Semi-linear response of energy absorption

Doron Cohen, Ben-Gurion University

Collaborations:

Tsampikos Kottos (Wesleyan)

Holger Schanz (Gottingen)

Michael Wilkinson (UK)

Bernhard Mehlig (Goteborg)

Swarnali Bandopadhyay (BGU)

Yoav Etzioni (BGU)

Alex Stotland (BGU)

Rangga Budoyo (Wesleyan)

Tal Peer (BGU)

Nir Davidson (Weizmann)

Discussions:

Yuval Gefen (Weizmann)

Shmuel Fishman (Technion)

$ISF, $GIF, $DIP, $BSF

Diffusion and Energy absorption

Driven chaotic system with Hamiltonian H(X(t))

X = some control parameter

X = rate of the (noisy) driving

; diffusion in energy space:

D = Gdiffusion X2

; energy absorption:

E = Gabsorption X2

[Ott, Brown, Grebogi, Wilkinson, Jarzynski, D.C.]

There is a dissipation-diffusion relation.

In the canonical case E = D/T .

Below we use for G scaled units.

Models

H = {En} −X(t){Vnm}

with:

Stotland

Davidson

deformed boundaryor point scattereror Gaussian bump

with:

Wilkinson

Mehlig

Radiation

grainMetallic

with:

Stotland

Budoyo

Peer

Kottos

Flux

Some results

Cold atoms in vibrating traps:

10-12

10-10

10-8

10-6

10-4

<X>g

10-1010-810-610-4s-scatterer

σ / Lx = 0.005

σ / Lx = 0.075

σ / Lx = 0.2

10-6

10-4

10-2

u

10-10

10-8

10-6

10-4

G

Metallic rings driven by EMF:

10-3

10-2

10-1

100

101

w

10-8

10-4

100

104

G

SLRT (Meso)LRT (Kubo)Drude

Digression: size distribution

Given a matrix that looks random {Vnm},

Consider the size distribution of the elements.

Histogram of log(x) where x = |Vnm|2

10-12

10-9

10-6

10-3

x10

-4

10-3

10-2

10-1

100

u = 10-2

u = 10-3

u = 10-4

u = 10-8

u = 10-12

Algebraic average: 〈〈x〉〉a = 〈x〉

Harmonic average: 〈〈x〉〉h = [〈1/x〉]−1

Geometric average: 〈〈x〉〉g = exp[〈log x〉]

〈〈x〉〉h � 〈〈x〉〉g � 〈〈x〉〉a

Digression: random walk

wnm = probability to hop from m to n per step.

Var(n) =∑n

[wnmt] (n−m)2 ≡ 2Dt

For n.n. hopping with rate w we get D = w.

The diffusion equation:

∂pn

∂t= − ∂

∂nJn = D

∂2

∂n2pn

Fick’s law:

Jn = −D∂

∂npn

If we have a sample of length N then

J = −D

N× [pN − p0]

D/N = inverse resistance of the chain

If the w are not the same:

D

N=

[N∑

n=1

1

wn,n−1

]−1

Hence

D = 〈〈w〉〉h for n.n. hopping

Digression: Fermi Golden rule

The Hamiltonian in the standard representation:

H = {En} −X(t){Vnm

}

The transformed Hamiltonian:

H = {En} − X(t){

iVnm

En − Em

}

The FGR transition rate for ω ∼ 0 driving:

wnm = 2π∣∣∣∣ Vnm

En − Em

∣∣∣∣2 |X|2 δΓ(En−Em)

Note that the spectral content of the driving is

S(ω) = |X|2 δΓ(ω − (En−Em))

Semi Linear Response Theory (SLRT)

H = {En} −X(t){Vnm}

dpn

dt= −

∑m

wnm(pn − pm)

wnm = const× gnm × X2

+J

−J

EJ

En

gnm = 2%−3 |Vnm|2

(En−Em)2δΓ(En−Em)

〈〈|Vmn|2〉〉 ≡ inverse resistivity of the network

D = π% 〈〈|Vmn|2〉〉 × X2 ≡ G X2

Example: cold atoms in vibrating trap

The Hamiltonian in the n = (nx, ny) basis:

H = diag{En}+ u{Unm}+ f(t){Vnm}

The matrix elements for the wall displacement:

Vnm = −δny ,my ×π2

ML3x

nxmx

The Hamiltonian in the En basis:

H = diag{En}+ f(t){Vnm}

〈〈|Vnm|2〉〉a ≈Mv3

E

2πL2xLy

〈〈|Vnm|2〉〉g ≈4M2v4

E

L3xLyω2

c

exp[−M2v2

E(σ2x + σ2

y)]× u2

The SLRT result:

GSLRT = q exp[2√− ln q

]×GLRT

SLRT vs LRT

X = some control parameter

X = rate of the (noisy) driving

The definition of the “conductance”:

D = G X2

LRT implies

D =∫

G(ω)|Xω|2 dω =∫

G(ω)S(ω) dω

Within the framework of LRT

S(ω) 7→ λS(ω) =⇒ D 7→ λD

S(ω) 7→∑

i

Si(ω) =⇒ D 7→∑

i

Di

But there are circumstance such that e.g.

