Supersymmetric non-Hermitian topological interface laser Motohiko Ezawa1Natsuko Ishida2Yasutomo Ota3and Satoshi Iwamoto2 1Department of Applied Physics The University of Tokyo 7-3-1 Hongo Tokyo 113-8656 Japan

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Supersymmetric non-Hermitian topological interface laser

Motohiko Ezawa,1Natsuko Ishida,2Yasutomo Ota,3and Satoshi Iwamoto2

1Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan

2Research Center for Advanced Science and Technology,

The University of Tokyo, 4-6-1 Komaba, Tokyo 113-8656, Japan

3Research Center for Department of Applied Physics and Physico-Informatics, Keio University, 3-14-1 Hiyoshi, Japan

(Dated: October 25, 2022)

We investigate laser emission at the interface of a topological and trivial phases with loss and gain. The system

is described by a Su-Schrieffer-Heeger model with site-dependent hopping parameters. We study numerically

and analytically the interface states. The ground state is described by the Jackiw-Rebbi mode with a pure

imaginary energy, reﬂecting the non-Hermiticity of the system. It is strictly localized only at the A sites. We

also ﬁnd a series of analytic solutions of excited states based on SUSY quantum mechanics, where the A and

B sites of the bipartite lattice form SUSY partners. We then study the system containing loss and gain with

saturation. The Jackiw-Rebbi mode is extended to a nonlinear theory, where B sites are also excited. The

relative phases between A and B sites are ﬁxed, and hence it will serve as a large area coherent laser.

I. INTRODUCTION

Topological physics is one of the most exciting ﬁelds[1,2].

The Su-Schrieffer-Heeger (SSH) model is a simplest example

of topological insulators[3]. The topological phase is charac-

terized by the emergence of zero-energy states at the edges of

a sample. A zero-energy state emerges also at an interface be-

tween a topological phase and a trivial phase, which is called

a topological interface state. The Jackiw-Rebbi solution[4] is

an analytic solution for the topological interface state. Now,

non-Hermitian topological physics is an emerging ﬁeld. The

Jackiw-Rebbi solution seems to be not valid because the en-

ergy of the topological interface state is nonzero in general.

Topological photonics is an ideal playground of studying

topological physics[5–22]. Topological laser is one of the

most successful applications of topological physics[23–33].

A strong lasing from a single coherent mode is possible due to

a topological edge or interface state. In topological photonics,

loss is inevitable and hence leading to non-Hermitian topolog-

ical physics[34,35]. We need to add a gain in order to obtain

a laser. Especially, a topological interface laser has enabled a

large area coherent lasing by using a smooth interface[36].

In this paper, in order to understand laser emission at the

interface between a topological and trivial phases, we ana-

lyze a non-Hermitian SSH model ﬁrst by including linear loss

and gain terms. We solve numerically a set of nonlinear dif-

ferential equations. We also make an analytical study of the

Jackiw-Rebbi mode to describe the topological interface state,

upon which we construct a series of excitation states at the in-

terface based on supersymmetric (SUSY) quantum mechanics

generalized to a non-Hermitian system. We call them SUSY

Jackiw-Rebbi modes because they preserve SUSY although

the original Jackiw-Rebbi mode breaks SUSY. Not only the

topological interface state but also the SUSY Jackiw-Rebbi

modes are shown to have pure imaginary energies. Here,

SUSY partners are formed by the A and B sites of the bi-

partite lattice, where only A sites are excited in the original

Jackiw-Rebbi mode. We conﬁrm that the analytical solutions

well coincide with numerical solutions. Next, we include a

saturation term to the gain, which is a nonlinear term. Such a

system well describes a large area stable laser emission from

an interface of a topological system. The Jackiw-Rebbi topo-

logical mode is solely stimulated in laser emission. We extend

the Jackiw-Rebbi mode to the nonlinear regime. Excitations at

B sites are induced in the Jackiw-Rebbi mode by a nonlinear

effect, where the wavefunction at B sites is ﬁxed to be pure

imaginary. The relative phases between the saturated wave-

functions at the A and B sites are ﬁxed. Since the Jackiw-

Rebbi mode extends over a wide region around the interface,

it will give a large area coherent laser.

II. MODEL

We investigate the dynamics of a laser system governed

by[23]

idψn

dt =X

Mnmψm−iγ 1−χ(1 −(−1)n)/2

1 + |ψn|2/η !ψn,

(1)

with a site dependent hopping matrix

Mnm =κA,n (δ2n,2m−1+δ2m,2n−1)

+κB(δ2n,2m+1 +δ2m,2n+1),(2)

where ψnis the amplitudes at the site n, where n=

1,2,3,··· , N in the system composed of Nsites; γrepre-

sents the loss in each resonator; γχ represents the amplitude

of the optical gain via stimulated emission induced only at the

odd site; ηrepresents the nonlinear saturation constant[23].

