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Transcript
12
The Integer Quantum Hall E↵ect
The Hall e↵ect has played already an interesting role in classical electrodynamics, since it provides a unique tool for determining the sign of the current
carriers. This task is challenging since the electric current produced by particles of charge q moving at velocity v, is the same as that produced by particles of the opposite charge ( q) moving at the opposite direction (at the same
speed and density of the moving charges). The two cases can however be distinguished by the sign of the Hall resistance which is measured in the presence
of the combination of crossed electric (E) and magnetic (B) fields.
The Lorentz force on moving charges F = qv ⇥ B deflects the current in a
transverse direction, e.g. towards a lateral edge of the conductor in which the
current flows. The charge accumulation there results in a transversal voltage
di↵erence between the lateral edges of the conductor, which is named VHall
after the discoverer of this e↵ect. In the steady state the corresponding electric
field counters the transversal drift, that is:
q v ⇥ B + q EHall = 0 .
(12.1)
Since from macroscopic-scale measurements one can determine the direction
of qv as well as of EHall , the above relation allows to deduce the sign of the
current carriers’ charge q.
The Hall, i.e., transversal resistance is:
⇢H =
VHall
Ichannel
(12.2)
According to the classical explanation of electrical conduction, at given density
of the current carriers the Hall resistance would be proportional to the applied
magnetic field (as is implied by (12.2) and the Drude picture of electrical conductance).
141
142
The Integer Quantum Hall E↵ect
In their 1975 paper on a quantum calculation of the Hall conductance, T. Ando,
Y. Matsumoto and Y. Uemura [21] noted (italics added here):
• “[The Green function G N or XN is determined by the self-consistency equa(1)
tion (2.8) of II.] From the above equation, one can conclude that
XY van2
ishes and XY becomes nec/H = e (N + 1)/2⇡~, when the Fermi level
lies in energy gaps between adjacent N-th and N + 1-th Landau levels at
zero temperature. ...”
Figure 12.1 A sketch of the Hall resistance (⇢ x,y ), plotted along with the direct
resistance (⇢ x,x ), for two dimensional samples at low temperature. The Hall conductance exhibits plateaux of integer multiples of e2 /h as a function of the perpendicular magnetic field. The direct conductivity ( xx = ⇢ x,x /(⇢2x,x +⇢2x,y )), which
is shown here in unrelated units, vanishes over the plateaux.
The significance of this point was realized in the 1980 experimental work K.
von Klitzing who, working with samples provided by G. Dorda and M. Pepper,
turned the quantization of the Hall conductance (within its plateaux) into a tool
for a very precise determination of e2 /h. This constant’s value was reportedly
determined by such means up to the relative precision of 10 9 . Within five
years the work was awarded the Nobel prize. More surprises were in the store,
in particular the discovery of the Fractional Quantum Hall E↵ect [242], i.e.
observation of plateaux with Hall conductance at fractional multiples of e2 /h.
We shall however not get to this subject here.
The successful observation of IQHE required to work at low temperatures
12.1 Laughlin’s charge pump
143
and high magnetic fields but not necessarily with ‘clean’ samples.1 In fact, in
was noted that the e↵ect is improved by disorder (in the form of impurities in
the sample).
Soon after the experiments have focused attention of the IQHE phenomenon
theorists started to grasp the beautiful mathematical structure behind it. In fact,
the quantization has topological aspect. Its robust explanation combines functional analysis, topology (in the space of operators), and probability. In particular, it was understood that in the absence of direct conductance the quantum
Hall conductance of two dimensional systems can be expressed both as a Chern
number and as a Fredholm index. The precision of integers is protected by their
topological nature! The e↵ect provides also an example of the interesting phenomenon that adiabatic transport [21] tends to be quantized in situations which
are enabled by the vanishing of the direct conductivity.
12.1 Laughlin’s charge pump
There have by now been a number of di↵erent derivations of the IQHE. Let us
present here a slightly formal one, which fits neatly with the general setup of
random operators discussed in these lecture notes. The argument starts from
Laughlin’s observation that Hall conductance can be probed in a charge pump
Gedankenexperiment [172] which is outlined in Figure 12.1.
