non-commutative geometry and materials science · 2021. 3. 1. · c -algebra of an etale grupoid...

Non-Commutative Geometry and Materials Science————————————————–

Emil Prodan

Yeshiva University, New York, USA

Global NCG SeminarJan 2021

Many thanks to my collaborators and mentors.

Work supported by U.S. NSF grant DMR-1823800 and Keck Foundation

Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 1 / 22

The Algebra of Local Observables

The algebra of local physical observables is: [Ax = MN×N(C)]

A = K(CN × `2(L)

)in the linear regime

A = lim−→⊗

x∈Lk⊂L Ax in the non-linear regime

The experimenter’s main task is to map the time evolution

α : R→ Aut(A).


Time Evolution of the Local Observables

The generators δα of the mapped evolutions should be almost-inner (or inner-limit)

In the linear regime, all bounded almost-inner derivations come from B(CN × `2(L)

):

δα(A) = ı[A,HL], HL ∈ B(CN × `2(L)

).

This, however, is hardly enough because

K0

(B(H)

)= 0.

Any two gapped Hamiltonians can be deformed into each other without closing the spectral gap.


Fundamental Insight from Bellissard and Kellendonk

Setting:

The experimenter deforms the lattice (without modifying the resonators) and maps

L 7→ HL =∑

x,x′∈Lwx,x′ (L)⊗ |x〉〈x ′|

Galilean invariance is at work: If L′ = L − y , then

HL′ = SyHLS∗y , Sy : `2(L)→ `2(L′).

Theorem (Bellissard & Kellendonk):

If L is a Delone set, then all HL’s can be generated from the left-regular representations of theC∗-algebra of an etale grupoid canonically associated to L.

No matter what type of resonators we work with, the dynamics is always associated to a

FIXED SEPARABLE C∗-algebra [denoted here as C∗(L)]


Computable Examples

Phason

Space

f

t(0,1)(f)

t(1,0)(f)

t(1,1)(f)

p(0,0)p(1,0)

p(0,1) p(1,1)

Td′ = Rd′/L′, τn(φ) =(φ +

d∑i=1

niai

)modL′, φ ∈ Td′ , ai ∈ Rd ⊆ Rd′

If F : Td′ → Rd is a continuous map and

Lφ = pn(φ)n∈Zd , pn(φ) = p0 +d∑

i=1

niai + F(τn(φ)

), φ ∈ Td′

then the algebra generated by the Galilean invariant almost-inner derivations reduces to

C∗(L) = C(Td′ ) o Zd ' AΘ [non-commutative (d+d’)-torus]


Examples (Spectra computed with HL =∑

x,x′∈L e−|x−x′|/Λ |x〉〈x ′|)

y (m

m)

x (mm) x (mm) x (mm)

a b c

q

c

q

b

Spec

tru

m

q

a


K-Theory in Action

q q

IDS

IDS(E) =# of eigenvalues below E

# of resonators7→ T

(χ(−∞,E ]

(HL))

Proposition (Elliot 1984): If [PG ]0 =∑

J⊆1,...,d+d′ nJ [eJ ]0 ∈ K0(C∗(L)), then:

IDS(E ∈ Gap) =∑J

nJ Pfaffian(ΘJ) [|J| = even]


Experimental Implementation [Chen et all, 2020]

a

b

e

f

g

3 mm

40 mm

10 mm

xy

c

d

0 1𝜃

5.8

6.0

6.2

6.4

6.6

6.8

7.0

7.2

Freq

uen

cy (

kHz)

Topological gap

Trivial gap

6.00

5.95

5.90

Freq

uen

cy (

kHz)

5.85

0

1 1

0𝜙1 𝜙2

min

max

Bulk5.80

5.90

5.95

6.00

6.05

6.10

5.85

10𝜙1= 𝜙2

Freq

uen

cy (

kHz)

Bulk5.80

5.90

5.95

6.00

6.05

6.10

5.85

10 𝜙2 (f1=0.5)

Freq

uen

cy (

kHz)

min

max

6.1

6.0

5.9

5.8

0

1 1

𝜙1 𝜙2

0

min

max

h

Bulk

Bulk


Moire patterns [Rosa et all, 2020]


Bulk-boundary correspondence: The spectral statement

Spec(h) Spec(ℎ)

Topological gap Topological boundary spectrum

Statement: Almost all gaps seen in our experimental designs are topological.


