Weak Localization for the alloy-type 3D Anderson Model

Transcription

1 Outline Statement of the result Proof of the Theorem Weak Localization in the alloy-type 3D Anderson Model Zhenwei Cao Math Department Virginia Tech Arizona School of Analysis and Mathematical Physics March 2012 Zhenwei Cao Weak Localization for the alloy-type 3D Anderson Model

2 Outline Statement of the result Proof of the Theorem Outline Background Statement of the result Proof of weak localization Wegner s estimate Proof of lemma Zhenwei Cao Weak Localization for the alloy-type 3D Anderson Model

3 Outline Statement of the result Proof of the Theorem Setup Background Assumptions Results Setup Consider the random Schrödinger operator of the following type (Hωψ)(x) λ := 1 2 ( ψ)(x) + λv ω(x)ψ(x). Here denotes the discrete Laplace operator, ( ψ)(x) = ψ(x + e) 6ψ(x), e Z 3, e =1 and V ω stands for a random multiplication operator of the form V ω (x) = ω i u(x i). i Z 3 Note the spectrum of the unperturbed operator H 0 w is absolutely continuous and is the interval [0, 6]. Zhenwei Cao Weak Localization for the alloy-type 3D Anderson Model

4 Background For 3D models, the first weak localization result for the Anderson model was proved by Frölich and Spencer(1983) through multiscale analysis. There are many results quantifying the weak localization region when the single cite potential is a delta function. For example, Aizenman(1994) shows there is localization in the region [ aλ, aλ + λ 5/4 ]. Klopp(2002) derives a upper bound on the of order λ 7/6. Elgart(2009) pushes the upper bound to λ 2 by using a Feynman diagrammatic techinque. This work extends the results in Elgart(2009) to a general single cite potential. Related work on diagramatic techniques includes Erdos and Yau(2000) and Chen(2005). Notice non-monotonicity of the single cite potential poses a problem in deriving Wegner s estimate. Also there are issues involving employing the diagramatic technique.

5 Outline Statement of the result Proof of the Theorem Setup Background Assumptions Results Assumptions u decays exponentially fast: u(x) Ce A x u is compactly supported. The random variables {ω i } are independent, identically distributed, even, and compactly supported on an interval J, with bounded probability density ρ. Moreover, function ρ is Lipschitz continuous: The moments of ω i satisfy ρ(x) ρ(y) K x y E[ω 2m i ] = c 2m (2m)!c v, c 2 = 1, i Z 3, m N Zhenwei Cao Weak Localization for the alloy-type 3D Anderson Model

6 Outline Statement of the result Proof of the Theorem Setup Background Assumptions Results Results Let û denotes the Fourier transform of u. û(p) = n Z 3 e i2πp n u(n), p T 3 = [ 1/2, 1/2] 3 Spectral localization E 0 = 2λ 2 û 2 2λ 4 û 4 (1) For any α > 0 there exists λ 0 (α) such that for all λ < λ 0 (α) the spectrum of H ω within the set E < E 0 λ 4 α is almost surely of the pure-point type, and the corresponding eigenfunctions are exponentially localized. Zhenwei Cao Weak Localization for the alloy-type 3D Anderson Model

7 for all x, y Z 3. Lemma 1 For any integer N and energies E that satisfy the condition of above theorem we have the decomposition R(x, y) = N 1 n=0 A n (x, y) + z Z 3 Ã N (x, z)r(z, y), with A 0 (x, y) = R r (x, y), and where the (real valued) kernels A n, Ã N satisfy bounds E A n (x, y) 2 (4n)! E (C(E ) E Ã N (x, y) (4N)! ( C(E ) where δ := E 0 E E /( 6π). The zero order contribution A 0 satisfies λ 2 E ) n e δ x y, n 1 ; λ 2 E ) N/2 e δ x y /2, N > 1 ; A 0 (x, y) 2 e δ 3 3 x y (2)

8 Proof of the theorem E Using above lemma, choosing (4N) 4 =, we get a bound C(E ) λ 2 ) E R(E + iɛ; x, k) C (e δ L/2 + e N. ɛ δ Hence we obtain E R ΛL,x (E + iɛ; x, w) E R(E + iɛ; x, w) + E R ΛL,x (E + iɛ; x, w) R(E + iɛ; x, w) C L2 ɛ max E R(E + iɛ; x, k) dist(k, Λ) 1 C L2 ɛ ] [e δ L/2 + e N. (3) ɛδ Let I = [E ɛ 1/4, E + ɛ 1/4 ], and consider two events, the first one is G ω (I ) := { ω Ω : σ(h ΛL,x ) I = }, the other one is σ(h ΛL,x ) I For the first part, since RΛL,x (E + iɛ; x, w) R ΛL,x (E; x, w) ɛ 1/2.

