Non-Hermitian exceptional boundary mode
and its application to topological laser
Kazuki Sone

University of Tokyo, Sagawa group
Collaborators: Yuto Ashida, Takahiro Sagawa
KS, Y. Ashida, and T. Sagawa, Nat. Commun. 11, 5745 (2020).
KS, Y. Ashida, and T. Sagawa, Phys. Rev. B 105, 235426 (2022).
1

Outline
Introduction- Non-Hermitian topology

Exceptional edge mode
- Fundamental concept- Numerical results on a toy model

Applications to topological lasers- Toy models and numerical results- Proposal of photonic realization

2Outline
Introduction- Non-Hermitian topology

Exceptional edge mode
- Fundamental concept- Numerical results on a toy model

Applications to topological lasers- Toy models and numerical results- Proposal of photonic realization3Band Topology and Bulk-Edge Correspondence
Bulk (periodic boundary)

Edge (open boundary)(Eigenequation of each wavenumber sector)

Bulk-edge
correpondence
Chern number

Edge modes• Localized states• Gapless dispersions• Backscattering-free(unidirectional) current

e.g. Quantum
Hall effect• Robustness against disorders

4Topology of Non-Hermitian Hamiltonians
Non-Hermitian 𝐻 ≠ 𝐻†• Description for systems with gain-loss and/or nonreciprocity
e.g. photonics, electrical circuits, cold atoms

C. E. Rüter et al. Nat. Phys. 6, 192 (2010)

J. Schindler et al. Phys. Rev. A 84, 040101 (2011)

Unique topological features• Novel classification utilizing
complex eigenvalues• Topological protection of
exceptional points

Z. Gong et al. PRX 8, 3 (2018)

5

Exceptional Points: Non-Hermitian Topological Gapless Points
Exceptional points (EPs)= Nondiagonalizable gapless points(cf. Jordan normal form)

cf)

exceptional point at 𝑘 = 0
H. Shen et al. PRL 120, 146402 (2018)

Topologically protected!!

Analogous to Weyl points→ Removed only via pair annihilations
Re E

Re ERe Ekkk
X. Wan et al., PRB 83, 205101 (2011).

Green points: exceptional points

6Topological Laser: Application of Non-Hermitian Topology
Topological laser⇒ Amplifying boundary modes
Advantage: robustness against disorders• Topological insulators + judicious gain at the edge of the sample-B. Bahari et al. Science
358, 636 (2017).
G. Harari et al. Science359, 1230 (2018).
M. A. Bandres et al.
Science 359, 1231 (2018).

• Symmetry protected lasing mode
PTsymmetry-

K. Kawabata, K. Shiozaki,M. Ueda, M. Sato
PRX 9, 041015 (2019).
A. Y. Song et al.
PRL 125, 033603 (2020).
7

Outline
Introduction- Non-Hermitian topology

Exceptional edge mode
- Fundamental concept- Numerical results on a toy model

Applications to topological lasers- Toy models and numerical results- Proposal of photonic realization

8

Exceptional Edge Modes Independent of Bulk Topology
Exceptional edge modes(unique to non-Hermitian systems)

NonHermiticity• Independent of the bulk topology→ Breakdown of the bulk-edgecorrespondence（different from skin effect）

Exceptional points protectedge modes like “glue”

Typical construction procedure

Non-Hermitiancoupling9

Toy Model for Exceptional Edge Modes
Toy model (tight-binding)

Chern insulator

Non-Hermitian coupling2 × 2 Hamiltonian of
the QWZ model (Chern insulator)

Curve: Hermitian coupling
Wave curve: non-Hermitian coupling

Ref. QWZ model: X. L. Qi, Y. S. Wu, & S. C. Zhang PRB 74, 085308 (2006).
BHZ model: B. A. Bernevig, T. L. Hughes, & S. C. Zhang Science 314, 1757 (2006).

10

Independence of the Bulk Topology
0𝜈=0

Non-Hermiticity

large

𝜈=011

Independence of the Bulk Topology
0𝜈=0

Non-Hermiticity

𝜈=0large

Exceptional points

𝜈=0

𝜈=0

Gapless edge modes
butno bulk gap closings12Topological Protection of Exceptional Points Under Symmetries
Exceptional points in 1D + symmetryex) PT (parity-time) symmetry
𝑃𝑇𝐻 𝒌 𝑃𝑇 −1 = 𝐻 ∗ 𝒌→ Eigenvalues are real or pairs of complex conjugates.

→ EPs are protected!

