非厄米体系中的量子反常和量子纠缠研究

在物理学领域中，放弃厄米性作为一种新的思维尝试备受关注。虽然在经典
物理中，这种做法相对来说比较自然，但推广到量子物理层面却不那么容易理解。
本文从三个不同的方面研究非厄米性在量子物理中带来的非平凡效应，探索其深
层次的物理内涵和应用。
首先，我们采用非厄米性作为一种不寻常的方式来解决手征费米子问题，这
是一个长期以来未能解决的难题。通过计算谱流、指标定理和手征模的含时演化，
我们详细地检查了该模型的手征反常和引力反常，并证明了我们的模型可以实现
手征费米子的格点化。
其次，我们探索了非厄米物理中一个重要而又奇特的概念——奇异点在能谱中所带来的效应。与一般的研究方式不同，我们采用能谱中存在奇异点时的纠缠熵以及纠缠谱作为探测奇异点行为的指标。我们发现，能谱中的奇异点往往与（实的）非幺正的共形场论以及复数共形场论有联系，它们分别标志着宇称-时间对称性自发破缺的二级相变和某种尚不明确的一级相变。这对我们理解非厄米的量子临界行为和一些一级相变理论提供了新的思路。
最后，我们探讨了非厄米模型的含时演化问题，这是在开放量子系统下进行
的。通过一个通用模型含时纠缠熵的计算，我们发现，即使在有相互作用的情况
下，一维费米液体的短时间热化过程具有比较普适的特性。而长时间的衰减行为
则取决于刘维尔谱的“能隙”，这对于开放量子多体系统的研究具有很强的指导意义。

The abandonment of Hermiticity as a novel approach has garnered significant attention
in the field of physics. While the abandonment of Hermiticity may seem relatively natural within classical physics, its extension to the quantum level poses nontrivial challenges to intuition. In this thesis, we investigate non-Hermiticity from three distinct perspectives, exploring the nontrivial effects that arise at the quantum level and uncovering the deeper implications and potential applications of non-Hermitian physics.
First, we employ non-Hermiticity as an unconventional means of addressing the long-standing problem of chiral fermions. Through the calculation of spectral flow, index theorems and the time-dependent evolution of chiral modes, we examine the chiral and gravitational anomalies of our model, demonstrating the lattice realization of chiral
fermions through our approach. Next, we investigate the effects of exceptional points in energy spectra, a particularly important and intriguing concept in non-Hermitian physics.
In contrast to conventional approaches, we utilize the entanglement entropy and spectrum as indicators of exceptional behavior. We find that exceptional points in energy spectra are often related to (real) non-unitary conformal field theories and complex conformal field theories, respectively marking second order phase transitions resulting from spontaneous breaking of parity-time symmetry and some as-yet unclear first-order phase transition.
Finally, we consider the issue of time-dependent evolution in non- Hermitian models within the context of open quantum systems. Through the calculation of time-dependent entanglement entropy in a generic model, we demonstrate that the short-time thermalization process of a one-dimensional fermi liquid has universal characteristics even with
interactions. The long-time decay behavior, however, depends on the ”gap” in the Liouvillian spectrum. This has strong guiding significance for the study of open quantum many-body systems.

[1] GAMOW G. Zur quantentheorie des atomkernes[J]. Zeitschrift für Physik, 1928, 51(3-4):204-212
[2] SIEGERT A J F. On the Derivation of the Dispersion Formula for Nuclear Reactions[J]. PhysRev, 1939, 56(8): 750-752
[3] MAJORANA E. Sulla formazione dello ione molecolare di elio[J]. Il Nuovo Cimento (1924-1942), 1931, 8: 22-28
[4] FESHBACH H. The Optical Model and Its Justification[J]. Annual Review of Nuclear Science,1958, 8(1): 49-104
[5] BREUER H P, PETRUCCIONE F. Oxford Graduate Texts: The Theory of Open QuantumSystems[M]. Oxford University Press, 2002
[6] BREUER H P, LAINE E M, PIILO J, et al. Colloquium : Non-Markovian dynamics in openquantum systems[J]. Rev Mod Phys, 2016, 88(2): 021002
[7] MANZANO D. A short introduction to the Lindblad master equation[J]. AIP Advances, 2020,10(2): 025106
[8] FEYNMAN R, VERNON F. Dynamical symmetries and conserved quantities[J]. Ann Phys,1963, 24: 118-173
[9] CALZETTA E, HU B L. Cambridge Monographs on Mathematical Physics: NonequilibriumQuantum Field Theory[M]. Cambridge University Press, 2008
[10] MAKRIS K G, EL-GANAINY R, CHRISTODOULIDES D N, et al. Beam Dynamics in P TSymmetric Optical Lattices[J]. Phys Rev Lett, 2008, 100(10): 103904
[11] MUSSLIMANI Z H, MAKRIS K G, EL-GANAINY R, et al. Optical Solitons in P T PeriodicPotentials[J]. Phys Rev Lett, 2008, 100(3): 030402
[12] REGENSBURGER A, BERSCH C, MIRI M A, et al. Parity–time synthetic photonic lattices[J]. Nature, 2012, 488(7410): 167-171
[13] WANG H, ZHANG X, HUA J, et al. Topological physics of non-Hermitian optics and photonics:a review[J]. J Opt, 2021, 23(12): 123001
[14] PARTO M, LIU Y G N, BAHARI B, et al. Non-Hermitian and topological photonics: optics atan exceptional point:[J]. Nanophotonics, 2021, 10(1): 403-423
[15] LONGHI S. Parity-time symmetry meets photonics: A new twist in non-Hermitian optics[J].EPL, 2018, 120(6): 64001
[16] STEGMAIER A, IMHOF S, HELBIG T, et al. Topological Defect Engineering and P T Symmetryin Non-Hermitian Electrical Circuits[J]. Phys Rev Lett, 2021, 126(21): 215302
[17] GHATAK A, BRANDENBOURGER M, VAN WEZEL J, et al. Observation of non-Hermitiantopology and its bulk–edge correspondence in an active mechanical metamaterial[J]. Proceedingsof the National Academy of Sciences, 2020, 117(47): 29561-29568
[18] SCHOMERUS H. Nonreciprocal response theory of non-Hermitian mechanical metamaterials:Response phase transition from the skin effect of zero modes[J]. Phys Rev Research, 2020, 2(1): 013058
[19] ZHU X, RAMEZANI H, SHI C, et al. P T -Symmetric Acoustics[J]. Phys Rev X, 2014, 4(3):031042
[20] ZHU W, FANG X, LI D, et al. Simultaneous Observation of a Topological Edge State andExceptional Point in an Open and Non-Hermitian Acoustic System[J]. Phys Rev Lett, 2018,121(12): 124501
[21] ZHANG Z, ROSENDO LóPEZ M, CHENG Y, et al. Non-Hermitian Sonic Second-Order TopologicalInsulator[J]. Phys Rev Lett, 2019, 122(19): 195501
[22] ZHANG L, YANG Y, GE Y, et al. Acoustic non-Hermitian skin effect from twisted windingtopology[J]. Nat Commun, 2021, 12(1): 6297
[23] LEE T E. Anomalous Edge State in a Non-Hermitian Lattice[J]. Phys Rev Lett, 2016, 116(13):133903
[24] KAWABATA K, BESSHO T, SATO M. Classification of Exceptional Points and Non-HermitianTopological Semimetals[J]. Phys Rev Lett, 2019, 123(6): 066405
[25] LEYKAM D, BLIOKH K Y, HUANG C, et al. Edge Modes, Degeneracies, and TopologicalNumbers in Non-Hermitian Systems[J]. Phys Rev Lett, 2017, 118(4): 040401
[26] YAO S, WANG Z. Edge States and Topological Invariants of Non-Hermitian Systems[J]. PhysRev Lett, 2018, 121(8): 086803
[27] DING K, MA G, XIAO M, et al. Emergence, Coalescence, and Topological Properties ofMultiple Exceptional Points and Their Experimental Realization[J]. Phys Rev X, 2016, 6(2):021007
[28] BERGHOLTZ E J, BUDICH J C, KUNST F K. Exceptional topology of non-Hermitian systems[J]. Rev Mod Phys, 2021, 93(1): 015005
[29] HELBIG T, HOFMANN T, IMHOF S, et al. Generalized bulk–boundary correspondence innon-Hermitian topolectrical circuits[J]. Nat Phys, 2020, 16(7): 747-750
[30] LUO X W, ZHANG C. Higher-Order Topological Corner States Induced by Gain and Loss[J].Phys Rev Lett, 2019, 123(7): 073601