9 Proof of the theorem Pairing this bound with (3) and using Chebyshev s inequality ] } Prob {ω G ω (I ) : R ΛL,x (E; x, w) C [e L2 δ L/2 + e N + ɛ 1/4 ɛδ ɛ 5/4 The Wegner estimate implies that for the second event Cɛ 1/4. (4) Prob { σ(h ΛL,x ) I } C I Λ L,x 2 D 3/2 = C ɛ 1/4 D 3/2 L 4. (5) Combining (4) and (5) we arrive at ] } Prob { R ΛL,x (E; x, w) C [e L2 δ L/2 + e N + ɛ 1/4 ɛδ ɛ 4/3 Cɛ 1/4 D 3/2 L 4. Choose E = λ 4 α /2, L = λ 2 and ɛ = e 4λ B α/2 we get the following initial volume estimate { } Prob R Λλ 2,x (E; x, w) e λ B α/2 e λ B α/2. (6)

10 Wegner s estimate Wegner s estimate Let I be an open interval of energies such that Then we have D := dist(i, σ(h 0 ω)) > 0. E trp I (H Λ,λ ω ) C I Λ 2 D 3/2, (7) where H Λ,λ ω denotes a natural restriction of H λ ω to Λ Z 3. This Wegner s estimate may not be the optimal one, as in the δ potential case, where it is Λ instead of Λ 2. But suffice for the proof of localization.

11 Proof of Wegner s estimate Notice EF ω = E tr Im e 2δ 1+δ 1 δ F ω ρ(ω i )dω i = 1 2δ i Λ ( H Λ,λ ω dv ) 1 E iη 1+δ 1 δ v Λ dv F v ˆω ρ(v ˆω i )d ˆω i. i Λ η v 2 tr ( (v 1 A + Vˆω ) 2 + (η/2) 2) 1 ρ(v ˆω i )d ˆω i. i Λ where A = Λ E is positive. Change variable u = v 1, and factor out A = A 1/2 A 1/2. We get Tr ( (ua + Vˆω ) 2 + (η/2) 2) 1 A 1 Tr ) 1 (E (u + A 1/2 Vˆω A 1/2 ) 2 + η2. (8) 4 A Integrate over u first and then dŵ i s. We finally get E trp I (H Λ,λ ω ) C I Λ 2 (E ) 3/2.

12 Outline Statement of the result Proof of the Theorem Outline of the proof of Lemma 1 Resolvent expansion An example An example An example General case Resolvent expansion near the self-energy, and the range of localization follows from the bound on the self-energy. Renormalization (cancellation of tadpoles). Extraction of exponential decay. Diagramatic estimation on the resulting integral. Zhenwei Cao Weak Localization for the alloy-type 3D Anderson Model

13 Resolvent expansion Decompose H λ ω as H λ ω = H r +Ṽ, H r := 1 2 σ(p, E+iɛ), Ṽ := λv ω +σ(p, E+iɛ), Let R r := (H r E iɛ) 1 We can expand R into resolvent series R = n ( R r Ṽ ) i R r + ( R r Ṽ ) n+1 R. (9) i=0 Stop expansion term by term at order N. An order of a term is the number of appearances of σ and V ω, where σ counts as 2. Thus R r σr r λv ω R r σr has order 5. The following is the expansion for N = 2 following this stopping rule. R = R r R r σr {λr r V ω R} = R r R r σr λr r V ω R r + λr r V ω R r σr + λ 2 R r V ω R r V ω R,

