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(1)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Some results on the Chalker Coddington model

Joachim Asch 1

Centre de Physique Th´ eorique, Marseille

Grenoble, november 2012

(2)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Setup

l 2 ( Z 2 )

U ω (ϕ) = D ω S (ϕ) - random unitary D ω = (. . . , ω µ , . . .) - random diagonal

one random phase ω µ per site µ ∈ Z 2 , independenly and uniformly distributed on T = {z ∈ C ; |z| = 1}.

Model parameter ϕ.

S (ϕ) := cos ϕ

| {z }

=:t

S + i sin ϕ

| {z }

=:t

S - deterministic shift.

H2j,2kL

Figure: Elementary action of S j,k and S j,k

J. Asch , Grenoble 2012

(3)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents r

r r r

r r r r

r r r r

r r r r

r r r r

r r r r

r r r r

r r r r

r r r r

1 t t 1

t t t 1

t t t 1

t t 1 1

1 t t t

t t t t

t t t t

t t 1 t

1 t t t

t t t t

t t t t

t t 1 t

1 1 t t

t 1 t t

t 1 t t

t 1 1 t

r r r

r r

r r

r r

r r r

H0,0L

Figure: Action of U

Question: Long time behavior of n → U n

in particular the localization ←→ delocalization question:

(4)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Quantum Walk Language

l 2 ( Z 2 ) ∼ = l 2 ( Z 2 ) ⊗ C 4 ∼ = l 2 ( Z 2 ) ⊗ l 2 ( Z 4 )

J. Asch , Grenoble 2012

(5)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Quantum Walk Language

l 2 ( Z 2 ) ∼ = l 2 ( Z 2 ) ⊗ C 4 ∼ = l 2 ( Z 2 ) ⊗ l 2 ( Z 4 )

Μ=Hi,jL

(6)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Quantum Walk Language

l 2 ( Z 2 ) ∼ = l 2 ( Z 2 ) ⊗ C 4 ∼ = l 2 ( Z 2 ) ⊗ l 2 ( Z 4 )

Μ=Hi,jL

ÈH2ΜL>@ÈΜ0>

J. Asch , Grenoble 2012

(7)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Quantum Walk Language

l 2 ( Z 2 ) ∼ = l 2 ( Z 2 ) ⊗ C 4 ∼ = l 2 ( Z 2 ) ⊗ l 2 ( Z 4 )

Μ=Hi,jL

ÈH2ΜL>@ÈΜ0> ÈΜ1>

ÈΜ2>

ÈΜ3>

(8)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Quantum Walk Language

l 2 ( Z 2 ) ∼ = l 2 ( Z 2 ) ⊗ C 4 ∼ = l 2 ( Z 2 ) ⊗ l 2 ( Z 4 )

Μ=Hi,jL

ÈH2ΜL>@ÈΜ0> ÈΜ1>

ÈΜ2>

ÈΜ3>

Τ

S ∼ = cos(ϕ) I ⊗ T + i . . .

J. Asch , Grenoble 2012

(9)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Quantum Walk Language

l 2 ( Z 2 ) ∼ = l 2 ( Z 2 ) ⊗ C 4 ∼ = l 2 ( Z 2 ) ⊗ l 2 ( Z 4 )

r r r r

r r r r

r r r r

r r r r

r r r r

r r r r

r r r r

r r r r

r r r r

t t t 1

t t t 1

t t 1 1

t t t t

t t t t

t t 1 t

t t t t

t t t t

t t 1 t

t 1 t t

t 1 t t

t 1 1 t

r r r

r r r

r r r

H0,0L

(10)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Quantum Walk Language

l 2 (Z 2 ) ∼ = l 2 (Z 2 ) ⊗ C 4 ∼ = l 2 (Z 2 ) ⊗ l 2 (Z 4 )

Μ=Hi,jL

ÈH2ΜL>@ÈΜ0> ÈΜ1>

ÈΜ2>

ÈΜ3>

Τ

T

D

T

R

T

U

T

L

S ∼ = cos(ϕ) I ⊗ T + i sin(ϕ) I ⊗ T −1 W

W = T D ⊗|0i h0|+T R ⊗|1i h1|+T U ⊗|2i h2|+T L ⊗|3i h3|

J. Asch , Grenoble 2012

(11)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Properties

Complete localization for ϕ ∈ π 4 Z:

Proposition

For rt = 0 the spectrum of U ω (ϕ) is always pure point and

|hµ, U ω n νi| = 0 |µ − ν| ≥ 2 Localization length is zero.