Toeplitz extension [Richter, Kellendonk, Schulz-Baldes (2002)]

Universal C∗-algebra AΘ = C∗(u1, . . . , us) [s = d + d ′]:

ui uj = eıθij uj ui , θij ∈ T1.

u∗1 u1 = 1, u1u∗1 = 1− e (e = projection)

Generic element:a =

∑n,m∈N

un1 (u∗1 )manm, anm ∈ AΘ Θ = Θ2,...,s

The principal ideal generated by e

AΘ = AΘ e AΘ ' K⊗AΘ

serves as the algebra of the boundary observables.

The canonical representation π:

H = `2(N⊗ Zd−1), uq 7→ ΠSqΠ∗, Π : Zd → N× Zd−1, q = 1, . . . , d ,

supplies all Galilean invariant physical models with a boundary.


The exact sequence

The exact sequence is split at the level of linear spaces:

0 - AΘ

i- AΘ

ev-i′

AΘ- 0

Hence we have the presentation:

AΘ 3 a = i ′(a) + a, with unique pair a ∈ AΘ, a ∈ AΘ.

Furthermore [φ ∈ Td′ ]:

πφ(i ′(a)

)= Ππ(a)Π (Dirichlet boundary condition)

πφ(a) = boundary perturbations

The data for the bulk-boundary problem is:

(h, h) such that ev(h) = h (h, h always from stabilized algebras)

Every statement (topological invariant) should be based on this data alone.


The K -Groups and Generators

K0(AΘ)i∗- K0(AΘ)

ev∗- K0(AΘ)

K1(AΘ)

Ind6

ev∗ K1(AΘ) i∗

K1(AΘ)

Exp

?

The generators of K0(AΘ) and K0(AΘ) can be indexed as:

[eJ ]0 and [eJ′ ]0, J, 1 ∪ J′ ⊆ 1, . . . , s, |J|, |J′| = even

The generators of K1(AΘ) and K1(AΘ) can be indexed as:

[vJ ]1 and [vJ′ ]1, J, 1 ∪ J′ ⊆ 1, . . . , s, |J|, |J′| = odd

and chosen such that:

Exp[e1∪J ]0 = [vJ ]1

Ind[v1∪J ]1 = [eJ ]0


The spectral statement explained

Consider (h, h,G) given and let pG = χ(−∞,g ](h) be the bulk gap-projection (g ∈ G).

Proposition: Exp mapping of [pG ]0 can be expressed in terms of this data:

Consider a continuous f : R→ R with f = 0/1 below/above G. Then:

Exp[pG ]0 =[e−2πıf (h)

]1

(1− f (h) supplies a lift for pG )

Bulk-Boundary correspondence (Spectral Statement): Let

K0(AΘ) 3 [pG ]0 =∑

|J|=even

nJ [eJ ]0

If any of nJ ’s with J ∩ 1 are nonzero, then:

G ⊂ Spec(h).

Proof.

If Spec(h) is not the whole G , then we can concentrate the variation of f in ρ(h).

But then e−2πıf (h) = 1 and we know this is not possible.


Bulk-Boundary correspondence: The dynamical statement

SourceSource

Localized De-Localized

Bulk-Boundary correspondence principle (dynamical form):

A “strong” topological gap leads to de-localized boundary modes. The statement is that suchstrong topological gaps exist.

Remark: Techniques for proving localization in the presence of disorder abound. The onlybroadly applicable technique known to me for proving de-localization relies on Index Theory.