14 An example The advantage of the above stopping rule is that all tadpoles will be cancelled. The following example illustrates this idea. Consider all the terms of order 4 in the expansion. λ 4 R r V ω R r V ω R r V ω R r V ω R r, λ 2 R r V ω R r V ω R r σr r, λ 2 R r V ω R r σr r V ω R r, λ 2 R r σr r V ω R r V ω R r, R r σr r σr r The expectation of the product of random variables will give us some delta functions. E[ω(x 1 )ω(x 2 )ω(x 3 )ω(x 4 )] = (1 δ(x 1 x 3 ))δ(x 1 x 2 )δ(x 3 x 4 ) + (1 δ(x 1 x 2 ))δ(x 1 x 3 )δ(x 2 x 4 ) + (1 δ(x 1 x 3 ))δ(x 1 x 4 )δ(x 2 x 3 ) + c 4 δ(x 1 x 2 )δ(x 3 x 4 )δ(x 1 x 3 ) = δ(x 1 x 2 )δ(x 3 x 4 ) + δ(x 1 x 3 )δ(x 2 x 4 ) + δ(x 1 x 4 )δ(x 2 x 3 ) + ( c 4 3)δ(x 1 x 2 )δ(x 3 x 4 )δ(x 1 x 3 ) Set c 4 = c 4 3, notice σ = λ 2 R r (x, x)

15 An example E λ 4 xr r V ω R r V ω R r V ω R r V ω R r y = x 1,x 2,x 3,x 4 xr r x 1 x 1 R r x 2 x 2 R r x 3 x 3 R r x 4 x 4 R r y E[ω x1 ω x2 ω x3 ω x4 ] = σ 2 xr 3 r y + λ 4 x 1,x 2 xr r x 1 x 1 R r x 2 x 2 R r x 1 x 1 R r x 2 x 2 R r y + λ 2 σ x 1 xr r x 1 x 1 R 2 r x 1 x 1 R r y + c 4 λ 4 x 1 xr r x 1 x 1 R r x 1 3 x 1 R r x 1 The tadpoles are the first term and third terms. The first term is equal to E xλ 2 R r V ω R r V ω R r σr r y, E xλ 2 R r σr r V ω R r V ω R r y, and E xr r σr r σr r y. The third term is equal to E xλ 2 R r V ω R r σr r V ω R r y. So they cancel out exactly. This is the case when the single cite function is the delta function.

16 An example When the single cite potential is of the more general form V ω (x) = λ i Z d q i (ω)u(x i), we represent it in its Fourier transform. Using R r (z, w) = e i2π(z p) d 3 p E(p), We get for E λ 4 xr r V ω R r V ω R r V ω R r V ω R r y (T 3 ) 5 e 2πi(p 1x p 5 y) 5 i=1 d 3 p i E(p i ) ˆV ω (p) = û(p)ˆω(p), 4 û(p i p i+1 ) i=1 E [ˆω(p 1 p 2 )ˆω(p 2 p 3 )ˆω(p 3 p 4 )ˆω(p 4 p 5 )] and the renormalization process goes through.

17 General case Some combinatorics will show the renormalization is true for general N. Without concerning too much details of the notation, we present the following identity. E N m = c Sj δ(x Sj ), where j Υ N,N ω xj m=1 π={s j } m j=1 j=1 δ(x S ) = δ xj y, y Z 3 and c 2l (cl) 2l+1, c 2 = E ω 2 x = 1. If S j in π Π has form S j = {i, i + 1}, we refer to it as a tadpole. Then the following lemma is a result of the following identity N ( 1) k k=0 E π Π: π=πk c {S} [ ] ω xi i S j S S l π c k δ(x Sl ) = π Π: π=π 0 c Sj δ(x Sj ). S j π

18 Lemma 2 Lemma 2 For A l defined in Lemma 1, the function E A l (x, y) 2 is a function of the variable x y. Let then we have E A l (x, y) 2 = λ 2l A l,e (x y) := E A l (x, y) 2, (T 3 ) 2l+2 e iα dp l+1 E(p l+1 ) i Υ l û(p j p j+1 ) π Π l : π=π 0 dp 2l+2 E(p 2l+2 ) S k π l dp j E(p j ) j=1 2l+1 j=l+2 dp j E (p j ) c Sk δ p i p i+1, (10) i Sk

