Time-Reversal Symmetric Bloch bundles: topology & differential geometry.

Time-Reversal Symmetric Bloch bundles:

topology & differential geometry

Giuseppe De Nittis

(FAU, Universität Erlangen-Nürnberg)

——————————————————————————–

Topological Phases of Quantum Matter

ESI, Wien, Austria • Aug 04 - Sep 12, 2014

——————————————————————————–

Collaborator: K. Gomi

Reference:

Motivations

K-theory

[

P

]

∈

K

(

B

)

k

Chern pairing [topology]

⇓

cohomology

ch

(

P

)

∈

H

(

B

,

Z

)

k

Chern-Weil theory [differential geometry]

⇓

differential cohomology

ω(

P

)

∈

H

(

B

)

k

homology-cohomology (Kronecker) pairing [integration]

⇓

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

Let

_B

a topological space, (“

Brillouin zone

”). Assume that:

- Bis compact, Hausdorff and path-connected;

- Badmits a CW-complex structure.

Definition (Topological Quantum System (

TQS

))

Let

H

be a separable Hilbert space and

K

(

H

)

the algebra of

compact operators. A

TQS

(of class

A) is a self-adjoint map

B

3

k

7−→

H

(

k

) =

H

(

k

)

∈

K

(

H

)

continuous with respect to the norm-topology.

Thespectrumσ(H(k)) ={Ej(k)|j∈I⊆Z} ⊂R, is a sequence of

eigenvalues ordered according to

. . .E−2(k)6E−1(k)<06E1(k)6E2(k)6. . . .

H

(

k

)

ψ

(

k

) =

E

(

k

)

ψ

(

k

)

,

k

∈

B

Gap condition and Fermi projection

-

An

isolated family

of energy bands is any (finite) collection

{

E

(·)

, . . . ,

E

(·)}

of energy bands such that

min

k∈B dist

 

m

[

s=1

{E_js(k)}, [ j∈I\{j1,...,jm}

{E_j(k)}



 = Cg >0.

This is usually called “

gap condition

”.

-

An isolated family is described by the “

Fermi projection

”

P

(

k

) :=

∑

s=1

|ψ

(

k

)ihψ

(

k

)|

.

This is a

continuous

projection-valued map

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

The Serre-Swan construction

For eachk∈B

Hk := RanP_F(k) ⊂ H is a subspace ofH offixeddimensionm. The collection

EF :=

G

k∈B

Hk

is a topological space (saidtotal space) and themap

π: EF −→B

defined byπ(k,v) =kis continuous (and open).

- Amorphismof rankmcomplex vector bundles overBis a mapf

E1

f

- E₂

B π2

π1

-such thatπ2◦f=π1.

- E₁'E2if there is anisomorphismf:E1→E2(equivalence relation).

Vecm_C(B) := set of equivalence classes of rankmvector bundles.

Definition (Topological phases)

Let

_B

3

k

7−→

H

(

k

)

be a TQS with an isolated family of

m

energy bands and associated

Bloch bundle

E

−→

B

. The

topological phase

of the system is specified by

“Ordinary” quantum system:

TQS in a

trivial

phase

m

Trivial

vector bundle,

0

≡

[

B

×

_C

_]

m

Exists a

global

frame of

continuous

Bloch functions

Allowed (adiabatic) deformations:

Transformations which doesn’t

alter the nature

of the system

m

Vector bundle

isomorphism

m

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

For a TQS in class

A

the classification of topological phases

agrees with the determination of

Vec

(

B

)

.

Homotopyclassification theorem:

Vecm_C(B) ' [B,Grm(CN)] (N1)

where

Grm(CN) := U(N)/ U(m)×U(N−m)

is theGrasmannianofm-planes inCN.

The computation of[B,Grm(CN)]is a difficult task (non algorithmic!!).

One introduces (Chern)characteristic classeswhich are thepullbackof the cohomology ring of theclassifying spaceGrm(CN)under the

classifying homotopy (algorithmic!!).

