Non-Commutative Geometry of the Complex Classes of ...imar.ro/ConfGN/lectures/EProdan.pdf · Emil Prodan Yeshiva University, New York, USA IMAR, July 2014 In honor of 70th birthday

Non-Commutative Geometry of the Complex Classes ofTopological Insulators

Emil ProdanYeshiva University, New York, USA

IMAR, July 2014

In honor of 70th birthday of Gheorghe Nenciu.

Work in collaboration with:

Jean Bellissard, Georgia Tech, USAHermann Schulz-Baldes, Universitat Erlangen-Nurnberg, Germany

This work was supported by the U.S. NSF grants DMS-1066045 and DMR-1056168.

0-0

NCG of the Complex Classes of TIs

Happy Birthday my Mentor!

IMAR, July 2014 Emil Prodan Page1


Symmetry and Topology

7

Table 1. Listed are the ten generic symmetry classes of single-particleHamiltonians H, classified according to their behavior under time-reversalsymmetry (T ), charge-conjugation (or particle–hole) symmetry (C), as wellas ‘sublattice’ (or ‘chiral’) symmetry (S). The labels T, C and S representthe presence/absence of time-reversal, particle–hole and chiral symmetries,respectively, as well as the types of these symmetries. The column entitled‘Hamiltonian’ lists, for each of the ten symmetry classes, the symmetric space ofwhich the quantum mechanical time-evolution operator exp(itH) is an element.The column ‘Cartan label’ is the name given to the corresponding symmetricspace listed in the column ‘Hamiltonian’ in Élie Cartan’s classification scheme(dating back to the year 1926). The last column entitled ‘G/H (ferm. NLM)’lists the (compact sectors of the) target space of the NLM describing Andersonlocalization physics at long wavelength in this given symmetry class.Cartan label T C S Hamiltonian G/H (ferm. NLM)

A (unitary) 0 0 0 U(N ) U(2n)/U(n) U(n)

AI (orthogonal) +1 0 0 U(N )/O(N ) Sp(2n)/Sp(n) Sp(n)

AII (symplectic) 1 0 0 U(2N )/Sp(2N ) O(2n)/O(n) O(n)

AIII (ch. unit.) 0 0 1 U(N + M)/U(N ) U(M) U(n)

BDI (ch. orth.) +1 +1 1 O(N + M)/O(N ) O(M) U(2n)/Sp(2n)

CII (ch. sympl.) 1 1 1 Sp(N + M)/Sp(N ) Sp(M) U(2n)/O(2n)

D (BdG) 0 +1 0 SO(2N ) O(2n)/U(n)

C (BdG) 0 1 0 Sp(2N ) Sp(2n)/U(n)

DIII (BdG) 1 +1 1 SO(2N )/U(N ) O(2n)

CI (BdG) +1 1 1 Sp(2N )/U(N ) Sp(2n)

The only case when the behavior under the combined transformation S = T · C is not determinedby the behavior under T and C is the case where T = 0 and C = 0. In this case, either S = 0or S = 1 is possible. This then yields (3 3 1) + 2 = 10 possible types of behavior of theHamiltonian.

The list of ten possible types of behavior of the first quantized Hamiltonian under T , Cand S is given in table 1. These are the ten generic symmetry classes (the ‘tenfold way’),which are the framework within which the classification scheme of topological insulators(superconductors) is formulated.

Let us first point out a very general structure seen in table 1. This is listed in the columnentitled ‘Hamiltonian’. When the first quantized Hamiltonian H is ‘regularized’ (or ‘put on’)a finite lattice, it becomes an N N matrix (as discussed above). The entries in the column‘Hamiltonian’ specify the type of N N matrix that the quantum mechanical time-evolutionoperator exp(itH) is. For example, for systems that have no time-reversal or charge-conjugationsymmetry properties at all, i.e. for which T = 0, C = 0, S = 0, which are listed in the first rowof the table, there are no constraints on the Hamiltonian except for Hermiticity. Thus, H is ageneric Hermitian matrix and the time-evolution operator is a generic unitary matrix, so thatexp(itH) is an element of the unitary group U(N ) of unitary N N matrices. By imposingtime-reversal symmetry (for a system that has, e.g., no other degree of freedom such as, e.g.,spin), there exists a basis in which H is represented by a real symmetric N N matrix. This,in turn, can be expressed as saying that the time-evolution operator is an element of the coset

New Journal of Physics 12 (2010) 065010 (http://www.njp.org/)

The 10 (and only 10) universal symmetry classes (Altland and Zirnbauer 1997)



Classification Table of Topological Insulators and superconductors

Table 3 in Ryu, Shnyder, Furusaki and Ludwig, New J. Phys. 2010

11

Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.

dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .

Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .

Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .

dimensions if and only if the target space of the NLM on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when d(G/H) = d1(G/H) = Z2, or(ii) a WZW term, which is the case when d(G/H) = d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.

A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.

24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).


v v



Elements of K-Theory for ˚-Algebras

The K0-Group: Let PnpAq denote the set of idempotents fromAˆMn,npCq and consider:

P8pAq “ Y8n“1PnpAq, and PnpAq Q p „ q P PmpAq ðñ

$

&

%

p “ vu

q “ uv.(1)

The following addition operation on P8: p` q “

¨

˝

p 0

0 q

˛

‚is compatible with the equivalence

relation „ and pP8pAq „,`q becomes a semigroup. KA0 pAq is defined as the Grothendieck

completion of this semigroup.

The K1-Group: LetUnpAq denote the set of invertibles fromAˆMn,npCq and consider:

U8pAq “ limnÑ8

UnpAq pinductive limitq. (2)

Definition: KA1 pAq is defined as:

KA1 pAlocq “U8pAlocq rU8pAlocq,U8pAlocqs, (3)

where rU8pAlocq,U8pAlocqs is the normal subgroup of commutators, generate by products ofthe form f g f´1g´1.



Elements of Non-Commutative Geometry: Cyclic Cohomology

Defined as the cohomology of the complex:

. . .bÑ Cn´1

λ pCqbÑ Cn

λpCqbÑ . . . (4)

where CnλpCq is the space of cyclic pn` 1q-linear functionals on C:

φpc1, c2, . . . , cn, c0q “ p´1qnφpc0, c1, . . . , cnq, (5)

and b : CnλpCq Ñ Cn`1

λ pCq is the Hochschild coboundary map:

bφpc0, c1, . . . , cn`1q “

nÿ

j“0

p´1q jφpc0, . . . , c jc j`1, . . . cn`1q

` p´1qn`1φpcn`1c0, . . . , cnq. (6)

1. An element ϕ from CnλpCq is said to be an n-cyclic cocycle if it satisfies bϕ “ 0.

2. The cohomology class of ϕ, which contains all ϕ1 with ϕ´ ϕ1 “ bφ, will be denoted by rϕs.

3. A cyclic cocycle is odd/even if n is an odd/even integer.



Elements of Non-Commutative Geometry

Pairing odd cyclic cocycles with K1: The map

U8pAq Q v Ñ pϕ#Trqpv´1 ´ 1, v´ 1, . . . , v´1 ´ 1, v´ 1q (7)

is constant on the equivalence class rvs of v in KA1 pAq. Furthermore, ϕ can be replaced by any

other representative from its cohomology class. As such, there exists a natural pairing betweenKA

1 pAq and the odd cohomology ofA:

xrvs, rϕsy “ pϕ#Trqpv´1 ´ 1, v´ 1, . . . , v´1 ´ 1, v´ 1q. (8)

The pairing is not necessarily integral.

Pairing even cyclic cocycles with K0: The map

P8pCq Q p Ñ pϕ#Trqpp, p, . . . , pq (9)

is constant on the equivalence class rps of p in KA0 pAq. Furthermore, ϕ can be replaced by any

other representative from its cohomology class. As such, there exists a natural pairing betweenKA

0 pAq and the even cohomology ofA:

xrps, rϕsy “ pϕ#Trqpp, p, . . . , pq. (10)

The pairing is not necessarily integral.



Elements of Non-Commutative Geometry: The odd Fredholm-modules pH ,Fq:

• A representation π ofA in a Hilbert spaceH ;

• An operator F onH with the properties:

1. F: “ F

2. F2 “ I

3. rF, πpcqs “ compact for any c P A.

A Fredholm-module is said to be q-summable overA if rF, πpcqs belongs to the q-th Schatten classfor all c P A.

