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Page 1: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

Edge current channels and Chern numbersin the integer quantum Hall e�ectJ. Kellendonk1, T. Richter1, H. Schulz-Baldes2;1,1 Fachbereich Mathematik, Technische Universit�at Berlin,Stra�e des 17. Juni 136, 10623 Berlin, Germany2 Department of Mathematics, University of California at Irvine,Irvine, California, 92697, USAAbstractA quantization theorem for the edge currents is proven for discrete magnetic half-plane operators. Hence the edge channel number is a valid concept also in presence ofa disordered potential. Under a gap condition on the corresponding planar model, thisquantum number is shown to be equal to the quantized Hall conductivity as given by theKubo-Chern formula. For the proof of this equality, we consider an exact sequence ofC�-algebras (the Toeplitz extension) linking the half-plane and the planar problem, anduse a duality theorem for the pairings of K-groups with cyclic cohomology.1 IntroductionIn quantum Hall e�ect (QHE) experiments, one observes the quantization of the Hall conduc-tance of an e�ectively two-dimensional semiconductor in units of the universal constant e2=h[35, 45]. As the Hall conductance is a macroscopic quantity, this e�ect is of completely dif-ferent nature than any quantization in atomic physics resulting from Bohr-Sommerfeld rules.Although also a pure quantum e�ect, the quantum numbers of the QHE rather turn out to beglobal topological invariants of the underlying magnetic Hamiltonian. These invariants becomeapparent only in the strong localization regime.For an explanation of the integer QHE, the only situation studied in this work, a one-particleframework is widely excepted to be su�cient. The three main existing theoretical approachesare respectively based on the Laughlin Gedankenexperiment [38], on the edge channel pictureintroduced by Halperin [31] and B�uttiker [17] and on the Kubo-Chern formula for the Hallconductivity �rst derived by TKN2 [50]. Laughlin's argument was rigorously analyzed by Avron,Seiler, Simon and Ya�e even for multiparticle Hamiltonians and in presence of a disorderedpotential [6, 8, 7]. Bellissard [10, 11] and Kunz [37] and others [13, 2] generalized the TKN2 work1

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in order to show quantization of the Hall conductivity also in presence of a disordered potentialas long as the Fermi level lies in a region of dynamically localized states. The mathematicalconnection between these two works is well understood [7, 13, 2]. Although several recent worksconcern boundary conditions for magnetic half-plane operators [4] as well as spectral questionsfor these operators [39, 24, 30], there was up to now no microscopic mathematical theory of edgechannel conduction, except for the case without disorder, of course. Furthermore, a conceptualunderstanding of the link of bulk and edge theory was lacking even though the beautiful work ofHatsugai [32, 33] was an important step in this direction. The results of this work, announcedin [49], �ll these gaps.For a description of our main results in a particular case, letHH : `2(Z2)! `2(Z2) denote theHarper Hamiltonian describing the motion of a tight-binding electron in a plane submitted to aconstant (but arbitrary) magnetic ux per unit cell (see Section 2.1 for its de�nition). Furtherlet V be an Anderson type disorder potential, then we consider the stochastic HamiltonianH = HH + V . In presence of an external electric �eld, all extended states below the Fermilevel undergo the Lorentz drift. The associated (bulk) Hall conductivity �b?(�; �) at inversetemperature � and Fermi energy � is calculated in linear response theory by the Kubo formula.The main result of [50, 13, 2] is that �b?(�) = �b?(1; �) is equal to a constant integer multipleof q2=h as long as the Fermi level � varies in a given interval � of dynamically localized states(q is the particle charge and h Planck's constant). In particular, if � is a gap of the spectrumof H, the conclusion holds. The appearing integer is actually given by the Chern number ofthe Fermi projection P� of H. It is hence a topological invariant of the planar model.In a macroscopic Hall bar, there is now another conduction mechanism by edge currents.For a classical two-dimensional electron gas (2DEG), the cyclotron orbits are intercepted bythe boundary and this leads to a net current along the boundary. In order to calculate thecorresponding quantum mechanical edge current, we will study the restriction H of H to thehalf-plane Hilbert space `2(Z �N), together with a given boundary condition. All operatorson the half-plane will carry a hat throughout this work. Let Jx = q{[X; H]=�h be the electricalcurrent operator along the boundary (here X is the position operator of the direction parallelto the boundary) and P� the spectral projection of H on an interval � � R. The edge currentcarried by the states in � is then given byje(�) = TryTx(P�Jx) , (1)where Tx the trace per unit volume parallel to the boundary and Try the trace in the directionperpendicular to the boundary. Note that the calcultion of edge currents is mere equilibriumquantum mechanics: no electric �eld, linear approximation or dissipation mechnism is needed.Our main result is then the following:Theorem Let � be a gap of the (almost sure) spectrum of H = HH+V . Then, for any interval�0 � �, one has almost surelyje(�0) = ��b?(�) 1q j�0j , 8 � 2 � ,where j�0j denotes the length of �0. 2

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x x−π +π

E E

k kFigure 1: l.h.s.: schematic representation of the spectrum of the half plane operator HL (solidlines), the dashed lines show the Landau bands; r.h.s.: spectrum of HH , the solid lines are theDirichlet bands and the shaded regions are the Bloch bands )for numerical results, see [32]).The theorem states that the edge channel number in the sense of [31, 17] of a magneticHamiltonian in the half-plane remains a valid concept in presence of a disordered potential,and that this number is moreover equal to the Chern number of the planar Hamiltonian. Inthe following we will call the limit of the quotient �qje(�)=j�j as � ! f�g also the edgeHall conductivity �e?(�) at �. It is well known [17] that its quantization can only hold fora magnetic operator on a semi-in�nite space because for a strip geometry the backscattering,notably tunneling from upper to lower edge, will destroy quantization.For the unperturbed Landau Hamiltonian HL, the continuous analogon of the theorem juststates that in the nth gap of HL, the half-plane operator HL has exactly n bands (compare Fig.1). In fact, if HL = RR dkx HL(kx) is the Bloch decomposition in the x-direction and Ej(kx)are the corresponding edge channels, then for an interval � in the nth gap, one hasje(�) = � qhXj�0 ZR dkx �(Ej(kx) 2 �) dEj(kx)dkx = � qh n j�j ,where � denotes the characteristic function. For the Harper Hamiltonian HH , the theoremasserts that the sum of the Chern numbers of the lowest n bands is equal to the number of edgechannels within the nth gap (also Dirichlet bands in [32]) of the half-plane operator multipliedby their orientation. In the commensurate case of rational ux per unit cell, this result is due toHatsugai [32, 33]. His proof is completely unrelated to ours though and cannot be generalizedto a situation with broken translation invariance due to either an irrational magnetic ux or toa disordered potential.Our proof of the above theorem splits into two parts. In the �rst purely analytic step, weconsider only half-plane operators and prove that the edge Hall conductivity is equal to theFredholm index of a certain unitary operator. This can be viewed at as an odd index theoremresulting from a pairing between an odd cyclic cohomology class with the K1-class of the above3

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unitary operator. Similarly, the quantized Kubo-Hall conductivity is given by a pairing of acertain even cyclic cohomology class with the K0-class de�ned by the Fermi projection. Thesetwo pairings are over two di�erent C�-algebras which are linked by a short exact sequence calledthe Toeplitz extension [44]. In the second step, we use the K-theoretic six-term exact sequenceof the Toeplitz extension as well as its dualK-homologic counterpart to prove the equality of thepairings. This duality theorem unveals that the equality of edge and bulk Hall conductivitiesis a consequence of a fundammental topological concept, namely of Bott periodicity. Hencethis work fully exploits (and thereby justi�es) the use of the non-commutative C�-algebraicframework developed by Bellissard [10].Let us now comment on variations and generalizations of the above theorem. First of all, itscontinuous conterpart is under preperation. It uses the Wiener-Hopf rather than the Toeplitzextension [20, 16] and the duality theorem corresponding to Connes' Thom isomorphism [27].Further we believe the consequences of the theorem to hold under the weaker condition that � isan interval containing only dynamically localized states of H. Under precisely this condition itis possible to prove quantization of the bulk Hall conductivity [13]. The gap condition imposedabove only allows to deal with the weak disorder regime (unsu�cient in order to explain theQHE). In this regime the Hamiltonian H has purely absolutely continuous spectrum as showpositive commutator estimates for the boundary current operator (the arguments of [24] directlytranspose to our discrete case as long as the edge bands of the free magnetic operator havea de�nite sign). In the regime of intermediate disorder, the gaps of H �ll with dynamicallylocalized states. In presence of a boundary, the Tunnel e�ect to the edge states should turnall these localized states into resonances so that we expect the spectrum to remain absolutelycontinuous. At the same time, the edge channel number should remain an integer equal to theChern number of the Fermi pojection of the planar model. In the high disorder regime, thesystem completely localizes [40, 3]; in between there is a cascade of metal-insulator transitions[13].The article is organized as follows. In the next Chapter 2, we develop the mathematicalframework, then state our main results more precisely in Chapter 3, and explain in detailwhy they lead to a consistent physical picture of the QHE in a Hall bar. It shows that themacroscopic Hall conductance is quantized no matter what proportion of the current ows bybulk or edge states respectively. We present furthermore a simple argument showing that in atypical QH experiment, at most 10% of the current is carried by edge states. In Chapter 4 wegive a complete direct proof of the edge current quantization. Only in Chapter 5 we then useK-theoretic arguments in order to show the equality between edge and bulk Hall conductivities.In order to clearly exhibit the concepts involved, we chose a slightly more general formulationthan needed in our context. A short Appendix resuming the most basic results of K-theory isincluded for the convenience of the reader.We would like to express our deep gratitude to our teachers Jean Bellissard and RuediSeiler. Their works set the stage for the present article which hereby inherits their spirit.We moreover pro�ted from discussions with many colegues, beneeth them Y. Avron, J.-M.Combes, S. DeBievre, G. Elliott, F. Germinet, S. Jitormirskaya, A. Klein and M. Seifert. Weacknowledge support of the SFB 288. 4

