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

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### Haruki Watanabe

University of Tokyo

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### symmetry & topology

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

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## Fu-Kane formula

2

### Pfaffian in k space

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

2

### Pfaffian in k space

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

2

### Pfaffian in k space

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

### Combination of inversion eigenvalues indicates Z2 QSH

Irreducible representations of a more general space group

(0,0) (π,0)

(0,π) (π,π)

−−

++

++

++

cf. Professor Tay-Rong Chang’s talk

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## Fu-Kane formula

2

### Pfaffian in k space

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

### 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)

−−

++

++

++

cf. Professor Tay-Rong Chang’s talk

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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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### 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

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## Plan of my talk

### 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

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### 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 u (α = 1, 2, …) of Gk

4. Count the number of times u appears in band structure {n}

※ Note compatibility relations among {n}

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

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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 u (α = 1, 2, …) of Gk

4. Count the number of times u appears in band structure {n}

※ Note compatibility relations among {n}

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

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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 u (α = 1, 2, …) of Gk

4. Count the number of times u appears in band structure {n}

※ Note compatibility relations among {n}

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

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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 u (α = 1, 2, …) of Gk

4. Count the number of times u appears in band structure {n}

※ Note compatibility relations among {n}

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

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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 u (α = 1, 2, …) of Gk

4. Count the number of times u appears in band structure {n}

※ Note compatibility relations among {n}

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

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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 u (α = 1, 2, …) of Gk

4. Count the number of times u appears in band structure {n}

※ Note compatibility relations among {n}

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

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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 = {n} = (nk11, nk12, … nk21, nk22, …) satisfying all compatibility relations at high-sym momenta

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

{BS} = { b = {n} | satisfying compatibility relations } ⊂ ZdBS

dBS

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## Atomic Insulators

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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) :

(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 = {n} = (nk11, nk12, … nk21, nk22, …)

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

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

{AI} = { a = {n} | corresoinding to AI} ⊂ ZdAI

### 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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## topology

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

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

(Γ,X,Y,M) :

(Γ,X,Y,M) :

Atomic Insulators Band Structures

Γ = (0,0) X = (π,0)

Y = (0,π) M = (π,π)

### {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

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

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

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}

2

2

### lattice of {AI}

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

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## Plan of my talk

### 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

(38)

### 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

(39)

2

2

2

4

### inversion &TRS with SOC in 3D

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

Chen Fang & Liang Fu, arXiv:1709.01929

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

Sum of inversion parities

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2

2

2

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### inversion &TRS with SOC in 3D

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

Chen Fang & Liang Fu, arXiv:1709.01929

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

Sum of inversion parities

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2

2

2

4

### inversion &TRS with SOC in 3D

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

Chen Fang & Liang Fu, arXiv:1709.01929

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

Sum of inversion parities

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### 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

n

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### 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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### 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

(46)

2

2

2

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

Song-Zhang-Fang PRX (2018)

for nodal semimetals in the ansence of SOC

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### 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

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### Honeycomb lattice with SOC

Wannier orbital Exponential decay of Wannier

M K

fragile topo.

trivial

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

Wannier orbital Exponential decay of Wannier

M K

fragile topo.

trivial

trivial

Po-Watanabe-Vishwanath PRL (2018)

Stability against interaction:

Else-Po-Watanabe arXiv:1809.02128

A

B

C

D

triangular

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### Honeycomb lattice with SOC

Wannier orbital Exponential decay of Wannier

M K

fragile topo.

trivial

trivial

Po-Watanabe-Vishwanath PRL (2018)

Stability against interaction:

Else-Po-Watanabe arXiv:1809.02128

A

B

C

D

honeycomb

triangular

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### Honeycomb lattice with SOC

Wannier orbital Exponential decay of Wannier

M K

fragile topo.

trivial

trivial

Po-Watanabe-Vishwanath PRL (2018)

Stability against interaction:

Else-Po-Watanabe arXiv:1809.02128

A

B

C

D

honeycomb

triangular

honeycomb

honeycomb

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### 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

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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

### Applications include

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

3. Weyl semimetals 4. Fragile topology

5. Topological superconductors

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## A useful fact

{BS} = {b = {n} | satisfying compatibility rels.} ⊂ ZdBS {AI} = {a = {n} | corresoinding to AI} ⊂ ZdAI

### 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}!

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

Chang, Sym- metric Mendelsohn triple systems and large sets of disjoint Mendelsohn triple systems, in Combinatorial de- signs and applications (Lecture Notes in Pure and

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