Topological non-Hermitian skin effect

doi:10.1007/s11467-023-1309-z

Front. Phys.

2023, Vol. 18

Issue (5) : 53605 https://doi.org/10.1007/s11467-023-1309-z

REVIEW ARTICLE

Topological non-Hermitian skin effect

Rijia Lin¹, Tommy Tai^2,³(

), Linhu Li¹(

), Ching Hua Lee³(

)

¹. Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing & School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China
². Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
³. Department of Physics, National University of Singapore, Singapore 117542, Singapore

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Abstract

This article reviews recent developments in the non-Hermitian skin effect (NHSE), particularly on its rich interplay with topology. The review starts off with a pedagogical introduction on the modified bulk-boundary correspondence, the synergy and hybridization of NHSE and band topology in higher dimensions, as well as, the associated topology on the complex energy plane such as spectral winding topology and spectral graph topology. Following which, emerging topics are introduced such as non-Hermitian criticality, dynamical NHSE phenomena, and the manifestation of NHSE beyond the traditional linear non-interacting crystal lattices, particularly its interplay with quantum many-body interactions. Finally, we survey the recent demonstrations and experimental proposals of NHSE.

Keywords non-Hermitian skin effect topological phases

Corresponding Author(s): Tommy Tai,Linhu Li,Ching Hua Lee

About author: ^* These authors contributed equally to this work.

Issue Date: 04 July 2023

Cite this article:

Rijia Lin,Tommy Tai,Linhu Li, et al. Topological non-Hermitian skin effect[J]. Front. Phys. , 2023, 18(5): 53605.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1309-z
https://academic.hep.com.cn/fop/EN/Y2023/V18/I5/53605

Fig.1 (a) Schematic of the non-Hermitian SSH model. (b) Squared wavefunction amplitude

| ψ (x) | 2

plots, showing that the bulk modes are localized near the left boundary. (c) Energy spectra of the non-Hermitian SSH model with open (purple) and periodic (red and blue solid lines denote different bands) boundary conditions. Parameters:

t 2 = 2, γ = 1, t 1 = 1; L = 160

. Figures reproduced from the model in Ref. [46].

Fig.2 The Brillouin zone (blue loop, a unit circle) and generalized Brillouin zone (pink loop) of the non-Hermitian SSH model, which lies within the Brillouin zone. In general, the generalized Brillouin zone is usually not circular for more general settings. Parameters are

t 2 = 2, γ = 1, t 1 = 1

. Figure reproduced from the non-Bloch band theory introduced in Refs. [46, 47].

Fig.3 Transition of topological localization direction and half quantized winding numbers in the non-Hermitian SSH model in Eq. (1). (a?c) The squared wavefunction amplitude

| ψ (x) | 2

of zero-energy modes. (d?f) The PBC (red, blue) and OBC (gray) spectra of the non-Hermitian SSH model with PBC spectra are seen to deform from two loops into one, while OBC ones remain gapped with a pair of zero-energy eigenmodes. (g?i) Winding trajectories of

h r (k) = (h x r, h y r)

. Pink star and red triangle represent EPs given by

h r (k) = (± h y i, ? h x i) = (± γ / 2, 0)

, for the model with

h x = t 1 + t 2 cos ? (k)

and

h y = t 2 sin ? (k) + i γ / 2

. Parameters are

t 2 = 2

γ = 1

L = 160

(number of unit cells), and

t 1 = 1

1.5

, and

1.8

from left to right respectively. Figures reproduced from the methods in Refs. [29, 91] for the model of Eq. (1).

Fig.4 Topological surface states of non-Hermitian nodal knot metals. (a) The disagreement between the surface state projections with (yellow) and without non-Hermiticity (blue) for the Hopf link. (b) Analytically solved surface state of the Hopf link with

k ⊥ = k 2

. (c) Schematic comparing the tight-binding implementations of the Hermitian and non-Hermitian nodal knots. (d) Similar to (a) but for the trefoil knot. (e) Non-reciprocal similarity transforms can rescale

z = e i k ⊥

, leading to fluctuations of

log ? | z | = 0

level, identified as the “sea level”. The skin gap closures correspond to the surface states and manifest as “tidal boundary”, whereas the band intersections with the sea level recover the Hermitian drumhead surface states, with a close analogy to “shore”. (f) More detailed construction of the tidal phase boundary of the trefoil knot. (g) Evolution of the PBC and OBC spectra with

k 1

, revealing intricate relations between the complex energy bands, vorticity and the “tidal surface states”. (h) The “tidal region” is demarcated by the half-integer vorticity boundary. (i) A direct relationship between the tidal islands and the Seifert surface of the dual nodal knot metal. Figures reproduced from Ref. [119].

Fig.5 Hybrid skin-topological effect in a non-Hermitian Benalcazar?Bernevig?Hughes model and its 3D extension. (a) The model contains four sublattices in each unit cell, with non-reciprocal real hopping parameters

δ 1, 2, 3, 4

along

x

and

y

directions. (b) Emergence of 1D boundary modes under

x

-OBC/

y

-PBC (cylinder geometry). (c) Boundary modes accumulate towards the corners under double OBC due to the hybrid skin-topological effect. (d, e) Spectra of the system with PBC (brown),

x

-OBC/

y

-PBC (gray), and double OBC (black). (f) A 3D lattice from stacks of the 2D model in (a), which are coupled through extra non-reciprocal hoppings along

z

direction. (g) STT corner modes, (h) SST corner modes, and (i) ST0 hinge modes in the 3D model, originated from different hybridizations of NHSE and topological localization along different directions. Reproduced from Ref. [95].

Fig.6 Hybrid skin-topological effect in a non-Hermitian Haldane model. (a) The model contains real nearest neighbor hopping

t 1

, and complex next-nearest neighbor hopping

t 2 e ± ?

, with arrows indicating the direction of hopping with a positive phase. Non-Hermiticity is introduced by the complex on-site potential

m ± i γ

on the two sublattices respectively. (b) Zigzag boundary of the system. Long solid and dashed arrows [also in (d)?(f)] represent the chiral edge current for the two sublattices, which are toward and opposite to the localization direction of NHSE respectively. (c) Armchair boundary of the system. Arrows connecting different sites in (b) and (c) represent the same property as that in (a). (d?f) Distribution of edge modes in the model with negative, zero, positive

γ

respectively. Reproduced from Ref. [97].

Fig.7 Schematics contrasting higher-order NHSE and higher-order Hermitian topological phases for 2D systems with

L × L

sites. (a) Hermitian first-order topological insulator.

O (L)

chiral or helical modes appear at the edges. (b) Hermitian second-order topological insulator.

O (1)

zero modes appear at the corners. (c) First-order non-Hermitian skin effect.

O (L 2)

skin modes appear at the edges. (d) Second-order non-Hermitian skin effect.

O (L)

skin modes appear at the corners. Reproduced from Ref. [99].

Fig.8 Second order non-Hermitian skin effect for the model of Eq. (33). The parameters are

λ = 1, γ = 0.5

. Open boundary conditions are imposed along both of the directions. (a) The complex spectrum, (b) corner skin modes (

E = ? 0.027 ? 0.0008 i

), and (c) delocalized bulk modes (

E = ? 1.64 ? 0.94 i

) of the model are demonstrated for

30 × 30

sites. Reproduced from Ref. [99].

Fig.9 Qualitatively different types of energy gaps in Hermitian and non-Hermitian systems. (a) Two gapped Hermitian bands can be flattened to two points at

E = ± 1

along the energy axis, with a flattened Hamiltonian

H 2 = 1

. (b) Complex energy bands with a point gap can be flattened to a unit circle in the complex energy plane, with the system’s Hamiltonian becoming a unitary one,

H ? H = 1

. (c) Hermitian and anti-Hermitian flattening of complex energy bands with a line gap. A non-Hermitian Hamiltonian with a real (an imaginary) line gap can be flattened to a Hermitian (an anti-Hermitian) Hamiltonian with

H 2 = + 1

(

H 2 = ? 1

). Reproduced from Ref. [221].

Fig.10 A 3D system with different bulk and surface behaviors under different boundary conditions. Gray color indicate the PBC spectrum, and blue and orange colors correspond to the spectra with boundary conditions specified in the figures. (a) The PBC spectrum has a point gap with a nontrivial 3D winding number

+ 1

. (b) When OBC is taken along

z

direction, surface states cover the point-gapped region with a nontrivial 3D winding number. (c) When OBC is taken along

y

direction, a sharp change of the spectrum indicates the occurrence of NHSE, and in-gap skin modes (orange) appear. Reproduced from Ref. [233].

Fig.11 Quantized response of non-Hermitian spectral winding topology. Results are obtained from the model of Eq. (16), with

t 1 = 1

t ? 1 = 0.5

t 2 = 2

, and

t j = 0

for other values of

j

. (a, b) the two quantities extracted from the Green’s function as functions of

β

, the parameters determining the boundary conditions. (c) Spectra at different values of

β

, corresponding to the five dashed lines in (b) respectively. Red star indicates the chosen reference energy

E r = ? 0.96 + i

for (a) and (b). (d) The quantized response quantity

ν = M a x [ν 1 ν 2]

, as spectral winding number ranges from

0

2

for the chosen system. Reproduced from Ref. [105].

Fig.12 Several examples of topologically distinct braids, their knot closures and their realizations in 1D non-Hermitian Bloch bands.

τ i

and

τ i ? 1

denote braid operators of the

i

th string crossing over and under the

(i + 1)

th string from the left, respectively. Braid diagrams in the second row are mapped to the knots in the third row when connected top to bottom. The fourth row demonstrate typical structures of energy bands for these braids and knots in the 3D space of momentum and complex energy. Reproduced from Ref. [249].

Fig.13 Emergent spectral graph topology. (a) An assortment of OBC spectra displaying a rich variety of spectral graph topology. (b) Eigenvalues of Eq. (39) flow into each other as a real boundary flux

R e (?)

is threaded over a period, for the model given in Eq. (39) [50]. The outermost loop describes the PBC loop; successively smaller inner loops represent the spectrum as the boundary coupling is decreased through increasing attenuation factors

e ? I m (?)

. (c) Illustration of PBC-OBC spectral flow of Eq. (39), with PBC bulk eigenvalues (red) flow along the blue magenta curves upon threading

κ ∝ I m (?)

, which eventually converge to the OBC eigenvalues (black) [50]. (d) An equivalent representation would be plotting the intersections of the inverse skin depth solution surfaces

| κ i |

[131]. (e,f) The PBC-OBC spectral flow can be generalized to all non-Hermitian lattices, where the intimate relationship between the complex energies

E

and the inverse spatial decay lengths

κ

in (e) draw a parallel with the electrostatic potential landscape in (f), with PBC and OBC spectral loci corresponding to grounded conductors (

V = 0

) and lines of induced charges respectively [89]. Figures (a, d), (b, c) and (e, f) are reproduced from Refs. [131], [50] and [89] respectively.

Fig.14 Classification table to illustrate the complexity of spectral graph topology, as reproduced from Ref. [131]. For each form of the canonical dispersion

P (E, z)

, one can 1) associate a non-unique minimal Hamiltonian

H (z)

, 2) identify emergent global symmetries of the spectral graph

E ˉ

not necessarily present in

H (z)

, and 3) characterize its spectral graph topology with its number of branches

N S, N L

and loops

N ?

, as well as its adjacency matrix.

Fig.15 Emergent criticality in finite non-Hermitian lattices, as reproduced from Ref. [62]. (a?c) Critical behaviour in asymmetrically coupled systems involving two non-Hermitian systems with different inverse skin lengths. As the size of the lattice increases, the open boundary spectrum transitions from the decoupled thermodynamic limit (green spectra) to the coupled thermodynamic limit (red spectra). (d?g) Similarly, a critical topological phase transition can be obtained with cross inter-chain non-reciprocal couplings as the size of the lattice increases. This is robust with exponentially weak, but non-zero inter-chain coupling.

Fig.16 Dynamical effects associated with the NHSE. (a, b) Wavepacket spreading and acceleration of an initially stationary localized wavepacket in a non-Hermitian lattice [287]. Upon traversing to the boundary, the wavepacket does not get reflected, suggesting an inelastic scattering. (c) More generally, the self-acceleration plotted is proportional to the area enclosed by the PBC spectrum [288]. (d) Self-healing of a wavepacket upon striking an arbitrary space-time potential. (e) Topological criterion to self-healing edge skin modes [57]. (f) Example of Floquet edge states in a periodically driven non-Hermitian lattice [293]. Figures (a, b), (c), (d, e) and (f) are reproduced from Refs. [287], [288], [57] and [293] respectively.

Fig.17 Unconventional manifestations of the NHSE. (a) Lattices with non-reciprocal impurities can result in the coexistence of scale-free accumulation and NHSE [61]. (b) Introducing non-Hermiticity, the imaginary vector potential can result in the NHSE even under PBC [319]. (c) With both nonlinearity and nonreciprocal non-Hermiticity, a novel oscillatory soliton, a topological end breather, is formed and is strongly localized to a self-induced topological domain near the end of the lattice [320]. (d) A combination of particle statistics and suitably engineered many-body interactions can result in boundaries (sites of disallowed occupation) in the many-body configuration space. Topological and skin states can thus form without physical boundaries [321]. (e) An example of a 1D chain with well-designed two-body interactions that result in a chiral propagating state along the diagonal boundary in the two-body configuration space [321]. (f) This principle can be generalized to the strongly interacting limit, which results in the formation of localized non-Hermitian skin clusters [322], shaped by the connectivity structure of the many-body Hilbert space instead of the real-space lattice. Figures (a), (b), (c), (d, e) and (f) are reproduced from Refs. [61], [319], [320], [321] and [322] respectively.

