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arXiv:1005.1540v1 [cond-mat.dis-nn] 10 May 2010

Between a metal and an insulator: the critical state of the Anderson transition

Gabriel Lemarié,1, ∗Hans Lignier,2, †Dominique Delande,1Pascal Szriftgiser,2and Jean-Claude Garreau2

1Laboratoire Kastler Brossel, UPMC-Paris 6, ENS, CNRS; 4 Place Jussieu, F-75005 Paris, France

2Laboratoire de Physique des Lasers, Atomes et Molécules, Université Lille 1 Sciences et Technologies,

UMR CNRS 8523; F-59655 Villeneuve d’Ascq Cedex, France‡

(Dated: May 11, 2010)

Using a three-frequency one-dimensional kicked rotor experimentally realized with a cold atomic

gas, we study the transport properties at the critical point of the metal-insulator Anderson transition.

We accurately measure the time-evolution of an initially localized wavepacket and show that it

displays at the critical point a scaling invariance characteristic of this second-order phase transition.

The shape of the momentum distribution at the critical point is found to be in excellent agreement

with the analytical form deduced from self-consistent theory of localization.

PACS numbers: 03.75.-b, 05.45.Mt, 72.15.Rn

Different phase transitions observed in various fields

of physics often share similar characteristics [1]. Of spe-

cial interest is the behavior of the system at the critical

point (for example scale invariance) and in its immedi-

ate vicinity (e.g. divergence of a characteristic length

scale). The advent of cold atom physics has offered new

possibilities of direct experimental observation of such

characteristics of quantum phase transitions. In this let-

ter, we show that the Anderson metal-insulator transi-

tion (which has only recently been observed with atomic

matter waves [2]) obeys scale invariance at the threshold,

defining a new state of matter between a metal and an

insulator.

The Anderson transition takes place in 3-dimensional

(3D) disordered non-interacting systems in the meso-

scopic regime (where the transport is coherent). It in-

volves a metallic phase at low disorder associated with an

essentially diffusive transport, and an insulating phase at

large disorder where transport over long distance is inhib-

ited by interference effects: this is the so-called Ander-

son localization phenomenon [3]. The Anderson transi-

tion is a second-order (continuous) phase transition: On

the insulating side, the localization characteristic length

ℓ diverges algebraically, ℓ ∝ |K − Kc|−νwhen K, the

control parameter, approaches the threshold Kc of the

transition. On the metallic side, similarly, the diffusion

constant vanishes algebraically D ∝ |K −Kc|s. The crit-

ical exponents s and ν are equal in 3D, and universal

(they do not depend on the microscopic details of the

system) [4]. Only recently have these theoretical pre-

dictions been confirmed experimentally and the value of

ν = s unambiguously determined [2, 5, 6]: ν = 1.4 ± 0.3

is found perfectly compatible with ν = 1.57 ± 0.02 ob-

tained from numerical simulations of the 3D Anderson

model [7, 8].

The state of a disordered system is, in this context,

characterized by its transport properties. One can con-

sider the behavior at large distances and long times of

the (disorder) averaged intensity Green function (AIGF)

which determines the probability P(r,r′;t) for a parti-

cle to go from r to r′in time t [9]. In the insulating

phase, the AIGF is a stationary function exponentially

localized:

P(r,r′;t) ∼ exp(−|r − r′|/2ℓ), [localized]

(1)

while in the metallic regime, it is a Gaussian expanding

diffusively:

P(r,r′;t) ∼ exp[−(r − r′)2/2Dt]. [diffusive] (2)

These two behaviors are however long time asymptotics.

Indeed, a localized AIGF is observed only for times

t ≫ tℓ, where tℓis the localization time (the time-scale

associated to localization). At the transition, tℓ∼ ℓ3di-

verges and the system becomes scale invariant. What is

the AIGF behavior at the critical point? In the following,

we show that it scales as:

P(r,r′;t) ∼ exp

?

−α|r − r′|3/2/t1/2?

,[critical] (3)

where α is a known (measurable) quantity. This defines

a new state, since the shape does not change with time.

Such a state of matter, intermediate between an insula-

tor and a metal at all scales, has never been directly ob-

served experimentally, although interesting results have

recently been published for ultrasound waves in the lo-

calized regime [10]. The purpose of this letter is to report

the first experimental characterization of such a critical

state of the Anderson transition.

