Quantization of Hall Currents in Presence of Disorder: Edge Currents Without Edges, Papers of Health sciences

Download Quantization of Hall Currents in Presence of Disorder: Edge Currents Without Edges and more Papers Health sciences in PDF only on Docsity! ON THE QUANTIZATION OF HALL CURRENTS IN PRESENCE OF DISORDER JEAN-MICHEL COMBES, FRANÇOIS GERMINET, AND PETER D. HISLOP Abstract. We review recent results of two of the authors concerning the quantization of Hall currents, in particular a general quantization formula for the difference of edge Hall conductances in semi-infinite samples with and without a confining wall. We then study the case where the Fermi energy is located in a region of localized states and discuss new regularizations. We also sketch the proof of localization for 2D-models with constant magnetic field with random potential located in a half-plane in two different situations: 1) with a zero potential in the other half plane and for energies away from the Landau levels and 2) with a confining potential in the other half plane and on an interval of energies that covers an arbitrary number of Landau levels. 1. The edge conductance and general invariance principles Quickly after the discovery of the integer quantum Hall effect (IQHE) by von Klitzing et. al. [vK1], then Halperin [Ha] put the accent on the crucial role of quantum currents flowing at the edges of the (finite) sample. Such edge cur- rents, carried by edge states, should be quantized, and the quantization should agree with the one of the transverse (Hall) conductance. While edge currents have been widely studied in the physics literature since the early eighties, e.g. [MDS, AS, FGK, ZMH, CFGP] (see also [PG, vK2] and references therein), it is only recently that a mathematical understanding of the existence of such edge cur- rents has been obtained [DBP, FGW, EJK1, EJK2, CHS1, FM, CHS2]. The study of the quantization of the edge Hall conductance at a mathematical level is even more recent [SBKR, KRSB, KSB, EG, CG, EGS]. We consider the simplest model for quantum devices exhibiting the IQHE. This consists of an electron confined to the 2-dimensional plane considered as the union of two complementary semi-infinite regions supporting potentials V1 and V2, re- spectively, and under the influence of a constant magnetic field B orthogonal to the sample. In the absence of potentials V1 and V2, the free electron is described by the free Landau Hamiltonian HL = p2x + (py −Bx)2. The spectrum of HL consists of the well-known Landau levels BN = (2N − 1)B, N ≥ 1, with the convention B0 = −∞. To introduce the half-plane potentials, we let 1− and 1+ be the char- acteristic functions of, respectively, {x ≤ 0} and {x > 0}. Then, if V1, V2 are two potentials bounded from below and in the Kato class [CFKS], the Hamiltonian of the system is given, in suitable units and Landau gauge, by H(V1, V2) := HL + V11− + V21+ , (1.1) as a self-adjoint operator acting on L2(R2,dxdy), where HL = H(0, 0) in this notation. For technical reasons it is convenient to assume that V1, respectively V2, does not grow faster than polynomially as x→ −∞, respectively, as x→ +∞. 1 2 JEAN-MICHEL COMBES, FRANÇOIS GERMINET, AND PETER D. HISLOP We shall say that V1 is a left confining potential with respect to the interval I = [a, b] ⊂ R if, in addition to the previous conditions, the following holds: There exists R > 0, s.t. ∀x ≤ −R, ∀y ∈ R, V1(x, y) > b. (1.2) The “hard wall” case, i.e. V1 = +∞ and H = HL + V2 acting on L2(R+×R,dxdy) with Dirichlet boundary conditions, can be considered as well. As typical examples for H(V1, V2) one may think of the right potential V2 as an impurity potential and the left potential V1 as either a wall, confining the electron to the right