Induced EMF
⊗
Hall current
B
Increasing
magnetic flux
Φ
(up to sign)
wire leads
Figure 12.2 Laughlin’s charge pump, based on the Hall e↵ect (drawing based
on [27]): A conducting sheet is wrapped into a cylinder bounded by a pair of
rings connected to lead wires. In the presence of non vanishing transversal magnetic field B, increase in the enclosed magnetic flux results in charge transport
between the rings which form the cylinder boundary.
Deforming this picture, the cylinder can be stretched and flattened to form
a ’Corbino disk’ whose boundaries consist of a pair of concentric rings. Thus,
1
The temperature range has since been brought up, in particular working with graphene - the
subject of another Nobel prize.
Random Operators c M. Aizenman & S. Warzel
DRAFT
144
The Integer Quantum Hall E↵ect
one considers a system in which the charges are confined to a plane and the
magnetic field is changed through an adiabatic process which results in an
increase of the flux through the smaller ring. As described by the Faraday law,
responding to changes in the magnetic flux is a time-dependent electric field E
whose integral along a loop C is:
I
d C
E · dl =
.
(12.3)
dt
C
where
C is the change in the flux enclosed by C. By Lenz’ law, the current
which would result from this field would tend to diminish the rate of change of
the enclosed flux. However, we are now interested in the transversal current.
The latter may be present if in addition to the above electromotive force there
is also a transversal magnetic field throughout the lattice (B , 0).
The induced current density is
!
D
H
j=
E.
(12.4)
H
D
where D and H (the direct and the Hall conductance) are elements of the
bulk homogenized conductivity tensor within the plane. Correspondingly, the
rate of the induced charge transport across the contour C is
I
dQ
=
j · n d`
dt
C
I
I
= D E · n d` + H
E · dl
(12.5)
The last integral on the right-hand side of (12.5) is the induced electromotive force and tied by Lenz’s law (12.3) to the flux change rate. The first term
vanishes in situations in which the direct conductivity D is zero. To keep this
condition, the process will be carried out adiabatically. Under this condition,
integrating (12.5) and (12.3) over time one obtains an expression for the Hall
conductance as the ratio of the transported charge to the flux change:
H
=
Q
B
.
(12.6)
12.2 The magnetic Hamiltonian
Let us recall from (3.7) that a natural way to incorporate the magnetic field into
the Hamiltonian is to replace the regular Laplacian by its magnetic version, i.e.
12.3 Charge transport as an index
145
consider
(A)
H =
with
(
(A)
)(x) =
X ⇣
+ V(!)
e
iqA(x,y)
(y)
|x y|=1
(12.7)
⌘
(x)
(12.8)
where A(x, y) = A(y, x) |x y|,1 corresponds to the vector potential integrated
over the edge from x to y. The magnetic flux through the lattice plaquettes
(P, which carry orientation) is given by the discrete version of the relation
B = Curl A, i.e.
X
BP =
A(b)
(12.9)
b2@P
where b is summed over the oriented edges which form the boundary of P.
Of relevance to the discussion which follows is the observation that the insertion of a unit flux (measured in units h/q) at a point a 2 R2 \Z2 in the plane
corresponds to a gauge transformation which is given by the unitary mapping:
(Ua ) (x) := e
i✓a (x)
(x) ,
where ✓a (x) := Arg(x a) 2 ( ⇡, ⇡] is the argument of the vector x
the latter being identified with C.
(12.10)
a 2 R2 ,
12.3 Charge transport as an index
ΔΦ
ΔQ
Figure 12.3 The Avron-Seiler-Simon Hilbert hotel based on the Hall e↵ect
Y. Avron, R. Seiler and B. Simon [26] stretched the Corbino disk picture
even further, and considered the setup in which the flux is inserted through
one of the plaquettes of a regular lattice and the outer ring is taken out to
Random Operators c M. Aizenman & S. Warzel
DRAFT
146
The Integer Quantum Hall E↵ect
infinity (at least on the lattice scale). Furthermore, they in e↵ect propose that
the lattice itself can serve as a charge reservoir and instead of looking at the
charge flow through the system one could try to make sense of the change in the
total charge within the infinite lattice due to a finite increase in the magnetic
flux. We now turn to a more explicit review of this approach to the IQHE,
assuming now that the system’s valance electrons (which have charge q) can
be treated as a Fermi gas at zero temperature.