Quantized calculus with Fredholm-module pH ,Fqwhich are pn` 1q-summable:

• Graded algebra pΩ, dq, where Ω “À

Ωk with:

Ωk “ spanntc0rF, c1s . . . rF, cks, c j P Au,

• The differentiation:Ωk Q ηÑ dη “ Fη´ p´1qkηF,

• And the closed graded trace:

Ωn Q ηÑ Tr1tηu “ 12 TrtFdηu.



Elements of Non-Commutative Geometry: The odd Chern character ĂCh˚pH ,Fq

Theorem:

• The cyclic pn` 1q-linear functional:

rτnpc0, c1, . . . , cnq “ p´1qn2n`1 Tr1pπpc0qrF, πpc1qs, . . . , rF, πpcnqsu (11)

is well defined due to the summability condition and represents an odd cyclic cocycle.

• Furthermore, ĂCh˚pH ,Fq pairs well with the algebraic KA1 pAq group (see Eq. 8) and the

paring is integral:

xrvs, ĂCh˚pH ,Fqy “ Index EπpvqE P Z, (12)

where E is the idempotent E “ 12 p1` Fq, as naturally imbedde in M8pAq “ AbM8pCq via

Eb 1.



Elements of Non-Commutative Geometry: The even Fredholm-modules pH ,F, γq:

• A representation π ofA in a Hilbert spaceH ;

• An operator F onH with the properties:

1. F: “ F

2. F2 “ I

3. rF, πpcqs “ compact for any c P C;

• A grading γ (γ: “ γ, γ2 “ 1) such that:

1. γπpcq “ πpcqγ for all c P C

2. γF “ ´Fγ.

An even Fredholm-module is said to be q-summable if rF, πpcqs belongs to the q-th Schatten class.

Quantized calculus with even Fredholm-module pH ,F, γq, which are pn` 1q-summable::

• The graded algebra pΩ, dq, where Ω “À

Ωk with:

Ωk “ spanntc0rF, c1s . . . rF, cks, c j P Cu,

• The differentiation:Ωk Q ηÑ dη “ Fη´ p´1qkηF P Ωk`1,

• And the closed graded trace:

Ωn Q ηÑ Tr1tηu “ 12 TrtγFdηu.



Elements of Non-Commutative Geometry: The even Chern character Ch˚pH ,Fq

Theorem:

• The cyclic pn` 1q-linear functional:

τnpc0, c1, . . . , cnq “ p´1qn2n`1 Tr1pπpc0qrF, πpc1qs, . . . , rF, πpcnqsu (13)

is well defined due to the summability condition and represents an even cyclic cocycle.

• Furthermore, Ch˚pH ,Fq pairs well with the algebraic KA0 pCq group (see Eq. 10) and the

pairing is integer:xrps,Ch˚pH ,Fqy “ Index π´ppqFπ`ppq P Z, (14)

where F is naturally extended over M8pAq “ AbM8pCq via Fb 1 and π˘ is thedecomposition with respect to the grading γ of the similarly extended representation π.



Homogeneous Aperiodic Quantum Lattice Systems (Following Bellissard 1990’)

I Let pΩ, dP , tq be a minimal classical dynamical system

- Ω “ (disorder) configuration space

- dP “ probability measure on Ω

- t : Ω Ñ Ω an action of Zd on Ω

- t acts ergodically and leave dP invariant.

I Let tHωuωPΩ be a covariant family of Hamiltonians, i.e:

UaHωU´1a “ Htaω, Ua “ magnetic translation by “a”

Then the quadruple pΩ, dP , t, tHωuωPΩq defines a homogeneous system.



Typical Examples from the Theory of Materials Science

Hω : `2pZd,CNq Ñ `2pZd,CNq

pHωψqpxq “ÿ

yPZd

eıx^y ptx,ypωqψpyq

Where:

• eıx^y is the Peierls factor due to a uniform magnetic field.

• Random hopping matrices: ptx,ypωq “ p1` λωx,yqptxý

• Disorder configuration space: ω ” tωx,yu, ω P Ω “ r´ 12 ,

12 sZdˆ r´ 1

2 ,12 sZDˆ . . .

• Natural probability measure: dP pωq “ś

x,y dωx,y

• Ergodic translations: ptaωqx,y “ ωxá,yá

These models:

• Can be generated from first-principle calculations or from purely empirical data.

• They can be tuned to accurately describe the physics of a material.



The algebra of covariant observables C˚pΩ o Zd,^q

• Elements: f : ΩˆZd Ñ C

• Addition: p f ` gqpω,xq “ f pω,xq ` gpω,xq.