All these parameters are taken positive semideﬁnite. The lat-

tice structure of the SSH model is bipartite, where the odd and

even sites are called the A and B sites, respectively. The sys-

tem turns out to be a linear model in the limit η→ ∞.On the

other hand, γcontrols the non-Hermicity, where the system is

Hermitian for γ= 0.

The hopping amplitudes are explicitly given by

κA,n =κ1 + λtanh n−nIF + 1/2

ξ, κB=κ, (3)

arXiv:2210.12592v1 [cond-mat.mes-hall] 23 Oct 2022

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9

N=10 N=9

N=100 N=99

trivial sector

(a1) (b1)

(a2) (b2)

interface stateedge state

eigen index eigen index

(edge state)

topological sector trivial sector topological sector

l ll l ll l ll ll l ll l

interface state

l l l

20 40 60 80 100

-2

-1

20 40 60

50 80 99

-2

-1

kBkB

FIG. 1. (a1), (b1) Illustration of the interface (marked in red) in

the SSH chain for N= 10 and 9. (a1) Topological edge states

(marked in red) appear at the two edges of a topological sector. (b1)

The topological edge state is absent at the edge of a sample when

Nis odd. The topological state emerges only at the interface. (a2),

(b2) Energy spectrum (vertical axis) of the SSH model as a function

of the eigen index (horizontal axis) for N= 100 and 99, where

the eigen index is sorted in the increasing order of the energy. Two

and one zero-energy topological states (marked in red) emerge in the

SSH chain with N= 10 and 9. The structure of kinks at p= 16

and p= 84 is due to the difference of the band width between the

topological and the trivial sectors. We have set λ= 0.5.

with λ > 0, where nIF is the smallest odd number larger than

or equal to N/2. Then, n−nIF + 1/2>0for n≥nIF,

and n−nIF + 1/2<0for n<nIF. We call the site n=

nIF the interface of the chain. See Fig.1(a1) and (b1) for an

illustration in the case of N= 10 and 9.

The explicit equations for a ﬁnite chain with length Nfol-

low from Eq.(1) as

idψ2n−1

dt =κBψ2n−2+κA,nψ2n

−iγ 1−χ

1 + |ψ2n−1|2/η !ψ2n−1,(4)

idψ2n

dt =κBψ2n+1 +κA,nψ2n−1−iγψ2n.(5)

We solve this set of equations together with the initial condi-

tion

ψn(t= 0) = δn,nIF .(6)

This is a quench dynamics starting from the interface site by

giving an input to it initially. The initial input triggers the gain

effect in Eq.(4) because nIF is an odd number.

III. LINEAR THEORY

We start with the linear model (η→ ∞). Then, Eq.(1) is

reduced to

idψn

dt =X

Mnmψm,(7)

where

Mnm =Mnm −iγ 1−χ

2δnm,(8)

with

Mnm =Mnm −iγχ (−1)n

2δnm.(9)

Since f

Mnm and Mnm are different only by a c-number term,

they describe the identical physics. Hereafter, we use f

Mnm

for the study of dynamics and Mnm for the analytical study

of the system.

A. Topological edge and interface states

1. SSH model

We analyze the SSH model Mnm by taking the negligible

penetration depth (ξ→0). Then, Eq.(3) amounts to

κA,n =κ(1 + λ)for n≥nIF,

κA,n =κ(1 −λ)for n<nIF.(10)

The hopping amplitudes are constant κA,n =κ(1 + λ)for

the segments with n≥nIF, while they are constant κA,n =

κ(1 −λ)for the segments with n<nIF, separately. Note that

κB=κ. The hopping matrix Mnm deﬁnes the SSH model in

each segment.

The SSH model with constant hopping amplitudes κAand

κBhas a topological phase for κA< κBand the trivial phase

for κA> κB. The topological phase is characterized by the

emergence of zero-energy states at the edges of a ﬁnite chain,

as demonstrated numerically in Fig.1(a2) for N= 100. This

is the standard bulk-edge correspondence. It is illustrated in

Fig.1(a1) for N= 10. See Appendix for details.

There is an intriguing phenomenon in the SSH model with

respect to the even-odd effect of the number of the sites within

the chain[30,36]. We may remove the edge site at n=N

from an SSH chain with even Nto obtain an SSH chain with

odd total number N−1. See an illustration in Fig.1(a1)

and (b1), where two chains with N= 10 and 9are shown.

We demonstrate numerically that there is only one zero-mode

state in the odd chain with N= 99 in Fig.1(b2), which is

the topological interface state illustrated in Fig.1(a2). This is

also a bulk-edge correspondence. Recall that the topological

number is deﬁned for the unit cell of the bulk.

In the rest of this work, we focus on the topological inter-

face state by taking an SSH chain with odd N. Furthermore,

we do not take the limit ξ→0any longer.

