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arXiv:cond-mat/9702135v1 14 Feb 1997

Emergence of Quantum Ergodicity in Rough Billiards

Klaus M. Frahm and Dima L. Shepelyansky∗

Laboratoire de Physique Quantique, UMR C5626 du CNRS, Universit´ e Paul Sabatier, F-31062 Toulouse Cedex 4, France

(14 February 1997)

By analytical mapping of the eigenvalue problem in rough billiards on to a band random matrix

model a new regime of Wigner ergodicity is found. There the eigenstates are extended over the

whole energy surface but have a strongly peaked structure. The results of numerical simulations

and implications for level statistics are also discussed.

PACS numbers: 05.45.+b, 72.15.Rn, 03.65.Sq

In 1974, Shnirelman [1] proved a theorem according to

which quantum eigenstates in chaotic billiards become

ergodic for sufficiently high level numbers. Later it was

demonstrated [2,3] that in this regime the level spacing

statistics p(s) is well described by random matrix the-

ory [4]. However, one can ask the question how this

quantum ergodicity emerges with increasing level num-

ber N? This question becomes especially important in

the light of recent results [5,6] for diffusive billiards where

the time of classical ergodicity τDdue to diffusion on the

energy surface is much larger than the collision time with

the boundary τb. In such a situation quantum localiza-

tion on the energy surface may break classical ergodicity

eliminating the level repulsion in p(s). The investiga-

tion of rough billiards [6] showed that this change of p(s)

happens when the localization length ℓ in the angular

momentum l-space becomes smaller than the size of the

energy surface characterized by the maximal l = lmaxat

given energy (ℓ < lmax). For ℓ > lmaxthe eigenfunctions

are extended over the whole surface but as we will see

they are not necessarily ergodic (Fig. 1).

In this situation which we will furthermore call Wigner

ergodicity the eigenstates are composed of rare strong

peaks distributed on the whole energy surface.

a case is very different from the Shnirelman ergodicity

where the eigenstates are uniformly distributed.

usual scenario of ergodicity breaking was based on the

image of quantum localization along the energy surface

[7]. Here we show that the transition between localized

and Shnirelman ergodic states can pass trough an inter-

mediate phase of Wigner ergodicity. Our description and

understanding of this phase is based on the mapping of

the billiard problem with weakly rough (random) bound-

ary on to a superimposed band random matrix (SBRM).

This model is characterized by strongly fluctuating di-

agonal elements corresponding to a preferential basis of

the unperturbed problem. Recently such type of matri-

ces was studied in the context of the problem of particle

interaction in disordered systems [8–11]. There it was

found that the eigenstates can be extended over the whole

matrix size while having a very peaked structure. The

origin of this behavior is due to the Breit-Wigner form

[12] of the local density of states according to which only

Such

The

unperturbed states in a small energy interval ΓE con-

tribute to the final eigenstate.

-50

0

50

10

20

30

0

0.1

0.2

-50

0

50

10

20

30

0

0.1

0.2

n

-50

0

50

10

20

30

0

0.1

0.2

l

a

c

b

FIG. 1. Transition from localization to Shnirelman ergod-

icity on energy surface for level number N ≈ 2250, lmax ≈ 95

and M = 20; shown are the absolute amplitudes |C(α)

eigenstate: (a) localization for D(lr = 0) = 20; (b) Wigner

ergodicity for D = 80; (c) Shnirelman ergodicity for D = 1000

(see text).

nl| of one

Recent optical experiments with micrometer size

droplets initiated new theoretical investigations of weakly

deformed circular billiards [13]. In this case the ray dy-

namics becomes chaotic leading to a strong directional-

ity of light emission [14]. Here we will consider another

1

Page 2

type of a weakly deformed circle [6], namely we chose

a random elastic boundary deformation which can be

represented by R(θ) = R0+ ∆R(θ) with ∆R(θ)/R0 =

Re

?M

efficients and M is large but finite.

formation seems to be very generic and may appear

in numerous different physical situations [6]. In small

droplets such boundary perturbations may be created

by temperature induced surface waves. We will restrict

ourself to the case of weak surface roughness given by

κ(θ) = (dR/dθ)/R0 ≪ 1 and all γm being of the same

order of magnitude. Then we have for the angle average

˜ κ2= ?κ2(θ)?θ∼ M2(∆R/R0)2.