Under the assumption that the Fermi energy falls within a spectral gap, or,
as emphasized by J. Bellisard [32], within a ’mobility gap’, i.e. a regime of
Anderson localization, one may expect that under an adiabatic flux insertion,
the system’s state would evolve from the Fermi projection
PEF = 1[H  E F ]
(12.11)
to the corresponding projection for the modified Hamiltonian. The final state
is particularly easy to write explicitly if the flux increment is h/e (or an integer
multiple of it) since in that case the final state is given by Ua PE F Ua† with Ua
the gauge transformation (12.10). Thus, it is tempting to argue that
⇣
⌘
Q = q tr Ua PE F Ua† PE F ,
(12.12)
to the extent that the di↵erence makes sense.
The latter question is not quite trivial: i) if the di↵erence is calculated by
evaluating the traces separately it becomes (1 1) which is ill defined, and
ii) had the di↵erence of the two projection operators been trace class the difference trace would vanish since locally the densities are equal:
, Ua PE F Ua† x i .
(12.13)
⌘
As it turns out, the di↵erence Ua PE F Ua† PE F is not trace class, so the
above trivial answer does not apply. However, if E F is in the localization
regime the di↵erence is a compact operator, and thus instead of (12.12) one
may take the index which is defined as follows.
h
x
, PE F
xi
⇣
= h
x
Definition 12.1 The index of two orthogonal projections P, Q on a Hilbert
space H whose di↵erence P Q is a compact operator is
Index(P, Q) := dim ker(P
Q
1)
dim ker(Q
P
1) .
(12.14)
It may be added that the dimensions are well defined since the assumed
compactness of P Q ensures that dim ker(P Q ± 1) < 1.
In the special case where P and Q are trace class the index is given by (12.12),
that is, it equals the di↵erence of the dimensions of PH and QH. A more generally useful formula is (12.17), which is explained within the proof of Proposition 12.2.
12.4 A calculable expression for the index
147
In this terminology, the tentative proposal (12.12) is revised to:
⇣
⌘
Q = q Index Ua PE F Ua† , PE F ,
(12.15)
When defined, this index takes integer values, and through (12.6) it yields H 2
(q2 /h) Z. Furthermore, the value which it yields for the Hall conductance H
coincides with that provided by the Kubo formula, as was shown in [26, 32],
and is explained below following [8].
12.4 A calculable expression for the index
It may be worth stressing that the notion of index comes with certain subtleties
which are actually of relevance in the present context:
1. When defined, the index takes only integer values, and it is robust in its independence of the flux insertion point, and by implication also independent
of the disorder (in the ’almost sure’ sense).
2. However, the relative index of two projections Index(P, Q) is defined only
under some strict
In
⇣ conditions.
⌘ the case of interest here, a necessary condi†
tion for Index Ua PE F Ua , PE F to make sense is that the Fermi projection P
has rapidly decaying matrix elements. For that, localization plays an essential enabling role. This caveat well reflects also the observed physics, with
the Hall conductance changing over regimes where it is not quantized.
3. Formal manipulations, in which questions of convergence are ignored, could
be misleading. Splitting traces or invoking cyclicity of the trace without
paying attention to trace-class restrictions (as is allowed in finite dimensions) may easily read to contradictions, or to the wrong impression that
the index is always zero.
From the index definition one may conclude the following elementary properties:
Index(P, Q) =
Index(Q, P) =
Index(P? , Q? )
= Index UPU 1 , UQU
1
,
(12.16)
where P? := 1 P and Q? := 1 Q stand for the projections onto the orthogonal
complement and U : H ! H is any invertible linear map.
