Non-Hermitian boundary spectral winding Zuxuan Ou1Yucheng Wang2 3 4and Linhu Li1 1Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing School of Physics and Astronomy

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Non-Hermitian boundary spectral winding

Zuxuan Ou,1Yucheng Wang,2, 3, 4 and Linhu Li1, ∗

1Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing &School of Physics and Astronomy,

Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China

2Shenzhen Institute for Quantum Science and Engineering,

Southern University of Science and Technology, Shenzhen 518055, China

3International Quantum Academy, Shenzhen 518048, China

4Guangdong Provincial Key Laboratory of Quantum Science and Engineering,

Southern University of Science and Technology, Shenzhen 518055, China

(Dated: October 25, 2022)

Spectral winding of complex eigenenergies represents a topological aspect unique in non-Hermitian systems,

which vanishes in one-dimensional (1D) systems under the open boundary conditions (OBC). In this work, we

discover a boundary spectral winding in two-dimensional non-Hermitian systems under the OBC, originating

from the interplay between Hermitian boundary localization and non-Hermitian non-reciprocal pumping. Such

a nontrivial boundary topology is demonstrated in a non-Hermitian breathing Kagome model with a triangle

geometry, whose 1D boundary mimics a 1D non-Hermitian system under the periodic boundary conditions with

nontrivial spectral winding. In a trapezoidal geometry, such a boundary spectral winding can even co-exist with

corner accumulation of edge states, instead of extended ones along 1D boundary of a triangle geometry. An

OBC type of hybrid skin-topological eﬀect may also emerge in a trapezoidal geometry, provided the boundary

spectral winding completely vanishes. By studying the Green’s function, we unveil that the boundary spectral

winding can be detected from a topological response of the system to a local driving ﬁeld, oﬀering a realistic

method to extract the nontrivial boundary topology for experimental studies.

Introduction.- Non-Hermitian systems can support not only

topological phases with boundary states protected by conven-

tional band topology [1, 2], but also spectral winding topol-

ogy of complex eigenenergies which has no Hermitian ana-

logue. Nontrivial spectral winding generally emerges in many

non-Hermitian lattices under the periodic boundary condi-

tions (PBC), and vanishes when the boundary is opened (i.e.

the open boundary condition, OBC), resulting in the non-

Hermitian skin eﬀect (NHSE) where bulk states become skin-

localized at the system’s boundary [3–7]. One of the most

noteworthy consequences of the interplay between conven-

tional and spectral winding topology is the breakdown of con-

ventional topological bulk-boundary correspondence [8, 9],

which has led to recent extensive investigations of its recovery

through several diﬀerent methods [3, 10–14] and many other

exciting phenomena induced by NHSE and spectral winding

topology [15–36].

In contemporary literature, spectral winding topology is

mainly studied in one dimensional (1D) systems, as by def-

inition it corresponds to 1D trajectories in the two dimen-

sional (2D) complex-energy plane, which cannot be straight-

forwardly generalized into higher spatial dimensions. On

the other hand, being a boundary phenomenon, NHSE in

2D or higher dimension are also far more sophisticated than

in 1D, possessing many variations associated with diﬀerent

boundaries and defects due to their richer geometric structures

[37–42]. In this work, we unveil an exotic aspect of spec-

tral winding in higher dimensions, namely nontrivial spec-

tral winding for 1D boundary states of 2D lattices under the

OBC, in sharp contrast to our knowledge of vanishing spec-

tral winding topology of OBC systems. Its emergence origi-

nates from the interplay between topological boundary local-

ization and a non-Hermitian chiral pumping along the bound-

ary, namely asymmetric hoppings with stronger amplitudes

toward a chiral direction [see Fig. 1(a) for an illustration].

A similar mechanism is known to be responsible for the hy-

brid skin-topological eﬀect (HSTE) [43–46], a type of higher-

order NHSE with topological protection [47–49], which in-

duces corner skin-topological localization in 2D lattices. In-

terestingly, our example model of non-Hermitian breathing

Kagome lattice can support both nontrivial boundary spectral

winding and an OBC type of HSTE under the same param-

eters, but with diﬀerent triangle and trapezoidal geometries,

where the chiral pumping is destroyed by geometric properties

of the latter case. In the intermedia regime between these two

scenarios, nontrivial boundary spectral winding may even co-

incide with a weak corner localization, indicating an anoma-

lous co-existence of the seemingly contradictory boundary

spectral winding and HSTE, which require rather diﬀerent ge-

ometric properties along boundaries. Moreover, we discover

that this boundary spectral winding can be detected from a

topological response to a driving ﬁeld when locally perturbing

the 2D system, providing a feasible method to extract the non-

trivial boundary topology from realistic non-Hermitian sys-

tems.

Non-Hermitian breathing Kagome model.- We consider a

non-Hermitian breathing Kagome model with asymmetric in-

tracell hoppings as shown in Fig.1(a), with its bulk Hamilto-

nian given by

H(k)=





0ta+t+

be−ik1ta+t−

beik3

ta+t−

beik10ta+t+

be−ik2

ta+t+

be−ik3ta+t−

beik20







,(1)

where k1=−kx/2−√3ky/2, k2=kxand k3=−kx/2+√3ky/2,

t±

b=tbe±αrepresent asymmetric intercell (upward trian-

gle) hopping parameters, and tais the amplitude of intracell

arXiv:2210.12178v1 [cond-mat.mes-hall] 21 Oct 2022

(a) (b)

corner 1

corner 2 corner 3

-2 -1 0 1 2

-5

0.4

0.8

1.2

1.6

non-Hermitian

chiral pumping

FIG. 1. (a) A skecth of the non-Hermitian breathing Kagome lattice

with L=5 rows of unit cells. Gray area represents the 1st-order

boundary of the system, which gives the eﬀective boundary Hamil-

tonian discussed latter. Blue arrows indicate the direction of non-

Hermitian chiral pumping along the boundary. (b) Energy spectrum

versus ta, for the system with a triangle geometry in the Hermitian

limit of α=0. Eigenenergies are marked by diﬀerent colors ac-

cording to their corresponding FD. Other parameters are tb=1 and

L=30.

