal of Statistical Mechanics: Theory and Experiment 2008, P07011 (2008).","DOI":"10.1088\/1742-5468\/2008\/07\/p07011"},{"key":"4","doi-asserted-by":"publisher","unstructured":"Y.-C. Tzeng and M.-F. Yang. ``Scaling properties of fidelity in the spin-1 anisotropic model''. Phys. Rev. A 77, 012311 (2008).","DOI":"10.1103\/PhysRevA.77.012311"},{"key":"5","doi-asserted-by":"publisher","unstructured":"Y.-C. Tzeng, H.-H. Hung, Y.-C. Chen, and M.-F. Yang. ``Fidelity approach to Gaussian transitions''. Phys. Rev. A 77, 062321 (2008).","DOI":"10.1103\/PhysRevA.77.062321"},{"key":"6","doi-asserted-by":"publisher","unstructured":"J. Ren, G.-H. Liu, and W.-L. You. ``Entanglement entropy and fidelity susceptibility in the one-dimensional spin-1 XXZ chains with alternating single-site anisotropy''. J. Phys.: Cond. Mat. 27, 105602 (2015).","DOI":"10.1088\/0953-8984\/27\/10\/105602"},{"key":"7","doi-asserted-by":"publisher","unstructured":"B. Wang, M. Feng, and Z.-Q. Chen. ``Berezinskii-Kosterlitz-Thouless transition uncovered by the fidelity susceptibility in the XXZ model''. Phys. Rev. A 81, 064301 (2010).","DOI":"10.1103\/PhysRevA.81.064301"},{"key":"8","doi-asserted-by":"publisher","unstructured":"H.-L. Wang, J.-H. Zhao, B. Li, and H.-Q. Zhou. ``Kosterlitz-Thouless phase transition and ground state fidelity: a novel perspective from matrix product states''. J. Stat. Mech.: Theo. Exp. 2011, L10001 (2011).","DOI":"10.1088\/1742-5468\/2011\/10\/l10001"},{"key":"9","doi-asserted-by":"publisher","unstructured":"G. Sun, A. K. Kolezhuk, and T. Vekua. ``Fidelity at Berezinskii-Kosterlitz-Thouless quantum phase transitions''. Phys. Rev. B 91, 014418 (2015).","DOI":"10.1103\/PhysRevB.91.014418"},{"key":"10","doi-asserted-by":"publisher","unstructured":"J. Zhang. ``Fidelity and entanglement entropy for infinite-order phase transitions''. Phys. Rev. B 104, 205112 (2021).","DOI":"10.1103\/PhysRevB.104.205112"},{"key":"11","doi-asserted-by":"publisher","unstructured":"Y. Ashida, Z. Gong, and M. Ueda. ``Non-Hermitian physics''. Advances in Physics 69, 249\u2013435 (2020).","DOI":"10.1080\/00018732.2021.1876991"},{"key":"12","doi-asserted-by":"publisher","unstructured":"Ramy El-Ganainy, Konstantinos G. Makris, Mercedeh Khajavikhan, Ziad H. Musslimani, Stefan Rotter, and Demetrios N. Christodoulides. ``Non-Hermitian physics and PT symmetry''. Nature Physics 14, 11\u201319 (2018).","DOI":"10.1038\/nphys4323"},{"key":"13","doi-asserted-by":"publisher","unstructured":"G. Lindblad. ``On the generators of quantum dynamical semigroups''. Comm. Math. Phys. 48, 119\u2013130 (1976).","DOI":"10.1007\/BF01608499"},{"key":"14","doi-asserted-by":"publisher","unstructured":"D. C. Brody. ``Biorthogonal quantum mechanics''. J. Phys. A 47, 035305 (2013).","DOI":"10.1088\/1751-8113\/47\/3\/035305"},{"key":"15","doi-asserted-by":"publisher","unstructured":"Bart\u0142omiej Gardas, Sebastian Deffner, and Avadh Saxena. ``Non-hermitian quantum thermodynamics''. Sci. Rep. 6, 23408 (2016).","DOI":"10.1038\/srep23408"},{"key":"16","doi-asserted-by":"publisher","unstructured":"D.-J. Zhang, Q.-H. Wang, and J. Gong. ``Time-dependent $\\mathcal{PT}$-symmetric quantum mechanics in generic non-hermitian systems''. Phys. Rev. A 100, 062121 (2019).","DOI":"10.1103\/PhysRevA.100.062121"},{"key":"17","doi-asserted-by":"publisher","unstructured":"C.-Y. Ju, A. Miranowicz, G.-Y. Chen, and F. Nori. ``Non-Hermitian Hamiltonians and no-go theorems in quantum information''. Phys. Rev. A 100, 062118 (2019).","DOI":"10.1103\/PhysRevA.100.062118"},{"key":"18","doi-asserted-by":"publisher","unstructured":"S. Yao and Z. Wang. ``Edge states and topological invariants of non-hermitian systems''. Phys. Rev. Lett. 121, 086803 (2018).","DOI":"10.1103\/PhysRevLett.121.086803"},{"key":"19","doi-asserted-by":"publisher","unstructured":"W. D. Heiss. ``The physics of exceptional points''. J. Phys. A. 45, 444016 (2012).","DOI":"10.1088\/1751-8113\/45\/44\/444016"},{"key":"20","doi-asserted-by":"publisher","unstructured":"Mohammad-Ali Miri and Andrea Alu. ``Exceptional points in optics and photonics''. Science 363, eaar7709 (2019).","DOI":"10.1126\/science.aar7709"},{"key":"21","doi-asserted-by":"publisher","unstructured":"J. Doppler, A. A. Mailybaev, J. B\u00f6hm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter. ``Dynamically encircling an exceptional point for asymmetric mode switching''. Nature 537, 76\u201379 (2016).","DOI":"10.1038\/nature18605"},{"key":"22","doi-asserted-by":"publisher","unstructured":"H. Xu, D. Mason, Luyao Jiang, and J. G. E. Harris. ``Topological energy transfer in an optomechanical system with exceptional points''. 