Interplay of symmetry & topology in noninteracting (and interacting) systems

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Interplay of symmetry & topology in noninteracting

(and interacting) systems

Haruki Watanabe

University of Tokyo

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•

Symmetry protects topology e.g. Quantum Spin Hall

•

Symmetry detects topology

π

1

(S

¹

)

Interplay of

symmetry & topology

+i +i

-i +i

(3)

•

Symmetry protects topology e.g. Quantum Spin Hall

•

Symmetry detects topology

π

1

(S

¹

)

Interplay of

symmetry & topology

+i +i

-i +i

(4)

•

Symmetry protects topology e.g. Quantum Spin Hall

•

Symmetry detects topology

π

1

(S

¹

)

Interplay of

symmetry & topology

+i +i

-i +i

(5)

•

Symmetry protects topology e.g. Quantum Spin Hall

•

Symmetry detects topology

π

1

(S

¹

)

Interplay of

symmetry & topology

+i +i

-i +i

W mod 2 can be seen from the product of two rotation eigenvalues!

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Fu-Kane formula

•

Z

2

index for Quantum Spin Hall insulators (2D, TR) Requires a careful gauge fixing and integration of

Pfaffian in k space

•

With additional inversion symmetry

Fu-Kane formula: ν = Πk=TRIMs ξk = ±1 Easy & helpful for material search!

Combination of inversion eigenvalues indicates Z2 QSH

(0,0) (π,0)

(0,π) (π,π)

−−

++

cf. Professor Tay-Rong Chang’s talk

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Fu-Kane formula

•

Z

2

index for Quantum Spin Hall insulators (2D, TR) Requires a careful gauge fixing and integration of

Pfaffian in k space

•

With additional inversion symmetry

Combination of inversion eigenvalues indicates Z2 QSH

(0,0) (π,0)

(0,π) (π,π)

−−

++

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Fu-Kane formula

•

Z

2

index for Quantum Spin Hall insulators (2D, TR) Requires a careful gauge fixing and integration of

Pfaffian in k space

•

With additional inversion symmetry

Combination of inversion eigenvalues indicates Z2 QSH

Irreducible representations of a more general space group

(0,0) (π,0)

(0,π) (π,π)

−−

++

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Fu-Kane formula

•

Z

2

index for Quantum Spin Hall insulators (2D, TR) Requires a careful gauge fixing and integration of

Pfaffian in k space

•

With additional inversion symmetry

Combination of inversion eigenvalues indicates Z2 QSH

Irreducible representations of a more general space group

more general topology including HOTI

(0,0) (π,0)

(0,π) (π,π)

−−

++

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“topological” insulators

1. Presense of protected gapless edge/surface states 2. Winding number (e.g. Chern number, Z2 QSH index) 3. Obstruction in adiabatically connectinng to trivial states Most general definition / applicable to interacting systems

A

B

C

D

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•

Basics of symmetry indicators

•

What can we “see” from it?

1. Conventional topological insulators (Chern, Z2 TI, etc) 2. Higher-order topological insulators

3. Weyl semimetals 4. Fragile topology

5. Topological superconductors

Plan of my talk

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Plan of my talk

•

Basics of symmetry indicators

•

What can we “see” from it?

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Characterizing band

structure by irreducible representations

Po-Vishwanath-Watanabe, Nat. Commun. (2017)

Related works:

Bradlyn-…-Bernevig (2017) Shiozaki-Sato-Gomi (2018) Song-…-Fang (2018)

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Representations in band structures

Hemstreet & Fong (1974)

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Representations in band structures

Hemstreet & Fong (1974)

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Step-by-step process

Given symmetry setting (e.g., space group G, TR, spin-orbit)

1. List up all different types of high-sym k (points, lines, planes) 2. For each k, find the little group Gk = { g in G | gk = k + G }

3. Find irreps ukα (α = 1, 2, …) of Gk

4. Count the number of times ukα appears in band structure {nkα}

※ Note compatibility relations among {nkα}

5. Form a vector b = (nk11, nk12, … nk21, nk22, …)

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Step-by-step process

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Step-by-step process

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Step-by-step process

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Step-by-step process

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Step-by-step process

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Example: 2D lattice with inversion symmetry

k

- k Γ = (0,0) X = (π,0)

