高次非エルミート表皮効果
川畑 幸平（東大理）ID rId2

2020 年 9 月 11 日
ID rId2

共同研究者：佐藤 昌利・塩崎 謙（京大基研）

arXiv: 2008.07237

Non-Hermitian skin effect

1/10☆ Bulk-boundary correspondence
・Zero modes in the Su-Schrieffer-Heeger model・Corner zero modes in the Benalcazar-Bernevig-Hughes model(higher-order topological insulator)
Benalcazar, Bernevig & Hughes, Science 357, 61 (2017)
Lee, PRL 116, 133903 (2016)☆ Non-Hermitian skin effectYao & Wang, PRL 121,
New type of BBC unique to non-Hermitian systems.
086803 (2018)
Kunst et al., PRL 121,
An extensive number of boundary (skin) modes.
026808 (2018)

ModelNumberTopology

Zero modes in SSH

O (1) from O (L)Hermitian (line gap)

Corner zero modes in BBH

O (1) from O (L2)

Hermitian (line gap)

Non-Hermitian skin effectO (L) from O (L)Non-Hermitian (point gap)

Skin effect in experiments
・mechanical metamaterials

Brandenbourger et al., Nat. Commun. 10, 4608 (2019)・quantum walk

Xiao et al., Nat. Phys. 16, 761 (2020)

2/10

・electrical circuits

Helbig et al., Nat. Phys. 16, 747 (2020)

・photonic lattice

Weidemann et al., Science 368, 311 (2020)

Various skin effects
・Symmetry-protected skin effectlocalized at left

3/10Okuma et al., PRL 124, 086801 (2020)

localized at rightreciprocal (Kramers) pair・2D skin effectlocalized at bothreciprocal skin effectZ2 topological invariant

・3D skin effectdefectOkuma et al., PRL 124, 086801 (2020)

Terrier & Kunst, PRR 2, 023364 (2020)
Denner et al., arXiv:2008.01090

Hermitian, 1st

Hermitian, 2nd

Non-Hermitian, conventional

Non-Hermitian, 2nd

Skin effect in one dimension
・Hatano-Nelson model

5/10

Hatano & Nelson, PRL 77, 570 (1996)☆ Topological origin of the skin effectintrinsic non-Hermitian topology
Gong et al., PRX 8, 031079 (2018)

Zhang, Yang & Fang, arXiv:1910.01131
Okuma et al., PRL 124, 086801 (2020)

HN modelSSH model(skin modes)(zero modes)

Model
SSH model

6/10
HN model2nd-order TI (BBH model)

2nd-order skin effect☆ Spatial symmetry

・inversion
・mirror
Benalcazar, Bernevig & Hughes,Science 357, 61 (2017)

・C4 rotation(unconventional symmetry unique to non-Hermitian systems)

Second-order skin effectPBCOBC

7/10
skinbulk☆ The emergence of O (L) corner skin modes.

Wess-Zumino term

8/10

☆ C4-rotation-type symmetry:

・ C4-rotation-type symmetry vanishes the conventional skin effects.

・ C4-rotation-type symmetry quantizes the Wess-Zumino term.

(not quantized in the absence of symmetry)

Wess & Zumino, Phys. Lett. B 37, 95 (1971)☆ The nontrivial WZ term is the origin of the corner skin effect.

our model:

forcf. zero modes at topological defects

Teo & Kane, PRB 82, 115120 (2010)

Non-Bloch band theory

9/10☆ The skin effect invalidates the Bloch band theory.

Non-Bloch band theory (Yao-Wang-Yokomizo-Murakami): solutions to
Yao & Wang, PRL 121, 086803 (2018)
Yokomizo & Murakami, PRL 123, 066404 (2019)

・Symmetry can restore the Bloch band theory. Kawabata et al., PRX 9, 041015 (2019)e.g. inversion
YWYM

If the YWYM condition is valid even in higher dimensions,the skin effect vanishes in the presence of inversion symmetry.
But the O (L) corner skin modes emerge!
☆ Breakdown of the YWYM condition in higher dimensions.
[but expected to describe O (Ld) modes]

Third-order skin effect
3rd-order TI (BBH model)

10/10
3rd-order skin effectskinbulk
Benalcazar, Bernevig & Hughes,
Science 357, 61 (2017)

Summary

arXiv:2008.07237

・We discover the higher-order non-Hermitian skin effect.
・2nd-order skin effect in 2D: O (L) corner skin modes.
・3rd-order skin effect in 3D: O (L) corner skin modes.
・Unique to non-Hermitian topological systems in higher dimensions.
(a) Hermitian, 1stedge modes(c) non-Hermitian, 1st(b) Hermitian, 2ndcorner modes(d) non-Hermitian, 2ndhigher-ordernon-Hermitianskin effectedge skin modes

corner skin modes