D =[∫

R(ω)[S(ω)

]−1dω]−1

Simplest illustration

ω

ω

2

1

ω ω1 2

E S( )ω

D � D1 + D2

Example: energy absorption by metallic grains

Linear response theory:

D = σ2h%∫ ∞0

dω ω2 R2(hω)S(ω)

Semi-linear response theory:

D =σ2

(%h)3

[∫ dx e−x2/2

(2π)N/2x2

]−1[∫ ∞0dω

P2(%hω)

S(ω)

]−1

Level spacing statistics:

P2(s) ≈ aβsβ exp(−cβs2) with β = 1, 2, 4

The LRT result of Gorkov and Eliashberg:

G = Cβσ2(h%)β+1 T β+2

Our SLRT result (large s statistics!):

G =σ2

2h%

1

(h%ω0)β−1exp

[− 1

π(h%T )2

]

The conductance of small mesoscopic

disordered rings

Doron Cohen, Ben-Gurion University

Collaborations:

Tsampikos Kottos (Wesleyan)

Holger Schanz (Gottingen)

Swarnali Bandopadhyay (BGU)

Yoav Etzioni (BGU)

Alex Stotland (BGU)

Rangga Budoyo (Wesleyan)

Tal Peer (BGU)

Discussions:

Michael Wilkinson (UK)

Bernhard Mehlig (Goteborg)

Yuval Gefen (Weizmann)

Shmuel Fishman (Technion)

$ISF, $GIF, $DIP, $BSF

The model

Non interacting “spinless” electrons in a ring.

H(r, p; Φ(t))

−Φ = electro motive force (RMS)

G Φ2 = rate of energy absorption

G = π(

e

L

)2

DOS2 〈〈|vmn|2〉〉

〈〈|vmn|2〉〉harmonic � 〈〈|vmn|2〉〉 � 〈〈|vmn|2〉〉algebraic

M mode ring of length L with disorder W

Flux

Numerical Results

Regimes: ballistic; diffusive; localizaion

10-3

10-2

10-1

100

101

w

10-8

10-4

100

104

G

SLRT (Meso)LRT (Kubo)Drude

10-4

10-2

100

102

L / l10

-6

10-4

10-2

100

GSL

RT

/GL

RT

results for the tight binding modelresults for untextured matricesresults for log-normal RMT ensambleresults for log-box RMT ensamble

Linear response theory (LRT)

H = {En} −e

LΦ(t){vnm}

G = π(

e

L

)2 ∑n,m

|vmn|2 δT (En−EF ) δΓ(Em−En)

G = π(

e

L

)2

DOS2 〈〈|vmn|2〉〉algebraic

applies if

EMF driven transitions � relaxation

otherwise

connected sequences of transitions are essential.

leading to

Semi Linear Response Theory (SLRT)

Semi Linear Response Theory (SLRT)

H = {En} −e

LΦ(t){vnm}

dpn

dt= −

∑m

wnm(pn − pm)

wnm = const× gnm × EMF2

+J

−J

EJ

En

gnm = 2%−3F

|vnm|2

(En−Em)2δΓ(En−Em)

〈〈|vmn|2〉〉 ≡ inverse resistivity of the network

G = π(

e

L

)2

DOS2 〈〈|vmn|2〉〉

Bandprofile, sparsity and texture

G = π(

e

L

)2

DOS2 〈〈|vmn|2〉〉

〈〈|vmn|2〉〉 ≡ inverse resistivity of the network

Bounds:

〈〈|vmn|2〉〉harmonic � 〈〈|vmn|2〉〉 � 〈〈|vmn|2〉〉algebraic

Analytical estimates:

• Mixed average scheme

• Variable range hopping scheme

Conductance versus disorder

1

GDrude

G

Diffusiveregimeregime

Ballisticregime

Anderson−Mott

∼(∆/Γ)

disorder strength (1/l)

Naive expectation (assuming Γ > ∆):

G =e2

2πhM `

L+O

(∆

Γ

)L = perimeter of the ring

` = mean free path ∝ W 2

`∞ = localization length ≈ M`

Ballistic regime: L � `

Diffusive regime: ` � L � `∞

Anderson regime: `∞ � L

Strategy of analysis

Given W ...