In such a billiard the dynamics is diffusive in or-

bital momentum due to collisions with the rough bound-

ary provided ˜ κ is above the chaos border κc ∼ M−5/2

[6].The diffusion constant is determined by the av-

erage change of orbital momentum per collision being

D =< (∆l)2>= 4(l2

sion rate for l close to lr. The quantum interference leads

to localization of this diffusion with the length ℓ = D for

M < ℓ < lmax while for ℓ > lmax the eigenstates are

extended over the energy surface [6]. The transition be-

tween these two regimes is illustrated in Fig. 1. Here

we present the absolute values of eigenfunction ampli-

tudes C(α)

nlin the eigenbasis |nl > of circular billiard as a

function of unperturbed radial and orbital quantum num-

bers n, l with α marking the eigenenergy Eα. For small

roughness ˜ κ (or D) the states are exponentially local-

ized (Fig. 1a) while for large ˜ κ they are homogeneously

distributed (Fig. 1c) on the energy surface. The case

of Fig. 1b corresponds to an unusual regime of Wigner

ergodicity where the eigenstate is extended over the sur-

face but is composed of rare strong peaks. The positions

of these peaks on the energy surface of the circular bil-

liard E = H(n,l) are shown in Fig. 2a. The equation of

the surface, projected on the action plane (n,l), can be

found from the Bohr-Sommerfeld quantization µl(E) =

?l2

where l2

the wave number. A part of the surface is shown in more

detail in Fig. 2b. Here it is clearly seen that the peaks

are large for those integer n, l which are close to the

line H(n,l) = Eα. Our understanding of the fact that

not all integer values of the (n,l)-lattice near to this line

are populated is based on the concept of Breit-Wigner

structure of eigenstates described below.

According to Refs. [6,15] the internal scattering at the

rough boundary can be described by the S-matrix

m=2γmeimθwhere γm are random complex co-

This type of de-

max−l2

r) ˜ κ2. This D is the local diffu-

max− l2− larctan(l−1?l2

max− l2) + π/4 = π(n + 1)

0E/¯ h2= k2R2

max= 4N = 2mR2

0with k being

Sl˜l(E) = eiµl(E)< l|eiV (θ)|˜l > eiµ˜l(E), (1)

with µl(E) being the scattering phases of the circle and

V (θ) = 2?l2

map [6] is defined with respect to amplitudes al in the

wave function expansion ψ(r,θ) = B?

max− l2

r∆R(θ)/R0. This quantum rough

lalJ|l|(kr)eilθ

with Bessel functions Jl and B being a normalization

constant. The S-matrix gives a local unitary description

for l close to a resonant value lr. The eigenvalue equation

reads?

are determined by det[1−S(Eα)] = 0. For V = 0, we re-

cover the Bohr-Sommerfeld quantization for eigenvalues

Enlof the ideal circle.

˜lSl,˜l(Eα)a(α)

˜l

= a(α)

l

so that the eigenvalues Eα

6

7

8

9

505560

b

0

10

20

30

-50050

a

l

n

n

l

FIG. 2. (a) Main peaks of eigenstate in Fig. 1b (squares

for |C(α)

(see text); (b) rescaled part of (a): diamonds show the integer

(n,l)-lattice, the errorbar size is 2|C(α)

nl| ≥ 0.1) shown on the energy surface H(n,l) = Eα

nl|.

The semiclassical regime of ray dynamics corresponds

to the limit V ≫ 1 where the θ-integral can be evalu-

ated in a saddle point approximation giving the classi-

cal limit of quantum rough map [6]. Here we are inter-

ested in a different regime where V < 1 corresponding

to D < M2. There, by the mapping on an effective solid

state Hamiltonian Heff introduced by Fishman, Grem-

pel, and Prange [16], the equation for eigenstates takes

the form

+1

2

˜l

tan[µl(Eα)]a(α)

l

?