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2 Observable algebrasIn this chapter, after having de�ned the (geometric) hull of a homogeneous tight-binding Hamil-tonian in absense of a magnetic �eld, we construct the observable algebras as the Toeplitzextension of a twisted crossed product, the twisting being introduced by the magnetic �eld.We then develop the non-commutative analysis tools of di�erentiation and integration on thesealgebras. All these constructions are done in the symmetric gauge, and we show in Section 2.6how a gauge transformation to the Landau gauge allows to work with an iterated crossed prod-uct without twisting, a more convenient formulation for the K-theoretic arguments in Chapter5. For further details and for physical motivation of the following de�nitions and subsequentconstructions, we refer to [10, 12, 13, 48, 14].2.1 Homogeneous operators and their hullAs in [14] we de�ne the (geometric) hull of a tight-binding operator in absense of a magnetic�eld. This seems more adapted than its de�nition in [13] because the use of magnetic transla-tions leads to a non-trivial hull even for a free magnetic operator as the Harper Hamiltonian(for the Landau Hamiltonian on continuous physical space this problem does not appear dueto its invariance w.r.t. the magnetic translations).De�nition 1 Let U0(a), a 2 Z2, denote the translations on `2(Z2) de�ned by U0(A) (n) = (n � a) for n 2 Z2 and 2 `2(Z2). Consider a self-adjoint bounded operator H : `2(Z2) !`2(Z2) of the form H = Xn;m2Z2Hn;m jnihmj , Hn;m 2 R ,where hmj and jni are the usual bra-ket notations for states localized at m 2 Z2 and n 2 Z2.Then H is called homogeneous if the set fU0(a)HU0(a)� j a 2 Z2g has a compact closure in the strong operator topology. By continuity, U0 extends to an action T of Z2 on . Thedynamical system (; T;Z2) is then called the hull of H.Example: Let H0 be translation invariant, that is U0(a)H0U0(a) = H0 for all a 2 Z2. A typicalexample is the discrete Laplacian. Let further = [�1; 1]Z2 be furnished with the Tychonovtopology as well as the shift action T of Z2. We �x an invariant and ergodic probability measureP on with the property P(f! 2 jV!(n) 2 Ig) > 0 for all n 2 Z2 and all intervals I � [�1; 1]of non-vanishing length. Finally we denote by V! : `2(Z2) ! `2(Z2) the Anderson-type onsitepotential corresponding the disorder con�guration ! 2 . Then, for P-almost all ! 2 , theoperator H! = H0 + �V!, � 2 R, is homogeneous and its hull is equal to (; T;Z2). �From a mathematical point of view, this article is about the analysis of certain topologicalinvariants of the dynamical system (; T;Z2). Associated to it is the C�-dynamical system(C(); �;Z2) where (�af)(!) = f(T�a!) of continuous functions f 2 C(). This gives riseto a natural C�-algebra, the twisted crossed product [42] with a twisting given by the group5

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cocycle Z2 � Z2 ! S1 : (a; b) 7! �a ^ b mod(2�) where a ^ b = axby � aybx and � = qB=�h isthe magnetic ux per unit cell. This algebra has a natural extension, the (twisted) Toeplitzextension. If moreover an ergodic probability measure P on is given, we have a W�-dynamicalsystem (L1(;P); �;Z2) with an associated von Neumann crossed product [42]. This also hasa Toeplitz extension. These exact sequences describe the link between magnetic operators inthe plane and in the half-plane.2.2 Algebraic twisted crossed product and its Toeplitz extensionThe magnetic translations in symmetric gauge U(a), a = (ax; ay) 2 Z2, are de�ned byU(a) (n) = exp� {2� a ^ n� (n� a) , n = (nx; ny) 2 Z2 , 2 `2(Z2) ,where a ^ n = axny � aynx and � = qB=�h is the ux per unit cell. U is a projective unitaryrepresentation of the group Z2 on `2(Z2), notably U(a+ b) = exp( {2�a^ b)U(a)U(b). We denoteUx = U(1; 0) and Uy = U(0; 1), then UyUx = e{�UxUy. The C�-algebra generated by Ux and Uyis called the rotation algebra A�.Now we explicitly construct the algebraic twisted crossed product and its Toeplitz extension.The formulas below are chosen such that the magnetic translation U(a) is the representation ofthe function (!; n) 7! �n;�a. Let A0 = CK(�Z2) be the continuous complex valued functionson � Z2 with compact support. With the following de�nitions, A0 becomes a �-algebra(A;B 2 A0, ! 2 , m 2 Z2):AB(!;m) = Xl2Z2A(!; l)B(T�l!;m� l) exp� {2�l ^m� , A�(!;m) = A(T�m!;�m) .(2)Further let E0 = CK(�Z�N�N) and T (A)0 = E0�A0. As E0 and T (A)0 will describehalf-plane operators, their elements will carry a hat. In order to simultaneously de�ne the�-algebra structure on E0 and T (A)0, we identify elements (A1; A2) 2 T (A)0 with a functionA 2 C(� Z�N�N) by means of the formula (mx 2 Z, ny; my 2 N):A(!;mx; ny; my) = A1(!;mx; ny; my) + A2(!;mx; my � ny) . (3)For A; B 2 T (A)0, the product and involution are then de�ned byAB(!;mx; ny; my) = X(lx;ly)2Z�N A(!; lx; ny; ly)B(T (�lx;ny�ly)!;mx � lx; ly; my) �� exp� {2�(lx(my � ny)�mx(ly � ny))� , (4)A�(!;mx; ny; my) = A(T (�mx;ny�my)!;�mx; my; ny) . (5)6

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Let us further introduce the inclusion i : E0 ! T (A)0 sending A 2 E0 to (A; 0) 2 T (A)0 as wellas the projection � : T (A)0 !A0 de�ned by�(A)(!;mx; my) = limk!1 A(!;mx; k;my + k) , A 2 T (A)0 . (6)One directly veri�es �(A) = A2 if A = (A1; A2) 2 T (A)0. Hence the operation � eliminates thecompact perturbation on the boundary by shifting the system away from it.The above constructions are made such that the following algebraic lemma holds.Lemma 1 We have the following exact sequence of �-algebras:0 ! E0 i! T (A)0 �! A0 ! 0 ,notably i and � are �-morphisms satisfying Im(i) = Ker(�).We point out that this exact sequence of �-algebras is never split exact even though it is splitexact as an exact sequence of vector spaces. More precisely, there does not exist a �-morphism� : A0 ! T (A)0 such that � � � = 1.Proof of Lemma 1. First of all, i : E0 ! T (A)0 is clearly a �-morphism because the operations(4) and (5) are the same in E0 and T (A)0 and moreover leave E0 invariant in T (A)0 = E0�A0.Let us check that � is also a �-morphism (A = (A1; A2), B = (B1; B2) 2 T (A)0):�(AB)(!;mx; my) = limk!1 Xlx2Z Xly�0 A(!; lx; k; ly)B(T�(lx;ly�k)!;mx � lx; ly; my + k) �� exp� {2�(lxmy �mx(ly � k))� .Now because both A1; B1 2 E0 have compact support, they will vanish for k su�ciently big.Then we make the change of variables ly $ ly � k and right out the identi�cation (3):�(AB)(!;mx; my) = limk!1 Xlx2Z Xly��kA2(!; lx; ly)B2(T�(lx;ly)!;mx � lx; ly �my) �� exp� {2�(lxmy �mxly)� ,Now because A2 is of compact support, we can replace the summution over ly � �k by thatover ly 2 Z for k su�ciently large. Comparing with (2) we see that �(AB) = �(A)�(B) as�(A) = A2 and �(B) = B2. The identity �(A)� = �(A�) can be immediately checked. Thismoreover shows that Im(i) =Ker(�). 2These algebras will be called the bulk algebra A0, its Toeplitz extension T (A)0 and theedge algebra E0. The algebra A0 and its various completions will describe operators which are7

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homogeneous in the plane, E0 models operators which are homogeneous along the boundary (x-direction), but compact in the y-direction perpendicular to the boundary, whereas T (A)0 con-tains both of these operators, notably homogeneous half-plane operators with compact bound-ary contributions.By means of the following formulas, we introduce two families of physical representations�! and �!, ! 2 , of the �-algebras A0 and T (A)0 on `2(Z2) and `2(Z�N) respectively (bothn = (nx; ny) and m = (mx; my) are in Z2 or Z�N respectively):hnj�!(A)jmi = A(T�n!;m� n) exp� {2� n ^m� , A 2 A0 , (7)and hnj�!(A)jmi = A(T�n!;mx � nx; ny; my) exp� {2� n ^m� , A 2 T (A)0 . (8)It is in fact elementary to verify that �! and �! are �-morphisms from A0 and T (A)0 tothe algebras of bounded operators on `2(Z2) and `2(Z �N) respectively and further that thefollowing covariance relations hold (a = (ax; ay) 2 Z2):U(a)�!(A)U(a)� = �Ta!(A) , U(ax; 0)�!(A)U(ax; 0)� = �T (ax;0)!(A) . (9)where U denotes the restriction of U to `2(Z �N). Covariance under U(0; ay) only holds foray � 0; for ay < 0 there are corrections by operators in E0.2.3 Exact sequence of C�-algebrasUsing these representations, we can now introduce norms on A0 and T (A)0 (A 2 A0 andA 2 T (A)0): kAk = sup!2 k�!(A)k , kAk = sup!2 k�!(A)k . (10)It is elementary to verify the C�-equation for these norms, hence they are actually the uniqueC�-norms on A0 and T (A)0. The C�-algebras A, T (A) and E are de�ned to be the closureof A0, T (A)0 and E0 with respect to these norms. Now, because any �-morphism betweenpre-C�-algebras can be extended by continuity to their C�-closures, the representations �! and�! extend to covariant representations of the C�-algebras and furthermore Lemma 1 leads toan exact sequence of C�-algebras:0 ! E i! T (A) �! A ! 0 . (11)We note that A the twisted crossed product C() �T Z2 of the dynamical system (; T;Z2)associated to the magnetic twisting introduced above. The structure of this exact sequence willbe further analysed in Section 2.6.Example: Let us consider this exact sequence explicitly for the case of the rotation algebraA�, namely the C�-algebra generated by the unitary operators Ux and Uy. Then the Toeplitz8

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extension is generated by Ux and Uy which still satisfy the same commutation relation UyUx =e{�UxUy, but while Ux remains unitary, Uy satis�es U�y Uy = 1 and UyU�y = 1 � �0 where�0 = Pl2Z jl; 0ihl; 0j. The edge C�-algebra is given by the C�-tensor product C(S1), notbly theC�-algebra generated by Ux, with the compact operators K in y-direction (not strictly, becausethe multiplication in this product contains the phases as in (4). The projection is given by�(Ux;y) = Ux;y. �The following proposition shows that the Hamiltonian which we started out with is in theC�-algebra A. For its proof we refer the reader to [12, 13].Proposition 1 Supose that ~H is a homogeneous bounded self-adjoint operator on `2(Z2) theo�-diagonal coe�cients of which fall o� like jhnj ~Hjmij � cjn�mj�2 for some constant c > 0.Let (; T;Z2) be its hull. Then there exists an H 2 A and a !0 2 such that �!0(H) = ~H.2.4 von Neumann algebra and its extensionLet P be a probability measure on which is invariant and ergodic with respect to both Tx andTy. We suppose that its support is all of . All our results hold for any such measure, but itschoice is of physical importance (in particular, the almost sure values of the Hall conductivitiesmay be di�erent for di�erent measures). We now construct the (twisted) von Neumann crossedproduct L1(A;P) of the W�-dynamical system (L1(;P); T;Z2) by the standard procedure[42, 22] using the direct integral representation:� = Z � dP(!) �! : A ! B(L2(� Z2;P �)) ,where � is the counting measure on Z2 and B(H) denotes the algebra of bounded operatorson the Hilbert space H = L2( � Z2;P �). Then L1(A;P) is de�ned as the weak closureof �(A) in B(L2( � Z2;P �)). Elements of L1(A;P) are weakly-measurable covariantoperator families and can hence be described by measurable functions on � Z2 satisfyingA(!; n)! 0 as jnj ! 1. We continue to use the notation �! for the (�berwise) representationof these functions on `2(Z2). The C�-norm on L1(A;P) is given bykAk1 = P-ess sup!2 k�!(A)k .Similarly, one can construct the W�-crossed product L1(C()��x Z;P) of the W�-dynamicalsystem (L1(;P); �x;Z) where (�xf)(!) = f(T�1x !) for f 2 L1(;P). Elements therein canbe represented by functions on � Z. Let L1(E ;P)0 denote the �nite dimensional matriceswith entries in L1(C() ��x Z;P). We can represent elements of L1(E ;P)0 as measurablefunctions on � Z � N � N. Set further L1(T (A);P)0 = L1(E ;P)0 � L1(A;P). Usingthe identi�cation (3), the formulas (4) and (5) introduce an algebraic structure on L1(E ;P)0and L1(T (A);P)0. Using again the �berwise representations �! identifying the functions on�Z�N�N with operators on B(L2(�Z�N;P�)), a C�-norm on both of these algebrasis given by 9

