Macroscopic index theory and geometricquantum Hall eect
1

Guo Chuan Thiang

BICMR, Peking University
CREST workshop

5-7 Feb 2024

1

Joint with M. Ludewig, Y. Kubota, J. Xia.

CMP'21,'22; arXiv:2308.02819,2401.07449

1 / 69

Geometric robustness of QHE• 1980 IQHE: Classical/macroscopic R-valued Hall conductancee2−10 error.
measurement, whose range is
h Z, with ∼ 10
• Most theory ∼ perfect Euclidean plane sample.
• Experimental samples are bumpy and nite sized.
• Quantized Hall conductance persists, ignores ne geometricdetails. This is why it is useful, e.g. redenition of Kilogram.

• Summary: Macroscopic index theory explains how
macroscopic quantization emerges from microscopic QM.

2 / 69QHE experiments

N. Mitchell et al, Nature Phys. (2018)3 / 69

One calculation to rule them all• Topological invariance: deform complicated calculations to
special simple ones.

• Non-trivial step is to justify deformation invariance of relevant
calculation.

• For IQHE, geometric invariances and basic calculation seem tobe missing.
4 / 69Outline• I. Dierential geometry of Landau quantization• Riemannian, spin, gauge.
• II. Functional analysis and traces of commutators• Cancelling innities in QM.
• III. Coarse geometry and index theory• Macroscopic quantization of Hall conductance.
Focus is on Geometry and Analysis, not on Topology and algebra.
No specialized condensed matter ideas, models, or backgroundassumed.

5 / 69

Geometry in quantum mechanics• Usually, Schrödinger operator on Riemannian M is
H = −∇2 + V ,where V is real-valued potential.

• H , and its spectral projections P , will commute with complex
conjugation.

• Such real Hamiltonians cannot describe QHE.
• Must replace ∇ with ∇ − iA, where A is connection 1-form
for U(1) line bundle.
• This is a fundamental feature of quantum mechanics!

6 / 69

Geometry in quantum mechanics

Quantum phenomena do not occur in a Hilbert space, they occur
in a laboratory.  A. Peres.
• A natural Hilbert space to represent position operators on M
2is L (M).
• Is quantum state a L2 wavefunction ψ : M → C?
• No! Even in spinless case, ψ is not a C-scalar eld, but a2
section of a U(1) line bundle L → M .
• Need connection, to compare copies of C living at dierent
points of M .
• On contractible M with no magnetic eld, this may beforgotten without consequence. (But recall AB-eect!)2

Consider what Galilean invariant momentum operator means. . .

7 / 69

Geometry in quantum mechanics• Generally, we have a Hermitian vector bundle V → M .
• A (local) gauge is an orthonormal frame, making the bundleN
look (locally) like M × C . Gauge group unitarily representedN
in C .
• Relative to a gauge choice, a section of V is described as a
CN -valued function.
• To dierentiate ψ gauge-covariantly, we use a connection,or parallel transport, dened by properties∇u+f ·v ψ = ∇u ψ + f · ∇v ψ,∇v (h · ψ) = v (h) · ψ + h · ∇v ψ,u, v vector elds, f , h smooth functions, ψ section.

8 / 69

Geometry in quantum mechanics

For simplicity, work with U(1) line bundles, L → M .

• After choosing a (local) gauge and coordinates, ψ is a
C-valued function of x 1 , . . . , x d .
• Correspondingly, ∇ becomes a covariant derivative,∇j ψ = (∂j − iAj )ψ,where A =P

j Aj dx

j is a R = −iu(1)-valued 1-form, called

the connection 1-form.

• Constant function with respect to gauge choice generallydiers from  ∇-constant. Mismatch is encoded by A.