• Multiplication: p f ˚ gqpω,xq “ř

yPZdeıpx^yq f pω,yqgpt´1

y ω,x´ yq.

• Covariant representation on `2pZdq:`

pπω f qψ˘

pxq “ř

yPZdeıy^x f pt´1

x ω,y ´ xqψpyq.

• Standard norm: f “ supωPΩ

πω f .

• Standard ˚-operation: f˚pω,xq “ f pt´xω,´xq:.

Then

C˚pΩ o Zd,^q “ pCcpΩˆZdq,^, ,˚ q

becomes a (non-commutative) C˚-algebra, which replaces the algebra of ordinary functions overthe Brillouin torus.



The Non-Commutative Differential Calculus over C˚pΩ o Zd,^q

The general philosophy is:

1) C˚pΩ o Zd,^q replaces the algebra of functions defined over the Brillouin torus.

2) The following trace replaces the classical integral in the momentum space:

ż

dk pFpkq Ñ T p f q “ż

Ωdω f pω, 0q

3) The k-derivations are replaced by their real-space representations:

Bk jpFpkq ÑpB j f qpω,xq “ ix j f pω,xq

πωpB j f q “ irX j, πω f s.

Then pC˚pΩ o Zd,^q,T , Bq becomes a non-commutative differential manifold,

called the Non-commutative Brillouin Torus.



The algebra of localized observables

Proposition. Consider the setAloc of elements f in the weak von-Neumann closure ofA,obeying the following condition:

ż

ΩdP pωq | f pω,xq| ď Ae´λ|x|, for some A, λ ą 0. (15)

Endow this set with the topology induced by the GNS norm:

|| f ||GNS “

b

T t f f˚u “b

T t| f |2u. (16)

Then:

1. The setAloc is a dense topological ˚-sub-algebra of the weak von-Neumann closure ofA.

2. The integralsT tB

a1 f1 . . . Bak fku (17)

are always finite for fi-s fromAloc.

3. Any functional of the type (and the natural generalizations):

Aloc Q f Ñ Tp f q “ T tpBa f qgu, pg P Alocq (18)

is continuous.



The natural Fredholm-modules

d=odd

• The Hilbert spaceH of the Fredholm-module: H “ `2pZdq b Cliffpdq.

• The representation: πω f b id.

• F is defined as the phase of the Dirac operator:

Dx0 “

dÿ

i“1

pXi ´ xi0q b σi, Fx0 “

Dx0

|Dx0 |. (19)

d=even

• The Hilbert spaceH of the Fredholm-module: H “ `2pLq b Cliffpdq

• Representation: πω b id.

• The grading operator: γ “ 1b γ0

• The operator F is defined again as the phase of the Dirac operator:

Dx0 “

dÿ

i“1

pXi ´ xi0q b γi, Fx0 “

Dx0

|Dx0 |. (20)



Summability

Proposition The family of Fredholm-modules introduced above, pH ,Fx0 , πω b idq andpH ,Fx0 , πω b idq, with ω P Ω and x0 P r0, 1sd, are pd` 1q-summable over the sub-algebraAloc, i.e.for any f P Aloc:

ż

r0,1sddx0

ż

Ωdω Trt|rπωp f q,Fs|d`1u ă 8. (21)

The Chern character for the odd family: The cohomology class (in the cyclic cohomology ofAloc)of the following pd` 1q-cyclic cocycle:

rτdp f0, f1, . . . , fdq “id`1

2d

ż

r0,1sd

dx0

ż

Ω

dP pωq Tr1tπωp f0qrFx0 , πωp f1qs . . . rFx0 , πωp fdqsu. (22)

The Chern character for the even family: The cohomology class (in the cyclic cohomology ofAloc) of the following pd` 1q-cyclic cocycle:

τdp f0, f1, . . . , fdq “ż

r0,1sd

dx0

ż

Ω

dω Tr1tπωp f0qrFx0 , πωp f1qs . . . rFx0 , πωp fdqsu. (23)



The fundamental result for d “odd

Theorem. The odd Chern character associated to the family pH ,Fx0 , πω b idq has the followingfundamental properties:

1. The pairing between ĂCh˚ and KA1 pAlocq remains integral:

xrvs, ĂCh˚y “ Index Ex0πωpvqEx0 P Z, (24)

where the Fredholm index on the right is almost surely independent of ω and x0.