Proof r = 0.

Μ=Hi,jL

ÈH2ΜL>@ÈΜ0> ÈΜ1>

ÈΜ2>

ÈΜ3>

Τ

(12)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Properties

If all the phases are equal to one (or periodic !?) the system propagates completely :

Proposition

For ϕ / ∈ π 4 Z the spectrum of S (ϕ) is purely absolutely continuous.

Proof.

Representation of S ◦ S by 2 × 2 matrix valued multiplication operator, Fourier representation

rt(e −iy − e −ix ) r 2 e −ix + t 2 e −iy t 2 e iy + r 2 e ix rt(e ix − e iy )

.

Bandfunctions are not flat, eigenfunctions are extended.

J. Asch , Grenoble 2012

(13)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents Φ

Spectrum of SHΦL

(14)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Scientific interest

Open problem: understand the dynamics of a 2d electron gas in crossed magnetic and electric fields; in particular the

“localization–delocalization” question of the Quantum Hall effect.

J. Asch , Grenoble 2012

(15)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

No electron-electron interactions, zero temperature, planar geometry, constant magnetic field

→ one particle Hamiltonian:

H = 1 2

−i∇ − 1 2 q

2

+ V (` c q) in L 2 ( R 2 ),

V ∈ C ( R 2 , R ). ` c magnetic length, small parameter

(16)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Effective Dynamics in the Lowest Landau Level

V = 0 = ⇒ (H − 1

2 I )P = 0

for P the infinite dimensional projection on the LLL . V 6= 0 = ⇒ the electron “follows the contour lines of V ”:

Theorem

For ` c small enough there exists a unitary U such that for H eff := U HU −1 : [H eff , P ] = 0.

The restriction of H eff to Ran P is in first order Op w

1 2π

Z 2π 0

V ((c 1 , c 2 ) + (cos t, sin t)) dt

,

with c := −p + q 2 the guiding center.

[Op W (c 2 ), Op W (c 1 )] = i

J. Asch , Grenoble 2012

(17)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

The Chalker Coddington model, step 1

use the effective Hamiltonian in one Landau band

Figure: Chalker Coddington (1988)

(18)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

The Chalker Coddington model, step 2

focus on tunneling at saddle points

Figure: Chalker Coddington (1988)

J. Asch , Grenoble 2012

(19)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

The Chalker Coddington model, step 2

and describe tunneling events by scattering matrices S (q) :=

q 1 q 2 0 0 q 1 q 2

t −r r t

q 3 0 0 q 3

∈ U (2) where

r, t ∈ [0, 1], such that, r 2 + t 2 = 1, are deterministic;

the 3 random phases per node q 1 , q 2 , q 3 account for the

impurities.

(20)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

The Chalker Coddington model, step 3

consider a network of tunneling events

(2j,2k) (2j+1,2k+1)

even node

odd node

x-axis

y-axis

with dynamics described by the family of random unitary operators (valleys: |t|: tunnel (turn left); |r|: stay (turn right))

J. Asch , Grenoble 2012

(21)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

The Chalker Coddington model, step 3

consider a network of tunneling events

(2j,2k) (2j+1,2k+1)

even node

odd node

x-axis

y-axis

t

r

t t t

r r

r

with dynamics described by the family of random unitary

(22)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Purpose of the model

Study the localization length as a method to understand the delocalization phenomenon:

CC’s numerical evidence:

-the localization length is finite for t 6= r -the localization length diverges as |t/r | → 1 as

1 ln | t r |

α

with critical exponent α = 2.5 ± 0.5.

