PHYSICAL REVIEW E 72, 056206 共2005兲

Distribution of the spacing between two adjacent avoided crossings
1

Manabu Machida1,* and Keiji Saito2,†

Institute of Industrial Science, The University of Tokyo, Meguro-ku, Tokyo 153-8505, Japan
Department of Physics, Graduate School of Science, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
共Received 29 July 2005; published 11 November 2005兲

2

We consider the frequency at which avoided crossings appear in an energy-level structure when an external
field is applied to a quantum chaotic system. The distribution of the spacing in the parameter between two
adjacent avoided crossings is investigated. Using a random matrix model, we find that the distribution of these
spacings is well fitted by a power-law distribution for small spacings. The powers are 2 and 3 for the Gaussianorthogonal ensemble and Gaussian unitary ensemble, respectively. We also find that the distributions decay
exponentially for large spacings. The distributions in concrete quantum chaotic systems agree with those of therandom matrix model.
DOI: 10.1103/PhysRevE.72.056206

PACS number共s兲: 05.45.Mt

I. INTRODUCTION

The avoided crossing is a ubiquitous structure in energy
levels of quantum chaotic systems with external perturbation
关1兴. As the time-dependent external perturbation is applied tothe system, the quantum state changes nonadiabatically. Especially when the perturbation changes linearly and slowly
in time, nonadiabatic transitions can be described by theLandau-Zener transitions 关2兴 at avoided crossings. Using thismicroscopic mechanism, the nonadiabatic change of the total
quantum state might be understood. Energy diffusion phenomena have been intensively studied from this point ofview 关3–6兴. To understand macroscopic phenomena such asenergy diffusion, it is important to study universal aspects of
avoided crossings in quantum systems. Thus, using randommatrices 关7,8兴, various distributions concerning avoided
crossings have been studied 关9兴. For example, the distributions of the energy-level curvature 关10–14兴, the difference
between the slopes of the asymptotic lines 关15,16兴, and theminimum-energy gap 关16,17兴 are derived at avoided crossings, and their universalities are confirmed in concrete quantum systems 关11,12,16–18兴.
In this paper, we consider another distribution related tothe structure of avoided crossings. We study the distributionof the spacing in a parameter space of perturbation, which ismeasured as the distance between two adjacent avoided
crossings involved in two neighboring energy levels 共examples of this spacing are depicted in Fig. 1兲. We call this
distribution the avoided-crossing spacing distribution
共ACSD兲. Only a few qualitative studies related to the ACSD
have been published. Goldberg and Schweizer obtained the
ACSD of the hydrogen atom in a magnetic field and that of
the Africa billiard to estimate the statistical error of the gap
distribution 关19兴. Wilkinson and Austin discussed the densityof avoided crossings in quantum systems with two free parameters, although the distribution of the spacing between
avoided crossings was not discussed 关20兴. The main aim of*Electronic address: machida@iis.u-tokyo.ac.jp†

Electronic address: saitoh@ap.t.u-tokyo.ac.jp1539-3755/2005/72共5兲/056206共5兲/$23.00

this paper is to obtain the quantitative properties of the
ACSD.
We first consider the Hamiltonian taken from random matrices to extract the general features of the ACSD. We calculate the ACSD’s for the Gaussian orthogonal ensemble
共GOE兲 and the Gaussian unitary ensemble 共GUE兲 by a precise numerical calculation. We find that, for small spacings,
the ACSD for the GOE 共GUE兲 shows a power-law behavior
with power 2 共3兲. For large spacings, the ACSD’s decay exponentially. We then consider two concrete quantum
systems—i.e., the coupled rotators model and the AharonovBohm billiard—and find that the ACSD’s of these systems
agree with those predicted by simulations using random matrices.
This paper is constructed as follows. In Sec. II, we obtain
the ACSD’s of a random matrix model for the GOE and
GUE, and discuss differences between them, focusing especially on the powers of the distributions. In Secs. III and IV,
we present the ACSD’s of two concrete models and showthat they are identical to those of the corresponding randommatrix models. Finally, in Sec. V, we give a summary.
II. RANDOM MATRIX MODEL