Some preliminaries

Bringing the boundary disorder inside the formalism:

For a given boundary, AΘ reduces to C∗(u2, . . . , us ,C(Ω)

)where

(Ω, t : Zd−1 → Homeo(Ω),dP) [Ergodicity and contractibility are assumed]

is the hull of the boundary (Example: Ω = [0, a]Zd−1

for a disordered boundary). It is thenconvenient to start right from beginning with a bulk algebra

AΘ → C∗(u1, . . . , us ,C(Ω)

)= AΘ(Ω)

Canonical trace:

T(∑

aquq

)=

∫dP(ω)a0(ω), uq = uq1

1 . . . uqdd .

Circle actions:

ui → λiui , λi ∈ T1

supply s-derivations ∂i .


Numerical invariants

Proposition (Nest, 1988):

ξJ (a0, . . . , an) = ΛJ

∑ρ∈SJ

(−1)ρ T(a0

|J|∏j=1

∂ρj aj

), J ⊂ 1, . . . , s,

generate the entire (periodic) cyclic cohomology of the smooth sub-algebra A∞Θ (Ω).

Theorem (E.P & Schulz-Baldes 2016): Using guidance from [see Carey et al, Mem. AMS 2014]

K∗(C,AΘ(Ω)

)×KK∗

(AΘ(Ω),AΘ

J(Ω)) Kasparov product- K0

(AΘ

J(Ω)) Tr- R

HP∗(AΘ(Ω)

)Ch∗

?× HP∗

(AΘ(Ω)

)Ch∗

?Connes′ pairing - R

?

the following generalized index theorems were derived:

〈ξJ , [p]0〉 = Tr(

Ind(πω(p)Fπω(p)

)), 〈ξJ , [u]1〉 = Tr

(Ind(Px0πω(u)Px0

))Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 17 / 22

Pushing into Sobolev

The natural domain of ξJ is DJ = L∞(AΘ(Ω), T

)∩WJ

(AΘ(Ω), T

)where

‖a‖WJ=∑j∈JT(|∂ja||J|

) 1|J|

Proposition: [Bellissard et all 1994, E.P. et all 2013, E.P & Schulz-Baldes 2016, Bourne & Rennie 2018]

Under a set of assumptions and if p, u ∈ DJ , the local index formulas hold P-almost surely. As aconsequence, if p(t) or u(t) are homotopies inside DJ , then the pairings are constant andquantized.

Definition: The boundary spectrum

Spec(h) := Spec(h) \ Spec(h)

is said to be localized in the J-directions if g(h) ∈ DJ for any piece-wise continuous function

g : Spec(h)→ C.


Let’s put the machinery at work [uG = eı2πf (h)]

〈ξ1∪J , [pG ]0〉 = Tr(

[pG ]0 × [Dirac1∪J ])

= Tr(

[uG ]1 × [DiracJ ])

= 〈ξJ , [uG ]1〉

Proof:[pG ]0 × [ext]1 = [uG ]1.

([pG ]0 × [ext]1)× [DiracJ

] = [uG ]1 × [DiracJ

]

[ext]1 × [DiracJ

] = [Dirac1∪J ] [Bourne et al, 2015]

Conclusion: if n1∪J 6= 0 in [pG ]0 =∑

J nJ [eJ ]0 then⟨ξJ ,[eı2πf (h)

]1

⟩6= 0

Punch line:

Assume that the boundary spectrum is localized in the J-directions. Then we can deform f to a

step function and such that eı2πft (h) ∈ DJ . But then

eı2πf1(h) = 1⇒⟨ξJ ,[eı2πf (h)

]1

⟩= 0 (A contradiction)


Structure of an NCG program in Materials Science

Compute the *-algebraof equivariant almost-inner derivations and itsextensions

K-Theory

Supplies a global pictureand complete set of invariants

Certain spectral statements can be made at this level

Cyclic cohomology

Pairing with the K-theorysupplies numerical invariants with physical interpretation.

