Localization Properties of the Chalker Coddington Model

Transkript

1 Ann. Henri Poincaré (200), c 200 Springer Basel AG /0/ published online November 3, 200 DOI 0.007/s Annales Henri Poincaré Localization Properties of the Chalker Coddington Model Joachim Asch, Olivier Bourget and Alain Joye We dedicate this work to the memory of our friend and colleague Pierre Duclos Abstract. he Chalker Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M. We prove first that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly, that this implies spectral localization. hirdly, we prove a houless formula and compute the mean Lyapunov exponent, which is independent of M.. Introduction We start with a mathematical then a physical description of the model. Fix the parameters r, t [0, ], such that, r 2 + t 2 =, denote by the complex numbers of modulus and for q =(q,q 2,q 3 ) 3 by S(q) the general unitary U(2) matrix depending on these three phases ( )( )( ) q q S(q) := 2 0 t r q q q 2 r t 0 q 3 Let ( Ω, F, P) be the probability space: Ω := ( 6) (2Z) 2, P := (2Z) 2d 6 l where dl is the normalized Lebesgue measure on, and F the σ-algebra generated by the cylinder sets. With p Ω, p(2j, 2k) =:(p,p 2,p 3, }{{} p 4,p 5,p 6 ) }{{} p e(2j,2k) p o(2j+,2k+) (j, k Z)

2 342 J. Asch et al. Ann. Henri Poincaré and the basis vectors e μ (ρ) :=δ μ,ρ (μ, ρ Z 2 ), the family of unitary operators Û(p) :l 2 (Z 2 ) l 2 (Z 2 ) is defined by its matrix elements Ûμ;ν = e μ, Ûe ν : Û μ;ν := 0 except for the blocks ( ) Û(p)(2j+,2k);(2j,2k) Û(p) (2j+,2k);(2j+,2k+) := S(p e (2j, 2k)) () Û(p) (2j,2k+);(2j,2k) Û(p) (2j,2k+);(2j+,2k+) ( Û(2j+2,2k+2);(2j+2,2k+) Û (2j+2,2k+2);(2j+,2k+2) Û (2j+,2k+);(2j+2,2k+) Û (2j+,2k+);(2j+,2k+2) ) := S(p o (2j +, 2k + )). Note that Û is an ergodic family of random unitary operators; indeed, Û Û = I = ÛÛ because of the unitarity of the blocks; further denote by Θ the action of Z 2 on functions f on Z 2 : ( Θ (l,m) f)(μ) :=f(μ +(2l, 2m)) (μ Z 2, (l, m) Z 2 ), and, by abuse of notation, the corresponding shift on Ω. hen Θ ismeasure preserving and ergodic on Ω and Û( Θp) = ΘÛ(p) Θ. his model was introduced in the physics literature by Chalker and Coddington [0], see [2] for a review, in order to study essential features of the quantum Hall transition in a quantitative way. Û describes the dynamics of a 2D electron in a strong perpendicular magnetic field and a smooth bounded random electric potential which is supposed to have some array of hyperbolic fixed points forming the nodes of a graph. In this picture, the electron moves on the directed edges of the graph whose nodes are even : {(/2, /2)+(2j, 2k),j,k Z} or odd : {(/2, /2)+ (2j +, 2k +),j,k Z} with edges connecting the even (odd) nodes to their nearest odd (even) neighbors. Û describes the evolution at time one of the electron. he edges are labeled by their midpoints. hey are directed in such a way that Û models the tunneling near the hyperbolic fixed points of the potential, see Fig.. he tunneling is described by the scattering matrices S associated with the even, respectively odd, nodes. he i.i.d. random phases associated with each node take into account the deviation of the random electric potential from periodicity. Following the literature on tunneling near a hamiltonian saddle point [,4], the parameter t is +e ε where ε is the distance of the electron s energy to the nearest Landau Level. An application of a finite size scaling

3 Vol. (200) Localization Properties of the Chalker Coddington Model 343 (2j,2k) (2j+,2k+) x-axis even node odd node y-axis Figure. he network model with its incoming (solid arrows) and outgoing links method to their numerical observations led Chalker and Coddington [0], see also [2], to conjecture that the localization length diverges as t/r as ( ) α ln t r, where the critical exponent α exceeds substantially the exponent expected when a classical percolation model is applied to the problem [25]; the values advocated for α are 2.5 ± 0.5 for the quantum and 4/3 for the classical case. Because of its importance for the understanding of the integer quantum Hall effect the one electron magnetic random model in two dimensions was and continues to be heavily studied in the mathematical literature. Mathematical results concerning the full Schrödinger Hamiltonian can be traced from the following contributions and their references: [27] for percolation, [6] for the existence of the localization delocalization transition [2, 8, 5] for the general theory of the quantum Hall effect. For results concerning a 2D electron in a magnetic field and periodic potential, which corresponds to the absence of phases here, see [8, 26]. For recent work on Lyapunov exponents on Hamiltonian strip models see [6, 7, 23]. Our results concern the restriction of the model to a strip of width 2M and periodic boundary conditions; they are presented as follows. In Sect. 2, we analyze the extreme cases, r =0andr =. hen, for the case where all phases are chosen to be, we give a description of the spectrum. Questions related to transfer matrix formalism are handled in Sects. 3, 4, 5. In Sect. 6, weprove simplicity of the Lyapunov spectrum and finiteness of the localization length. In Sect. 7, we prove a houless formula and show that the density of states is flat, which implies our results on the mean Lyapunov exponent. In Sect. 8, we prove complete spectral localization.

4 344 J. Asch et al. Ann. Henri Poincaré 2. Some Properties of the Model 2.. Extreme Cases Note that in case of complete reflection or transmission the system localizes completely: Proposition 2.. Let rt =0. hen, for any p Ω, the spectrum of Û(p) is pure point. Proof. Assume r =0,p Ω and define the family of subspaces (H j,k ) (j,k) Z 2 as: H j,k =Ran(e 2j,2k,e 2j+,2k,e 2j+,2k,e 2j,2k ). hese subspaces are invariant under Û(p) and (j,k) Z 2 H j,k = l 2 (Z 2 ), (2) which means the operator Û(p) is pure point. he case t = 0 is treated similarly. On the other hand, one has complete propagation if all the phases are equal to one; define Ω p =(...,,,,...)byp(2j, 2k) :=(,,,,, ) then we have: Proposition 2.2. Let rt 0. hen, the spectrum of Û(...,,,,...) is purely absolutely continuous. Proof. We make use of a decomposition similar to (2) and define the unitary V from l 2 (Z 2 )tol 2 (Z 2 ) C 4 by Ve 2j,2k := e j,k e,ve 2j+,2k+ := e j,k e 2,Ve 2j,2k+ := e j,k e 3,Ve 2j+,2k := e j,k e 4.LetP be the projection P := I ( e e + e 2 e 2 ). From the definition of Û in () one reads that V Û 2 V commutes with P and that PVÛ2 V P is equivalent to ( ) rt(0,,0 ) r 2,0 + t 2 0, t 2 0, + r 2,0 rt(,0 0, ) with the translations on l 2 (Z 2 ) defined by n,m ψ(j, k) :=ψ(j + n, k + m) (n, m Z). he Fourier transform F : l 2 (Z 2 ) L 2 ( 2 ) transforms the translations to multiplication operators: F n,m F =exp( i(nx + my)), thus the restriction to the range of P of FPVÛ2 V P F is equivalent to a matrix-valued multiplication operator ( ) rt(e iy e ix ) r 2 e ix + t 2 e iy. (3) t 2 e iy + r 2 e ix rt(e ix e iy ) he trace of this matrix is not constant, its determinant is hence the spectral bands are not flat, thus the spectrum of the restriction of Û 2 is purely absolutely continuous. By an analogous argument this also holds for the restriction to P.