1	M. Bender C. , Boettcher S. . Real spectra in non-Hermitian Hamiltonians having PT symmetry. Phys. Rev. Lett., 1998, 80(24): 5243 https://doi.org/10.1103/PhysRevLett.80.5243
2	M. Bender C. . Making sense of non-Hermitian Hamiltonians. Rep. Prog. Phys., 2007, 70(6): 947 https://doi.org/10.1088/0034-4885/70/6/R03
3	Rotter I. . A non-Hermitian Hamilton operator and the physics of open quantum systems. J. Phys. A Math. Theor., 2009, 42(15): 153001 https://doi.org/10.1088/1751-8113/42/15/153001
4	Yoshida T. , Peters R. , Kawakami N. . Non-Hermitian perspective of the band structure in heavy-Fermion systems. Phys. Rev. B, 2018, 98(3): 035141 https://doi.org/10.1103/PhysRevB.98.035141
5	Shen H. , Fu L. . Quantum oscillation from in-gap states and a non-Hermitian Landau level problem. Phys. Rev. Lett., 2018, 121(2): 026403 https://doi.org/10.1103/PhysRevLett.121.026403
6	Yamamoto K. , Nakagawa M. , Adachi K. , Takasan K. , Ueda M. , Kawakami N. . Theory of non-Hermitian Fermionic superfluidity with a complex-valued interaction. Phys. Rev. Lett., 2019, 123(12): 123601 https://doi.org/10.1103/PhysRevLett.123.123601
7	Ma G. , Sheng P. . Acoustic metamaterials: From local resonances to broad horizons. Sci. Adv., 2016, 2(2): e1501595 https://doi.org/10.1126/sciadv.1501595
8	C. S. A. Cummer J. , Alù A. . Controlling sound with acoustic metamaterials. Nat. Rev. Mater., 2016, 1(3): 16001 https://doi.org/10.1038/natrevmats.2016.1
9	Zangeneh-Nejad F. , Fleury R. . Active times for acoustic metamaterials. Reviews in Physics, 2019, 4: 100031 https://doi.org/10.1016/j.revip.2019.100031
10	Feng L. , El-Ganainy R. , Ge L. . Non-Hermitian photonics based on parity–time symmetry. Nat. Photonics, 2017, 11(12): 752 https://doi.org/10.1038/s41566-017-0031-1
11	El-Ganainy R. , G. Makris K. , Khajavikhan M. , H. Musslimani Z. , Rotter S. , N. Christodoulides D. . Non-Hermitian physics and PT symmetry. Nat. Phys., 2018, 14(1): 11 https://doi.org/10.1038/nphys4323
12	Longhi S. . Parity−time symmetry meets photonics: A new twist in non-Hermitian optics. Europhys. Lett., 2017, 120(6): 64001 https://doi.org/10.1209/0295-5075/120/64001
13	Ozawa T. , M. Price H. , Amo A. , Goldman N. , Hafezi M. , Lu L. , C. Rechtsman M. , Schuster D. , Simon J. , Zilberberg O. , Carusotto I. . Topological photonics. Rev. Mod. Phys., 2019, 91(1): 015006 https://doi.org/10.1103/RevModPhys.91.015006
14	Xiao L. , Deng T. , Wang K. , Zhu G. , Wang Z. , Yi W. , Xue P. . Non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys., 2020, 16(7): 761 https://doi.org/10.1038/s41567-020-0836-6
15	Mostafazadeh A. . Pseudo-hermiticity versus PT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian. J. Math. Phys., 2002, 43(1): 205 https://doi.org/10.1063/1.1418246
16	Günther U. , Rotter I. , F. Samsonov B. . Projective Hilbert space structures at exceptional points. J. Phys. A Math. Theor., 2007, 40(30): 8815 https://doi.org/10.1088/1751-8113/40/30/014
17	E. F. Foa Torres L. . Perspective on topological states of non-Hermitian lattices. Journal of Physics: Materials, 2019, 3: 014002 https://doi.org/10.1088/2515-7639/ab4092
18	Lin Z. , Ramezani H. , Eichelkraut T. , Kottos T. , Cao H. , N. Christodoulides D. . Unidirectional invisibility induced by PT-symmetric periodic structures. Phys. Rev. Lett., 2011, 106(21): 213901 https://doi.org/10.1103/PhysRevLett.106.213901
19	Feng L. , L. Xu Y. , S. Fegadolli W. , H. Lu M. , E. B. Oliveira J. , R. Almeida V. , F. Chen Y. , Scherer A. . Experimental demonstration of a unidirectional reflectionless parity−time metamaterial at optical frequencies. Nat. Mater., 2013, 12(2): 108 https://doi.org/10.1038/nmat3495
20	Wiersig J. . Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection. Phys. Rev. Lett., 2014, 112(20): 203901 https://doi.org/10.1103/PhysRevLett.112.203901
21	P. Liu Z. , Zhang J. , K. Özdemir Ş. , Peng B. , Jing H. , Y. Lü X. , W. Li C. , Yang L. , Nori F. , Liu Y. . Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition. Phys. Rev. Lett., 2016, 117(11): 110802 https://doi.org/10.1103/PhysRevLett.117.110802
22	Hodaei H. , U. Hassan A. , Wittek S. , Garcia-Gracia H. , El-Ganainy R. , N. Christodoulides D. , Khajavikhan M. . Enhanced sensitivity at higher-order exceptional points. Nature, 2017, 548(7666): 187 https://doi.org/10.1038/nature23280
23	Chen W. , K. Özdemir Ş. , Zhao G. , Wiersig J. , Yang L. . Exceptional points enhance sensing in an optical microcavity. Nature, 2017, 548(7666): 192 https://doi.org/10.1038/nature23281
24	Dembowski C. , D. Gräf H. , L. Harney H. , Heine A. , D. Heiss W. , Rehfeld H. , Richter A. . Experimental observation of the topological structure of exceptional points. Phys. Rev. Lett., 2001, 86(5): 787 https://doi.org/10.1103/PhysRevLett.86.787
25	Gao T. , Estrecho E. , Y. Bliokh K. , C. H. Liew T. , D. Fraser M. , Brodbeck S. , Kamp M. , Schneider C. , Höfling S. , Yamamoto Y. , Nori F. , S. Kivshar Y. , G. Truscott A. , G. Dall R. , A. Ostrovskaya E. . Observation of non-Hermitian degeneracies in a chaotic exciton-polariton billiard. Nature, 2015, 526(7574): 554 https://doi.org/10.1038/nature15522
26	A. Mailybaev A. , N. Kirillov O. , P. Seyranian A. . Geometric phase around exceptional points. Phys. Rev. A, 2005, 72(1): 014104 https://doi.org/10.1103/PhysRevA.72.014104
27	E. Lee T. . Anomalous edge state in a non-Hermitian lattice. Phys. Rev. Lett., 2016, 116(13): 133903 https://doi.org/10.1103/PhysRevLett.116.133903
28	Leykam D. , Y. Bliokh K. , Huang C. , D. Chong Y. , Nori F. . Edge modes, degeneracies, and topological numbers in non-Hermitian systems. Phys. Rev. Lett., 2017, 118(4): 040401 https://doi.org/10.1103/PhysRevLett.118.040401
29	Yin C. , Jiang H. , Li L. , Lü R. , Chen S. . Geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems. Phys. Rev. A, 2018, 97(5): 052115 https://doi.org/10.1103/PhysRevA.97.052115
30	Shen H. , Zhen B. , Fu L. . Topological band theory for non-Hermitian Hamiltonians. Phys. Rev. Lett., 2018, 120(14): 146402 https://doi.org/10.1103/PhysRevLett.120.146402
31	Li L. , H. Lee C. , Gong J. . Geometric characterization of non-Hermitian topological systems through the singularity ring in pseudospin vector space. Phys. Rev. B, 2019, 100(7): 075403 https://doi.org/10.1103/PhysRevB.100.075403
32	Hu W. , Wang H. , P. Shum P. , D. Chong Y. . Exceptional points in a non-Hermitian topological pump. Phys. Rev. B, 2017, 95(18): 184306 https://doi.org/10.1103/PhysRevB.95.184306
33	U. Hassan A. , Zhen B. , Soljačić M. , Khajavikhan M. , N. Christodoulides D. . Dynamically encircling exceptional points: Exact evolution and polarization state conversion. Phys. Rev. Lett., 2017, 118(9): 093002 https://doi.org/10.1103/PhysRevLett.118.093002
34	Z. Hasan M. , L. Kane C. . Colloquium: Topological insulators. Rev. Mod. Phys., 2010, 82(4): 3045 https://doi.org/10.1103/RevModPhys.82.3045
35	L. Qi X. , C. Zhang S. . Topological insulators and superconductors. Rev. Mod. Phys., 2011, 83(4): 1057 https://doi.org/10.1103/RevModPhys.83.1057
36	A. Bernevig B.L. Hughes T., Topological Insulators and Topological Superconductors, Princeton University Press, 2013
37	S. Rudner M. , S. Levitov L. . Topological transition in a non-Hermitian quantum walk. Phys. Rev. Lett., 2009, 102(6): 065703 https://doi.org/10.1103/PhysRevLett.102.065703
38	C. Hu Y. , L. Hughes T. . Absence of topological insulator phases in non-Hermitian PT-symmetric Hamiltonians. Phys. Rev. B, 2011, 84(15): 153101 https://doi.org/10.1103/PhysRevB.84.153101
39	Esaki K. , Sato M. , Hasebe K. , Kohmoto M. . Edge states and topological phases in non-Hermitian systems. Phys. Rev. B, 2011, 84(20): 205128 https://doi.org/10.1103/PhysRevB.84.205128
40	Diehl S. , Rico E. , A. Baranov M. , Zoller P. . Topology by dissipation in atomic quantum wires. Nat. Phys., 2011, 7(12): 971 https://doi.org/10.1038/nphys2106
41	Zhu B. , Lü R. , Chen S. . PT symmetry in the non-Hermitian Su−Schrieffer−Heeger model with complex boundary potentials. Phys. Rev. A, 2014, 89(6): 062102 https://doi.org/10.1103/PhysRevA.89.062102
42	Malzard S. , Poli C. , Schomerus H. . Topologically protected defect states in open photonic systems with non-Hermitian charge-conjugation and parity−time symmetry. Phys. Rev. Lett., 2015, 115(20): 200402 https://doi.org/10.1103/PhysRevLett.115.200402
43	K. Harter A. , E. Lee T. , N. Joglekar Y. . PT-breaking threshold in spatially asymmetric Aubry−André and Harper models: Hidden symmetry and topological states. Phys. Rev. A, 2016, 93(6): 062101 https://doi.org/10.1103/PhysRevA.93.062101
44	Xiong Y. . Why does bulk boundary correspondence fail in some non-Hermitian topological models. J. Phys. Commun., 2018, 2(3): 035043 https://doi.org/10.1088/2399-6528/aab64a
45	M. Martinez Alvarez V. , E. Barrios Vargas J. , E. F. Foa Torres L. . Non-Hermitian robust edge states in one dimension: Anomalous localization and eigenspace condensation at exceptional points. Phys. Rev. B, 2018, 97(12): 121401 https://doi.org/10.1103/PhysRevB.97.121401
46	Yao S. , Wang Z. . Edge states and topological invariants of non-Hermitian systems. Phys. Rev. Lett., 2018, 121(8): 086803 https://doi.org/10.1103/PhysRevLett.121.086803
47	Yokomizo K. , Murakami S. . Non-Bloch band theory of non-Hermitian systems. Phys. Rev. Lett., 2019, 123(6): 066404 https://doi.org/10.1103/PhysRevLett.123.066404
48	H. Lee C. , Li L. , Thomale R. , Gong J. . Unraveling non-Hermitian pumping: Emergent spectral singularities and anomalous responses. Phys. Rev. B, 2020, 102(8): 085151 https://doi.org/10.1103/PhysRevB.102.085151
49	Fu Y. , Wan S. . Degeneracy and defectiveness in non-Hermitian systems with open boundary. Phys. Rev. B, 2022, 105(7): 075420 https://doi.org/10.1103/PhysRevB.105.075420
50	H. Lee C. , Thomale R. . Anatomy of skin modes and topology in non-Hermitian systems. Phys. Rev. B, 2019, 99(20): 201103 https://doi.org/10.1103/PhysRevB.99.201103
51	K. Kunst F. , Edvardsson E. , C. Budich J. , J. Bergholtz E. . Biorthogonal bulk-boundary correspondence in non-Hermitian systems. Phys. Rev. Lett., 2018, 121(2): 026808 https://doi.org/10.1103/PhysRevLett.121.026808
52	Y. Zou Y.Zhou Y.M. Chen L.Ye P., Measuring non-unitarity in non-Hermitian quantum systems, arXiv: 2208.14944 (2022)
53	Song F. , Yao S. , Wang Z. . Non-Hermitian skin effect and chiral damping in open quantum systems. Phys. Rev. Lett., 2019, 123(17): 170401 https://doi.org/10.1103/PhysRevLett.123.170401
54	C. Wanjura C. , Brunelli M. , Nunnenkamp A. . Topological framework for directional amplification in driven-dissipative cavity arrays. Nat. Commun., 2020, 11(1): 3149 https://doi.org/10.1038/s41467-020-16863-9
55	C. Wanjura C. , Brunelli M. , Nunnenkamp A. . Correspondence between non-Hermitian topology and directional amplification in the presence of disorder. Phys. Rev. Lett., 2021, 127(21): 213601 https://doi.org/10.1103/PhysRevLett.127.213601
56	T. Xue W. , R. Li M. , M. Hu Y. , Song F. , Wang Z. . Simple formulas of directional amplification from non-Bloch band theory. Phys. Rev. B, 2021, 103(24): L241408 https://doi.org/10.1103/PhysRevB.103.L241408
57	Longhi S. . Self-healing of non-Hermitian topological skin modes. Phys. Rev. Lett., 2022, 128(15): 157601 https://doi.org/10.1103/PhysRevLett.128.157601
58	T. Xue W. , M. Hu Y. , Song F. , Wang Z. . Non-Hermitian edge burst. Phys. Rev. Lett., 2022, 128(12): 120401 https://doi.org/10.1103/PhysRevLett.128.120401
59	C. Budich J. , J. Bergholtz E. . Non-Hermitian topological sensors. Phys. Rev. Lett., 2020, 125(18): 180403 https://doi.org/10.1103/PhysRevLett.125.180403
60	X. Guo C. , H. Liu C. , M. Zhao X. , Liu Y. , Chen S. . Exact solution of non-Hermitian systems with generalized boundary conditions: Size-dependent boundary effect and fragility of the skin effect. Phys. Rev. Lett., 2021, 127(11): 116801 https://doi.org/10.1103/PhysRevLett.127.116801