CC’s method: finite size scaling based on the restriction of the model to a cylinder of perimeter M.

Our aim here: analytic proof of finiteness of localization length; bounds.

J. Asch , Grenoble 2012

(23)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Spectral analysis

Translation invariance

Θ ν U ω (ϕ)Θ −1 ν = U Θ

ν

ω (ϕ)

for Θ ν (ψ)(µ) := ψ(µ + 2ν) ψ ∈ l 2 (Z 2 ) and ergodicity of the i.u.d. phases −→ ∃ Σ, Σ pp such that :

Σ = σ(U ω ), Σ pp = σ pp (U ω ) ω a.s .

Spectral properties are almost surely deterministic.

(24)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization far from transition

Theorem

There exists a ϕ 0 > 0 such that for |ϕ mod π 2 | ≤ ϕ 0 it holds :

1. Σ(ϕ) = Σ pp (ϕ);

2. p ≥ 0 ω almost surely:

sup

n∈ Z

k|X | p U ω n (ϕ)ψk < ∞;

ψ ∈ l 2 ( Z 2 ) of compact support, |X | p e µ := |µ|e µ ; 3. ∃`, c > 0, ∀µ, ν ∈ Z 2

E

"

sup

f ∈C( T ),kf k

≤1

|he µ , f (U ω (ϕ)) e ν i|

#

≤ ce

1`

|µ−ν| .

J. Asch , Grenoble 2012

(25)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization far from transition

Theorem

There exists a ϕ 0 > 0 such that for |ϕ mod π 2 | ≤ ϕ 0 it holds :

1. Σ(ϕ) = Σ pp (ϕ);

2. p ≥ 0 ω almost surely:

sup

n∈ Z

k|X | p U ω n (ϕ)ψk < ∞;

ψ ∈ l 2 ( Z 2 ) of compact support, |X | p e µ := |µ|e µ ; 3. ∃`, c > 0, ∀µ, ν ∈ Z 2

E

"

sup

f ∈C( T ),kf k

≤1

|he µ , f (U ω (ϕ)) e ν i|

#

≤ ce

1`

|µ−ν| .

(26)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization far from transition

Theorem

There exists a ϕ 0 > 0 such that for |ϕ mod π 2 | ≤ ϕ 0 it holds :

1. Σ(ϕ) = Σ pp (ϕ);

2. p ≥ 0 ω almost surely:

sup

n∈ Z

k|X | p U ω n (ϕ)ψk < ∞;

ψ ∈ l 2 ( Z 2 ) of compact support, |X | p e µ := |µ|e µ ; 3. ∃`, c > 0, ∀µ, ν ∈ Z 2

E

"

sup

f ∈C( T ),kf k

≤1

|he µ , f (U ω (ϕ)) e ν i|

#

≤ ce

1`

|µ−ν| .

J. Asch , Grenoble 2012

(27)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization far from transition

Theorem

There exists a ϕ 0 > 0 such that for |ϕ mod π 2 | ≤ ϕ 0 it holds :

1. Σ(ϕ) = Σ pp (ϕ);

2. p ≥ 0 ω almost surely:

sup

n∈ Z

k|X | p U ω n (ϕ)ψk < ∞;

ψ ∈ l 2 ( Z 2 ) of compact support, |X | p e µ := |µ|e µ ; 3. ∃`, c > 0, ∀µ, ν ∈ Z 2

E

"

sup

f ∈C( T ),kf k

≤1

|he µ , f (U ω (ϕ)) e ν i|

#

≤ ce

1`

|µ−ν| .

(28)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization for cylinders

restrict to a cylinder of perimeter 2M → new random unitary U ω,M on l 2 ( Z × Z 2M );

Figure: The cylinder Z × Z 2M

J. Asch , Grenoble 2012

(29)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization for cylinders

Theorem

Let M ∈ N , ϕ / ∈ π 4 Z . Then 1. Σ(ϕ, M ) = Σ pp (ϕ, M ) = T ;

2. the eigenfunctions decay exponentially, almost surely.