5 Vol. (200) Localization Properties of the Chalker Coddington Model 345 Remark that a more general periodic distribution of phases leads to matrix-valued translation operators with periodic coefficients thus to nontrivial Hofstadter like problems. 3. Restriction to a Cylinder, ransfer Matrices Let M N. Use the notation Z 2M := Z/(2MZ) for the discrete circle of perimeter 2M. Consider the restriction of the model to the cylinder Z Z 2M : U(p) :l 2 (Z Z 2M ) l 2 (Z Z 2M ) defined by its matrix elements with respect to the canonical basis U(p) μ,ν := Û(μ,μ 2 mod 2M);(ν,ν 2 mod 2M). (4) Remark that U(p) has the same spectral properties for the extreme cases as Û(p), the model on the full lattice: Proposition 3.. Let rt =0. hen, for any p Ω, the spectrum of U(p) is pure point. Proof. Similar to the proof of Proposition 2.. Proposition 3.2. Let rt 0. hen the spectrum of U(...,,,,...) is purely absolutely continuous. Proof. In the proof of Proposition 2.2 note that V now acts from l 2 (Z Z 2M ) to l 2 (Z Z M ) C 4 and replace F by the Fourier transform from l 2 (Z Z M ) to L 2 ( Z M ) defined by Fψ(x, κ) = M κk j Z,k Z M ψ j,k e ixj e i 2π which diagonalizes the translations. hen, setting y = 2π M κ (κ Z M ), the matrix-valued multiplication operator obtained in (3) is understood as a family of matrix-valued operators indexed over Z M. he spectral bands are not flat by the same argument. From now on we restrict the discussion to the case rt 0. In the following z denotes a complex number; also, unless otherwise stated, all indices in the second variable are to be understood mod 2M, e.g.: ψ 2j,2k+ = ψ 2j,2k+ mod 2M = ψ 2j,2k+[2M]. A standard approach to the spectral problem of U is the transfer matrix method. hough this is well known, we wish to recall the construction explicitly for the model at hand.

6 346 J. Asch et al. Ann. Henri Poincaré Proposition 3.3. For z 0, q =(q,q 2,q 3 ) 3 define ( ) ( )( ) q q eo (z,q) := 2 0 z r q3 0, 0 q 3 t r z 0 q q 2 ( ) ( )( ) q3 0 z t q q oe (z,q) := q q 2 r t z. 0 q 3 hen. For ψ : Z Z 2M it holds: U μν ψ ν = zψ μ μ Z Z 2M ν Z Z 2M ( ψ2j+,2k ψ 2j+,2k+ ) = eo (z,p e (2j, 2k)) and ( ) ψ2j+2,2k+ = oe (z,p o (2j +, 2k + )) ψ 2j+2,2k+2 ( ψ2j,2k ψ 2j,2k+ ) ( ψ2j+,2k+ ψ 2j+,2k+2 2. For z, it holds that oe, eo U(, ), the Lorentz group defined as a subset of the complex 2 2 matrices by { ( )} U(, ) := B M 2,2 (C); B 0 JB = J, J := 0 Proof. By definition of U, we have for the even nodes: ( ) ( ) (Uψ)2j+,2k ψ2j,2k = S (p e (2j, 2k)) = z (Uψ) 2j,2k+ and, for the odd nodes: For a matrix it holds: ( ) a = S b with S (p o (2j +, 2k + )) Ŝ = S 22 Now ( t r r t ( ψ2j+2,2k+ ( ) S S S = 2 S 2 S 22 ( ) x y ψ 2j+,2k+2 ( ) a = y Ŝ ψ 2j+,2k+ ) = z with S 22 S 2 0 ( ) x b ). ( ψ2j+,2k ( ψ2j+2,2k+2 ψ 2j+,2k+ ( ) x = a S ψ 2j,2k+ ). ( ) b y ( ) ( ) det S S2 S22, S =. S 2 S 2 S det S ) = t ( ) r ; r ( ) t r = ( ) t r t r t ),

7 Vol. (200) Localization Properties of the Chalker Coddington Model 347 so ( ) ( )( )( )( ) a q2 0 t r q3 0 x z = q b 0 q 2 r t 0 q 3 y ( ) ( ) ( )( )( ) a q q = 2 0 z r q3 0 x y 0 q 3 t r z 0 q q 2 b ( ) ( ) ( )( )( ) x q3 0 z t q q = 2 0 b a 0 q q 2 r t z 0 q 3 y from which the first claim follows. Denote by I the identity matrix in C 2. S is a unitary matrix if( and only ) I if the pullback of the quadratic form in C 4 associated with Q = I (blanks stand for 0 entries) to the graph of S: { (u, Su) C 4,u ( C 2} ) is zero. J he mapping from (x, y, a, b) to(x, b, a, y) transforms Q to.he J pullback of the corresponding form to the graph of eo being zero, it follows that eo and, by the analogous argument, oe,belongtothelorentzgroup. For later use we fix the following notation Definition 3.4. Denote by J the 2M 2M block diagonal matrix consisting of M non-zero diagonal blocks equal to J and by U M (, ) := {B M 2M,2M (C); B JB = J}. the unitary group of the hermitian form defined by J. Note that U M (, ) is isomorphic to the classical unitary group U(M,M) of the hermitian form z z M 2 z M+ 2 z 2M Relevant Phases Because of the uniform distribution, it is possible to reduce the number of relevant phases in the model to two phases per node. Before proceeding we do this reduction. We shall repeatedly make use of Lemma 4.. Let ϕ,...,ϕ n be independent and uniformly distributed random variables on R/Z and let A M m,n (Z). hen,θ,...,θ m defined by θ = A ϕ are independent and uniformly distributed if and only if Rank A is maximal. Proof. For k Z m it holds E(e i k, θ )=E(e i k,a ϕ )=E(e i At k, ϕ )=δ A t k,0. hus, the θ are independent and uniformly distributed if and only if E(e i k, θ )= δ k,0 if and only if KerA t = {0}, equivalently, if and only if Rank A is maximal.

8 348 J. Asch et al. Ann. Henri Poincaré Proposition 4.2. here exists g : Ω Z2 such that for p Ω the evolution Û(p) defined by () is unitarily equivalent to D(g(p))S on l 2 (Z 2 ) where D(q) is diagonal, D(q) (j,k);(j,k) = q j,k,ands = Û(...,,,,...). Moreover, the image measure of (2Z) 2d 6 l by g is Z 2dl. Proof. By (), Û(p) isoftheformû(p) =D() (p)sd (2) (p) where D (j) (p) are diagonal, and defined by their diagonal elements: D () (p) 2j+,2k = p p 2 (2j, 2k), D () (p) 2j+2,2k+2 = p 4 p 5 (2j +, 2k +), D (2) (p) 2j,2k = p 3 (2j, 2k), D (2) (p) 2j+2,2k+ = p 6 (2j +, 2k +), D () (p) 2j,2k+ = p p 2 (2j, 2k), D () (p) 2j+,2k+ = p 4 p 5 (2j +, 2k +), D (2) (p) 2j+,2k+ = p 3 (2j, 2k), D (2) (p) 2j+,2k+2 = p 6 (2j +, 2k +). Hence, Û(p) is unitarily equivalent to D (2) (p)d () (p)s which has the asserted shape. Define q = g(p) by q(2j +, 2k) :=p 6 (2j +, 2k )p p 2 (2j, 2k), q(2j, 2k +):=p 6 (2j, 2k +)p p 2 (2j, 2k), q(2j +2, 2k +2):=p 3 (2j +2, 2k +2)p 4 p 5 (2j +, 2k +), q(2j +, 2k +):=p 3 (2j, 2k)p 4 p 5 (2j +, 2k +). Now, an application of Lemma 4. shows the q s are i.i.d. and uniformly distributed. Remark 4.3. Note that the unitary transformation just constructed is diagonal and thus does not affect the localization properties of the model. In the following, we abuse notations and call for q Z2 the matrix operator D(q)S again Û(q); same abuse for the restriction to the cylinder. 5. Characteristic Exponents We now define and analyze the transfer matrices and in particular the localization length. Consider U(p) =D(p)S on l 2 (Z Z 2M ) (5) with identically distributed uniformly distributed phases in Z Z2M, Z Z2M dl and the cylinder set algebra.