2. The odd Chern cocycle accepts the following local formula:

rτdp f0, f1, . . . , fdq “ rΛd

ÿ

ρPSd

p´1qρT

˜

f0dź

i“1

Bρi fi

¸

, (25)

with rΛd “ipíπq

d´12

d!! . Sd is the permutations group.

3. The odd Chern cocycle rτd is continuous onAloc. As such, the topological KT1 pAlocq group can

be used instead of the algebraic KA1 pAlocq group.

Key Identity:

ż

Rddx trσ

#

dź

i“1

pxi ` x´ xi`1 ` xq ¨ σ

+

“2

d`12 p2πq

d´12

ıd´1

2 d!!

ÿ

ρPSd

p´1qρdź

i“1

xρi

i (26)



The fundamental result for d “even

Theorem. The even Chern character associated to the family pH ,Fx0 , γ, πω b idq has thefollowing fundamental properties:

1. The pairing between Ch˚ and KA0 pAlocq remains integral:

xrps,Ch˚y “ Index π´ω ppqFx0π`ω ppq P Z, (27)

where the Fredholm index on the right is almost surely independent of ω and x0.

2. The even Chern cocycle accepts the following local formula:

τdp f0, f1, . . . , fdq “ Λd

ÿ

ρPSd

p´1qρT

˜

f0dź

i“1

Bρi fi

¸

, (28)

with Λd “p2πıq

d2

pd2q! .

3. The even Chern cocycle τd is continuous onAloc. As such, the topological KT0 pAlocq group

can be used instead of the algebraic KA0 pAlocq group.

Key identity:

ż

Rddx trγ

#

γ0

dź

i“1

pxi ` x´ xi`1 ` xq ¨ γ

+

“ ´p2πq

d2

ıd2 pd2q!

ÿ

ρPSd

p´1qρdź

i“1

xρi

i (29)



Stability of the AIII-class at strong disorder

A homogeneous aperiodic system is in AIII-symmetry class if there exists a Hermitean N ˆNmatrix S acting on the orbital degrees of freedom, such that S2 “ 1 and:

SHωS´1 “ ´Hω for all ω P Ω. (30)

Corollary. Consider a homogeneous topological insulator from the AIII-symmetry class. Then:

1. The ground state is completely determined by the unitary operator Uω “ S`signpHωqS´.

2. If the Fermi level resides in a region of localized energy spectrum, the non-commutativeodd-Chern number,

ĂChdpuq “ rΛd

ÿ

ρPSd

p´1qρ T

#

dź

i“1

u´1Bρi u

+

(31)

corresponding to its ground state is finite and quantized:

ĂChdpuq “ Index Ex0 UωEx0 P Z. (32)

3. Let Hωptq “ Hω ` tδHω, with sup |peαx|δHω|eβyq| ď 8, be a deformation of the homogeneous

system preserving the AIII-symmetry. Assume that the Fermi level stays in a region oflocalized spectrum at all times. Then the odd-Chern number corresponding to the groundstate of Hωptq, that is ĂChdputq, remains pinned at a quantized value for all t’s.



Stability of the A-class at strong disorder

In this case, Hω has no symmetry other than being self-adjoint.

Corollary. Consider a homogeneous topological insulator from the A-symmetry class. Then:

1. If the Fermi level resides in a region of localized energy spectrum, the non-commutativeeven-Chern number corresponding to its ground state:

Chdppqdef“ Λd

ÿ

ρPSd

p´1qρ T

#

pdź

i“1

Bρi p

+

. (33)

is finite and quantized:Chdppq “ Index P´ωFx0 P`ω P Z. (34)

2. Let Hωptq “ Hω ` tδHω, with sup |peαx|δHω|eβyq| ď 8, be a deformation of the homogeneous

system. Assume that the Fermi level stays in a region of localized spectrum at all times.Then the even-Chern number corresponding to the ground state of Hωptq, that is Chdpptq,remains pinned at a quantized value for all t’s.