half-plane and generating an edge current near x = 0, or as the zero potential. In this latter case, the issue is to determine whether or not V2 is strong enough to create edge currents by itself. We will discuss this in Section 3.1. Another example is the strip geometry, where both V1 and V2 are confining potentials outside of a strip x ∈ [−R,R], where the electron is localized. We define a “switch” function as a smooth real valued increasing function equal to 1 (resp. 0) at the right (resp. left) of some bounded interval. Following [SBKR, KRSB, EG, CG], we define the (Hall) edge conductance as follows. Definition 1.1. Let X ∈ C∞(R2) be a x-translation invariant switch function with suppX ′ ⊂ R × [− 14 , 1 4 ], and let −g ∈ C ∞(R) be switch a function with supp g′ ⊂ I = [a, b] a compact interval. The edge conductance of H(V1, V2) in the interval I is defined as σe(g, V1, V2) = −tr(g′(H(V1, V2))i[HL,X ]) (1.3) whenever the trace is finite (we shall also use the notation σe(g,H) = σe(g, V1, V2) if H = H(V1, V2)). Note that in the situations of interest σe(g, V1, V2) will turn out to be independent of the particular shape of the switch function X and also of the switch function g, provided supp g′ does not contain any Landau level. We turn to the description of the results of [CG]. Let us assume that I lies in between two successive Landau levels, say the N th and the (N+1)th. While clearly σe(g, 0, 0) = 0, for any g as above since g′(HL) = 0, a straightforward computation shows that σe(g, V1, 0) = N , provided V1(x1, x2) = V1(x1) is such that limx1→−∞ V1(x1) > b (see, for example, [CG, Proposition 1]). The first result tells us that the edge conductance is stable under a perturbation by a potential W located in a strip [L1, L2]× R of finite width. Theorem 1.2. ([CG, Theorem 1]) Let H = H(V1, V2) be as in (1.1), and let W be a bounded potential supported in a strip [L1, L2]× R, with −∞ < L1 < L2 < +∞. Then the operator (g′(H +W )− g′(H))i[HL,X ] is trace class, and tr((g′(H +W )− g′(H))i[HL,X ]) = 0. (1.4) As a consequence: (i) σe(g,HL +W ) = 0. (ii) Assume V1 is a y-invariant potential, i.e. V1(x, y) = V1(x), that is left confining with respect to I ⊃ supp g′. If I ⊂]BN , BN+1[, for some N ≥ 0, then σe(g,HL + V1 +W ) = N. (1.5) We note that Theorem 1.2 extends perturbations W that decay polynomially fast in the x-direction. In particular, it allows for more general confining potentials than y-invariant ones. But, it is easy to see that Theorem 1.2 does not hold for ON THE QUANTIZATION OF HALL CURRENTS IN PRESENCE OF DISORDER 5 next two sections we investigate new regularizations that commute with Hω only asymptotically (as R→∞). 2.2. A time averaged regularization for a dynamically localized system. We assume that the operator H = H(0, V ) exhibits dynamical localization in an open interval I ∈]BN , BN+1[. This means that for any p ≥ 0, there exists a nonnegative constant Cp <∞ such that for any Borel function f on I, with |f | ≤ 1, and for any X1, X2 ∈ R2, sup t∈R ‖1X1f(H)e−itH1X2‖2 ≤ Cp min(1, |X1 −X2|)−p). (2.6) We used the Hilbert-Schmidt norm. For random Schrödinger operators Hω, this assumption is one of the standard conclusions of multiscale analysis [GDB, GK1]. We show in Section 3.1 that as long as I ∈]BN , BN+1[ such an analysis applies to the Hamiltonian Hω = H(0, Vω) as in (2.4). Note that it follows from (2.6) that if X ∈ R2 and A is a subset of R2 (A may contain X) then, for any p > 0, there exists a (new) constant 0 ≤ Cp < ∞, such that sup t∈R ‖1Af(H)e−itH1X‖2 ≤ Cp min(1,dist({X}, A)−p). (2.7) For R <∞, η > 0 and γ > 0, we set, with H = H(0, V ) and X = (x, y), JR = η ∫ ∞ 0 EH(I)eitH1x≤Re−itHEH(I)e−ηtdt, with R = η−γ . (2.8) Theorem 2.1. Let JR as in (2.8), with γ ∈]0, 1[. Assume that