Download --Fundamental Problems and Application to Material Science-

Progress of Theoretical Physics Supplement No. 98, 1989
383
Quantum Chaos
--Fundamental Problems and Application to Material ScienceKatsuhiro NAKAMURA*>
The fames Franck Institute, The University of Chicago
Chicago, Illinois 60637, U.S.A.
(Received October 28, 1988)
We investigate quantum mechanics of nonintegrable and chaotic systems. Two realistic
examples of quantum chaos in magnetic phenomena are given: (1) Quantum billiard in a
magnetic field; (2) quantum dynamics of a pulsed spin system. In these examples, we discuss
salient aspects of irregular energy spectra and complicated quantum diffusion. Then, funda·
mental problems of quantum chaos are examined from a viewpoint of integrable dynamical
systems. Singularities at degenerate points, which abound in the generic case with several
nonintegrability parameters being included, are incorporated into a high·dimensional field
theory.
Contents
§1. Introduction
§2. Quantum oilliard in a magnetic field--avoided level-crossings and diamag-
netism
§3. Quantum dynamics of a pulsed spin system--anoma1ous behavior of semiclassical wavefunctions
§4. Universal dynamical system behind the quantum chaos--a single parameter
case
§5. Reduction to a field-theoretical complex-Grassmannian sigma model---several
parameters case
§6. Summary and discussion
§ 1.
Introduction
The subject of quantum chaos has received a growing interest in contemporary
physics. 1> The terminology of quantum chaos is currently used for the quantum
mechanics of systems which exhibit dynamical chaos in the classical limit. Strictly
speaking, the linear nature of Schrodinger equation suppresses a chaotic diffusion and
eventually leads to the vanishing of Kolmogorov-Simii entropy and of other characteristic exponents. Nevertheless, many fundamental problems remain unsolved, e.g.,
those of irregular energy spectra and of complicated quantum diffusion even in the
recurrent time region. Theoretical tools borrowed from a field of random systems
*> Present and permanent address: Fukuoka Institute of Technology, Fukuoka 811-02, Japan.
384
K. Nakamura
are not always effective here. The Wigner-Dyson level-spacing distribution in the
random matrix theory2l explains only a very limited aspect of energy spectra. The
concept of localization of wavefunctions3l loses its significance when the localization
length ( ocn- 2 ) is larger than the dimensionality of Hilbert space. Recent challengers4 l
have derived coupled dynamical equations for energy eigenvalues and matrix elements following a philosophy due to Dyson. 5 l Most of their efforts, however, have
been directed towards confirming the effectiveness of the random matrix theory in the
subject of quantum chaos. But, this new subject should contain much more fruitful
informations than those due to the random matrix theory.
On the other hand, the spirit of natural science requires to find realization of
quantum chaos and thereby to establish the connection between chaotic dynamics and
experimentally-accessible macroscopic quantum observables. In this paper, we
provide two realistic examples of quantum chaos in magnetic phenomena. In § 2, we
discuss single-electron energy spectra for the free electron gas cofined in a billiard in
the presence of magnetic field. In § 3, the ergodic behavior of semiclassical wavefunctions of a pulsed spin system is examined in the long-time regime. In §§ 4 and 5, we
seek for a universal theory behind the quantum chaos. In §4, we derive a universal
dynamical system from generic quantum systems whose Hamiltonians possess only a
singl~ nonintegrability parameter. In § 5, a high-dimensional field theory is constructed, corresponding to adiabatic-ansatz induced eigenvalue problems which are
characterized by several nonintegrability parameters. We emphasize there the role
of topological singularities at degenerate points.6 l The final section will be devoted
to a summary and discussion.
§ 2. Quantum billiard in a magnetic field
--avoided level-crossings and diamagnetism
We choose here a typical autonomous system found in condensed matter physics.
Let us consider the quantum mechanics of noninteracting electrons in a planar billiard
(e.g., a thin conducting disk) in a uniform magnetic field normal to the plane.1l The
shape of the boundary is taken as elliptic. Unless the boundary effects are taken into
consideration, quantum-mechanical treatment merely yields the Landau diamagnetic
susceptibility. 8l Recent analysis of single-electron classical dynamics, 9l however,
elucidated the onset of chaos in the case when the Larmor radius is comparable to the
linear dimension of the billiard, indicating the crucial role of the convex boundary.
Quantum aspects of chaos will be captured by incorporating Dirichlet-type boundary
conditions. In the following, we shall first solve the Dirichlet eigenvalue problem for
a single-electron system. Then, global aspects of the spectra, i.e., the sensitivities of
energies and their average over "occupied" levels- the diamagnetic susceptibility at
absolute zero- will be investigated by changing the magnetic field.