9 Vol. (200) Localization Properties of the Chalker Coddington Model 349 We use the unitary equivalence l 2 (Z Z 2M ) = l 2 (Z {0,...,2M }) l 2 (Z; l 2 (Z 2M )) = l 2 (Z, C 2M ) ψ Ψ (6) Ψ j := (ψ j,0,...,ψ j,2m ). Note that with the reduced phases the building blocks of the transfer matrices read with phases p,q ( ) ( )( ) p 0 z r 0, 0 t r z 0 q ( ) ( )( ) 0 z t q 0 0 p r t z. 0 As we shall explain below, the previous analysis leads us to deal with the following random dynamical system: Consider the probability space defined by Ω = ( 4M ) 2Z, P = Z d 4M l, and F, the cylinder set algebra. he shift Θ:Ω Ω, Θp(2m) :=p(2(m + )) (m Z) is measure preserving and ergodic. For p Ω define the following elements of ( ) 2M 2Z p r =(,p,,p 3,...,,p 2M ) p l =(p 0,,p 2,,...,p 2M 2, ) p m =(p 2M,p 2M+,...,p 4M ). Denote for q 2M the unitary diagonal matrix D(q) := q... q 2M, (where 0 valued matrix entries are represented by blanks) and for z 0the 2M 2M matrices z r M (z) := r z..., t z r r z z t z t M 2 (z) := t z.... r z t t z t z

10 350 J. Asch et al. Ann. Henri Poincaré Define for a fixed z 0 A z :Ω U M (, ) (7) A z (p) :=D (p l ) M 2 (z)d (p m ) M (z)d (p r ). hen A generates the cocycle Φ over the ergodic dynamical system (Ω, F, P, (Θ n ) n Z ) defined by Φ z : Z Ω U M (, ) A(Θ n p)...a(p) n>0 Φ z (n, p) := I n =0. A (Θ n p)...a (Θ p) n<0 Oseledets theorem holds for Φ, see [], heorem 3.4. and Remark (ii): Definition 5.. Let z 0. here exists an invariant subset of full measure of p Ω such that the limits lim n (Φ z(n, p)φ z (n, p)) /2n = lim n (Φ z(n, p)φ z (n, p)) /2 n =: Ψ z (p) exist. Denote by γ k (p, z), k {,...,2M}, the eigenvalues of Ψ z (p) arranged in decreasing order. Due to ergodicity there exists γ k (z) 0 such that γ k (p, z) = γ k (z) on an invariant subset of full measure. he characteristic exponents are defined by λ k (z) := log γ k (z). Due to the Lorentz symmetry of the transfer matrices for z we have Proposition Let B U M (, ). hen for the singular values SV (B) it holds γ SV (B) γ SV (B). 2. For λ j := log γ j,γ j SV (B) arranged in decreasing order it holds: λ j+m = λ M j+ j {0,...,M}. Proof. We have B JB=J. In particular, det B 0,soγ 0andJ B = B J as well as BJ = JB.Now, det ( B B z 2) =0 det ( J B BJ z 2) =0 det ( (B B) z 2) ( ) =0 z 4M det(b B) det z 2 B B =0. From which the two claims follow. hus, we restrict our discussions to the first M non-negative Lyapunov exponents λ λ 2 λ M 0 which we shall call for simplicity the Lyapunov exponents in the sequel. We show that due to the translation invariance of the uniform distribution, the exponents are independent of z:

11 Vol. (200) Localization Properties of the Chalker Coddington Model 35 Lemma 5.3. For any w, A wz (p) =A z (w p), where w p is defined by w p 2j := w p 2j,andw p 2j+ := wp 2j+. Proof. Write D(p m )in(7) asd(p l )D(p r), where p r := (,p 2M+,,p 2M+3,...,,p 4M ) p l := (p 2M,,p 2M+2,,...,p 4M 2, ). hus, A z (p) is the product of the block diagonal matrices D(p l )M 2 (z)d(p l ) and D(p r)m (z)d(p r ) whose blocks are ( ) ( )( ) a l 0 z t 0 z(p, q) := 0 p r t z = ( ) z qt 0 q r pt pqz ( ) ( )( ) a r 0 z r 0 z(p, q) := = ( ) z qr. 0 p t r z 0 q r pr pqz For any w, these matrices satisfy a l wz(p, q) =wa l z(w p, w q), a r wz(p, q) =w a l z(wp, wq), from which the result follows. herefore, for any fixed w, the matrices A z (w p) have the same distribution as A wz (p). As a consequence, Corollary 5.4. All characteristic exponents λ k (z) = λ k are independent of z. Proof. λ k = E (log(γ k (z,p))) = E ( log(γ k (, z p))) = E (log(γ k (,p))). Definition 5.5. he localization length ξ M [0, ] is defined as ξ M :=. λ M Remark 5.6. In the physics literature, see [2], ξ M in assumed to be finite for all parameters; a change of the asymptotic behavior as M is conjectured when the parameters of the model approach the critical point t = r. his conjecture is supported by a numerical finite size scaling method and is supposed to reflect the divergence of the localization length of the full system at the critical point. hus a first step to support these heuristics is to prove finiteness of ξ M and to establish precise information of its behavior as a function of M. he announced equivalence to the propagation problem is the content of the following Proposition 5.7. Let U(p) be the ergodic family of unitary operators defined in (5) over the probability space Γ:= Z Z2M, Z Z2M dl and the cylinder set algebra. Let f :Γ Ω be defined for j Z by f(p)(2j) :=(p 2j,0,p 2j+2,,p 2j,2,p 2j+2,3...p 2j,2M 2,p 2j+2,2M p 2j+,0,p 2j+,...p 2j+,2M ).