Classification Table of Topological Insulators and superconductors

Table 3 in Ryu, Shnyder, Furusaki and Ludwig, New J. Phys. 2010

11

Table 3. Classification of topological insulators and superconductors as afunction of spatial dimension d and symmetry class, indicated by the ‘Cartanlabel’ (first column). The definition of the ten generic symmetry classes of singleparticle Hamiltonians (due to Altland and Zirnbauer [29, 30]) is given in table 1.The symmetry classes are grouped into two separate lists, the complex andreal cases, depending on whether the Hamiltonian is complex or whether one(or more) reality conditions (arising from time-reversal or charge-conjugationsymmetries) are imposed on it; the symmetry classes are ordered in such a waythat a periodic pattern in dimensionality becomes visible [27]. (See also thediscussion in subsection 1.1 and table 2.) The symbols Z and Z2 indicate thatthe topologically distinct phases within a given symmetry class of topologicalinsulators (superconductors) are characterized by an integer invariant (Z) or aZ2 quantity, respectively. The symbol ‘0’ denotes the case when there exists notopological insulator (superconductor), i.e. when all quantum ground states aretopologically equivalent to the trivial state.

dCartan 0 1 2 3 4 5 6 7 8 9 10 11 . . .

Complex case:A Z 0 Z 0 Z 0 Z 0 Z 0 Z 0 . . .AIII 0 Z 0 Z 0 Z 0 Z 0 Z 0 Z . . .

Real case:AI Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 . . .BDI Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 0 . . .D Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z 0 . . .DIII 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 Z . . .AII 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 Z2 . . .CII 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 Z2 . . .C 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z 0 . . .CI 0 0 0 2Z 0 Z2 Z2 Z 0 0 0 2Z . . .

dimensions if and only if the target space of the NLM on the d-dimensional boundary allowsfor either (i) a Z2 topological term, which is the case when d(G/H) = d1(G/H) = Z2, or(ii) a WZW term, which is the case when d(G/H) = d+1(G/H) = Z. By using this rule inconjunction with table 2 of homotopy groups, we arrive with the help of table 1 at table 3 oftopological insulators and superconductors24.

A look at table 3 reveals that in each spatial dimension there exist five distinct classes oftopological insulators (superconductors), three of which are characterized by an integral (Z)topological number, while the remaining two possess a binary (Z2) topological quantity25.

24 To be explicit, one has to move all the entries Z in table 2 into the locations indicated by the arrows, and replacethe column label d by d = d + 1. The result is table 3.25 Note that while d = 0, 1, 2, 3-dimensional systems are of direct physical relevance, higher-dimensionaltopological states might be of interest indirectly, because, for example, some of the additional components ofmomentum in a higher-dimensional space may be interpreted as adiabatic parameters (external parameters onwhich the Hamiltonian depends and which can be changed adiabatically, traversing closed paths in parameterspace—sometimes referred to as adiabatic ‘pumping processes’).


v v



The Concept of Approximating Non-Commutative Spaces

(C*(Ω x Zd),∂,T) (C*(ΩΛ x ZD),∂,Tper)

(C*(ΩΛ x TD),∂,T)

Λ

per

~ ~



Foundation for a computational NCG program

Theorem. (E.P., Appl. Math. Res. Express 2013) Let h be compact supported from C˚pΩˆZDq

and let hper from C˚pΩNper ˆZDq obtained by restricting h to ΩNper. Then, for any analytic functions

Φ j around σphq:

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

T

¨

˝

Mź

j“1

Bα jΦ jphq

˛

‚´ Tper

¨

˝

Mź

j“1

Bα jΦ jphperq

˛

‚

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ď B1pξ, tαuq

¨

˝

Mź

j“1

Φ j

˛

‚N´1e´ξN .

Theorem. (E.P., Appl. Math. Res. Express 2013) Let h be a compactly supported Hamiltonianand h “ Pphperq. Let Φ j ( j “ 1, . . . ,M) be analytic functions in the neighborhood of σphq. Then

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

Tper

¨

˝

Mź

j“1

Bα jΦ jphperq

˛

‚´ TT

¨

˝

Mź

j“1

Bα jΦ jphq

˛

‚

ˇ

ˇ

ˇ

ˇ

ˇ

ˇ

ă Bpξ, tαuq

¨

˝

Mź

j“1

Φ j

˛

‚e´ξN .



Numerical Evaluationspin-up sector of the Kane Mele model (zero Rashba), strong disorder regime (spectral gap closed!!)