One uses theChern-Weil theoryto represent characteristic classes as

Classification: topology

Theorem (

Peterson, 1959

)

If

dim(

B

)

6

4

then

Vec

(

B

)

'

H

(

B

,

Z

)

Vec

(

B

)

'

H

(

B

,

Z

)

⊕

H

(

B

,

Z

)

(

m

>

2

)

and the

isomorphism

Vec

(

B

)

3

[

E

]

7−→

(

c

,

c

)

∈

H

(

B

,

Z

)

⊕

H

(

B

,

Z

)

is given by the first two Chern classes (notice

c

=

0

if m

=

1

).

The computation of the groupsHk_{(B,Z)isalgorithmic.}

In many cases the conditiondim(B)64 covers all the physical interesting situations.

Classification: differential geometry

Let assume now that

B

has a manifold structure

.

As a consequence of thede Rham theorem

Hk(B,Z) ,→ Hk(B,R) ' H_dRk (B)

one can interpret Chern classes as (classes of)differential forms. TheChern-Weil theoryallows to construct thedifferential expressionof

Chern classes starting from thecurvature2-formΩby means of the invariant polynomial

det

is

2πΩ+1m

=

_∑

sjc_j.

The curvature 2-formΩcan be computed from aconnection1-form ω:={Aα}by means of the structure equation

Ω = dω + 1 2[ω;ω].

The natural candidate is theBerry-(Bott-Chern) connectiongiven by

Ai,j

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

Definition (Involution on the Brillouin zone)

A homeomorphism

τ

:

B

→

B

is called

involution

if

τ

=

Id

.

The pair

(

B

,

τ

)

is called an

involutive space

.

Definition (TQS with time-reversal symmetry)

Let

(

B

,

τ

)

be an involutive space,

H

a separable Hilbert space

endowed with a

complex conjugation

C

. A TQS

B

3

k

7→

H

(

k

)

has a

time-reversal symmetry

(TRS) of parity

ε

∈ {±}

if there is

a continuous unitary-valued map

k

7→

U

(

k

)

such that

U

(

k

)

H

(

k

)

U

(

k

)

=

C

H

(τ(

k

))

C

,

C U

(τ

(

k

))

C

=

ε

U

(

k

)

.

A TQS of class

AI

(resp.

AII) is endowed with an

even

(resp.

“Real” and “Quaternionic” structures

The equivariance relation for the Fermi projection

U

(

k

)

P

(

k

)

U

(

k

)

=

C

P

(τ(

k

))

C

∀

k

∈

B

induces an additional structure on the Bloch-bundle

E

→

B

.

Definition (

Atiyah, 1966 - Dupont, 1969

)

Let

(

B

,

τ

)

be an involutive space and

E

→

B

a

complex

vector

bundle. Let

Θ

:

E

→

E

an

homeomorphism

such that

Θ

:

E

|

−→

E

|

is

anti

-linear

.

The pair

(

E

,

Θ)

is a

“Real”

-bundle over

(

B

,

τ)

if

Θ

:

E

|

−→

E

|

∀

k

∈

_B

;

The pair

(

E

,

Θ)

is a

“Quaternionic”

-bundle over

(

B

,

τ

)

if

- Amorphismof rankm“Real” or “Quaternionic” bundles over(B,τ)is a mapf

(E₁,Θ₁) f - (E₂,Θ₂)

(B,τ)

π2

π1

-such that π2◦f=π1, Θ2◦f=f◦Θ1.

- (E₁,Θ₁)'(E₂,Θ₂)if there is anisomorphismf(equivalence relation)

Vecm_R(B,τ) := set of equivalence classes of rankm“Real” bundles

Vecm_Q(B,τ) := set of equivalence classes of rankm“Quaternionic” bundles.

These names are justified by the following isomorphisms:

Vecm_R(B,Id_B) ' Vecm_R(B)

Definition (Topological phases in presence of TRS)

Let

(

B

,

τ

)

be an involutive space and

B

3

k

7−→

H

(

k

)

a TQS

with a TRS of parity

ε

and an isolated family of

m

energy bands.

Let

E

−→

B

be the associated

Bloch bundle

with equivariant

structure

Θ

. The

topological phase

of the system is specified by

[(

E

,

Θ)]

∈

Vec

(

B

)

,

T

≡

R

,

T

≡

Q

.