In computing the index we shall rely on:
Proposition 12.2
Let P, Q be a orthogonal projects on a Hilbert space H.
Random Operators c M. Aizenman & S. Warzel
DRAFT
148
The Integer Quantum Hall E↵ect
Then for any n with which (P
Q)2n+1 is trace class one has
Index(P, Q) = tr (P
Q)2n+1 .
(12.17)
Proof The strange independence from n of the the value in (12.17) is explained by the observation that the spectrum of P Q is antisymmetric, counting multiplicity, except possibly for the values 1 and ( 1). This is established
in [26] by considering the pair of anticommuting operators C := P Q and
S := P Q? , which are easily seen to satisfy:
S 2 + C2 = 1
and
S C + CS = 0 .
(12.18)
For any eigenvalue 2 (C)\{0} and a corresponding (normalized) eigenfunction ' 2 H one has:
CS ' = S C' =
S',
(12.19)
i.e., unless S ' = 0, the functions S ' is an eigenfunction corresponding to the
reflected eigenvalue. In the exceptional case S ' = 0 implies 2 ' = C 2 ' =
' S 2 ' = ' and hence 2 {±1}.
Since with the exception of the eigenvalues ±1 the spectrum of C (counted
with its multiplicity) is invariant under reflections, at any odd power at which
the eigenvalues are summable one gets cancellations, and the contribution left
standing refers to just the term which are counted in the definition of the index.
Thus:
X
2n+1
tr C 2n+1 =
dim ker(C
) = dim ker(C 1) dim ker(C + 1)
2 (C)
= Index(P, Q) .
(12.20)
⇤
Guided by the criterion provided⇣ by Proposition ⌘12.2, one would want to
know under what conditions is T = Ua PE F Ua† PE F a compact operator, and
more precisely under what conditions and for what powers p > 0 is
kT k p := (tr |T | p )1/p < 1
?
(12.21)
In this case T is said to be in the Schatten-p class.
The following lemma spells a sufficient condition for that.
Lemma 12.3 Let T : `2 (Zd ) ! `2 (Zd ) be a bounded linear operator with
kernel T (x, y) = h x , T y i. Then for any p 2 [1, 1):
X⇣X
⌘ 1p
kT k p 
(12.22)
|T (x, y)| p .
y2Zd
x2Zd
12.4 A calculable expression for the index
149
P
Proof We rewrite T = ↵2Zd T ↵ in terms of the operators defined by: T ↵ (x, y) :=
T (x, y) x ↵,y . The triangle inequality yields
X
X
kT k p 
kT ↵ k p =
kT ↵† T ↵ k1/2
p/2
↵2Zd
=
X⇣X
↵2Zd
x2Zd
↵2Zd
|T (x + ↵, x)| p
⌘ 1p
(12.23)
.
⇤
To answer the question posed in (12.21) let us note that
⇣
h x , Ua PE F Ua† y i PE F (x, y) = h x , PE F y i ei[✓a (y) ✓a (x)]
⌘
1 .
(12.24)
Assuming the o↵-diagonal matrix elements h x , PE F y i decay rapidly, but
terms with bounded distance |x y| do not, the factor involving |ei[✓a (y) ✓a (x)]
1| = 2| sin( 12 \(x, a, y))| yields decay at the rate of O(1/|x|). Thus, one is led
to expect that in the regime of exponential localization the condition (12.21)
would be met in the two dimensional system considered here for all k > 2. In
⇣
⌘3
particular, the index should be computable through tr Ua PE F Ua† PE F .
In addition to the above estimates we shall find useful also the following
property of the unitary gauge transformations Ua .
Lemma 12.4 The unitary transformations Ua : `2 (Z2 ) ! `2 (Z2 ), defined by
the gauge transformation (Ua )(x) = e i✓a (x) (x), with a 2 R2 \Z2 , satisfy:
1. The operator di↵erences Ub Ua are in the Schatten-3 class, and furthermore for all a, b 2 R2 \Z2 with |a b|  1:
kUb
Ua k3 := (tr |Ub
Ua |3 )1/3  C |a
b|1/3 ,
(12.25)
with a uniform constant C < 1.