H(0, V ) exhibits dynamical localization (i.e. (2.6)) in I ⊂]BN , BN+1[ for some N ≥ 0. Then JR regularizes H(0, V ), and thus also H(V0, V ), in the sense that C1 and C2 hold. Moreover the edge conductances take the quantized values: σrege (g, 0, V ) = 0 and σrege (g, V0, V ) = N . Remark 2.2. In [EGS], a similar regularization is considered, where γ = 1 and H is the bulk Hamiltonian H(V, V ). We also note that if R and η are inde- pendent variables, then one recovers the regularization [CG, Eq. (7.13)], see [CG, Remark 13]. Proof. Since ‖JR‖ ≤ 1, to get C1 it is enough to check limR→∞ JREH(I)ψEH(I)ψ for compactly supported states, and it is thus enough to note that by (2.7), ‖(1− JR)EH(I)10‖ ≤ η ∫ ∞ 0 sup t ‖1x>Re−itHEH(I)10‖e−ηtdt ≤ CpR−p. (2.9) We turn to C2. As in [CG], we write i[H,X ] = i[H,X ]1|y|≤ 12 , with 1|y|≤ 12 =∑ x2∈Z 1(x2,0). Note that, by hypothesis on I, g ′(H) = g′(H) − g′(HL), so that terms that are far in the left half plane will give small contributions. To see this, we develop ∥∥(g′(H)− g′(HL))i[HL,X ]EH(I)e−itH1x≤−R∥∥1 (2.10) using the Helffer-Söjstrand formula [HeSj, HuSi] and the resolvent identity with RL(z) = (HL − z)−1 and R(z) = (H − z)−1. It is thus enough to control terms of the form, with Imz 6= 0, x1, y1, x2 ∈ Z, X1 = (x1, y1),∥∥∥R(z)V 1x≥0RL(z)i[HL,X ]1|y|≤ 12EH(I)e−itH1x≤−R∥∥∥1 (2.11) ≤ ∑ x1≥0,y1,x2 ‖R(z)V 1X1‖2‖1X1RL(z)i[HL,X ]1(x2,0)‖‖1(x2,0)EH(I)e −itH1x≤−R‖2 6 JEAN-MICHEL COMBES, FRANÇOIS GERMINET, AND PETER D. HISLOP But, as a well-known fact, R(z)1X1 is Hilbert-Schmidt in dimension 2, e.g. [GK2, Lemma A.4], and its Hilbert-Schmidt norm is bounded by C[dist(z, σ(H))]−1 ≤ C=z−1, uniformly in X1. Then, for some κ ≥ 1, (2.11) ≤ C (=z)κ ∑ x1≥0,y1,x2≤−R2 ‖1X1RL(z)i[HL,X ]1(x2,0)‖ (2.12) + C (=z)κ ∑ x2≥−R2 ‖1(x2,0)EH(I)e −itH1x≤−R‖2 (2.13) ≤ C (=z)κ R−p, (2.14) where the latter follows from (2.7) and from the fact that ‖1X1RL(z)i[HL,X ]1(x2,0)‖ decays faster than any polynomial in |X1−X2|, as can be seen by a Combes-Thomas estimate together with standard computations (e.g. [CG, Lemma 3]). As a con- sequence the trace norm (2.10) is finite and goes to zero as R → ∞, uniformly in η. The next step is to control contributions coming from terms living far from the support of X ′, i.e. terms s.t. |y| ≥ Rν with ν ∈]0, 1]. Set S(R, ν) = {(x, y) ∈ R2, |x| ≤ R, |y| ≥ Rν}. Then, using (2.7),∥∥g′(H)i[HL,X ]EH(I)e−itH1S(R,ν)∥∥1 (2.15) ≤ ∑ x2∈Z ∥∥g′(H)i[HL,X ]1(x2,0)∥∥2 ∥∥1(x2,0)EH(I)e−itH1S(R,ν)∥∥2 (2.16) ≤ C ∑ x2∈Z min(1,dist({(x2, 0)}, S(R, ν))−p) ≤ C(2R)(Rν)−p. (2.17) Contributions from (2.15) are thus negligible as R → ∞, uniformly in η. Letting KR,ν denote the compact set KR,ν = {(x, y) ∈ R2, |x| ≤ R, |y| ≤ Rν}, we are left so far with the evaluation of η ∫ ∞ 0 e−ηtdt g′(H)i[HL,X ]EH(I)eitH1KR,νe−itHEH(I), (2.18) which is clearly now a trace class operator (the integral is absolutely convergent in trace norm). In other terms g′(H)i[HL,X ]JR is thus trace class. It remains to show that its trace goes to zero as R goes to infinity. But on the account of (2.10) and (2.15), it remains to show that the trace of (2.18) goes to zero. By cyclicity, tr(2.18) = −η ∫ ∞ 0 e−ηttr { g′(H)X e−itHi[HL, EH(I)1KR,νEH(I)]eitH } dt = η ∫ ∞ 0 e−ηt d dt tr { g′(H)X e−itHEH(I)1KR,νEH(I)eitH } dt = ηtr { g′(H)XEH(I)1KR,νEH(I) } −η2 ∫ ∞ 0 e−ηttr { g′(H)X e−itHEH(I)1KR,νEH(I)eitHEH(I) } dt. Thus |tr(2.18)| ≤ Cη|KR,ν | = CηR1+ν . Since R = η−γ , the trace goes to zero if γ < 11+ν .  