CAZ TRS Category VB

A 0 complex Vecm_C(B)

AI + “Real” Vecm_R(B,τ)

AII − “Quaternionic” Vecm_Q(B,τ)

Problem :

1)Classification viatopology;

Relevant physical situations

Periodic

quantum systems (

e.g.

absence of

disorder

):

- Rd-translations ⇒ free(Dirac) fermions; - Zd-translations ⇒ crystal(Bloch) fermions.

The Bloch-Floquet (or Fourier) theory exploits the

invariance under translations

of a periodic structure to

describe the state of the system in terms of the

quasi-momentum

k

on the

Brillouin zone

B

.

Complex conjugation (TRS) endows

_B

with an involution

τ

.

Examples are:

- Gapped electronic systems, - BdG superconductors,

Continuous case

B

=

S

S

S

(+

k

,

+

k

, . . . ,

+

k

)

τd

-

(+

k

,−

k

, . . . ,−

k

)

˜

Periodic case

_B

=

T

T

=

S

×

. . .

×

_S

-

T

=

S

×

. . .

×

S

˜

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

For a TQS in class

AI

the classification of topological phases

agrees with the determination of

Vec

(

B

,

τ

)

.

Homotopyclassification theorem[Edelson, 1971]:

Vecm_R(B,τ) ' [(B,τ),(Grm(CN),ϑR)]Z2 (N1)

Z2-homotopy classes ofequivariantmapsf:B→Grm(CN)such that

f◦τ=ϑR◦f. The involutionϑRacts by the complex conjugation,i.e.

ϑ_R : span(v₁, . . . ,vm) 7−→span(v1, . . . ,vm).

The computation of[(B,τ),(Grm(CN),ϑR)]Z2is a difficult task (non

algorithmic!!).

One introduces“Real” Chern classes[Kahn, 1987]which are the

pullbackof theequivariant Borel cohomology ringof theinvolutive

classifying space(Grm(CN),ϑ_R)under theequivariantclassifying homotopy (algorithmic!!).

The Borel’s construction

(

B

,

τ

)

any involutive space and

(

S

,

θ

)

the

infinite

sphere

(contractible space) with the

antipodal

(free) involution:

B

:=

S

_×

_B

θ

×

τ

(

homotopy quotient

)

.

Z

any

abelian

ring (module, system of coefficients, ...)

H

2

(

B

,

Z

) :=

H

₍

B

,

Z

)

(

eq. cohomology groups

)

.

Z

(

m

)

is the

Z

-local system

on

B

based on the module

Z

Z

(

m

)

'

_B

×

_Z

endowed with

(

k

, `

)

7→

(τ

(

k

)

,

(−

1

)

`

)

.

More precisely:Z(m)is the module withunderlyinggroup (isomorphic to)Z

and scalars in thegroup ringZ[π1(B∼τ)]induced by the (two) possible

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

Theorem (

D. & Gomi, 2014

)

Let

(

B

,

τ

)

such that

d

:=

dim(

B

)

6

3

and the fixed point set

B

has only

isolated points

. There is an

isomorphism

Vec

(

B

,

τ

)

'

H

(

B

,

Z

(1))

if

dim(

B

)

6

3

given by the

“Real” (first) Chern class

[

E

]

7→

c

˜

(

E

)

.

Proof.

- “Real”stable-range condition

Vec_Rm(B,τ) ' Vec_R[d/2](B,τ), [·] =integer part.

- “Real”line bundles description[Kahn, 1987]

Vec1_R(B,τ) =:Pic(B,τ) ' H_Z2

2(B,Z(1)).

Gapped fermionic systems

Theorem (

D. & Gomi, 2014

)

H

2

(

˜

S

,

Z

(

1

)) =

H

2

(

˜

T

,

Z

(

1

)) =

0

∀

d

∈

_N

.

The proof requires anequivariantgeneralization of theGysin sequenceand thesuspension periodicity.