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B 99, 121101 (2019).","DOI":"10.1103\/PhysRevB.99.121101"},{"key":"26","doi-asserted-by":"publisher","unstructured":"K. Kawabata, T. Bessho, and M. Sato. ``Classification of exceptional points and non-Hermitian topological semimetals''. Phys. Rev. Lett. 123, 066405 (2019).","DOI":"10.1103\/PhysRevLett.123.066405"},{"key":"27","doi-asserted-by":"publisher","unstructured":"J. Xu, Y.-X. Du, W. Huang, and D.-W. Zhang. ``Detecting topological exceptional points in a parity-time symmetric system with cold atoms''. Opt. Express 25, 15786\u201315795 (2017).","DOI":"10.1364\/OE.25.015786"},{"key":"28","doi-asserted-by":"publisher","unstructured":"Bo Peng, \u015eahin Kaya \u00d6zdemir, Matthias Liertzer, Weijian Chen, Johannes Kramer, Huzeyfe Y\u0131lmaz, Jan Wiersig, Stefan Rotter, and Lan Yang. ``Chiral modes and directional lasing at exceptional points''. 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Chen. ``Topological invariants and phase diagrams for one-dimensional two-band non-Hermitian systems without chiral symmetry''. Phys. Rev. A 98, 052116 (2018).","DOI":"10.1103\/PhysRevA.98.052116"},{"key":"52","doi-asserted-by":"publisher","unstructured":"D.-J. Zhang, Q.-H. Wang, and J. Gong. ``Quantum geometric tensor in $\\mathcal{PT}$-symmetric quantum mechanics''. Phys. Rev. A 99, 042104 (2019).","DOI":"10.1103\/PhysRevA.99.042104"},{"key":"53","doi-asserted-by":"publisher","unstructured":"G. Sun, J.-C. Tang, and S.-P. Kou. ``Biorthogonal quantum criticality in non-Hermitian many-body systems''. Frontiers of Physics 17, 33502 (2022).","DOI":"10.1007\/s11467-021-1126-1"},{"key":"54","doi-asserted-by":"publisher","unstructured":"C. Chen, L. Jin, and R.-B. Liu. ``Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system''. New J. Phys. 21, 083002 (2019).","DOI":"10.1088\/1367-2630\/ab32ab"},{"key":"55","doi-asserted-by":"publisher","unstructured":"Y. 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Rev. Lett. 42, 1698\u20131701 (1979).","DOI":"10.1103\/PhysRevLett.42.1698"},{"key":"63","doi-asserted-by":"publisher","unstructured":"Y.-C. Tzeng, L. Dai, M. Chung, Luigi Amico, and Leong-Chuan Kwek. ``Entanglement convertibility by sweeping through the quantum phases of the alternating bonds XXZ chain''. Sci. Rep. 6, 26453 (2016).","DOI":"10.1038\/srep26453"},{"key":"64","doi-asserted-by":"publisher","unstructured":"Y.-C. Tzeng and M.-F. Yang. ``Fate of Fermi-arc states in gapped Weyl semimetals under long-range interactions''. Phys. Rev. B 102, 035148 (2020).","DOI":"10.1103\/PhysRevB.102.035148"},{"key":"65","doi-asserted-by":"publisher","unstructured":"M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A. Bandres, J. Ren, M. C. Rechtsman, M. Segev, D. N. Christodoulides, and M. Khajavikhan. ``Edge-mode lasing in 1D topological active arrays''. Phys. Rev. Lett. 120, 113901 (2018).","DOI":"10.1103\/PhysRevLett.120.113901"},{"key":"66","doi-asserted-by":"publisher","unstructured":"M. Pan, H. Zhao, P. Miao, S. Longhi, and L. Feng. ``Photonic zero mode in a non-hermitian photonic lattice''. Nat. Comm. 9, 1\u20138 (2018).","DOI":"10.1038\/s41467-018-03822-8"},{"key":"67","doi-asserted-by":"publisher","unstructured":"W. Song, W. Sun, C. Chen, Q. Song, S. Xiao, S. Zhu, and T. Li. ``Breakup and recovery of topological zero modes in finite non-Hermitian optical lattices''. Phys. Rev. Lett. 123, 165701 (2019).","DOI":"10.1103\/PhysRevLett.123.165701"},{"key":"68","doi-asserted-by":"publisher","unstructured":"L. Herviou, N. Regnault, and J. H. Bardarson. ``Entanglement spectrum and symmetries in non-Hermitian fermionic non-interacting models''. SciPost Physics 7, 69 (2019).","DOI":"10.21468\/SciPostPhys.7.5.069"},{"key":"69","doi-asserted-by":"publisher","unstructured":"S. Lieu. ``Topological phases in the non-Hermitian Su-Schrieffer-Heeger model''. Phys. Rev. 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