61	Li L. , H. Lee C. , Gong J. . Impurity induced scale-free localization. Commun. Phys., 2021, 4(1): 42 https://doi.org/10.1038/s42005-021-00547-x
62	Li L. , H. Lee C. , Mu S. , Gong J. . Critical non-Hermitian skin effect. Nat. Commun., 2020, 11: 5491 https://doi.org/10.1038/s41467-020-18917-4
63	Yokomizo K. , Murakami S. . Scaling rule for the critical non-Hermitian skin effect. Phys. Rev. B, 2021, 104(16): 165117 https://doi.org/10.1103/PhysRevB.104.165117
64	H. Liu C. , Zhang K. , Yang Z. , Chen S. . Helical damping and dynamical critical skin effect in open quantum systems. Phys. Rev. Res., 2020, 2(4): 043167 https://doi.org/10.1103/PhysRevResearch.2.043167
65	Q. Sun X. , Zhu P. , L. Hughes T. . Geometric response and disclination-induced skin effects in non-Hermitian systems. Phys. Rev. Lett., 2021, 127(6): 066401 https://doi.org/10.1103/PhysRevLett.127.066401
66	A. Bhargava B. , C. Fulga I. , van den Brink J. , G. Moghaddam A. . Non-Hermitian skin effect of dislocations and its topological origin. Phys. Rev. B, 2021, 104(24): L241402 https://doi.org/10.1103/PhysRevB.104.L241402
67	Schindler F. , Prem A. . Dislocation non-Hermitian skin effect. Phys. Rev. B, 2021, 104(16): L161106 https://doi.org/10.1103/PhysRevB.104.L161106
68	Panigrahi A. , Moessner R. , Roy B. . Non-Hermitian dislocation modes: Stability and melting across exceptional points. Phys. Rev. B, 2022, 106(4): L041302 https://doi.org/10.1103/PhysRevB.106.L041302
69	Manna S.Roy B., Inner skin effects on non-Hermitian topological fractals, arXiv: 2202.07658 (2022)
70	Zhang K. , Yang Z. , Fang C. . Universal non-Hermitian skin effect in two and higher dimensions. Nat. Commun., 2022, 13(1): 2496 https://doi.org/10.1038/s41467-022-30161-6
71	Helbig T. , Hofmann T. , Imhof S. , Abdelghany M. , Kiessling T. , W. Molenkamp L. , H. Lee C. , Szameit A. , Greiter M. , Thomale R. . Generalized bulk-boundary correspondence in non-Hermitian topolectrical circuits. Nat. Phys., 2020, 16(7): 747 https://doi.org/10.1038/s41567-020-0922-9
72	Hofmann T. , Helbig T. , Schindler F. , Salgo N. , Brzezińska M. , Greiter M. , Kiessling T. , Wolf D. , Vollhardt A. , Kabaši A. , H. Lee C. , Bilušić A. , Thomale R. , Neupert T. . Reciprocal skin effect and its realization in a topolectrical circuit. Phys. Rev. Res., 2020, 2(2): 023265 https://doi.org/10.1103/PhysRevResearch.2.023265
73	Liu S. , Shao R. , Ma S. , Zhang L. , You O. , Wu H. , J. Xiang Y. , J. Cui T. , Zhang S. . Non-Hermitian skin effect in a non-Hermitian electrical circuit. Research, 2021, 2021: 5608038 https://doi.org/10.34133/2021/5608038
74	Zou D. , Chen T. , He W. , Bao J. , H. Lee C. , Sun H. , Zhang X. . Observation of hybrid higher-order skin-topological effect in non-Hermitian topolectrical circuits. Nat. Commun., 2021, 12(1): 7201 https://doi.org/10.1038/s41467-021-26414-5
75	Zhang X. , Tian Y. , H. Jiang J. , H. Lu M. , F. Chen Y. . Observation of higher-order non-Hermitian skin effect. Nat. Commun., 2021, 12(1): 5377 https://doi.org/10.1038/s41467-021-25716-y
76	Shang C. , Liu S. , Shao R. , Han P. , Zang X. , Zhang X. , N. Salama K. , Gao W. , H. Lee C. , Thomale R. , Manchon A. , Zhang S. , J. Cui T. , Schwingenschlögl U. . Experimental identification of the second-order non-Hermitian skin effect with physics-graph-informed machine learning. Adv. Sci. (Weinh.), 2022, 9(36): 2202922 https://doi.org/10.1002/advs.202202922
77	Zhang L. , Yang Y. , Ge Y. , J. Guan Y. , Chen Q. , Yan Q. , Chen F. , Xi R. , Li Y. , Jia D. , Q. Yuan S. , X. Sun H. , Chen H. , Zhang B. . Acoustic non-Hermitian skin effect from twisted winding topology. Nat. Commun., 2021, 12(1): 6297 https://doi.org/10.1038/s41467-021-26619-8
78	Gao H.Xue H.Gu Z.Li L.Zhu W.Su Z.Zhu J.Zhang B.D. Chong Y., Non-Hermitian skin effect in a ring resonator lattice, arXiv: 2205.14824 (2022)
79	Weidemann S. , Kremer M. , Helbig T. , Hofmann T. , Stegmaier A. , Greiter M. , Thomale R. , Szameit A. . Topological funneling of light. Science, 2020, 368(6488): 311 https://doi.org/10.1126/science.aaz8727
80	Song Y. , Liu W. , Zheng L. , Zhang Y. , Wang B. , Lu P. . Two-dimensional non-Hermitian skin effect in a synthetic photonic lattice. Phys. Rev. Appl., 2020, 14(6): 064076 https://doi.org/10.1103/PhysRevApplied.14.064076
81	Xiao L. , Deng T. , Wang K. , Wang Z. , Yi W. , Xue P. . Observation of non-Bloch parity-time symmetry and exceptional points. Phys. Rev. Lett., 2021, 126(23): 230402 https://doi.org/10.1103/PhysRevLett.126.230402
82	Wang K. , Li T. , Xiao L. , Han Y. , Yi W. , Xue P. . Detecting non-Bloch topological invariants in quantum dynamics. Phys. Rev. Lett., 2021, 127(27): 270602 https://doi.org/10.1103/PhysRevLett.127.270602
83	Brandenbourger M. , Locsin X. , Lerner E. , Coulais C. . Non-reciprocal robotic metamaterials. Nat. Commun., 2019, 10(1): 4608 https://doi.org/10.1038/s41467-019-12599-3
84	Ghatak A. , Brandenbourger M. , van Wezel J. , Coulais C. . Observation of non-Hermitian topology and its bulk-edge correspondence in an active mechanical metamaterial. Proc. Natl. Acad. Sci. USA, 2020, 117(47): 29561 https://doi.org/10.1073/pnas.2010580117
85	Gou W. , Chen T. , Xie D. , Xiao T. , S. Deng T. , Gadway B. , Yi W. , Yan B. . Tunable non-reciprocal quantum transport through a dissipative Aharonov−Bohm ring in ultracold atoms. Phys. Rev. Lett., 2020, 124(7): 070402 https://doi.org/10.1103/PhysRevLett.124.070402
86	Liang Q. , Xie D. , Dong Z. , Li H. , Li H. , Gadway B. , Yi W. , Yan B. . Dynamic signatures of non-Hermitian skin effect and topology in ultracold atoms. Phys. Rev. Lett., 2022, 129(7): 070401 https://doi.org/10.1103/PhysRevLett.129.070401
87	S. Scheurer M. , J. Slager R. . Unsupervised machine learning and band topology. Phys. Rev. Lett., 2020, 124(22): 226401 https://doi.org/10.1103/PhysRevLett.124.226401
88	W. Yu L. , L. Deng D. . Unsupervised learning of non-Hermitian topological phases. Phys. Rev. Lett., 2021, 126(24): 240402 https://doi.org/10.1103/PhysRevLett.126.240402
89	Yang R. , W. Tan J. , Tai T. , M. Koh J. , Li L. , Longhi S. , H. Lee C. . Designing non-Hermitian real spectra through electrostatics. Sci. Bull. (Beijing), 2022, 67(18): 1865 https://doi.org/10.1016/j.scib.2022.08.005
90	Oztas Z. , Candemir N. . Su−Schrieffer−Heeger model with imaginary gauge field. Phys. Lett. A, 2019, 383(15): 1821 https://doi.org/10.1016/j.physleta.2019.02.037
91	Zhu W. , X. Teo W. , Li L. , Gong J. . Delocalization of topological edge states. Phys. Rev. B, 2021, 103(19): 195414 https://doi.org/10.1103/PhysRevB.103.195414
92	Cheng J. , Zhang X. , H. Lu M. , F. Chen Y. . Competition between band topology and non-Hermiticity. Phys. Rev. B, 2022, 105(9): 094103 https://doi.org/10.1103/PhysRevB.105.094103
93	Okuma N. , Sato M. . Non-Hermitian topological phenomena: A review. Annu. Rev. Condens. Matter Phys., 2022, 14: 83 https://doi.org/10.1146/annurev-conmatphys-040521-033133
94	R. Wang X. , X. Guo C. , P. Kou S. . Defective edge states and number-anomalous bulk-boundary correspondence in non-Hermitian topological systems. Phys. Rev. B, 2020, 101(12): 121116 https://doi.org/10.1103/PhysRevB.101.121116
95	H. Lee C. , Li L. , Gong J. . Hybrid higher-order skin-topological modes in nonreciprocal systems. Phys. Rev. Lett., 2019, 123(1): 016805 https://doi.org/10.1103/PhysRevLett.123.016805
96	Li L. , H. Lee C. , Gong J. . Topological switch for non-Hermitian skin effect in cold-atom systems with loss. Phys. Rev. Lett., 2020, 124(25): 250402 https://doi.org/10.1103/PhysRevLett.124.250402
97	Li Y. , Liang C. , Wang C. , Lu C. , C. Liu Y. . Gain-loss-induced hybrid skin-topological effect. Phys. Rev. Lett., 2022, 128(22): 223903 https://doi.org/10.1103/PhysRevLett.128.223903
98	Zhu W. , Gong J. . Hybrid skin-topological modes without asymmetric couplings. Phys. Rev. B, 2022, 106(3): 035425 https://doi.org/10.1103/PhysRevB.106.035425
99	Kawabata K. , Sato M. , Shiozaki K. . Higher-order non-Hermitian skin effect. Phys. Rev. B, 2020, 102(20): 205118 https://doi.org/10.1103/PhysRevB.102.205118
100	Okugawa R. , Takahashi R. , Yokomizo K. . Second- order topological non-Hermitian skin effects. Phys. Rev. B, 2020, 102(24): 241202 https://doi.org/10.1103/PhysRevB.102.241202
101	Fu Y. , Hu J. , Wan S. . Non-Hermitian second-order skin and topological modes. Phys. Rev. B, 2021, 103(4): 045420 https://doi.org/10.1103/PhysRevB.103.045420
102	Zhang K. , Yang Z. , Fang C. . Correspondence be- tween winding numbers and skin modes in non-Hermitian systems. Phys. Rev. Lett., 2020, 125(12): 126402 https://doi.org/10.1103/PhysRevLett.125.126402
103	S. Borgnia D. , J. Kruchkov A. , J. Slager R. . Non-Hermitian boundary modes and topology. Phys. Rev. Lett., 2020, 124(5): 056802 https://doi.org/10.1103/PhysRevLett.124.056802
104	Okuma N. , Kawabata K. , Shiozaki K. , Sato M. . Topological origin of non-Hermitian skin effects. Phys. Rev. Lett., 2020, 124(8): 086801 https://doi.org/10.1103/PhysRevLett.124.086801
105	Li L. , Mu S. , H. Lee C. , Gong J. . Quantized classical response from spectral winding topology. Nat. Commun., 2021, 12(1): 5294 https://doi.org/10.1038/s41467-021-25626-z
106	Q. Liang H. , Mu S. , Gong J. , Li L. . Anomalous hybridization of spectral winding topology in quantized steady-state responses. Phys. Rev. B, 2022, 105(24): L241402 https://doi.org/10.1103/PhysRevB.105.L241402
107	Longhi S. . Topological phase transition in non-Hermitian quasicrystals. Phys. Rev. Lett., 2019, 122(23): 237601 https://doi.org/10.1103/PhysRevLett.122.237601
108	Jiang H. , J. Lang L. , Yang C. , L. Zhu S. , Chen S. . Interplay of non-Hermitian skin effects and Anderson localization in nonreciprocal quasiperiodic lattices. Phys. Rev. B, 2019, 100(5): 054301 https://doi.org/10.1103/PhysRevB.100.054301
109	Longhi S. . Metal−insulator phase transition in a non-Hermitian Aubry−André−Harper model. Phys. Rev. B, 2019, 100(12): 125157 https://doi.org/10.1103/PhysRevB.100.125157
110	B. Zeng Q. , Xu Y. . Winding numbers and generalized mobility edges in non-Hermitian systems. Phys. Rev. Res., 2020, 2(3): 033052 https://doi.org/10.1103/PhysRevResearch.2.033052
111	B. Zeng Q. , B. Yang Y. , Xu Y. . Topological phases in non-Hermitian Aubry−André−Harper models. Phys. Rev. B, 2020, 101(2): 020201 https://doi.org/10.1103/PhysRevB.101.020201
112	Liu Y. , P. Jiang X. , Cao J. , Chen S. . Non-Hermitian mobility edges in one-dimensional quasicrystals with parity−time symmetry. Phys. Rev. B, 2020, 101(17): 174205 https://doi.org/10.1103/PhysRevB.101.174205
113	J. Zhai L. , Yin S. , Y. Huang G. . Many-body localization in a non-Hermitian quasiperiodic system. Phys. Rev. B, 2020, 102(6): 064206 https://doi.org/10.1103/PhysRevB.102.064206
114	Cai X. . Boundary-dependent self-dualities, winding numbers, and asymmetrical localization in non-Hermitian aperiodic one-dimensional models. Phys. Rev. B, 2021, 103(1): 014201 https://doi.org/10.1103/PhysRevB.103.014201
115	Liu Y. , Wang Y. , J. Liu X. , Zhou Q. , Chen S. . Exact mobility edges, PT-symmetry breaking, and skin effect in one-dimensional non-Hermitian quasicrystals. Phys. Rev. B, 2021, 103(1): 014203 https://doi.org/10.1103/PhysRevB.103.014203
116	Liu Y. , Zhou Q. , Chen S. . Localization transition, spectrum structure, and winding numbers for one-dimensional non-Hermitian quasicrystals. Phys. Rev. B, 2021, 104(2): 024201 https://doi.org/10.1103/PhysRevB.104.024201
117	Claes J. , L. Hughes T. . Skin effect and winding number in disordered non-Hermitian systems. Phys. Rev. B, 2021, 103(14): L140201 https://doi.org/10.1103/PhysRevB.103.L140201