VB CAZ d=1 d=2 d=3 d=4

Vecm_C(Sd) A 0 Z 0 0 (m=1)

Z (m>2) Free

Vecm_R(˜Sd) AI 0 0 0 0 (m=1)

2_Z (m_>2) systems

Vecm_C(Td) A 0 Z Z3 Z

6 _(m=1)

Z7 (m_>2) Periodic

Vecm_R(T˜d) AI 0 0 0 0 (m=1)

The case

d

=

4 is interesting for the

magneto-electric response

(space-time variables)

Theorem (

D. & Gomi, 2014

)

Class

AI

topological insulators in d

=

4

are completely

classified by the

2-nd “Real” Chern class

. These classes are

representable as

even

integers and the isomorphisms

Vec

( ˜

T

)

'

2

Z

,

Vec

(˜

S

)

'

2

Z

are given by the (usual)

2-nd Chern number

.

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

Just a summary of the main results

On can introduce consistently the notions of“Real” principal bundleand

“Real” connection(i.e.τ∗ω=ω).

Let(B,τ)be asmooth manifoldwith asmooth involution. The de Rham cohomology groupsH_dRk (B)split as

H_dRk (B) = H_dRk _,+(B,τ)⊕H_dRk _,−(B,τ)

whereφ∈H_dRk _,±(B,τ)if and only ifτ∗φ=±φ.

One has the following isomorphisms

H_Zk₂(B,R(0)) ' H_dRk _,+(B,τ), H_Zk₂(B,R(1)) ' H_dRk _,−(B,τ).

There is a map

H_Zk

2(B,Z(j)) −→H

k

Z2(B,R(j)) ' H

k

Z2(B,Z(j)) ⊗Z R

Letωa “Real” connection andΩthe related curvature. The differential formsc_j obtained from the invariant polynomial

det

is

2πΩ+1m

=

_∑

sjc_j.

verifyτ∗cj= (−1)jcj, hence

c_j ∈ H_dR2j_,(−)j(B,τ).

Theorem (

D. & Gomi, 2014

)

The de Rham cohomology class

c

agrees with the image of the

“Real” Chern class

˜

c

∈

H

(

B

,

Z

(

j

))

unde the homomorphism

H

2

(

B

,

Z

(

j

))

−→

H

k

Z2

(

B

,

R

(

j

))

'

H

2j

dR,(−)j

(

B

,

τ

)

.

The

torsion

part can be reconstructed by the

“Real” Holonomy

.

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

For a TQS in class

AII

the classification of topological phases

agrees with the determination of

Vec

(

B

,

τ

)

.

If thefixed point setBτ_{6=/0the rank of a}Q_{-bundle has to beeven}

(Kramers degeneracy).

Homotopyclassification theorem (adaption of[Edelson, 1971]): Vec2_Qm(B,τ) ' [(B,τ),(Gr_2m(C2N),ϑ_Q)]_Z2 (N1)

Z2-homotopy classes ofequivariantmapsf:B→Grm(C2N)such that

f◦τ=ϑ_Q◦f. The involutionϑ_Qacts as

ϑQ : span(v1, . . . ,v2m) 7−→ span(Q·v1, . . . ,Q·v2m)

whereQ:=1N⊗iσ2verifiesQ2:=−12N.

The computation of[(B,τ),(Gr_2m(C2N),ϑ_Q)]_Z2is a difficult task (non algorithmic!!).

One introduces theFKMM-invariant[Furuta, Kametani, Matsue,

Minami, 2000]which can be interpreted as the pullback of auniversal

classfor(Gr2m(C2N),ϑ_Q)under theequivariantclassifying homotopy (algorithmic!!).

Construction and interpretation of the FKMM-invariant

Let(B,τ)be an involutive space andY⊂Bτ-invariant. Therelative

equivariant cohomology groups

H_Zk

2(B|Y,Z(m))

can be consistently defined.

One has the geometric interpretations

H_Z1

2(B,Z(1)) ' [(B,τ),(U(1),c)]Z2 [Gomi, 2013]

H_Z2₂(B,Z(1)) ' Vec1_R(B,τ) [Kahn, 1987]

wherecis the complex conjugation.

We have also the geometric interpretation[D. & Gomi, 2013]

H_Z2

2(B|Y,Z(1))' Vec

1

R(B|Y,τ) =:Pic(B|Y,τ)

whereVec1_R(B|Y,τ)is the (Picard) group of isomorphism classes of pairs(L,s)whereL→Bis a “Real” line bundle ands:Y→L|_Yis a

Theorem (

D. & Gomi, 2014

)

Each

[(

E

,

Θ)]

∈

Vec

(

B

,

τ

)

defines a unique element in

Vec

(

B

|

_B

_,

τ

)

and the map

Vec

(

B

,

τ

)

3

[(

E

,

Θ)]

7−→

κ[

E

]

∈

H

(

B

|

B

,

Z

(

1

))

(defined by the above isomorphism) is the

FKMM-invariant

.