2. The unitary operators Ua transform covariantly, U x(A) † Ua U x(A) = Ua x ,
under the magnetic translations U x(A) , x 2 Z2 that are given by U x(A) (y) =
e iBq(y1 x2 x1 y2 )/2 (y x).
Proof For a proof of the first assertion we note that Ub Ua is diagonal in
the basis of position eigenfunctions x , and thus
X
3
kUb Ua k33 =
e i✓a (x) e i✓b (x) .
(12.26)
x2Z2
By elementary geometric considerations:
(
)
|b a|
|ei✓a (x) ei✓b (x) |  min 2, p
,
|x a||x b|
Random Operators c M. Aizenman & S. Warzel
(12.27)
DRAFT
150
The Integer Quantum Hall E↵ect
for all a, b 2 R2 and x 2 Z2 . Therefore, for min{|x a|, |x b|} |b a|/2
the summand in (12.26) is of the order O(1/|x|3 ), which is summable in d = 2
dimensions. The claimed bound (12.25) follows by summation over unit increments in |b a|.
The second assertion is a direct consequence of the way the magnetic translations are set up.
⇤
12.5 Evaluating the charge transport index under the
mobility gap assumption
The above preparatory statements enable the following general result which
allows to conclude quantization of the quantum Hall conductance at T = 0 and
Fermi energy E F within the regime of Anderson localization.
Theorem 12.5 (Existence of charge transport index) Let P 2 Kmc be a covariant family of orthogonal projections on `2 (Z2 ) and assume that its kernel
P(x, y) = h x , P y i satisfies:
X
x2Z2
⇣ h
i⌘1/3
|x| E |P(0, x)|3
< 1.
(12.28)
Then there is a set ⌦0 of full measure P(⌦0 ) = 1 such that for all ! 2 ⌦0 :
1. P(!) Ua P(!)Ua† is Schatten-3 class for all a 2 R2 \Z2 , so that the charge
transport index
Ca (P)(!) = Index(Ua P(!)Ua† , P(!))
(12.29)
is well defined and takes integer values,
2. the value of Ca (P)(!) does not change with a 2 R2 \Z2 , and
3. it is almost surely equal to its mean value.
Proof 1. For n 2 Z2 let Bn = {a 2 R2 : ka
nk1 < 1}. Lemma 12.3 allows to
12.5 Evaluating the charge transport index under the mobility gap assumption
151
deduce the uniform estimate:
2
3
66
77
E 664 sup kP Ua PUa† k3 775
a2B0 \Z2


=
X h
⇣X
⇣
E sup
P(x + y, x) 1
y2Z2
a2B0 \Z2
X⇣X
y2Z2
x2Z2
ei(✓a (x)
✓a (x+y))
x2Z2
h
i
E |P(x, x + y)|3 sup e
i✓a (x+y)
e
a2B0 \Z2
X h
i1 ⇣ X
E |P(0, y)|3 3
sup e
y2Z2
x2Z2
i✓a (x+y)
a2B0 \Z2
⌘ 3 ⌘ 13 i
i✓a (x)) 3
i✓a (x)) 3
e
⌘ 13
⌘ 13
(12.30)
< 1
where the second step is by Jensen’s inequality. By an estimate similar to (12.27)
the last term in the right side is shown to be bounded by C(1 + |y|). This proves
the finiteness of the right side.
As a consequence, there is a full measure set of ! for which P(!) Ua P(!)Ua†
is Schatten-3 class for all a 2 B0 \Z2 . Let ⌦0 be the countable intersection of
lattice shifts of this set. Then ⌦0 is still of full measure and for all ! 2 ⌦0 , the
operator di↵erence P(!) Ua P(!)Ua† is Schatten-3 class for all a 2 R2 \Z2 . On
this set, Index(Ua P(!)Ua† , P(!)) is therefore well defined and integer valued
for all a 2 R2 \Z2 .