12 352 J. Asch et al. Ann. Henri Poincaré he image measure by f is the measure on Ω and it holds U(p)ψ = zψ Ψ 2N =Φ z (N,f(p))Ψ 0 for Ψ defined in (6). Proof. he construction of f follows from Proposition (3.3). he image measure follows from Lemma (4.). 6. Finiteness of the Localization Length Using the methods exposed in [5], see also [7], we prove that all Lyapunov exponents are distinct and in particular that the localization length for the cylinder in finite. heorem 6.. For rt 0,z it holds λ >λ 2 > >λ M > 0. Proof. We follow the strategy exposed in [5] and prove the theorem in several steps making use of lemmata to be proven below. Denote by G := the smallest subgroup of U M (, ) generated by {A(p),p Ω}. By Lemma 6.2 G = U M (, ). In particular it is then known: G is connected. Furthermore, see also [23], G is isomorphic to the complex symplectic group. Indeed : denote by ( ג the 2M ) 2M block diagonal matrix consisting of 0 = ג write: M non-zero blocks σ = ; we 0 M σ for short; denote by {ג = Bג B Sp(M,C) :={B M 2M,2M (C); the complex symplectic group. From ( ) i J ( ) i = iσ 2 i 2 i it follows defining C := ( ) M i 2 that i G = U M (, ) = CSp(M,C)C. In order to freely use results in [5] we shall do our argument for real matrices. o this end we separate real and imaginary parts and consider τ : M 2M,2M (C) M 4M,4M (R) ( ) a b x = a + ib. b a

13 Vol. (200) Localization Properties of the Chalker Coddington Model 353 It holds: τ(x + y) =τ(x) +τ(y); τ(xy) =τ(x)τ(y); τ(x )=τ(x) t ; ker{τ} = {0}; det τ(x) = det x 2,thus with the real symplectic group τ (C GC) Sp(2M,R) {ג = Bג Sp(2M,R) :={B M 4M,4M (R); B t for = ג 2M σ. det τ(x) = det x 2 implies that τ(x) shares its eigenvalues with x with the degeneracies doubled. So the Lyapunov exponents γ defined by the τ C transformed products of transfer matrices are γ = γ 2 = λ γ 2p = γ 2p = λ p γ 2M = γ 2M = λ M. As τ (C GC) is connected one can infer from [5] heorem 3.4 and Exercice 2.9 for p {,...,M}: τ (C GC) L 2p irreducible and τ (C = γ 2p = λ p >λ p+ = γ 2p+ GC) 2p contracting in particular for p = M: λ M > 0. Now by Lemmas 6.3 and 6.4, the group τ (C GC) is2p irreducible and 2p contracting for all p {,...,M} so all Lyapunov exponents are distinct and λ M > 0. he following lemmata complete the proof of heorem 6., we use the notations introduced in the above proof. Lemma 6.2. G = U M (, ). Proof. By definition G U M (, ) is a closed subgroup of Gl(2M,C) thusg is a Lie group. By connectedness of U M (, ) it is sufficient to show that the Lie algebras g and u M (, ) coincide. Now u M (, ) = {A M 2M,2M (C); A jk = A kj ( ) k+j } whose dimension as a real vector space equals 4M 2. Denote by D j (t) = diag(,,...,,e it,,...,) the unitary matrix where the phase sits at the j th slot, for j =, 2,...,2M and use the M j as defined in Sect. 5. Forz the matrices i j j, im 2 (z) j j M 2 (z), im (z) j j M (z) (8) belong to g, forj =, 2,...,2M as they are the generators of the curves D j (t), M 2 (z)d j (t)m2 (z), M (z)d j(t)m (z), which lie in G as D j (t) =D j (t)m 2 (z)m (z)(m 2 (z)m (z)), M 2 (z)d j (t)m 2 (z) =M 2(z)D j (t)m (z)(m 2 (z)m (z)), M (z)d j(t)m (z) =(M 2 (z)m (z)) M 2 (z)d j (t)m (z).

14 354 J. Asch et al. Ann. Henri Poincaré he generators in (8) have the same block structure as the M j. We compute the relevant blocks. For im 2 (z) j j M 2 (z) we get ( )( ) i z t 0 )( z t r 2 = i ( ) tz t z 0 0 t z r 2 t z t 2 ( )( ) i z t 0 0 )( z t r 2 = i ( ) t 2 tz t z 0 t z r 2. t z Similarly, for im (z) j j M (z), the blocks take the form ( )( )( ) i z r 0 z r t 2 = i ( ) rz r z 0 0 r z t 2 r z r 2 ( )( )( ) i z r 0 0 z r t 2 = i ( ) r 2 rz r z 0 r z t 2. r z Now using these matrices for z / R and the diagonal matrix i j j, j =, 2 one gets by taking suitable real linear combinations of the matrices above that, in both cases, the relevant blocks are generated by { ( ) ( ) ( ) ( )} i,i,i, For real z, use ( the curves ) D j (t), D e M ( 2 D j (t)m ) 2 D e, Do M D j(t) 0 w 0 M D o, with D e = and D 0 w o = for w, which amounts 0 to perform the change z w z. aking into account the shift in the blocks and the period 2M of the indices in the matrices, we get that the restrictions of g and u M (, ) to their tridiagonal elements, mod 2M coincide. o go off the diagonals we use commutators, i.e. we exploit that X, Y g implies [X, Y ] g. Let A k = k + k + k k + g, fork Z 2M. Considering [A j,a k ]for all values of j, k, we generate a basis of all anti self-adjoint matrices that have non-zero real matrix elements at distance two away from the diagonal (and in the corners, by periodicity). By commuting A k with Ãj = i( j + j j j + ) g, we get a basis of self-adjoint matrices with non-zero purely imaginary elements on the same upper and lower diagonals (plus corners) only. hese matrices correspond to the restriction of all matrices in u M (, ) to these diagonals. We generalize the argument as follows: Assume we already generated a basis of all matrices A u M (, ) such that A jk =0if j k >m, m fixed. Again, periodicity is implicit here. Let u M (, ) B ± j (m) = j + m j ± j j + m. We compute [A j+m,b ± j (m)] = B j (m +). his way we generate all matrices A u M (, ) such A jk =0if j k > m +. Hence by induction, we see that g = u M (, ), so that G = U M (, ).

15 Vol. (200) Localization Properties of the Chalker Coddington Model 355 Lemma 6.3. τ (C GC) =τ (Sp(M,C)) is L 2p irreducible for p {,...,M}. Proof. Denote e i, i {,...,4M} the canonical basis vectors of R 4M.By definition (see [5] with adaptation to our symplectic form) L q := span { v Λ q R 4M ; v = Me Me 3 Me 2q,M Sp(2M,R) } for q 2M. Remark that the set of directions in L q corresponds to the set of isotropic subspaces of R 4M. τ (C GC) isl q -irreducible if there is no proper linear subspace V L q invariant under Λ q τ (C GC). Consider M real numbers a >a 2 > a M >. he 4M 4M diagonal matrix ( A = diag a,,a 2,,...,a M,,a,,a 2,,...,a M, a a 2 a M a a 2 belongs to τ (C GC) ande e 3 e 2q is an eigenvector of (Λ q A) n for all n with simple dominant eigenvalue >. hus for an invariant subspace V of L q either e e 3 e 2q V which implies Λ q M(e e 3 e 2q ) V, M, thus V = L q,ore e 3 e 2q V, which implies for all w V 0= Λ q M t w, e e 3 e 2q = w, Λ q Me e 3 e 2q thus V = L q V = {0}. hus we conclude the claimed irreducibility for q =2p. Lemma 6.4. τ (C GC)=τ (Sp(M,C)) is 2p contracting for p {,...,2M }. ( Proof. For ) any a R\0 there exist x, y R with x 2 y 2 = such that x y, which belongs to U(, ), has eigenvalues a, /a. aking such matrices as blocks one sees that there exists an element of U M (, ) whose singular y x values are distinct: a >a 2 > >a M > > /a M and thus an element of τ (C GC) with 2M distinct singular values b = b 2 = a > >b 2p = b 2p = a p > >b 2M = b 2M = a M > 0. hus b n 2p+/b n 2p n 0 and it follows from Proposition 2., p. 8 of [5] that τ (C GC) is2p contracting. Remark 6.5. o summarize, we have proved that if the transfer matrices generate the complex symplectic group Sp(M,C) then the results of [5] apply, i.e.: the Lyapunov spectrum is simple. he results in [5] are stated for real groups only. While it is remarked in their introduction that these results should hold in the complex case, this seems not to be obvious to specialists in the field. a M )

