Non-Hermitian topological band structuresprotected by inversion symmetry
Ryo Okugawa (Tokyo University of Science)Phys. Rev. B 102, 241202(R) (2020)
Phys. Rev. B 103, 205205 (2021)

1

Outline
1. Introduction
Non-Hermitian skin effect

2. Non-Hermitian band topology with generalized inversion
symmetry
3. Higher-order skin effect protected by inversion symmetry2Non-Hermitian (NH) physics
NH physics can appear in

Photonic system

Electronic circuit• Open quantum systems• Photonic systems• Electronic circuits• Active matter• Robotic metamaterials• Bosonic Bogoliubov-de Gennes systems• Strongly correlated systems

H. Zhou, et al.,Science, 359, 1009 (2018)

Active matter

T. Helbig et al.,Nat. Phys. 16, 757 (2020)…
They can be described as effective non-Hermitian
Hamiltonians with the band structures.

L. S. Palacious et al,Nat. Commun. 12 4691 (2021)
3

Non-Hermitian band topology
Non-Hermiticity can realize novel topological band structures that is not allowed
to happen in Hermitian physics.
• Non-Hermitian skin effect• Exceptional point• Exceptional topological insulator⋮

NHSEs have been experimentally observed in ultracold atoms, electric circuits, active matter etc.
Q. Liang et al., PRL 129 070401 (2022)
T. Helbig et al., Nat. Phys. 16, 757 (2020)
L. S. Palacious et al, Nat. Commun. 12 4691 (2021)
4

Non-Hermitian skin effect (NHSE)
In non-Hermitian systems, energy spectra strongly depend on boundary conditions (even in thethermodynamic limit).
The strong dependence is called a non-Hermitian skin effect.
Ex. Hatano-Nelson model††𝐻 = ෍(𝑡𝐿 𝑐𝑖−1𝑐𝑖 + 𝑡𝑅 𝑐𝑖+1𝑐𝑖 )𝑖Skin effectSkin modes
Periodic boundary condition(PBC)

Open boundary condition
(OBC)

5

Non-Hermitian skin effect (NHSE)
When the NHSE happens, many eigenmodes (skin modes) are localized at the edgein the finite system.
††
𝐻 = ෍(𝑡𝐿 𝑐𝑖−1
𝑐𝑖 + 𝑡𝑅 𝑐𝑖+1𝑐𝑖 )
𝑖

OBC

𝜓𝑛 2

Localization

Skin modes
1

𝑁

The NHSE is important to bulk-boundary correspondence in non-Hermitian systems.
6Topology of the NHSE
The topological origin of the NHSE is a winding number of the band structure under the PBC
due to non-Hermiticity.
One-band caseN. Okuma et al., PRL 124, 086801 (2020)
K. Zhang et al., PRL 125, 126402 (2020)

1 𝜋𝑑The winding number is described as 𝑊 𝐸 =න 𝑑𝑘log det[𝐻 𝑘 − 𝐸] .
2𝜋𝑖 −𝜋 𝑑𝑘

The NHSE can happen without any symmetries.
7Motivation
・How do crystal symmetries affect skin effects?
There are some difficulties in numerical calculations for non-Hermitian Hamiltonians.

・It is difficult to identify skin effects in high-dimensional systems
under the full open BC.
・Skin modes should be calculated in a open system,
which gives rise to large numerical errors.

We provide a topological approach using inversion symmetry.
8Outline
1. Introduction
Non-Hermitian skin effect

2. Non-Hermitian band topology with generalized inversion
symmetry
3. Higher-order skin effect protected by inversion symmetry
9Why winding number?
Topological characterization of NH Hamiltonians

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

Point-gap topology at a reference energy 𝐸
Non-Hermitian Hamiltonian 𝐻
Point-gap

det 𝐻 − 𝐸 ≠ 0

Extended Hermitian Hamiltonian

෩=𝐻0𝐻† − 𝐸 ∗𝐻−𝐸0

Topological correspondenceBand topology of 𝐻 at 𝐸 corresponds to that of෩ with chiral symmetry Γ.
the Hermitian Hamiltonian 𝐻

Γ=

100−1

෩ can be characterized by the same winding number in 1D.
Both 𝐻 and 𝐻
10

Topological characterization
Non-Hermitian Hamiltonians have the same topology as chiral-symmetric Hermitian Hamiltonians.