118	Longhi S. . Non-Hermitian topological mobility edges and transport in photonic quantum walks. Opt. Lett., 2022, 47(12): 2951 https://doi.org/10.1364/OL.460484
119	Zhang X. , Li G. , Liu Y. , Tai T. , Thomale R. , H. Lee C. . Tidal surface states as fingerprints of non-Hermitian nodal knot metals. Commun. Phys., 2021, 4(1): 47 https://doi.org/10.1038/s42005-021-00535-1
120	Herviou L. , H. Bardarson J. , Regnault N. . Defining a bulk-edge correspondence for non-Hermitian Hamiltonians via singular-value decomposition. Phys. Rev. A, 2019, 99(5): 052118 https://doi.org/10.1103/PhysRevA.99.052118
121	G. Zirnstein H. , Refael G. , Rosenow B. . Bulk-boundary correspondence for non-Hermitian Hamiltonians via Green functions. Phys. Rev. Lett., 2021, 126(21): 216407 https://doi.org/10.1103/PhysRevLett.126.216407
122	P. Su W. , R. Schrieffer J. , J. Heeger A. . Solitons in polyacetylene. Phys. Rev. Lett., 1979, 42(25): 1698 https://doi.org/10.1103/PhysRevLett.42.1698
123	Creutz M. . End states, ladder compounds, and domain-wall Fermions. Phys. Rev. Lett., 1999, 83(13): 2636 https://doi.org/10.1103/PhysRevLett.83.2636
124	Q. Liang H. , Li L. . Topological properties of non-Hermitian Creutz ladders. Chin. Phys. B, 2022, 31(1): 010310 https://doi.org/10.1088/1674-1056/ac3991
125	Edvardsson E. , K. Kunst F. , Yoshida T. , J. Bergholtz E. . Phase transitions and generalized biorthogonal polarization in non-Hermitian systems. Phys. Rev. Res., 2020, 2(4): 043046 https://doi.org/10.1103/PhysRevResearch.2.043046
126	Masuda S. , Nakamura M. . Relationship between the electronic polarization and the winding number in non-Hermitian systems. J. Phys. Soc. Jpn., 2022, 91(4): 043701 https://doi.org/10.7566/JPSJ.91.043701
127	Masuda S. , Nakamura M. . Electronic polarization in non-Bloch band theory. J. Phys. Soc. Jpn., 2022, 91(11): 114705 https://doi.org/10.7566/JPSJ.91.114705
128	S. Deng T. , Yi W. . Non-Bloch topological invariants in a non-Hermitian domain wall system. Phys. Rev. B, 2019, 100(3): 035102 https://doi.org/10.1103/PhysRevB.100.035102
129	Liu H.Lu M.Q. Zhang Z.Jiang H., Modified generalized-Brillouin-zone theory with on-site disorders, arXiv: 2208.03013 (2022)
130	Yang Z. , Zhang K. , Fang C. , Hu J. . Non-Hermitian bulk-boundary correspondence and auxiliary generalized Brillouin zone theory. Phys. Rev. Lett., 2020, 125(22): 226402 https://doi.org/10.1103/PhysRevLett.125.226402
131	Tai T.H. Lee C., Zoology of non-Hermitian spectra and their graph topology, arXiv: 2202.03462 (2022)
132	Q. Zhang Z. , Liu H. , Liu H. , Jiang H. , C. Xie X. . Bulk-boundary correspondence in disordered non-Hermitian systems. Sci. Bull. (Beijing), 2023, 68(2): 157 https://doi.org/10.1016/j.scib.2023.01.002
133	Chen R. , Z. Chen C. , Zhou B. , H. Xu D. . Finite-size effects in non-Hermitian topological systems. Phys. Rev. B, 2019, 99(15): 155431 https://doi.org/10.1103/PhysRevB.99.155431
134	M. Rafi-Ul-Islam S. , B. Siu Z. , Sahin H. , H. Lee C. , B. A. Jalil M. . Unconventional skin modes in generalized topolectrical circuits with multiple asymmetric couplings. Phys. Rev. Res., 2022, 4(4): 043108 https://doi.org/10.1103/PhysRevResearch.4.043108
135	Wang W. , Wang X. , Ma G. . Non-Hermitian morphing of topological modes. Nature, 2022, 608(7921): 50 https://doi.org/10.1038/s41586-022-04929-1
136	Wang W. , Wang X. , Ma G. . Extended state in a localized continuum. Phys. Rev. Lett., 2022, 129(26): 264301 https://doi.org/10.1103/PhysRevLett.129.264301
137	Longhi S. . Non-Hermitian gauged topological laser arrays. Ann. Phys., 2018, 530(7): 1800023 https://doi.org/10.1002/andp.201800023
138	Tang M.Wang J.Valligatla S.N. Saggau C.Dong H., et al.., Symmetry induced selective excitation of topological states in SSH waveguide arrays, arXiv: 2211.06228 (2022)
139	D. M. Haldane F. . Model for a quantum Hall effect without landau levels: Condensed-matter realization of the “parity anomaly”. Phys. Rev. Lett., 1988, 61(18): 2015 https://doi.org/10.1103/PhysRevLett.61.2015
140	L. Qi X. , L. Hughes T. , C. Zhang S. . Topological field theory of time-reversal invariant insulators. Phys. Rev. B, 2008, 78(19): 195424 https://doi.org/10.1103/PhysRevB.78.195424
141	Z. Chang C. , Zhang J. , Feng X. , Shen J. , Zhang Z. , Guo M. , Li K. , Ou Y. , Wei P. , L. Wang L. , Q. Ji Z. , Feng Y. , Ji S. , Chen X. , Jia J. , Dai X. , Fang Z. , C. Zhang S. , He K. , Wang Y. , Lu L. , C. Ma X. , K. Xue Q. . Experimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. Science, 2013, 340(6129): 167 https://doi.org/10.1126/science.1234414
142	H. Lee C. , L. Qi X. . Lattice construction of pseu-dopotential Hamiltonians for fractional Chern insulators. Phys. Rev. B, 2014, 90(8): 085103 https://doi.org/10.1103/PhysRevB.90.085103
143	Neupert T. , Chamon C. , Iadecola T. , H. Santos L. , Mudry C. . Fractional (Chern and topological) insulators. Phys. Scr., 2015, T164: 014005 https://doi.org/10.1088/0031-8949/2015/T164/014005
144	Yao S. , Song F. , Wang Z. . Non-Hermitian Chern bands. Phys. Rev. Lett., 2018, 121(13): 136802 https://doi.org/10.1103/PhysRevLett.121.136802
145	Kawabata K. , Shiozaki K. , Ueda M. . Anomalous helical edge states in a non-Hermitian Chern insulator. Phys. Rev. B, 2018, 98(16): 165148 https://doi.org/10.1103/PhysRevB.98.165148
146	X. Xiao Y. , T. Chan C. . Topology in non-Hermitian Chern insulators with skin effect. Phys. Rev. B, 2022, 105(7): 075128 https://doi.org/10.1103/PhysRevB.105.075128
147	Liu H. , S. You J. , Ryu S. , C. Fulga I. . Supermetal−insulator transition in a non-Hermitian network model. Phys. Rev. B, 2021, 104(15): 155412 https://doi.org/10.1103/PhysRevB.104.155412
148	M. Philip T. , R. Hirsbrunner M. , J. Gilbert M. . Loss of Hall conductivity quantization in a non-Hermitian quantum anomalous Hall insulator. Phys. Rev. B, 2018, 98(15): 155430 https://doi.org/10.1103/PhysRevB.98.155430
149	Sayyad S. , D. Hannukainen J. , G. Grushin A. . Non-Hermitian chiral anomalies. Phys. Rev. Res., 2022, 4(4): L042004 https://doi.org/10.1103/PhysRevResearch.4.L042004
150	Bessho T. , Sato M. . Nielsen−Ninomiya theorem with bulk topology: Duality in Floquet and non-Hermitian systems. Phys. Rev. Lett., 2021, 127(19): 196404 https://doi.org/10.1103/PhysRevLett.127.196404
151	Wang C. , R. Wang X. . Hermitian chiral boundary states in non-Hermitian topological insulators. Phys. Rev. B, 2022, 105(12): 125103 https://doi.org/10.1103/PhysRevB.105.125103
152	H. Lee C. , Wang Y. , Chen Y. , Zhang X. . Electromagnetic response of quantum Hall systems in dimensions five and six and beyond. Phys. Rev. B, 2018, 98(9): 094434 https://doi.org/10.1103/PhysRevB.98.094434
153	Petrides I. , M. Price H. , Zilberberg O. . Six-dimensional quantum Hall effect and three-dimensional topological pumps. Phys. Rev. B, 2018, 98(12): 125431 https://doi.org/10.1103/PhysRevB.98.125431
154	Shao K. , T. Cai Z. , Geng H. , Chen W. , Y. Xing D. . Cyclotron quantization and mirror-time transition on nonreciprocal lattices. Phys. Rev. B, 2022, 106(8): L081402 https://doi.org/10.1103/PhysRevB.106.L081402
155	Deng K. , Flebus B. . Non-Hermitian skin effect in magnetic systems. Phys. Rev. B, 2022, 105(18): L180406 https://doi.org/10.1103/PhysRevB.105.L180406
156	M. Denner M.Schindler F., Magnetic flux response of non-Hermitian topological phases, arXiv: 2208.11712 (2022)
157	Lu M. , X. Zhang X. , Franz M. . Magnetic suppression of non-Hermitian skin effects. Phys. Rev. Lett., 2021, 127(25): 256402 https://doi.org/10.1103/PhysRevLett.127.256402
158	Zhen B. , W. Hsu C. , Igarashi Y. , Lu L. , Kaminer I. , Pick A. , L. Chua S. , D. Joannopoulos J. , Soljačić M. . Spawning rings of exceptional points out of Dirac cones. Nature, 2015, 525(7569): 354 https://doi.org/10.1038/nature14889
159	Xu Y. , T. Wang S. , M. Duan L. . Weyl exceptional rings in a three-dimensional dissipative cold atomic gas. Phys. Rev. Lett., 2017, 118(4): 045701 https://doi.org/10.1103/PhysRevLett.118.045701
160	Liu J. , Li Z. , G. Chen Z. , Tang W. , Chen A. , Liang B. , Ma G. , C. Cheng J. . Experimental realization of Weyl exceptional rings in a synthetic three-dimensional non-Hermitian phononic crystal. Phys. Rev. Lett., 2022, 129(8): 084301 https://doi.org/10.1103/PhysRevLett.129.084301
161	Cerjan A. , Huang S. , Wang M. , P. Chen K. , Chong Y. , C. Rechtsman M. . Experimental realization of a Weyl exceptional ring. Nat. Photonics, 2019, 13(9): 623 https://doi.org/10.1038/s41566-019-0453-z
162	Tang W.Ding K.Ma G., Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces, arXiv: 2211.15921 (2022)
163	Bzdušek T. , S. Wu Q. , Rüegg A. , Sigrist M. , A. Soluyanov A. . Nodal-chain metals. Nature, 2016, 538: 75 https://doi.org/10.1038/nature19099
164	Huh Y. , G. Moon E. , B. Kim Y. . Long-range Coulomb interaction in nodal-ring semimetals. Phys. Rev. B, 2016, 93(3): 035138 https://doi.org/10.1103/PhysRevB.93.035138
165	Li L. , A. N. Araujo M. . Topological insulating phases from two-dimensional nodal loop semimetals. Phys. Rev. B, 2016, 94(16): 165117 https://doi.org/10.1103/PhysRevB.94.165117
166	Li L. , Yin C. , Chen S. , A. N. Araujo M. . Chiral topological insulating phases from three-dimensional nodal loop semimetals. Phys. Rev. B, 2017, 95(12): 121107 https://doi.org/10.1103/PhysRevB.95.121107
167	Li L. , Chesi S. , Yin C. , Chen S. . 2π-flux loop semimetals. Phys. Rev. B, 2017, 96(8): 081116 https://doi.org/10.1103/PhysRevB.96.081116
168	Yan Z. , Bi R. , Shen H. , Lu L. , C. Zhang S. , Wang Z. . Nodal-link semimetals. Phys. Rev. B, 2017, 96(4): 041103 https://doi.org/10.1103/PhysRevB.96.041103
169	Li S. , M. Yu Z. , Liu Y. , Guan S. , S. Wang S. , Zhang X. , Yao Y. , A. Yang S. . Type-II nodal loops: Theory and material realization. Phys. Rev. B, 2017, 96(8): 081106 https://doi.org/10.1103/PhysRevB.96.081106
170	Li L. , H. Yap H. , A. N. Araújo M. , Gong J. . Engineering topological phases with a three-dimensional nodal-loop semimetal. Phys. Rev. B, 2017, 96(23): 235424 https://doi.org/10.1103/PhysRevB.96.235424
171	Zhou Y. , Xiong F. , Wan X. , An J. . Hopf-link topological nodal-loop semimetals. Phys. Rev. B, 2018, 97(15): 155140 https://doi.org/10.1103/PhysRevB.97.155140
172	Pezzini S. , R. van Delft M. , M. Schoop L. , V. Lotsch B. , Carrington A. , I. Katsnelson M. , E. Hussey N. , Wiedmann S. . Unconventional mass enhancement around the Dirac nodal loop in ZrSiS. Nat. Phys., 2018, 14(2): 178 https://doi.org/10.1038/nphys4306
173	Zhang X. , M. Yu Z. , Zhu Z. , Wu W. , S. Wang S. , L. Sheng X. , A. Yang S. . Nodal loop and nodal surface states in the Ti3Al family of materials. Phys. Rev. B, 2018, 97(23): 235150 https://doi.org/10.1103/PhysRevB.97.235150
174	H. Lee C. , W. Ho W. , Yang B. , Gong J. , Papić Z. . Floquet mechanism for non-Abelian fractional quantum Hall states. Phys. Rev. Lett., 2018, 121(23): 237401 https://doi.org/10.1103/PhysRevLett.121.237401
175	N. Ünal F. , Bouhon A. , J. Slager R. . Topological Euler class as a dynamical observable in optical lattices. Phys. Rev. Lett., 2020, 125(5): 053601 https://doi.org/10.1103/PhysRevLett.125.053601
176	Bouhon A. , S. Wu Q. , J. Slager R. , Weng H. , V. Yazyev O. , Bzdušek T. . Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe. Nat. Phys., 2020, 16(11): 1137 https://doi.org/10.1038/s41567-020-0967-9
177	H. Lee C. , H. Yap H. , Tai T. , Xu G. , Zhang X. , Gong J. . Enhanced higher harmonic generation from nodal topology. Phys. Rev. B, 2020, 102(3): 035138 https://doi.org/10.1103/PhysRevB.102.035138
178	S. Ang Y.H. Lee C.K. Ang L., Universal scaling and signatures of nodal structures in electron tunneling from two-dimensional semimetals, arXiv: 2003.14004 (2020)