9 / 69

Geometry in quantum mechanics• Connection 1-form A is gauge-dependent description of ∇.
• Rotate gauge choice by applyingg = e iΛ : M → U(1),A ⇝ A + dΛ.
• If A happens to be exact, A = dΛ, we can apply g = e −iΛ tond a better gauge in which the connection looks trivial,∇j = ∂j .
• Since d 2 = 0, an obstruction to A being exact isdA =X(∂i Aj − ∂j Ai ) dx i ∧ dx j ≡ F ∇ .

i,j

10 / 69

Geometry in quantum mechanics• F ∇ is called the curvature 2-form of ∇. It is

gauge-independent.
• Minimal coupling of charge-q particle to E&M Faraday 2-form
F means: ∇ is locallyq∇ j = ∂ i − i Aj ,ℏ

dA = F ∇ = F.

• Free Hamiltonian is
Hfree =

ℏ2 ∗
∇ ∇.
2m

• On M = R2 and uniformly magnetic eld F = b · dx ∧ dy ,this is called theLandau Hamiltonian. But it is dened formuch more realistic geometries.

11 / 69

Landau operatorse g ), say Euclidean R3 .
• Lab is a Riemannian 3-manifold (M,e• Charge q is conned to 2D orientable submanifold ι : M ,→ M(the sample).

e is set up.
• Magnetic eld Fe ∈ Ω2 (M)• Since velocity vector v ∈ TM , only restricted eldF := ι∗ Feaects tangential motion (Lorentz force).

• Pick an orientation on M , thus volM . ThenF = B · volMformagnetic eld strength B ∈ C ∞ (M).
12 / 69

Landau operators• If M is contractible, B · volM is the curvature of someconnection ∇, unique up to gauge equivalence. Then we havetheLandau operator

HB = ∇∗ ∇.
• Simplest example,
M = R2 ,B ≡ b ∈ R \ { 0}has spectrum (Landau '30),

Spec(Hb ) = (2N + 1)|b|.

• I will explain the dierential geometry behind Landau
quantization.

13 / 69

Dirac operator

On a Riemannian manifold, LeviCivita
connection parallel transports tangent
vectors.

Spin connection ∇Spin on spinor elds (fermions) gives Dirac
operator,D=dXc(ei ) · ∇Spinei(physics : −iγ µ ∇µ ,γ µ = eµi e i )i=1where {ei }i=1,...,d is orthonormal frame and {c(ei ), c(ej )} = −2δij .
14 / 69

Dirac operator• On 2D spin Riemannian manifold M , spinor bundle is,
S = S+ ⊕ S−
where S

± → M is a line bundle with connection ∇Spin,± .

• ± corresponds to
2 :1
U(1) ∼= Spin(2) −→ SO(2)e iθ 7→ e ±iθ/2 .
• Curvature of ∇Spin,± isR∓ volM4

where R ∈ C∞ (M) is Riemannian scalar curvature function.

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Landau quantization and Dirac index• Dirac operator is an odd operator on S = S + ⊕ S − ,


0
D−+
DS =
.
D+−
0
• Couple charged fermion to E&M, twist by LB → M withcurvature B · volM .
±Twisted spin connection on S ⊗ LB has curvature
Spin,±F ∇B= (B ∓ R4 ) · volM .

• Two possible Laplacians on S ⊗ LB .
• Dirac2 .
• ∇SpinLaplacian.
B

16 / 69

Landau quantization and Dirac index• The Laplacian for ∇Spin
B ,Spin ∗(∇B

Spin ∼) ∇B=
HB−R/400

HB+R/4 ,is direct sum of two Landau operators.

• Related to Dirac2 by SchrödingerLichnerowicz identity:


R
−B 0
Spin ∗ Spin
2.
DS⊗LB = (∇B ) ∇B + +0
B4Equivalently,

2
DS⊗L

B+ R
4

=

HB − B

00

HB+ R + R2 + B!
.

217 / 69

Landau quantization and Dirac index• For large uniform magnetic eld strength, B ≡ b > − R2 ,

2
DS⊗L

b+ R

4


=


Hb − b00

Hb+ R +
2

R


+ b
 ≥ 0.
|2 {z }
>0

Only Hb − b can have zero eigenvalues.