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kAk1 = P-ess sup!2 k�!(A)k ,and the closure w.r.t. this norm de�ne the C�-algebras L1(E ;P) and L1(T (A);P). Finallythe inclusion i and projection � on the pre-C� algebras L1(E ;P)0 and L1(T (A);P)0 can bede�ned as in (6) and the counterpart of Lemma 1 can be veri�ed, so that the �-morphisms iand � can be extended by continuity to the C�-closures giving the the following exact sequenceof C�-algebras: 0 ! L1(E ;P) i! L1(T (A);P) �! L1(A;P) ! 0 . (12)2.5 Non-commutative analysis toolsThe algebra L1(A;P) admits a di�erential structure generated by a two-parameter group �k,k = (kx; ky) 2 T 2 = [��; �)�2, of �-automorphisms (A 2 L1(A;P) and m = (mx; my) 2 Z2):�k(A)(!;m) = exp({(kxmx + kymy)) A(!;m) . (13)The associated �-derivations ~r = (rx;ry) are densly de�ned and explicitely given byrxA(!;m) = {mxA(!;m) , ryA(!;m) = {myA(!;m) , (14)whenever the right hand sides are in L1(A;P). If ~X = (X; Y ) denotes the position operatorson `2(Z2), then one can verify for such di�erentiable operators the identities:�!(~r(A)) = {[�!(A); ~X] . (15)Similarly, there is a two-parameter family of �-automorphisms �k on L1(T (A);P) given by�k(A)(!;mx; ny; my) = exp({(kxmx + ky(my � ny)) A(!;mx; ny; my) . (16)The associated �-derivations are denoted ~r = (rx; ry). They satisfy identities similar to (15).For n 2 N, we de�ne Cn(A) to be those elements for which all nth order gradients are in theC�-algebra. An operator in A the o�-diagonal coe�cients fall o� exponentially is in any Cn(A),n � 0.Given an invariant and ergodic probability measure P with support , a normalized faithfultrace T on L1(A;P) is de�ned by [42]T (A) = Z dP(!) A(!; 0) , A 2 L1(A;P) . (17)For any increasing sequence (�L)L2N of squares centered at the origin, Birkho�'s ergodic the-orem implies T (A) = limL!1 1j�Lj Xm2�Lhmj�!(A)jmi , (18)10

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for P-almost all ! 2 showing that T is the trace per unit volume.For p 2 [1;1), the Banach space Lp(A;P) is de�ned as the closure of A0 under the normkAkp = (T (jAjp))1=p. If �GNS is the GNS representation of A on L2(A; T ) associated to thestate T , then the von Neumann algebra �GNS(A)00 where 00 is the bicommutant can easily be seento be isomorphic to L1(A;P) [22]. A trace T on L1(T (A);P) is given by T (A) = T (�(A)).However, T is obviously not faithful.Let now L1(T (A);P)+ denote the cone of positive operators in L1(T (A);P) and de�neTE : L1(T (A);P)+! [0;1] byTE(A) = Xmy2N Z dP(!) A(!; 0; my; my) . (19)It follows again from Birkho�'s ergodic theorem that TE is the trace per unit volume Tx inthe x-direction, followed by the usual trace in the y-direction. Clearly TE is a weight, namelyTE(A + �B) = TE(A) + �TE(B) for all A; B 2 L1(T (A);P)+ and � � 0. Moreover one candirectly check that TE(AA�) = TE(A�A) for all A 2 L1(T (A);P). Hence TE is a trace [25]. Inparticular, we dispose of its unitary invariance, that is for all A 2 L1(T (A);P)+ and unitaryU 2 L1(T (A);P) one has TE(UAU�) = TE(A).As usual (e.g. [25, 42]), one now says that an operator A 2 L1(T (A);P) is TE -traceclass ifTE(jAj) <1. The set of all TE -traceclass operators form an ideal in L1(T (A);P) onto whichTE can be extended. Further let us de�ne Lp(E ;P) as the closure of E0 with respect to the normkAkp = �TE(jAjp)�1=p. The ideal of TE -traceclass operators is then L1(E ;P) \ L1(E ;P).It can be directly veri�ed that the traces T and TE are invariant under the automorphismgroups �k and �k respectively. Thus we have for any A 2 C1(A) and any A 2 L1(E ;P)\C1(E):T (rx;yA) = 0 , TE(rxA) = 0 . (20)2.6 Iterated crossed product and Landau gaugeLet C() ��x Z be the C�-crossed product without twisting [42] of the C�-dynamical system(C(); �x;Z) where (�xf)(!) = f(T�1x !) for f 2 C(). Elements of C() ��x Z can berepresented by continuous functions on � Z. This C�-algebra inherits a dynamics by theaction �y de�ned by(�yf)(!;mx) = exp(�{�mx) f(T�1y !;mx) , f 2 C()��x Z .Hence we obtain an iterated crossed product C�-algebra C() ��x Z ��y Z. Its elements aregiven by functions on � Z2 and the multiplication is explicitely given byAB(!;m) = Xl2Z2A(!; l)B(T�l!;m� l) exp ({�ly(lx �mx)) ,11

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mimicing the Landau gauge. The iterated crossed product has a Toeplitz extension w.r.t. theaction �y in the precise sense of Pimsner and Voiculescu [44] and this extension turns out tobe isomorphic to the exact sequence (11):0 ! C()��x ZK ! T �! C()��x Z��y Z ! 0� # � # � #0 ! E i! T (A) �! A ! 0 (21)Here K denotes the C�-algebraic tensor product with the compact operators on `2(N) and ,� and T are the mappings and the Toeplitz extension as de�ned in [44]. In fact, it is elementaryto verify that(�A)(!;mx; ny; my) = exp� {2�mx(my + ny)� A(T (0;ny)!;mx; ny; my) , (22)and (�A)(!;m) = exp�� {2�mxmy� A(!;m) , (23)are (gauge) isomorphisms. As it is not used later on, we do not write out a explicit formula for� here.3 Main results and their interpretation3.1 Edge Hall conductivityLet H = (H; K) 2 T (A) be some lift of the planar (stochastic) self-adjoint HamiltonianH 2 A.For the representation corresponding to the disorder con�guration !, this means:�!(H) = � �!(H) + �!(K) ,where � denotes the projection from `2(Z2) onto `2(Z �N). We suppose K to have non-zeroentries only a �nite distance from the boundary (see Sect. 4.1 for mor precise and generalhypothesis). Choosing such a lift means that we �x some boundary condition for the operatoron the half-plane which is homogenous in the x-direction. For example, if one wishes to imposethe �-boundary condition cos(�) (mx; 0) + sin(�) (mx; 1) = 0 for all mx 2 Z, this gives riseto a projectin �� on the subspace of all states in `2(Z � N) satisfying this condition; then�!(H) = �� �!(H), so that K 2 E has to satisfy �!(K) = (�� � �)�!(H).Following the discussions in the introduction, let us now de�ne the edge Hall conductivityas follows: �e?(�) = �q2�h lim�!f�g 1j�j TE(P�rxH) . (24)12

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Recall (see e.g. [23]) that the spectrum of �!(H) is P-almost surely independent of !. Ourresults below show that, under the two hypothesis that (i) H 2 C3(A) and (ii) � is withina P-almost sure gap of the spectrum of �!(H) (equivalently, � is in a gap of the density ofstates), the operator P� 2 L1(E ;P) is in the ideal of TE -traceclass operators. If � is not ina gap, the operator P� is de�nitely not TE -traceclass. Nevertheless, the edge current may bewell-de�ned and �nite if � is a dynamically localized region of the spectrum of H (cf. Remark2 in Section 4.1).Further we let �x denote the projection in `2(Z�N) onto `2(N�N), namely the subspacespanned by the states jmx; myi with mx � 0.Theorem 1 Suppose that H 2 C3(T (A)). Let the interval � � R be P-almost surely in a gapof the spectrum of �!(H). SetU(�) = exp �2�{ H � E 0j�j P�! , E 0 = inf(�) . (25)Then the operator �x�!(U(�))��x is P-almost surely a Fredholm operator on `2(N�N) andits index is P-almost surely constant. For any � 2 �, the almost sure value is equal to the edgeHall conductivity: �e?(�) = q2h Ind ��x �!(U(�)�) �x j`2(N�N)� .The index formula can only hold P-almost surely because there are always exceptionaldisorder con�gurations.3.2 Link to the Kubo-Chern bulk Hall conductivityIn relaxation time approximation, the Kubo formula for the bulk Hall conductivity �b?(�; �; �)of a two-dimensional electron gas in the plane at inverse temperature �, chemical potential �and relaxation time � reads as follows [13, 48]:�b?(�; �; �) = q2h T rx(f�;�(H)) 11=� � LH (ryH)! ,where f�;� is the Fermi-Dirac function and LH the Liouville (super) operator de�ned byLH(A) = {[H;A]=�h for A 2 A. The direct conductivity �b==(�; �; �) is given by the sameformula after replacement of ry by rx. The zero dissipation limit � ! 1 of the direct con-ductivity can only give a �nite value (other than 0 or1) when the quantum motion is di�usive(weak localization regime) [47, 48]. However, as the Lorentz drift in a magnetic �eld is a trans-verse di�usive motion, the Hall conductivity is due to non-dissipative transport. Consequentlythe zero dissipation limit always gives a �nite value. This can actually be proven as shows thefollowing proposition. 13

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Proposition 2 [13, Corollary 2] Suppose that H 2 C1(A) and that the Fermi level � is in agap of the spectrum of H. Then the bulk Hall conductivity �b?(�) at zero temperature (� =1)and zero dissipation (� =1) is�b?(�) = q2h 2�{ T (P�[rxP�;ryP�]) . (26)The second main result of this work is the following theorem.Theorem 2 Suppose that H 2 C3(A) and that the interval � is in a gap of the spectrum ofH. Then for any � 2 �, �b?(�) = �e?(�) , (27)and their common value is constant in � and equal to an integer multiple of q2=h.The Hall conductivity can either be calculated by the index in Theorem 1 or by that ofreference [13]. Hence this work gives the 1002nd proof of the quantization of the bulk Hallconductivity, once one acceptes (26) as its de�nition.3.3 Physical interpretationIn this section, we explain how our mathematical results allow to develop a physically consis-tent, but not mathematically rigorous picture of the QHE. We focus on a so-called Hall barexperiment pictured schematically in Fig. 2. We also use the model with continuous physicalspace for these explanations, although the mathematical results of this work are restricted tothe discret case.As stressed by B�uttiker [17], the size of such a bar has to be much larger than the magneticlength l � 100�A in order to suppress tunneling between the edge states at the upper and loweredge, otherwise the QH regime cannot be attained. In von Klitzing's experiments [35], thesample size was 400�m� 50�m, but nowadays samples 20 times larger are in use [55]. Thereare six contacts on the Hall bar. The contacts 1 and 4 are the source and drain contacts forthe current I. If Vk;l denotes the measured tension between contacts k and l, the associatedresistance is denoted by Rk;l = Vk;l=I. The Hall tensions are then R2;5, R2;6, R3;5 and R3;6, thelongitudinal or direct resistances R2;3 and R5;6, while R1;4 is called the two-terminal (contact)resistance. The QH regime is characterized by quantization of the Hall tension and vanishingof the longitudinal resistance (it is quantized to zero). Typically the experiment is made atabout 1K, but for the explanations below we suppose the temperature to be zero.An answer to the question what the tensions measured really are was given by B�uttiker[17] using the Landauer conductance formalism. The contacts are metallic reservoirs close tothermodynamic equilibrium and Vk;l is the di�erence of the electrochemical potentials of thetwo reservoirs k and l. These contacts interact with the 2DEG. Let us �rst suppose that thecontacts are ideal in the following sense [17], a hypothesis justi�ed later on:14