179	Yang E. , Yang B. , You O. , C. Chan H. , Mao P. , Guo Q. , Ma S. , Xia L. , Fan D. , Xiang Y. , Zhang S. . Observation of non-Abelian nodal links in photonics. Phys. Rev. Lett., 2020, 125(3): 033901 https://doi.org/10.1103/PhysRevLett.125.033901
180	H. Lee C. , Sutrisno A. , Hofmann T. , Helbig T. , Liu Y. , S. Ang Y. , K. Ang L. , Zhang X. , Greiter M. , Thomale R. . Imaging nodal knots in momentum space through topolectrical circuits. Nat. Commun., 2020, 11: 4385 https://doi.org/10.1038/s41467-020-17716-1
181	Tai T. , H. Lee C. . Anisotropic nonlinear optical response of nodal-loop materials. Phys. Rev. B, 2021, 103(19): 195125 https://doi.org/10.1103/PhysRevB.103.195125
182	M. Lenggenhager P. , Liu X. , S. Tsirkin S. , Neupert T. , Bzdušek T. . From triple-point materials to multiband nodal links. Phys. Rev. B, 2021, 103(12): L121101 https://doi.org/10.1103/PhysRevB.103.L121101
183	Wang M. , Liu S. , Ma Q. , Y. Zhang R. , Wang D. , Guo Q. , Yang B. , Ke M. , Liu Z. , T. Chan C. . Experimental observation of non-Abelian earring nodal links in phononic crystals. Phys. Rev. Lett., 2022, 128(24): 246601 https://doi.org/10.1103/PhysRevLett.128.246601
184	J. Slager R.Bouhon A.N. Ünal F., Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase, arXiv: 2208.12824 (2022)
185	Peng B. , Bouhon A. , Monserrat B. , J. Slager R. . Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates. Nat. Commun., 2022, 13(1): 423 https://doi.org/10.1038/s41467-022-28046-9
186	Bouhon A.J. Slager R., Multi-gap topological conversion of Euler class via band-node braiding: Minimal models, PT-linked nodal rings, and chiral heirs, arXiv: 2203.16741 (2022)
187	Park H. , Wong S. , Bouhon A. , J. Slager R. , S. Oh S. . Topological phase transitions of non-Abelian charged nodal lines in spring-mass systems. Phys. Rev. B, 2022, 105(21): 214108 https://doi.org/10.1103/PhysRevB.105.214108
188	Yang X. , Cao Y. , Zhai Y. . Non-Hermitian Weyl semimetals: Non-Hermitian skin effect and non-Bloch bulk-boundary correspondence. Chin. Phys. B, 2022, 31(1): 010308 https://doi.org/10.1088/1674-1056/ac3738
189	Li L. , H. Lee C. , Gong J. . Realistic Floquet semimetal with exotic topological linkages between arbitrarily many nodal loops. Phys. Rev. Lett., 2018, 121(3): 036401 https://doi.org/10.1103/PhysRevLett.121.036401
190	Yan Z. , Wang Z. . Floquet multi-Weyl points in crossing-nodal-line semimetals. Phys. Rev. B, 2017, 96(4): 041206 https://doi.org/10.1103/PhysRevB.96.041206
191	Carlström J. , J. Bergholtz E. . Exceptional links and twisted Fermi ribbons in non-Hermitian systems. Phys. Rev. A, 2018, 98(4): 042114 https://doi.org/10.1103/PhysRevA.98.042114
192	Chen R. , Zhou B. , H. Xu D. . Floquet Weyl semimetals in light-irradiated type-II and hybrid line-node semimetals. Phys. Rev. B, 2018, 97(15): 155152 https://doi.org/10.1103/PhysRevB.97.155152
193	Carlström J. , Stålhammar M. , C. Budich J. , J. Bergholtz E. . Knotted non-Hermitian metals. Phys. Rev. B, 2019, 99(16): 161115 https://doi.org/10.1103/PhysRevB.99.161115
194	W. Kim K. , Kwon H. , Park K. . Floquet topological semimetal with a helical nodal line in 2+1 dimensions. Phys. Rev. B, 2019, 99(11): 115136 https://doi.org/10.1103/PhysRevB.99.115136
195	Stålhammar M. , Rødland L. , Arone G. , C. Budich J. , Bergholtz E. . Hyperbolic nodal band structures and knot invariants. SciPost Phys., 2019, 7: 019 https://doi.org/10.21468/SciPostPhys.7.2.019
196	Salerno G. , Goldman N. , Palumbo G. . Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric. Phys. Rev. Res., 2020, 2(1): 013224 https://doi.org/10.1103/PhysRevResearch.2.013224
197	Meng H. , Wang L. , H. Lee C. , S. Ang Y. . Terahertz polarization conversion from optical dichroism in a topological Dirac semimetal. Appl. Phys. Lett., 2022, 121(19): 193102 https://doi.org/10.1063/5.0122299
198	Qin F. , H. Lee C. , Chen R. . Light-induced phase crossovers in a quantum spin Hall system. Phys. Rev. B, 2022, 106(23): 235405 https://doi.org/10.1103/PhysRevB.106.235405
199	Wu H. , H. An J. . Non-Hermitian Weyl semimetal and its Floquet engineering. Phys. Rev. B, 2022, 105(12): L121113 https://doi.org/10.1103/PhysRevB.105.L121113
200	Wang K. , Xiao L. , C. Budich J. , Yi W. , Xue P. . Simulating exceptional non-Hermitian metals with single-photon interferometry. Phys. Rev. Lett., 2021, 127(2): 026404 https://doi.org/10.1103/PhysRevLett.127.026404
201	A. Benalcazar W. , A. Bernevig B. , L. Hughes T. . Quantized electric multipole insulators. Science, 2017, 357(6346): 61 https://doi.org/10.1126/science.aah6442
202	Edvardsson E. , K. Kunst F. , J. Bergholtz E. . Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence. Phys. Rev. B, 2019, 99(8): 081302 https://doi.org/10.1103/PhysRevB.99.081302
203	K. Ghosh A. , Nag T. . Non-Hermitian higher-order topological superconductors in two dimensions: Statics and dynamics. Phys. Rev. B, 2022, 106(14): L140303 https://doi.org/10.1103/PhysRevB.106.L140303
204	M. Kim K. , J. Park M. . Disorder-driven phase transition in the second-order non-Hermitian skin effect. Phys. Rev. B, 2021, 104(12): L121101 https://doi.org/10.1103/PhysRevB.104.L121101
205	T. Lei Z.H. Lee C.H. Li L., PT-activated non-Hermitian skin modes, arXiv: 2304.13955v1 (2023)
206	Note that asymmetric couplings are not always need to break reciprocity [98].
207	Li J. , K. Harter A. , Liu J. , de Melo L. , N. Joglekar Y. , Luo L. . Observation of parity-time symmetry breaking transitions in a dissipative Floquet system of ultracold atoms. Nat. Commun., 2019, 10(1): 855 https://doi.org/10.1038/s41467-019-08596-1
208	Wu H. , Q. Wang B. , H. An J. . Floquet second-order topological insulators in non-Hermitian systems. Phys. Rev. B, 2021, 103(4): L041115 https://doi.org/10.1103/PhysRevB.103.L041115
209	W. Zhou L. , W. Bomantara R. , L. Wu S. . qth-root non-Hermitian Floquet topological insulators. SciPost Phys., 2022, 13: 015 https://doi.org/10.21468/SciPostPhys.13.2.015
210	Cao Y. , Li Y. , Yang X. . Non-Hermitian bulk-boundary correspondence in a periodically driven system. Phys. Rev. B, 2021, 103(7): 075126 https://doi.org/10.1103/PhysRevB.103.075126
211	Ezawa M. . Non-Hermitian higher-order topological states in nonreciprocal and reciprocal systems with their electric-circuit realization. Phys. Rev. B, 2019, 99(20): 201411 https://doi.org/10.1103/PhysRevB.99.201411
212	Ezawa M. . Non-Hermitian boundary and interface states in nonreciprocal higher-order topological metals and electrical circuits. Phys. Rev. B, 2019, 99(12): 121411 https://doi.org/10.1103/PhysRevB.99.121411
213	Song Z. , Fang Z. , Fang C. . (d-2)-dimensional edge states of rotation symmetry protected topological states. Phys. Rev. Lett., 2017, 119(24): 246402 https://doi.org/10.1103/PhysRevLett.119.246402
214	Khalaf E. . Higher-order topological insulators and superconductors protected by inversion symmetry. Phys. Rev. B, 2018, 97(20): 205136 https://doi.org/10.1103/PhysRevB.97.205136
215	Trifunovic L. , W. Brouwer P. . Higher-order topological band structures. physica status solidi (b), 2021, 258: 2000090 https://doi.org/10.1002/pssb.202000090
216	A. Li C.Trauzettel B.Neupert T.B. Zhang S., Enhancement of second-order non-Hermitian skin effect by magnetic fields, arXiv: 2212.14691v1 (2022)
217	Jiang H.H. Lee C., Dimensional transmutation from non-hermiticity, arXiv: 2207.08843 (2022)
218	Zhu P.Q. Sun X.L. Hughes T.Bahl G., Higher rank chirality and non-Hermitian skin effect in a topolectrical circuit, arXiv: 2207.02228 (2022)
219	Song F.Y. Wang H.Wang Z., Non-Bloch PT symmetry breaking: Universal threshold and dimensional surprise, A Festschrift in Honor of the C. N. Yang Centenary, arXiv: 2102.02230 (2022)
220	Jiang H. , H. Lee C. . Filling up complex spectral regions through non-Hermitian disordered chains. Chin. Phys. B, 2022, 31(5): 050307 https://doi.org/10.1088/1674-1056/ac4a73
221	Kawabata K. , Shiozaki K. , Ueda M. , Sato M. . Symmetry and topology in non-Hermitian physics. Phys. Rev. X, 2019, 9(4): 041015 https://doi.org/10.1103/PhysRevX.9.041015
222	Theiler J. . Estimating fractal dimension. J. Opt. Soc. Am. A, 1990, 7(6): 1055 https://doi.org/10.1364/JOSAA.7.001055
223	L. Qi X., Exact holographic mapping and emergent space−time geometry, arXiv: 1309.6282 (2013)
224	H. Lee C. , L. Qi X. . Exact holographic mapping in free Fermion systems. Phys. Rev. B, 2016, 93(3): 035112 https://doi.org/10.1103/PhysRevB.93.035112
225	Gu Y. , H. Lee C. , Wen X. , Y. Cho G. , Ryu S. , L. Qi X. . Holographic duality between (2+1)-dimensional quantum anomalous Hall state and (3+1)-dimensional topological insulators. Phys. Rev. B, 2016, 94(12): 125107 https://doi.org/10.1103/PhysRevB.94.125107
226	Yoshida T. , Mizoguchi T. , Hatsugai Y. . Mirror skin effect and its electric circuit simulation. Phys. Rev. Res., 2020, 2(2): 022062 https://doi.org/10.1103/PhysRevResearch.2.022062
227	H. Liu C. , Chen S. . Topological classification of defects in non-Hermitian systems. Phys. Rev. B, 2019, 100(14): 144106 https://doi.org/10.1103/PhysRevB.100.144106
228	H. Liu C. , Hu H. , Chen S. . Symmetry and topological classification of Floquet non-Hermitian systems. Phys. Rev. B, 2022, 105(21): 214305 https://doi.org/10.1103/PhysRevB.105.214305
229	Okugawa R. , Takahashi R. , Yokomizo K. . Non-Hermitian band topology with generalized inversion symmetry. Phys. Rev. B, 2021, 103(20): 205205 https://doi.org/10.1103/PhysRevB.103.205205
230	M. Vecsei P. , M. Denner M. , Neupert T. , Schindler F. . Symmetry indicators for inversion-symmetric non-Hermitian topological band structures. Phys. Rev. B, 2021, 103(20): L201114 https://doi.org/10.1103/PhysRevB.103.L201114
231	Gong Z. , Ashida Y. , Kawabata K. , Takasan K. , Higashikawa S. , Ueda M. . Topological phases of non-Hermitian systems. Phys. Rev. X, 2018, 8(3): 031079 https://doi.org/10.1103/PhysRevX.8.031079
232	M. Denner M. , Skurativska A. , Schindler F. , H. Fischer M. , Thomale R. , Bzdušek T. , Neupert T. . Exceptional topological insulators. Nat. Commun., 2021, 12: 5681 https://doi.org/10.1038/s41467-021-25947-z
233	Nakamura D.Bessho T.Sato M., Bulk-boundary correspondence in point-gap topological phases, arXiv: 2205.15635 (2022)
234	Song F. , Yao S. , Wang Z. . Non-Hermitian topological invariants in real space. Phys. Rev. Lett., 2019, 123(24): 246801 https://doi.org/10.1103/PhysRevLett.123.246801
235	Z. Tang L. , F. Zhang L. , Q. Zhang G. , W. Zhang D. . Topological Anderson insulators in two-dimensional non-Hermitian disordered systems. Phys. Rev. A, 2020, 101(6): 063612 https://doi.org/10.1103/PhysRevA.101.063612
236	A. A. Ghorashi S. , Li T. , Sato M. , L. Hughes T. . Non-Hermitian higher-order Dirac semimetals. Phys. Rev. B, 2021, 104(16): L161116 https://doi.org/10.1103/PhysRevB.104.L161116
237	A. A. Ghorashi S. , Li T. , Sato M. . Non-Hermitian higher-order Weyl semimetals. Phys. Rev. B, 2021, 104(16): L161117 https://doi.org/10.1103/PhysRevB.104.L161117
238	Edvardsson E. , Ardonne E. . Sensitivity of non-Hermitian systems. Phys. Rev. B, 2022, 106(11): 115107 https://doi.org/10.1103/PhysRevB.106.115107
239	Böttcher A.M. Grudsky S., Spectral Properties of Banded Toeplitz Matrices, SIAM, 2005
240	Bartlett J. , Hu H. , Zhao E. . Illuminating the bulk-boundary correspondence of a non-Hermitian stub lattice with Majorana stars. Phys. Rev. B, 2021, 104(19): 195131 https://doi.org/10.1103/PhysRevB.104.195131
241	X. Teo W. , Li L. , Zhang X. , Gong J. . Topological characterization of non-Hermitian multiband systems using Majorana’s stellar representation. Phys. Rev. B, 2020, 101(20): 205309 https://doi.org/10.1103/PhysRevB.101.205309
242	S. Pan J. , Li L. , Gong J. . Point-gap topology with complete bulk-boundary correspondence and anomalous amplification in the Fock space of dissipative quantum systems. Phys. Rev. B, 2021, 103(20): 205425 https://doi.org/10.1103/PhysRevB.103.205425