ON THE QUANTIZATION OF HALL CURRENTS IN PRESENCE OF DISORDER 7 2.3. Regularization under a stronger form of dynamical localization. In this section, we consider JR = EH(0,V )(I)1x≤REH(0,V )(I). (2.19) Note that the regularization (2.8) studied in Section (2.2) is the time average of (2.19). The effect of the time averaging is to provide a control on the cross terms arising in (2.19) if one expands EH(0,V )(I) over a basis of eigenfunctions. In [CG, Eq. (7.13)], cross terms were suppressed from the very definition of JR. By showing that JR, given in (2.19), regularizes H(0, Vω) under, basically, the same assumption as in [CG, Theorem 3], we strengthen [CG]’s result. Let T be the multiplication operator by T (X) = 〈X〉ν , ν > d2 = 1, with 〈X〉 = (1 + |X|2) 12 , for X ∈ R2. It is well known for Schrödinger operators that tr(T−1EH(0,V )(I)T−1) <∞, if I is compact (e.g. [GK2]). Definition 2.3 (SUDEC). Assume H has pure point spectrum in I with eigenvalues En and corresponding normalized eigenfunctions ϕn, listed with multiplicities. We say that H has Summable Uniform Decay of Eigenfunction Correlations (SUDEC) in I, if there exist ζ ∈]0, 1[ and a finite constant c0 > 0 such that for any En ∈ I and X1, X2 ∈ Z2, ‖1X1ϕn‖‖1X2ϕn‖ ≤ c0αn‖T1X1‖2‖T1X2‖2e−|X1−X2| ζ , (2.20) where αn = ‖T−1ϕn‖2. Note that, ∑ n αn = tr(T−1EH(0,V )(I)T−1) <∞. (2.21) Remark 2.4. Property (2.20) (or a modified version of it) was called (WULE) in [CG] and was introduced in [Ge]. The more accurate acronym (SUDEC) comes from [GK3] and Property (SUDEC) is used in [GKS] as a very natural signature of localization in order to get the quantization of the bulk conductance. Theorem 2.5. Assume that H(0, V ) has (SUDEC) in I ⊂]BN , BN+1[ for some N ≥ 0. Then JR, given in (2.19), regularizes H(0, V ), and thus also H(V0, V ), in the sense that C1 and C2 hold. Moreover the edge conductances take the quantized values: σrege (g, 0, V ) = 0 and σ reg e (g, V0, V ) = N . Proof. That the operator g′(H(0, V ))i[H(0, V ),X ]JR is trace class follows from the comparison g′(H(0, V )) = g′(H(0, V ))− g′(HL). In order to control the region x ≤ 0, and the immediate estimate, let Pn be the eigenprojector on the eigenfunction ϕn, and write ‖1XEH(0,V )(I)1Y ‖2 ≤ ∑ n ‖1XPn1Y ‖2 = ∑ n ‖1Xϕn‖‖1Y ϕn‖ (2.22) ≤ c0( ∑ n αn)‖T1X‖2‖T1Y ‖2e−|X−Y | ζ , (2.23) where we used the assumption (2.20) (and recall (2.21)). We proceed and set Λ2,R = 1x≤R. We are looking at σ (reg) E (g,R) = tr(g ′(H(0, V ))i[H(0, V ),X ]JR). (2.24) 10 JEAN-MICHEL COMBES, FRANÇOIS GERMINET, AND PETER D. HISLOP λ > 0, B > 0 given, the Hamiltonian Hω = H(0, Vω), with Vω = λ ∑ i∈Z+×Z ωiu(X − i). (3.40) The assumptions on the random variables ωi, i ∈ Z+ × Z, and on the single site potential u are the one considered in [CH, GK2]. Namely, the ωi’s are i.i.d. random variables with a common law µ(dt) = g(t)dt, where g is an even bounded function with support in [−M,M ], M > 0, with, in addition, the condition µ([0, t]) ≥ cmin(t,M)ζ , for some ζ > 01. In order to apply the percolation estimate as in [CH] we require that supp u ∈ B(0, 1/ √ 2). (3.41) With no loss we assume that ‖u‖∞ = 1, so that the spectrum of Hω satisfies σ(Hω) ⊂ ⋃ n≥1 [Bn −M,Bn +M ]. We note that thanks to the ergodicity of Hω with respect to integer transla- tions in the y-direction, the spectrum equals a deterministic set for almost all ω = (ωi)i∈Z+×Z. For convenience we shall extend the ωi’s to the left half plane by setting ωi = 0 if i ∈ Z− × Z. The only difference between the present model and that of [CH, GK2] is that the random potential in the left half-plane is replaced by a zero potential. This absence of a potential creates a classically forbidden region in the spectral sense for energies between Landau levels. This situation is different from a classically forbidden region created by a wall. The intuition is that looking at a given distance of a Landau level (in the energy axis), the absence of potential should help for localization. One may think of [CH, Wa, GK2]’s result as a weak disorder result. The disorder is kept fixed and localization is obtained for