Proof.

- Determinant construction:L:=det(E)is a line bundle overBandΘ induces a“Real”-structuredetΘ:L→L.

- Because_Bτ_{is made by fixed pointsL|}

Bτ has a uniquecanonical

trivializationh:L|_Bτ→Bτ×Cwhich define acanonical(nowhere

vanishing, equivariant)sections(k) :=h−1_(k,1).

- [(E,Θ)]defines[(L,s)]∈Vec1_R(B|Bτ_,τ).

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

B

is a

FKMM-involutive-space

if:

-

(

B

,

τ

)

Hausdorff, path-connected,

. . .

;

-

B

₌

_{

_k

1

, . . . ,

k

}

;

-

H

2

(

B

,

Z

(

1

)) =

0.

Theorem (

D. & Gomi, 2014

)

Let

(

B

,

τ

)

be a FKMM-involutive-space:

κ

:

Vec

(

B

,

τ

)

→

H

(

B

|

B

,

Z

(

1

))

if

dim(

B

)

6

3

κ

, called

FKMM-invariant

, is an

injective

map .

Gapped fermionic systems

Theorem (

D. & Gomi, 2014

)

For

B

= ˜

S

or

B

= ˜

T

with

d

6

3

the FKMM-invariant

κ

is an

isomorphism

.

VB CAZ d=1 d=2 d=3 d=4

Vecm_C(Sd) A 0 Z 0 0 (m=1)

Z (m>2) Free

Vec2_Qm(˜Sd) AII 0 _{Z2 Z2 Z} systems

Vecm_C(Td) A 0 _{Z Z}3 Z6 (m=1)

Z7 (m_>2) Periodic

Vec2_Qm(T˜d) AII 0 Z2 Z42 Z102 ⊕Z systems

Remark:

In the case

dim(

B

) =

4 we consider the pair of maps

(κ

,

c

) :

Vec

(

B

,

τ

)

−→

H

(

B

|

B

,

Z

(

1

))

⊕

H

(

B

,

Z

)

where

c

is the

second

Chern class.

Theorem (

D. & Gomi, 2014

)

Class

AII

topological insulators in d

=

4

are classified by the

2-nd Chern class

c

and the

FKMM-invariant

κ

. The maps

(κ

,

c

) :

Vec

(˜

S

)

−→

Z

⊕

Z

(κ

,

c

) :

Vec

( ˜

T

)

−→

Z

⊕

Z

are injective.

Ran(κ

,

c

)

is fixed by the constraint

(−

1

)

=

∏

j=1

κ

(

E

)

,

E

is a

Q

−

vector bundle

Outline

1

Topological Quantum Systems & Time-reversal Symmetry

What is a Topological Quantum System?

Bloch bundle and topological phases

The problem of the classification of topological phases

Time-reversal topological quantum systems

2

Classification of “Real” Bloch-bundles

“Real” Chern classes and equivariant cohomology

Topological classification

“Real” Chern-Weil theory

3

Classification of “Quaternionic” Bloch-bundles

The FKMM-invariant

Topological classification

Just a summary of the main results

On can introduce consistently the notions of“Quaternionic” principal bundleand“Quaternionic” connection, (i.e.τ∗ω=−Q·ω·Q).

Let(B,τ)be asmooth manifoldwith asmooth involutionand a finite number of fixed points. Letωa “Quaternionic” connection associated with the “Quaternionic” vector bundle(E,Θ).

Letdim(B) =3and consider theChern-Simons form

CS(E) := 1 8π2

Z

B Tr

ω∧dω+2

3ω∧ω∧ω

then

κstrong(E) = ei2πCS(E)

is a topological invariant of(E,Θ)which provides thestrongpart of the FKMM-invariant (already known forB=T3!!).

Ifdim(B) =2then

κstrong(E) = ei2πWS(E)