2. The independence of Ca (P)(!) from a is implied by its quantization and
continuity in a over the connected domain R2 \Z2 . The continuity relies on the
continuity of the Schatten-3 norm k·k3 = (tr |·|3 )1/3 of the gauge transformations
which yields
Ub P(!)Ub†
d(a, b; !) := Ua P(!)Ua†
 (Ua
 2 kUa
Ub )P(!)Ua† 3
3
+ Ub P(!)(Ua†
Ub k3  C |a
Ub† )
b|1/3 .
3
(12.31)
Here the last inequality is by Lemma 12.4 and holds for |a b|  1 and almost
all !. The proof of the continuity of the index is completed by the estimate
tr(P(!)
+ 3 kP
Ua P(!)Ua† )3
Ua P(!)Ua† k3
tr(P(!)
2
Ub P(!)Ub† )3  d(a, b; !)3
d(a, b; !) + 3 kP(!)
Ua P(!)Ua† k23
(12.32)
d(a, b; !) ,
which is a consequence of an elementary algebraic identity for the di↵erence
of (P Ua PUa† )3 (P Ub PUb† )3 and Hölder’s inequality for Schatten norms.
3. The invariance of Ca (P) under lattice shifts of a, and the covariance of the
Schrödinger operators under magnetic shifts, implies that as a function of !
Ca (P) is invariant under the ergodic action of this group. It is therefore almost
surely constant, and hence also equal to its mean value.
⇤
Random Operators c M. Aizenman & S. Warzel
DRAFT
152
The Integer Quantum Hall E↵ect
12.6 Correspondence with and quantization of the
Streda-Kubo Hall conductance
The next result will show that under the localization assumption (12.28) on the
Fermi projection PE F , the explicit formula for the charge-transport index,
C(PE F ) = Index(Ua PE F Ua† , PE F )
(12.33)
with a 2 (Z2 )⇤ an arbitrary point on the dual lattice, agrees also with the
Streda-Kubo formula (11.21) for Hall conductance (at zero temperature). That
is somewhat reassuring since the justification of the above approach did include a certain creative ansatz. The relation can also be used to explain the
Hall conductance plateaux.
The following statement, in a slightly di↵erent form, dates back to [32]. The
present version is based on [8]. Use is make here also of the continuity of the
integrated density of states, which is known for all ergodic random operators
of the subsequent form [70].
Theorem 12.6 (Quantization and plateaux of Hall conductance) Let H(!) =
(A)
+ V(!) be a random operator on `2 (Z2 ) with a constant perpendicular
magnetic field B , 0 and a bounded ergodic random potential V. Then over
any value of the Fermi energy E F at which
⇠(E F ) :=
X
x2Z2

|x| E h
0
, PE F (H)
xi
3 1/3
< 1
(12.34)
the zero-temperature Streda-Kubo Hall conductance 1,2 (E F ) defined by (11.21)
coincides with the charge transport index described by Theorem 12.5 times
q2 /2⇡ ⌘ q2 /h, i.e. almost surely:
1,2 (E F )
=
q2
C(PE F ) .
h
(12.35)
Furthermore, the value of 1,2 (E F ), which in this case is an integer multiple of
e2 /h, stays constant as E F ranges over any interval I in which (12.34) holds
with supE2I ⇠(E) < 1.
Proof 1. In the setting of Theorem 12.5 with P = PE F one gets for the charge
12.6 Correspondence with and quantization of the Streda-Kubo Hall conductance
153
transport index:
 ⇣
⌘3
C(P) := E tr Ua PUa† P
X
=
E [P(x, u) P(u, v) P(v, x)]
x,u,v2Z2
⇣
⌘⇣
⇥ ei(✓a (u) ✓a (x)) 1 ei(✓a (v)
X
= 2i
E [P(x, u) P(u, v) P(v, x)]
x,u,v2Z2
=
2i
X
✓a (u))
⌘⇣
1 ei(✓a (x)
✓a (v))
⌘
1
⇥ sin \(u, a, x) + sin \(v, a, u) + sin \(x, a, v)
E [P(0, u) P(u, v) P(v, 0)]
u,v2Z2
a2(Z2 )⇤
⇥ sin \(u, a, 0) + sin \(v, a, u) + sin \(0, a, v) .