Let us consider an N ⫻ N Hamiltonian matrix H共⌳兲 including a parameter ⌳ as an external field. In order to cap-

FIG. 1. The scaled energy spectrum for HRM共⌳兲 of the GOE as
a function of ␭. Three examples of avoided-crossing spacings arealso shown.
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PHYSICAL REVIEW E 72, 056206 共2005兲

M. MACHIDA AND K. SAITO

ture general features of the ACSD, we take H共⌳兲 from random matrix ensembles 关1,7兴. We prepare random matrices
and VRM of N = 1000. We consider the following ranHRM
0
dom matrix model:
RM
HRM共⌳兲 = HRM
sin ⌳.
0 cos ⌳ + V

共1兲

Note that this model keeps the variance of matrix elements
unchanged as the parameter varies 关16兴.
and VRM from the GOE, the HamilIf we take both HRM
0tonian HRM共⌳兲 represents quantum systems invariant under
an antiunitary transformation such as the time-reversal transformation. On the other hand, if we take both of them fromthe GUE, the Hamiltonian represents quantum systems thathave no symmetries for antiunitary transformations 关1,8,21兴.
2of diagoWe let ⌳ move from 0.5 to 1.5. The variance ␴RM
RM
RM
nal elements of H0 and V is 2.
First of all, we need to scale ⌳ and E j共⌳兲 关the jth eigenvalue of the instantaneous Hamiltonian H共⌳兲兴 in order toeliminate the dependence of the parameters on the external
field and energy 关22–26兴. Following Simons and Altshuler关22兴, we define the scaled parameter ␭ and scaled energy ⑀ j
asj−1

Ei+1共⌳兲 − Ei共⌳兲
,
⌬i共⌳兲
i=1

⑀ j共⌳兲 = ⑀1共⌳兲 + 兺k−1

␭kj = ␭0j + 兺 共⌳k⬘+1 − ⌳k⬘兲k⬘=1

冑冬冢冏 冏 冣 冭⳵⌳FIG. 2. Log-log plots of the cumulative distributions 共a兲
共1兲
共2兲
IRM
共S␭兲 and 共b兲 IRM
共S␭兲.

2

⳵⑀ j共⌳兲

,

⌳ ⬘
k

共2兲

where ⌬ j共⌳兲 is the mean level spacing at E j共⌳兲 and 具¯典
denotes a statistical average over a typical range of levels
and ⌳. We put ⑀1共⌳兲 = 0共=const兲, which is followed by具⳵⑀ j共⌳兲 / ⳵⌳典 = 0. ␭0j is set to ⌳0. Thus, we obtain the scaledenergy level flow ⑀ j共␭兲. We show the scaled energy spectrum
⑀ j共␭兲 for HRM共⌳兲 of the GOE in Fig. 1. Some avoided crossings look like crossings indeed, but this is due solely to thelinewidth. In Fig. 1, three distances 共the right and left arrows兲 are shown as examples of avoided-crossing spacing.
An avoided-crossing spacing is defined by the spacing
between two neighboring avoided crossings on the same energy level flow. Now, ␭kj is identified as the position of an
avoided crossing on the jth energy level flow if
⑀ j+1共␭k−1,j+1兲 − ⑀ j共␭k−1,j兲 ⬎ ⑀ j+1共␭k,j+1兲 − ⑀ j共␭k,j兲
and⑀ j+1共␭k+1,j+1兲 − ⑀ j共␭k+1,j兲 ⬎ ⑀ j+1共␭k,j+1兲 − ⑀ j共␭k,j兲 are satisfied.
By taking the differences in positions between two adjacent
avoided crossings, avoided-crossing spacings S␭ are obtained. We thereby obtain the ACSD P共S␭兲—i.e., the probability that the spacing in the parameter space of the adjacent
avoided crossings is S␭. Here P共S␭兲 satisfies the following
normalization conditions:

冕

⬁

0

P共S␭兲dS␭ = 1,

冕

⬁

0

S␭ P共S␭兲dS␭ = 1.