243	Wang K. , Dutt A. , Y. Yang K. , C. Wojcik C. , Vučković J. , Fan S. . Generating arbitrary topological windings of a non-Hermitian band. Science, 2021, 371(6535): 1240 https://doi.org/10.1126/science.abf6568
244	H. Lee C. , Imhof S. , Berger C. , Bayer F. , Brehm J. , W. Molenkamp L. , Kiessling T. , Thomale R. . Topolectrical circuits. Commun. Phys., 2018, 1: 39 https://doi.org/10.1038/s42005-018-0035-2
245	Liu Y. , Zeng Y. , Li L. , Chen S. . Exact solution of the single impurity problem in nonreciprocal lattices: Impurity-induced size-dependent non-Hermitian skin effect. Phys. Rev. B, 2021, 104(8): 085401 https://doi.org/10.1103/PhysRevB.104.085401
246	Ou Z. , Wang Y. , Li L. . Non-Hermitian boundary spectral winding. Phys. Rev. B, 2023, 107: L161404 https://doi.org/10.1103/PhysRevB.107.L161404
247	Okuma N. . Boundary-dependent dynamical instability of bosonic Green’s function: Dissipative Bogoliubov–de Gennes Hamiltonian and its application to non-Hermitian skin effect. Phys. Rev. B, 2022, 105(22): 224301 https://doi.org/10.1103/PhysRevB.105.224301
248	Mao L. , Deng T. , Zhang P. . Boundary condition independence of non-Hermitian Hamiltonian dynamics. Phys. Rev. B, 2021, 104(12): 125435 https://doi.org/10.1103/PhysRevB.104.125435
249	Hu H. , Zhao E. . Knots and non-Hermitian Bloch bands. Phys. Rev. Lett., 2021, 126(1): 010401 https://doi.org/10.1103/PhysRevLett.126.010401
250	Li Y. , Ji X. , Chen Y. , Yan X. , Yang X. . Topological energy braiding of non-Bloch bands. Phys. Rev. B, 2022, 106(19): 195425 https://doi.org/10.1103/PhysRevB.106.195425
251	Louis H., Kauffman, Knots and Physics, Vol. 1, World Scientific, 1991
252	Hu H. , Sun S. , Chen S. . Knot topology of exceptional point and non-Hermitian no−go theorem. Phys. Rev. Res., 2022, 4(2): L022064 https://doi.org/10.1103/PhysRevResearch.4.L022064
253	Li L. , H. Lee C. . Non-Hermitian pseudo-gaps. Sci. Bull. (Beijing), 2022, 67(7): 685 https://doi.org/10.1016/j.scib.2022.01.017
254	Wang K. , Dutt A. , C. Wojcik C. , Fan S. . Topological complex-energy braiding of non-Hermitian bands. Nature, 2021, 598(7879): 59 https://doi.org/10.1038/s41586-021-03848-x
255	Tang W. , Ding K. , Ma G. . Experimental realization of non-Abelian permutations in a three-state non-Hermitian system. Natl. Sci. Rev., 2022, 9(11): nwac010 https://doi.org/10.1093/nsr/nwac010
256	Tang W. , Jiang X. , Ding K. , X. Xiao Y. , Q. Zhang Z. , T. Chan C. , Ma G. . Exceptional nexus with a hybrid topological invariant. Science, 2020, 370(6520): 1077 https://doi.org/10.1126/science.abd8872
257	Fu Y.Zhang Y., Anatomy of open-boundary bulk in multiband non-Hermitian systems, arXiv: 2212.13753 (2022)
258	Kohn W. . Analytic properties of Bloch waves and Wannier functions. Phys. Rev., 1959, 115(4): 809 https://doi.org/10.1103/PhysRev.115.809
259	He L. , Vanderbilt D. . Exponential decay properties of Wannier functions and related quantities. Phys. Rev. Lett., 2001, 86(23): 5341 https://doi.org/10.1103/PhysRevLett.86.5341
260	Brouder C. , Panati G. , Calandra M. , Mourougane C. , Marzari N. . Exponential localization of Wannier functions in insulators. Phys. Rev. Lett., 2007, 98(4): 046402 https://doi.org/10.1103/PhysRevLett.98.046402
261	H. Lee C. , P. Arovas D. , Thomale R. . Band flatness optimization through complex analysis. Phys. Rev. B, 2016, 93(15): 155155 https://doi.org/10.1103/PhysRevB.93.155155
262	Monaco D. , Panati G. , Pisante A. , Teufel S. . Optimal decay of Wannier functions in Chern and quantum Hall insulators. Commun. Math. Phys., 2018, 359(1): 61 https://doi.org/10.1007/s00220-017-3067-7
263	Longhi S. . Non-Bloch-band collapse and chiral Zener tunneling. Phys. Rev. Lett., 2020, 124(6): 066602 https://doi.org/10.1103/PhysRevLett.124.066602
264	Qin F.Ma Y.Shen R.H. Lee C., Universal competitive spectral scaling from the critical non-Hermitian skin effect, arXiv: 2212.13536 (2022)
265	M. Rafi-Ul-Islam S. , B. Siu Z. , Sahin H. , H. Lee C. , B. A. Jalil M. . Critical hybridization of skin modes in coupled non-Hermitian chains. Phys. Rev. Res., 2022, 4(1): 013243 https://doi.org/10.1103/PhysRevResearch.4.013243
266	M. Rafi-Ul-Islam S. , B. Siu Z. , Sahin H. , H. Lee C. , B. A. Jalil M. . System size dependent topological zero modes in coupled topolectrical chains. Phys. Rev. B, 2022, 106(7): 075158 https://doi.org/10.1103/PhysRevB.106.075158
267	Sun G. , C. Tang J. , P. Kou S. . Biorthogonal quantum criticality in non-Hermitian many-body systems. Front. Phys., 2022, 17(3): 33502 https://doi.org/10.1007/s11467-021-1126-1
268	X. Bao X. , F. Guo G. , P. Du X. , Q. Gu H. , Tan L. . The topological criticality in disordered non-Hermitian system. J. Phys.: Condens. Matter, 2021, 33(18): 185401 https://doi.org/10.1088/1361-648X/abee3d
269	Arouca R. , H. Lee C. , M. Smith C. . Unconventional scaling at non-Hermitian critical points. Phys. Rev. B, 2020, 102(24): 245145 https://doi.org/10.1103/PhysRevB.102.245145
270	Aquino R.Lopes N.G. Barci D., Critical and non-critical non-Hermitian topological phase transitions in one dimensional chains, arXiv: 2208.14400 (2022)
271	Rahul S. , Sarkar S. . Topological quantum criticality in non-Hermitian extended Kitaev chain. Sci. Rep., 2022, 12(1): 1 https://doi.org/10.1038/s41598-022-11126-7
272	K. Jian S. , C. Yang Z. , Bi Z. , Chen X. . Yang−Lee edge singularity triggered entanglement transition. Phys. Rev. B, 2021, 104(16): L161107 https://doi.org/10.1103/PhysRevB.104.L161107
273	Shen R.Chen T.Qin F.Zhong Y.H. Lee C., Proposal for observing Yang−Lee criticality in Rydberg atomic arrays, arXiv: 2302.06662 (2023)
274	Zhou B. , Wang R. , Wang B. . Renormalization group approach to non-Hermitian topological quantum criticality. Phys. Rev. B, 2020, 102(20): 205116 https://doi.org/10.1103/PhysRevB.102.205116
275	Yin S. , Y. Huang G. , Y. Lo C. , Chen P. . Kibble−Zurek scaling in the Yang−Lee edge singularity. Phys. Rev. Lett., 2017, 118(6): 065701 https://doi.org/10.1103/PhysRevLett.118.065701
276	Okuma N. , Sato M. . Quantum anomaly, non-Hermitian skin effects, and entanglement entropy in open systems. Phys. Rev. B, 2021, 103(8): 085428 https://doi.org/10.1103/PhysRevB.103.085428
277	Kawabata K.Numasawa T.Ryu S., Entanglement phase transition induced by the non-Hermitian skin effect, arXiv: 2206.05384 (2022)
278	H. Lee C. . Exceptional bound states and negative entanglement entropy. Phys. Rev. Lett., 2022, 128(1): 010402 https://doi.org/10.1103/PhysRevLett.128.010402
279	Peschel I. , Eisler V. . Reduced density matrices and entanglement entropy in free lattice models. J. Phys. A Math. Theor., 2009, 42(50): 504003 https://doi.org/10.1088/1751-8113/42/50/504003
280	L. Qi X. . Generic wave-function description of fractional quantum anomalous Hall states and fractional topological insulators. Phys. Rev. Lett., 2011, 107(12): 126803 https://doi.org/10.1103/PhysRevLett.107.126803
281	L. Hughes T. , Prodan E. , A. Bernevig B. . Inversion-symmetric topological insulators. Phys. Rev. B, 2011, 83(24): 245132 https://doi.org/10.1103/PhysRevB.83.245132
282	Alexandradinata A. , L. Hughes T. , A. Bernevig B. . Trace index and spectral flow in the entanglement spectrum of topological insulators. Phys. Rev. B, 2011, 84(19): 195103 https://doi.org/10.1103/PhysRevB.84.195103
283	H. Lee C. , Ye P. . Free-Fermion entanglement spectrum through Wannier interpolation. Phys. Rev. B, 2015, 91(8): 085119 https://doi.org/10.1103/PhysRevB.91.085119
284	Herviou L. , Regnault N. , H. Bardarson J. . Entanglement spectrum and symmetries in non-Hermitian Fermionic non-interacting models. SciPost Phys., 2019, 7: 069 https://doi.org/10.21468/SciPostPhys.7.5.069
285	M. Chen L. , A. Chen S. , Ye P. . Entanglement, non-hermiticity, and duality. SciPost Phys., 2021, 11: 003 https://doi.org/10.21468/SciPostPhys.11.1.003
286	Y. Chang P. , S. You J. , Wen X. , Ryu S. . Entanglement spectrum and entropy in topological non-Hermitian systems and nonunitary conformal field theory. Phys. Rev. Res., 2020, 2(3): 033069 https://doi.org/10.1103/PhysRevResearch.2.033069
287	Li H. , Wan S. . Dynamic skin effects in non-Hermitian systems. Phys. Rev. B, 2022, 106(24): L241112 https://doi.org/10.1103/PhysRevB.106.L241112
288	Longhi S. . Non-Hermitian skin effect and self-acceleration. Phys. Rev. B, 2022, 105(24): 245143 https://doi.org/10.1103/PhysRevB.105.245143
289	Li T. , Z. Sun J. , S. Zhang Y. , Yi W. . Non-Bloch quench dynamics. Phys. Rev. Res., 2021, 3(2): 023022 https://doi.org/10.1103/PhysRevResearch.3.023022
290	H. Lee C. , Longhi S. . Ultrafast and anharmonic Rabi oscillations between non-Bloch bands. Commun. Phys., 2020, 3: 1
291	Li L. , X. Teo W. , Mu S. , Gong J. . Direction reversal of non-Hermitian skin effect via coherent coupling. Phys. Rev. B, 2022, 106(8): 085427 https://doi.org/10.1103/PhysRevB.106.085427
292	Peng Y. , Jie J. , Yu D. , Wang Y. . Manipulating the non-Hermitian skin effect via electric fields. Phys. Rev. B, 2022, 106(16): L161402 https://doi.org/10.1103/PhysRevB.106.L161402
293	Zhang X. , Gong J. . Non-Hermitian Floquet topological phases: Exceptional points, coalescent edge modes, and the skin effect. Phys. Rev. B, 2020, 101(4): 045415 https://doi.org/10.1103/PhysRevB.101.045415
294	Longhi S. . Probing non-Hermitian skin effect and non-Bloch phase transitions. Phys. Rev. Res., 2019, 1(2): 023013 https://doi.org/10.1103/PhysRevResearch.1.023013
295	Yang F. , D. Jiang Q. , J. Bergholtz E. . Liouvillian skin effect in an exactly solvable model. Phys. Rev. Res., 2022, 4(2): 023160 https://doi.org/10.1103/PhysRevResearch.4.023160
296	Longhi S. . Unraveling the non-Hermitian skin effect in dissipative systems. Phys. Rev. B, 2020, 102(20): 201103 https://doi.org/10.1103/PhysRevB.102.201103
297	Longhi S. . Stochastic non-Hermitian skin effect. Opt. Lett., 2020, 45(18): 5250 https://doi.org/10.1364/OL.403182
298	Longhi S. . Bulk-edge correspondence and trapping at a non-Hermitian topological interface. Opt. Lett., 2021, 46(24): 6107 https://doi.org/10.1364/OL.445437
299	Longhi S. . Non-Hermitian Hartman effect. Ann. Phys., 2022, 534(10): 2200250 https://doi.org/10.1002/andp.202200250
300	Stegmaier A. , Imhof S. , Helbig T. , Hofmann T. , H. Lee C. , Kremer M. , Fritzsche A. , Feichtner T. , Klembt S. , Höfling S. , Boettcher I. , C. Fulga I. , Ma L. , G. Schmidt O. , Greiter M. , Kiessling T. , Szameit A. , Thomale R. . Topological defect engineering and PT symmetry in non-Hermitian electrical circuits. Phys. Rev. Lett., 2021, 126(21): 215302 https://doi.org/10.1103/PhysRevLett.126.215302
301	Longhi S. . Phase transitions in a non-Hermitian Aubry−André−Harper model. Phys. Rev. B, 2021, 103(5): 054203 https://doi.org/10.1103/PhysRevB.103.054203
302	Suthar K. , C. Wang Y. , P. Huang Y. , H. Jen H. , S. You J. . Non-Hermitian many-body localization with open boundaries. Phys. Rev. B, 2022, 106(6): 064208 https://doi.org/10.1103/PhysRevB.106.064208
303	Zhang B. , Li Q. , Zhang X. , H. Lee C. . Real non-Hermitian energy spectra without any symmetry. Chin. Phys. B, 2022, 31(7): 070308 https://doi.org/10.1088/1674-1056/ac67c6
304	C. Rechtsman M. , M. Zeuner J. , Plotnik Y. , Lumer Y. , Podolsky D. , Dreisow F. , Nolte S. , Segev M. , Szameit A. . Photonic Floquet topological insulators. Nature, 2013, 496: 196 https://doi.org/10.1038/nature12066
305	Fläschner N. , S. Rem B. , Tarnowski M. , Vogel D. , S. Lühmann D. , Sengstock K. , Weitenberg C. . Experimental reconstruction of the Berry curvature in a Floquet Bloch band. Science, 2016, 352(6289): 1091 https://doi.org/10.1126/science.aad4568
306	Zhou L. , Gong J. . Non-Hermitian Floquet topological phases with arbitrarily many real-quasienergy edge states. Phys. Rev. B, 2018, 98(20): 205417 https://doi.org/10.1103/PhysRevB.98.205417