40 x 40

EF EF

EF

60 x 60

EF

80 x 80 100 x 100

Che

rn N

umbe

rC

hern

Num

ber

Che

rn N

umbe

rC

hern

Num

ber

Thursday, September 19, 2013IMAR, July 2014 Emil Prodan Page25


Quantization with MACHINE PRECISION

Table 1: Numerical values for average Chern numbers

Energy 40ˆ 40 60ˆ 60 80ˆ 80 100ˆ 100

-2.0000000000000000 0.0293885304649968 0.0183147848896676 0.0134785966919230 0.0055726403061233

-1.8999999999999999 0.0442301583027775 0.0274502505545331 0.0200229343621875 0.0112501012411246

-1.8000000000000000 0.0563736772645283 0.0416811880195335 0.0285382576963500 0.0259995275657507

-1.7000000000000000 0.0868202901241971 0.0612803850743208 0.0506852078002088 0.0377798251819264

-1.6000000000000001 0.1121154018269069 0.0905166860071905 0.0781754600177580 0.0554182299457663

-1.5000000000000000 0.1617580454580226 0.1291516191502659 0.1133966598848624 0.0977984662347778

-1.3999999999999999 0.2093536896403097 0.1883311262238442 0.1733092018533850 0.1386844139850113

-1.3000000000000000 0.2687556358733589 0.2575144956897765 0.2146703753513447 0.2040079233029510

-1.2000000000000000 0.3565352143319771 0.3333569482253110 0.3319133571108642 0.3066419928551302

-1.1000000000000001 0.4646789224167249 0.4444784219466996 0.4310440221933989 0.4427699238861748

-1.0000000000000000 0.5479958396159215 0.5561471440680733 0.5442615536532044 0.5738596277941682

-0.9000000000000000 0.6624275864985472 0.6798953821199148 0.7086514094234754 0.7228749266484203

-0.8000000000000000 0.7742005453064691 0.8124137607528051 0.8270271100278364 0.8487923693232788

-0.7000000000000000 0.8672349391630054 0.9079791895178040 0.9301639459675241 0.9432234611493278

-0.6000000000000000 0.9392873717233425 0.9636994770114942 0.9802652381114992 0.9872940741308633

-0.5000000000000000 0.9784417158133359 0.9935074963179980 0.9974987656403326 0.9988846769813913

-0.4000000000000000 0.9958865415757685 0.9992024708366942 0.9998527876642247 0.9999656328302596

-0.3000000000000000 0.9998184404341747 0.9999824660477071 0.9999988087144891 0.9999996457562911

-0.2000000000000000 0.9999952917010211 0.9999977443008894 0.9999999997655000 0.9999999999862120

-0.1000000000000000 0.9999999046002306 0.9999999998972079 0.9999999999998473 0.9999999999999849

0.0000000000000000 0.9999999963422543 0.9999999999988873 0.9999999999999996 0.9999999999999999



Finite-Size scaling of the Chern number near transition

Che

rn N

umbe

r

Che

rn N

umbe

rEF EF

The Chern lines overlap almost perfectly after a rescaling of the energy axis

E Ñ Ec ` pE´ Ecq ˚ pLL0qν

(ν “ 2.6, as it should for the unitary class).



(Mondragon, Hughes, Song and E.P, to appear PRL)

0 1 2 3 40

0.5

1

1.5

2

m10−1100101102103104105106

(b)

ν Λ

0 10 20 300

0.5

1

1.5

2

W10−1100101102103104105

(c)

ν Λ

0 1 2 3 40

0.5

1

1.5

2

m10−1100101102103104105106

(e)

ν Λ

0 10 20 300

0.5

1

1.5

2

W100

101

102

103

104

105

(f)

ν Λ

Monday, January 27, 14

AIII (D=1) BDI (D=1) AIII (D=1) BDI (D=1)



List of Relevant Publications• E. Prodan and B. Leung and J. Bellissard, The non-commutative n-th Chern number (n ě 1), J. Phys. A: Math. Theor. 46, 485202

(2013)

• E. Prodan and H. Schulz-Baldes, Non-commutative odd Chern numbers and topological phases with chiral symmetry,http://arxiv.org/abs/1402.5002. (2014)

• E. Prodan, The Non-Commutative Geometry of the Complex Classes of Topological Insulators, http://arxiv.org/abs/1402.5002 (2014)

Thank You and Happy Birthday GheorgheNenciu!


Documents

Non-Commutative Geometry of the Complex Classes of ...imar.ro/ConfGN/lectures/EProdan.pdf · Emil Prodan Yeshiva University, New York, USA IMAR, July 2014 In honor of 70th birthday