共3兲Hereafter, we distinguish between the GOE and GUE by
the superscript ␤. We have ␤ = 1 for the GOE and ␤ = 2 for共␤兲
the GUE. That is, PRM
共S␭兲 denotes the ACSD of model 共1兲
for the GOE and GUE. We take the parameter slice ⌬⌳⬅ ⌳k+1 − ⌳k to be 0.001, which is small enough. We find4.0⫻ 105 共3.5⫻ 105兲 avoided-crossing spacings in 50
samples of HRM共S␭兲 for the GOE 共GUE兲.
共␤兲
共S␭兲 of theWe also define the cumulative distribution IRM
共␤兲
共S␭兲ACSD of HRM共␭兲 for the GOE and GUE as IRM
共␤兲S␭ 共␤兲
= 兰0 PRM共S␭⬘ 兲dS␭⬘ . Figure 2 shows log-log plots of IRM共S␭兲.
The solid lines in the figures are obtained by fitting the datain the small-S␭ region by the least-squares method. Conse共1兲
quently, we find that IRM
共S␭兲 behaves as 1.1⫻ S␭3.0 and that共2兲
IRM共S␭兲 behaves as 1.7⫻ S␭4.0. Therefore, we may write
共␤兲
共S␭兲 ⬃ S␭␤+1
PRM

共S␭ Ⰶ 1兲.

共4兲共␤兲To study how the ACSD decays, we compare PRM
共S␭兲

with the following trial function:
共␤兲
共S␭兲 =
Ptriala␤S␭␤+2.
sinh b␤S␭

共5兲

Here, the coefficients a␤ and b␤ are determined from thenormalization conditions 共3兲: a1 = 共8 / ␲4兲b41, b1 = 372␨共5兲 / ␲4,
a2 = 关2 / 93␨共5兲兴b52, and b2 = ␲6 / 186␨共5兲. In Figs. 3共a兲 and共␤兲共␤兲共S␭兲 is well approximated by Ptrial
共S␭兲
3共b兲, we see that PRM
共␤兲
for large S␭. Since Eq. 共5兲 implies that PRM共S␭兲 decays ex-

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DISTRIBUTION OF THE SPACING BETWEEN TWO…

共␤兲共␤兲
FIG. 3. 共a兲 For ␤ = 1, PRM
共S␭兲 is compared with Ptrial
共S␭兲. 共b兲The same as 共a兲 except ␤ = 2.

ponentially for large S␭, we may conclude that no correlation
exists between avoided crossings for large S␭. Note that, for
共1兲
共2兲
small S␭, we have Ptrial
共S␭兲 ⯝ 5.1S␭2 and Ptrial
共S␭兲 ⯝ 64S␭3 ,which do not reproduce the numerical data.
We also obtained the ACSD of the following random matrix Hamiltonian:RM.
HRM0共⌳兲 = HRM
0 + ⌳V共6兲

We checked that the ACSD of this model is the same as thatof HRM共⌳兲 in Eq. 共1兲.
III. COUPLED ROTATORS MODEL

As an example of quantum systems for the GOE, we consider the coupled-rotators model 关27兴. We shall show that the
共1兲
ACSD of this model is well approximated by PRM
共S␭兲. The
Hamiltonian of the system is given by
HCRT共⌳兲 = Lx1Lx2 + ⌳共Lz1 + Lz2兲,

共7兲

where L1 and L2 are angular momenta with l1 and l2—i.e.,
the eigenvalues of L21 and L22 are l1共l1 + 1兲ប2 and l2共l2 + 1兲ប2,
respectively. Since ប is a dimensionless parameter in thepresent unit, we put ប = 0.158 75 so that the correspondingclassical system becomes strongly chaotic 关28兴. Noting that
HCRT共⌳兲 changes the z component of the total angular momentum 共Jz = Lz1 + Lz2兲 by 0, ±2ប 关29兴, the Hilbert space isdivided into two subspaces corresponding to even Jz / ប and
odd Jz / ប. Here, we take the subspace in which Jz / ប is even.
Furthermore, we divide this subspace into two more sub-