Let us consider an ellipse with area 1rV=1rab where a and bare semi-major and
semi-minor axes in x- and y-directions, respectively. The eigenvalue problem is
given by H¢=E¢ with ¢=0 at the boundary. Here H=(l/2m)((n/i)f7+(e/c)A) 2•
We take a symmetric gauge: A=( -(1/2)yB, (1/2)xB) with B being the magnetic field.
The present system has C2-symmetry. Using the map (x, y)~(r,8) via x=arcos8, y
=brsin8, the eigenvalue problem is reduced to fi(r, 8)(f(r,8)=E¢(r,8) with ¢=0
Quantum Chaos
385
at the boundary of the unit disk. Basis
functions are now constructed in terms
of integer-order Bessel functions. For 40
convenience, we introduce the dimensionless parameters a=b/a, a=a/L, li
=b/L and B=B(ctt/(eV))- 1• Using
computed integrals in the matrix elements, we have solved the eigenvalue
problem. We present the results for a
nonintegrable case a=0.5 (elliptic bil0.
20.8 40.
liard) and the integrable case (circular
billiard). Comparison of the two cases Fig. 1. .8-dependent energy spectra for the evenparity manifold; (a) circle billiard (a=l); (b)
helps to elucidate the effects of noninteellipse billiard (a=O_S).
grability.
In Fig. 1, the even-parity part of the energy spectra is shown. In both of Figs.
1(a) and (b), most of the levels are found not to be well bunched into Landau levels.
Level repulsion leading to avoided crossings is widely seen in the case a=0.5 (Fig.
1(b)), while true crossings between levels with different k-values predominate in the
case a=1 (Fig. 1(a)). The presence of many avoided crossings corresponds to chaos
in the underlying classical dynamics. 9> For a=F1, chaos around the unstable
diametral orbit and/ or flyaway chaos are reported to dominate phase space, provided
the Larmor radius rc=mvc/eB satisfies
rc/L';::; Pmln
for the smallest curvature radius Pm1n= b2 /a.
ity, we find
(2·1)
Noting E=1/2mv 2 for electron veloc(2·2)
In consistence with the classical findings, we clearly observe in Fig. 1(b) that avoided
crossings dominate the spectra in the region E/B 2 ';::;0.35 for a=0.5.
We consider a sample containing 2N electrons, and neglect the Zeeman splitting
of the spin states. Then, in the free-electron ground state at a given value of the
applied field, the N lowest levels Ej(B) (j = 1, · · ·, N) are filled with two electrons each,
the Fermi level cF lying between EN(B) and EN+I(B). The isothermal susceptibility
per electron at absolute zero is given by the 2nd order derivative of the total energy
as
(2·3)
The contributions to the sum in Eq. (2·3) are computed separately for each manifold
of different symmetry (parity for a=F 1, k-values for a= 1). Singularities of .tP Ej/L1B 2
at true crossings are thus removed. In Fig. 2, the negative of x is shown as a function
of B for the case N = 100 occupied levels, where approximately 50 levels are of even
parity and the remaining ones are of odd parity.
For a= 1, x is found to retain the essential features of Landau diamagnetism in
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K. Nakamura
2D systems : - x takes the largest value
aa~------------~
in the vicinity of B=O, and decreases
monotonically with increasing B. Let
us trace back to Fig. 1(a): All crossings
appearing in this figure are true crossings between levels with different k.
Therefore, each energy level shows a
0.1
smooth variation with B with positive
curvature L12 l{/L1B 2 > 0, which decreases
with increasing B. The characteristics
ao~4H-~:H--------L-~
of the diamagnetic susceptibility for 11
= 1 in Fig. 2 are thus well explained by
the regular behavior of the spectrum
-0.1
which is a direct consequence of the
integrability of this case.
For 11=0.5, on the contrary, x shows
-0.2l-------'-'r--~---~~
remarkably different features: The value
20.
40.