[31] YANG Z, CHIU C K, FANG C, et al. Jones Polynomial and Knot Transitions in Hermitian andnon-Hermitian Topological Semimetals[J]. Phys Rev Lett, 2020, 124(18): 186402
[32] BORGNIA D S, KRUCHKOV A J, SLAGER R J. Non-Hermitian Boundary Modes and Topology[J]. Phys Rev Lett, 2020, 124(5): 056802
[33] XUE H, WANG Q, ZHANG B, et al. Non-Hermitian Dirac Cones[J]. Phys Rev Lett, 2020, 124(23): 236403
[34] SONG F, YAO S, WANG Z. Non-Hermitian Skin Effect and Chiral Damping in Open QuantumSystems[J]. Phys Rev Lett, 2019, 123(17): 170401
[35] SONG A Y, SUN X Q, DUTT A, et al. P T -Symmetric Topological Edge-Gain Effect[J]. PhysRev Lett, 2020, 125(3): 033603
[36] ZHOU H, LEE J Y. Periodic table for topological bands with non-Hermitian symmetries[J].Phys Rev B, 2019, 99(23): 235112
[37] LIU T, ZHANG Y R, AI Q, et al. Second-Order Topological Phases in Non-Hermitian Systems[J]. Phys Rev Lett, 2019, 122(7): 076801
[38] KAWABATA K, HIGASHIKAWA S, GONG Z, et al. Topological unification of time-reversaland particle-hole symmetries in non-Hermitian physics[J]. Nat Commun, 2019, 10(1): 297
[39] LIANG S D, HUANG G Y. Topological invariance and global Berry phase in non-Hermitiansystems[J]. Phys Rev A, 2013, 87(1): 012118
[40] SATO M, HASEBE K, ESAKI K, et al. Time-Reversal Symmetry in Non-Hermitian Systems[J]. Progress of Theoretical Physics, 2012, 127(6): 937-974
[41] YANG C N, LEE T D. Statistical Theory of Equations of State and Phase Transitions. I. Theoryof Condensation[J]. Phys Rev, 1952, 87(3): 404-409
[42] LEE T D, YANG C N. Statistical Theory of Equations of State and Phase Transitions. II. LatticeGas and Ising Model[J]. Phys Rev, 1952, 87(3): 410-419
[43] FISHER M E. Yang-Lee Edge Singularity and ϕ 3 Field Theory[J]. Phys Rev Lett, 1978, 40(25): 1610-1613
[44] CARDY J L. Conformal Invariance and the Yang-Lee Edge Singularity in Two Dimensions[J].Phys Rev Lett, 1985, 54(13): 1354-1356
[45] MAASSARANI Z, SERBAN D. Non-unitary conformal field theory and logarithmic operatorsfor disordered systems[J]. Nuclear Physics B, 1997, 489(3): 603-625
[46] BENDER C M, BOETTCHER S, JONES H F, et al. Bound states of non-Hermitian quantumfield theories[J]. Physics Letters A, 2001, 291(4): 197-202
[47] CASTRO-ALVAREDO O A, DOYON B, RAVANINI F. Irreversibility of the renormalizationgroup flow in non-unitary quantum field theory*[J]. J Phys A: Math Theor, 2017, 50(42):424002
[48] ALEXANDRE J, BENDER C M, MILLINGTON P. Light neutrino masses from a non-Hermitian Yukawa theory[J]. J Phys: Conf Ser, 2017, 873(1): 012047
[49] BIANCHINI D, CASTRO-ALVAREDO O A, DOYON B. Entanglement entropy of non-unitaryintegrable quantum field theory[J]. Nuclear Physics B, 2015, 896: 835-880
[50] BENDER C M, MILTON K A. A nonunitary version of massless quantum electrodynamicspossessing a critical point[J]. J Phys A: Math Gen, 1999, 32(7): L87
[51] FRING A, TAIRA T. Pseudo-Hermitian approach to Goldstone’s theorem in non-Abeliannon-Hermitian quantum field theories[J]. Phys Rev D, 2020, 101(4): 045014
[52] MOFFAT J W. Noncommutative quantum gravity[J]. Physics Letters B, 2000, 491(3): 345-352
[53] ARAGEORGIS A, EARMAN J, RUETSCHE L. Weyling the time away: the non-unitaryimplementability of quantum field dynamics on curved spacetime[J]. Studies in History andPhilosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2002, 33(2): 151-184
[54] MANNHEIM P. Astrophysical evidence for the non-Hermitian but PT-symmetric Hamiltonianof conformal gravity[J]. Fortschritte der Physik, 2013, 61(2-3): 140-154
[55] YEşILTAş ￿. Non-Hermitian Dirac Hamiltonian in Three-Dimensional Gravity and Pseudosupersymmetry[J]. Advances in High Energy Physics, 2015, 2015: e484151
[56] LV C, ZHANG R, ZHAI Z, et al. Curving the space by non-Hermiticity[J]. Nat Commun, 2022,13(1): 2184
[57] JU C Y, MIRANOWICZ A, MINGANTI F, et al. Einstein’s quantum elevator: Hermitizationof non-Hermitian Hamiltonians via a generalized vielbein formalism[J]. Phys Rev Research,2022, 4(2): 023070
[58] ASHIDA Y, GONG Z, UEDA M. Non-Hermitian physics[J]. Advances in Physics, 2020, 69(3): 249-435
[59] BENDER C M, BOETTCHER S. Real Spectra in Non-Hermitian Hamiltonians Having PTSymmetry[J]. Phys Rev Lett, 1998, 80: 5243-5246
[60] CHIU C K, TEO J C Y, SCHNYDER A P, et al. Classification of topological quantum matterwith symmetries[J]. Rev Mod Phys, 2016, 88(3): 035005
[61] CHEN X, GU Z C, LIU Z X, et al. Symmetry-Protected Topological Orders in InteractingBosonic Systems[J]. Science, 2012, 338(6114): 1604-1606
[62] CHEN X, GU Z C, LIU Z X, et al. Symmetry protected topological orders and the groupcohomology of their symmetry group[J]. Phys Rev B, 2013, 87(15): 155114
[63] GONG Z, ASHIDA Y, KAWABATA K, et al. Topological Phases of Non-Hermitian Systems[J]. Phys Rev X, 2018, 8: 031079
[64] KAWABATA K, SHIOZAKI K, UEDA M, et al. Symmetry and Topology in Non-HermitianPhysics[J]. Phys Rev X, 2019, 9: 041015
[65] LIU C H, JIANG H, CHEN S. Topological classification of non-Hermitian systems with reflectionsymmetry[J]. Phys Rev B, 2019, 99: 125103
[66] XI W, ZHANG Z H, GU Z C, et al. Classification of topological phases in one dimensionalinteracting non-Hermitian systems and emergent unitarity[J]. Science Bulletin, 2021, 66(17):1731-1739
[67] BERRY M. Physics of Nonhermitian Degeneracies[J]. Czechoslovak Journal of Physics, 2004,54(10): 1039-1047
[68] LEE C H. Exceptional Bound States and Negative Entanglement Entropy[J]. Phys Rev Lett,2022, 128: 010402
[69] YANG Z, SCHNYDER A P, HU J, et al. Fermion Doubling Theorems in Two-DimensionalNon-Hermitian Systems for Fermi Points and Exceptional Points[J]. Phys Rev Lett, 2021, 126:086401
[70] KOZII V, FU L. Non-Hermitian Topological Theory of Finite-Lifetime Quasiparticles: Predictionof Bulk Fermi Arc Due to Exceptional Point[A]. 2017. arXiv: 1708.05841
[71] TZENG Y C, JU C Y, CHEN G Y, et al. Hunting for the non-Hermitian exceptional points withfidelity susceptibility[J]. Phys Rev Res, 2021, 3: 013015
[72] MANDAL I, BERGHOLTZ E J. Symmetry and Higher-Order Exceptional Points[J]. Phys RevLett, 2021, 127: 186601
[73] TANG W, JIANG X, DING K, et al. Exceptional nexus with a hybrid topological invariant[J].Science, 2020, 370(6520): 1077-1080
[74] TANG W, DING K, MA G. Experimental realization of non-Abelian permutations in a threestatenon-Hermitian system[J]. National Science Review, 2022, 9(11)
[75] GUO C X, CHEN S, DING K, et al. Exceptional Non-Abelian Topology in Multiband Non-Hermitian Systems[A]. 2022. arXiv: 2210.17031
[76] ZHANG Q, ZHAO L, LIU X, et al. Experimental Characterization of Three-Band Braid Relationsin Non-Hermitian Acoustic Systems[A]. 2022. arXiv: 2212.07609
[77] DING K, MA G, XIAO M, et al. Emergence, Coalescence, and Topological Properties of MultipleExceptional Points and Their Experimental Realization[J]. Phys Rev X, 2016, 6: 021007