19 Proof of the Lemma 1 We first want to establish the exponential decay of A l,e (x y) in x y We will show that for a general value of l, where A l,e (x) û 2l,R e δ/3 x  l,e (0), (11)  l,e (0) := λ 2l e iα dp l+1 dp 2l+2 (T 3 ) 2l+2 e(p l+1 ) + E e(p 2l+2 ) + E π Π l : π=π 0 S k π j Υ l dp j e(p j ) + E c Sk δ p i p i+1. (12) i Sk The expression  l,e (0) has been studied in Elgart(2009) before and shown that  l,e (0) (4l)! E (C ln 9 (E ) λ 2 E ) l. (13)

20 Proof of the Lemma 1 Using the delta function introduced before and integrate over the tree momenta. Let E(p) = e(p) E iɛ σ(p, E + iɛ). A l,e (x) = λ r π 2l c π dw 1 e i2πw 1 s π 1 x E û(w 1 +Q j ) (w π 1 + q i ) e i2πw 2 x t Φ dw t 2n+2 i=r π+1 i=1 1 E (q i ) 2n j=s π+1 j=1 û(q j ), where E (p) stands for either E(p) or E (p), Φ is a set of indices of loop momentum that does not include w 1, and q i, Q j are some linear combinations of the loop variables in Φ. Note now that the integral with respect to w 1 becomes dw 1 e i2π(w 1 x (w 1 e γ)x γ) 1/2 1/2 d(w 1 e γ ) r π i=1 1 E (w 1 + q i ) e i2π(w 1 e γ)x γ s π j=1 û(w 1 + Q j ) (14)

21 Proof of the Lemma 1 The exponential decay is established by extending the second part of (14) to complex coordinate. Let R denote { 1/2 ± i δ; 1/2 ± i δ}. This integral is periodic over vertical segments of R and therefore r π d(w 1 e γ ) T i=1 r π = T i d(w 1 e γ ) δ i=1 û sπ,e e x E /3 Now we have arrived at (11). 1 E (w 1 + q i ) e i2πxγ (w 1 e γ) s π j=1 1 E (w 1 + q i ) e i2πxγ (w 1 e γ) T d(w 1 e γ ) r π i=1 û(w 1 + Q j ) s π û(w 1 + Q j ) j=1 1 e(w 1 + q i ) + E, (15)

22 Properties of self-energy σ(p, E) Recall the self energy term σ, associated with H λ ω, is given by the solution of the self-consistent equation σ(p, E + iɛ) = λ 2 T 3 d 3 q û(p q) 2 e(q) E iɛ σ(q, E + iɛ). (16) We need existence, periodicity, and analyticity of the self energy operator σ(p, E + iɛ). Consider space L(T 3 ) = {f : T 3 C f <, f is real analytic}. and define map T ɛ : L(T 3 ) L(T 3 ) pointwise as (T ɛ f )(p) = λ 2 T 3 d 3 q û(p q) 2 e(q) E iɛ f (q). (17) Then T ɛ is a contraction on the ball B β (0) where β = 2λ 2 û 2 for all p, ɛ and E E 0 = 2λ 2 û 2 2λ 4 û 4.

23 References M. Aizenman, Localization at weak disorder: some elementary bounds, Rev. Math. Phys.,6: , T. Chen, Localization Lengths and Boltzmann Limit for the Anderson Model at Small Disorders in Dimension 3, J. Stat. Phys., 120: , A. Elgart, Lifshitz tails and localization in the three-dimensional Anderson model, Duke Math. J., 146: , L. Erdos and H.-T. Yau, Linear Bolzmann equation as the weak coupling limit of the random Schrödinger equation, Commun. Pure. Appl. Math., LIII: , J. Fröhlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys., 88: , F. Klopp, Localization for some continuous random Schrödinger operators, Comm. Math. Phys., 167: , 1995.

24 Thank you Outline Statement of the result Proof of the Theorem Proof of the Lemma 1 Proof of the Lemma 1 Proof of the Lemma 1 Properties of self-energy σ(p, E) References Thank you Thank you! Zhenwei Cao Weak Localization for the alloy-type 3D Anderson Model