16 356 J. Asch et al. Ann. Henri Poincaré 7. houless Formula and the Mean Lyapunov Exponent In this section we shall prove the announced identity in a series of lemmata. heorem 7.. Let M N. For the first M Lyapunov exponents associated with U defined in Definition (5.) with z it holds: M M λ i (z) = 2 log rt 2 log 2 i= Proof. Let z. Denoting by P 2L (z) the propagator i P 2L (z) :Ω U M (, ) P 2L (z)(p) :=Φ z (L, p)(φ z ( L, p)) we have for m {,...,M} m λ i = lim L 4L log m P 2L (z)(p) p a.e. (9) where m denotes the mth exterior product (cf. [], ch. 3). We analyze the above limit in Proposition 7.2 below and show: M M i λ i =2 log z x dl(x)+ 2 log rt. he assertion follows by an explicit calculation proving that log z x dl(x) =0. Proposition 7.2. (houless formula) Let M N,z C\0 then M λ i (z) =2 log z x dl(x)+ M 2 log log z rt i Proof. Will be done in Appendix. Remark 7.3. We prove in particular that the density of states is the Lebesgue measure, see Lemma 9.3 below. 7.. Bounds on the Localization Length We now use the houless formula and an M independent bound on the largest Lyapunov exponent to derive a bound on the localization length. We remark that this bound is very crude and that more involved techniques should be established to get more detailed information; cf. [23] and references therein. First observe that a lower bound on the mean Lyapunov exponent together with a tight upper bound on the largest, implies a lower bound on all.

17 Vol. (200) Localization Properties of the Chalker Coddington Model 357 Lemma 7.4. Let κ>0,δ >0 such that M N,z M λ j κ, and λ κ + δ, M j= then, for all j =0,,...,M, λ j+ κ jδ M j. (0) Proof. First note that λ M M j= λ j.husλ κ, which corresponds to (0) forj = 0. Similarly, using also the upper bound on λ, we have for any j M, j M M Mκ + λ k j(κ + δ)+ so that k= k=j+ λ j+ M j M k=j+ λ k κ k=j+ jδ M j. Remark. In view of localization properties, the estimate is useful only if κ>(m )δ. () We now estimate the cocycle to derive an upper bound on the largest Lyapunov exponent, which is uniform in the quasienergy and width of the strip M. Proposition 7.5. Let M N. For the generator of the cocycle defined in (7), it holds A(p) ( + r)( + t); rt 2. It follows: 2λ log ( rt) + log (( + r)( + t)). 3. here exists a c>0 such that for M N it holds: dist(r, {0, }) <e cm = ξ M = λ M log ( rt λ k 2 ). (M ) log (( + r)( + t)) Proof. he estimate on A follows from its definition. he estimate on λ is obtained using the equality (9). Finally, from the estimate (0) it follows ξ M = λ M log ( rt 2 ). (M ) log (( + r)( + t)) he bound is symmetric around t = r = 2 and finite for r sufficiently away from the critical point 2 because of the singularity of log /rt.

18 358 J. Asch et al. Ann. Henri Poincaré 8. Spectral Localization We follow the strategy which was successfully employed for the case of one dimensional Schrödinger operators: polynomial boundedness of generalized eigenfunctions, positivity of the Lyapunov exponent and spectral averaging. We lean on the work of [4,9]. Our result is: heorem 8.. Let M N, rt 0. hen, the Chalker Coddington model on the cylinder exhibits spectral localization throughout the spectrum, almost surely. More precisely,. the almost sure : spectrum Σ, continuous spectrum Σ c and pure point spectrum Σ pp of U(p) satisfy Σ=Σ pp = and Σ c = ; 2. the eigenfunctions decay exponentially, almost surely. Proof. We prove the theorem in Appendix 2. Acknowledgements We should like to thank the referee for his constructive criticism and H. Schulz Baldes and H. Boumaza for enlightening discussions. We acknowledge gratefully support from the grants Fondecyt Grant ; Anillo PBC-AC3; MAH-AmSud, 09MAH05; Scientific Nucleus Milenio ICM P F. 9. Appendix We follow the strategy of [3] and first prove the lower bound M λ i (z) 2 log z x dl(x)+ M 2 log log z (2) rt i for 0 z C\ which follows from Lemma 9.3 Eq. 5 below in the limit L. Lemma 9.. Denote by U D the unitary operator defined by restriction of U to l ( 2 { 2L,...,2L},l 2 (Z 2M ) ) with reflecting boundary conditions: the scattering picture for the links which are incoming to walls at (2L+) and 2L+ reads U D e 2L,2k+ = e 2L,2k+2, U D e 2L,2k = e 2L,2k+. For z C let F z := { ψ l 2 } (Z 2M ); ψ 2k+ = zψ 2k+2,k Z M, G z := { ψ l 2 } (Z 2M ); zψ 2k+ = ψ 2k,k Z M and denote by Q F the orthogonal projection to a subspace F. It holds: z is an eigenvalue of U D Ψ 2L = P 2L (z)ψ 2L and Ψ 2L F z and Ψ 2L G z Ker ( Q G z P 2L (z)q Fz ) {0}

19 Vol. (200) Localization Properties of the Chalker Coddington Model 359 Proof. It holds U D ψ = zψ = ψ 2L,2k+ = zψ 2L,2k+2 and zψ 2L,2k+ = ψ 2L,2k so Ψ 2L F z and Ψ 2L G z. he identity Ψ 2L = P 2L (z)ψ 2L holds by construction of the transfer matrices so U D ψ = zψ Q G z P 2L (z)q Fz Ψ=0. Lemma 9.2. Denote the even subspace of l 2 (Z 2M ) by E := span{e 2k ; k Z M }. For z 0 there exist invertible operators V z,w z on l 2 (Z 2M ) such that W z (E) =F z and V z (E) =G z such that. z is an eigenvalue of U D det ( Q E Vz ) P 2L (z)w z Q E =0 where we understand the determinant to apply to the restriction to E. 2. For z 0; {z,...,z (4L+)2M } the eigenvalues of U D it holds: z (4L+)M det ( Q E Vz ) P 2L W z Q E = Π(4L+)2M (rt) 2LM i= z z i. (3) Proof. Fix 0 z C. In the following N j,d j denote generic, z independent matrices whose precise values may change from line to line. he D j are diagonal. he transfer matrix A z defined in (7) isoftheform A z = ( z 2 D Q O + z 2 D 2 Q E + zn + N 2 + z ) N 3 rt where O denotes the odd subspace defined by O + E = l 2 (Z 2M ). hus Note that (rt) 2L P 2L = z 4L D Q O + z 4L D 2 Q E + 4L j= 4L+ z j N j. { } 2 F z = span (ze 2k+ + e 2k+2 );k Z M { } ( G z = span e2k + z ) e 2k+ ; k ZM. 2

20 360 J. Asch et al. Ann. Henri Poincaré On l 2 (Z 2M ) define the operators W z := ze 2k+ + e 2k+2 e 2k+2 + e 2k+ + z e 2k+2 e 2k+ 2 k Z M V z := e 2k+ + ze 2k e 2k + e 2k + z e 2k+ e 2k+. 2 k Z M hen W z Q E = Q Fz W z and V z Q E = Q G z V z. Moreover, one checks that Wz 2 = I, Vz = KV z K (4) with K := k Z M e 2k+ e 2k + e 2k e 2k+. It follows: Now z is eigenvalue of U D ker ( Q E V z P 2L W z Q E ) {0}. Q E Vz P 2L W z Q E = e 2k e 2k+ + 2 z e 2k,P 2L (ze 2m+ + e 2m+2 ) e 2m+2 k,m = z 4L+ (rt) 2L D Q O + z 4L D 2 Q E + z j N j. j <4L+ Multiplication by z 4L+ implies that for some a j C z (4L+)M det ( (8L+2)M Q E Vz ) P 2L W z Q E = (rt) 2LM z j a j. D is unitary thus, in particular, a (8L+2)M =.z (4L+)M det...being a polynomial of degree (8L +2)M whose leading coefficient has modulus (rt) 2LM and which is zero on the (4L + )2M eigenvalues of U D the formula for the determinant follows. We now prove convergence of the finite volume (L < ) density of states μ M L as L to a non-random measure: the density of state. hen we show that this measure equals the Lebesgue measure. Lemma 9.3. Denote μ (M) L the measure defined by (4L + )2M trf(u D )=: f(x)dμ M L (x). hen μ M L the Lebesgue measure on. vaguely L dl 0 (f C()).