• Lowest Landau level is Dirac kernel,
PLLL ≡ ker(Hb − b) = ker D+− ,0 = ker D−+ ,b > 0.

Breaks SUSY. . . but by how much?

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Landau quantization and Dirac index• Original Landau quantization is R ≡ 0 case,
Hb Spectrum :

•b

•3b

•5b

•7b

...

• For almost constant B ≈ b , small R and potentials, getLandau band.

HB Spectrum· ■■ ·· · ■■■■• We need completeness for essential self-adjointness.
Very dierent story when M has a boundary!
• How does Landau quantization of spectrum relate toquantized Hall conductance?!

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Landau quantization and Dirac index• LLL is innitely-degenerate when M is non-compact.
• Its size is the K -theoretic Dirac index,[PLLL ] = Index(D) ∈ K0 (C ∗ (M)),
which is acoarse geometric invariant of M .

• Higher-dimensional Landau quantization exists (L+T'24).
• K -theory index is an abstract abelian group element.
• Numerical macroscopic trace homomorphism
K0 (C ∗ (M)) → R, which is Kubo formula for Hall
conductance from physics.

Values are quantized!

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Landau quantization and Dirac index• By abstract invariances (later), it suces to deform to basic
Euclidean plane case, R ≡ 0, and B = b .
• In complex coordinates,
DS⊗Lb = −2i0∂¯ + b4 z∂ − b4 z̄0
,and lowest Landau level isb2ker(∂¯ + z) = spanC {z m e −b|z| /4 : m ≥ 0}
4

= Bargmann − Fock space.
We will calculate the Hall conductance of this space.

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End of Lecture 1

22 / 69

Last time/today• Last time, geometric mechanism for Landau level phenomenon
was given, and LLL was identied with twisted Dirac kernel.

• They have abstract K-theoretic indices, which I will turn intoquantized numbers today.

• These are renormalized numbers hidden in intricate
traces-of-commutators (e.g. Kubo formula).

• Explicit calculation for Euclidean Landau levels.
• New technique of CareyPincusHeltonHowe theory will be
introduced.

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Commutators[Position, Momentum] = [X , P] = iℏ.
• Can you measure ℏ this way?
⟨ψ| − i[X , P]|ψ⟩ = ℏ ??
• Textbook QM: Density matrix ρ, bounded observable
A = A∗ ,⟨A⟩ρ = Tr(ρA) ∈ R.
Need ρ to be trace-class here.

• Local observables in Dirac sea state?

24 / 69

Trace• For operators on CN ,(P
e-values
Tr = Pdiagonal
Tr(AB) = Tr(BA) ⇒ Tr[A, B] = 0.

• What about(
Tr[X , P] = Tr(iℏ) =0?

∞?

• Correct answer isTr[Xreg , Preg ] =i.
2π
25 / 69

Crash course on trace• On a Hilbert space H , a bounded operator A is trace class if
X⟨ek ||A||ek ⟩ < ∞,{ek }k∈N some O.N.B.

k√A∗ A. ThenX
Tr(A) :=
⟨ek |A|ek ⟩ ∈ C,

where |A| ={ek }k∈N any O.N.B.

k• Set of trace class ops. is denoted L1 (H ).
Finite-rank ⊂ Trace class ⊂ Compact operators.

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Lidskii theorem '59• A deep result:
Tr(A) =

X

e-values(A)Theorem: If AB and BA are trace class, then
Tr[A, B] = Tr(AB) − Tr(BA) = 0.

• Interesting situation is almost commuting pair,[A, B] ∈ L1 but Tr[A, B] ̸= 0.
In such a situation,
0 ̸= Tr[A, B] = “Tr(AB) − Tr(BA)” = “∞ − ∞”.