[78] HATANO N, NELSON D R. Vortex pinning and non-Hermitian quantum mechanics[J]. PhysRev B, 1997, 56: 8651-8673
[79] HATANO N, NELSON D R. Non-Hermitian delocalization and eigenfunctions[J]. Phys RevB, 1998, 58: 8384-8390
[80] LI L, LEE C H, GONG J. Topological Switch for Non-Hermitian Skin Effect in Cold-AtomSystems with Loss[J]. Phys Rev Lett, 2020, 124: 250402
[81] GUO C X, LIU C H, ZHAO X M, et al. Exact Solution of Non-Hermitian Systems with GeneralizedBoundary Conditions: Size-Dependent Boundary Effect and Fragility of the Skin Effect[J]. Phys Rev Lett, 2021, 127: 116801
[82] SONG F, YAO S, WANG Z. Non-Hermitian Topological Invariants in Real Space[J]. Phys RevLett, 2019, 123(24): 246801
[83] SU W P, SCHRIEFFER J R, HEEGER A J. Soliton excitations in polyacetylene[J]. Phys RevB, 1980, 22: 2099-2111
[84] SU W P, SCHRIEFFER J R, HEEGER A J. Solitons in Polyacetylene[J]. Phys Rev Lett, 1979,42: 1698-1701
[85] KUNST F K, EDVARDSSON E, BUDICH J C, et al. Biorthogonal Bulk-Boundary Correspondencein Non-Hermitian Systems[J]. Phys Rev Lett, 2018, 121: 026808
[86] YANG Z, ZHANG K, FANG C, et al. Non-Hermitian Bulk-Boundary Correspondence andAuxiliary Generalized Brillouin Zone Theory[J]. Phys Rev Lett, 2020, 125: 226402
[87] REITHMAIER J P, SęK G, LöFFLER A, et al. Strong coupling in a single quantum dot–semiconductor microcavity system[J]. Nature, 2004, 432(7014): 197-200
[88] ZHANG W, OUYANG X, HUANG X, et al. Observation of Non-Hermitian Topology withNonunitary Dynamics of Solid-State Spins[J]. Phys Rev Lett, 2021, 127: 090501
[89] WU Y, LIU W, GENG J, et al. Observation of parity-time symmetry breaking in a single-spinsystem[J]. Science, 2019, 364(6443): 878-880
[90] NAKAMURA Y, PASHKIN Y A, TSAI J S. Coherent control of macroscopic quantum statesin a single-Cooper-pair box[J]. Nature, 1999, 398(6730): 786-788
[91] GUTHRIE A, SATRYA C D, CHANG Y C, et al. Cooper-Pair Box Coupled to Two Resonators:An Architecture for a Quantum Refrigerator[J]. Phys Rev Applied, 2022, 17(6): 064022
[92] FITZPATRICK M, SUNDARESAN N M, LI A C Y, et al. Observation of a Dissipative PhaseTransition in a One-Dimensional Circuit QED Lattice[J]. Phys Rev X, 2017, 7: 011016
[93] HUANG Y, YIN Z Q, YANG W L. Realizing a topological transition in a non-Hermitian quantumwalk with circuit QED[J]. Phys Rev A, 2016, 94: 022302
[94] CALDEIRA A O, LEGGETT A J. Quantum tunnelling in a dissipative system[J]. Annals ofPhysics, 1983, 149(2): 374-456
[95] COULE D H, MARTIN J. Quantum cosmology and open universes[J]. Phys Rev D, 2000, 61(6): 063501
[96] SHANDERA S, AGARWAL N, KAMAL A. Open quantum cosmological system[J]. Phys RevD, 2018, 98(8): 083535
[97] GORINI V, KOSSAKOWSKI A, SUDARSHAN E C G. Completely positive dynamical semigroupsof N‐level systems[J]. Journal of Mathematical Physics, 1976, 17(5): 821-825
[98] LINDBLAD G. On the generators of quantum dynamical semigroups[J]. Comm Math Phys,1976, 48(2): 119-130
[99] WEIMER H, KSHETRIMAYUM A, ORúS R. Simulation methods for open quantum manybodysystems[J]. Rev Mod Phys, 2021, 93(1): 015008
[100] BENGURIA R, KAC M. Quantum Langevin Equation[J]. Phys Rev Lett, 1981, 46(1): 1-4
[101] FORD G W, LEWIS J T, O’CONNELL R F. Quantum Langevin equation[J]. Phys Rev A, 1988,37(11): 4419-4428
[102] PAN L, CHEN X, CHEN Y, et al. Non-Hermitian linear response theory[J]. Nature Physics,2020, 16(7): 767–771
[103] WISEMAN H, MILBURN G. Quantum Measurement and Control[M]. Cambridge: CambridgeUniversity Press, 2010
[104] BELLOMO B, LO FRANCO R, COMPAGNO G. Non-Markovian Effects on the Dynamics ofEntanglement[J]. Phys Rev Lett, 2007, 99(16): 160502
[105] PIILO J, MANISCALCO S, HÄRKÖNEN K, et al. Non-Markovian Quantum Jumps[J]. PhysRev Lett, 2008, 100(18): 180402
[106] DE VEGA I, ALONSO D. Dynamics of non-Markovian open quantum systems[J]. Rev ModPhys, 2017, 89(1): 015001
[107] HORODECKI R, HORODECKI P, HORODECKI M, et al. Quantum entanglement[J]. RevMod Phys, 2009, 81: 865-942
[108] LAFLORENCIE N. Quantum entanglement in condensed matter systems[J]. Physics Reports,2016, 646: 1-59
[109] ZENG B, CHEN X, ZHOU D L, et al. Quantum information meets quantum matter[M].Springer, 2019
[110] PEREZ-GARCIA D, VERSTRAETE F, WOLF M M, et al. Matrix Product State Representations[M]. arXiv, 2007
[111] ORúS R. A practical introduction to tensor networks: Matrix product states and projectedentangled pair states[J]. Annals of Physics, 2014, 349: 117-158
[112] NISHIOKA T. Entanglement entropy: Holography and renormalization group[J]. Rev ModPhys, 2018, 90: 035007
[113] EISERT J, CRAMER M, PLENIO M B. Colloquium: Area laws for the entanglement entropy[J]. Rev Mod Phys, 2010, 82: 277-306
[114] NIELSEN M A, CHUANG I L. Quantum Computation and Quantum Information: 10th AnniversaryEdition[M]. Cambridge University Press, 2010
[115] WOLF M M, VERSTRAETE F, HASTINGS M B, et al. Area Laws in Quantum Systems:Mutual Information and Correlations[J]. Phys Rev Lett, 2008, 100(7): 070502
[116] EISERT J, CRAMER M, PLENIO M B. Colloquium : Area laws for the entanglement entropy[J]. Rev Mod Phys, 2010, 82(1): 277-306
[117] BRANDãO F G S L, HORODECKI M. An area law for entanglement from exponential decayof correlations[J]. Nature Phys, 2013, 9(11): 721-726
[118] BRAVYI S, HASTINGS M B, VERSTRAETE F. Lieb-Robinson Bounds and the Generationof Correlations and Topological Quantum Order[J]. Phys Rev Lett, 2006, 97(5): 050401
[119] VERSTRAETE F, MURG V, CIRAC J. Matrix product states, projected entangled pair states,and variational renormalization group methods for quantum spin systems[J]. Advances inPhysics, 2008, 57(2): 143-224
[120] CIRAC J I, PéREZ-GARCíA D, SCHUCH N, et al. Matrix product states and projected entangledpair states: Concepts, symmetries, theorems[J]. Rev Mod Phys, 2021, 93(4): 045003
[121] VERSTRAETE F, WOLF M M, PEREZ-GARCIA D, et al. Criticality, the Area Law, andthe Computational Power of Projected Entangled Pair States[J]. Phys Rev Lett, 2006, 96(22):220601
[122] CINCIO L, DZIARMAGA J, RAMS M M. Multiscale Entanglement Renormalization Ansatzin Two Dimensions: Quantum Ising Model[J]. Phys Rev Lett, 2008, 100(24): 240603
[123] EVENBLY G, VIDAL G. Quantum Criticality with the Multi-scale Entanglement RenormalizationAnsatz[M]//AVELLA A, MANCINI F. Springer Series in Solid-State Sciences: StronglyCorrelated Systems: Numerical Methods. Berlin, Heidelberg: Springer, 2013: 99-130
[124] SWINGLE B. Entanglement Entropy and the Fermi Surface[J]. Phys Rev Lett, 2010, 105(5):050502
[125] OGAWA N, TAKAYANAGI T, UGAJIN T. Holographic Fermi surfaces and entanglemententropy[J]. J High Energ Phys, 2012, 2012(1): 125
[126] LUITZ D J, PLAT X, ALET F, et al. Universal logarithmic corrections to entanglement entropiesin two dimensions with spontaneously broken continuous symmetries[J]. Phys Rev B, 2015, 91:155145