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1

2 3

4

56Figure 2: Schematic representation of the Hall bar sample with 6 contacts.� the contacts absorb any incoming electron independent of its energy and phase;� the contacts emit into all those states of the sample that carry a current away from thecontact and that have an energy less than the electro-chemical potential of the contactreservoir.These conditions are supposed to hold for all contacts, under the supplementary conditionthat there is a net current I from contact 1 to 4, but no current leaving any of the other fourcontacts. Hence the 2DEG is an open conductor with considerable particle exchange with thecontacts.In the QH regime, the 2DEG is in a steady non-equililibrium state interacting with theideal contacts according to the above two rules and showing quantized Hall and longitudinalresistances. The theoretical picture developed here is based on the following assumptions widelyaccepted in the physical community, but possibly not that of mathematical physicists:1. the macroscopic sample can be devided into a (continuously indexed) family of mesoscopicvolumes in such a way that the equilibrium physics in each of these mesoscopic volumescan be described by an in�nite volume one-particle Hamiltonian;2. each mesoscopic volume is near thermodynamical equilibrium; it hence has a well-de�nedlocal particle density;3. the properties of the non-equilibrium state can locally be described by the equilibriumHamiltonian; in particular, we suppose that the response to an external electro-static �eldis linear and given by Kubo's formula.Thus we suppose that, for every point ~r = (x; y) in the sample, there are a homogenous one-particle Hamiltonian H(~r) and a local particle density n(~r).An important physical question concerns the size of the mesoscopic volumes. Several con-ditions should be satis�ed. Their size should be much larger that the magnetic length, butsmall enough for the particle density to be assumed to be constant. Furthermore they should15

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be smaller than the inelastic scattering length, notably the typical length scale over which theelectrons do not collide with phonons or other electrons. Otherwise the one-particle descriptionbreaks down (of course, this does not exclude elastic collisions with static disorder due to im-purities). We believe these conditions to hold for mesoscopic volumes of about 1�m diameter.Now we clearly have to distinguish two di�erent in�nite volume models for the mesocscopicvolumes:� The bulk model at position ~r is given by a Hamiltonian H(~r) = H + qVE(~r) acting onL2(R2). Here H is the sum of the magnetic Landau Hamiltonian HL and a disorderedpotential Vdis, and VE(~r) is the external electro-static potential which is supposed to beconstant within the mesoscopic volume at ~r. Its origin will be discussed below. Whensubmitted to an external electro-static �eld E(~r) = dVE(~r)=d~r causing a non-equilibriumsituation within the mesoscopic volume, the current-carrying states aquire a drift velocitydue to the Lorentz force. This non-dissipative current is linear in the electric �eld witha proportionality constant given by the Kubo-Chern formula. Note that we describehere the non-equilibrium state due to a small electric �eld E(~r) by a local equilibriumHamiltonian H(~r) containing a constant electro-static potential VE(~r).� The edge Hamiltonian at position ~r is H(~r) = H + VE(~r) acting on L2(R � R+) whereH is the restriction of H to the right half-plane. The basic characteristic of this model isthe presence of n current-carrying edge channels at the Fermi level whenever the latter isbetween the nth and (n+1)th Landau level. We further note that due to the compressiblenature of the 2DEG near the boundary, screening does not allow an electric �eld to bebuild up within several magnetic lengths o� the edge [19].To study the total current through the sample, we next have to discuss the form of theelectrical potential VE(~r), the associated electric �eld E(~r) = dVE(~r)=d~r and the local chemicalpotential �(~r) de�ned by the relationn(~r) = T (P�(~r)) = N ((�1; �(~r)) ,where P�(~r) is the Fermi projector of H to the energy �(~r), T the trace per unit volume andN the density of states of H. Note that this de�nition of �(~r) is not with respect to H(~r), butto H. We next suppose that the Hall bar is su�ciently long so that all these quantities onlydepend on y. Furthermore the projection of the electric �eld in the x-direction then vanishes.Fig. 3 shows a possible pro�le of qVE(x; y) and qVE(x; y)+�(x; y). Before discussing the originof these curves, let us calculate the associated currents. In each mesoscopic volume of the bulk,the Hall current density j(~r) in y-direction can be calculated by Kubo's formula and is givenby jb(~r) = �b?(~r) E(~r) ,where �b?(~r) is the Hall conductivity at ~r calculated using the Hamiltonian H and chemicalpotential �(~r). In order to calculate the total bulk current Ib, we keep x �xed and have tointegrate jb(x; y) over all y between the left and right edge. Under the hypothesis that �(~r)16

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qV + µ

qV

0 y L

V26

V26- ∆ µ

Figure 3: Plot of the electrostatic potential and its sum with the local chemical potential as afunction of the position of the position y in the Hall bar.varies in the same region of dynamically localized states of H, �b?(~r) is constant and equal to�b? [13]. Thus, if we set V b = R L0 dy E(x; y) as in Fig. 3,Ib = Z L0 dy �b?(x; y) E(x; y) = �b? V b .Next the total edge current Ie is equal to the sum of the left edge and right edge currents. Bothcan be calculated with the Hamiltonian H if one takes account for the di�erent orientation ofthese currents by a minus sign. Supposing again that the interval � = [�(0); �(L)] is containedin an interval of dynamically localized states of H, edge current quantization then impliesIe = �qTE(P�(L)rxH) + qTE(P�(0)rxH) = �qTE(P�rxH) = �e? j�j .Finally, according to Fig. 3 and the above assumptions of the ideal contacts, the measured Halltension V2;6 (or one of the others) is equal to the sum of V b and �=q. As now �b? = �e? = �?,because the Fermi level stays in the same region of dynamically localized states of H, so thatone obtains the desired quantization:R2;6 = V2;6I = V2;6Ib + Ie = V2;6�b?V b + �e?j�j = 1�? .The proportion of current carried by the bulk states is equal to V b=(V b + j�j) and theslope of VE gives the bulk current distribution. Hence the pro�le of the electro-static potentialand the chemical potential in Fig. 3 is of great interest. Considerable e�ort has been madeto experimentally measure and numerically calcuate these curves. In early experiments [45],supplementary leads were introduced into the sample in order to measure the tension pro�leaccross the sample. It turned out to be linear so that one conclude that the current wasuniformly distributed through the bulk. However, the contacts modify the electronic structurewithin the sample too heavily as that this conclusion can be drawn. More recent Pockels e�ect17

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measurements [29] and inductive coupling measurements [55] (in a sample with back gate) showthat about 80% of the Hall tension falls o� within a strip of about 400�m width close to theboundaries. Hence an important portion of the bulk current ows close to the boundaries, butthis does not mean that an important portion ows by the edge states in the above sense. Infact, edge states are localized within a few magnetic lengths (l � 100�A) o� the boundary andthe measured current ows in a strip three orders of magnitude wider. The authors of [55]further conclude that, within their experimental resolution, at most 5% of the total current ows by edge states.We obtain a similar result from the following simple theoretical reasoning. In the aboveargument, it was crucial for the quantization to hold that the local chemical potential �(~r)stays within the same gap of extended states of H for all ~r in the sample. For the existenceof the plateaux, it is furthermore necessary that this condition holds for all �lling factorscorresponding to the plateaux. This imposes that the energetic interval � = [�(0); �(L)] hasto be smaller than the energetic distance �h!c(1� q) (here !c is the cyclotron frequency so that�h!c is the distance between two Landau levels, and q is the quotient of the energetic width ofthe plateaux and �h!c). As the size of � determines the proportion of the edge currents, we canestimate this condition using the experimental data for a typical QH regime from [45, Chapter2] for the �? = 4 plateau: B � 6T , V2;6 � 170mV and q � 0:6. Using the data for the e�ectiveelectron mass (m� � 0:07me) and the electron charge, we obtain �h!c � 48meV and a maximalproportion of edge currents of 10%.Next we discuss the physical origin of the electrostatic potential VE and of the variationsof local chemical potential �(~r). In several numerical and theoretical works [34, 54, 15], thescreening within the 2DEG (without impurities) is analyzed in a self-consistent way. Screeningin the vicinity of the edge was studied separately in [19]. On the other hand, the experimentalresults of [55] clearly indicate that an important screening e�ect can be due to the back gatesused in many QHE experiments. Such a back gate is usually a metallic plate on which the 3Dsemiconductor (doped GaAs) is mounted and it can be used to vary the particle density of the2DEG. These gates are within a mm of the 2DEG, therefore the screening of their electronscan lead to an important electric �eld in the 2DEG. Another important e�ect not su�cientlyanalyzed to our knowledge is the electric �eld build up by the occupied localized states in thesample. In fact, if the chemical potential varies from left to right border within a region oflocalized states, the corresponding variation of the occupation rate of the localized states mustlead to important electric �elds within the 2DEG.We conclude this section with a brief discussion of the role of the contacts. It is basedon the works of B�uttiker [17, 18]. If the contacts are ideal as in the above sense, then thechemical potential at the whole left edge is equal to the chemical potential �1 of contact 1,hence �2 = �3 = �1. Similarly, �5 = �6 = �4. Thus not only the Hall and direct resistance arequantized, but also the two-terminal contact resistance R1;4 is quantized and equal to the Hallresistance if all the contacts are ideal. This quantization of R1;4 was actually observed [28].However, B�uttiker further argued that the Hall and direct resistance are also quantized if eitheronly the contact 1 and 4 or only the contacts 2,3,5 and 6 are ideal. The other contacts in thesesituations can then be disordered or dirty in the sense that they only absorb a portion of the18