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Regularized CCR

Consider f: R → [− 12 , 12 ], t 7→ 2√1t+t 2 . The regularized
Xreg = f (X ),Preg = f (P),are almost-commuting, and it can be calculated that

Tr[Xreg , Preg ] =After the lectures, think abouti.
2π

why this is robust!

Xreg , Preg is called an almost-commuting self-adjoint pair.

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CareyPincusHeltonHowe '70s

• Suppose [A, B] ∈ L1 . Then polynomials pi (A, B) also
almost-commute, e.g.,[A, AB] = A[A, B],[A, BA] = [A, B]A∈ L1 .

• No ordering ambiguity in p(A, B) under the trace,
Tr[A, AB] = Tr(A[A, B])

Lidskii=

Tr([A, B]A) = Tr[A, BA].

• Conclude that ∃ bilinear, antisymmetric map(p1 , p2 ) 7→ Tr[p1 (A, B), p2 (A, B)].

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CPHH Theorem

Theorem

Let A, B be self-adjoint a.c. pair. Then there exists principal
function GA,B , such that•Z

{p1 , p2 } GA,B = Tr[p1 (A, B), p2 (A, B)]C

holds for any polynomials pi ≡ pi (x, y ) ≡ pi (x + iy ).
• Whenever λ is not an essential spectral value of A + iB ,2πi · GA,B (λ) = −Index(A + iB − λ) ∈ Z.

Cf. quantization of Poisson brackets,

30 / 69

CPHH Therem• Each a.c. pair A, B gives an exact quantization of phasespace weighted by principal function GA,B .
• GA,B is basically unknown on essential spectrum, so trace
formula had little practical utility.

• New interesting examples come from Kubo formula for
Landau levels!

Reminder: A bounded operator S is Fredholm if
Index(S) := dim ker S − dim coker S < ∞.
Away from essential spectrum, S − λ is Fredholm.

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Preview of QHE application• Kubo formula for Hall conductance has the form
σHall = i Tr[A, B].
• Furthermore, A + iB has essential spectrum being the unit
square.

• Away from unit square,(−1 λ inside square,
Index(A + iB − λ) =0,
λ outside square.
⇒ GA,B =

1
2πi

χ■

• Thus σHall = i Tr[A, B] is quantized.

32 / 69

BargmannFock space index = LLL Hall conductanceWrite P : L (C) → HLLL for the orthogonal projection onto
2

LLL/Fock space,
2

HLLL := spanC {z m e −|z| /2 : m ∈ N} ⊂ L2 (C).

f ∈ L∞ (C) acts on L2 (C) by multiplication operators. Compress it
to the Landau level,

Pf := P f P,
to getToeplitz operator on HLLL with symbol f .

33 / 69

Switch functions• Dene switch functions,(η : R → [0, 1],

η(t) =t ≤ −a,1, t ≥ a,0,

for some interpolation interval [−a, a]. Also

C → [0, 1]
x -switch(x + iy ) 7→ η1 (x)y -switch(x + iy ) 7→ η2 (y )• These are half-plane position observable.

34 / 69

Chiral asymmetry of Landau levels• Unlike L2 (C), subspace HLLL prefers anticlockwise.
• Let f1 , f2 by x -switch and y -switch respectively. When
projected to HLLL ,Pf1 Pf2 −| {z}clockwise

Pf2 Pf1
| {z
}

̸= 0.

anticlockwise• Individually, Pf1 Pf2 and Pf2 Pf1 are not trace class.
Theorem (T; T+Xia '24)

For any x -switch f1 and y -switch f2 (at any angle 0 < θ < π ),Tr[Pf1 , Pf2 ] =

1
2πi

.

(⋆)

35 / 69Proof ofTr[Pf , Pf ] = πi
1

2

1
2

• Rewrite in Kubo form,[Pf1 , Pf2 ] = P [f1 , P], [f2 , P]
• Integral kernel of P is smooth and rapidly decaying,p(z, w ) =

1

π

e−

|z|2 +|w |22

e z w̄ =

1

π

1

2

e − 2 |z−w | e iw ∧z .