In cold atomic gases, it is possible to prepare the sys-

tem in a localized state and follow its evolution over

time; this constitutes an experimental measurement of

the (A)IGF [2, 11, 12], which is impossible to achieve in

state of art solid state physics. Observing the 3D An-

derson transition in configuration space with cold atoms

requires a disordered potential with a correlation length

comparable to the de Broglie wavelength [13] which has

not yet been achieved. We have recently shown [2, 5] that

it is nevertheless possible to observe Anderson localiza-

tion and the Anderson transition in momentum space by

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-200

p

20

0

0.05

0.1

0.15

t=20

t=40

t=80

t=160

-200

p

20

0

0.05

0.1

t=20

t=40

t=80

t=120

t=160

-1000

p

100

0

0.005

0.01

0.015

0.02

0.025

t=20

t=80

t=40

t=160

-20 0

p

20

0

0.05

0.1

0.15

t=20

t=40

t=80

t=160

-10

-5

0

5

10

0

0.1

0.2

0.3

t=20

t=40

t=80

t=120

t=160

-20-100 1020

0

0.05

0.1

t=20

t=40

t=80

t=160

Localized CriticalDiffusive

p/t1/3

p/t1/2

Π(p,t)

Π(p,t)

Figure 1: First row: Measured AIGF of the quasiperiodic atomic kicked rotor at different times t in the (left to right) localized

(K,ε) = (6,0.1), critical (K = Kc,ε) = (8,0.38) and diffusive (K,ε) = (11,0.8) regimes. Second row: Appropriate rescalings

of the momentum by t0(localized), t1/3(critical) or t1/2(diffusive), bring the curves at different times into coincidence (the

vertical scales are also rescaled in order to preserve normalization). The shapes are different in the three regimes: exponential

localization, Eq. (1), in the localized regime, Gaussian shape, Eq. (2), in the diffusive regime, and the new “Airy shape”, Eq. (6),

at the critical point. Parameters are: ¯ k = 2.89, ω2 = 2π√7, ω3 = 2π√17. Time is measured in number of kicks and momentum

in units of 2?kL.

using a different system, the atomic kicked rotor (de-

scribed below), where the chaotic nature of the classical

motion replaces the disordered potential.

Our atom-optics system (see [5] for a detailed descrip-

tion) consists in a cloud of laser-cooled cesium atoms

(FWHM of the momentum distribution of 8?kL) inter-

acting with a pulsed (period T1 = 27.778 µs), far de-

tuned (∆ = −11.3 GHz) standing wave (wavenumber

kL = 7.4 × 106m−1and one way intensity I0 = 150

mW). The amplitude of the kicks is modulated with two

frequencies ω2and ω3. The Hamiltonian reads:

H =p2

2+ K cosx[1 + εcos(ω2t)cos(ω3t)]

N−1

?

n=0

δ(t − n) ,

(4)

where time is measured in units of T1, space in units

of (2kL)−1, momentum in units of 2?kL/¯ k, with ¯ k =

4?k2

LT1/M = 2.89 (M is the atom mass) playing the role

of an effective Planck constant ([x,p] = i¯ k) and K is the

average kick amplitude. The kicks are short enough (du-

ration τ = 0.8 µs) as compared to the atom dynamics

so that they can be considered as Dirac delta functions.

Decoherence processes, analyzed in detail in [5] are neg-

ligible for the typical duration of the experiment t ≃ 160

kicks.

If ω2, ω3, π and ¯ k are incommensurate, this 1D

quasiperiodic kicked rotor has been shown to be equiva-

lent to a 3D disordered anisotropic system [6, 14, 15] and

to display an Anderson metal-insulator transition, as ev-

idenced by the fact that it belongs to the universality

class of the 3D Anderson model [6, 7], i.e. has the same

critical exponent ν. Here, the localization manifests it-

self in momentum space instead of configuration space.

We thus expect the AIGF to take simpler forms in mo-

mentum space, with expressions similar to Eqs. (1)-(3)

(simply replacing position r by momentum p). In order

to avoid confusion, we will use the notation Π(p,p′;t) for

the AIGF in momentum space. Thus, an initial momen-

tum distribution W(p,t = 0) will on average evolve at

time t to:

W(p,t) =

ˆ

Π(p,p′;t) W(p′,0) dp′.