[127] KITAEV A, PRESKILL J. Topological Entanglement Entropy[J]. Phys Rev Lett, 2006, 96(11):110404
[128] LEVIN M, WEN X G. Detecting Topological Order in a Ground State Wave Function[J]. PhysRev Lett, 2006, 96(11): 110405
[129] LUNDGREN R, CHUA V, FIETE G A. Entanglement entropy and spectra of the onedimensionalKugel-Khomskii model[J]. Phys Rev B, 2012, 86(22): 224422
[130] WOLF M M. Violation of the Entropic Area Law for Fermions[J]. Phys Rev Lett, 2006, 96(1):010404
[131] LAFLORENCIE N. Scaling of entanglement entropy in the random singlet phase[J]. Phys RevB, 2005, 72(14): 140408
[132] REFAEL G, MOORE J E. Entanglement Entropy of Random Quantum Critical Points in OneDimension[J]. Phys Rev Lett, 2004, 93(26): 260602
[133] FAGOTTI M, CALABRESE P, MOORE J E. Entanglement spectrum of random-singlet quantumcritical points[J]. Phys Rev B, 2011, 83(4): 045110
[134] LIU H, SUH S J. Entanglement Tsunami: Universal Scaling in Holographic Thermalization[J].Phys Rev Lett, 2014, 112(1): 011601
[135] D’ALESSIO L, KAFRI Y, POLKOVNIKOV A, et al. From quantum chaos and eigenstatethermalization to statistical mechanics and thermodynamics[J]. Advances in Physics, 2016, 65(3): 239-362
[136] KAUFMAN A M, TAI M E, LUKIN A, et al. Quantum thermalization through entanglementin an isolated many-body system[J]. Science, 2016, 353(6301): 794-800
[137] ABANIN D A, ALTMAN E, BLOCH I, et al. Colloquium : Many-body localization, thermalization,and entanglement[J]. Rev Mod Phys, 2019, 91(2): 021001
[138] PERES A. Separability Criterion for Density Matrices[J]. Phys Rev Lett, 1996, 77: 1413-1415
[139] PLENIO M B. Logarithmic Negativity: A Full Entanglement Monotone That is not Convex[J].Phys Rev Lett, 2005, 95: 090503
[140] CHUNG M C, PESCHEL I. Density-matrix spectra of solvable fermionic systems[J]. Phys RevB, 2001, 64(6): 064412
[141] PESCHEL I. Calculation of reduced density matrices from correlation functions[J]. Journal ofPhysics A: Mathematical and General, 2003, 36(14): L205-L208
[142] PESCHEL I, EISLER V. Reduced density matrices and entanglement entropy in free latticemodels[J]. Journal of Physics A: Mathematical and Theoretical, 2009, 42(50): 504003
[143] GRAY R M, DAVISSON L D. The Society for Industrial and Applied Mathematics: Introductionto Toeplitz and Circulant Matrices[M]. SIAM, 2006
[144] JIN B Q, KOREPIN V E. Quantum Spin Chain, Toeplitz Determinants and the Fisher—HartwigConjecture[J]. Journal of Statistical Physics, 2004, 116(1): 79-95
[145] BASOR E L, TRACY C A. The Fisher-Hartwig conjecture and generalizations[J]. Physica A:Statistical Mechanics and its Applications, 1991, 177(1): 167-173
[146] LUTTINGER J M. An Exactly Soluble Model of a Many‐Fermion System[J]. Journal ofMathematical Physics, 1963, 4(9): 1154-1162
[147] COLEMAN S. Quantum sine-Gordon equation as the massive Thirring model[J]. Phys Rev D,1975, 11(8): 2088-2097
[148] MATTIS D C. Theory of quasimagnetic impurity in a metal: An exactly soluble model ofinteracting electrons[J]. Annals of Physics, 1975, 89(1): 45-67
[149] HALDANE F D M. ’Luttinger liquid theory’ of one-dimensional quantum fluids. I. Propertiesof the Luttinger model and their extension to the general 1D interacting spinless Fermi gas[J].Journal of Physics C: Solid State Physics, 1981, 14(19): 2585
[150] EFETOV K B, LARKIN A I. Nonzero spin pairing in layer superconductors and in quasi-onedimensionalsuperconductors[J]. Zh Eksp Teor Fiz, v 68, no 1, pp 155-163, 1975
[151] SéNéCHAL D. An Introduction to Bosonization[M]//SéNéCHAL D, TREMBLAY A M,BOURBONNAIS C. CRM Series in Mathematical Physics: Theoretical Methods for StronglyCorrelated Electrons. New York, NY: Springer, 2004: 139-186
[152] TSVELIK A M. Cambridge Monographs on Mathematical Physics: Bosonization of StronglyCorrelated Systems[M]. Cambridge University Press, 1998
[153] 易伟柱, 奚文杰, 陈伟强. 重整化平均场理论及其在铜氧化物高温超导材料中的应用[J].低温物理学报, 2020, 42(2): 59-67
[154] DI FRANCESCO P, MATHIEU P, SéNéCHAL D. Graduate Texts in Contemporary Physics:Conformal Field Theory[M]. Springer, 1997
[155] BELAVIN A A, POLYAKOV A M, ZAMOLODCHIKOV A B. Infinite conformal symmetryin two-dimensional quantum field theory[J]. Nuclear Physics B, 1984, 241(2): 333-380
[156] CARDY J L. Conformal invariance and surface critical behavior[J]. Nuclear Physics B, 1984,240(4): 514-532
[157] CARDY J L. Conformal invariance and universality in finite-size scaling[J]. Journal of PhysicsA: Mathematical and General, 1984, 17(7): L385
[158] CARDY J. Cambridge Lecture Notes in Physics: Scaling and Renormalization in StatisticalPhysics[M]. Cambridge University Press, 1996
[159] BAXTER R J. Exactly Solved Models in Statistical Mechanics[M]. Academic Press, 1982
[160] MUSSARDO G. Oxford Graduate Texts: Statistical Field Theory: An Introduction to ExactlySolved Models in Statistical Physics[M]. Oxford University Press, 2010
[161] POLCHINSKI J. String Theory[M]. Cambridge University Press, 1998
[162] NIELSEN H, NINOMIYA M. A no-go theorem for regularizing chiral fermions[J]. PhysicsLetters B, 1981, 105(2): 219 - 223
[163] ADLER S L. Axial-Vector Vertex in Spinor Electrodynamics[J]. Phys Rev, 1969, 177: 2426-2438
[164] BELL J S, JACKIW R W. A PCAC puzzle: 𝜋0 → 𝛾𝛾 in the 𝜎-model[J]. Nuovo Cimento, 1969,60(CERN-TH-920): 47-61
[165] ALVAREZ-GAUMé L, WITTEN E. Gravitational anomalies[J]. Nuclear Physics B, 1984, 234(2): 269 - 330
[166] STONE M. Edge waves in the quantum Hall effect[J]. Annals of Physics, 1991, 207(1): 38 -52
[167] WILSON K. Erice lecture notes, 1975[J]. Susskind, Lectures at Les Houches Summer School,1976
[168] SUSSKIND L. Lattice fermions[J]. Phys Rev D, 1977, 16: 3031-303993参考文献
[169] DRELL S D, WEINSTEIN M, YANKIELOWICZ S. Strong-coupling field theory. I. Variationalapproach to 𝜑4 theory[J]. Phys Rev D, 1976, 14: 487-516
[170] DEMARCO M, WEN X G. A Single Right-Moving Free Fermion Mode on an Ultra-Local1 + 1 d Spacetime Lattice[A]. 2018. arXiv:1805.03663
[171] WANG J, WEN X G. Solution to the 1 + 1 dimensional gauged chiral Fermion problem[J].Phys Rev D, 2019, 99: 111501
[172] KIKUKAWA Y. Why is the mission impossible? Decoupling the mirror Ginsparg–Wilsonfermions in the lattice models for two-dimensional Abelian chiral gauge theories[J]. Progressof Theoretical and Experimental Physics, 2019, 2019(7)
[173] WANG J, WEN X G. Non-Perturbative Regularization of 1+ 1D Anomaly-Free Chiral Fermionsand Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules[A]. 2013. arXiv:1307.7480
[174] DEMARCO M, WEN X G. A Novel Non-Perturbative Lattice Regularization of an Anomaly-Free 1 + 1𝑑 Chiral 𝑆𝑈(2) Gauge Theory[A]. 2017. arXiv:1706.04648
[175] DEMARCO M A. Chiral Phases on the Lattice[A]. 2022. arXiv:2203.01427
[176] LEE J Y, AHN J, ZHOU H, et al. Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics[J]. Phys Rev Lett, 2019, 123: 206404