共1兲FIG. 4. 共a兲 Histogram of PCRT共S␭兲 with PRM
共S␭兲. 共b兲 ICRT共S␭兲
共1兲
共2兲
together with IRM共S␭兲 and IRM共S␭兲. The inset shows the magnified
figure with a log-log plot near the origin.

spaces corresponding to the states symmetric and antisymmetric under the exchange of Lz1 and Lz2. Here, we choose thesubspace corresponding to the symmetric states. In this subspace, we have 1024 levels with no degeneracies when we
set l1 = l2 = 31. We have confirmed that the energy-level spacing distribution of the coupled rotators model is well approximated by the eigenvalue distribution of random matrices taken from the GOE in the whole range of ⌳
苸 关0.5, 1.5兴.
After the scaling 共2兲, we find 9094 avoided-crossing spacings. Thus, we obtain the ACSD of the coupled-rotatorsmodel PCRT共S␭兲 and the cumulative distribution ICRT共S␭兲⫻关=兰S0␭ PCRT共S␭⬘ 兲dS␭⬘ 兴. In Fig. 4共a兲, we show PCRT共S␭兲 to共1兲
共S␭兲. We see that PCRT共S␭兲 is roughly equal
gether with PRM
共1兲
共1兲
共S␭兲 andto PRM共S␭兲. In Fig. 4共b兲, we show ICRT共S␭兲 with IRM
共2兲
IRM共S␭兲. The inset, which shows the magnified figure with a
共1兲
共S␭兲. Therefore,
log-log plot, shows that ICRT共S␭兲 follows IRM
共1兲
共S␭兲.
we classify PCRT共S␭兲 as PRM
IV. AHARONOV-BOHM BILLIARD

Now we consider the Aharonov-Bohm billiard as an example of a quantum system corresponding to the GUE
关30–32兴. We shall show that the ACSD of this system is well
共2兲
approximated by PRM
共S␭兲. In the billiard, a charged particle
with mass m and charge q moves inside the boundary ⳵D.
The domain D is threaded by a magnetic flux ⌽ at the origin.
The Schrödinger equation of this system is written as

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M. MACHIDA AND K. SAITO

FIG. 5. Boundaries of the Aharonov-Bohm
billiard with ⌳= 共a兲 0.15, 共b兲 0.2, and 共c兲 0.25.

1关− iប⵱uv − qA共r兲兴2␺共r兲 = E␺共r兲
2m
⵱uv ⫻ A共r兲 = n̂⌽␦共u兲␦共v兲,共8兲

where r = 共u , v兲, ␺共r兲 = 0 on ⳵D, and n̂ is the unit vector perpendicular to the billiard. The domain D is determined by thefollowing conformal transformation in the complex planew = u + iv:
w = z + ⌳z2 + ⌳ei␲/3z3 .

共9兲Here, z共=x + iy兲 is a complex value in the unit disk 共兩z兩 艋 1兲.
The parameter ⌳ determines the shape of the billiard. Figure
5 shows the boundaries of the billiard when ⌳ = 0.15, 0.2,
and 0.25.
Let us introduce two new parameters:

␣⬅

q⌽
,
2␲ប

k2 ⬅

2mE
.
ប

共10兲

The value of ␣ is restricted to the range 关0 , 21 兴 from thesymmetry of the system 关30兴. We put ␣ = 41 , with which thecorresponding classical system becomes most strongly chaotic 关30兴. Now, the problem with getting the energy E of thesystem in the Schrödinger equation 共8兲 results in the following eigenvalue problem by expanding the wave function with
the Bessel function 关30兴:c兺 cn⬘l⬘M n⬘l⬘nl共⌳兲 = knl2 .