So P is• Approx. nite propagation,• Locally trace class.

• Product [f1 , P][f2 , P] is supported near origin,
Supp(f2′ )Supp(f1′ )36 / 69Proof ofTr[Pf , Pf ] = πi
1

2

1
2

• So P[f1 , P][f2 , P] and P[f1 , P][f2 , P] are trace class, and we
can integrate their kernels along diagonal.

• Magnetic translational symmetry + geometry of switches +residue theorem gives exact integral as

1
.
2πi

(arXiv:2401.06660, cf. AvronSeilerSimon '90s)

Relation to CPHH:• Write π : B → B/K.
• Pf1 , Pf2 are self-adjoint a.c., so π(Pf1 + iPf2 ) is normal in B/K.
• π(Pf1 ) and π(Pf2 ) have spectrum inside [0, 1], so
ess-Spec(Pf1 +if2 ) = Spec(π(Pf1 +if2 )) ⊂ ■.

37 / 69

Tr[Pf , Pf ] = πi and CPHH
1

2

1
2

In T+Xia, we constructed Fredholm inverses for Pf1 +if2 − λ when λis in interior of square.

e iθ Re i(π+θ)/2 R

e iθ/2 R
A

R-

This shows ess Spec(Pf1 +if2 ) =.
38 / 69

Fractional quantization• As discussed earlier, principal function is therefore
GPf1 ,Pf2 =

in
χ■ ,
2π

˚
n = Index(Pf1 +if2 − λ), λ ∈ ■.

• CPHH base formula is
2πσHall (P) = −2πi Tr[Pf1 , Pf2 ] = −2πi

|

{z

−1

}

Z

in
= n,
■ 2π

with n = −1 due to calculation (⋆).
• For polynomials pi ∈ Z[x, y ], get rational quantization,
2πi Tr[p1 (Pf1 , Pf2 ), p2 (Pf1 , Pf2 )] ∈ Q.

Roughly: contribution along indirect paths, e.g., 1212 − 2121.
39 / 69

Summary of LLL integer conductance• LLL is a classical function space, whose chiral asymmetry is
detected by pairing its P with switch functions f1 , f2 .
• By abstract functional analytic reasons + plane geometry, thispairing is index-theoretic and must be integral.

• Explicitly computable in LLL case. Works for higher LL(T+Xia)!

• This gives (rst?) explicit calculation that every LL has one
unit of Hall conductance.

• It seems that there are no nite-propagation projections
which can exhibit non-zero Hall conductance. The analysis ismandatory!

40 / 69

Kubo formula vs experiment• Rigorous derivation of Kubo formula for σHall ?
e.g. De RoeckElgartFraas, Inventiones '23.

• Such σHall describes a thought experiment on innite-sized
sample, where we measure transverse current induced byslow introduction of voltage.

• Actual QHE experiments measure an aggregate resistivity on
nite-sized sample with boundary.

• Meaning of spectral gap, mobility gap, etc., is not sharp.
• Basic mystery: why quantization of 1/resistivity is almost
exact in suciently large nite-sized sample.

41 / 69

Topological category?

• The phrase topological phase is arguably a misnomer.
• Microscopically, dynamics ↔ geometry.
• Analysis is crucial  estimating complicated system(nite/with boundary) against clean limiting model (innite).

• Algebraic topology, roughly Top → Algebra has great
functorial properties, but its locality principle (MayerVietoris)
is much too abby.

• Rigidity via imposition of symmetry by hand defeats the
purpose of genericity.

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Coarse perspective of index• Index or zero modes is what's left behind in the low
energy/large-scale limit.

• For bounded M , large-scale limit is a point. Only topological
data remains, and determine the index = # zero-modes.
• For unbounded M , the idea is to label dierent types of∞-degenerate zero mode spaces by its homotopy class in asuitable operator algebra.