Characterization of the eigenstates:

• participation ratio (PR)

Characterization of vnm and RMT modeling

• bandwidth

• sparsity (p)

• texture

Approximation schemes for G

• Mixed average

• Variable range hopping estimate

Ergodicity of the eigenstates

• Weak disorder (ballistic rings):

Wavefunctions are localized in mode space.

• Strong disorder (Anderson localization):

Wavefunctions are localized in real space.

10−2

10−1

100

0

5

10

15

20

25

30

1−gT

PR

The PR of eigenstates of a ring

with a single scatterer. The

horizontal axis is the reflection

of the scatterer.

10-2

100

102

w10

0

102

PR

mode spaceposition spacemode-pos. space

0

0.1

0.2

0.3

0.4

0.5

0.6

p

p

The PR of eigenstates of a ring

with disorder. The horizontal

axis is W .

The sparsity (p) of the pertur-

bation matrix is related to the

ergodicity of the eigenstates.

{|vnm|2} as a random matrix {X}

The fraction of

”large” elements:

p ≡ F(〈X〉)

Sparsity: p � 1. log(X)

p

<X> = p X

log(<X>)XX0 1

1

X

XX0 X1

BiModal distribution

LogBox distribution

10-4

10-2

100

102

X/<X>

0

0.2

0.4

0.6

0.8

1

F(X)

W=0.05W=0.35W=7.50

Histograms of X:

Ballistic:

X ∼ LogNormal

Localization:

X ∼ LogBox

RMT based prediction for GSLRT/GLRT

RMT implied dependence on p

Log-normal distribution:

0 0.05 0.1 0.15 0.2 0.25 0.3p

10-6

10-4

10-2

100

GM

eso

/GK

ubo

results for the analytical VRH calculationresults for the resistor network calculationresults for the analytical geometric average calculationresults for the analytical harmonic average calculationresults for the mixed average calculation

Log-box distribution:

0 0.05 0.1 0.15 0.2 0.25 0.3p

10-6

10-4

10-2

100

GM

eso

/GK

ubo

results for the analytical VRH calculationresults for the resistor network calculationresults for the analytical geometric average calculationresults for the analytical harmonic average calculationresults for the mixed average calculation

The VRH estimate

G = πh(

e

L

)2 ∑n,m

|vmn|2 δT (En−EF ) δΓ(Em−En)

G =1

2

(e

L

)2

%F

∫Cqm(ω) δΓ(ω) dω

Cqm-LRT(ω) ≡ 2π%F 〈X〉Cqm-SLRT(ω) ≡ 2π%F X

where by definition:(

ω

∆

)Prob

(X > X

)∼ 1

For strong disorder we get:

X ≈ v2F exp

(−∆`

ω

)

G ∝∫

exp

(−∆`

|ω|

)exp

(−|ω|

ωc

)dω

LRT, SLRT and beyond

−Φ = electro motive force (RMS)

G Φ2 = rate of energy absorption

Semi linear response theory

[1] D. Cohen, T. Kottos and H. Schanz,

“Rate of energy absorption by a closed ballistic ring”,

(JPA 2006)

[2] S. Bandopadhyay, Y. Etzioni and D. Cohen,

The conductance of a multi-mode ballistic ring,

(EPL 2006)

[3] M. Wilkinson, B. Mehlig, D. Cohen,

The absorption of metallic grains,

(EPL 2006)

[4] D. Cohen,

“From the Kubo formula to variable range hopping”,

(PRB 2007)

[5] A. Stotland, R. Budoyo, T. Peer, T. Kottos and D. Cohen,

The conductance of disordered rings,

(JPA / FTC 2008)

Beyond (semi) linear response theory

[6] D. Cohen and T. Kottos,

“Non-perturbative response of Driven Chaotic Mesoscopic Systems”,

(PRL 2000)

[7] A. Stotland and D. Cohen,

”Diffractive energy spreading and its semiclassical limit”,

(JPA 2006)

[8] A. Silva and V.E. Kravtsov,

Beyond FGR,

(PRB 2007)

[9] D.M. Basko, M.A. Skvortsov and V.E. Kravtsov,

Dynamical localization,

(PRL 2003)

Conclusions

(*) Wigner (∼ 1955):

The perturbation is represented by

a random matrix whose elements are

taken from a Gaussian distribution.

Not always...

1. Ballistic ring =⇒ log-normal distribution.

2. Strong localization =⇒ log-box distribution.

3. Resistors network calculation to get GSLRT.

4. Generalization of the VRH estimate

5. SLRT is essential whenever the distribution

of matrix elements is wide (“sparsity”)

or if the matrix has “texture”.

6. Other applications of SLRT...