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  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 . 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  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
. 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
 of the local density of states according to which only
unperturbed states in a small energy interval ΓE con-
tribute to the final eigenstate.
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
nl| of one
Recent optical experiments with micrometer size
droplets initiated new theoretical investigations of weakly
deformed circular billiards . In this case the ray dy-
namics becomes chaotic leading to a strong directional-
ity of light emission . Here we will consider another
type of a weakly deformed circle , namely we chose
a random elastic boundary deformation which can be
represented by R(θ) = R0+ ∆R(θ) with ∆R(θ)/R0 =
efficients and M is large but finite.
formation seems to be very generic and may appear
in numerous different physical situations . 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
.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 . The transition be-
tween these two regimes is illustrated in Fig. 1. Here
we present the absolute values of eigenfunction ampli-
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) =
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-
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  is defined with respect to amplitudes al in the
wave function expansion ψ(r,θ) = B?
r∆R(θ)/R0. This quantum rough
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
are determined by det[1−S(Eα)] = 0. For V = 0, we re-
cover the Bohr-Sommerfeld quantization for eigenvalues
Enlof the ideal circle.
so that the eigenvalues Eα
FIG. 2. (a) Main peaks of eigenstate in Fig. 1b (squares
(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
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 . 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 , the equation for eigenstates takes
< l|V |˜l > a(α)
= 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
we used the relation between phase average of V2(θ) and
D. This expression is valid  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 .
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-
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
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
determine the local density of states by
max− l2) = 2¯ h/τb
ρW(E − Enl) =
δ(E − Eα)|C(α)
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
(E − Enl)2+ Γ2
ΓE= EbΓµ= Eb
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. 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).
Γ ρ (E-E )
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 ) 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
sin2[(E − Enl)/Eb]
(E − Enl)2
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=
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)
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 ,
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 . 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  which
should be sensitive to the above energy scales.
For E < Ec as usual we expect gaus-
However, such lo-
Σ ( )
5 1015 20
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.
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