Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Welcome !
International workshop”Variety and universality of Bulk-Edge Correspondence in topological phases:
From solid state physics to transdisciplinary concepts”
BE/BC2022 (Bulk-Edge/Boundary Correspondence)(BE/BC2022)

Yasuhiro Hatsugai
Department of Physics
University of TsukubafiGrant-in-Aid for Scienti c Research (S) Project No.17H06138

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Welcome !
Bulk-edge correspondence :From Quantum Hall effects

Yasuhiro Hatsugai
Department of Physics
University of TsukubafiGrant-in-Aid for Scienti c Research (S) Project No.17H06138

Our project

BEC

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Bulk-Edge CorrespondenceWith edges
Observables
In gap states
Low energy modes

Solid states

As a bulk

bulk-edge
correspondence
TopologicalstatesPhotonics & Newton’s“Math “Cold atoms

Topology is hidden
GappedUnconventional

Our project

BEC

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Bulk-Edge Correspondence

Solid statesbulk-edge
correspondence
TopologicalstatesPhotonics & Newton’sCold atoms

NTTI2019 & BEC2019 Hiroshima, July.14-19 (2019)

Tsukuba, Jan. 6-8 (2018)

BEC2016 Kyoto, Sep. 27-30 (2016)
BEC2015

Tokyo, Sep. 27-29 (2015)

Project 1

BEC2018 Jan

Supports by

2014–2017.3

BEC2018X Tokyo, Dec.9-13 (2018)2017–2022.3

BE/BC2020F Tsukuba (virtualized due to COVID19), Feb.28-29 (2020)

Project 2

History (international workshops)
We are now 7-th !
BE/BC2022 Zoom, Feb.11-13 (2022)

BE/BC2022 Zoom, Feb.11-13 (2022)

We are now 7-th !

BE/BC2020F Tsukuba (virtualized due to COVID19), Feb.28-29 (2020)
NTTI2019 & BEC2019 Hiroshima, July.14-19 (2019)2017–2022.3

Surprises/Big events !

BEC2018 Jan

Tsukuba, Jan. 6-8 (2018)

BEC2016 Kyoto, Sep. 27-30 (2016)
BEC2015

Tokyo, Sep. 27-29 (2015)

2014–2017.3

BEC2018X Tokyo, Dec.9-13 (2018)

Started from here

Quantum Hall effects

Ix =<latexit sha1_base64="VdctMb6oOnEEO3fl6fW3/1LvOP8=">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</latexit>

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

electrons in a magnetic eldxy Vy

i = 1, 2, 3 · · · Integer QHE<latexit sha1_base64="2OR1FF+6O3cyExfh3oPk6V1WH4k=">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</latexit>

fi

fi

Grant-in-Aid for Scienti c Research (S) Project No.17H06138

Quantum Hall effects

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

TKNN

B<latexit sha1_base64="2lc23l7VaKYTF46+oXaeiayehOs=">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</latexit>

Iy =<latexit sha1_base64="737Hvg0sgIBAXTNYWA6lajZ6gWs=">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</latexit>

yx Vx

e2
⇥n
yx =
h
n = 1, 2, 3, · · ·<latexit sha1_base64="JEJhggw8g7qapK/9MLS3pU7CEd8=">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</latexit><latexit sha1_base64="SWD4pSCpE12weXhqp+yRKFmH7BY=">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</latexit>

TKNN integers

fi

Grant-in-Aid for Scienti c Research (S) Project No.17H06138

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Quantum Hall effects

TKNN
Topological numbers
for condensed matter physics
Cool !
(exceptional ?)I was a grad. student in the late 80’.

fi

Grant-in-Aid for Scienti c Research (S) Project No.17H06138

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Quantum Hall effects

TKNN
Topological numbers
for condensed matter physics
Cool !

Universal !

fiGrant-in-Aid for Scienti c Research (S) Project No.17H06138

Quantum Hall effects

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

electrons in a magnetic eld

F = ev ⇥ B<latexit sha1_base64="Y8lmM/3NkcjJA0N51Leyxz2Pzhk=">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</latexit>

Circular (cyclotron) motion
due to Lorentz forcee<latexit sha1_base64="J2AtWaG7Koy//bxldUiKJWQv1gU=">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</latexit>

fi

fi

Grant-in-Aid for Scienti c Research (S) Project No.17H06138

Quantum Hall effects

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

With boundaries

One way going !!
Cannot stop !

“skipping orbit”

fiGrant-in-Aid for Scienti c Research (S) Project No.17H06138

Quantum Hall effects

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

With boundaries

One way going !!
Cannot stop !
No back scattering
Stable for impurities !!