< l|V |˜l > a(α)

˜l

= 0 . (2)

In this way the eigenvalue equation is reduced to a solid

state problem with 2M coupled sites. The Heff-matrix

is of SBRM type with strongly fluctuating diagonal ele-

ments produced by scattering phases µl. The investiga-

tions of such matrices [9–11] showed that the local density

of states has the Breit-Wigner width given by the Fermi

golden rule Γµ = 2πρµ < (V (θ)/2)2>≈ 3D/(2M2)

where ρµ= 1/π is the density of diagonal elements and

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Page 3

we used the relation between phase average of V2(θ) and

D. This expression is valid [17] when Γµ exceeds the

mean level spacing (∼ 1/M) in the band width M. In

the opposite limit ΓµM < 1 the eigenstates are given

by standard perturbation theory. Together with the con-

dition V < 1 we find that the Breit-Wigner regime ex-

ists for M < D < M2near zero energy of Heff. In

this regime the localization length is ℓ = D [6,8–11,15].

However, the Breit-Wigner structure remains in both lo-

calized (ℓ < lmax) and delocalized (ℓ > lmax) cases if

M < D < M2. Therefore for lmax< D < M2the states

are extended but only l with |tan(µl)| < Γµ < 1 are

mixed leading to a peaked structure of eigenstates [18].

The fraction of peaks in max(ℓ,lmax) is Γµ.

The above properties of scattering amplitudes a(α)

low to understand the behavior of eigenfunction coeffi-

cients C(α)

nl=< ψα|nl >. For this one has to compute the

expansion Jl(kαr)eilθin terms of |nl >. Since ∆R ≪ R0

the angular and radial integrals factorize and can be eval-

uated using the radial eigenvalue equation and the semi-

classical expression for Jl(kr). As a result we obtain

l

al-

C(α)

nl≈˜Ba(α)

l

(l2

max− l2)1/4sin∆µ

∆µ

, (3)

with ∆µ = µl(Eα) − µl(Enl) ≈ (Eα− Enl)/Eb and

Eb = dE/dµl(E) = ¯ h2l2

being the energy scale related to the ballistic collision

time τb,˜B is a normalization constant. The amplitudes

C(α)

nl

determine the local density of states by

max/(mR2

0

?l2

max− l2) = 2¯ h/τb

ρW(E − Enl) =

??

α

δ(E − Eα)|C(α)

nl|2?

(4)

The averaging is performed with respect to different

roughness realizations and/or over a sufficiently large en-

ergy interval. Due to the Breit-Wigner distribution for

tan(µl) in (2) we obtain

ρBW(E − Enl) =1

π

ΓE/2

(E − Enl)2+ Γ2

E/4

(5)

with

ΓE= EbΓµ= Eb

N

NW

?

1 −

l2

l2

max

?

, NW=

M2

24˜ κ2. (6)

The equations (5), (6) are valid for ∆E < EB(∆µ < 1)

and Γµ < 1 or N < NW. Here NW is the border of

Breit-Wigner regime in level number N. We remind that

the eigenstates are localized for N < Ne= 1/(64˜ κ4) cor-

responding to ℓ < lmax[6]. As a result the Breit-Wigner

structure can exist both in the localized and delocalized

cases. An example of Breit-Wigner distribution is shown

in Fig. 3. Our numerical data confirm the theoretical

expression (6) for variation of Γµby more than one order

of magnitude (inset).

µ

Γ /

E

b

E

(E-E )/Γ

nl

Γ

E

0.1

1

0.11

Γ ρ (E-E )

E

nl

W

1

0.01

0.0001

-1001020

FIG. 3. Breit-Wigner distribution for eigenstates of rough

billiard (diamonds) with the parameters of Fig. 1b (5 eigen-

states for each of 10 roughness realizations are used). Full

curve gives the distribution (5) with the theoretical ΓE value

(6), ΓE/Eb = Γµ = 0.3.Insert shows the variation of

ΓE/Eb (diamonds) as a function of Γµ for parameter range

10 ≤ M ≤ 40 and M ≤ D ≤ M2(Eb and Γµ are taken at

l = 0). The theory (6) is shown by straight line.

For N > NW the kick amplitude V in (1) is larger

than 1 and the mapping on to equation (2) is not valid.