• Roe achieved such an index theory of Dirac operators, at thelevel of K -theory of Roe-algebras.

43 / 69

End of Lecture 2.

44 / 69

Recap• Kubo formula has Tr[Pf1 .Pf2 ] form, makes sense for general
spectral projections P , and manifolds M . Trace converges
under mild conditions given later.

• Quantized Kubo formula is explicitly computed in basic LLL
case, and proved in limited settings using topology.

• Real life quantization persists far beyond topology setting.
• Quantization has functional analytic origin, valid once very
mild geometric conditions are satised (given later). Can
deform from LLL case to these.

• Finite, but macroscopic, aspect becomes apparent.

45 / 69

Coarse geometry• Let M be a Riemannian manifold, or some good
discretization of it.

• On Hilbert space L2 (M), a subset A ⊂ M is regraded as
multiplication operator by χA . For example,
AAc = A ∩ Ac = 0.
• Roe algebra Bn (M) comprises operators L satisfying,
1 Locally trace class,
KL, LK ∈ L1 when K ⊂ M compact.
2 Finite propagation,∃R , such that KLK ′ = 0 whend(K , K ′ ) > R .

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2000s: Coarse techniques were anticipated

47 / 69

Roe algebras• Bn (M) is too naïve  does not even contain the mostimportant example, e.g., LLL projection.

• To x this, one usually takes the norm completion, C ∗ (M).
• Convenient for K -theory calculations.
• But Kubo formula typically diverges!
• We constructed a Fréchet Roe algebra B(M):• Contains all reasonable low energy spectral projections ofsensible gapped Hamiltonians on M ;• Kubo formula always converges, homotopy invariant, always
quantized.

48 / 69

Measuring Roe algebra invariantsIdea: Non-trivial Roe algebra projections P are necessarily
3

delocalized .
We detect delocalization by partitioning M and correlations of P
along various paths connecting the partitioning sets.

The Kubo formula σHall (P) = −i Tr[Pf1 , Pf2 ] is precisely such a
projection-partition pairing.

Precise choice of partition/knowledge of P is unimportant, due toautomatic cancellation of small-scale contributions.

3

Physicists: No localized Wannier representation

49 / 69

Coarse partitions

In 2000, Kitaev proposedreal-space Chern
number.
Recently, amorphous
practitioners use this.

The idea is to consider hopping amplitudes along triangles, with
one vertex in each partitioning set. Sum over all clockwise
triangles minus anticlockwise triangles.

Miracle: Something can survive (but only on unbounded sample!?)50 / 69

Coarse partitions

B

A

B

C

A

B

C

Coarse, non-trivial

Coarse but trivial

AC

Not coarse.

Write Br (A) for set of points within distance r of A.
A partition M =F

i Ai is coarse (transverse) if\

Br (Ai ) bounded ∀r > 0.

i

51 / 69

Coarse partitions pair with Roe algebra operators• If Li ∈ Bn (M), thenA0 L0 A1 L1 . . . Aq Lq
is supported within some coarsened intersection

Ti Br (Ai ),which is a compact set K .

• Since Lq locally trace class,A0 L0 A1 L1 . . . Aq Lq = A0 L0 A1 L1 . . . Aq Lq K ∈ L1 .
|{z}
t .c .

• In particular, for projection P = P 2 ∈ Bn (M), dene
Tr[A0 , . . . , Aq ]P :=Xsgn(σ)·Tr PAσ(0) PAσ(1) . . . PAσ(q) P .

σ∈Sq+152 / 69

Coarse partitions pair with Roe algebra projections

Tr[A0 , . . . , Aq ]P :=Xsgn(σ) · Tr PAσ(0) PAσ(1) . . . PAσ(q)σ∈Sq+1
Trace of totally antisymmetrized product of Toeplitz operators.