“Topological” stability“skipping orbit”
Edge states are essential ! Halperin ’82
fiGrant-in-Aid for Scienti c Research (S) Project No.17H06138

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Quantum Hall effects

QHE due to edge states

Laughlin’s argument ‘81“cylinder”⇡<latexit sha1_base64="(null)">(null)</latexit><latexit

fi

Grant-in-Aid for Scienti c Research (S) Project No.17H06138

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Quantum Hall effects

QHE due to edge states

Cj = Ij

Ij 1

YH ’93fi

Grant-in-Aid for Scienti c Research (S) Project No.17H06138

BEC

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Bulk-Edge Correspondence

Consider bulk with edges
Topological number :mostly hidden

fiGrant-in-Aid for Scienti c Research (S) Project No.17H06138

Bulk-edge correspondence
Bulk state(scattering state)
Bulk Gap
Non trivial VacuumControlwitheach other

Edge state(Bound state)

Particles in the gap

They can not be independent

QHE

History…

⇡<latexit sha1_base64="(null)">(null)</latexit><latexit

Laughlin ’81 : Gauge invariance on cylinder
Halperin ’82: Role of edge states in QHE
X.G.Wen ’90 : Gauge invariance with edges
YH ’93 : BEC of IQHE (Hofstadter)
Kitaev ’01 Majorana boundary states of superconductors
Ryu-YH ’02 BEC of graphene, d-wave superconductivity
Qi-Wu-Zhang ’06, BEC : general theorem… more & more…

History…
Haldane model
Quantum Spin Hall effects

Zoom & Univ. Tsukuba, Feb.11-13 (2022)

Haldane ’88
Kane-Mele ‘05
Bernevig-Hughes-Zhang ’06

Topological number is hidden !
Observe edge states !
predicted by the non-trivial bulk

Topological phases & Bulk-edge correspondence
Quantum Hall effects
Quantum spin Hall effects
Topological insulators

Before2005 （Millennium）
After
Surprise in this Millennium‘Quantum”
Schrödinger eq.

Do NOT need quantum mechanics
Topological photonics

Haldane-Raghu ’05
Wang-Chong-Joannopoulos-Soljačić ’08

Edge states of photons

Hafezi-Demler-Lukin-Taylor ‘11

Topological mechanics

Prodan-Prodan ’09
Kane-Lubensky ’14
V. Vitteli et al. ‘14

Topological circuit

Albert et al. ‘15

‘Quantum”
Schrödinger eq.

Before2005 （Millennium）
After
Surprise in this Millennium

2017–2022.3

Quantum Hall effects
Quantum spin Hall effects
Topological insulators

Project 2

Topological phases & Bulk-edge correspondence

Do NOT need quantum mechanics

Hafezi-Demler-Lukin-Taylor ‘11

Topological mechanics

Prodan-Prodan ’09
Kane-Lubensky ’14
V. Vitteli et al. ‘14

Topological circuit

Albert et al. ‘15

Try to explore more in our project from 2017…

2014–2017.3

N

!
e
v
a
Edge states of photons
ew w

Project 1

Topological photonics

Haldane-Raghu ’05
Wang-Chong-Joannopoulos-Soljačić ’08

2005 （Millennium）
After
Surprise in this Millennium
Do NOT need quantum mechanics
Topological photonics

N

!
e
v
a
Edge states of photons
ew w

r
e
h
t
r
Fu

Haldane-Raghu ’05
Wang-Chong-Joannopoulos-Soljačić ’08
Hafezi-Demler-Lukin-Taylor ‘11

Topological mechanics

Prodan-Prodan ’09
Kane-Lubensky ’14
V. Vitteli et al. ‘14

Topological circuit

Albert et al. ‘15

s
e
s
i
r
surp

Talks in the workshop…

Quantum to classical”Quantization

Newtonx·p=p·x

Schrödinger

x · p6=p · x<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>

Classical mechanics/EM elds

Quantum mechanics

Newton eq.
Maxwell eq.

Schrödinger eq.

fi”Bulk-edge correspondence”

Massive Dirac fermions (level crossing) is essential

Key ingredients
3
2
1
0
-1
-2
-3

Bulk

Dirac fermions/gap closing
in parameter space

If gapped, they are intrinsically hidden !

Massless Dirac fermions = gap closing
2×2 eigen value problem

R = (x, y, z),H= ✏ , ✏ = ±R, R = |R|

A=

†

r

Maxwell eq.

Berry connection

i R
B =r⇥A =±2 R3<latexit sha1_base64="xJku47fnO1wTEDbl6F0VxWj2GW4=">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</latexit>r · B = ±2⇡i (R)<latexit sha1_base64="6ytSeeRsUDQ1PixwO2ICQe7xLsI=">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</latexit>(magnetic) Dirac mono pole

e R
EEM=
4⇡✏0 R3
e
r·E =
(R)
EM
✏0
point charge

Non trivial Berry connection

Gauge elds (synthetic in general )+=Laughlin argument

Bulk-edge correspondenceslicing
1D

2D is essential
quantization
intrinsically topologicalembeddingfiTopology & BE/BC are everywhere

3DEdge states are everywhere : topological origin
Topological phases
Quantum Hall effects
Newton eq.
Topological insulators
Cold atoms…
Photonic crystals Mech. graphene
Maxwell eq.
Game theory
Topological circuits
Geophysics

Biology

Diffusion phenomena
non hermitian phenomena

Bulk-edge correspondenceTwo years ago

Tokyo Campus, Univ. Tsukuba, Feb.28-29 (2020)(Virtual)

Exchange idea between theorists & experimentalists
among different elds

BE/BC2020F

Enjoy this virtual workshop !
Feb. 28-29 (2020)

fi+ Lots of new results of the participants todayZoom & Univ. Tsukuba, Feb.11-13 (2022)

Exchange idea between theorists & experimentalists
among different elds

BE/BC2022

Enjoy this workshop !
By zoom (new wisdom)“Science of Bulk-edge correspondence”
Beyond topological phasesfiLet’s try to establish