[177] LEE T E. Anomalous Edge State in a Non-Hermitian Lattice[J]. Phys Rev Lett, 2016, 116:133903
[178] LEYKAM D, BLIOKH K Y, HUANG C, et al. Edge Modes, Degeneracies, and TopologicalNumbers in Non-Hermitian Systems[J]. Phys Rev Lett, 2017, 118: 040401
[179] YAO S, WANG Z. Edge States and Topological Invariants of Non-Hermitian Systems[J]. PhysRev Lett, 2018, 121: 086803
[180] KAWABATA K, BESSHO T, SATO M. Classification of Exceptional Points and Non-HermitianTopological Semimetals[J]. Phys Rev Lett, 2019, 123: 066405
[181] ZHANG X X, FRANZ M. Non-Hermitian Exceptional Landau Quantization in Electric Circuits[J]. Phys Rev Lett, 2020, 124: 046401
[182] BORGNIA D S, KRUCHKOV A J, SLAGER R J. Non-Hermitian Boundary Modes and Topology[J]. Phys Rev Lett, 2020, 124: 056802
[183] LONGHI S. Non-Bloch-Band Collapse and Chiral Zener Tunneling[J]. Phys Rev Lett, 2020,124: 066602
[184] LEE T E, CHAN C K. Heralded Magnetism in Non-Hermitian Atomic Systems[J]. Phys RevX, 2014, 4: 041001
[185] PESKIN M E. An introduction to quantum field theory[M]. CRC press, 2018
[186] BERTLMANN R A. Anomalies in quantum field theory: Vol. 91[M]. Oxford university press,2000
[187] FUJIKAWA K. Path-Integral Measure for Gauge-Invariant Fermion Theories[J]. Phys Rev Lett,1979, 42: 1195-1198
[188] BLUMENHAGEN R, PLAUSCHINN E. Introduction to conformal field theory: with applicationsto string theory: Vol. 779[M]. Springer Science & Business Media, 2009
[189] KARSTEN L H. Lattice fermions in euclidean space-time[J]. Physics Letters B, 1981, 104(4):315 - 319
[190] CHERNODUB M N. The Nielsen–Ninomiya theorem, PT-invariant non-Hermiticity and single8-shaped Dirac cone[J]. Journal of Physics A: Mathematical and Theoretical, 2017, 50(38):385001
[191] BRODY D C. Biorthogonal quantum mechanics[J]. Journal of Physics A: Mathematical andTheoretical, 2013, 47(3): 035305
[192] FRANCESCO P, MATHIEU P, SÉNÉCHAL D. Conformal field theory[M]. Springer Science& Business Media, 2012
[193] GONG Z, ASHIDA Y, KAWABATA K, et al. Topological Phases of Non-Hermitian Systems[J]. Phys Rev X, 2018, 8: 031079
[194] EZAWA M. Non-Hermitian higher-order topological states in nonreciprocal and reciprocalsystems with their electric-circuit realization[J]. Phys Rev B, 2019, 99: 201411
[195] LIU S, MA S, YANG C, et al. Gain- and Loss-Induced Topological Insulating Phase in a Non-Hermitian Electrical Circuit[J]. Phys Rev Applied, 2020, 13: 014047
[196] KATO T. Perturbation theory for linear operators[M]. Springer Science & Business Media,2013
[197] BERRY M. Physics of Nonhermitian Degeneracies[J]. Czechoslovak Journal of Physics, 2004,54(10): 1039-1047
[198] HEISS W D. Exceptional points of non-Hermitian operators[J]. J Phys A: Math Gen, 2004, 37(6): 2455-2464
[199] ASHIDA Y, GONG Z, UEDA M. Non-Hermitian physics[J]. Advances in Physics, 2020, 69(3): 249-435
[200] HEISS W D. The physics of exceptional points[J]. J Phys A: Math Theor, 2012, 45(44): 444016
[201] DING K, MA G, XIAO M, et al. Emergence, Coalescence, and Topological Properties of MultipleExceptional Points and Their Experimental Realization[J]. Phys Rev X, 2016, 6: 021007
[202] WU Q, SOLUYANOV A A, BZDUšEK T. Non-Abelian band topology in noninteracting metals[J]. Science, 2019, 365(6459): 1273-1277
[203] MIRI M A, ALù A. Exceptional points in optics and photonics[J]. Science, 2019, 363(6422):eaar7709
[204] ÖZDEMIR ￿ K, ROTTER S, NORI F, et al. Parity–time symmetry and exceptional points inphotonics[J]. Nat Mater, 2019, 18(8): 783-798
[205] KAWABATA K, SHIOZAKI K, UEDA M, et al. Symmetry and Topology in Non-HermitianPhysics[J]. Phys Rev X, 2019, 9(4): 041015
[206] LAI Y H, LU Y K, SUH M G, et al. Observation of the exceptional-point-enhanced Sagnaceffect[J]. Nature, 2019, 576(7785): 65-69
[207] KAWABATA K, ASHIDA Y, UEDA M. Information Retrieval and Criticality in Parity-Time-Symmetric Systems[J]. Phys Rev Lett, 2017, 119(19): 190401
[208] DóRA B, HEYL M, MOESSNER R. The Kibble-Zurek mechanism at exceptional points[J].Nat Commun, 2019, 10(1): 2254
[209] LIAO Q, LEBLANC C, REN J, et al. Experimental Measurement of the Divergent QuantumMetric of an Exceptional Point[J]. Phys Rev Lett, 2021, 127(10): 107402
[210] HU H, ZHAO E. Knots and Non-Hermitian Bloch Bands[J]. Phys Rev Lett, 2021, 126: 010401
[211] YANG Z, CHIU C K, FANG C, et al. Jones Polynomial and Knot Transitions in Hermitian andnon-Hermitian Topological Semimetals[J]. Phys Rev Lett, 2020, 124: 186402
[212] TANG W, JIANG X, DING K, et al. Exceptional nexus with a hybrid topological invariant[J].Science, 2020, 370(6520): 1077-1080
[213] ZHANG G Q, CHEN Z, XU D, et al. Exceptional Point and Cross-Relaxation Effect in a HybridQuantum System[J]. PRX Quantum, 2021, 2: 020307
[214] GAO T, ESTRECHO E, BLIOKH K Y, et al. Observation of non-Hermitian degeneracies in achaotic exciton-polariton billiard[J]. Nature, 2015, 526(7574): 554-558
[215] HEISS D. Circling exceptional points[J]. Nature Phys, 2016, 12(9): 823-824
[216] HEISS W D, MüLLER M, ROTTER I. Collectivity, phase transitions, and exceptional pointsin open quantum systems[J]. Phys Rev E, 1998, 58(3): 2894-2901
[217] DOPPLER J, MAILYBAEV A A, BöHM J, et al. Dynamically encircling an exceptional pointfor asymmetric mode switching[J]. Nature, 2016, 537(7618): 76-79
[218] HASSAN A U, ZHEN B, SOLJAčIć M, et al. Dynamically Encircling Exceptional Points:Exact Evolution and Polarization State Conversion[J]. Phys Rev Lett, 2017, 118(9): 093002
[219] DEMBOWSKI C, GRÄF H D, HARNEY H L, et al. Experimental Observation of the TopologicalStructure of Exceptional Points[J]. Phys Rev Lett, 2001, 86: 787-790
[220] LI A, DONG J, WANG J, et al. Hamiltonian Hopping for Efficient Chiral Mode Switching inEncircling Exceptional Points[J]. Phys Rev Lett, 2020, 125: 187403
[221] MILBURN T J, DOPPLER J, HOLMES C A, et al. General description of quasiadiabaticdynamical phenomena near exceptional points[J]. Phys Rev A, 2015, 92: 052124
[222] YU F, ZHANG X L, TIAN Z N, et al. General Rules Governing the Dynamical Encircling ofan Arbitrary Number of Exceptional Points[J]. Phys Rev Lett, 2021, 127: 253901
[223] LIU Q, LI S, WANG B, et al. Efficient Mode Transfer on a Compact Silicon Chip by EncirclingMoving Exceptional Points[J]. Phys Rev Lett, 2020, 124: 153903
[224] KLAIMAN S, GüNTHER U, MOISEYEV N. Visualization of Branch Points in P T -SymmetricWaveguides[J]. Phys Rev Lett, 2008, 101(8): 080402
[225] ZHEN B, HSU C W, IGARASHI Y, et al. Spawning rings of exceptional points out of Diraccones[J]. Nature, 2015, 525(7569): 354-358
[226] CERJAN A, RAMAN A, FAN S. Exceptional Contours and Band Structure Design in Parity-Time Symmetric Photonic Crystals[J]. Phys Rev Lett, 2016, 116(20): 20390296参考文献
[227] LONGHI S. Spectral singularities and Bragg scattering in complex crystals[J]. Phys Rev A,2010, 81(2): 022102