21 Vol. (200) Localization Properties of the Chalker Coddington Model 36 For M N there exists c M > 0 such that for all L N, 0 z C\ M 4L log M P 2L 2 log ( rt + 2+ ) log z x dμ M L (x) 2L ( + ) log z c M 4L L. (5) Proof.. We first prove the existence of a non-random limit measure. he first step consists in showing that p a.e, lim L f(u D (p))dμ M L = 4 {E ( e 0,0,f(U)e 0,0 )+E ( e,,f(u)e, ) + E ( e,0,f(u)e,0 )+E ( e 0,,f(U)e 0, )} =: fdμ M, for all f C(). his follows from a classical argument based on ergodicity, separability of C() and from the fact that U U D has norm and rank uniformly bounded in L, see e.g. [20] for the details for the unitary case. In order to identify μ M recall that the normalized Lebesgue measure dl on is uniquely characterized by : p n dl = δ n,0 n Z where p n (x) :=x n. Consider the space of loops of euclidean length n starting at (0, 0) : Γ (0,0) = {γ : {0,...,n} { 2L,...,2L} Z 2M,γ(0) = γ(n) =(0, 0)}. hen because of the structure of U e 0,0,p n (U)e 0,0 = γ Γ (0,0) e 0,0,Ue γ()... e γ(n ),Ue 0,0. Now e γ(j),u(p)e γ(j+) = l(p)t α r β for some α, β 0andl a uniformly distributed random variable. hus, E ( e 0,0,p n (U)e 0,0 )=δ n,0. Applying the same argument to e,,f(u)e,, e 0,..., we conclude: dμ M = dl

22 362 J. Asch et al. Ann. Henri Poincaré 2. By formula (3): 4LM log ( ) det QE Vz P 2L W z Q E = 2 log ( rt + 2+ ) log z x dμ M L (x) 2L ( + ) log z 4L 4LM log ( M Q E Vz ) P 2L W z Q E M 4L log M P 2L + ( log M Q E Vz + log M W z Q E ) L } 4M {{} =:c M where we used the identity det Q E AQ E = e 0 e 2M 2, M Ae 0 e 2M 2. From this, the claim follows. We turn now to the proof of the opposite inequality: M λ i (z) 2 log z x dl(x)+ M 2 log log z (6) rt for 0 z C\: i Proposition { 9.4. Suppose that for any choice of sets of vectors d 0,d } { 2,...,d 2M 2 and d + 0,d + } 2,...,d+ 2M 2 in the odd subspace O lim sup L M(4L +) log (e 0 + d 0 ) (e 2M 2 + d 2M 2 ), M (Vz P 2L (z)w z )(e 0 + d + 0 ) (e 2M 2 + d + 2M 2 ) 2 ln rt +2 log x z dl(x) log z (7) i then, for all 0 z C\, M λ i (z) M 2 log rt +2 log x z dl(x) log z Proof. he vectors of the form {(e 0 +d 0 ) (e 2M 2 +d 2M 2 ); d 0,...,d 2M 2 O} span M C 2M. On the other hand, given any spanning sets S and S 2 in M C 2M, the mapping S defined by A S sup φ, Aψ φ S,ψ S 2

23 Vol. (200) Localization Properties of the Chalker Coddington Model 363 defines a norm over the algebra of operators in M C 2M. It follows that there exists c>0, which depends on S and S 2, such that for any matrix A, A S c A, hence that M M λ i (z) 2 log rt +2 log ζ z dl(ζ) log z. i We now prove that the inequality (7) is satisfied. his will be achieved in two steps. In order to keep track of the L dependence denote by UL D the former U D. Now reinterpret the left hand side of Eq. 7 as the characteristic polynomial of a deformation of UL D denoted V L D ; more precisely: we aim at Eq. 8 below. he problem is then reduced to the proof of the weak convergence of the associated sequence of counting measures (νl,z M ) towards μm. 9.. Deformation of U D L Let L in N and define the matrix V D L+ on l2 ({ 2L 2,...,2L +2},l 2 (Z 2M )) by: ψ l 2 ({ 2L 2,...,2L +2},l 2 (Z 2M )), (V D L+ψ) 2L+2,2k = M l=0 (V D L+ψ) 2L+,k = ψ 2L+,k (V D L+ψ) 2L,2k+ = (V D L+ψ) 2L,2k = M l=0 M l=0 (V D L+ψ) 2L,k = ψ 2L,k (V D L+ψ) 2L 2,2k+ = M l=0 B + 2k,2l ψ 2L,2l C + 2k+,2l+ ψ 2L+2,2l+, B 2k,2l ψ 2L 2,2l C 2k+,2l+ ψ 2L,2l+, with the same reflecting boundary conditions as UL+ D and (V L+ D ψ) μ,ν = (UL+ D ψ) μ,ν for any values of (μ, ν) which were not described previously. he matrix VL+ D is a deformation of the matrix U L+ D, but its structure remains close to the structure of UL+ D. Note that span{e 2L+,k,e 2L,j ; j, k Z 2M } belongs to ker ( VL+ D I).Forψ an eigenvector of VL+ D associated with the eigenvalue z, V D L+ψ = zψ. his implies that either ψ span{e 2L+,k,e 2L,j ; j, k Z 2M } and z =,or ψ 2L 2 F z and ψ 2L+2 G z and

24 364 J. Asch et al. Ann. Henri Poincaré ψ 2L = P 2L (z)ψ 2L ψ 2L+2 = A + (z)ψ 2L = z Q E B + Q E ψ 2L + zq O C + Q O ψ 2L ψ 2L = A (z)ψ 2L 2 = z Q E B Q E ψ 2L 2 + zq O C Q O ψ 2L 2. he transfer matrices A + (z) anda (z) are deformations of the matrices A z. his construction is useful to establish the following lemma. In the following, z will be fixed as a parameter. Lemma 9.5. Let (d + 2k ) k {0,...,M } and (d 2k ) k {0,...,M } two families of vectors belonging to the odd subspace O. hese families are the columns of two corresponding matrices denoted ( D + and ) D respectively. Assume that for z 0, max( D, D + ) max z, z and consider the matrix VL D parametrized by z B + z = z + D + C + z B z C z = z( z D + ) = z + z 2 D =(z D ) with hen, 0 0 = (e 0 + d 0 ) (e 2M 2 + d 2M 2 ), M (Vz P 2L (z)w z )(e 0 + d + 0 ) (e 2M 2 + d + 2M 2 ) = e 0 e 2M 2, M (Vz A P 2L (z)a + W z )e 0 e 2M 2. (8) Proof. By (4), Wz 2 {0,...,M } : = I, V z = KV z K, z 0. It follows for all k A + W z e 2k = W z (e 2k + d + 2k ) A Vz e 2k = Vz (e 2k + d 2k ). Given z, D + and D and the associated matrix VL+ D, we consider the corresponding eigenvalue problem: V D L+ψ = z ψ