(downward triangle) Hermitian hopping. Here we set tb=1

as the unit energy. In the Hermitian scenario with α=0,

the Kagome lattice model supports both 1st-order edge states

and 2nd-order corner states in certain parameter regimes, as

shown in Fig. 1(b) where diﬀerent bulk and boundary states

are characterized by diﬀerent values of their fractal dimension

(FD), deﬁned as

FD =−ln[X

r|ψn,r|4]/ln √3N,(2)

with ψn,rthe wave amplitude at position rof the n-th eigen-

state, and Nthe total number of unit cells. In a triangle ge-

ometry, N=(1 +L)L/2 with Lthe number of rows of unit

cells in the lattice. A 2D bulk state and a 1D edge state in

our system shall have their FD close to 2 and 1 respectively.

As seen in Fig. 1(b), the breathing Kagome lattice supports

1D edge states (represented by green color for FD '1.2) in a

large parameter regime.

Destructive interference of non-reciprocity and boundary

spectral winding.- By construction, the model can be viewed

as a combination of three sets of non-Hermitian Su-Schrieﬀer-

Heeger (SSH) chains [3, 50–52] along diﬀerent directions.

Speciﬁcally, the three non-Hermitian SSH chains are chosen

to be identical, resulting in a C3rotation symmetry of the

system, as shown in Fig. 1(a). In this way, the asymmetric

hoppings along the three directions form a closed loop and

balance out in each unit cell, leading to a destructive interfer-

ence of non-reciprocity in the bulk. The system is thus net-

reciprocal even in the presence of asymmetric hoppings. As

seen in Fig. 2(a), FD is close to 2 for eigenstates in three bulk

bands (yellow color) for the system with a triangle geometry,

indicating the absence of NHSE for bulk states. Consistently,

the summed bulk distribution, deﬁned as

ρbulk(r)=X

n∈bulk |ψn,r|2

with summation runs over all eigenstates in the bulk bands,

edge

effective

corner 1 corner 2 corner 3

(c)

(b)

(e)

(a)

boundary

effective

-0.1

0.0

0.1

(d)

-2 0 2

-1

-1 1

ρbulk, ρedge

012

FIG. 2. (a) Energy spectrum under the OBC, colors indicate the

FD of each eigenstate. (b) and (c) Summed distribution of bulk

states and edge states respectively.(d) Spectra of edge states for the

2D lattice (cyan dots) as in (a), and 1D eﬀective boundary sys-

tem corresponding to the gray area in Fig. 1(a) (red circles). (e)

Summed distribution of edge states for the 2D lattice (cyan), and

of all eigenstates for the 1D boundary system (red). Parameters are

ta=0.25,tb=1, α =0.5,L=30.

also distributes uniformly in the 2D bulk [Fig. 2(b)]. On

the other hand, 1st-order edge states distribute mostly along

1D edges, and are subjected to a net non-Hermitian non-

reciprocal pumping. The same mechanism is known to in-

duce the HSTE in diﬀerent square and honeycomb lattices.

However, in our model with a triangle geometry, the de-

structive interference of non-reciprocity limits the choices of

the non-reciprocal directions and forbids the HSTE. Namely,

in the presence of destructive interference of non-reciprocity

along three directions, a triangle lattice must have chiral non-

reciprocal pumping along its 1D boundary, hence it is im-

possible to have two edges with non-reciprocity toward their

shared corner. Consequently, we anticipate no hybrid skin-

topological corner mode to appear in our system.

As veriﬁed in our numerical calculations, 1st-order edge

states are indeed extended along 1D edges, as shown in Fig.

2(c) by the summed edge distribution

ρedge(r)=X

n∈edge |ψn,r|2

with summation runs over all edge states. Interestingly, a non-

trivial spectral winding is seen to emerge for edge states, even

when the system is under OBC [Fig. 2(a)]. To understand its

emergence, we note that these edge states inhabit within the

edges of a 2D lattice, which form an eﬀective 1D boundary

system without an open boundary, analogous to a 1D non-

reciprocal system under PBC. By taking the edges of the 2D

lattice as a 1D system decoupled from the 2D bulk, we ﬁnd

that its spectrum is almost identical to that of the 1st-order

edge states of the original 2D system, as shown in Fig. 2(d).

As seen in Fig. 2(e), their eigenstates display a slightly diﬀer-

ent but still extended distribution, as shown by ρeﬀ=Pn|ψ1D

n,x|2

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Non-HermitianboundaryspectralwindingZuxuanOu,1YuchengWang,2,3,4andLinhuLi1,1GuangdongProvincialKeyLaboratoryofQuantumMetrologyandSensing&SchoolofPhysicsandAstronomy,SunYat-SenUniversity(ZhuhaiCampus),Zhuhai519082,China2ShenzhenInstituteforQuantumScienceandEngineering,SouthernUniversityofScienceandT...

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Non-Hermitian boundary spectral winding Zuxuan Ou1Yucheng Wang2 3 4and Linhu Li1 1Guangdong Provincial Key Laboratory of Quantum Metrology and Sensing School of Physics and Astronomy

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