[228] FENG L, WONG Z J, MA R M, et al. Single-mode laser by parity-time symmetry breaking[J].Science, 2014, 346(6212): 972-975
[229] PENG B, ÖZDEMIR ￿ K, ROTTER S, et al. Loss-induced suppression and revival of lasing[J]. Science, 2014, 346(6207): 328-332
[230] LIU W, WU Y, DUAN C K, et al. Dynamically Encircling an Exceptional Point in a RealQuantum System[J]. Phys Rev Lett, 2021, 126: 170506
[231] DING L, SHI K, ZHANG Q, et al. Experimental Determination of 𝒫𝒯-Symmetric ExceptionalPoints in a Single Trapped Ion[J]. Phys Rev Lett, 2021, 126: 083604
[232] NAGHILOO M, ABBASI M, JOGLEKAR Y N, et al. Quantum state tomography across theexceptional point in a single dissipative qubit[J]. Nat Phys, 2019, 15(12): 1232-1236
[233] ASHIDA Y, UEDA M. Full-Counting Many-Particle Dynamics: Nonlocal and Chiral Propagationof Correlations[J]. Phys Rev Lett, 2018, 120(18): 185301
[234] DEMBOWSKI C, DIETZ B, GRÄF H D, et al. Encircling an exceptional point[J]. Phys RevE, 2004, 69: 056216
[235] WONG Z J, XU Y L, KIM J, et al. Lasing and anti-lasing in a single cavity[J]. Nature Photon,2016, 10(12): 796-801
[236] HASSANI GANGARAJ S A, MONTICONE F. Topological Waveguiding near an ExceptionalPoint: Defect-Immune, Slow-Light, and Loss-Immune Propagation[J]. Phys Rev Lett, 2018,121(9): 093901
[237] CHEN W, KAYA ÖZDEMIR Ş, ZHAO G, et al. Exceptional points enhance sensing in anoptical microcavity[J]. Nature, 2017, 548(7666): 192-196
[238] HODAEI H, HASSAN A U, WITTEK S, et al. Enhanced sensitivity at higher-order exceptionalpoints[J]. Nature, 2017, 548(7666): 187-191
[239] ZHANG M, SWEENEY W, HSU C W, et al. Quantum Noise Theory of Exceptional PointAmplifying Sensors[J]. Phys Rev Lett, 2019, 123: 180501
[240] XU Y, WANG S T, DUAN L M. Weyl Exceptional Rings in a Three-Dimensional DissipativeCold Atomic Gas[J]. Phys Rev Lett, 2017, 118(4): 045701
[241] TANG W, DING K, MA G. Direct Measurement of Topological Properties of an ExceptionalParabola[J]. Phys Rev Lett, 2021, 127: 034301
[242] DING K, MA G, ZHANG Z Q, et al. Experimental Demonstration of an Anisotropic ExceptionalPoint[J]. Phys Rev Lett, 2018, 121: 085702
[243] VIDAL G, LATORRE J I, RICO E, et al. Entanglement in Quantum Critical Phenomena[J].Phys Rev Lett, 2003, 90(22): 227902
[244] CALABRESE P, CARDY J. Entanglement entropy and quantum field theory[J]. J Stat Mech:Theor Exp, 2004, 2004(06): P06002
[245] CALABRESE P, CARDY J. Entanglement entropy and conformal field theory[J]. Journal ofPhysics A: Mathematical and Theoretical, 2009, 42(50): 504005
[246] HERVIOU L, REGNAULT N, BARDARSON J H. Entanglement spectrum and symmetries innon-Hermitian fermionic non-interacting models[J]. SciPost Phys, 2019, 7(5): 069
[247] GUO Y B, YU Y C, HUANG R Z, et al. Entanglement entropy of non-Hermitian free fermions[J]. J Phys: Condens Matter, 2021, 33(47): 475502
[248] BIANCHINI D, CASTRO-ALVAREDO O, DOYON B, et al. Entanglement entropy of nonunitaryconformal field theory[J]. Journal of Physics A: Mathematical and Theoretical, 2014,48(4): 04FT01
[249] LOOTENS L, VANHOVE R, HAEGEMAN J, et al. Galois Conjugated Tensor Fusion Categoriesand Nonunitary Conformal Field Theory[J]. Phys Rev Lett, 2020, 124(12): 120601
[250] COUVREUR R, JACOBSEN J L, SALEUR H. Entanglement in Nonunitary Quantum CriticalSpin Chains[J]. Phys Rev Lett, 2017, 119(4): 040601
[251] XU W T, SCHUCH N. Characterization of topological phase transitions from a non-Abeliantopological state and its Galois conjugate through condensation and confinement order parameters[J]. Phys Rev B, 2021, 104(15): 155119
[252] BIANCHINI D, RAVANINI F. Entanglement entropy from corner transfer matrix in Forrester–Baxter non-unitary RSOS models[J]. Journal of Physics A: Mathematical and Theoretical,2016, 49(15): 154005
[253] CHANG P Y, YOU J S, WEN X, et al. Entanglement spectrum and entropy in topological non-Hermitian systems and nonunitary conformal field theory[J]. Phys Rev Research, 2020, 2(3):033069
[254] CHEN C Y, LAO B X, YU X Y, et al. Galois conjugates of string-net model[J]. Phys Rev D,2022, 105: 065009
[255] YARKONY D R. Diabolical conical intersections[J]. Rev Mod Phys, 1996, 68: 985-1013
[256] KAUSCH H G. Curiosities at c=-2[A]. 1995
[257] SCAFFIDI T, PARKER D E, VASSEUR R. Gapless Symmetry-Protected Topological Order[J]. Phys Rev X, 2017, 7: 041048
[258] VERRESEN R, THORNGREN R, JONES N G, et al. Gapless Topological Phases andSymmetry-Enriched Quantum Criticality[J]. Phys Rev X, 2021, 11: 041059
[259] YIN C, JIANG H, LI L, et al. Geometrical meaning of winding number and its characterizationof topological phases in one-dimensional chiral non-Hermitian systems[J]. Phys Rev A, 2018,97: 052115
[260] AROUCA R, LEE C H, MORAIS SMITH C. Unconventional scaling at non-Hermitian criticalpoints[J]. Phys Rev B, 2020, 102(24): 245145
[261] FRUCHART M, HANAI R, LITTLEWOOD P B, et al. Non-reciprocal phase transitions[J].Nature, 2021, 592(7854): 363-369
[262] BENINI F, IOSSA C, SERONE M. Conformality Loss, Walking, and 4D Complex ConformalField Theories at Weak Coupling[J]. Phys Rev Lett, 2020, 124(5): 051602
[263] GORBENKO V, RYCHKOV S, ZAN B. Walking, weak first-order transitions, and complexCFTs[J]. J High Energ Phys, 2018, 2018(10): 108
[264] KAPLAN D B, LEE J W, SON D T, et al. Conformality lost[J]. Phys Rev D, 2009, 80(12):125005
[265] MA H, HE Y C. Shadow of complex fixed point: Approximate conformality of Q > 4 Pottsmodel[J]. Phys Rev B, 2019, 99(19): 195130
[266] MICHISHITA Y, PETERS R. Equivalence of Effective Non-Hermitian Hamiltonians in theContext of Open Quantum Systems and Strongly Correlated Electron Systems[J]. Phys RevLett, 2020, 124(19): 196401
[267] CHEN W Q, WU Y S, XI W, et al. Fate of Quantum Anomalies for 1d lattice chiral fermionwith a simple non-Hermitian Hamiltonian[J]. Journal of High Energy Physics, 2023, 2023(5):90
[268] ZHANG C, LEVIN M. Exactly Solvable Model for a Deconfined Quantum Critical Point in1D[J]. Phys Rev Lett, 2023, 130(2): 026801
[269] HUSE D A, FISHER M E. Commensurate melting, domain walls, and dislocations[J]. PhysRev B, 1984, 29(1): 239-270
[270] NYCKEES S, MILA F. Commensurate-incommensurate transition in the chiral Ashkin-Tellermodel[J]. Phys Rev Res, 2022, 4: 013093
[271] WHITSITT S, SAMAJDAR R, SACHDEV S. Quantum field theory for the chiral clock transitionin one spatial dimension[J]. Phys Rev B, 2018, 98: 205118
[272] SENTHIL T, VISHWANATH A, BALENTS L, et al. Deconfined Quantum Critical Points[J].Science, 2004, 303(5663): 1490-1494
[273] JIANG S, MOTRUNICH O. Ising ferromagnet to valence bond solid transition in a onedimensionalspin chain: Analogies to deconfined quantum critical points[J]. Phys Rev B, 2019,99(7): 075103
[274] WANG C, NAHUM A, METLITSKI M A, et al. Deconfined Quantum Critical Points: Symmetriesand Dualities[J]. Phys Rev X, 2017, 7(3): 031051