25 Vol. (200) Localization Properties of the Chalker Coddington Model 365 he complex number z is an eigenvalue of V D L+ iff (z ) 4M e 0 e 2M 2, M (V z A z (z )P 2L (z )A + z (z )W z ) e 0 e 2M 2 =0 where A ± z (z )=z Q E B z ± Q E + z Q O C z ± Q O. Once multiplied by z (4L+5)M, the left-hand side is a polynomial of degree 2M(4L +5)inz. Following the houless argument, we get for the logarithm of the modulus divided by 4ML: 2 log ( rt log z + 2+ ) ( ) log x z dν M 2L L,z(x)+O, L B(0,R z) where the family of measures ν M L,z are supported on some closed ball B(0,R z), due to the fact that sup L U D L V D L <. Note that if z = z, A + z (z) =A + and A z (z) =A End of Proof of Inequality (7) We split the proof in the two following lemmas, whose proof is an adaptation of the argument given in [3]. Lemma 9.6. If (ν L ) L N and μ are measures supported on B(0,R) for some R>0, andif(ν L ) converge weakly to μ, then for any z C log ζ z dν L (ζ) log ζ z dμ(ζ). Proof. Given z C, letf ɛ be defined by: { fɛ (ζ) = log ζ z if ζ z ɛ f ɛ (ζ) = log ɛ L if ζ z ɛ Since the support is compact, lim f ɛ (ζ)dν L (ζ) = f ɛ (ζ)dμ(ζ). On the other hand, for any ζ in, so that: lim sup L log ζ z f ɛ (ζ) log ζ z dν L (ζ) lim sup f ɛ (ζ)dν L (ζ) L = f ɛ (ζ)dμ(ζ). he result follows by monotone convergence theorem when ɛ goes to zero.

26 366 J. Asch et al. Ann. Henri Poincaré Remark. Let us note that the -algebra of trigonometric polynomials F defined by: F = {f C(B(0,R)); f(r, θ) = a k,k 2 r k e ik2θ, N N 0,k N 0,k 2 Z} k + k 2 N separates points and contains the constants. Its closure under the supremum norm is C(B(0,R)). he weak convergence of the measures is equivalent to have for all f in F, lim f(ζ)dν L (ζ) = f(ζ)dμ(ζ). L B(0,R) B(0,R) Lemma 9.7. As a Borel measure on C, the sequence of measures (ν M L,z ) parametrized by z,m converges almost surely weakly to dl as L tends to infinity. Proof. Let (r j,z e iξj,z ) 2M(4L+) j= and (e iλj ) 2M(4L+) j= be the eigenvalues of the problems with reflecting boundary conditions for V and U D, which correspond respectively to the modified and unmodified potentials. We have that: ν M L,z = μ M L = 2M(4L +) 2M(4L +) j δ rj,ze iξ j,z δ e iλ j. hese measures are supported on some B(0,R z ). We will drop the z subscript in the sequel. Since we already know that (μ M L ) converges almost surely weakly to dl, we only need to show that for any non-negative integer k and any integer k 2, lim L 2M(4L+) 2M(4L +) j= j e ik2λj r k j eik2ξj =0. Actually, it is enough to prove it for non-negative integers k, k 2. Let us fix such a couple (k,k 2 ) and decompose the term on the left-hand side as follows: 2M(4L+) e ik2λj r k j 2M(4L +) eik2ξj = (L)+ 2 (L) where (L) = 2 (L) = j= 2M(4L+) (r k2 j 2M(4L +) j= k2 r(ud 2M(4L +) V k2 ). r k j )eik2ξj

27 Vol. (200) Localization Properties of the Chalker Coddington Model 367 If k = min(k,k 2 )andl = max(k,k 2 ), we have (L) Following [3], we first prove that: lim L 2M(4L+) r k r l k j 2M(4L +) j= Rz k 2M(4L+) r l k j. 2M(4L +) j= 2M(4L+) 2M(4L +) j= r j =0. We know that there exists two orthonormal bases (φ j ) 2M(4L+) j= and (φ j )2M(4L+) j= such that: V D L = 2M(4L+) j= μ j (V D L ) φ j φ j, where (μ j (VL D)) are the singular values of the operator V L D. Actually, μ j(vl D)= r j and we assume them to be ordered: μ j+ (VL D) μ j(vl D ) 0. Note that: {μ 2 j(vl D ); j {,...,2M(4L +)}} = σ(vl D V D L )\{0}. Since for each j {,...,2M(4L +)}, μ j (UL D ) = we deduce from the remark following heorem.20 in [24] that: 2M(4L+) j= r j = 2M(4L+) j= μ j (V D L ) μ j (U D L ) 2M(4L+) j= μ j (V D L U D L ). Since VL D U L D has rank and norm uniformly bounded in L, we obtain that: lim (L) =0. L he term 2 (L) will be treated in a similar way. he operator UL D V L D has rank and norm uniformly bounded in L. his implies that for all integer k 2, UL D k 2 VL D k 2 has also rank and norm uniformly bounded in L. So, lim 2(L) =0, L which concludes the proof. he above lemmata together with Eq. 8 establish the inequality (7) which implies (6). We finish with the proof of the houless formula on : Lemma 9.8. For all z, M λ i (z) = M 2 log rt +2 i log x z dl(x)

28 368 J. Asch et al. Ann. Henri Poincaré Proof. We note with [2] that lim L 4L log M (P 2L )(z) is subharmonic in C\{0} and log z x dl(x) subharmonic on C\. he two exceptional sets are of measure zero in C, these quantities must agree everywhere. Remark 9.9. We note that the above proof does not depend on the specific form of the density of states. 0. Appendix 2 Now we prove heorem 8. in several steps. By heorem 6. the localization length is finite for all values of the parameters. Note that the spectrum is characterized by the existence of generalized eigenfunctions: Suppose that the support of E p ( ), the spectral resolution of U(p), is the whole circle. Proposition 0.. For M N, p Ω the spectrum of U(p) is the closure of the set S p = {z ; U(p)φ = zφ has a non-trivial polynomially bounded solution} and E p (\S p )=0. Proof. he stated behaviour at infinity of the generalized eigenvectors and the spectrum of U(p) are related by Sh nol s heorem. his well known deterministic fact for self-adjoint operators was proven in [4] to hold in the unitary setup for band matrices on l 2 (Z). It is straightforward to check that the result holds for band matrices on l 2 (Z, C 2M ), with M finite. Secondly we prove the existence of a finite cyclic subspace: Lemma 0.2. Let M N,rt 0. Denote I 0 := {0} Z 2M. he vectors {e μ ; μ I 0 } span a cyclic subspace of l 2 (Z Z 2M ). Proof. he only non-vanishing elements in U are the blocks given in Eq.. Denoting generically the elements of S by ( ) α β S =: γ δ and observing that Uμ,ν = U ν,μ we have ( ) ( ) U(2j+,2k);(2j,2k) U (2j+,2k);(2j+,2k+) α β = U (2j,2k+);(2j,2k) U (2j,2k+);(2j+,2k+) γ δ ( ) U(2j+2,2k+2);(2j+2,2k+) U = (2j+2,2k+2);(2j+,2k+2) U (2j+,2k+);(2j+2,2k+) U (2j+,2k+);(2j+,2k+2) and ( U (2j,2k);(2j+,2k) U (2j+,2k+);(2j+,2k) = ( U 2j+2,2k+;2j+2,2k+2 U 2j+,2k+2;2j+2,2k+2 U (2j,2k);(2j,2k+) U (2j+,2k+);(2j,2k+) U 2j+2,2k+;2j+,2k+ U 2j+,2k+2;2j+,2k+ ) ( ) α γ = β δ ).