Y = (0,π) M = (π,π)

−

+

+ +

Γ = (0,0)

X = (π,0) Y = (0,π) M = (π,π)

(Γ,X,Y,M) : (−,+,+,+) Inversion I²= +1

→ eigenvalues +1 or -1

b = (0,1,1,0,1,0,1,0)

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Band structure space {BS}

• Consider a vector b = {nkα} = (nk11, nk12, … nk21, nk22, …) satisfying all compatibility relations at high-sym momenta

• Form a set b’s (band structure space) :

{BS} = { b = {nkα} | satisfying compatibility relations } ⊂ Z^dBS

lattice of {BS} ⊂ Z

^dBS

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Trivial subset of {BS}

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Atomic Insulators

TB model but no hopping (trivial flat bands)

Product state in real space (trivial) Wannier orbitals

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Example: 2D lattice with inversion symmetry

unit cell

We have to specify the position x and the orbital type 1. Choose x in unit cell. e.g. x =

2. Find little group (site-symmetry) Gx. Gx = {e, I} at x =

3. Choose an orbit (an irrep of Gx). (I = +1) (I = −1)

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(+,+,+,+) (+,−,+,−) (+,+,−,−) (+,−,−,+)

(−,−,−,−) (−,+,−,+) (−,−,+,+) (−,+,+,−)

Irrep contents of AI

Γ = (0,0) X = (π,0)

Y = (0,π) M = (π,π)

(Γ,X,Y,M) :

Momentum space

Real space

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k = (0, π) I = +1

k = (0, 0) I = +1

k = (π, π) I = +1

k = (π, 0) I = +1

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k = (0, π) I = +1

k = (0, 0) I = +1

k = (π, π) I = −1

k = (π, 0) I = −1

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Atomic insulator space {AI}

• Consider a vector a = {nkα} = (nk11, nk12, … nk21, nk22, …)

corresponding to atomic insulators. They automatically satisfy all compatibility relations.

• Form the set a’s (atomic insulator space) :

{AI} = { a = {nkα} | corresoinding to AI} ⊂ Z^dAI

lattice of {AI}

(+,+,+,+) (+,−,+,−)

a1 = (1,0,1,0,1,0,1,0) a2 = (1,0,0,1,0,1,1,0)

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Diagnosing the

topology

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(+,+,+,+) (+,−,+,−) (+,+,−,−) (+,−,−,+)

(−,−,−,−) (−,+,−,+) (−,−,+,+) (−,+,+,−)

(Γ,X,Y,M) :

Atomic Insulators Band Structures

Γ = (0,0) X = (π,0)

Y = (0,π) M = (π,π)

{BS}: set of b’s

{AI}: set of a’s

(Γ,X,Y,M) : b1 = (+,−,−,+)

b3 = (+,+,+,−)

b2 = (++,+−,+−,++)

b4 = (++,++,++,−−)

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Compare {BS} and {AI}

•

{BS} \ {AI}: subtraction of two sets

poor mathematical structure. like vector-bundle classification.

•

{BS} / {AI}: quotient of Abelian groups {BS} < {AI}

symmetry indicators: stable topology like K-theory.

need to allow “negative integers” in {BS}, {AI}

lattice of {BS} lattice of {AI}

Po-Vishwanath-Watanabe Nat. Commun. (2017)

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Compare {BS} and {AI}

•

{BS} \ {AI}: subtraction of two sets

•

{BS} / {AI}: quotient of Abelian groups {BS} < {AI}

lattice of {BS} lattice of {AI}

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Compare {BS} and {AI}

•

{BS} \ {AI}: subtraction of two sets

•

{BS} / {AI}: quotient of Abelian groups {BS} < {AI}

lattice of {BS} lattice of {AI}

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Compare {BS} and {AI}

•

{BS} \ {AI}: subtraction of two sets

•

{BS} / {AI}: quotient of Abelian groups {BS} < {AI}

lattice of {BS}

X = Z

2

× Z

2

lattice of {AI}

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Plan of my talk

•

Basics of symmetry indicators

•

What can we “see” from it?