・ Hermitian Hamiltonians in 1D class AIII
1 𝜋𝑑𝑤=
න 𝑑𝑘
log det[𝑞 𝑘 ]2𝜋𝑖 −𝜋 𝑑𝑘

෩=𝐻0𝑞†

𝑞0

・ Non-Hermitian Hamiltonians without any symmetries
1 𝜋𝑑
𝑊 𝐸 =
න 𝑑𝑘log det[𝐻 𝑘 − 𝐸]
2𝜋𝑖 −𝜋 𝑑𝑘෩=𝐻0
𝐻† − 𝐸 ∗𝐻−𝐸0

How does inversion symmetry affect the topology?
11

Inversion symmetry and topological invariant
One can easily know whether the system is topological by using crystal symmetryin Hermitian systems.
Ex. Fu-Kane-Mele formula for inversion-symmetric Z2 topological insulators𝑛− 𝚪𝜈−1 = ෑ −1 2𝚪Γ: time-reversal-invariant momenta (TRIM)Γ ≡ −Γ (mod 𝐆)
𝑛− (Γ): the number of occupied states with
odd-parity eigenvalues at TRIM

Can we simplify a winding number for non-Hermitian systems
using crystal symmetry?
12

Inversion symmetry and winding numberT. Hughes et al., PRB 83, 245132 (2011)

Inversion symmetry gives a simple formula of the topological invariant in 1D class AIII.
Expression

Parity formula1 𝜋𝑑𝑤=
න 𝑑𝑘log det[𝑞 𝑘 ]2𝜋𝑖 −𝜋 𝑑𝑘−1 𝑤 = ෑ −1 𝑛− (Γ)

1D class AIII0෩
𝐻= †𝑞𝑞0

Γ=0,𝜋𝑛− (Γ): the number of occupied
Γ: inversion-invariant momenta
states with negative parity eigenvaluesΓ ≡ −Γ (mod 𝐆)

We generalize this parity formula to calculate 𝑊 𝐸 for non-Hermitian systems.
13Conventional inversion symmetry
When does inversion symmetry appear in an extended Hermitian Hamiltonian?
Conventional inversion symmetry
𝑃𝐻 𝒌 𝑃−1 = 𝐻(−𝒌)𝑃𝑃† = 𝑃2 = 1

However, because the winding number is always trivial, the skin effect does not happen…
1 𝜋𝑑1 𝜋𝑑𝑊 𝐸 =න 𝑑𝑘
log det[𝐻 𝑘 − 𝐸] =න 𝑑𝑘
log det[𝐻 −𝑘 − 𝐸] = 0
2𝜋𝑖 −𝜋𝑑𝑘
2𝜋𝑖 −𝜋 𝑑𝑘(This inversion symmetry is incompatible with localized modes at one edge in 1D.)
14

Generalized inversion symmetry
Extended Hermitian Hamiltonians can acquire some symmetries which the original non-Hermitian Hamiltonians
do not have.

Generalized inversion symmetry

𝑈𝐼 𝐻 † 𝒌 𝑈𝐼 −1 = 𝐻(−𝒌)𝑈𝐼 𝑈𝐼† = 𝑈𝐼2 = 1

In the presence of generalized inversion symmetry,the extended Hermitian Hamiltonian with real 𝐸 can obtain inversion symmetry.

෩=𝐻0
𝐻† − 𝐸

𝐻−𝐸00𝐼ሚ =𝑈𝐼𝑈𝐼0

Γ=

100−115

Parity formula for NH systems
Reformulate the parity formula −1 𝑤 = −1 𝑛− 0 −𝑛− (𝜋) for non-Hermitian Hamiltonians 𝐻෩ Γ =𝐻0𝐻† Γ − 𝐸0𝐻 Γ −𝐸=𝑈𝐼 𝐻 Γ 𝑈𝐼−1 − 𝐸0𝐻 Γ −𝐸0(Γ = 0, 𝜋)𝑛− (Γ): the number of occupied
states with negative parity eigenvalues𝑛− Γ = 𝑁+ (Γ) : # of positive eigenvalues of 𝑈𝐼 (𝐻 Γ − 𝐸)
This formula can connect non-Hermitian band topology with Hermitian band topology.