⫻关=兰S0␭ PAB共S␭⬘ 兲dS␭⬘ 兴. In Fig. 6共a兲, we show PAB共S␭兲 together
共2兲
共S␭兲. We see that PAB共S␭兲 is roughly equal towith PRM
共2兲
共1兲
共S␭兲. In Fig. 6共b兲, we show IAB共S␭兲 with IRM
共S␭兲 and
PRM
共2兲
IRM共S␭兲. The inset, which shows the magnified figure with a
共2兲
共S␭兲. Therefore,
log-log plot, reveals that IAB共S␭兲 follows IRM
we conclude that the ACSD of the Aharonov-Bohm billiard
共2兲
共S␭兲.
is classified as PRM

V. SUMMARY

We have studied the ACSD in quantum chaotic systems.
共␤兲The distribution PRM
共S␭兲 of the random matrix model was
numerically obtained. We found power-law behavior forsmall spacings and exponential decay for large spacings. For
small spacings, we found that the powers for the GOE andGUE are 2 and 3, respectively. The exponential decay implies that the correlation between two avoided crossings vanishes for large spacings. Although these properties are established numerically, an analytical proof for them remains to
be completed. In Secs. III and IV, we have shown by accurate numerical calculations that the distributions predicted by

共11兲

n⬘l⬘

Here, 兵cnl其 are components of the eigenvectors andM n⬘l⬘nl共⌳兲 =NnlNn⬘l⬘anlan⬘l⬘

冕 冕
1

2␲

drr

0

0

d␾ei共l−l⬘兲␾J兩l−␣兩共anlr兲J兩l⬘−␣兩

⫻共an⬘l⬘r兲兩w⬘共z兲兩 ,

共12兲

2

where anl is the nth zero of the Bessel function J兩l−␣兩共x兲 and
Nnl =

1

⬘ ␣兩共anl兲兩 .
冑␲ 兩J兩l−−1

共13兲

Thus, we obtain the energy eigenvalues of the system by
diagonalizing the Hermitian matrix M共⌳兲.
We make a 2000⫻ 2000 matrix as M共⌳兲 and use the lowest 550 levels 关31兴. We have confirmed that the energy-levelspacing distribution of the system is well approximated by
the eigenvalue distribution of the GUE in the whole range of⌳ 苸 关0.15, 0.25兴.
After the scaling 共2兲, we find 1050 avoided-crossing spacings. Thus, we obtain the ACSD of the Aharonov-Bohm billiard PAB共S␭兲 and the cumulative distribution IAB共S␭兲

共2兲
FIG. 6. 共a兲 Histogram of PAB共S␭兲 with PRM
共S␭兲. 共b兲 IAB共S␭兲
共1兲
共2兲
together with IRM共S␭兲 and IRM共S␭兲. The inset shows the magnified
figure with a log-log plot near the origin.

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the random matrix model are indeed realized in concrete
quantum chaotic systems.
The ACSD may play an important role in the quantum
dynamics of finite fermion systems such as quantum billiards
关4兴. It is especially worthwhile to consider the time duration
until the system starts diffusing after the external perturbation is applied at zero temperature. As the perturbation isapplied, pairs of neighboring levels in the energy level flow
form avoided crossings. The quantum state at the Fermi levelchanges over time by making a nonadiabatic transition at the
avoided crossing. The distribution of the time duration until
the Fermi level encounters the first avoided crossing would
be related to the ACSD. It is interesting to experimentallyobserve the time duration and to understand it in the context
of the ACSD.

The authors thank Professor Seiji Miyashita for his continuous encouragement. One of the authors 共M.M.兲 also
thanks Professor Marko Robnik for his valuable comments
about the Aharonov-Bohm billiard. M.M. would like to acknowledge useful discussions with Professor Akira Shimizu
and the warm encouragement of Professor Naomichi Hatano.
The simulations were partially carried out by using the computational facilities of the Super Computer Center at the Institute for Solid State Physics, the University of Tokyo.

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ACKNOWLEDGMENTS

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