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incoming particles and that the distribution of the outgoing particles can be a non-equilibriumone. For example, if contacts 1 and 4 are dirty in this sense, there would be no equilibriumdistribution beneath the edge states between contacts 1 and 2 as well as 4 and 5. Here one couldnot work with Fermi projections as in all the arguments above. However, the ideal contacts 2and 5 would then absorb the whole current and reemit it in an equilibrium distribution. Hencethe Hall and direct tension are quantized as before. Furthermore, all contacts could be dirtyif the inelastic scattering within the sample is e�ective. This inelastic scattering would thenequilibrate all the edge channels, for example, along the path between contacts 1 and 2, nomatter what the occupation of the edge states at contact 1 is. This latter picture is, however,physically incorrect. Several experiments [18] have shown that the equilibration length betweenthe edge channels is very long and typically of the size of the sample. Such a long equilibrationlength is also theoretically consistent [19]. Furthermore, in the situation with dirty contacts,the two terminal resistance R1;4 should not be quantized as it experimentally is. Hence theQHE is due to ideal contacts and as all of them are more or less identical, the above picturewith six ideal contacts is probably justi�ed.4 Edge current theory4.1 Traceclass property near the Fermi surfaceThe following summability result will be used in the next section. It also implies that the edgecurrent is actually a well-de�ned mathematical quantity under a gap condition on the planeoperator. The precise conditions on the boundary condition of H = (H; K) are the following:Hypothesis: K = K1 + K2 2 C3(E) and there is an L <1 such that for all ! 2 :hnx; nyj�!(K1)jmx; myi = 0 for ny > L , hnx; nyj�!(K2)jmx; myi = 0 for my > L .Let further the x-sign operator on `2(Z�N) be de�ned byXjXj jmx; myi = sgn(mx)jmx; myi = ( jmx; myi , mx � 0 ,�jmx; myi , mx < 0 , (mx; my) 2 Z�N .Proposition 3 Suppose that H = (H; K) 2 C3(T (A)) where K satis�es the above hypothesis.Let the interval � � R be contained in a gap of the spectrum of �!(H) for all ! 2 g whereP(g) = 1. Let f be a real C4-function supported in �. Then f(H) 2 C3(E) \ L1(E ; TE) andmoreover the Hilbert-Schmidt norm of the operator [ XjXj ; �!(f(H))] on `2(Z �N) is uniformlybounded for all ! 2 g.Proof. First recall that every operator A 2 C (A) satis�esjhnj�!(A)jmij � c1 + jn�mj , (28)19

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where we use the notation jn�mj = jnx�mxj +jny�myj . A similar bound holds for operatorsin C (T (A)). As H 2 C3(T (A)) and f is a C4-function, we deduce that f(H) 2 C3(T (A))(this follows from a straight-forward generalization of [46, Theorem 3.3.7]) so that (28) holdsfor f(H) with = 3.Furthermore, given E 2 �, let E 0 7! gE(E 0) be some C4-function equal to (E � E 0)�1outside of �. By the same argument as above, the operator gE(H) is in C3(A) as long asE 2 �. Maximising over E 2 �, we conclude that the bound (28) holds for A = (E � H)�1for a constant c which can be chosen uniform in E 2 � and ! 2 g. As last preliminary letus note that the operator (H ��H) = K1 + (K2 ��H(1��)) from `2(Z2) to `2(Z�N) alsosatis�es (28) for all ! 2 and thathnj�!(K2 � �H(1� �))jmi = 0 for my > L .The main idea is now to use these bounds in Stone's formula for f(H) in which the resolvantis developed according to the geometric resolvant formula. As all the above bounds hold for all! 2 g, we suppress the index ! 2 g throughout the rest of the proof.We replace the geometric resolvant identity1z�� H = � 1z �H + 1z�� H (H � �H) 1z �H . (29)twice into Stone's formula for f(H) and develop (E � {� � H)�1 into a power series in �. Ofthis series only the constant term, namely (E �H)�1, remains after the limit �! 0:f(H) = 12�{ s-lim�!0 Z� dE f(E) 1E � {�� H � 1E + {�� H! (H � �H) 1E �H , (30)and thereforejhnjf(H)jmij � c Xl2Z�N Xk2Z2 jhnjf(H)jlij jhljH � �Hjkij supE2� jhkj 1E �H jmij . (31)As pointed out above, the operator H � �H is localized near the boundary. We decompose itinto two parts K1 and K2 � �H(1� �) for which we do the following bounds seperately. Thecontribution C2 of K2 � �H(1� �) can be bounded byC2 � c Xl2Z�N Xkx2Z;ky�w 11 + jn� lj3 11 + jl � kj3 11 + jk �mj3� c Xlx;kx2Z Xly�0 11 + jn� lj3 11 + jlx � kxj2 11 + jkx �mxj3 +m3y� c Xlx;kx2Z 11 + jnx � lxj2 11 + jly � kyj2 11 + jkx �mxj3 +m3y20

Page 21: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

� c 11 + jnx �mxj3 +m3y + 11 + jnx �mxj2 11 +m2y! , (32)where the constants c vary, and we have used various elementary estimations such as (�; > 1):Xk2Z 1jaj+ jkj� 1jbj+ jk �mj � c a�1+1=� 1jbj+ jmj + b�1+1= 1jbj+ jmj�! .Similarly, one can bound C1. Hence hnjf(H)jmi can be bounded by the r.h.s. of (32). This inparticular implies that for all ! 2 g:jh0; myjf(H)j0; myij � c1 +m2y .As this holds P-almost surely with a uniform constant, it directly implies that f(H) is TE -traceclass. Replacing (32) twice intoTr`2(Z�N) 0@�����" XjXj ; f(H)#�����21A = Xn;m2Z�N(sgn(mx)� sgn(mx + nx))2 jhnjf(H)jmij2 ,the second assertion follows from a short calculation. 2Remark 1 The above prove can be slightly modi�ed in order to prove that P� 2 L1(E ;P)and is TE -traceclass under the slightly stronger hypothesis that H 2 C4(A). For that purposeone has to replace f in (30) by the indicator function on � and then bound hnjP�jli in (31)trivially by 1. However, the proof of Theorem 1 below shows that the TE -traceclass property ofP� holds whenever H 2 C3(A). �Remark 2 Replacing the geometric resolvant identity into the Stone formula for P� just asabove shows that in general P� = �P� + R where P� is the spectral projection of H onto �and R is a rest. Under the hypothesis that � is a dynamically localized region of the spectrum(or that in � the localization bound [2, Lemma 4] holds), we expect R to be TE -traceclass.Furthermore note that the term �P� does note contribute to the current becauseTE(�P�rxH) = Xmy�0 T (P�rxH) = 0 ,where in the last step we used [13, Proposition 3] showing that T (P�rxH) = 0. Note that itis essential in this calculation to take the trace per unit volume in the x-direction before takingthe usual trace in the y-direction. If R is TE -traceclass, the de�nition of the edge current isthus meaningful in this sense. �Remark 3 In a more general context one can show that, if A 2 E is TE -traceclass and anelement of C2(E), then [ XjXj ; �!(A)] is a Hilbert-Schmidt operator on `2(Z �N) for P-almostall ! 2 . This follows from the remarkable identity21

Page 22: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

14 Z dP(!) Tr`2(Z�N)0@�����" XjXj ; �!(A)#�����21A = 1{ TE((�A)�rxA) , (33)holding for any A 2 E0 where(�A)(!;mx; ny; my) = sgn(mx) A(!;mx; ny; my) .As A 2 C2(E) implies that (�A) 2 E , the formula (33) can be extended to C2(E) \ L1(E ; TE),implying hence the above result. Note that this does not give us a uniform bound in ! thoughand is thus a weaker result (not su�cient for our purposes below). A proof of (33) can be givenalong the lines of that of Proposition 4. �4.2 1-cocycles over the edge algebraWe will show in the next section that the edge Hall conductivity is given by TE((U��1)rxU) forsome unitary operator U 2 ~E . Hence it can be interpreted as a non-commutative winding num-ber. In order to explain and further develop this analogy, let us recall that the winding numberof a continuous unitary function f :S1!C is given by the similar expression RS1 f �df=(2�{); itis furthermore equal to the Fredholm index of the associated Toeplitz operator on Hardy space.There is a corresponding index theorem for the edge Hall conductivity, once we have linkedit to a 1-cocycle of a standard Hilbert space framework. This is done in Proposition 4 below.From then on, one only has to follow the various steps of Connes' index calculation [22].The strategy in all these calculations is to �rst restrict oneself to a dense sub-algebra whereall algebraic manipulations can be easily justi�ed, and then to extend the index theorem tosuitable unitary elements of the C�-algebra.The following expression is well-de�ned and �nite for A; B 2 E0:�1(A; B) = { TE(ArxB) . (34)Actually �1 can be de�ned on a much wider class of operators. For the purpose of this work, itis su�cient to consider elements in D = C2(E) \ L1(E ; TE). Note that the product rule for rxand the ideal property of the TE -traceclass operators imply that D is an algebra. It becomes anormed algebra when furnished with the norm:kAkD = kAk+ kr2xAk+ TE(jAj) .Lemma 2 �1 is a cyclic 1-cocycle on D, notably it is cyclic and closed under the Hochschildboundary operator b:(i) �1(A; B) = ��1(B; A) for all A; B 2 D.(ii) 0 = b�1(A; B; C) = �1(AB; C)� �1(A; BC) + �1(CA; B) for all A; B; C 2 D.22

Page 23: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

Proof. Both algebraic identities can be veri�ed using the product rule for the derivation rxand the invariance of the trace TE under rx. 2Next we introduce another 1-cocycle on the unitization ~D of D by setting�1(A; B) = Z dP(!) �!1 (A; B) , A; B 2 ~D ,where �!1 (A; B) = 14 Tr`2(Z�N) XjXj " XjXj ; �!(A)# " XjXj ; �!(B)#! . (35)According to Remark 3 of the last section, this cocycle is actually well-de�ned and �nite on D.Proposition 4 On D, we have �1 = �1.Proof. It is su�cient to show the equality for the dense subalgebra E0 � D because both �1and �1 are continuous with respect to k : kD. A direct calculation shows that for A; B 2 E0:�1(A; B) = �14 Z dP(!) Xm2Z�N Xl2Z�N sgn(mx)(sgn(mx)� sgn(lx))2hmj�!(A)jli hlj�!(B)jni .Because A 2 E0, the sum over mx 2 Z actually only contains a �nite number of elements,and can thus be exchanged with the integral over P. Then we make the change of variablesnx = lx �mx and use the covariance relation (9) in order to obtain:�1(A; B) = �14 Xmx2N Z dP(!) Xmy ;ly2N Xnx2Z sgn(mx) (sgn(mx)� sgn(mx + nx))2 �� h0; myj�T (�mx;0)!(A)jnx; lyi hnx; lyj�T (�mx;0)!(B)j0; myi .Next, by invariance of the measure P, we can replace T (�mx;0)! by !. Then we change the sumover mx and the integral over P again and use the identityXmx2N sgn(mx) (sgn(mx)� sgn(mx + nx))2 = � 4nx .By de�nition of rx, one therefore has�1(A; B) = { Z dP(!) Xmy2Nh0; myj�!(A)�!(rxB)j0; myi ,which is precisely �1(A; B). 2Finally let us introduce a further 1-cocyle on ~D using the projection operator �x = (1 +X=jXj)=2 on `2(Z�N): 23