(30)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Analysis of Lyapunov exponents

–for U M write the spectral- as a transfer matrix problem;

–define the Lyaponov exponents λ j (M) for the cocycle defined by the transfer matrices;

–define the localization length as the inverse of the smallest positive Lyapunov exponent;

Definition

` M ∈ [0, ∞] is defined as

` M := 1

λ min (M ) .

J. Asch , Grenoble 2012

(31)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Analysis of Lyapunov exponents

–for U M write the spectral- as a transfer matrix problem;

–define the Lyaponov exponents λ j (M) for the cocycle defined by the transfer matrices;

–define the localization length as the inverse of the smallest positive Lyapunov exponent;

Definition

` M ∈ [0, ∞] is defined as

` M := 1

λ min (M ) .

(32)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Analysis of Lyapunov exponents

–for U M write the spectral- as a transfer matrix problem;

–define the Lyaponov exponents λ j (M) for the cocycle defined by the transfer matrices;

–define the localization length as the inverse of the smallest positive Lyapunov exponent;

Definition

` M ∈ [0, ∞] is defined as

` M := 1

λ min (M ) .

J. Asch , Grenoble 2012

(33)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Proposition 1. ` M < ∞;

2. ∃ c > 0, ∀M ∈ N it holds:

dist(r, {0, 1}) < e −cM = ⇒

` M ≤ 2

log rt 1

− (M − 1) log ((1 + r)(1 + t)) .

(34)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders Credits Proofs

Localization Exponents

Credits

Collaboration with : Olivier Bourget, C´ edric Meresse, Alain Joye

This material in

J. Asch, O. Bourget, A. Joye, Dynamical Localization of the Chalker-Coddington Model far from Transition J. of Stat.

Phys: Volume 147, Issue 1 (2012), Page 194-205

J. Asch, O. Bourget, A. Joye, Localization Properties of the Chalker-Coddington Model Annales Henri Poincare, 11 (2010), 1341–1373

and

J. Asch, C. Meresse, A constant of quantum motion in two dimensions in crossed magnetic and electric fields, J. Phys.

A: Math. Theor. 43 (2010) 474002.

J. Asch , Grenoble 2012

(35)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization by fractional powers of the resolvent

L := (L 1 , L 2 ) ∈ N 2 . Unitary restriction U ω Λ

L

(ϕ) = D ω Λ

L

S Λ

L

(ϕ) on L 2L ) to 4L 1 L 2 blocks. volΛ L := 4L 1 L 2

r r r r

r r r r

r r r r

1 t t 1

t t 1 1

1 t t t

t t 1 t

1 t t t

t t 1 t

1 1 t t

t 1 1 t

r

r r

r r

r r

r

H0,0L

r

Figure: The action of S Λ

L

for L = (2, 1)

(36)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization by fractional powers of the resolvent

L := (L 1 , L 2 ) ∈ N 2 . Unitary restriction U ω Λ

L

(ϕ) = D ω Λ

L

S Λ

L

(ϕ) on L 2L ) to 4L 1 L 2 blocks. volΛ L := 4L 1 L 2

r r r r

r r r r

r r r r

1 t t 1

t t 1 1

1 t t t

t t 1 t

1 t t t

t t 1 t

1 1 t t

t 1 1 t

r

r r

r r

r r

r

H0,0L

Figure: The action of S Λ

L

for L = (2, 1)

J. Asch , Grenoble 2012

(37)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization, scaling of spectral gaps

Proposition

For any L ∈ N 2 , η > 0 it holds P

dist

z , σ

U ω Λ

L

+v (0)

≤ η

= O (ηvolΛ L ))

for ηvolΛ L small enough, uniformly in v ∈ (2Z) 2 and

z ∈ C \ T .