[275] ROBERTS B, JIANG S, MOTRUNICH O I. Deconfined quantum critical point in one dimension[J]. Phys Rev B, 2019, 99(16): 165143
[276] HUANG R Z, LU D C, YOU Y Z, et al. Emergent symmetry and conserved current at a onedimensionalincarnation of deconfined quantum critical point[J]. Phys Rev B, 2019, 100(12):125137
[277] LIU W Y, HASIK J, GONG S S, et al. Emergence of Gapless Quantum Spin Liquid fromDeconfined Quantum Critical Point[J]. Phys Rev X, 2022, 12(3): 031039
[278] YI W Z, LIN H J, LIN Z X, et al. Temporal evolution of one-dimensional fermion liquid withparticle loss[A]. 2021
[279] TURNER A M, POLLMANN F, BERG E. Topological phases of one-dimensional fermions:An entanglement point of view[J]. Phys Rev B, 2011, 83: 075102
[280] TURNER A M, ZHANG Y, VISHWANATH A. Band Topology of Insulators via the EntanglementSpectrum[A]. 2010. arXiv: 0909.3119
[281] WYBO E, POLLMANN F, SONDHI S L, et al. Visualizing quasiparticles from quantum entanglementfor general one-dimensional phases[J]. Phys Rev B, 2021, 103(11): 11512099参考文献
[282] LEE C H. Exceptional Bound States and Negative Entanglement Entropy[J]. Phys Rev Lett,2022, 128: 010402
[283] GEHLEN G V. Critical and off-critical conformal analysis of the Ising quantum chain in animaginary field[J]. J Phys A: Math Gen, 1991, 24(22): 5371-5399
[284] GEHLEN G V. Non-hermitian tricriticality in the blume?capel model with imaginary field[M]//Perspectives on Solvable Models. WORLD SCIENTIFIC, 1995: 59-81
[285] KORFF C. PTsymmetry of the non-Hermitian XX spin-chain: non-local bulk interaction fromcomplex boundary fields[J]. J Phys A: Math Theor, 2008, 41(29): 295206
[286] CARLSTRÖM J, STÅLHAMMAR M, BUDICH J C, et al. Knotted non-Hermitian metals[J].Phys Rev B, 2019, 99: 161115
[287] ASHIDA Y, FURUKAWA S, UEDA M. Parity-time-symmetric quantum critical phenomena[J]. Nat Commun, 2017, 8(1): 15791
[288] LI L, LEE C H, GONG J. Topological Switch for Non-Hermitian Skin Effect in Cold-AtomSystems with Loss[J]. Phys Rev Lett, 2020, 124(25): 250402
[289] PAN L, CHEN X, CHEN Y, et al. Non-Hermitian linear response theory[J]. Nat Phys, 2020,16(7): 767-771
[290] ALTLAND A, FLEISCHHAUER M, DIEHL S. Symmetry Classes of Open Fermionic QuantumMatter[J]. Phys Rev X, 2021, 11(2): 021037
[291] LIEU S, MCGINLEY M, COOPER N R. Tenfold Way for Quadratic Lindbladians[J]. PhysRev Lett, 2020, 124(4): 040401
[292] YU L W, DENG D L. Unsupervised Learning of Non-Hermitian Topological Phases[J]. PhysRev Lett, 2021, 126(24): 240402
[293] GONG Z, ASHIDA Y, KAWABATA K, et al. Topological Phases of Non-Hermitian Systems[J]. Phys Rev X, 2018, 8(3): 031079
[294] HARARI G, BANDRES M A, LUMER Y, et al. Topological insulator laser: Theory[J]. Science,2018, 359(6381): eaar4003
[295] HATANO N, NELSON D R. Localization Transitions in Non-Hermitian Quantum Mechanics[J]. Phys Rev Lett, 1996, 77: 570-573
[296] WANG X R, GUO C X, KOU S P. Defective edge states and number-anomalous bulk-boundarycorrespondence in non-Hermitian topological systems[J]. Phys Rev B, 2020, 101: 121116
[297] HU B, ZHANG Z, ZHANG H, et al. Non-Hermitian topological whispering gallery[J]. Nature,2021, 597(7878): 655-659
[298] WANG K, DUTT A, WOJCIK C C, et al. Topological complex-energy braiding of non-Hermitian bands[J]. Nature, 2021, 598(7879): 59-64
[299] BANDRES M A, WITTEK S, HARARI G, et al. Topological insulator laser: Experiments[J].Science, 2018, 359(6381): eaar4005
[300] XIAO L, DENG T, WANG K, et al. Non-Hermitian bulk–boundary correspondence in quantumdynamics[J]. Nat Phys, 2020, 16(7): 761-766100参考文献
[301] ZHAO H, QIAO X, WU T, et al. Non-Hermitian topological light steering[J]. Science, 2019,365(6458): 1163-1166
[302] ZHANG W, OUYANG X, HUANG X, et al. Observation of Non-Hermitian Topology withNonunitary Dynamics of Solid-State Spins[J]. Phys Rev Lett, 2021, 127(9): 090501
[303] CHEN W, ABBASI M, JOGLEKAR Y N, et al. Quantum Jumps in the Non-Hermitian Dynamicsof a Superconducting Qubit[J]. Phys Rev Lett, 2021, 127(14): 140504
[304] OZAWA T, PRICE H M, AMO A, et al. Topological photonics[J]. Rev Mod Phys, 2019, 91(1):015006
[305] LI L, LEE C H, MU S, et al. Critical non-Hermitian skin effect[J]. Nat Commun, 2020, 11(1):5491
[306] SUN X Q, ZHU P, HUGHES T L. Geometric Response and Disclination-Induced Skin Effectsin Non-Hermitian Systems[J]. Phys Rev Lett, 2021, 127(6): 066401
[307] LEE C H, LI L, GONG J. Hybrid Higher-Order Skin-Topological Modes in NonreciprocalSystems[J]. Phys Rev Lett, 2019, 123(1): 016805
[308] MARTINEZ ALVAREZ V M, BARRIOS VARGAS J E, FOA TORRES L E F. Non-Hermitianrobust edge states in one dimension: Anomalous localization and eigenspace condensation atexceptional points[J]. Phys Rev B, 2018, 97(12): 121401
[309] OKUMA N, KAWABATA K, SHIOZAKI K, et al. Topological Origin of Non-Hermitian SkinEffects[J]. Phys Rev Lett, 2020, 124: 086801
[310] YIN S, HUANG G Y, LO C Y, et al. Kibble-Zurek Scaling in the Yang-Lee Edge Singularity[J]. Phys Rev Lett, 2017, 118(6): 065701
[311] ZEUNER J M, RECHTSMAN M C, PLOTNIK Y, et al. Observation of a Topological Transitionin the Bulk of a Non-Hermitian System[J]. Phys Rev Lett, 2015, 115(4): 040402
[312] GE L, STONE A D. Parity-Time Symmetry Breaking beyond One Dimension: The Role ofDegeneracy[J]. Phys Rev X, 2014, 4(3): 031011
[313] ASHIDA Y, FURUKAWA S, UEDA M. Quantum critical behavior influenced by measurementbackaction in ultracold gases[J]. Phys Rev A, 2016, 94(5): 053615
[314] LONGHI S. Topological Phase Transition in non-Hermitian Quasicrystals[J]. Phys Rev Lett,2019, 122(23): 237601
[315] OKUMA N, SATO M. Topological Phase Transition Driven by Infinitesimal Instability: MajoranaFermions in Non-Hermitian Spintronics[J]. Phys Rev Lett, 2019, 123(9): 097701
[316] MIRI M A, ALù A. Exceptional points in optics and photonics[J]. Science, 2019, 363(6422):eaar7709
[317] DENNER M M, SKURATIVSKA A, SCHINDLER F, et al. Exceptional topological insulators[J]. Nat Commun, 2021, 12(1): 5681
[318] MOLINA R A, GONZáLEZ J. Surface and 3D Quantum Hall Effects from Engineering ofExceptional Points in Nodal-Line Semimetals[J]. Phys Rev Lett, 2018, 120(14): 146601
[319] YOKOMIZO K, MURAKAMI S. Non-Bloch Band Theory of Non-Hermitian Systems[J]. PhysRev Lett, 2019, 123(6): 066404
[320] MCDONALD A, PEREG-BARNEA T, CLERK A. Phase-Dependent Chiral Transport andEffective Non-Hermitian Dynamics in a Bosonic Kitaev-Majorana Chain[J]. Phys Rev X, 2018,8(4): 041031
[321] RUDNER M S, LEVITOV L S. Topological Transition in a Non-Hermitian Quantum Walk[J].Phys Rev Lett, 2009, 102(6): 065703
[322] LEE J Y, AHN J, ZHOU H, et al. Topological Correspondence between Hermitian and Non-Hermitian Systems: Anomalous Dynamics[J]. Phys Rev Lett, 2019, 123(20): 206404
[323] SHEN H, ZHEN B, FU L. Topological Band Theory for Non-Hermitian Hamiltonians[J]. PhysRev Lett, 2018, 120(14): 146402