Page 24: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

�1(A; B) = Z dP(!) �!1 (A; B) , A; B 2 ~D ,where �!1 (A; B) = Tr`2(Z�N)(�x[�!(B); �!(A)]�x � [�x�!(B)�x;�x�!(A)�x]) . (36)Proposition 5 On ~D, we have �!1 = �!1 for all ! 2 .Proof. Some algebra shows�x�!(A)�!(B)�x � �x�!(A)�x�!(B)�x = �14�x " XjXj ; �!(A)# " XjXj ; �!(B)# . (37)Note that if h XjXj ; �!(A)i h XjXj ; �!(B)i is traceclass, so is the left-hand side (this is the case forA; B 2 ~D by Proposition 3). Hence using the same identity with A and B exchanged, weobtain:�!1 (A; B) = 14 Tr`2(Z�N) �x " XjXj ; �!(A)# " XjXj ; �!(B)#� �x " XjXj ; �!(B)# " XjXj ; �!(A)#!= 12 Tr`2(Z�N) XjXj " XjXj ; �!(A)# " XjXj ; �!(B)#� XjXj " XjXj ; �!(B)# " XjXj ; �!(A)#! :Note here that the second equality holds because both h XjXj ; �!(A)i and h XjXj ; �!(B)i are Hilbert-Schmidt [42]. From the above we deduce that�!1 (A; B) = 12 (�!1 (A; B)� �!1 (B; A)) ,and the cyclicity property of �!1 allows to conclude. 2Corollary 1 On D, we have �1 = �1 = �1.It is now immediate to deduce the two following odd index theorems for stochastic operators.Proposition 6 Suppose (only for the purpose of this proposition) that P is ergodic w.r.t. theZ-action Tx. Let A 2 ~D be invertible. Then �x�!(A)�x is P-almost surely a Fredholm operatoron `2(N �N) the index of which is P-almost surely independent of ! 2 . The almost surevalue is equal to �1(A� �; A�1 � ��1) whenever A� � 2 D, � 2 C, � 6= 0.24

Page 25: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

Proof. Because A 2 ~D, Proposition 3 implies that �!1 (A; A�1) < 1 for P-almost all ! 2 .Thus (36) and the Fedosov formula (e.g. [13, Prop. 8]) imply that �x�!(A)�x is a Fredholmoperator on �x`2(Z�N) = `2(N�N) and that its index is equal to �!1 (A; A�1). Because�x�T (a;0)!(A)�x j`2(N�N) = �x�!(A)�x +K jU(a;0)`2(N�N)�=`2(N�N) ,where K is a compact operator on `2(N�N) and the Fredholm index is invariant under compactperturbations, we see that the index is Tx-translation invariant in ! 2 . Hence it is P-almostsurely constant by the ergodicity ofP with respect to Tx. As �!1 (A; A�1) = �!1 (A��; A�1���1),Corollary 1 implies that the almost sure index is equal to �1(A� �; A�1 � ��1). 2In our context, the measure P is only ergodic w.r.t. the Z2-action T . However, this issu�cient to given an almost sure index for certain elements in ~D, notably those in the imageof the exponential map.Proposition 7 Let H satisfy the hypothesis of Proposition 3 and let g be a real C4 functionwith values in [0; 1], equal to 0 or 1 outside of �. Set U = exp(�2�{ g(H)). Then �x�!(U)�xis P-almost surely a Fredholm operator on `2(N � N) the index of which is P-almost surelyindependent of ! 2 . The almost sure value is equal to �1(U � 1; U�1 � 1).Proof. By Proposition 3, U 2 ~D. Thus from the proof of Proposition 6 follows that �x�!(U)�xis P-almost surely a Fredholm operator and that its index is Tx-invariant. To conclude, we haveto show its Ty-invariance.Herefore, let Q : `2(Z �N) ! `2(Z �N) denote the projection on the subspace spannedby f jmx; 0i j mx 2 Zg. Now, if H = (H; K) 2 T (A), we can use the equation �!(H) =��!(H) + �!(K) and the covariance relation Uy�T�1y !(H)U�y = �!(H) in order to obtain�!(H) = Uy�T�1y !(H)U�y + R! ,where R! is an operator family, covariant in the x-direction and compact in the y-direction andgiven byR! = �!(H)Q+Q�!(H)(1�Q) + (1�Q)�!(K)(1�Q) + Uy�T�1y !(K)U�y .Now R is actually a boundary operator satisfying the hypothesis of Section 4.1. Hence theoperator Uy�T�1y !(H)U�y + �R! is a lift of �!(H) and by Proposition 3U!(�) = exp(�2�{ f(Uy�T�1y !(H)U�y + �R!)) .is P-almost surely a norm-continuous (in �) family of Fredholm operators. Clearly U!(1) =�!(U) and moreover, because U�y Uy = 1, we have U!(0) = Uy�T�1y !(U)U�y . Therefore thehomotopy invariance of the Fredholm index implies:Ind(�x�!(U)�x) = Ind(�xUy�T�1y !(U)U�y�x) = Ind(�x�T�1y !(U)�x) ,25

Page 26: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

which concludes the proof. 2Remark. (Not needed for the understanding of the rest of this work.) There are underlyingFreholm modules in the above calculations. Recall [22] that a Fredholm module over an algebraA is a family (A; ~H; ~�; F ) where ~� is a trivially graded representation ofA on a graded seperableHilbert space ~H and where F is a self-adjoint operator on ~H such that(i) FG = �GF , (ii) F 2 = 1 , (iii) [~�(A); F ] 2 K 8 A 2 A .Here G is the graduation operator, and K denotes the compact operators on ~H. One thende�nes the di�erential (with help of the graded commutator [ ; ]s) and the supertrace of abounded operator T on ~H as usual:dT = [F; T ]s , Trs(T ) = 12 Tr(GFdT ) ,Let us consider the Fredholm module (D; `2(Z�N) `2(Z�N); ~�!; F ) where~�!(A) = �!(A) 00 �!(A) ! , F = { 0 XjXj� XjXj 0 ! .It is now a matter of calculation to verify that the 1-cocycle associated to this Fredholm modulecoincides with the one-cocycle �!1 de�ned in (35):�!1 (A; B) = �14 Trs(~�!(A)d~�!(B)) .4.3 Calculation of the edge currentThis section contains the proof of the edge current quantization and some comments.Proof of Theorem 1. Let G 2 C2(R) be a positive function supported on an interval �0 � �.We set g(E) = 1� 1RR dE 0G(E 0) Z E�1 dE 0G(E 0) ,and the unitary operator W = exp(�2�{ g(H)). By Proposition 7, �!(W ) and �!(W �) areP-almost surely Fredholm operators with a P-almost sure index. Changing �0 within � (orchanging G) does not change their index by the homotopy invariance of the Fredholm index.Hence their indices are actually associated to the gap � and can moreover be calculated as theindex of U(�) and U(�)� respectively. We denote that of U(�)� by Ind so thatInd = { TE((W � � 1)rxW ) . (38)Now recall Duhamel's formula for rxW �:rxW = �2�{ Z 10 ds W 1�s rxg(H) W s ,26

Page 27: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

where the integral is de�ned as a norm convergent Riemann sum in E . We replace this into(38), use the continuity of TE in order to exchange the trace with the integral over s and �nallyuse the unitary invariance of TE to deduceInd = 2� TE((1� W )rxg(H)) .To calculate this, we develop g(E) = Pk ckTk(E) in a series of Tchebychev polynomials over� (slightly enlarged if necessary). Because g 2 C4(R), this series converges absolutely anduniformly on �0 � �, so the sum over k can be exchanged with TE . On each polynomial in H,we now apply the gradient, and then use the cyclicity of the trace and [W ; H] = 0 in order toregroup all polynomials in H to the left of rxH. Because G(E) = �Pk ckT 0k(E)�(RR dE 0G(E 0)),we conclude that Ind = 2�RR dE 0G(E 0) TE((W � 1)G(H) rxH) .Now let Gj 2 C3(R) be a sequence of functions converging from below to the indicator functiononto �0. If Wj denotes the associated unitary, then (Wj � 1)Gj(H) converges in norm toU(�0)� 1. Therefore, for any �0 � �,Ind = 2�j�0j TE((U(�0)� 1) rxH) .For a given k 2 N, k � 1, we devide � into k intervals �j = [Ej�1; Ej], j = 1 : : : k, of equallength j�j=k. Note that E0 = E 0 = inf(�). Let us setUj � 1 = exp �2�{ H � Ejj�jj P�j!� 1 = (U(�)k � 1) P�j .Now the Fredholm index of each U�j is equal to that of U(�)�, namely Ind. Hence it followsthat Ind = 1k kXj=1 2�j�jj TE((Uj � 1) rxH) = 2�j�j TE((U(�)k � 1) rxH) . (39)Introducing a sign in the above de�nition of the Uj's, it can be veri�ed that (39) also holds fornegative integers, so it holds for all k 2 Z except k = 0.Finally we make a Fourier analysis on the interval �. Let �0 � � be an interval such thatinf(�0) > E 0 and let �00 = �n�0. The indicator function ��00 on �00 (taking the value 1=2 onthe two boundary points of �00) can now be written as:��00(E) = Xk2Z ak exp �2�{j�j k(E � E 0)! , ak = Z� dEj�j ��00(E) exp �2�{j�j k(E � E 0)! ,where the convergence of the Fourier series is pointwise. We have:27

Page 28: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

Xk2Z ak = 1 , a0 = j�00jj�j , Xk 6=0 ak = j�0jj�j ,so that (except for the unimportant boundary points):P� Xk 6=0 ak U(�)k = j�0jj�j P� � P�0 .Finally, summing (39) we thus obtain:Ind = j�jj�0j Xk 6=0 ak 2�j�j TE(P� U(�)k rxH)� 2�j�j TE(P� rxH) = � 2�j�0j TE(P�0 rxH) .�0 being an arbitrary interval in �, the result is thus proven. 2We conclude this section with a brief comment on why there actually is an edge current inpresence of a magnetic �eld. Although physically clear, it also follows from the following formalalgebraic calculation with the Harper Hamiltonian H = Ux + U�x + Uy + U�y . It is an elementof the rotation algebra A� where � is the ux per unit cell. First note that rxH = {(Ux� U�x).Thus: TE(HrxH) = Xmy�0 h0; myj(UyUx + U�y Ux � UyU�x � U�y U�x)j0; myi = 0 .Similarly, one can check with a little more e�ort and using the algebraic relations of Ux and Uythat TE(H2rxH) = 0, but TE(H3rxH) = sin(�) 6= 0 for non-vanishing magnetic �eld. Otherhigher powers of H and hence functions of H also give non-vanishing results. Actually, it isnot surprising that the third power gives the �rst contribution because it corresponds to thewalk from one edge side via the interior to the next edge site. The probability (or quantumexpectation) for this (either to the right or the left depending on the value of �) is positive.5 Duality of pairings of K-theory with cyclic cohomologyGiven a Banach algebra B, one can de�ne abelian groups which are dual to the K-groups of Bin the sense that there is a natural pairing between the two which takes values in the integers.The dual of K1(B) is e.g. given by the group Ext(B) of extensions of B and the pairing isquite simple [16]. Using suspension, one can then also obtain a six-term exact sequence of theExt-groups from a short exact sequence of algebras [16]. But when it comes to pair K0(B) withExt(SB), one realizes that it would be better to work with a di�erent picture of the dual ofK0(B) which is not based on suspension. Such a theory is fully developped in the topologicalcontext and goes under the name KK-theory. On the other hand, there is a cohomologytheory for algebras, cyclic cohomology, which is de�ned on the subalgebra of so-called smoothelements of B and which also pairs with K0(B), but not always integrally. It is related to the28