Quantized calculus

Supplies cocycles with quantized range.

Local index formulas

Enable the push into the Sobolev.

Dynamical statements can be made at this level.

NCG Arrow

Interesting open problems. Complete the program for:

the case when Zd is replaced by other discrete sub-groups of the Euclidean group

same as above but for hyperbolic spaces

for fractal patterns

the case when the local algebra is CAR or spin algebras (i.e. the non-linear regime)

create completely new dynamical patterns in meta-materials starting from abstractC∗-algebras


References (Formalism):

J. Bellissard, K-theory of C∗-algebras in solid state physics, Lect. Notes Phys. 257, 99–156(1986).

J. Bellissard, Gap labeling theorems for Schroedinger operators, in: M. Waldschmidt, P. Moussa,J.-M. Luck, C. Itzykson (Eds.), From Number Theory to Physics, Springer, Berlin, 1995.

J. Kellendonk, Noncommutative geometry of tilings and gap labelling, Rev. Math. Phys. 7,1133–1180 (1995).

J. Kellendonk, T. Richter, H. Schulz-Baldes, Edge current channels and Chern numbers in theinteger quantum Hall effect, Rev. Math. Phys. 14, 87–119 (2002).

E. Prodan, B. Leung and J. Bellissard, The non-commutative n-th Chern number, J. Phys. A:Math. Theor. 46, 485202 (2013).

E. Prodan and H. Schulz-Baldes, Non-commutative odd Chern numbers and topological phasesof disordered chiral systems, J. Func. Anal. 271, 1150–1176 (2016).

E. Prodan and H. Schulz-Baldes, Bulk and Boundary Invariants for Complex TopologicalInsulators: From K-Theory to Physics, (Mathematical Physics Studies, Springer, 2016).

E. Prodan and H. Schulz-Baldes, Generalized Connes-Chern characters in KK-theory with anapplication to weak topological invariants, Rev. Math. Phys. 28, 1650024 (2016).

E. Prodan, A Computational Non-Commutative Geometry Program for Disordered TopologicalInsulators, Springer Briefs in Mathematical Physics, Springer, 2017


References (Materials Science):

D. J. Apigo, K. Qian, C. Prodan, E. Prodan, Topological Edge Modes by Smart Patterning,Phys. Rev. Materials 2, 124203 (2018).

E. Prodan, Y. Shmalo, The K-Theoretic Bulk-Boundary Principle for Dynamically PatternedResonators, Journal of Geometry and Physics 135, 135-171 (2019).

D. J. Apigo, W. Cheng, K. F. Dobiszewski, E. Prodan, C. Prodan, Observation of TopologicalEdge Modes in a Quasi-Periodic Acoustic Waveguide, Phys. Rev. Lett. 122, 095501 (2019).

X. Ni, K. Chen, M. Weiner, D. J. Apigo, C. Prodan, A. Alu, E. Prodan, A. B. Khanikaev,Observation of Hofstadter Butterfly and Topological Edge States in ReconfigurableQuasi-Periodic Acoustic Crystals, Commun. Physics 2, 55 (2019).

W. Cheng, E. Prodan, C. Prodan, Experimental demonstration of dynamic topological pumpingacross incommensurate acoustic meta-crystals, Phys. Rev. Lett. 125, 224301 (2020).

M. Rosa, M. Ruzzene, E. Prodan, Topological gaps by twisting, (arXiv:2012.05130).

H. Chen, H. Zhang, Q. Wu, Y. Huang, H. Nguyen, E. Prodan, X. Zhou, G. Huang, Physicalrendering of synthetic spaces for topological sound transport, (arXiv:2012.11828).

W. Chen, E. Prodan, C. Prodan, Mapping the boundary Weyl singularity of the 4D Hall effectvia phason engineering in metamaterials, ( arXiv:2012.11828).


non-commutative geometry and materials science · 2021. 3. 1. · c -algebra of an etale grupoid...

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