{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,30]],"date-time":"2026-07-30T14:13:58Z","timestamp":1785420838611,"version":"3.56.0"},"reference-count":59,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T00:00:00Z","timestamp":1779148800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12575010"],"award-info":[{"award-number":["12575010"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"award":["12575010"],"award-info":[{"award-number":["12575010"]}],"id":[{"id":"https:\/\/ror.org\/01h0zpd94","id-type":"ROR","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>A remarkable feature of non-Hermitian topological systems with skin effects is that their spectra and eigenstates are strongly dependent on the choice of boundary conditions. Here, we investigate a system where the impurity couples to a nonreciprocal Su\u2013Schrieffer\u2013Heeger (SSH) chain at two points with nonreciprocal coupling. We first study the spectrum of the system and demonstrate that nonreciprocal couplings between the impurity and the chain alter its spectral structure. Particularly, this effect becomes particularly prominent in the limit of unidirectional coupling, inducing a shift in the parameter regime for the zero mode. Meanwhile, the impurity\u2013chain couplings give rise to two effective boundary conditions and determine the spatial distribution of the zero mode. In addition, the localization of bulk states is significantly altered by tuning the nonreciprocity of the impurity\u2013chain coupling. Notably, in the unidirectional coupling regime, two distinct types of bulk states coexist near the same boundary, one differing from the other in both spatial distribution and degree of localization. We also find that the bulk states undergo significant skin phase transitions as the coupling strength varies, characterized by a transition from conventional skin states to bipolar skin states. Our findings establish the feasibility of controlling non-Hermitian topological systems by coupling an impurity.<\/jats:p>","DOI":"10.3390\/e28050572","type":"journal-article","created":{"date-parts":[[2026,5,20]],"date-time":"2026-05-20T14:12:54Z","timestamp":1779286374000},"page":"572","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Anomalous Behavior Induced by a Single Impurity in Non-Hermitian Topological Systems with Nonreciprocal Coupling"],"prefix":"10.3390","volume":"28","author":[{"given":"Junjie","family":"Wang","sequence":"first","affiliation":[{"name":"College of Mechanical Engineering, Ningbo University of Finance & Economics, Ningbo 315175, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhenyan","family":"Wang","sequence":"additional","affiliation":[{"name":"College of Mechanical Engineering, Ningbo University of Finance & Economics, Ningbo 315175, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xie","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Artificial Intelligence, Ningbo University of Finance & Economics, Ningbo 315175, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xuexi","family":"Yi","sequence":"additional","affiliation":[{"name":"Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun 130024, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,5,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1039","DOI":"10.1023\/B:CJOP.0000044002.05657.04","article-title":"Physics of non-Hermitian degeneracies","volume":"54","author":"Berry","year":"2004","journal-title":"Czech. J. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Moiseyev, N. (2011). Non-Hermitian Quantum Mechanics, Cambridge University Press.","DOI":"10.1017\/CBO9780511976186"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"444016","DOI":"10.1088\/1751-8113\/45\/44\/444016","article-title":"The physics of exceptional points","volume":"45","author":"Heiss","year":"2012","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"035305","DOI":"10.1088\/1751-8113\/47\/3\/035305","article-title":"Biorthogonal quantum mechanics","volume":"47","author":"Brody","year":"2013","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1080\/00018732.2021.1876991","article-title":"Non-Hermitian physics","volume":"69","author":"Ashida","year":"2020","journal-title":"Adv. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"012118","DOI":"10.1103\/PhysRevA.87.012118","article-title":"Topological invariance and global Berry phase in non-Hermitian systems","volume":"87","author":"Liang","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"040401","DOI":"10.1103\/PhysRevLett.118.040401","article-title":"Edge modes, degeneracies, and topological numbers in non-Hermitian systems","volume":"118","author":"Leykam","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_8","first-page":"031079","article-title":"Topological phases of non-Hermitian systems","volume":"8","author":"Gong","year":"2018","journal-title":"Phys. Rev. X"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"076801","DOI":"10.1103\/PhysRevLett.122.076801","article-title":"Second-Order Topological Phases in Non-Hermitian Systems","volume":"122","author":"Liu","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"052115","DOI":"10.1103\/PhysRevA.97.052115","article-title":"Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems","volume":"97","author":"Yin","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"045106","DOI":"10.1103\/PhysRevB.97.045106","article-title":"Topological phases in the non-Hermitian Su-Schrieffer-Heeger model","volume":"97","author":"Lieu","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"146402","DOI":"10.1103\/PhysRevLett.120.146402","article-title":"Topological band theory for non-Hermitian Hamiltonians","volume":"120","author":"Shen","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"133903","DOI":"10.1103\/PhysRevLett.116.133903","article-title":"Anomalous edge state in a non-Hermitian lattice","volume":"116","author":"Lee","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"035043","DOI":"10.1088\/2399-6528\/aab64a","article-title":"Why does bulk boundary correspondence fail in some non-Hermitian topological models","volume":"2","author":"Xiong","year":"2018","journal-title":"J. Phys. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"026808","DOI":"10.1103\/PhysRevLett.121.026808","article-title":"Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems","volume":"121","author":"Kunst","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"086803","DOI":"10.1103\/PhysRevLett.121.086803","article-title":"Edge States and topological invariants of non-Hermitian systems","volume":"121","author":"Yao","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"015005","DOI":"10.1103\/RevModPhys.93.015005","article-title":"Exceptional topology of non-Hermitian systems","volume":"93","author":"Bergholtz","year":"2021","journal-title":"Rev. Mod. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"136802","DOI":"10.1103\/PhysRevLett.121.136802","article-title":"Non-Hermitian Chern bands","volume":"121","author":"Yao","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"170401","DOI":"10.1103\/PhysRevLett.123.170401","article-title":"Non-Hermitian Skin Effect and Chiral Damping in Open Quantum Systems","volume":"123","author":"Song","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"246801","DOI":"10.1103\/PhysRevLett.123.246801","article-title":"Non-Hermitian Topological Invariants in Real Space","volume":"123","author":"Song","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"226402","DOI":"10.1103\/PhysRevLett.125.226402","article-title":"Non-Hermitian Bulk-Boundary Correspondence and Auxiliary Generalized Brillouin Zone Theory","volume":"125","author":"Yang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"126402","DOI":"10.1103\/PhysRevLett.125.126402","article-title":"Correspondence between Winding Numbers and Skin Modes in Non-Hermitian Systems","volume":"125","author":"Zhang","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"186802","DOI":"10.1103\/PhysRevLett.125.186802","article-title":"Non-Hermitian Skin Modes Induced by On-Site Dissipations and Chiral Tunneling Effect","volume":"125","author":"Yi","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"086801","DOI":"10.1103\/PhysRevLett.124.086801","article-title":"Topological Origin of Non-Hermitian Skin Effects","volume":"124","author":"Okuma","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"056802","DOI":"10.1103\/PhysRevLett.124.056802","article-title":"Non-Hermitian Boundary Modes and Topology","volume":"124","author":"Borgnia","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"121401","DOI":"10.1103\/PhysRevB.97.121401","article-title":"Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points","volume":"97","author":"Alvarez","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"201103","DOI":"10.1103\/PhysRevB.99.201103","article-title":"Anatomy of skin modes and topology in non-Hermitian systems","volume":"99","author":"Lee","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"250402","DOI":"10.1103\/PhysRevLett.124.250402","article-title":"Topological Switch for Non-Hermitian Skin Effect in Cold-Atom Systems with Loss","volume":"124","author":"Li","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"016805","DOI":"10.1103\/PhysRevLett.123.016805","article-title":"Hybrid Higher-Order Skin-Topological Modes in Nonreciprocal Systems","volume":"123","author":"Lee","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"5491","DOI":"10.1038\/s41467-020-18917-4","article-title":"Critical non-Hermitian skin effect","volume":"11","author":"Li","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"123029","DOI":"10.1088\/1367-2630\/ac3e9f","article-title":"The topological counterparts of non-Hermitian SSH models","volume":"23","author":"Han","year":"2021","journal-title":"New J. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"205403","DOI":"10.1103\/PhysRevB.105.205403","article-title":"Chiral metals and entrapped insulators in a one-dimensional topological non-Hermitian system","volume":"105","author":"Banerjee","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_33","first-page":"5608038","article-title":"Non-Hermitian Skin Effect in a Non-Hermitian Electrical Circuit","volume":"9","author":"Liu","year":"2021","journal-title":"Research"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"023265","DOI":"10.1103\/PhysRevResearch.2.023265","article-title":"Reciprocal skin effect and its realization in a topolectrical circuit","volume":"2","author":"Hofmann","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"761","DOI":"10.1038\/s41567-020-0836-6","article-title":"Non-Hermitian bulk-boundary correspondence in quantum dynamics","volume":"16","author":"Xiao","year":"2020","journal-title":"Nat. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1038\/s41567-020-0922-9","article-title":"Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits","volume":"16","author":"Helbig","year":"2020","journal-title":"Nat. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"4608","DOI":"10.1038\/s41467-019-12599-3","article-title":"Non-reciprocal robotic metamaterials","volume":"10","author":"Brandenbourger","year":"2019","journal-title":"Nat. Commun."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1038\/s42005-021-00547-x","article-title":"Impurity induced scale-free localization","volume":"4","author":"Li","year":"2021","journal-title":"Commun. Phys."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"165117","DOI":"10.1103\/PhysRevB.104.165117","article-title":"Scaling rule for the critical non-Hermitian skin effect","volume":"104","author":"Yokomizo","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"116801","DOI":"10.1103\/PhysRevLett.127.116801","article-title":"Exact Solution of Non-Hermitian Systems with Generalized Boundary Conditions: Size-Dependent Boundary Effect and Fragility of the Skin Effect","volume":"127","author":"Guo","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"085401","DOI":"10.1103\/PhysRevB.104.085401","article-title":"Exact solution of the single impurity problem in nonreciprocal lattices: Impurity-induced size-dependent non-Hermitian skin effect","volume":"104","author":"Liu","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"075404","DOI":"10.1103\/PhysRevB.102.075404","article-title":"Diagnosis of bulk phase diagram of nonreciprocal topological lattices by impurity modes","volume":"102","author":"Liu","year":"2020","journal-title":"Phys. Rev. B"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"022215","DOI":"10.1103\/PhysRevA.104.022215","article-title":"Non-Hermitian skin effect as an impurity problem","volume":"104","author":"Roccati","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"103016","DOI":"10.1088\/1367-2630\/13\/10\/103016","article-title":"Non-magnetic impurities and in-gap bound states in topological insulators","volume":"13","author":"Lu","year":"2011","journal-title":"New J. Phys."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"085126","DOI":"10.1103\/PhysRevB.92.085126","article-title":"Impurity-bound states and Green\u2019s function zeros as local signatures of topology","volume":"92","author":"Slager","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"033058","DOI":"10.1103\/PhysRevResearch.5.033058","article-title":"Anomalous skin effects in disordered systems with a single non-Hermitian impurity","volume":"5","author":"Molignini","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"013003","DOI":"10.1103\/PhysRevResearch.5.013003","article-title":"Radiation-free and non-Hermitian topology inertial defect staes of on-chip magnons","volume":"5","author":"Zeng","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"032219","DOI":"10.1103\/622m-d2q6","article-title":"Interplay between defects and the non-Hermitian skin effect","volume":"112","author":"Huang","year":"2025","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"180403","DOI":"10.1103\/PhysRevLett.125.180403","article-title":"Non-Hermitian Topological Sensors","volume":"125","author":"Budich","year":"2020","journal-title":"Phys. Rev. Lett."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"5382","DOI":"10.1038\/s41467-020-19090-4","article-title":"Exponentially-enhanced quantum sensing with non-Hermitian lattice dynamics","volume":"11","author":"McDonald","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"013113","DOI":"10.1103\/PhysRevResearch.4.013113","article-title":"Quantum non-Hermitian topological sensors","volume":"4","author":"Koch","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"2301128","DOI":"10.1002\/advs.202301128","article-title":"Non-Hermitian topolectrical circuit sensor with high sensitivity","volume":"10","author":"Yuan","year":"2023","journal-title":"Adv. Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"106601","DOI":"10.1103\/f6wd-gljq","article-title":"Non-Hermitian Floquet topological sensors for ultrasensitive detection of dynamic signals","volume":"135","author":"Zhou","year":"2025","journal-title":"Phys. Rev. Lett."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"043602","DOI":"10.1103\/PhysRevLett.126.043602","article-title":"Tunable Chiral Bound States with Giant Atoms","volume":"126","author":"Wang","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"033522","DOI":"10.1103\/PhysRevA.106.033522","article-title":"Topology and retardation effect of a giant atom in a topological waveguide","volume":"106","author":"Cheng","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"455306","DOI":"10.1088\/1751-8121\/acfbc9","article-title":"Giant atom induced zero modes and localization in the nonreciprocal Su-Schrieffer-Heeger chain","volume":"56","author":"Wang","year":"2023","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Wang, J.J., Li, F.D., and Cheng, W.J. (2025). Anomalous behavior of the non-Hermitian topological system with an asymmetric coupling impurity. Entropy, 27.","DOI":"10.3390\/e27010078"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"033133","DOI":"10.1103\/h9b6-md2f","article-title":"Multiple physical quantities sensors based on non-Hermitian topological systems with an impurity","volume":"7","author":"Wang","year":"2025","journal-title":"Phys. Rev. Res."},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1038\/nmat4811","article-title":"Topologically protected bound states in photonic parity-time-symmetric crystals","volume":"16","author":"Weimann","year":"2017","journal-title":"Nat. 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