• Cobordism invariance:
B

A

B

C

A
C

=

53 / 69

Coarse partitions pair with Roe algebra projections• Boundary prole invariance: Can replace Ai = χAi by
partition of unity,XÃi = 1,iwith Supp(Ai ) coarsely transverse.

• Using these invariances, we can show that
2Tr[A, B, C ]P = Tr[Pf1 , Pf2 ].

BA← f2 →↑
f1↓

C

54 / 69

Kubo formula is a coarse projection-partition pairing• Mathematically, Kubo formula for σHall (P) is a pairing
between P and the coarse partition {A, B, C }.
• The pairing has invariances, meaning that it descends to abilinear pairing

K0 (Bn (M)) × HX 2 (M) → C[P], [{A, B, C }] 7→ Tr[A, B, C ]P .
• Important technical lie: Any realistic P does not have nitepropagation!

• Analysis (trace estimates) needed to make above work, and tond any interesting examples.

55 / 69

Universal quantization• Functional analytic way to see that pairing lands inside 21πi Z.
→ Universal quantization!
• The range is possibly 0, so we must calculate one non-trivial
example.

• For M = R2 , the K0 (C ∗ (R2 )) ∼= Z, generated by Diracindex. LLL realizes this concretely.

• Calculation is robust against deformations of M preserving thecoarse geometry, perturbations of Hamiltonian.

“topological, but not Top′′56 / 69

General quantization theorem

We outlined a principal function approach in last lecture. Today,
I'll proceed more coarse geometrically.

• Operators A with A − 1 ∈ L1 have a Fredholm determinant.
• Intuitively, the innite product of eigenvalues converges, sincemost of them are 1.

• If U − 1 ∈ L1 and V − 1 ∈ L1 individually, thendet(UVU −1 V −1 ) = det(U) det(V ) det(U)−1 det(V )−1 = 1.
• Generally, determinant of UVU −1 V −1 need not vanish: If[C , D] ∈ L1 , thendet(e C e D e −C e −D ) = exp(Tr[C , D]).

(Pincus + HH)

57 / 69

General quantization theorem• Kitaev '00 conjecture: For unitaries U, V ,(U − 1)(V − 1) ∈ L1 ⇒ det(UVU −1 V −1 ) = 1.
This was proved by ElgartFraas '23.

• The motivation is that if C , D are self-adjoint a.c., and weknow that(e 2πiC − 1)(e 2πiD − 1) ∈ L1 ,

(1)

then
1 = det

e 2πiC e 2πiD e −2πiC e −2πiD= exp(Tr[2πiC , 2πiD]) ⇒ 2πi Tr[C , D] ∈ Z.

58 / 69

General quantization theorem• Condition (1) is hard to use in practice.
• We argued that if(C − C 2 )(D − D 2 ) ∈ L1 ,then Condition (1) holds, so Tr[C , D] ∈

1
2πi

(2)

Z.

• Proof: The holomorphic function φ : z 7→ e 2πiz − 1 has zeros
at z = 0, 1, so
φ(z) = ψ(z)(z − z 2 ).
(e 2πiC − 1)(e 2πiD − 1) ≡ φ(C )φ(D)= ψ(C ) (C − C 2 )(D − D 2 ) ψ(D) ∈ L1 .
|{z}t.c. by assumption

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General quantization theorem• Our Condition (2) holds by physics:• For (magnetic) Schrödinger operators H = H ∗ with spectral
gap, Fermi projection P lies in B(M). Furthermore,(Pf1 − Pf21 )(Pf2 − Pf22 )algebra=Pf1 P(1 − f1 )Pf2 P(1 − f2 )Pis supported near the compact set

Supp(f1 ) ∩ Supp(1 − f1 ) ∩ Supp(f2 ) ∩ Supp(1 − f2 ),so we deduce it's trace class.

• Thus σHall (P) = −i Tr[Pf1 , Pf2 ] ∈ 21π Z.

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Exact quantization• Hall conductance is quantized to 21π Z, without need fortranslation symmetry, homogeneity, Euclidean geometry, etc.