(38)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization, smallness of localized resolvent

Proposition

For s ∈ (0, 1), a ≥ 0 and b >

4 + a(1+s) (1−s)s

it holds

E

he µ , R ω Λ

L

+v (ϕ, z)e ν i

s

= O 1

|L| a

uniformly in the region L ∈ N 2 , z ∈ C \ T ,

|ϕ| ≤ |L| 1

b

, v ∈ Z 2 , µ, ν ∈ Λ L + v , |µ − ν| ≥ 2.

J. Asch , Grenoble 2012

(39)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization, non propagation estimate

Proposition

For s ∈ (0, 1/3) there exist ` 0 > 0, ϕ 0 > 0 and q < 1 such that

E (|he µ , R(ϕ, z )e ν i| s ) ≤ q max

N∂(Λ

L

+[µ])

c

E (|he β , R(ϕ, z )e ν i| s ) for all L ∈ N 2 , |L| > ` 0 ,

z ∈ C \ T , |ϕ| < ϕ 0 , µ ∈ Z 2 , ν ∈

z }| {

L + [µ]) c .

(40)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Localization, exponential decay of resolvent powers

Theorem

For s ∈ (0, 1/3) there exist positive numbers ϕ 0 , c, g such that for all |ϕ| ≤ ϕ 0 and µ, ν ∈ Z 2 , z ∈ C \ T

E (|he µ , R(ϕ, z )e ν i| s ) ≤ ce

1`

|µ−ν|

Proof.

Use the n.p. estimate iteratively, n ≥ b |µ−ν| `

0

+2 c times and E

he µ , R ω # (ϕ, z)e ν i

s

≤ c

for the last step to conclude that for a c > 0 E (|he µ , R(ϕ, z )e ν i| s ) ≤ cq n .

Define 1 ` := | ` log q|

0

+2 .

J. Asch , Grenoble 2012

(41)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents, construction

X

ν∈ Z × Z

2M

U µν ψ ν = z ψ µ ∀µ ∈ Z × Z 2M

⇐⇒

ψ 2j+1,2k

ψ 2j+1,2k+1

= T eo (z , p)

ψ 2j ,2k

ψ 2j,2k+1

,

ψ

ψ

(42)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents, construction

T ∈

B ∈ M 2,2 ( C ); B JB = J, J :=

1 0 0 −1

| {z }

=:U(1,1)= Lorentzgroup T eo =

p 0 0 1

1 t

z −1 −r

−r z

1 0 0 q

,

T oe =

1 0 0 p

1 r

z −t t −z −1

q 0 0 1

.

J. Asch , Grenoble 2012

(43)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents, construction

D(q ) :=

 q 1

. ..

q 2M

 ,

M 1 (z) := 1 t

z −1 −r

−r z . ..

z −1 −r

−r z

 ,

M 2 (z) := 1 r

−z −1 t

z −t

t −z −1 . ..

z −t

.

(44)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents, construction

Define for a fixed z 6= 0 (Ω probability space of phases p := ω)

A z : Ω → U M (1, 1)

A z (p) := D (p l ) M 2 (z )D (p m ) M 1 (z)D (p r ) .

Then 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 (1, 1)

Φ z (n, p ) :=

A(Θ n−1 p) . . . A(p) n > 0

I n = 0

A −1n p) . . . A −1−1 p) n < 0 with Ω and Θ defined according to the two slice.

J. Asch , Grenoble 2012

(45)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents, construction

U (p)ψ = z ψ ⇐⇒ Ψ 2N = Φ z (N, f (p ))Ψ 0 . Oseledets for Φ = ⇒

for z 6= 0 there exists an invariant subset of full measure of p ∈ Ω such that the limits

n→±∞ lim (Φ z (n, p)Φ z (n, p)) 1/2|n| =: Ψ z (p) exist.

Denote by γ k (p, z) k ∈ {1, . . . , 2M } the eigenvalues of Ψ z (p) arranged in decreasing order. By ergodicity γ k (p, z ) = γ k (z) p a.e .