29 Vol. (200) Localization Properties of the Chalker Coddington Model 369 Computing Ue (0,2k) = αe (,2k) + γe (0,2k+) and the corresponding expressions for U e (,2k),Ue (0,2k+),U e (,2k+) we infer: e (,2k) = ( ) Ue(0,2k) γe (0,2k+) α e (,2k+) = β α e (0,2k) γ β U e (0,2k+) e (,2k+) = γ ( Ue(0,2k+) αe (0,2k+2) ) e (,2k+2) = δ γ e (0,2k+) α δ U e (0,2k+2). hus vectors with indices in {±} Z 2M belong to the subspace generated by U ± (I 0 ). he lemma follows by induction. Let I = {0, } Z 2M, Ω= Z4M \I, P = k Z 4M \I dl, p = {p j } j Z 4M \I Ω, and Θ I = {θ j } j I. We shall use the notation Ω p =(p, Θ I ). Denote λ M (p, z) := lim L ( 4L log( M P 2L (z)(p) ) ) 4L log( M P 2L (z)(p) ), if the limit exists. By construction, 5., it holds for almost every p λ M = λ M (p) By definition λ M (p, z) is independent of the finitely many Θ I,ifp = (p, Θ I ). By heorem 6. there exists Ω(z) Ω with P(Ω(z)) = such that for any z \R λ M ((p, Θ I ),z)=λ M > 0, for all θ j Θ I and all p Ω(z). We can apply Fubini to the measure P dl to get the existence of Ω 0 Ω with P(Ω 0 ) = such that for every p Ω 0 there is B p with l(b p )=0and λ M ((p, Θ I ),z) > 0 for all θ j Θ I, and all z B p C. (9) hen we show that for p Ω 0, B p C is a support of the spectral resolution of U((p, Θ I )) for almost every θ j Θ I w.r.t. d I l on I. For any fixed j I, we introduce the spectral measures μ j p associated with U(p) = xde p(x) defined for all Borel sets Δ by μ j p(δ) = e j E p (Δ) e j. Since U(p) = D(p)S, where D(p) is diagonal, the variation of a random phase at one site is described by a rank one perturbation. More precisely, dropping the variable p temporarily, we define D by taking θ j = in the definition of D: D = D + e j e j ( θ j )=e log(θj) ej ej D,

30 370 J. Asch et al. Ann. Henri Poincaré so that, with the obvious notations, Ũ = DS =e log(θj) ej ej U. he unitary version of the spectral averaging formula, see [9] and [3], reads in our case: for any f L (), dl(θ j ) f(x)dμ j (p,θ I ) (x) = f(x)dl(x). Applied to f = χ Bp, the characteristic function of B p, this yields 0=l(B p )= μ j (p,θ I ) (B p), dl(θ j ). (20) Consequently, μ j (p,θ I ) (B p)=0, for every θ k Θ I,k jand Lebesgue-a.e. θ j. herefore, for all p Ω 0, there exists J p I s.t. l(j p C )=0and Θ I J p μ j (p,θ I ) (B p)=0, j I. (2) Now fix p Ω 0 and Θ I J p and consider p =(p, Θ I ). By Lemma 0.2 and (2) we deduce that E p (B p )=0.IfS p is the set from Sh nol s heorem 0., then the set S p B C p is a support for E p ( ). Now take z S p B C p. By heorem 0., U(p)ψ = zψ has a non-trivial polynomially bounded solution ψ. On the other hand, by (9), λ M (p, z) > 0. hus, by Osceledec s heorem, every solution which is polynomially bounded necessarily has to decay exponentially both at + and, and therefore it is an eigenfunction of U(p). In other words, every z S p B C p is an eigenvalue of U(p), hence S p B C p is countable. herefore E p ( ) has countable support thus U(p) has pure point spectrum. With Ω 0 := {(p, {θ j } j I ) s.t. p Ω 0, {θ j } j I Θ I J p }, we have p Ω 0 σ c (U(p)) =. (22) Also, from l(j C p )=0wehave ( j I dl)(j p )=( j I dl)( I )=. (23) As P(Ω 0 ) =, we conclude from (22) and (23) that P(σ c (U(p)) = ) P(Ω 0 )= dp(p)( j Idl)(J p )=, Ω 0 which proves that U(p) has almost surely pure point spectrum. he fact that the support of the density of state coincides with the almost sure spectrum, see [20], shows that Σ pp =. We finally show that almost surely all eigenfunctions decay exponentially. Note that we actually have shown above that the event all eigenvectors of

31 Vol. (200) Localization Properties of the Chalker Coddington Model 37 U(p) decay at the rate of the smallest Lyapunov exponent has probability one, since this is true for all p Ω 0. Measurability of this event was proven for the case of ergodic one-dimensional Schrödinger operators by Kotani and Simon in heorem A. of [22]. he proof of this fact provided in [22] carries over to the CC model as well. It is enough to note that, due to Lemma 0.2, we may use ρ p = j I μj p as spectral measures in their argument. References [] Arnold, L.: Random dynamical systems. Springer Monographs in Mathematics. Springer, Berlin (998) [2] Avron, J.E., Seiler, R., Simon, B.: Charge deficiency, charge transport and comparison of dimensions. Comm. Math. Phys. 59, (994) [3] Bourget, O.: Singular continuous Floquet operator for periodic quantum systems. J. Math. Anal. Appl. 30, (2005) [4] Bourget, O., Howland, J.S., Joye, A.: Spectral analysis of unitary band matrices. Commun. Math. Phys. 234, (2003) [5] Bougerol, P., Lacroix, J.: Products of Random Matrices with Applications to Schrödinger Operators. Progress in Probability and Statistics, vol. 8. Birkhäuser, Boston (985) [6] Boumaza, H.: Hölder continuity of the IDS for matrix-valued Anderson models. Rev. Math. Phys. 20, (2008) [7] Boumaza, H., Stolz, G.: Positivity of Lyapunov exponents for Anderson-type models on two coupled strings. Elecron. J. Differ. Equ. 47, 8 (2007) [8] Bellissard, J., van Elst, A., Schulz-Baldes, H.: he noncommutative geometry of the quantum Hall effect. J. Math. Phys. 35, (994) [9] Combescure, M.: Spectral properties of a periodically kicked quantum Hamiltonian. J. Stat. Phys. 59, (990) [0] Chalker, J.., Coddington, P.D.: Percolation, quantum tunneling and the integer Hall effect. J. Phys. C 2, (988) [] Colin de Verdière, Y., Parisse, B.: Équilibre instable en régime semi-classique. I. Concentration microlocale. Commun. Partial Differ. Equ. 9, (994) [2] Craig, W., Simon, B.: Subharmonicity of the Lyaponov index. Duke Math. J. 50, (983) [3] Craig, W., Simon, B.: Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices. Commun. Math. Phys. 90(2), (983) [4] Fertig, H.A., Halperin, B.I.: ransmission coefficient of an electron through a saddle-point potential in a magnetic field. Phys. Rev. B 36, (987) [5] Graf, G.M.: Aspects of the integer quantum Hall effect. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon s 60th birthday. In: Proceedings of symposium of Pure Mathematics vol. 76, Part, pp American Mathematical Society, Providence, RI (2007) [6] Germinet, F., Klein, A., Schenker, J.: Dynamical delocalization in random Landau Hamiltonians. Ann. Math. 66, (2007) [7] Goldsheĭd, I.Ya., Margulis, G.A.: Lyapunov exponents of a product of random matrices. Uspekhi Mat. Nauk 44, 3 60 (989)