Re E, Im Ek

Untransformablewithout EPs
13

Robustness Against the Disorder

PT-sym. preserved→ robust

Symmetries broken→ unstable

Consistent with thesymmetry protectionof EPs

14

Outline
Introduction- Non-Hermitian topology

Exceptional edge mode
- Fundamental concept- Numerical results on a toy model

Applications to topological lasers- Toy models and numerical results- Proposal of photonic realization

15

Remained Problems on Topological Lasers
Q1. Propagation of lasing wave packets
without judicious gain
→ Without need of the knowledgeof boundary configurations

A1. Modified model of exceptional edge modes!

Q2. Topological lasers withoutboth juditicious gain and symmetries→ More robust topological lasers

A2. Extension of exceptional edge modes
to 3D systems!

16

Remained Problems on Topological Lasers
Q1. Propagation of lasing wave packets
without judicious gain
→ Without need of the knowledgeof boundary configurations

A1. Modified model of exceptional edge modes!

Q2. Topological lasers withoutboth juditicious gain and symmetries→ More robust topological lasers

A2. Extension of exceptional edge modes
to 3D systems!

17

Application to 2D Topological Laser
Hamiltonian

Edge band structure

Strong hopping→ Satisfying prerequisitesfor a topological laser
18

Application to 2D Topological Laser
Hamiltonian

Edge band structure

Strong hopping→ Satisfying prerequisites
for a topological laser
19Remained Problems on Topological Lasers
Q1. Propagation of lasing wave packets
without judicious gain
→ Without need of the knowledgeof boundary configurations

A1. Modified model of exceptional edge modes!

Q2. Topological lasers withoutboth juditicious gain and symmetries→ More robust topological lasers

A2. Extension of exceptional edge modes
to 3D systems!

20Construction Procedure of Three-Dimensional Topological LaserNo symmetry, 3D → (Line-gap) topology is trivial.
(cf. K. Kawabata et al., PRX 9, 041015 (2019).)

Exceptional points in boundary bands (2D) are stable.
→ extension of exceptional edge modes
Weak topological insulator(accumulation ofquantum spin Hall systems)

Non-Hermitiancoupling→ two Dirac cones+
Non-Hermitian spin coupling→ splitting into exceptional points→ large imaginary parts of eigenvalues(⇒amplification)

cf. One pair of EPs→ unstable under OBC

21

Toy Model: Weak Topological Insulator + Non-Hermitian Coupling1.

Chern insulator model2.

Coupling with its time-reversal
counterpart

X. QI, Y. S. Wu, S. C. Zhang PRB 74, 085308 (2006).

(quantum spin Hall (QSH) system)

3.

Non-Hermitian couplings

4.

Accumulation

QSH system

Interlayer coupling

Non-Hermitian coupling
22Band Structure of the Surface Modes

w/o noise

w/ noise

Exceptional
points⇒ topologicallyprotected

Large
imaginary parts⇒ amplification

23

Nonzero Group Velocity: Transfer of Lasing Wave Packets
Asymmetry between (effective) spins→ Nonzero group velocity → Transfer of lasing wave packets(cf. KS, Y. Ashida, and T. Sagawa, Nat. Commun. 11, 5745 (2020).)𝑏: strength of asymmetryProtection
by exceptional points
Boundary modes
with nonzero group velocity
24Real-Space Dynamics

DefectLasing wave packets along the edge
・Nonzero group velocity
・Backscattering-free at the defect= Robustness against disorders at the boundaries
25Real-Space Dynamics

DefectLasing wave packets along the edge
・Nonzero group velocity
・Scattering-free at the defect= Robustness against disorders at the boundaries
26

Possible Optical Setup Using Ring-Resonator Arrays
Ring-resonator arrays⇒ quantum spin Hall systems• Effective spin: clockwise andcounterclockwise modes• Hopping: via evanescent light• Artificial gauge field: phase-differenceof propagating lightcf. M. Hafezi et al. Nat. Photonics 7, 1001 (2013).

Accumulation⇒ Counterpart ofweak topological insulator
Waveguides with mirrors⇒ Non-Hermitian coupling(diminishing imbalance
between CW and CCW modes)

cf. A. Y. Song et al. PRL 125, 033603 (2020).
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Numerical Results: Exceptional Surface Modes in Proposed Setup

Real-space dynamics

Dispersion relation

Exceptional
points28

Numerical Results: Exceptional Surface Modes in Proposed Setup

Real-space dynamics

Dispersion relation

Exceptionalpoints

29Summary
Exceptional edge modes→ Protected by exceptional points• Robust even with a topologically trivial bulk
→ Breakdown of the bulk-edge correspondence
Applications to topological laser• Without judicious gains• In 3D, no needs of symmetry protections• Proposal of photonic systems
Refs: KS, Y. Ashida, and T. Sagawa, Nat. Commun. 11, 5745 (2020).
KS, Y. Ashida, and T. Sagawa, Phys. Rev. B 105, 235426 (2022).
30