−1 𝑊(𝐸) = −1 𝑁+ 0 −𝑁+ (𝜋) = ෑ sgn det[𝐻 Γ − 𝐸]
Γ=0,𝜋16

Model calculation
𝑀=3Ex. Generalized Hatano-Nelson model

The Hamiltonian under the PBCℎ† −𝑘 = ℎ 𝑘 , 𝑈𝐼 = 1The parity formula17

Summary (1)Phys. Rev. B 103, 205205 (2021)

If generalized inversion symmetry is present, the parity of the winding number for the NHSE
can be evaluated from the eigenvalues at the inversion-symmetric points.
generalized inversion symmetry

𝑈𝐼 𝐻 † 𝒌 𝑈𝐼 −1 = 𝐻(−𝒌)𝑈𝐼 𝑈𝐼† = 𝑈𝐼2 = 1−1 𝑊(𝐸) = −1 𝑁+ 0 −𝑁+ (𝜋) = ෑ sgn det[𝐻 Γ − 𝐸]
Γ=0,𝜋𝑁+ (Γ) : # of positive eigenvalues of 𝑈𝐼 (𝐻 Γ − 𝐸)
(This idea can be generalized to a winding number for exceptional points in the momentum space.)
Phys. Rev. B 105, 085109 (2022)18

Outline
1. Introduction
Non-Hermitian skin effect

2. Non-Hermitian band topology with generalized inversion
symmetry
3. Higher-order skin effect protected by inversion symmetry
Phys. Rev. B 102, 241202(R) (2020)
Phys. Rev. B 103, 205205 (2021)

K. Kawabata, M. Sato, and K. Shiozaki
PRB 102, 205118 (2020)19

Higher-order topological phases (HOTPs)
Higher-order topological phases have robust corner and hinge modes.
Higher-order topology
3D
2D
Corner states

Hinge states
2nd-order

Can higher-order topology coexist with non-Hermitian band topology?
20

Higher-order topology and NH topology
A NH Hamiltonian can inherit band topology of a chiral-symmetric Hermitian Hamiltonian.

Chiral-symmetric Hermitian Hamiltonian
with second-order topology

෩𝐻𝑂𝑇𝑃 =𝐻0𝐻†

𝐻0

Non-Hermitian Hamiltonian𝐻 (≠ 𝐻 † )Can we realize a NHSE due to second-order topology?

21

HOTP protected by inversion symmetry
Hermitian higher-order topology can be realized in 2D class AIII with inversion symmetry,
where corner zero modes appear.
E. Khalaf, PRB 97 205136 (2017)
A. Matsugatani and H. Watanabe, PRB 98 205129 (2018)
R. Takahashi et al., PRR 2, 013300 (2020)

Topological invariants in the systems
𝜇𝑥 = 𝑛− 0,0 − 𝑛− 𝜋, 𝜋 + 𝑛− 𝜋, 0 − 𝑛− (0, 𝜋)

𝜇𝑦 = 𝑛− 0,0 − 𝑛− 𝜋, 𝜋 − 𝑛− 𝜋, 0 + 𝑛− (0, 𝜋)𝑛− (Γ): the number of occupied
states with negative parity eigenvalues

When 𝜇𝑥(𝑦) = 2 (mod 4), the system is second-order topological.
22

The HOTP and winding numbers
R. Takahashi et al., PRR 2, 013300 (2020)

The topological invariants 𝜇𝑥,𝑦 are related to winding numbers for a ribbon geometry
as long as the ribbon retains inversion symmetry.

𝑤𝑥(𝑦)−𝑂𝐵𝐶 ≡𝜇𝑥 ≡ 2 (𝑚𝑜𝑑 4)𝜇𝑥(𝑦)2(mod 2)𝑤𝑥−𝑂𝐵𝐶 ≡ 1 (𝑚𝑜𝑑 2)

Corner modes

We can apply the topological invariant to a NHSE due to second-order topology!
23

Second-order NHSE
If a 2D NH Hamiltonian gives an extended-Hermitian Hamiltonian with the nontrivial 𝜇𝑥(𝑦) = 2 (mod 4),
the NH system shows a skin effect under the full OBC.

෩𝐻𝑂𝑇𝑃 =𝐻𝑈𝐼 𝐻† 𝒌 𝑈𝐼 −1 = 𝐻(−𝒌)0𝐻†

0𝐼ሚ =𝑈𝐼

𝐻0

𝑤𝑥(𝑦)−𝑂𝐵𝐶 ≡𝜇𝑥(𝑦)2𝑈𝐼0(mod 2)

Because 𝐻 reflects second-order topology with the nonzero winding number,the NH Hamiltonian can be characterized by the same winding number with real E.

