Mixed higher-order topology boundary non-Hermitian skin effect induced by a Floquet bulk Hui Liu1and Ion Cosma Fulga1

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Mixed higher-order topology: boundary non-Hermitian skin eﬀect induced by a

Floquet bulk

Hui Liu1and Ion Cosma Fulga1

1IFW Dresden and W¨urzburg-Dresden Cluster of Excellence ct.qmat, Helmholtzstrasse 20, 01069 Dresden, Germany

We show that anomalous Floquet topological insulators generate intrinsic, non-Hermitian topol-

ogy on their boundary. As a consequence, removing a boundary hopping from the time-evolution

operator stops the propagation of chiral edge modes, leading to a non-Hermitian skin eﬀect. This

does not occur in Floquet Chern insulators, however, in which boundary modes continue propagat-

ing. The non-Hermitian skin eﬀect on the boundary is a consequence of the nontrivial topology

of the bulk Floquet operator, which we show by designing a real-space topological invariant. Our

work introduces a form of ‘mixed’ higher-order topology, providing a bridge between research on

periodically-driven systems and the study of non-Hermiticity. It suggests that periodic driving,

which has already been demonstrated in a wide range of experiments, may be used to generate

non-Hermitian skin eﬀects.

I. INTRODUCTION

The Hermiticity of Hamiltonians is a principal feature

of quantum physics. As a consequence, time-evolution

operators are unitary, with eigenvalues constrained to

be phase factors, thus maintaining probability conser-

vation. In periodically-driven systems, commonly char-

acterized in terms of their Floquet operators, this means

that quasienergies εare only deﬁned modulo 2π/T , with

Tbeing the driving period.

The periodicity of Floquet eigenphases can have im-

portant consequences in the context of band topology [1–

3], since it provides an extra bulk quasienergy gap (at

ε=±π/T ) to topological boundary modes. When this

gap is nontrivial, the resulting Floquet topological insu-

lators are completely induced by the time-periodic driv-

ing, and thus have no counterpart in static systems [4–

11]. An example of such a phase is the so-called anoma-

lous Floquet topological insulator (AFTI) [12], a two-

dimensional (2D) system in which each gap contains

the same, nonzero number of chiral edge states, despite

all bulk bands being topologically trivial. As a result,

the chiral edge modes wind around the [−π/T, π/T )

quasienergy zone, or equivalently, around the unit circle

in the complex plane.

When Hermiticity is broken, the emergence of complex

Hamiltonian eigenvalues can alter the well-established,

fundamental concepts of band topology and yield new

phenomena [13–15]. An interesting example is the non-

Hermitian skin eﬀect, in which an extensive number of

modes pile up at the boundaries of a system [16–29].

One of the simplest models showing this behavior is the

Hatano-Nelson model [30], a one-dimensional chain with

nearest-neighbor hoppings that are nonreciprocal. The

skin modes that occur in this system when open bound-

ary conditions (OBCs) are imposed have a topological

origin. They are protected by the winding number of the

inﬁnite system spectrum [31].

The topology of unitary operators is mainly stud-

ied in a Hermitian context, with a primary focus

on periodically-driven, Hermitian Hamiltonians [32–39].

Here we examine unitary operators from the point of view

of non-Hermiticity, showing that this leads to an alter-

nate form of higher-order topology. Intuitively, both the

Hatano-Nelson model and the AFTI are characterized by

a nonzero spectral winding, of the bulk states in the ﬁrst

case and of the chiral edge modes in the second case.

We show that these two windings are in fact connected:

the bulk Floquet topology induces the formation of a

non-Hermitian topological chain at the system boundary,

without the need for any additional perturbation.

Working with one of the most well-known AFTI mod-

els [12], we found that removing a hopping from the

boundary of the Floquet operator stops the propagation

of chiral edge modes; they pile up at the defect posi-

tion instead. This is not a property of all Floquet chiral

edge modes, however, but of those which have a nonzero

spectral winding. When chiral edge modes exist without

spectral winding, such as in a Floquet Chern insulator

phase, the skin eﬀect is not robust, and the edge modes

can continue propagating around the defect. These two

diﬀerent behaviors can be predicted by a real-space in-

variant computed directly from the full, 2D Floquet op-

erator, conﬁrming the presence of a mixed higher-order

topology.

The rest of the work is organized as follows. In Sec.

II, we introduce the Floquet system and relate it to non-

Hermitian topology. In Sec. III, we cut the edge to show

the boundary skin eﬀect in an two-dimensional anoma-

lous Floquet topological phase. The topological protec-

tion of this phenomenon is studied in Sec. IV. We con-

clude in Sec. V.

II. NON-HERMITIAN TOPOLOGY IN A

FLOQUET SYSTEM

We start with the Rudner-Lindner-Berg-Levin

model [12], a two-dimensional bipartite lattice with

hopping strength varied in diﬀerent time steps [see Fig.

1(a)]. Its Hamiltonian reads H(t) = Hons +Hhop(t) with

arXiv:2210.03097v2 [cond-mat.mes-hall] 6 Jul 2023

Figure 1. (a) Periodically-driven Hamiltonian model, with

red and black dots representing the sublattices A and B. The

dotted, dashed, dot-dashed, and solid lines correspond to the

hopping terms of the time step 1, 2, 3, and 4, respectively. The

red (black) arrows indicate the evolutions of bulk modes on

sublattice A (B) at resonant driving. (b) Real-space Floquet

operator at resonant driving. The black arrows denote uni-

directional hoppings, identical to those of the Hatano-Nelson

model in the limit of maximal nonreciprocity. The edge cut-

ting corresponds to removing the hopping from the shaded

area. (c) The smallest of the two bulk gaps (at ε= 0 and

π/T ), ∆ = min(∆0,∆π/T ), is plotted as a function of J

and δAB. (d) Map of ρxc, the real-space probability density

summed over all states, corresponding to the site adjacent

to the cut hopping [top site enclosed in the ellipse of panel

(b)]. Larger densities (darker colors) suggest the appearance

of a non-Hermitian skin eﬀect. The white and blue dotted

lines show the approximate location of gap closings at ε= 0

and π/T . The trivial (Tri), Chern insulator (CI), and AFTI

phases are indicated. White arrows show which phases re-

place the CIs at δAB = 0. See SM for numerical details [40].