In this case the scattering phases (eigenphases of S) are

homogeneously distributed in the interval (0,2π). If in

addition N > Ne then as in the case of kicked rotator

(see B. V. Chirikov in [3]) the amplitudes alare homoge-

neous in l-space with |al|2≈ 1/(2lmax). Using Eq. (3),

we obtain that the local density of states is given by

ρW(E − Enl) =Eb

π

sin2[(E − Enl)/Eb]

(E − Enl)2

.(7)

This density is normalized to one and as a result

the probability |C(α)

the energy surface shown in Fig.

regime of Shnirelman ergodicity which emerges for N >

max(NW, Ne). For fixed roughness ˜ κ > κEW =√6/4M

we have NW > Neand the transition to Shnirelman er-

godicity with the increasing level number N crosses the

region of Wigner ergodicity for which an eigenfunction is

ergodic only inside the Breit-Wigner width ΓE< Eb. In

the opposite case κc< ˜ κ < κEWthe Shnirelman ergodic-

ity emerges directly from the localized phase (the Breit-

Wigner regime exists only in the localized phase). The

averaging of equation (7) over different l-values gives the

distribution (5) with ΓE ∼ Eb. This explains why also

for the case Γµ< 1 in Fig. 3 the distribution (5) remains

valid even for ∆E > Eb(note that πEb/ΓE≈ 10).

The above analysis shows that in the regime of Wigner

ergodicity there are four relevant energy scales: level

spacing ∆, Thouless energy for diffusion in l-space Ec=

¯ hD/(l2

and bouncing energy Eb = 2¯ h/τb which are ordered as

∆ < Ec < ΓE < Eb.These scales should appear

in the level statistics namely for the number variance

nl|2is ergodically distributed along

2a.This is the

maxτb), the Breit-Wigner width ΓE= 3¯ hD/(M2τb)

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Σ2(E) [3,4].

sian orthogonal ensemble (GOE) statistics to be valid

while in the interval Ec < E < ΓE/2 the behavior

should be modified, due to the diffusive dynamics [19],

being Σ2(E) ∼ (E/Ec)1/2. The first investigations of the

regime with ΓE/2 < E < Eb/2 for SBRM were done only

recently [20]. They showed that level rigidity is strongly

suppressed with a nearly linear energy behavior in Σ2(E)

due to disappearance of correlations between levels with

energy differences larger than ΓE.

cal characteristics as p(s) are still described by GOE if

Ec≫ ∆. Our numerical data qualitatively confirm these

expectations (see Fig. 4), however, quantitative numeri-

cal and analytical verifications are still required. In Fig.

4 the above energy scales are not separated by strong in-

equalities but parametrically it is possible to have them.

In this unusual regime it would be interesting to study

other physical properties. We mention for example the

frequency dependence of dielectrical response [21] which

should be sensitive to the above energy scales.

For E < Ec as usual we expect gaus-

However, such lo-

c

∆

/2

ΓE

Eb/2

Σ ( )

2

E

GOE

E

0

2

4

6

5 1015 20

Poisson

rough billiard

FIG. 4. Dependence of number variance Σ2(E) on energy

for rough billiard compared to Poisson and GOE for M = 50,

D = 800, lmax ≈ 95; 6 roughness realizations in the interval

2150 < N < 2350 are used for the average. The energy scales

are also shown in units of level spacing ∆.

In conclusion, we studied the parameter dependence

of the quantum energy surface width in rough billiards.

In the limiting case of Shnirelman ergodicity with high

level numbers, this width is determined by the typical

frequency of collisions with the boundary (ΓE ∼ Eb).

This means that all integer points on the (n,l)-lattice of

quantum numbers with a distance ∆l = ∆n ≈ 1 from

the energy line Eα= H(n,l) are occupied by one eigen-

function ψα(N > NW and N > Ne). We have found a

new regime of Wigner ergodicity where ΓE≪ Ebso that

only points with ∆l = ∆n ≤ Γµ≪ 1 contribute to ψα.

As a result a lot of holes appear in the energy surface

and ψαhas a strongly peaked structure on (n,l)-lattice.

However, at the same time all states in the Breit-Wigner

width ΓE are populated so that the eigenfunction is er-

godic inside this energy band. It would be interesting

to understand if the arithmetical properties of the lattice

will play an important role.

∗

Also Budker Institute of Nuclear Physics, 630090 Novosi-

birsk, Russia

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[18] Note that for the case studied in [6] with γm ∼ 1/m this

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