(5)

Experimentally, we are able to measure the momen-

tum distribution at the end of a pulse sequence (up

to 160 kicks), using Raman stimulated transitions (see

[16, 17] for details).The initial state W(p,t = 0) is

a thermal momentum distribution whose width is much

smaller than the width of the final distribution, and can

thus be approximated by a δ-function δ(p) in Eq. (5). As

a consequence, the final momentum distribution W(p,t)

faithfully measures the intensity Green function Π(p,t) ≡

Π(p,0;t).

Figure 1 shows the experimentally measured AIGF

Π(p,t) at various times in the localized, critical and dif-

fusive regimes. The three regimes obey different scaling

laws. In the localized regime (left column), the momen-

tum distribution is localized – i.e. it is time-independent

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Figure 2: Numerically simulated time-evolution of an ini-

tially localized momentum distribution (log scale), for the

quasiperiodic kicked rotor at the critical point of the Ander-

son transition. The spreading follows an anomalous diffusion,

with ?p2(t)? ∝ t2/3, and the shape is preserved (i.e. scale in-

variant), being neither exponentially (as it is in the localized

regime), nor Gaussian (as it is in the diffusive regime). The

analytic prediction, Eq. (6) is shown as the red thick curve

at 1 million kicks. The agreement is excellent, without any

adjustable parameter. Parameters are those of fig. 1. Time is

measured in millions of kicks and momentum in units of two

recoil momenta 2?kL.

– at long times and thus scales as t0. In the diffusive

regime, the average kinetic energy ?p2(t)? increases lin-

early with time, so that the typical momentum scales as

t1/2; this is manifest in the broadening of the distribu-

tion with time seen in the right column. At the critical

point of the Anderson transition, we observe [2, 5], as

predicted by the one parameter scaling theory [18, 19],

an anomalous diffusion ?p2?(t) ∼ t2/3. This implies that

the typical momentum scales as t1/3leading to a slower

broadening of the distribution (middle column). If the

raw experimental data are rescaled according to these

laws (lower row in Fig. 1), i.e. plotted vs. pt0, pt−1/3and

pt−1/2in the localized, critical and diffusive regimes re-

spectively, curves taken at various times coincide, which

constitutes an experimental proof of the validity of the

scaling laws [24]. The shapes of the distributions are dif-

ferent in the various regimes: exponential shape in the

localized regime, Gaussian shape in the diffusive regime.

The intermediate shape at the critical point is discussed

below.

Figure 1 is a clear manifestation of the scale invari-

ance at the critical point. The anomalous diffusion is

not a transient behavior and the AIGF keeps the same

shape at the critical point. However, slightly off the crit-

ical point, the AIGF tends gradually to either a localized

or diffusive behavior, following the anomalous diffusion

only for short times. To confirm this observation of scale

0

0.1

0.2

0.3

0

0.02

-0.02

0.02

0

0.02

-10

0

10

-0.02

0

(a)

(b)

(c)

(d)

p/t1/3

Π(p,t) t1/3

Figure 3: (color online) (a) Experimental data for the rescaled

critical AIGF (see fig. 1) averaged over time (black circles

with error bars) and a fit given by Eq. (6), with ρ the only

fitting parameter. The agreement is clearly excellent. The

fitted value ρ is found compatible with ρ = Γ(2/3)Λc/3. The

residual does not significantly differ from zero [panel (b)]. Fits

by an exponentially localized (c) or a Gaussian (d) distribu-

tion show significant deviations.

invariance over a time scale larger than 160 kicks, we per-

formed numerical simulations of the critical dynamics up

to t = 106kicks. The result is shown in Fig. 2. The ad-

vantage of numerical simulations is that it is possible to

explore the tails of the momentum distributions (hidden

by noise in a real experiment). The anomalous diffusion

– with the characteristic sub-diffusive t1/3scaling – is

clearly visible. Obviously, the distribution is neither ex-

ponentially shaped (which would result in straight lines

in the logarithmic plot), nor has a Gaussian shape (a

parabola in the logarithmic plot).