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•

Basics of symmetry indicators

•

What can we “see” from it?

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X = Z

2

× Z

2

× Z

2

× Z

4

Symmetry indicator for

inversion &TRS with SOC in 3D

Chen Fang & Liang Fu, arXiv:1709.01929

Khalaf-Po-Vsiwanath-Watanabe, PRX (2018)

Sum of inversion parities

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X = Z

2

× Z

2

× Z

2

× Z

4

weak TI

Symmetry indicator for

inversion &TRS with SOC in 3D

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X = Z

2

× Z

2

× Z

2

× Z

4

weak TI strong TI + α

Symmetry indicator for

inversion &TRS with SOC in 3D

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Symmetry indicator for

rotation symmetric systems in 2D

•

n-fold rotation → Chern number C mod n

•

Extention to interacting systems using twisted boundary condition

Matsugatani-Ishiguro-Shiozaki-Watanabe PRL (2018) Fang-Gilbert-Bernevig PRB (2012)

Γ = (0,0) X = (π,0) Y = (0,π) M = (π,π)

−

+

+ (−1)^C = product of rotation eigenvalues

In our language, X = Z

n

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Symmetry indicator for

rotation symmetric systems in 2D

•

n-fold rotation → Chern number C mod n

•

Extention to interacting systems using twisted boundary condition

Matsugatani-Ishiguro-Shiozaki-Watanabe PRL (2018) Fang-Gilbert-Bernevig PRB (2012)

Γ = (0,0) X = (π,0) Y = (0,π) M = (π,π)

−

+

+ (−1)^C = product of rotation eigenvalues

In our language, X = Z

n

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Prof. Tay-Rong Chang’s talk

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•

Basics of symmetry indicators

•

What can we “see” from it?

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X = Z

2

× Z

2

× Z

2

× Z

4

Weyl SM

A. Turner, …, A. Vishwanath (2010)

{BS}: “band structure” can be semimetal (band touching at generic points in BZ)

Symmetry indicator for TR breaking inversion symmetric system in 3D

Basics of symmetry indicators

•

What can we “see” from it?

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Honeycomb lattice with SOC

Wannier orbital Exponential decay of Wannier

M K

fragile topo.

trivial

Po-Watanabe-Vishwanath PRL (2018)

Stability against interaction:

Else-Po-Watanabe arXiv:1809.02128

A

B

C

D

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Honeycomb lattice with SOC

M K

fragile topo.

trivial

A

B

C

D

a

triangular

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Honeycomb lattice with SOC

M K

fragile topo.

trivial

A

B

C

D

a

honeycomb

a

triangular

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Honeycomb lattice with SOC

M K

fragile topo.

trivial

A

B

C

D

a

honeycomb

a

triangular

a

honeycomb

− a

honeycomb

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•

Basics of symmetry indicators

•

What can we “see” from it?

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Weak-coupling assumption

We can extract indicators for SCs

from the band structure in the normal phase!

Ono-Yanase-Watanabe, arXiv:1811.08712

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p+ip SC with nodes

(SC version of Weyl semimetal)

Ono-Yanase-Watanabe, arXiv:1811.08712

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Summary

PRX (2018)

Nat. Commun. (2017) Sci. Adv. (2018)

PRL (2018), arXiv:1809.02128

PRL (2018)

PRB (2018), arXiv:1811.08712

•

Extract band topology by comparing {BS} and {AI}

•

Applications include

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A useful fact

{BS} = {b = {nkα} | satisfying compatibility rels.} ⊂ Z^dBS {AI} = {a = {nkα} | corresoinding to AI} ⊂ Z^dAI

dBS = dAI

(+,+,+,−) = 1/2 [ (+,+,+,+) + (+,+,−,−) + (+,−,+,−) − (+,−,−,+) ]

b a1 a2 a3 a4

lattice of {BS} lattice of {AI}

We do not have to solve compatibility relations to find out {BS}!