𝑊𝑥(𝑦)−𝑂𝐵𝐶 (𝐸) ≡𝜇𝑥,𝑦 (𝐸)2(mod 2)24

Second-order NHSE
We can calculate topological invariants 𝜇𝑥(𝑦) from eigenvalues of the non-Hermitian Hamiltonian.

𝑛− Γ = 𝑁+ (Γ) : # of positive eigenvalues of 𝑈𝐼 (𝐻 Γ − 𝐸)
Winding number for ribbon geometry

𝑊𝑥(𝑦)−𝑂𝐵𝐶 (𝐸) ≡𝜇𝑥(𝑦) (𝐸)
2(mod 2)𝜇𝑥 ≡ 2 (𝑚𝑜𝑑 4)

𝜇𝑥 = 𝑁+ 0,0 − 𝑁+ 𝜋, 𝜋 + 𝑁+ 𝜋, 0 − 𝑁+ (0, 𝜋)
𝜇𝑦 = 𝑁+ 0,0 − 𝑁+ 𝜋, 𝜋 − 𝑁+ 𝜋, 0 + 𝑁+ (0, 𝜋)

𝑊𝑥−𝑂𝐵𝐶 ≡ 1 (𝑚𝑜𝑑 2)

Skin effect!

25Model calculation
𝐻2𝐷 𝒌 = (𝑚 − 𝑐 cos 𝑘𝑥 − 𝑐 cos 𝑘𝑦 )𝑠0 + 𝑖𝑡 sin 𝑘𝑦 𝑠𝑥 + 𝑖𝑡 sin 𝑘𝑥 𝑠𝑦 − 𝐵𝑥 𝑠𝑥 − 𝐵𝑦 𝑠𝑦

𝑚 = 𝑐 = 1.0, 𝑡 = 0.8, 𝐵𝑥 = 𝐵𝑦 = 0.15
26

Summary (2)A NHSE can be realized from second-order topology protected by inversion symmetry.
Topological invariants for the second-order NHSE
𝜇𝑥 = 𝑁+ 0,0 − 𝑁+ 𝜋, 𝜋 + 𝑁+ 𝜋, 0 − 𝑁+ 0, 𝜋
𝜇𝑦 = 𝑁+ 0,0 − 𝑁+ 𝜋, 𝜋 − 𝑁+ 𝜋, 0 + 𝑁+ 0, 𝜋
𝑁+ (Γ) : # of positive eigenvalues of 𝑈𝐼 (𝐻 Γ − 𝐸)

When the topological invariants 𝜇𝑥(𝑦) = 2 mod 4 ,the NHSE can happen under the full OBC.

𝑊𝑥(𝑦)−𝑂𝐵𝐶 (𝐸) ≡𝜇𝑥 ≡ 2

𝑊𝑥−𝑂𝐵𝐶 ≡ 1

Skin effect!

𝜇𝑥,𝑦 (𝐸)2(mod 2)27

Conclusion
PRB 102, 241202(R) (2020) PRB 103, 205205 (2021)

We derived simple formulas to evaluate winding numbers for NHSEs
if generalized inversion symmetry is present.

𝑈𝐼 𝐻 † 𝒌 𝑈𝐼 −1 = 𝐻(−𝒌)・1D skin effect

𝑈𝐼 𝑈𝐼† = 𝑈𝐼2 = 1−1 𝑊(𝐸) = −1 𝑁+ 0 −𝑁+ (𝜋) = ෑ sgn det[𝐻 Γ − 𝐸]
Γ=0,𝜋𝑁+ (Γ): # of positive eigenvalues of 𝑈𝐼 (𝐻 Γ − 𝐸)・2D second-order skin effect𝑊𝑥(𝑦)−𝑂𝐵𝐶 (𝐸) ≡𝜇𝑥(𝑦) (𝐸)2(mod 2)𝜇𝑥 = 𝑁+ 0,0 − 𝑁+ 𝜋, 𝜋 + 𝑁+ 𝜋, 0 − 𝑁+ (0, 𝜋)

𝜇𝑦 = 𝑁+ 0,0 − 𝑁+ 𝜋, 𝜋 − 𝑁+ 𝜋, 0 + 𝑁+ (0, 𝜋)28