The form of the critical AIGF can be deduced from

the self-consistent theory of localization [9]. This mean-

field theory describes quantum transport in disordered

systems at large distances and for long times [20]. It has

been shown to be relevant for the 1D periodically kicked

rotor [21] and correctly predicts a metal-insulator tran-

sition in three dimensions and the anomalous diffusion

at the threshold: D(ω) ∼ (−iω)1/3with ω the frequency

conjugated to time [22]. (the 1/3 exponent is the counter-

part in the frequency domain of the anomalous diffusion

?p2?(t) ∼ t2/3in the time domain). Using this critical be-

havior, we can compute the AIGF for the quasiperiodic

kicked rotor [23]. The details of the calculation will be

published elsewhere; we obtain:

Π(p,t) =3

2

?

3ρ3/2t

?−1/3

Ai

??

3ρ3/2t

?−1/3

|p|

?

,

(6)

where ρ is a parameter directly related to the criti-

cal quantity Λc = lim

Γ(2/3)Λc/3, where Γ is the Gamma function and Ai(x) is

t→∞?p2?/t2/3(see [2, 5]) via ρ =

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the Airy function. This expression is used for the plot in

Fig. 2. The asymptotic form Eq. (3) comes simply from

the limiting behavior of the Airy function for large x and

is found perfectly intermediate between the exponential

(localized) and the Gaussian (diffusive) shapes.

The analytic prediction, Eq. (6), matches very well

the shape obtained from numerical simulations of the

quasiperiodically kicked rotor shown in Fig. 2. The only

noticeable difference is near p = 0, where the result of the

numerical simulation is slightly larger than the analytic

prediction. Note that this is only observed at very long

times, beyond 1000 kicks; this phenomenon is currently

under study. On the time scale of the experiment (160

kicks), this effect is invisible.

Figure 3 shows the comparison between the experimen-

tally measured critical AIGF and the analytic prediction,

Eq. (6). The only fitting parameter is the global scale

ρ (found in excellent agreement with ρ = Γ(2/3)Λc/3).

Although it is visually not obvious to distinguish the ob-

served shape from either an exponential shape or a Gaus-

sian shape, a careful fitting procedure gives a clear cut

result. The residual between the observed distribution

and the analytic prediction (6), shown in panel (b), is

consistently zero (within the error bars), while a fit with

an exponential shape, panel (c), of a Gaussian shape,

panel (d), displays significant deviations. This is fully

confirmed by a quantitative check of the quality of the

fit. The fit by the Airy function gives a χ2per degree

of freedom equal to 1.09 – i.e. perfectly acceptable –

while the exponential fit gives 4.5 per degree of freedom

and the Gaussian fit 8.8, two unacceptably large values.

This clearly shows that the self-consistent theory of lo-

calization accounts for the critical AIGF and its scale

invariance.

In conclusion, we have studied experimentally the

transport at the threshold of the Anderson transition.

It obeys scale invariance, one fundamental property of

this second-order phase transition, and this defines a new

state, between an insulator and a metal.

form can be deduced from the self-consistent theory of

localization. Work is in progress to allow experimental

observations at longer times, which should allow us to

characterize the small deviations observed numerically,

whose origin could be multifractality.

The authors acknowledge Narei Martínez for her help

with the experiment. This work was partially financed

by Ministry of Higher Education and Research, Nord-

Pas de Calais Regional Council and FEDER through the

“Contrat de Projets Etat Region (CPER) 2007-2013” and

was granted access to the HPC resources of IDRIS un-

der the allocation 2009-96089 made by GENCI (Grand

Equipement National de Calcul Intensif).

Its analytic

∗Present address: Service de Physique de l’Etat Condensé

(CNRS URA 2464), IRAMIS/SPEC, CEA Saclay, F-

91191 Gif-sur-Yvette, France

†Present address: Laboratoire Aimé Cotton, Université

Paris-Sud, Bat. 505, Campus d’Orsay, F-91405 Orsay

Cedex, France

‡URL: www.phlam.univ-lille1.fr/atfr/cq

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[24] Extracting the precise location of the critical point is

not straightforward, especially because the experimental

data are limited to 160 kicks. As explained in [5], finite-

time-scaling makes it possible to determine it with a rea-

sonably small uncertainty, of the order of 0.3 on the value

of K. The experimentally observed value agrees very well

with the one extracted from numerical simulations with-

out any adjustable parameter.