large B. In this spirit putting ωi = 0 should be even better, for one creates fewer states at a given distance from the Landau level. One might think that the interface at x = 0 between the random potential in the right half-plane and the absence of potential in the left one would create some current along the interface. It could be so for energies very close to the Landau level where the above reasoning breaks down. To get localization, one has to investigate how the Wegner estimate, the multi- scale analysis (MSA), and the starting estimates of the MSA are affected by the new geometry of the random potential. In particular, since we broke translation invariance in the x direction, we have to check things for all boxes, regardless of the position with respect to the interface x = 0. The Wegner estimate: It is immediately seen that the proof of the Wegner estimate given in [CH] is still valid with this geometry. Indeed, if a box ΛL(x, y) is such that x < L/2, then ΛL(x, y) overlaps the left half-plane (it may even be contained in it). Then, in [CH, (3.8)] the sum is restricted to sites i = (i1, i2) where ωi 6= 0 (i.e. i1 > 0). The rest of the proof is unchanged, and as a result the volume factor one gets at the end is |ΛL(x, y) ∩ (R+ × R)| rather than |ΛL(x, y)|. In particular one gets zero if ΛL(x, y) ⊂ R− ×R, as expected. So (W) and (NE) of [GK1] hold. 1this last hypothesis is not necessary to prove localization at a given fixed distance, independent of B, from the Landau levels ON THE QUANTIZATION OF HALL CURRENTS IN PRESENCE OF DISORDER 11 The multiscale analysis: The deterministic part of the MSA (properties (SLI) and (EDI) in [GK1]) is not sensitive to changes of the random variables. Indepen- dence of far separated boxes (property (IAD) in [GK1]) is still true. In fact, what happens in the probabilistic estimates that appear in the MSA is that we shall estimate probabilities of bad events related to boxes which have an overlap with the left half-plane as if they where contained in the right half-plane, and thus by a bigger (thus worse) probability. In particular, if a box is totally included in the left half-plane, the probability of having a singular box is zero, and we shall estimate it by a polynomially (or sub-exponentially) small factor in the size of the box. The starting estimate: We follow the argument given in [CH]. Let us focus on energies E ∈]Bn, Bn +M ], the other case E ∈ [Bn−M,Bn[ being similar. We thus set E = Bn + 2a, a > 0. We say that a site i ∈ Z2 is occupied if ωi ∈ [−M,a], in other words, dist(E,Bn +ωi) ≥ a (recall ‖u‖∞=1). Note that by hypothesis on ωi, for any a > 0, P(ωi ∈ [−M,a]) ≥ 1 2 + caζ . In particular, the probability is P(ωi ∈ [−M,a]) = 1, if i ∈ Z− × Z. We are thus above the critical bond percolation threshold pc = 12 (in dimension 2) for all i ∈ Z 2. Consequently, bonds percolate, and [CH, Proposition 4.1] follows. The rest of the proof leading to the initial length scale estimate [CH, Proposition 5.1] is the same. At this stage Theorem 4.1 in [GK2] applies, and one has Anderson localization, (SULE), and strong Hilbert-Schmidt dynamical localization as described in [GK1], as well as (SUDEC) (following the proof of [Ge]; see also [GK3]). We note that the above arguments are not restricted to the particular half-plane geometry of the random potential we discussed here. Any random potential of the form Vω = ∑ i∈J ωiu(X − i), where J ⊂ Z2 has an infinite cardinal would yield the same localization result. 