2. Non-Hermitian SSH model

We investigate the system Mnm with a ﬁnite loss (γ6= 0)

and gain (γχ 6= 0). Diagonalizing the hopping matrix Mnm

in Eq.(8) numerically, we obtain the energy spectrum Eas

a function of χwhile setting γ= 0.1. We show the results

in the (χ, Re[E],Im[E]) space for ξ= 10 in Fig.2(a). See

also Fig.2(c1) and (c2) for its cross section at Im[E] = 0 and

FIG. 2. (a) Energy spectrum in the (γχ,Re[E],Im[E]) space for ξ=

10, where γχ stands for the gain (0 < γχ < 1.5). (b1), (c1),

(d1) Energy spectrum in the (γχ,Re[E]) plane for ξ= 1,10,100.

(b2), (c2), (d2) Energy spectrum in the (χ,Im[E]) plane for ξ=

1,10,100. The red line represents the topological interface state,

whose energy is pure imaginary. The width of the line is proportional

to the local density of states. The interface state is well separated

from (almost touched to) the bulk spectrum for ξ= 1,10 (ξ= 100).

We have set γ= 0.1and λ= 0.5. We have used the chain with

N= 99.

Re[E] = 0, respectively. We clearly observe a straight line

passing through the point (0,0,0) in the (χ, Re[E],Im[E])

space, which represents the energy of the topological inter-

face state we have just discussed.

Similarly, we show the energy spectrum for ξ= 1 and 100

in Fig.2(b1), (b2), (d1) and (d2). We also ﬁnd a straight line

passing through the point (0,0,0) in the (χ, Re[E],Im[E])

space.

The energy of the topological interface state is well ﬁtted

for any system parameters by the formula

EIF =i¯γwith ¯γ=γχ/2.(11)

The eigenvalue (11) and the associated eigenfunction are de-

rived as a Jackiw-Rebbi solution later in Section IV: See

Eq.(29).

In addition, we observe a band-edge mode[36] between the

interface mode and the bulk spectrum for ξ= 10. In the case

of ξ= 100, in addition to the band-edge mode, there are many

modes with almost equal spacing and characterized by their

pure imaginary energies. We call them SUSY Jackiw-Rebbi

modes, with respect to which we discuss based on the SUSY

quantum mechanics in Section IV.

B. Dynamics

The quench dynamics is a powerful tool to distinguish topo-

logical phase even for nonlinear systems[37–40]. Before an-

alyzing the dynamics of the system, it is convenient to study

FIG. 3. (a1), (b1), (c1) Energy Epand (a2), (b2), (c2) the component

|cp|as functions of the eigen index p. A red large disk indicates

the topological interface state. On the other hand, cyan small disks

indicate the bulk states. The size of a disk is proportional to the local

density of states. It becomes smaller for larger ξbecause the interface

mode becomes broader. The horizontal axis is the eigen index. (a1),

(a2) ξ= 1; (b1), (b2) ξ= 10; (c1), (c2) ξ= 100. We have set

N= 399,nIF = 199 and γ= 0.

the eigenvalues and the eigenfunctions of the hopping matrix

Mnm given by Eq.(8). We diagonalize it as

Mφp=e

Epφp,(12)

where plabels the eigen index, 1≤p≤N, and φpis the

eigenfunction. We show the eigenvalues e

Epin Fig.3(a1), (b1)

and (c1). Let the wavefunction of the topological interface

state be φIF. Its eigenvalue is

EIF =EIF −iγ 1−χ

2δnm =iγ (χ−1) ,(13)

with the use of Eq.(8) and Eq.(11).

Decoupled equations follow from Eq.(7) for the eigenfunc-

tions,

idφp

dt =e

Epφp,(14)

whose solutions are given by

φp(t) = exp h−it e

Epiφp.(15)

In particular, for the topological interface state, we have

φIF (t) = exp [γ(χ−1) t]φIF,(16)

with the use of Eq.(13). It has no dynamics for γ= 0 or

χ= 1. On the other hand, it grows exponentially for χ > 1.

The initial state (6) is expanded in terms of the eigenfunc-

tions as

ψn(t= 0) = δn,nIF =X

cpφp.(17)

We show the coefﬁcient |cp|in Fig.3(a2), (b2) and (c2), which

is determined by

cp=X

δn,nIF φp.(18)

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Supersymmetricnon-HermitiantopologicalinterfacelaserMotohikoEzawa,1NatsukoIshida,2YasutomoOta,3andSatoshiIwamoto21DepartmentofAppliedPhysics,TheUniversityofTokyo,7-3-1Hongo,Tokyo113-8656,Japan2ResearchCenterforAdvancedScienceandTechnology,TheUniversityofTokyo,4-6-1Komaba,Tokyo113-8656,Japan3Research...

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Supersymmetric non-Hermitian topological interface laser Motohiko Ezawa1Natsuko Ishida2Yasutomo Ota3and Satoshi Iwamoto2 1Department of Applied Physics The University of Tokyo 7-3-1 Hongo Tokyo 113-8656 Japan

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