(12.36)
Here \(u, a, x) = ✓a (x) ✓a (u) is the angle of the segment enclosed by the halflines anchoring in a and passing through u and x respectively, cf. Figure ??. In
the last line, we have used the homogeneity which allows us to shift x to the
origin and sum over a 2 (Z2 )⇤ instead.
In the last expression the sum over a 2 Z2 can be carried out with the help
of a striking formula of A. Connes [60] which is presented in Proposition 12.7
below. Applying it (with g(↵) = sin ↵) one has:
X
a2(Z2 )⇤
sin \(u, a, 0) + sin \(v, a, u) + sin \(0, a, v) = ⇡ u ^ v = ⇡(u2 v1
u1 v2 ) .
(12.37)
Therefore the charge transport index is given by
C(P) =
2⇡i
X
u,v2Z2
=
=
E [P(0, u) P(u, v) P(v, 0)] u ^ v
(12.38)
2⇡i E [h 0 , PX2 PX1 P 0 i
h 0 , PX1 PX2 P 0 i]
2⇡
2⇡i E [h 0 , P [[X2 , P] , [X1 , P]] P 0 i] = 2 1,2 ,
q
where the last expression relates directly to the transversal conductance as
given the Streda-Kubo formula (11.21).
2. The constancy of the Hall conductance over I under the uniformity assumption, follows from its quantization and continuity. To prove the latter,
let E
E 0 in I. Then applying the representation (12.38), an elementary
telescopic decomposition, translation invariance and Hölder’s inequality with
Random Operators c M. Aizenman & S. Warzel
DRAFT
154
2q
The Integer Quantum Hall E↵ect
1
+r
1
= 1, one gets:
1,2 (E)
 6
1,2 (E
X
u,v2Z2
0
)
h
E P# (0, u)
q i1/q
h
q i1/q
E P# (u, v)
E [| P(v, 0)|]1/r |u| |u
0
12
BBB X
h
i1/q CCC
q
CCC E [| P(0, 0)|]1/r
 6 BBB@
|u| E P# (0, u)
A
v|
(12.39)
u2Z 2
where P# is either P( 1,E] or P( 1,E 0 ] and P := P( 1,E] P( 1,E 0 ] = P(E 0 ,E] .
For the last inequality also use was made of the Cauchy-Schwarz inequality
in Hilbert space, | P(v, 0)|2  | P(0, 0)| | P(v, v)|, followed by the CauchySchwarz inequality for the expected value and translation invariance.
By the continuity of the integrated density of states [70] the expected value
E [| P(0, 0)|] tends to zero as E # E 0 . Thus, under the assumption that ⇠(E) is
uniformly bounded over I the quantized Hall conductance is also continuous
there.
⇤
12.7 Appendix: Connes’ area formula
In relating the charge-transport index to the Hall conductivity as given by the
Kubo formula, a vital role is played by Connes’ area formula. Following is
its formulation and derivation, using the streamlined argument of Colin de
Verdière, cf. [25].
Proposition 12.7 For a fixed triplet x1 , x2 , x3 2 Z2 , let ↵ j (a) 2 ( ⇡, ⇡) be
the angle of view from a 2 (Z2 )⇤ of x j+1 relative to x j in the positive angular
orientation, with the index interpreted cyclicly and ↵ j (a) = 0 if a lies between
them. Then the following equality holds
3
X X
g(↵ j (a)) = 2⇡ Area(x1 , x2 , x3 )
(12.40)
a2(Z2 )⇤ j=1
for any antisymmetric bounded function g : ( ⇡, ⇡) ! R which for small ↵ = 0
behaves as:
g(↵) = ↵ + O(↵3 ) .
with Area(·) the triangle’s oriented area.