307	Ma N. , Gong J. . Unsupervised identification of Floquet topological phase boundaries. Phys. Rev. Res., 2022, 4(1): 013234 https://doi.org/10.1103/PhysRevResearch.4.013234
308	Wu H. , H. An J. . Floquet topological phases of non-Hermitian systems. Phys. Rev. B, 2020, 102(4): 041119 https://doi.org/10.1103/PhysRevB.102.041119
309	N. Zhang Y. , Xu S. , D. Liu H. , X. Yi X. . Floquet spectrum and dynamics for non-Hermitian Floquet one-dimension lattice model. Int. J. Theor. Phys., 2021, 60(1): 355 https://doi.org/10.1007/s10773-020-04699-4
310	Zhou L. , Pan J. . Non-Hermitian Floquet topological phases in the double-kicked rotor. Phys. Rev. A, 2019, 100(5): 053608 https://doi.org/10.1103/PhysRevA.100.053608
311	Zhou L. . Dynamical characterization of non-Hermitian Floquet topological phases in one dimension. Phys. Rev. B, 2019, 100(18): 184314 https://doi.org/10.1103/PhysRevB.100.184314
312	S. Rudner M. , H. Lindner N. , Berg E. , Levin M. . Anomalous edge states and the bulk-edge correspondence for periodically driven two-dimensional systems. Phys. Rev. X, 2013, 3(3): 031005 https://doi.org/10.1103/PhysRevX.3.031005
313	Y. Wang H. , M. Zhao X. , Zhuang L. , M. Liu W. . Non-Floquet engineering in periodically driven dissipative open quantum systems. J. Phys.: Condens. Matter, 2022, 34(36): 365402 https://doi.org/10.1088/1361-648X/ac7c4e
314	Zhou L. , Gu Y. , Gong J. . Dual topological characterization of non-Hermitian Floquet phases. Phys. Rev. B, 2021, 103(4): L041404 https://doi.org/10.1103/PhysRevB.103.L041404
315	Zhang Z. , Delplace P. , Fleury R. . Superior robustness of anomalous non-reciprocal topological edge states. Nature, 2021, 598(7880): 293 https://doi.org/10.1038/s41586-021-03868-7
316	Zhou L. . Non-Hermitian Floquet topological superconductors with multiple Majorana edge modes. Phys. Rev. B, 2020, 101(1): 014306 https://doi.org/10.1103/PhysRevB.101.014306
317	Pan J. , Zhou L. . Non-Hermitian Floquet second order topological insulators in periodically quenched lattices. Phys. Rev. B, 2020, 102(9): 094305 https://doi.org/10.1103/PhysRevB.102.094305
318	Liu H.C. Fulga I., Mixed higher-order topology: Boundary non-Hermitian skin effect induced by a Floquet bulk, arXiv: 2210.03097 (2022)
319	Longhi S. . Non-Hermitian skin effect beyond the tight- binding models. Phys. Rev. B, 2021, 104(12): 125109 https://doi.org/10.1103/PhysRevB.104.125109
320	J. Lang L. , L. Zhu S. , D. Chong Y. . Non-Hermitian topological end breathers. Phys. Rev. B, 2021, 104(2): L020303 https://doi.org/10.1103/PhysRevB.104.L020303
321	H. Lee C. . Many-body topological and skin states without open boundaries. Phys. Rev. B, 2021, 104(19): 195102 https://doi.org/10.1103/PhysRevB.104.195102
322	Shen R. , H. Lee C. . Non-Hermitian skin clusters from strong interactions. Commun. Phys., 2022, 5(1): 1 https://doi.org/10.1038/s42005-022-01015-w
323	Sarkar R. , S. Hegde S. , Narayan A. . Interplay of disorder and point-gap topology: Chiral modes, localization, and non-Hermitian Anderson skin effect in one dimension. Phys. Rev. B, 2022, 106(1): 014207 https://doi.org/10.1103/PhysRevB.106.014207
324	Yuce C. , Ramezani H. . Coexistence of extended and localized states in the one-dimensional non-Hermitian Anderson model. Phys. Rev. B, 2022, 106(2): 024202 https://doi.org/10.1103/PhysRevB.106.024202
325	Roccati F. . Non-Hermitian skin effect as an impurity problem. Phys. Rev. A, 2021, 104(2): 022215 https://doi.org/10.1103/PhysRevA.104.022215
326	Wang C. , R. Wang X. . Chiral hinge transport in disordered non-Hermitian second-order topological insulators. Phys. Rev. B, 2022, 106(4): 045142 https://doi.org/10.1103/PhysRevB.106.045142
327	Longhi S. . Spectral deformations in non-Hermitian lattices with disorder and skin effect: A solvable model. Phys. Rev. B, 2021, 103(14): 144202 https://doi.org/10.1103/PhysRevB.103.144202
328	M. Chen L. , Zhou Y. , A. Chen S. , Ye P. . Quantum entanglement of non-Hermitian quasicrystals. Phys. Rev. B, 2022, 105(12): L121115 https://doi.org/10.1103/PhysRevB.105.L121115
329	Chakrabarty A.Datta S., Skin effect and dynamical delocalization in non-Hermitian quasicrystals with spin-orbit interaction, arXiv: 2208.10359 (2022)
330	Wang W.Hu M.Wang X.Ma G.Ding K., Experimental realization of geometry-dependent skin effect in a reciprocal two-dimensional lattice, arXiv: 2302.06314 (2023)
331	Lv C. , Zhang R. , Zhai Z. , Zhou Q. . Curving the space by non-hermiticity. Nat. Commun., 2022, 13(1): 2184 https://doi.org/10.1038/s41467-022-29774-8
332	X. Wang S. , Wan S. . Duality between the generalized non-Hermitian Hatano−Nelson model in flat space and a Hermitian system in curved space. Phys. Rev. B, 2022, 106(7): 075112 https://doi.org/10.1103/PhysRevB.106.075112
333	W. Lv C.Zhou Q., Emergent spacetimes from Hermitian and non-Hermitian quantum dynamics, arXiv: 2205.07429 (2022)
334	Gao W. , Lawrence M. , Yang B. , Liu F. , Fang F. , Béri B. , Li J. , Zhang S. . Topological photonic phase in chiral hyperbolic metamaterials. Phys. Rev. Lett., 2015, 114(3): 037402 https://doi.org/10.1103/PhysRevLett.114.037402
335	J. Kollár A. , Fitzpatrick M. , A. Houck A. . Hyperbolic lattices in circuit quantum electrodynamics. Nature, 2019, 571(7763): 45 https://doi.org/10.1038/s41586-019-1348-3
336	M. Lenggenhager P. , Stegmaier A. , K. Upreti L. , Hofmann T. , Helbig T. , Vollhardt A. , Greiter M. , H. Lee C. , Imhof S. , Brand H. , Kießling T. , Boettcher I. , Neupert T. , Thomale R. , Bzdušek T. . Simulating hyperbolic space on a circuit board. Nat. Commun., 2022, 13(1): 4373 https://doi.org/10.1038/s41467-022-32042-4
337	Ezawa M. . Dynamical nonlinear higher-order non-Hermitian skin effects and topological trap-skin phase. Phys. Rev. B, 2022, 105(12): 125421 https://doi.org/10.1103/PhysRevB.105.125421
338	Yuce C. . Nonlinear non-Hermitian skin effect. Phys. Lett. A, 2021, 408: 127484 https://doi.org/10.1016/j.physleta.2021.127484
339	Tuloup T. , W. Bomantara R. , H. Lee C. , Gong J. . Nonlinearity induced topological physics in momentum space and real space. Phys. Rev. B, 2020, 102(11): 115411 https://doi.org/10.1103/PhysRevB.102.115411
340	Hadad Y. , C. Soric J. , B. Khanikaev A. , Alù A. . Self-induced topological protection in nonlinear circuit arrays. Nat. Electron., 2018, 1(3): 178 https://doi.org/10.1038/s41928-018-0042-z
341	Wang Y. , J. Lang L. , H. Lee C. , Zhang B. , D. Chong Y. . Topologically enhanced harmonic generation in a nonlinear transmission line metamaterial. Nat. Commun., 2019, 10(1): 1102 https://doi.org/10.1038/s41467-019-08966-9
342	Kotwal T. , Moseley F. , Stegmaier A. , Imhof S. , Brand H. , Kießling T. , Thomale R. , Ronellenfitsch H. , Dunkel J. . Active topolectrical circuits. Proc. Natl. Acad. Sci. USA, 2021, 118(32): e2106411118 https://doi.org/10.1073/pnas.2106411118
343	Hohmann H. , Hofmann T. , Helbig T. , Imhof S. , Brand H. , K. Upreti L. , Stegmaier A. , Fritzsche A. , Müller T. , Schwingenschlögl U. , H. Lee C. , Greiter M. , W. Molenkamp L. , Kießling T. , Thomale R. . Observation of Cnoidal wave localization in nonlinear topolectric circuits. Phys. Rev. Res., 2023, 5(1): L012041 https://doi.org/10.1103/PhysRevResearch.5.L012041
344	A. Dobrykh D. , V. Yulin A. , P. Slobozhanyuk A. , N. Poddubny A. , S. Kivshar Y. . Nonlinear control of electromagnetic topological edge states. Phys. Rev. Lett., 2018, 121(16): 163901 https://doi.org/10.1103/PhysRevLett.121.163901
345	H. Fu P.Xu Y.H. Lee C.S. Ang Y.F. Liu J., Gate-tunable topological Josephson diode, arXiv: 2212.01980 (2022)
346	Ezawa M. . Nonlinear topological Toda quasicrystal. J. Phys. Soc. Jpn., 2022, 91(8): 084703 https://doi.org/10.7566/JPSJ.91.084703
347	Hofmann T. , Helbig T. , H. Lee C. , Greiter M. , Thomale R. . Chiral voltage propagation and calibration in a topolectrical Chern circuit. Phys. Rev. Lett., 2019, 122(24): 247702 https://doi.org/10.1103/PhysRevLett.122.247702
348	Mu S. , H. Lee C. , Li L. , Gong J. . Emergent Fermi surface in a many-body non-Hermitian fermionic chain. Phys. Rev. B, 2020, 102(8): 081115 https://doi.org/10.1103/PhysRevB.102.081115
349	B. Zhang S. , M. Denner M. , Bzdušek T. , A. Sentef M. , Neupert T. . Symmetry breaking and spectral structure of the interacting Hatano−Nelson model. Phys. Rev. B, 2022, 106(12): L121102 https://doi.org/10.1103/PhysRevB.106.L121102
350	Alsallom F. , Herviou L. , V. Yazyev O. , Brzezińska M. . Fate of the non-Hermitian skin effect in many-body fermionic systems. Phys. Rev. Res., 2022, 4(3): 033122 https://doi.org/10.1103/PhysRevResearch.4.033122
351	Dóra B. , P. Moca C. . Full counting statistics in the many-body Hatano−Nelson model. Phys. Rev. B, 2022, 106(23): 235125 https://doi.org/10.1103/PhysRevB.106.235125
352	Yoshida T. , Kudo K. , Hatsugai Y. . Non-Hermitian fractional quantum Hall states. Sci. Rep., 2019, 9(1): 16895 https://doi.org/10.1038/s41598-019-53253-8
353	N. Wang Y. , L. You W. , Sun G. . Quantum criticality in interacting bosonic Kitaev−Hubbard models. Phys. Rev. A, 2022, 106(5): 053315 https://doi.org/10.1103/PhysRevA.106.053315
354	Mao L.Hao Y.Pan L., Non-Hermitian skin effect in one-dimensional interacting Bose gas, arXiv: 2207.12637 (2022)
355	Chen G.Song F.L. Lado J., Topological spin excitations in non-Hermitian spin chains with a generalized kernel polynomial algorithm, arXiv: 2208.06425 (2022)
356	Yoshida T. , Hatsugai Y. . Reduction of one-dimensional non-Hermitian point-gap topology by interactions. Phys. Rev. B, 2022, 106(20): 205147 https://doi.org/10.1103/PhysRevB.106.205147
357	Zhang W. , Di F. , Yuan H. , Wang H. , Zheng X. , He L. , Sun H. , Zhang X. . Observation of non-Hermitian aggregation effects induced by strong interactions. Phys. Rev. B, 2022, 105(19): 195131 https://doi.org/10.1103/PhysRevB.105.195131
358	Kawabata K. , Shiozaki K. , Ryu S. . Many-body topology of non-Hermitian systems. Phys. Rev. B, 2022, 105(16): 165137 https://doi.org/10.1103/PhysRevB.105.165137
359	Yoshida T. . Real-space dynamical mean field theory study of non-Hermitian skin effect for correlated systems: Analysis based on pseudospectrum. Phys. Rev. B, 2021, 103(12): 125145 https://doi.org/10.1103/PhysRevB.103.125145
360	I. Arkhipov I. , Minganti F. . Emergent non-Hermitian localization phenomena in the synthetic space of zero-dimensional bosonic systems. Phys. Rev. A, 2023, 107(1): 012202 https://doi.org/10.1103/PhysRevA.107.012202
361	Qin F. , Shen R. , H. Lee C. . Non-Hermitian squeezed polarons. Phys. Rev. A, 2023, 107(1): L010202 https://doi.org/10.1103/PhysRevA.107.L010202
362	Micallo T.Lehmann C.C. Budich J., Correlation-induced sensitivity and non-Hermitian skin effect of quasiparticles, arXiv: 2302.00019 (2023)
363	Yang K. , C. Morampudi S. , J. Bergholtz E. . Exceptional spin liquids from couplings to the environment. Phys. Rev. Lett., 2021, 126(7): 077201 https://doi.org/10.1103/PhysRevLett.126.077201
364	Žnidarič M. . Solvable non-Hermitian skin effect in many body unitary dynamics. Phys. Rev. Res., 2022, 4(3): 033041 https://doi.org/10.1103/PhysRevResearch.4.033041
365	Gong Z. , Bello M. , Malz D. , K. Kunst F. . Anomalous behaviors of quantum emitters in non-Hermitian baths. Phys. Rev. Lett., 2022, 129(22): 223601 https://doi.org/10.1103/PhysRevLett.129.223601
366	Roccati F. , Lorenzo S. , Calajò G. , M. Palma G. , Carollo A. , Ciccarello F. . Exotic interactions mediated by a non-Hermitian photonic bath. Optica, 2022, 9(5): 565 https://doi.org/10.1364/OPTICA.443955
367	K. Cao, Q. Du, X. R. Wang, and S. P. Kou, Physics of many-body nonreciprocal model: From non-Hermitian skin effect to quantum Maxwell’s pressure − Demon effect, arXiv: 2109.03690 (2021)