λ k (z) := log γ k (z ).

(46)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents, construction

Scaling = ⇒

z λ i = 0.

Lorentz symmetry = ⇒

λ 1 ≥ λ 2 ≥ . . . ≥ λ M ≥ 0 ≥ −λ M ≥ . . . ≥ −λ 1 .

J. Asch , Grenoble 2012

(47)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: Results

–use “Bougerol - Lacroix” theory to show that the Lyapunov spectrum is simple thus

ξ M < ∞;

–proof a Thouless formula and compute the mean exponent 1

M

M

X

i

λ i ;

–use the mean exponent to get a lower bound on the localization length;

–use “Sh’nol” theory to show that the spectrum of U M is

pure point with exponentially decaying eigenfunctions.

(48)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: Results

–use “Bougerol - Lacroix” theory to show that the Lyapunov spectrum is simple thus

ξ M < ∞;

–proof a Thouless formula and compute the mean exponent 1

M

M

X

i

λ i ;

–use the mean exponent to get a lower bound on the localization length;

–use “Sh’nol” theory to show that the spectrum of U M is pure point with exponentially decaying eigenfunctions.

J. Asch , Grenoble 2012

(49)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: Results

–use “Bougerol - Lacroix” theory to show that the Lyapunov spectrum is simple thus

ξ M < ∞;

–proof a Thouless formula and compute the mean exponent 1

M

M

X

i

λ i ;

–use the mean exponent to get a lower bound on the localization length;

–use “Sh’nol” theory to show that the spectrum of U M is

pure point with exponentially decaying eigenfunctions.

(50)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: Results

–use “Bougerol - Lacroix” theory to show that the Lyapunov spectrum is simple thus

ξ M < ∞;

–proof a Thouless formula and compute the mean exponent 1

M

M

X

i

λ i ;

–use the mean exponent to get a lower bound on the localization length;

–use “Sh’nol” theory to show that the spectrum of U M is pure point with exponentially decaying eigenfunctions.

J. Asch , Grenoble 2012

(51)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: simplicity

Theorem

For rt 6= 0, z ∈ T it holds λ 1 > λ 2 > . . . > λ M > 0.

Proof. Follow the strategy of Bougerol Lacroix.

G := the smallest subgroup of U (M , M) gen. by {A(p)}.

Show that

G ∼ = U(M, M) ∼ = Sp(M , C ),

in particular that G is connected. Then show irreducibility

and contraction properties by hand.

(52)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: Thouless formula

Theorem It holds:

1 M

M

X

i=1

λ i = 1 2 log 1

rt ≥ 1 2 log 2

Proof. Follow Craig Simon to proof Thouless formula for the unitaries a hand, use that the density of states is flat because of the uniformly distributed phases.

J. Asch , Grenoble 2012

(53)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: Estimate

We estimate the cocycle to derive an upper bound on the largest Lyapunov exponent, which is uniform in the quasienergy and the width M of the strip.

Proposition

1. rt 6= 0. For the generator of the cocycle it holds kA(p)k ≤ 1

rt (1 + r )(1 + t);

2. it follows: 2λ 1 ≤ log rt 1

+ log ((1 + r)(1 + t)).

3. There exists a c > 0 such that for M ∈ N it holds:

dist(r, {0, 1}) < e −cM = ⇒ ξ M = 1

≤ 2

1 .

(54)

Chalker Coddington Joachim Ascha

a

joint work with Alain Joye and Olivier Bourget

Introduction The model

Setup Quantum Walk Language Properties Scientific Interest Results

Localization far from transition Localization for cylinders

Credits Proofs

Localization Exponents

Exponents: Localization

Proof.

Follow the strategy known for discrete Schr¨ odinger operators:

polynomial boundedness of generalized eigenfunctions, positivity of the Lyapunov exponent and spectral averaging.

Use the work of Bourget, Hamza, Howland, Joye, Stolz.

J. Asch , Grenoble 2012

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