Hons =δAB Pi,j (c†

A,i,j cA,i,j −c†

B,i,j cB,i,j ) and

Hhop(t) = JX

i,j











(c†

A,i,j cB,i,j + h.c.), t ∈T

5[0,1);

(c†

A,i,j cB,i−1,j+1 + h.c.), t ∈T

5[1,2);

(c†

A,i,j cB,i−1,j + h.c.), t ∈T

5[2,3);

(c†

A,i,j cB,i,j−1+ h.c.), t ∈T

5[3,4);

0, t ∈T

5[4,5).

(1)

Here, tis time, Jis the hopping strength, and c†

A,i,j

(cB,i,j ) denotes the creation (annihilation) operator for

sublattice A(B) in unit cell (i, j) [see Fig. 1(a)]. Set-

ting ℏ= 1, the dynamics of the system is governed by

a Floquet operator F=Te−RT

0iH(t)dt, with Tdenoting

time ordering. The quasienergy spectrum εis contained

in the fundamental domain [−π/T, π/T ) and can be ob-

tained from the eigenvalue equation det[F − e−iεT ] = 0.

In a ﬁnite-sized geometry, when switching oﬀ the sub-

lattice potential (δAB = 0), there are two limits in which

the Floquet operator takes a particularly simple form.

For the trivial limit with JT = 0, we have F= 1

and no particles can propagate. In the AFTI limit with

JT = 5π/2 (also known as the resonant driving point

[39]), all bulk states come back to their original sites after

one driving cycle, forming two degenerate, dispersionless

Floquet bulk bands at ε= 0. Even if states do not propa-

gate throughout the bulk, an extra quantized conducting

channel is formed at the edge, allowing particles to prop-

agate unidirectionally. This is the chiral edge mode of the

AFTI, which winds in quasienergy from −π/T to π/T .

At resonant driving, we draw a connection between

Floquet and non-Hermitian topology by using a dual-

ity that identiﬁes time-evolution operators with non-

Hermitian Hamiltonians. The latter has been used to

study the bulk spectral properties of non-Hermitian sys-

tems [41,42], and also for the purpose of topological clas-

siﬁcation [43,44]. Here, instead, we focus on its boundary

manifestations. We obtain a unitary, static tight-binding

model by treating the real-space Floquet operator as a

Hamiltonian:

Hu=F.(2)

Mathematically, this means that we represent the Flo-

quet operator of the ﬁnite-sized sytem as a matrix in

the real-space basis corresponding to the site positions of

the lattice in Fig. 1(a). Thus, in matrix representation,

[Hu]jk ≡ Fjk =⟨xk| F |xj⟩, with |xj⟩the position ket

of site j. The diagonal entries, Fjj , become (possibly

complex-valued) on-site terms, whereas the oﬀ-diagonal

terms, Fjk with j̸=k, are (possibly nonreciprocal) hop-

pings.

At δAB = 0 and JT = 5π/2, the real-space structure

of Huis shown in Fig. 1(b). Its bulk contains decou-

pled sites with unit on-site potentials, Fjj = 1, con-

sistent with the existence of dispersionless bulk bands

at energy E= 1 (meaning quasienergy ε= 0 in Flo-

quet language). On the boundary, however, the unidirec-

tional propagation of particles leads to one-way hoppings,

Fjk = 1, where jand kcorrespond to sites connected

by an arrow in Fig. 1(b). The AFTI boundary is iden-

tical to a maximally-nonreciprocal Hatano-Nelson chain

with periodic boundary conditions (PBC) [30]. As such,

it will show the same phenomenology as the Hatano-

Nelson chain: all states become localized at one end

when changing from periodic to open boundary condi-

tions. We achieve the latter by removing one hopping

from the chain (setting the oﬀ-diagonal term correspond-

ing to the arrow in the shaded ellipse to Fjk = 0). The

propagation of the Floquet chiral edge modes stops, lead-

ing to the formation of a non-Hermitian skin eﬀect.

Note that this behavior is diﬀerent from the recently-

introduced ‘hybrid skin-topological modes’ of Refs. [45–

48], which are generated by adding gain and loss to either

static or periodically-driven systems. Here, the 1D non-

Hermitian topology is intrinsic to the AFTI phase, it is

the boundary manifestation of a 2D AFTI bulk, and re-

moving one hopping simply serves to change the Hatano-

Nelson chain from PBC to OBC.

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Mixedhigher-ordertopology:boundarynon-HermitianskineffectinducedbyaFloquetbulkHuiLiu1andIonCosmaFulga11IFWDresdenandW¨urzburg-DresdenClusterofExcellencect.qmat,Helmholtzstrasse20,01069Dresden,GermanyWeshowthatanomalousFloquettopologicalinsulatorsgenerateintrinsic,non-Hermitiantopol-ogyontheirboundar...

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Mixed higher-order topology boundary non-Hermitian skin effect induced by a Floquet bulk Hui Liu1and Ion Cosma Fulga1

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