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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).
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 2 / 22
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.
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 3 / 22
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)]
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 4 / 22
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]
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 5 / 22
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
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 6 / 22
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]
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 7 / 22
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
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 8 / 22
Moire patterns [Rosa et all, 2020]
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 9 / 22
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.
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 10 / 22
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.
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 11 / 22
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.
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 12 / 22
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
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 13 / 22
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.
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 14 / 22
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.
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 15 / 22
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 .
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 16 / 22
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.
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 18 / 22
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)
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 19 / 22
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
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 20 / 22
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
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 21 / 22
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).
Emil Prodan (Yeshiva University) NCG and Materials Science Jan 2021 22 / 22
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