368	G. Zhou T. , N. Zhou Y. , Zhang P. , Zhai H. . Space-time duality between quantum chaos and non-Hermitian boundary effect. Phys. Rev. Res., 2022, 4(2): L022039 https://doi.org/10.1103/PhysRevResearch.4.L022039
369	Xu K. , Zhang X. , Luo K. , Yu R. , Li D. , Zhang H. . Coexistence of topological edge states and skin effects in the non-Hermitian Su−Schrieffer−Heeger model with long-range nonreciprocal hopping in topoelectric realizations. Phys. Rev. B, 2021, 103(12): 125411 https://doi.org/10.1103/PhysRevB.103.125411
370	Deng W. , Chen T. , Zhang X. . nth power root topological phases in Hermitian and non-Hermitian systems. Phys. Rev. Res., 2022, 4(3): 033109 https://doi.org/10.1103/PhysRevResearch.4.033109
371	Zhang H. , Chen T. , Li L. , H. Lee C. , Zhang X. . Electrical circuit realization of topological switching for the non-Hermitian skin effect. Phys. Rev. B, 2023, 107(8): 085426 https://doi.org/10.1103/PhysRevB.107.085426
372	Li L. , H. Lee C. , Gong J. . Emergence and full 3d-imaging of nodal boundary Seifert surfaces in 4D topological matter. Commun. Phys., 2019, 2(1): 135 https://doi.org/10.1038/s42005-019-0235-4
373	Lin Q. , Li T. , Xiao L. , Wang K. , Yi W. , Xue P. . Topological phase transitions and mobility edges in non-Hermitian quasicrystals. Phys. Rev. Lett., 2022, 129(11): 113601 https://doi.org/10.1103/PhysRevLett.129.113601
374	Lin Q. , Li T. , Xiao L. , Wang K. , Yi W. , Xue P. . Observation of non-Hermitian topological Anderson insulator in quantum dynamics. Nat. Commun., 2022, 13(1): 3229 https://doi.org/10.1038/s41467-022-30938-9
375	S. Palacios L. , Tchoumakov S. , Guix M. , Pagonabarraga I. , Sánchez S. , G. Grushin A. . Guided accumulation of active particles by topological design of a second-order skin effect. Nat. Commun., 2021, 12(1): 4691 https://doi.org/10.1038/s41467-021-24948-2
376	F. Yu Y. , W. Yu L. , G. Zhang W. , L. Zhang H. , L. Ouyang X. , Q. Liu Y. , L. Deng D. , Duan L.-M. . Experimental unsupervised learning of non-Hermitian knotted phases with solid-state spins. npj Quantum Inform., 2022, 8: 116 https://doi.org/10.1038/s41534-022-00629-w
377	J. Bergholtz E. , C. Budich J. , K. Kunst F. . Exceptional topology of non-Hermitian systems. Rev. Mod. Phys., 2021, 93(1): 015005 https://doi.org/10.1103/RevModPhys.93.015005
378	Zhang X. , Zhang T. , H. Lu M. , F. Chen Y. . A review on non-Hermitian skin effect. Adv. Phys. X, 2022, 7(1): 2109431 https://doi.org/10.1080/23746149.2022.2109431
379	Zhu B. , Wang Q. , Leykam D. , Xue H. , W. Qi J. , D. Chong Y. . Anomalous single-mode lasing induced by nonlinearity and the non-Hermitian skin effect. Phys. Rev. Lett., 2022, 129(1): 013903 https://doi.org/10.1103/PhysRevLett.129.013903
380	Mandal S. , Banerjee R. , C. H. Liew T. . From the topological spin-Hall effect to the non-Hermitian skin effect in an elliptical micropillar chain. ACS Photonics, 2022, 9(2): 527 https://doi.org/10.1021/acsphotonics.1c01425
381	Xu X. , Bao R. , C. H. Liew T. . Non-Hermitian topological exciton−polariton corner modes. Phys. Rev. B, 2022, 106(20): L201302 https://doi.org/10.1103/PhysRevB.106.L201302
382	F. Yu Z. , K. Xue J. , Zhuang L. , Zhao J. , M. Liu W. . Non-Hermitian spectrum and multistability in exciton−polariton condensates. Phys. Rev. B, 2021, 104(23): 235408 https://doi.org/10.1103/PhysRevB.104.235408
383	Yang M. , Wang L. , Wu X. , Xiao H. , Yu D. , Yuan L. , Chen X. . Concentrated subradiant modes in a one-dimensional atomic array coupled with chiral waveguides. Phys. Rev. A, 2022, 106(4): 043717 https://doi.org/10.1103/PhysRevA.106.043717
384	Lin Z. , Ding L. , Ke S. , Li X. . Steering non-Hermitian skin modes by synthetic gauge fields in optical ring resonators. Opt. Lett., 2021, 46(15): 3512 https://doi.org/10.1364/OL.431904
385	Zhong J. , Wang K. , Park Y. , Asadchy V. , C. Wojcik C. , Dutt A. , Fan S. . Nontrivial point-gap topology and non-Hermitian skin effect in photonic crystals. Phys. Rev. B, 2021, 104(12): 125416 https://doi.org/10.1103/PhysRevB.104.125416
386	Price H. , Chong Y. , Khanikaev A. , Schomerus H. , J. Maczewsky L. . et al.. Roadmap on topological photonics. J. Phys.: Photonics, 2022, 4: 032501 https://doi.org/10.1088/2515-7647/ac4ee4
387	Yan Q.Chen H.Yang Y., Non-Hermitian skin effect and delocalized edge states in photonic crystals with anomalous parity−time symmetry, arXiv: 2111.08213 (2021)
388	C. Xie L.Jin L.Song Z., Antihelical edge states in two-dimensional photonic topological metals, arXiv: 2302.05842 (2023)
389	Lin Q.Yi W.Xue P., Manipulating non-reciprocity in a two-dimensional magnetic quantum walk, arXiv: 2212.00387 (2022)
390	Li T. , S. Zhang Y. , Yi W. . Two-dimensional quantum walk with non-Hermitian skin effects. Chin. Phys. Lett., 2021, 38(3): 030301 https://doi.org/10.1088/0256-307X/38/3/030301
391	G. Pyrialakos G. , Ren H. , S. Jung P. , Khajavikhan M. , N. Christodoulides D. . Thermalization dynamics of nonlinear non-Hermitian optical lattices. Phys. Rev. Lett., 2022, 128(21): 213901 https://doi.org/10.1103/PhysRevLett.128.213901
392	Fleckenstein C. , Zorzato A. , Varjas D. , J. Bergholtz E. , H. Bardarson J. , Tiwari A. . Non-Hermitian topology in monitored quantum circuits. Phys. Rev. Res., 2022, 4(3): L032026 https://doi.org/10.1103/PhysRevResearch.4.L032026
393	H. Lin S. , Dilip R. , G. Green A. , Smith A. , Pollmann F. . Real- and imaginary-time evolution with compressed quantum circuits. PRX Quantum, 2021, 2(1): 010342 https://doi.org/10.1103/PRXQuantum.2.010342
394	Liu T. , G. Liu J. , Fan H. . Probabilistic non-unitary gate in imaginary time evolution. Quantum Inform. Process., 2021, 20(6): 1 https://doi.org/10.1007/s11128-021-03145-6
395	Kamakari H. , N. Sun S. , Motta M. , J. Minnich A. . Digital quantum simulation of open quantum systems using quantum imaginary–time evolution. PRX Quantum, 2022, 3(1): 010320 https://doi.org/10.1103/PRXQuantum.3.010320
396	Smith A. , S. Kim M. , Pollmann F. , Knolle J. . Simulating quantum many-body dynamics on a current digital quantum computer. npj Quantum Inform., 2019, 5: 106 https://doi.org/10.1038/s41534-019-0217-0
397	M. Koh J. , Tai T. , H. Phee Y. , E. Ng W. , H. Lee C. . Stabilizing multiple topological Fermions on a quantum computer. npj Quantum Inform., 2022, 8: 16 https://doi.org/10.1038/s41534-022-00527-1
398	M. Koh J. , Tai T. , H. Lee C. . Simulation of interaction-induced chiral topological dynamics on a digital quantum computer. Phys. Rev. Lett., 2022, 129(14): 140502 https://doi.org/10.1103/PhysRevLett.129.140502
399	M. Koh J.Tai T.H. Lee C., Observation of higher-order topological states on a quantum computer, arXiv: 2303.02179 (2023)
400	Chen T.Shen R.H. Lee C.Yang B., High-fidelity realization of the Aklt state on a NISQ-era quantum processor, arXiv: 2210.13840 (2022)
401	Schomerus H. . Nonreciprocal response theory of non-Hermitian mechanical metamaterials: Response phase transition from the skin effect of zero modes. Phys. Rev. Res., 2020, 2(1): 013058 https://doi.org/10.1103/PhysRevResearch.2.013058
402	Braghini D. , G. G. Villani L. , I. N. Rosa M. , R. de F Arruda J. . Non-Hermitian elastic waveguides with piezoelectric feedback actuation: Non-reciprocal bands and skin modes. J. Phys. D Appl. Phys., 2021, 54(28): 285302 https://doi.org/10.1088/1361-6463/abf9d9
403	Jin Y. , Zhong W. , Cai R. , Zhuang X. , Pennec Y. , Djafari-Rouhani B. . Non-Hermitian skin effect in a phononic beam based on piezoelectric feedback control. Appl. Phys. Lett., 2022, 121(2): 022202 https://doi.org/10.1063/5.0097530
404	Wen P. , Wang M. , L. Long G. . Optomechanically induced transparency and directional amplification in a non-Hermitian optomechanical lattice. Opt. Express, 2022, 30(22): 41012 https://doi.org/10.1364/OE.473652
405	Ren Z. , Liu D. , Zhao E. , He C. , K. Pak K. , Li J. , B. Jo G. . Chiral control of quantum states in non-Hermitian spin–orbit-coupled Fermions. Nat. Phys., 2022, 18(4): 385 https://doi.org/10.1038/s41567-021-01491-x
406	Guo S. , Dong C. , Zhang F. , Hu J. , Yang Z. . Theoretical prediction of a non-Hermitian skin effect in ultracold-atom systems. Phys. Rev. A, 2022, 106(6): L061302 https://doi.org/10.1103/PhysRevA.106.L061302
407	Zhou L. , Li H. , Yi W. , Cui X. . Engineering non-Hermitian skin effect with band topology in ultracold gases. Commun. Phys., 2022, 5(1): 252 https://doi.org/10.1038/s42005-022-01021-y
408	Qin Y.Zhang K.H. Li L., Geometry-dependent skin effect and anisotropic Bloch oscillations in a non-Hermitian optical lattice, arXiv: 2304.03792v1 (2023)
409	Li H. , Cui X. , Yi W. . Non-Hermitian skin effect in a spin−orbit-coupled Bose−Einstein condensate. JUSTC, 2022, 52(8): 2 https://doi.org/10.52396/JUSTC-2022-0003
410	Yoshida T. , Mizoguchi T. , Hatsugai Y. . Non-Hermitian topology in Rock–Paper–Scissors games. Sci. Rep., 2022, 12(1): 560 https://doi.org/10.1038/s41598-021-04178-8
411	Dobrinevski A. , Frey E. . Extinction in neutrally stable stochastic Lotka−Volterra models. Phys. Rev. E, 2012, 85(5): 051903 https://doi.org/10.1103/PhysRevE.85.051903
412	Knebel J. , Krüger T. , F. Weber M. , Frey E. . Coexistence and survival in conservative Lotka−Volterra networks. Phys. Rev. Lett., 2013, 110(16): 168106 https://doi.org/10.1103/PhysRevLett.110.168106
413	Knebel J. , M. Geiger P. , Frey E. . Topological phase transition in coupled Rock−Paper−Scissors cycles. Phys. Rev. Lett., 2020, 125(25): 258301 https://doi.org/10.1103/PhysRevLett.125.258301
414	Yoshida T. , Mizoguchi T. , Hatsugai Y. . Chiral edge modes in evolutionary game theory: A kagome network of Rock−Paper−Scissors cycles. Phys. Rev. E, 2021, 104(2): 025003 https://doi.org/10.1103/PhysRevE.104.025003
415	Umer M. , Gong J. . Topologically protected dynamics in three-dimensional nonlinear antisymmetric Lotka−Volterra systems. Phys. Rev. B, 2022, 106(24): L241403 https://doi.org/10.1103/PhysRevB.106.L241403
416	Scheibner C. , T. M. Irvine W. , Vitelli V. . Non-Hermitian band topology and skin modes in active elastic media. Phys. Rev. Lett., 2020, 125(11): 118001 https://doi.org/10.1103/PhysRevLett.125.118001
417	Fruchart M. , Hanai R. , B. Littlewood P. , Vitelli V. . Non-reciprocal phase transitions. Nature, 2021, 592(7854): 363 https://doi.org/10.1038/s41586-021-03375-9
418	Yu T. , Zeng B. . Giant microwave sensitivity of a magnetic array by long-range chiral interaction driven skin effect. Phys. Rev. B, 2022, 105(18): L180401 https://doi.org/10.1103/PhysRevB.105.L180401
419	Zeng B. , Yu T. . Radiation-free and non-Hermitian topology inertial defect states of on-chip magnons. Phys. Rev. Res., 2023, 5(1): 013003 https://doi.org/10.1103/PhysRevResearch.5.013003
420	Franca S. , Könye V. , Hassler F. , van den Brink J. , Fulga C. . Non-Hermitian physics without gain or loss: The skin effect of reflected waves. Phys. Rev. Lett., 2022, 129(8): 086601 https://doi.org/10.1103/PhysRevLett.129.086601
421	Zhang X.Zhang B.Zhao W.H. Lee C., Observation of non-local impedance response in a passive electrical circuit, arXiv: 2211.09152 (2022)
422	Geng H. , Y. Wei J. , H. Zou M. , Sheng L. , Chen W. , Y. Xing D. . Nonreciprocal charge and spin transport induced by non-Hermitian skin effect in mesoscopic heterojunctions. Phys. Rev. B, 2023, 107(3): 035306 https://doi.org/10.1103/PhysRevB.107.035306
423	Ghaemi-Dizicheh H. , Schomerus H. . Compatibility of transport effects in non-Hermitian nonreciprocal systems. Phys. Rev. A, 2021, 104(2): 023515 https://doi.org/10.1103/PhysRevA.104.023515
424	Schomerus H. . Fundamental constraints on the observability of non-Hermitian effects in passive systems. Phys. Rev. A, 2022, 106(6): 063509 https://doi.org/10.1103/PhysRevA.106.063509

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