[324] GORINI V, KOSSAKOWSKI A, SUDARSHAN E C G. Completely positive dynamical semigroupsof N‐level systems[J]. J Math Phys, 1976, 17(5): 821-825
[325] AMICO L, FAZIO R, OSTERLOH A, et al. Entanglement in many-body systems[J]. Rev ModPhys, 2008, 80: 517-576
[326] BRANDãO F G S L, HORODECKI M. An area law for entanglement from exponential decayof correlations[J]. Nature Phys, 2013, 9(11): 721-726
[327] DÓRA B, HAQUE M, ZARÁND G. Crossover from Adiabatic to Sudden Interaction Quenchin a Luttinger Liquid[J]. Phys Rev Lett, 2011, 106: 156406
[328] CAZALILLA M A. Effect of Suddenly Turning on Interactions in the Luttinger Model[J]. PhysRev Lett, 2006, 97(15): 156403
[329] FAGOTTI M, CALABRESE P. Evolution of entanglement entropy following a quantumquench: Analytic results for the X Y chain in a transverse magnetic field[J]. Phys Rev A,2008, 78(1): 010306
[330] CALABRESE P, CARDY J. Time Dependence of Correlation Functions Following a QuantumQuench[J]. Phys Rev Lett, 2006, 96(13): 136801
[331] BARDARSON J H, POLLMANN F, MOORE J E. Unbounded Growth of Entanglement inModels of Many-Body Localization[J]. Phys Rev Lett, 2012, 109(1): 017202
[332] SERBYN M, PAPIć Z, ABANIN D A. Universal Slow Growth of Entanglement in InteractingStrongly Disordered Systems[J]. Phys Rev Lett, 2013, 110(26): 260601
[333] MOCA C U U U U P M C, DÓRA B. Universal conductance of a 𝒫𝒯-symmetric Luttingerliquid after a quantum quench[J]. Phys Rev B, 2021, 104: 125124
[334] DÓRA B, MOCA C U U U U P M C. Quantum Quench in 𝒫𝒯-Symmetric Luttinger Liquid[J].Phys Rev Lett, 2020, 124: 136802
[335] BÁCSI A, MOCA C U U U U P M C, DÓRA B. Dissipation-Induced Luttinger Liquid Correlationsin a One-Dimensional Fermi Gas[J]. Phys Rev Lett, 2020, 124: 136401
[336] BÁCSI A, DÓRA B. Dynamics of entanglement after exceptional quantum quench[J]. PhysRev B, 2021, 103: 085137
[337] CAROLLO F, ALBA V. Dissipative quasiparticle picture for quadratic Markovian open quantumsystems[J]. Phys Rev B, 2022, 105: 144305
[338] BÁCSI A, MOCA C U U U U P M C, ZARÁND G, et al. Vaporization Dynamics of a DissipativeQuantum Liquid[J]. Phys Rev Lett, 2020, 125: 266803102参考文献
[339] ALBERTON O, BUCHHOLD M, DIEHL S. Entanglement Transition in a Monitored Free-Fermion Chain: From Extended Criticality to Area Law[J]. Phys Rev Lett, 2021, 126(17):170602
[340] PTASZYńSKI K, ESPOSITO M. Entropy Production in Open Systems: The Predominant Roleof Intraenvironment Correlations[J]. Phys Rev Lett, 2019, 123(20): 200603
[341] OKUMA N, SATO M. Quantum anomaly, non-Hermitian skin effects, and entanglement entropyin open systems[J]. Phys Rev B, 2021, 103(8): 085428
[342] ALBA V, CAROLLO F. Spreading of correlations in Markovian open quantum systems[J].Phys Rev B, 2021, 103(2): L020302
[343] ASHIDA Y, SAITO K, UEDA M. Thermalization and Heating Dynamics in Open GenericMany-Body Systems[J]. Phys Rev Lett, 2018, 121(17): 170402
[344] LI Y, CHEN X, FISHER M P A. Quantum Zeno effect and the many-body entanglement transition[J]. Phys Rev B, 2018, 98: 205136
[345] BENSA J, ŽNIDARIč M. Fastest Local Entanglement Scrambler, Multistage Thermalization,and a Non-Hermitian Phantom[J]. Phys Rev X, 2021, 11(3): 031019
[346] NAHUM A, ROY S, SKINNER B, et al. Measurement and Entanglement Phase Transitionsin All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory[J]. PRXQuantum, 2021, 2(1): 010352
[347] SKINNER B, RUHMAN J, NAHUM A. Measurement-Induced Phase Transitions in the Dynamicsof Entanglement[J]. Phys Rev X, 2019, 9(3): 031009
[348] APOLLARO T J G, PALMA G M, MARINO J. Entanglement entropy in a periodically drivenquantum Ising ring[J]. Phys Rev B, 2016, 94(13): 134304
[349] MASKARA N, MICHAILIDIS A, HO W, et al. Discrete Time-Crystalline Order Enabled byQuantum Many-Body Scars: Entanglement Steering via Periodic Driving[J]. Phys Rev Lett,2021, 127(9): 090602
[350] BERDANIER W, KOLODRUBETZ M, VASSEUR R, et al. Floquet Dynamics of Boundary-Driven Systems at Criticality[J]. Phys Rev Lett, 2017, 118(26): 260602
[351] PONTE P, PAPIć Z, HUVENEERS F, et al. Many-Body Localization in Periodically DrivenSystems[J]. Phys Rev Lett, 2015, 114(14): 140401
[352] ALBA V, CAROLLO F. Noninteracting fermionic systems with localized losses: Exact resultsin the hydrodynamic limit[J]. Phys Rev B, 2022, 105: 054303
[353] CALABRESE P. Entanglement and thermodynamics in non-equilibrium isolated quantum systems[J]. Physica A: Statistical Mechanics and its Applications, 2018, 504: 31-44
[354] PROSEN T. Third quantization: a general method to solve master equations for quadratic openFermi systems[J]. New Journal of Physics, 2008, 10(4): 043026
[355] CAI Z, BARTHEL T. Algebraic versus Exponential Decoherence in Dissipative Many-ParticleSystems[J]. Phys Rev Lett, 2013, 111(15): 150403
[356] HILL S, WOOTTERS W K. Entanglement of a Pair of Quantum Bits[J]. Phys Rev Lett, 1997,78(26): 5022-5025
[357] WOOTTERS W K. Entanglement of Formation of an Arbitrary State of Two Qubits[J]. PhysRev Lett, 1998, 80: 2245-2248
[358] CHEN L M, CHEN S A, YE P. Entanglement, non-hermiticity, and duality[J]. SciPost Physics,2021, 11(1): 003
[359] MAITY S, BANDYOPADHYAY S, BHATTACHARJEE S, et al. Growth of mutual informationin a quenched one-dimensional open quantum many-body system[J]. Phys Rev B, 2020, 101(18): 180301
[360] SHARMA A, RABANI E. Landauer current and mutual information[J]. Phys Rev B, 2015, 91(8): 085121
[361] BRODY D C. Biorthogonal quantum mechanics[J]. Journal of Physics A: Mathematical andTheoretical, 2013, 47(3): 035305
[362] HOLZHEY C, LARSEN F, WILCZEK F. Geometric and renormalized entropy in conformalfield theory[J]. Nuclear Physics B, 1994, 424(3): 443-467
[363] CALABRESE P, CARDY J. Entanglement entropy and conformal field theory[J]. Journal ofPhysics A: Mathematical and Theoretical, 2009, 42(50): 504005
[364] ZHOU Y, ZHANG Z, YIN Z, et al. Rapid and unconditional parametric reset protocol fortunable superconducting qubits[J]. Nature Communications, 2021, 12(1): 5924
[365] ABANIN D A, DEMLER E. Measuring Entanglement Entropy of a Generic Many-Body Systemwith a Quantum Switch[J]. Phys Rev Lett, 2012, 109(2): 020504
[366] DALEY A J, PICHLER H, SCHACHENMAYER J, et al. Measuring Entanglement Growth inQuench Dynamics of Bosons in an Optical Lattice[J]. Phys Rev Lett, 2012, 109(2): 020505
[367] CHOO K, VON KEYSERLINGK C W, REGNAULT N, et al. Measurement of the EntanglementSpectrum of a Symmetry-Protected Topological State Using the IBM Quantum Computer[J]. Phys Rev Lett, 2018, 121(8): 086808
[368] DALMONTE M, VERMERSCH B, ZOLLER P. Quantum simulation and spectroscopy ofentanglement Hamiltonians[J]. Nature Physics, 2018, 14(8): 827-831
[369] VARMA A V, DAS S. Simulating many-body non-Hermitian PT -symmetric spin dynamics[J].Phys Rev B, 2021, 104(3): 035153