• Identication as a coarse geometry pairing shows its
invariances against small-scale perturbations/imperfections• Flatness of sample• Uniformity of magnetic eld
• Holes in sample• Geometric assumptions in deriving Kubo formula• Gap-closing is of course necessary for transitions, but whatkind of gap closing?

• Large-scale geometric changes drive the transitions.

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Finite-sized sample and approximate quantization• In real world, M has nite size, so H has discrete spectrum,and P is nite-rank. Then [Pf1 , Pf2 ] is traceless, so
σHall (P) = 0 . . .??
• There is no paradox: non-trivial exact quantization holds inthought experiment with innite-sized sample.

• Hall conductivity is a bulk property.
• Let r be the (approximate) propagation of P . Choose a bulk
subset K ⊂ M which, for some R ≫ r ,• Contains the R -thickened intersection of the partition• Stays at least distance R from the sample boundary ∂M62 / 69

Finite-sized sample and approximate quantization

AR ∩ BR ∩ CR

K = bulk• The bulk contributes
σHall,K (P) = −2i Tr KAKPKBKPKCKPK + antisymm

Dependence on cut-o R ≫ r is small, because P has rapid
decay.
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Finite-sized sample and approximate quantization• Now σHall,K (P) need not vanish, and is not exactly quantized,
because KPK is not a projection.
b . True Hamiltonian• Sample M is embedded in innite-sized M
b , with some
H on M is a restriction of the ctitious Hb on M
boundary conditions imposed.

• For the K -truncated Fermi projections, we haveb .
KPK ≈ K PK
• We get approximate quantization,b ≈ σHall (P)b ∈σHall,K (P) ≈ σHall,K (P)

1
2π

Z.

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Finite-sized sample and approximate quantizationb approximates P ,Precision of quantization depends on how well Pb (spectral gap size), volume growth rate of M
b. . .
decay rate of P
Numerical convergence [Mitchell et al 2018].
Basic requirement is R ≫ r ∼ B−1/2 , i.e. macroscopic sample!

Note: The boundary part, K c PK c is not approximated byb c , it must be removed!
K c PK

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Quantum Hall experiment• Material has electron density µ, non-interacting (∼ 0K ).
• Each Landau level has eb
h electrons/unit area.
µh
1• Filling fraction ν = e b is varied by controlling b .

−1• At large b , it was found that Hall resistivity ρHall = σHall
isρHall (ν) =h 1,
e 2 Int(ν)∀ν ≈ integer.
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Quantum Hall experiment

2• So σHall ≈ eh Int(ν) when ν is near an integer.
• Geometric imperfections broaden the Landau levels, and allow
ν to vary continuously.
■■■ · · · ■■■ · · · ■■■ · ··• ν ≈ integer corresponds to gaps between idealized Landaulevels; states are localized (maybe Anderson). Varying ν here
doesn't change integral conductance.

• ν ≈ half-integer corresponds to core of Landau band; statesare very delocalized. Interpolation of integral conductanceoccurs here.

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Macroscopic Planck constant• Magnetic length scale is
26nm/p#Tesla ∼ 10nm.

• Size of M ≫ 10nm, so quantization would be very close toexact. Experimentally, 10−10 error.

2• QHE gives access to eh .
• Josephson junction (macroscopic) gives 2he .
• Ironically, macroscopic measurements give best access to h.

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Quantum kilograms(econd) = 9192631770 ticks of Cs-133 atomic clock(speed of light) × s
m(etre) =299792458• Prior to 2019, kilogram was locally dened,
Old kg = prototype in Paris.

• For > 100 years, we've known about Planck's constant, andcould dene quantum kilogram byh =: |6.626{z. . .??} ×10−34 (kgh ) · m2 s −1 .
x convention• After QHE,
newh =: 6.62607015 × 10−34 · (kgh ) · m2 s −1 ,69 / 69