### Interplay of symmetry & topology in noninteracting

### (and interacting) systems

### Haruki Watanabe

University of Tokyo

•

### Symmetry protects topology e.g. Quantum Spin Hall

•

### Symmetry detects topology

### π

1### (S

^{1}

### )

### Interplay of

### symmetry & topology

+i +i

-i +i

•

### Symmetry protects topology e.g. Quantum Spin Hall

•

### Symmetry detects topology

### π

1### (S

^{1}

### )

### Interplay of

### symmetry & topology

+i +i

-i +i

•

### Symmetry protects topology e.g. Quantum Spin Hall

•

### Symmetry detects topology

### π

1### (S

^{1}

### )

### Interplay of

### symmetry & topology

+i +i

-i +i

•

### 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!*

## 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

## 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

## 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

Irreducible representations of a more general space group

(0,0) (π,0)

(0,π) (π,π)

−−

++

++

++

cf. Professor Tay-Rong Chang’s talk

## 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

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

## “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

•

### 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

## Plan of my talk

•

### 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

### Characterizing band

### structure by irreducible representations

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

Related works:

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

### Representations in band structures

Hemstreet & Fong (1974)

### Representations in band structures

Hemstreet & Fong (1974)

## 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 G****k**** = { g in G | gk = k + G }**

*3. Find irreps u***kα*** (α = 1, 2, …) of G***k**

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

*※ Note compatibility relations among {n** kα*}

**5. Form a vector b = (n****k11***, n***k12***, … n***k21***, n** k22*, …)

## 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 G****k**** = { g in G | gk = k + G }**

*3. Find irreps u***kα*** (α = 1, 2, …) of G***k**

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

*※ Note compatibility relations among {n** kα*}

**5. Form a vector b = (n****k11***, n***k12***, … n***k21***, n** k22*, …)

## 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 G****k**** = { g in G | gk = k + G }**

*3. Find irreps u***kα*** (α = 1, 2, …) of G***k**

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

*※ Note compatibility relations among {n** kα*}

**5. Form a vector b = (n****k11***, n***k12***, … n***k21***, n** k22*, …)

## 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 G****k**** = { g in G | gk = k + G }**

*3. Find irreps u***kα*** (α = 1, 2, …) of G***k**

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

*※ Note compatibility relations among {n** kα*}

**5. Form a vector b = (n****k11***, n***k12***, … n***k21***, n** k22*, …)

## 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 G****k**** = { g in G | gk = k + G }**

*3. Find irreps u***kα*** (α = 1, 2, …) of G***k**

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

*※ Note compatibility relations among {n** kα*}

**5. Form a vector b = (n****k11***, n***k12***, … n***k21***, n** k22*, …)

## 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 G****k**** = { g in G | gk = k + G }**

*3. Find irreps u***kα*** (α = 1, 2, …) of G***k**

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

*※ Note compatibility relations among {n** kα*}

**5. Form a vector b = (n****k11***, n***k12***, … n***k21***, n** k22*, …)

### 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 I*^{2 }= +1

→ eigenvalues +1 or -1

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

### Band structure space {BS}

**• Consider a vector b = {n****kα***} = (n***k11***, n***k12***, … n***k21***, n** k22*, …)
satisfying all compatibility relations at high-sym momenta

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

{BS} = { b = {n* kα*} | satisfying compatibility relations } ⊂ Z

^{dBS}

### lattice of {BS} ⊂ Z

^{dBS}

## Trivial subset of {BS}

## Atomic Insulators

### TB model but no hopping (trivial flat bands)

### Product state in real space (trivial) Wannier orbitals

### 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) G** x*.

*G*

**x**

**= {e, I} at x =***3. Choose an orbit (an irrep of G***x***). (I = +1) (I = −1) *

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

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

## 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**

k = (0, π)
*I = +1*

k = (0, 0)
*I = +1*

k = (π, π)
*I = +1*

k = (π, 0)
*I = +1*

k = (0, π)
*I = +1*

k = (0, 0)
*I = +1*

k = (π, π)
*I = −1*

k = (π, 0)
*I = −1*

### Atomic insulator space {AI}

**• Consider a vector a = {n****kα***} = (n***k11***, n***k12***, … n***k21***, n** k22*, …)

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

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

{AI} = { a = {n* kα*} | corresoinding to AI} ⊂ Z

^{dAI}

### lattice of {AI}

(+,+,+,+) (+,−,+,−)

* a*1 = (1,0,1,0,1,0,1,0)

*2 = (1,0,0,1,0,1,1,0)*

**a**## Diagnosing the

## topology

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

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

(Γ,X,Y,M) :

(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) : b**1**** = (+,−,−,+)**

**b****3**** = (+,+,+,−)**

**b****2**** = (++,+−,+−,++)**

**b****4**** = (++,++,++,−−)**

## 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)

## 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)

## 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)

## 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}

### X = Z

2### × Z

2### lattice of {AI}

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

## Plan of my talk

•

### 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

•

### 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

### X = Z

2### × Z

2### × Z

2### × Z

4### Symmetry indicator for

### 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

### X = Z

2### × Z

2### × Z

2### × Z

4### weak TI

### Symmetry indicator for

### 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

### X = Z

2### × Z

2### × Z

2### × Z

4### weak TI strong TI + α

### Symmetry indicator for

### 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

### 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### 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

nProf. Tay-Rong Chang’s talk

•

### 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

### 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

See also

Song-Zhang-Fang PRX (2018)

for nodal semimetals in the ansence of SOC

•

### 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

### 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 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

**a **

triangular
**a**

### 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

**a **

honeycomb
**a**

**a **

triangular
**a**

### 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

**a **

honeycomb
**a**

**a **

triangular
**a**

**a **

honeycomb **a**

**− a **

honeycomb
**− a**

•

### 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

### Weak-coupling assumption

We can extract indicators for SCs

from the band structure in the normal phase!

Ono-Yanase-Watanabe, arXiv:1811.08712

## p+ip SC with nodes

### (SC version of Weyl semimetal)

Ono-Yanase-Watanabe, arXiv:1811.08712

## 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

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

3. Weyl semimetals 4. Fragile topology

5. Topological superconductors

## A useful fact

{BS} = {b = {n* kα*} | satisfying compatibility rels.} ⊂ Z

^{dBS}{AI} = {a = {n

*} | corresoinding to AI} ⊂ Z*

**kα**^{dAI}

### dBS = dAI

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

**b*** a*1

*2*

**a***3*

**a***4*

**a**lattice of {BS} lattice of {AI}

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

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