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dual of K0(B) in the sense of KK-theory and produces the same result when both apply. Butdespite the two apparent disadvantages (only de�ned on smooth elements, pairing not alwaysinteger valued), its advantages overwhelm: �rst, we �nd it technically and computationallymuch simpler to handle in our case, and second, it is not tight to the topological category andthe pairing extends to certain projections which are no longer in B. This allowed [13] to provequantization in the intermediate disorder regime and we hope that it will eventually lead toa proof of Theorem 2 under a similar hypothesis. Our presentation here is adopted from theworks of Pimsner [43] and Nest [41], with simpli�cations adequate to our context.5.1 De�nition of cyclic cohomologyGiven a complex algebra B, let Cn� (B) be the set of n + 1-linear functionals on B which arecyclic in the sense that '(A1; � � � ; An; A0) = (�1)n'(A0; � � � ; An) .De�nition 2 The cyclic cohomology HC(B) of B is the cohomology of the complex0! C0�(B)! � � � ! Cn� (B) b! Cn+1� (B)! � � �with b'(A0; � � � ; An+1) = nXj=0(�1)j'(A0; � � � ; AjAj+1; � � � ; An+1)+(�1)n+1'(An+1A0; � � � ; An): (40)An element ' 2 Cn� (B) satisfying b' = 0 is called a cyclic n-cocycle.Example 1 HC0(B) is the set of traces on B. In fact, let T : B ! C be a linear functional,then bT (A0; A1) = T (A0A1)� T (A1A0) .Example 2 Let rj, j = 1 � � � ; n be n commuting derivations on B (not necessarily fullyde�ned) and T an invariant trace (T � rj = 0), not necessarily �nite. Then �n : An+1 ! Cde�ned by�n(A0; � � � ; An) = X�2Sn(�1)sgn(�)T (A0r�(1)A1 � � �r�(n)An) = T (A0r[1A1 � � �rn]An) ,(the second expression is an abbreviation) is an n-cocycle. This will be proved below. Thedomain of de�nition of �n depends on the domains of de�nition of the derivations and that ofT . 29

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5.2 Pairing with K-groupsNow let B be a Banach algebra. The elements of cyclic cohomology of degree j pair with Kj(B)(j mod 2 in the latter case) in the following way. Let � be a 2k-cocycle, [�] its class. Thenh[P ]0 � [Pm]0; [�]i = c2k Tr �(P; : : : ; P ) , P 2Mm+l(B) ,is a well-de�ned pairing between K0(B) and HCev(B) [22]. If � is a 2k + 1-cocycle, thenh[V ]1; [�]i = c2k+1 Tr �(V �1 � 1; V � 1; : : : ; V � 1) , V 2 Mm(B) ,(the entries alternate) is a well-de�ned pairing between K1(B) and HCodd(B). Here the nor-malization constants are (chosen as in [21] and [43], but not as in [22]):c2k = 1(2�{)k 1k! , c2k+1 = 1(2�{)k+1 122k+1 1(k + 12)(k � 12) � � � 12 .This lets cyclic cohomology appear as a dual theory to K-theory, but there are fundamentaldi�erences between the two. The category of algebras for which these invariants are de�ned isdi�erent. K-theory is topological, whereas cyclic cohomology is rather a di�erential theory. Inparticular, one often considers functionals (cyclic cocycles) which are merely densely de�nedon a (local) Banach algebra (or not even densely). Nevertheless, HC has functorial propertieswhich are analogous and compatible with those of K.5.3 CyclesThere is a convenient reformulation of the description of cyclic cocycles in terms of gradeddi�erential algebras (; d) with graded closed traces (we use the widely spread notation here,because misinterpretation as the hull of Section 2.1 seems excluded). A graded algebra is agraded vectorspace = Ln2Zn (we denote by @A the degree of a homogeneous element A)such that @(AB) = @(A) + @(B) for homogeneous elements. It is called a graded di�erentialalgebra if there exists a graded di�erential d (a derivation whose square vanishes) of degree 1.A graded trace on n is a linear functional R which satis�es R w1w2 = (�1)@w1+@w2 R w2w1. Itis closed if R vanishes on d(n�1).De�nition 3 A n-dimensional cycle over B is a graded di�erential algebra (; d) of highestnon-trivial degree n together with a closed graded trace R on n and an algebra homomorphismB ! 0.We will assume here that the homomorphism B ! 0 is injective and hence identify B witha subalgebra of 0. See [22] for the proof of the following result.Proposition 8 Any cycle of dimension n over B de�nes a cyclic n-cocycle through what iscalled its character: �(A0; � � � ; An) = Z A0dA1 � � �dAn .30

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Example 3 Consider an action of Rn on the algebra B by commuting derivations rj (thusview Rn as Lie algebra) and suppose that T is an invariant trace, i.e. T (exp(trj)A) = T (A)for all t. Let be the tensor product B �Cn of B with the Grassmann algebra �Cn withgenerators ej, j = 1; � � � ; n and de�ned(A v) = nXj=1rjA ejv .It is straightforwardly checked that (; d) is a di�erential algebra. Note that �nCn�=C (asvector spaces) and any isomorphism is a graded trace. Taking {(e1 � � � en) = 1 as such anisomorphism, we de�ne R = � {, that isZ A0dA1 � � �dAn = T (A0r[1A1 � � �rn]An) ,a cycle of dimension n associated to our second example. Let us prove that this is actuallyclosed and graded. The graded cyclicity follows directly from the graded cyclicity of { and thecyclicity of T . To prove closedness write T { = { � (T id) on the highest degree. Since dacts trivially on C�Cn, the invariance of T implies (T id) � d = 0. This proves the claim.Note that the situation of this example arrises always if we have a dynamical system (B; G)with an abelian Lie group G.One might wonder whether one cannot always �nd a cycle for a cyclic cocycle. The con-struction of the universal di�erentiable algebra (B) over B gives a positive answer to thatquestion.5.4 Constructing new cycles from old onesWe next extend Example 3. Suppose we have the same situation as in that example exceptthat in place of an invariant trace there is an invariant k-cycle (; �; R ). By this we mean thatRn acts on by derivations such that � commutes with this action and the graded trace R isinvariant. The aim is to construct a k + n-cycle from these data. Before we do so let us recallthat the graded tensor product � of two graded algebras is the graded tensor product of thevector spaces with product (w1v1)(w2v2) = (�1)@w2@v1w1w2v1v2. In particular, it reducesto the usual tensor product if one of the algebras is trivially graded.Proposition 9 Let (; �; R ) be a k-cycle over the algebra B with an action of Rn by n deriva-tions ri which commute with � and leave R invariant. Taking 0 = �Cn, the graded tensorproduct, d0 = �1 + d with d(wv) = (�1)@wPj rjwejv, and R 0 = R { one obtains ak + n-cycle (0; d0; R 0) over B.Proof: 0 is naturally bigraded and one de�nes the total degree to be the sum of the compo-nents of the bidegree. A straightforward calculation shows that that � and d are anti-commutinggraded di�erentials which are of total degree 1. Therefore d0 = d + � is a di�erential of totaldegree one and (0; d0) a graded di�erential algebra with respect to the total degree. It remains31

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thus to check that R 0 is a closed graded trace. First note, that R 0 is non-zero precisely at thehighest non-trivial (total) degree. FurtherZ 0(w1v1)(w2v2) = (�1)@w2@v1 �Z w1w2� {(v1v2)= (�1)@w2@v1+@w1@w2+@v1@v2 �Z w2w1� {(v2v1)= (�1)(@w1+@v1)(@w2+@v2) Z 0(w2v2)(w1v1)shows that R 0 is a graded trace. We haveZ 0 d0(wv) = �Z �w� {(v) +Xj (�1)@w �Z rjw� {(ejv):The �rst term vanishes due to closedness of R and the second due to its invariance. Thus R 0 isclosed. 2There are two applications of the above which are important for us.5.4.1 Suspension of cyclic cocyclesIn the �rst, we consider a Banach algebra B with an n-cycle (; d; R ) and look at C(S1;B) thealgebra of continuous functions over S1 = R=2�Z with values in B. S1 is an abelian Lie groupthus acting on itself and we take this action and extend it trivially to C(S1;B). Then the Liealgebra R of S1 acts by di�erentiation along the parameter of S1 which we denote t. We de�nefor C(S1;B) an n-cycle (0; d0; R 0) as follows. 0 = C(S1;), d is the trivial extension of d andfor R 0 we combine the old n-cycle for B with integration (Lebesgue measure) over S1, namelyR 0 = RS1 dt R . Now Proposition 9 provides us with an n + 1-cycle (00; d00; R 00) for C(S1;B).Its restriction to the suspensions SB, sub-algebra of C(S1;B), plays an important role for us.Concretely, for Ai : S ! B, we getZ 00A0d00A1 � � �d00An+1 = n+1Xj=1(�1)n+1�j ZS1 dt Z A0dA1 � � �dAj�1 _AjdAj+1 � � �dAn+1 .If � denotes the character of (; d; R ) in this example then we denote by �s the character onthe suspension SB obtained from (00; d00; R 00) in the above way. Note that this is a suspensionconstruction that di�ers from Connes' [21, 22]. The following di�cult theorem is due Pimsner[43].Theorem 3 The map HCn(B) ! HCn+1(SB): � 7! �s is dual to the Bott map � for evencycles and dual to the suspension map � for odd cocyles, i.e.h[P ]0 � [Pm]0; [�]i = h�([P ]0 � [Pm]); [�s]i32

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for even cocycles � over B and h[V ]1; [�]i = �h�([V ]1); [�s]ifor odd ones.5.4.2 Cyclic cocycles for crossed productsIn the second application, we consider a n-cycle (; d; R ) over an algebra B on which is givena Z-action (such that d commutes with this action and R is invariant under it). The aim is toconstruct a n+ 1-cycle over B �� Z, the algebraic crossed product. To do so we �rst constructa n-cycle over B �� Z and then apply Proposition 9. The n-cycle over B is given by (0; d0; R 0)where 0 = �� Z, the algebraic crossed product, d0 the trivial extension of d to functionsA : Z ! , i.e. (d0A)(m) = dA(m), and R 0A = R A(0). To show that d0 is a derivation oneneeds indeed that it commutes with the Z-action � and the latter is also essential to show thatR 0 is a graded trace. Now on B and on we have an action of the dual group of Z which is S1.Its Lie algebra acts by the derivationrA(m) = {mA(m) .This action commutes with d0 and leaves R 0 invariant, the latter simply because rA(0) = 0.We thus can apply Proposition 9 to obtain an n + 1-cycle (00; d00; R 00). More explicitly, forA : Z! B we getZ 00A0d00A1 � � �d00An+1 = n+1Xj=1(�1)n+1�j Z (A0dA1 � � �dAj�1rAjdAj+1 � � �dAn+1) (0) .Now, if � is the character of (; d; R ), we denote by #�� the character corresponding to theabove n+ 1-cycle (00; d00; R 00). The proof of the following lemma is a straightforward algebraiccalculation.Lemma 3 Let B be an algebra furnished with a Z-action �, further let (; d; R ) be a �-invariantn-cycle over B and � its character. Then #�(�s) = (#��)s.5.5 Duality theorem for crossed productsThe following theorem appears in Nest's article [41, Prop. 12.6], but with a partially erronousproof. We use the homotopy invariance of cyclic cohomology in an analogous way to that of[27] in order to correct it.33