(12.41)
12.7 Appendix: Connes’ area formula
155
x
2
x
3
a
x1
Figure 12.4 The three angles in the sum (12.40)
Proof We may assume the triangle to be positively oriented. The statement
(12.40) is true for g(↵) = ↵. Indeed, for each ↵ 2 (Z2 )⇤ :
8
9
>
1 >
>
>
>
>
>
< 1 >
=
↵ j (a) = 2⇡ >
>
>
>
2
>
>
>
: 0 >
;
j=1
3
X
8
>
inside
>
>
>
<
for a >
on
the
boundary
>
>
>
:
outside
9
>
>
>
>
=
the triangle.
>
>
>
>
;
Thus, for g(↵) = ↵ the left side of (12.40) is 2⇡ times the number of dual
lattice sites inside within the triangle (counting a boundary site with weight 12 ).
This number is the same for triangles obtained by the lattice translation and
reflection symmetry operations. Since this set of triangles tiles the plane, the
number of enclosed dual sites must equal the triangle’s area.
The above observation reduces (12.40) to the statement that for f (↵) =
g(↵) ↵
3
X X
f (↵ j (a)) = 0 .
(12.42)
a2(Z2 )⇤ j=1
A significant di↵erence between f and g is that the individual terms f (↵ j (a))
are summable in a 2 (Z2 )⇤ , since by assumption f (↵ j (a)) = O(|a| 3 ) for |a| !
P
1. This allows to split the sum into three terms: a2(Z2 )⇤ f (↵ j (a)), j = 1, 2, 3.
Each term is antisymmetric with respect to a reflection which is a symmetry
of the lattice Z2 . Explicitly: the reflection with respect to the midpoint of the
corresponding edge, (x j+1 + x j+2 )/2 2 (Z/2)2 . Thus the individual sums (at
given j) vanish, and hence (12.42) holds.
⇤
Random Operators c M. Aizenman & S. Warzel
DRAFT
156
The Integer Quantum Hall E↵ect
Exercises
12.1 Let P be an orthogonal projection and U be a unitary operator on some
Hilbert space H and set Q := UPU † . Assume that P Q is Schatten2n + 1 class for some n 2 N0 .
1. Show that (P PQP)n+1 = (P Q)2n+2 P and similarly (Q QPQ)n+1 =
(P Q)2n+2 Q and prove that
Index(P, Q) = tr(P
PQP)n+1
tr(Q
QPQ)n+1 .
2. Consider F := PUP on the Hilbert space PH. Conclude that 1 F † F
and 1 FF † are Schatten-n + 1 class on PH and that Index(Q, P) =
tr(1 F † F)n+1 tr(1 FF † )n+1 , where the trace extends over PH.
3. Use the isospectrality of 1 F † F on (ker F)? with 1 FF † on (ker F † )?
to prove
Index(Q, P) = dim ker F
dim ker F † := IndexFN (F) .
[The right side defines the Fredholm-Noether index of F.]
12.2 To be expanded
Notes
We used the notations H = 1,2 = xy for the Hall conductance and ⇢H =
⇢1,2 = ⇢ x,y for the Hall resistance. The relation x,y = ⇢ x,y1 is valid only
over the regime where the direct conductance vanishes D = xx = yy = 0,
which coincides with the Hall plateaux. Transport quantization is present also
in other situations of quantum charge pumps, which are driven adiabatically
by a cyclic increase in some of a quantum system’s parameters, cf. [240]. In
this more general context it is the analog of the Hall resistance, rather than
conductance, which is expected to be quantized.
Exercise 12.1 concerns the relation, which was introduced in [26], of the
charge-transport index with the Fredholm-Noether index of the operator PUa P
on the Hilbert space P`2 (Z2 ):
Ca (P) = IndexFN (PUa P) .
The independence of Ca (P) from a can be viewed as homotopy invariance
within a norm-continuous family of operators of finite index, cf. [174]. This
can also be deduced from the invariance of the Fredholm-Noether index under
compact perturbations (since by Lemma 12.4 Ua Ub are compact operators).