3.2. A large disorder regime. We next consider the random Landau Hamiltonian defined in (3.40) with a left constant confining potential V0(x, y) = V01− (see (1.2)) so that H(V0, Vω) = HL + V01−+λVω1+, for large values of the disorder parameter λ. The random potential in the right half-plane Vω, as in (3.40), has i.i.d. random variables ω′is with a common positively supported distribution, say on [0, 1]. We also impose the condition that the single site potential u ∈ C∞c (R) satisfies the following covering condition: If Λ ⊂ R+ × R, ∑ i∈Λ u(X − i) ≥ C01Λ. (3.42) We show that if the disorder is large enough, then at low energy, no edge current will exist along the interface x = 0 in the sense that the regularized edge conductance σrege (g, V0, λVω) of H(V0, Vω) will be zero. As consequence of (2.2), however, the regularized edge conductance of H(0, Vω) will be quantized to a non zero value, i.e. σrege (g, 0, λVω) = −N . In other terms the random potential λVω is strong enough to create “edge currents without edges” (as in [EJK1]). Such a situation is similar to the model studied by S. Nakamura and J. Bellissard [NB], and revisited in [CG] from the “edge” point of view. The strategy to prove localization for H(V0, Vω) is the same as the one exposed in Section 3.1, i.e. use a modified multiscale analysis taking into account the new 12 JEAN-MICHEL COMBES, FRANÇOIS GERMINET, AND PETER D. HISLOP geometry of the problem. Here the potential in the left half-plane is no longer zero but a constant V0 > b, if I = [0, b] is the interval where we would like to prove localization. As in Section 3.1, the modifications of the Wegner estimate, of the starting estimates of the the multiscale analysis (MSA), and of the MSA itself, have to be checked separately. While the comments made in Section 3.1 concerning the MSA are still valid, the new geometry requires new specific arguments for the Wegner estimate and the starting estimate. The Wegner estimate: Its proof can no longer be borrowed from [CH] as in Section 3.1, and one has to explicitly take into account the effect of the confining potential V01−. We shall modify the argument given in [CHN] as follows. The only case we have to discuss is the one of a box ΛL(X), with |x| < L2 so that it overlaps both types of potentials. We set ṼL = ∑ i∈ΛL(X)∩Z2 ωiu(X − i). Let HL denote the restriction of H(V0, Vω) to the box ΛL(X) with self-adjoint boundary conditions (e.g. [GKS]). By Chebychev’s inequality, the proof of the Wegner estimate is reduced to an upper bound on the expectation of the trace of the spectral projector EHL(I) for the interval I. Following [CHK], we write trEHL(I) = tr1ΛL(X)EHL(I) (3.43) ≤ 1 V0 trV01ΛL(X)1−EHL(I) + λ C0 trṼLEHL(I) (3.44) ≤ 1 V0 tr1−HLEHL(I) + λ C0 trṼLEHL(I) (3.45) ≤ b V0 trEHL(I) + λ C0 trṼLEHL(I), (3.46) so that, with V0 > b by assumption, trEHL(I) ≤ λ C0 ( 1− b V0 )−1 trṼLEHL(I). (3.47) At this point, the proof follows the usual strategy, as in [CHN, CHKN, CHK]. The starting estimate: The initial estimate follows from the analysis, at large disorder, given in [GK2, Section 3]. Since in the left half-plane, the potential is already very high (V0 > b), it is enough to estimate the probability that all the random variables ωi in the right part of the box is higher than say b/2. Doing this creates a gap in the spectrum of the finite volume operator HL. This spectral gap, occurring with good probability, can be used to obtain the exponential decay of the (finite volume) resolvent thanks to a Combes-Thomas argument. References [AS] Avron, J.E., Seiler, R.: Quantization of the Hall conductance for general, multiparticle Schrödinger operators, Phys. Rev. Lett. 54 259-262 (1985). [BESB] Bellissard, J., van Elst, A., Schulz-Baldes, H.: The non commutative geometry of the quantum Hall effect. J. Math. Phys. 35, 5373-5451 (1994). [CG] Combes, J.M., Germinet, F.: Edge and Impurity Effects on Quantization of Hall Currents, to appear in Commun. Math. Phys. [CGH] Combes, J.M., Germinet, F., Hislop, P.D: in preparation.