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Theorem 4 Let (B;Z; �) be a C�-dynamical system, (; d; R ) be a �-invariant n-cycle over Band � its character. The map HCn(B)! HCn+1(B �� Z): � 7! #�� is dual to the connectingmap of the Pimsner-Voiculescu exact sequence, i.e.hexp([P ]0 � [Pm]0); [�]i = 2� h[P ]0 � [Pm]0; [#��]i ,for odd n, and hind([V ]1); [�]i = �2� h[V ]1; [#��]i ,for even n.Proof: We denote the unitary generating the action � in B �� Z by U , that is �(A) = UAU�for all A 2 B.Due to the Lemma 3 and Theorem 3, it su�ces to consider only the case of even n becausethe crossed product commutes with the suspension, notably SB �� Z = S(B �� Z).Now let [P ]0 2 K0(B) be in the image of the index map. From the Pimsner-Voiculescuexact sequence we then deduce that [�(P )]0 = [P ]0. Thus there exists a unitary W 2 B whichis homotopic to the identity and such that �(P ) = WPW � (if necessary, one passes to matrixalgebras over B). According to Proposition 10 of the Appendix, the preimage of [P ]0 under theindex map is then [UP +W (1 � P )]1 2 K1(B �� Z). Therefore the second statement of thetheorem is equivalent toh[P ]0 � [Pm]0; [�]i = �2� h[UP +W (1� P )]1; [#��]i ,for an n = 2k-cocycle � of B. Suppose �rst that W = 1. Thenh[UP + 1� P ]1; [#��]i = c2k+1 Z 00 P (U� � 1)d00((U � 1)P ) � � �d00((U � 1)P )with the cycle for the character as above. Since the action � commutes with d0 and [U; P ] = 0,we can collect all U 's to the left. Furthermore, P (d0P )mP vanishes if m is odd and is equal toP (d0P )m if m is even. Using this the r.h.s. becomes{c2k+1(k + 1) R 0 U(U � 1)k(U�1 � 1)k+1P (d0P )2k= �{c2k+1(k + 1) kXl=0 kl ! k + 1l + 1 !Z P (dP )2k= �{c2k+1(k + 1) 2k + 1k !Z P (dP )2k= � 12� h[P ]0; [�]i .To treat now the case W 6= 1, we use the new action �0(A) = W ��(A)W and the associatedcrossed product B ��0 Z. Further let us introduce an automorphism of M2(B) by34

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A BC D ! = �(A) �(B)WW ��(C) W ��(D)W ! .Then there are two embeddings�1 : B �� Z!M2(B)� Z , �2 : B ��0 Z!M2(B)� Z ,onto the upper left and lower right corner of the 2�2-matrix respectively. Finally we introducethe C1-path AdVt :M2(B)� Z!M2(B)� Z, t 2 [0; 1], whereVt = cos(�2 t) � sin(�2 t)sin(�2 t) cos(�2 t) ! .As AdV1 interchanges the upper left and lower right corners, we can use the following commu-tative diagram to de�ne an isomorphism i : B �� Z! B ��0 Z:B �� Z �1! M2(B)� Zi # # AdV1B ��0 Z �2! M2(B)� ZOne can now check that i(A) = A for all A 2 B and i(U) = WU 0 where U 0 is the generator ofthe �0-action in B ��0 Z. Hence i(UP +W (1� P )) = W (U 0P + (1� P )). Further one checksthat #�� = ��1(# (Tr2 �)) , #�0� = ��2(# (Tr2 �)) .To verify that the pairings with [#��]1 and [#�0�]1 coincide, we now use Connes' homotopyinvariance of cyclic cohomology [22] extended to non-unital algebras in [27]. As V0 = 1, it tellsus that [# (Tr2 �)]1 = [Ad�V1# (Tr2 �)]1 in HCodd(M2(B)� Z). Therefore we obtain:h[UP +W (1� P )]1; [#��]1i = h[�1(UP +W (1� P ))]1; [# (Tr2 �)]i= h[�1(UP +W (1� P ))]1; [Ad�V1(# (Tr2 �))]i= h[i(UP +W (1� P ))]1; [��2(# (Tr2 �))]i= h[U 0P + (1� P )]1; [#�0�]i ,where in the last step we use [W (U 0P + (1�P ))]1 = [U 0P + (1�P )]1 because W is homotopicto the identity in B. Now this last pairing is in B ��0 Z where the above calcuation can beapplied in order to conclude. 235

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5.6 Duality theory for quantum Hall systemsAccording to the results of Chapter 4 and of Proposition 2, the de�nitions in Section 5.2 showthat edge and bulk Hall conductivity result from the following pairings of cyclic cohomologywith K-theory over E and A respectively:�e?(�) = �q2h 2�{ h[U(�)]1; [�1]i , �b?(�) = q2h 4�2{ h[P�]0; [�2]i .Here U(�) and �1 are de�ned in (25) and (34), P� is the Fermi projection of the plane operatoron a Fermi level � 2 � and �2 is the 2-cocylce over A de�ned by�2(A;B;C) = { T (ArxBryC � AryBrxC) .We now use the isomorphisms � and � de�ned in (22) and (23) in order to transform thesetwo pairings to pairings over the algebras C() ��x Z K and C() ��x Z ��y Z of thePimsner-Voiculescu exact sequence (21):�e?(�) = �q2h 2�{ h[��1(U(�))]1; [���1]i , �b?(�) = q2h 4�2{ h[��1(P�)]0; [���2]i .Now we may apply the duality Theorem 4 for Pimsner-Voiculesu exact sequences by settingB = C()��x Z on which the action is given by � = �y. Hence Theorem 2 is proven once wehave veri�ed that exp([��1(P�)]0) = [��1(U(�))]1 and ���2 = #�y���1. Concerning the latterequality one can immediately verify that, if one de�nes �1 and �2 by the same formulas on thecorresponding algebras of the Pimsner-Voiculescu exact sequence, then ���2 = �2 and ���1 = �1(use herefore that � and � are isomorphisms leaving the traces invariant); then going back tothe de�nition of #�y in Subsection 5.4.2, the identity �2 = #�y�1 can be directly checked.For the second equality, it is clearly su�cient to show that exp([P�]0) = [U(�)]1. We�rst note that P� is equal to the continuous function of the Hamiltonian g(H) = PE0 � (H �E 0)P�=j�j where E 0 = inf(�). A selfadjoint lift of P� is thus given by g(H). Now using[PE0; P�] = 0 and the fact that exp(2�{P ) = 1 for any projection P , we thus obtain from thede�nition of the exponential map recalled in the Appendix:exp([P�]0) = [exp(�2�{ g(H))]1 = [U(�)]1 .This concludes the proof of Theorem 2.Appendix: Some help for cooking with K-theoryThis is only a very brief summary of the main K-theoretic notions used in this work. We refer to[16, 53] for all the details and even for the de�nition of the groups K0(B) and K1(B) of a giveninvolutive Banach algebra B. Roughly, K0 is the Grothendieck group generated by homotopyclasses of idempotents in matrix algebras over B, and K1(B) is formed by homotopy classes of36

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invertibles (or equivalently unitaries) therein. However, great care needs to be taken when Bdoes not have a unit.The suspension SB of B is the algebra of continuous functions f : S1 ! B which vanish ata distinguished point that we choose to be 1. This construction provides a link between theK-groups. First, the following map is a group isomorphism:� : K1(B)! K0(SB) , [V ]1 7! "Wt 1 00 0 !W �t #0 � " 1 00 0 !#0 .Here V 2 Mn(B) is an unitary representative and Wt 2 M2n(B) a homotopy from V 00 V � !to 1. The blocksize in the matrix is n� n. One of the key properties of K-theory is that� : K0(B)! K1(SB) , [P ]0 7! [e2�{tP ]1 = [(e2�{t � 1)P + (1� P )]1 ,is also an isomorphism called the Bott map. It shows thatK0(B) �= K0(SSB), makingK-theorya periodic theory.Another crucial property of K-theory is that any short exact sequence of Banach algebras,0 ! J i! B �! B=J ! 0 ,gives rise to a six-term exact sequence of K-groups:K0(J ) i�! K0(B) ��! K0(B=J )ind " # expK1(B=J ) �� K1(B) i� K1(J )The de�nition of the push-forward maps i� and �� is immediate. Let us recall those of theindex and exponential map. Given a unitary V 2 Mn(J ) de�ning a class in K1(B=J ), letW 2M2n(B) be a unitary lift of V 00 V � !. Thenind([V ]1) = "W 1 00 0 !W �#0 � " 1 00 0 !#0 .The exponential map is de�ned by exp := ��1 � ind � �. Explicitly, if P 2Mn(B=J ) de�nes aclass in K0(B=J ) and L(P ) 2 B a self-adjoint lift for it, thenexp([P ]0) = [exp(�2�{ L(P ))]1 .We furthermore recall some facts about the Toeplitz extension (see [44]) of the C�-crossedproduct B �� Z associated to the C�-dynamical system (B;Z; �) (for unital B):0 ! B K ! T (B) �! B �� Z ! 0 . (41)37

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Here K are the compact operators on `2(N) generated by the �nite rank operators en;m, n;m 2N, the matrix units, and is the homorphism de�ned in [44]. We have met such an extensionin Section 2.6. The imbedding j : B ! B K, j(b) = b e11 induces an isomorphismK(B) j��= K(B K) and in [44] Pimsner and Voiculescu have shown that K(T (B))�=K(B) aswell. Keeping carefully track of these isomorphisms, they obtained from the six-term exactsequence associated to (41) the nowadays called Pimsner-Voiculescu six-term exact sequence:K0(B) i����1�! K0(B) i�! K0(B �� Z)ind " # expK1(B �� Z) i� K1(B) i����1� K1(B)Strictly speaking the maps ind and exp in this diagram are not the same as the ones in theoriginal sequence, but have to be composed with j�1� .For the convenience of the reader, we now produce a direct proof of the following result byElliott and Natsume [26] used in Section 5.5, which is a discrete (one-way) analogon of Connes'Thom isomorphism [20]. We denote the unitary generator of the action in B��Z by U , notably�(A) = UAU� for all A 2 B.Proposition 10 Let P 2 Mn(B) be a projection de�ning a class in K0(B) which is in theimage of the index map, and let W 2 Mn(B) be a unitary such that �(P ) = WPW �. Then apreimage of [P ]0 is the class de�ned by the unitary X = UP +W (1� P ).Proof: First note that the unitaryW exists (possibly after enlarging n) because [P ]0 = [�(P )]0by the Pimsner-Voiculescu exact sequence. In order to compute ind([X]1), we have to �nd alift W 2M2n(T (B)) of X 00 X� ! and to compute j�1� "W 1 00 0 !W �#0 � " 1 00 0 !#0!.There exists a lift U 2 T (B) of U with the following properties: U U� = 1, U�U = 1�P0, whereP0 = i(1 e11) and UAU� = �(A) for all A 2 B. In particular, P commutes with P0 and it isnot di�cult to verify thatW = UP +W (1� P ) 0PP0 PU� + (1� P )W � !is a lift of X 00 X� !. Furthermore,"W 1 00 0 !W �#0 � " 1 00 0 !#0 = " 0 00 PP0 !#0 = j�([P ]0) . 238

Page 39: x x - pdfs.semanticscholar.org...dk x (k) is the Blo c h decomp osition in x-direction and E j are the corresp onding edge c hannels, then for an in terv al n th gap, one has j e =

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