of - x is greatly reduced at B=O as
compared to the Landau value. It Fig.2. -x as a function of B for N=lOO. Circles
and squares indicate a=l and 0.5 cases, reincreases
on the average with B,
spectively. Heavy symbols and lines denote
recovering the value for 11= 1 only for
combined contributions from both even- and
E10o/B 2 ~ Pmln, i.e., for B ~50. This
odd-parity manifolds, while fine cc;mnterparts
increase is accompanied by large fluctuadenote the contribution from the even-parity
manifold alone with energies below cF. x is
tions and anomalous dips. These feascaled by (2mL"/h 2)J.lB2•
tures can be traced back to the behavior
of the spectrum, which shows a multitude of avoided crossings (AC) (see Fig.1 (b)):
The rapid variation of the two levels with JJ near an AC gives rise to anomalous
contributions of L1 2Ei/L1iJ2 of opposite signs. If the AC is narrow and lies below cF,
the two contributions cancel in Eq. (2·3). But most ACs have width~their mutual
distance. This leads to a rather flat variation of each level with B, with a greatly
reduced average curvature and large fluctuations due to the nonuniform distribution
of ACs. With increasing B most ACs become extremely narrow, and the Bdependence of the levels approaches that for 11= 1. Thus, the anomalous features of
the diamagnetic susceptibility for !1=0.5 shown in Fig. 2 reflect the effects of level
repulsion and avoided crossings typical for a nonintegrable system. In the case of
ellipse billiards, in general, there exists a critical field Be: For B<Be the anomalous
feature of diamagnetic susceptibility is found and for B >Be Landau-like behaviors
are recovered. Be is determined by using Eqs. (2 ·1) and (2 · 2) as:
o.
e
(2·4)
Our findings cannot simply be interpreted in terms of the concept of bulk states
and edge states, since such a distinction of states is not possible for the case re/L
= 0(1). Figure 3 shows a typical pair of wavefunctions at an avoided crossing
indicated in Fig. 1(b). Indeed, they can be attributed neither to bulk nor to edge
Quantum Chaos
387
states.
Our prediction will be tested in thin
conducting disks containing a lowdensity 2D electron gas. Let us consider, for example, the interface layer in
semiconductor heterojunctions, where
current works concentrate on the quantum Hall effect. In these systems, the
electron concentration n is very low
-typically, n~l0 12cm- 2 • Noting that
the Fermi wave number kF ~ n 112 in 2D,
we have kF~l0 6 cm- 1 • For the above n,
N=l0 2 corresponds to L~lo-s em.
Then EN~ 8.0 X 102 for a= 0.5 and the
interesting aspect of quantum chaos is
seen for O<B~50, i.e., for O<B~2.5T.
The de Broglie wavelength ~ kF - 1 is
large enough for the results to be insensitive to the lattice discreteness both
within and along the boundary of the
interface layer. In ordinary metal Fig. 3. Wavefunctions 1¢1 at the avoided crossing
indicated by arrow in Fig. l(b). o-=0.5 and B
disks, on the other hand, n amounts to
=10: (a) Esz=284.0230; (b) Eaa=285.1459.
1016cm-I, which corresponds to kF~l0 8
cm- 1 • Then the de Broglie wavelength is comparable to the lattice constant so that
the results will be more or less modified. So our predictions due to the deterministic
chaos will be verified in real experiments for small 2D devices so long as both the
temperatures and the concentrations of atomic-scale defects are low enough.
In this way, chaotic dynamics in the nonintegrable elliptic billiard in a uniform
magnetic field is found to result in a multitude of avoided level-crossings, thereby
inducing a remarkable reduction and large fluctuations of diamagnetic susceptibility.
On the other hand, the integrable circular billiard yields results close to Landau
diamagnetism in 2D.
§ 3. Quantum dynamics of a pulsed spin system
-anomalous behavior of semiclassical wavefunctions
We now move to another example of quantum chaos in a driven nonautonomous
system. Chaos in driven spin system has recently become realized and its study
constitutes a very active research field, 10' though experiments to date have been
limited to dissipative spin-wave dynamics.11l Fully nonlinear dynamics for noninteracting spins, both classical and quantum, is also an attractive candidate by which to
study classical chaos and its quantum counterpart. The advantage of spin systems is
that, as a result of the finite-dimensional Hilbert space, we can readily tune the value
of n without artificial truncation procedures in quantum-mechanical treatments.
We consider wavefunctions for a periodically kicked spin system and demonstrate their fractal structures in semiclassically large-spin regions. 12 '' 13 ' Let us con-
388
K. Nakamura
sider the single-spin Hamiltonian, common to both classical and quantum spin variables S,
co
H=A(Sz)2-pBSx 1: 8(t-2;rn),
(3·1)
n==-oo
where A(>O) and B(>O) are an easy-plane anisotropy and magnetic field along the x
axis, respectively. In Eq. (3 ·1), we have chosen a tentative model Hamiltonian. One
may replace A(Sz)2 and pB by ASz and pBcos(mt), respectively. This kind of
replacement will not alter the essential part of the outcome described below. Further, in the experiment of spin echoes, for instance, an assembly of spin 1/2 systems
behaves coherently and constitutes actually a single large quantum-spin system.
Before proceeding to the quantum-mechanical treatment, we shall present brief
results for classical dynamics. Then Sis a three-component vector S=Sxex+Syey
+ Szez ( e is a unit vector), which obeys the equation of motion
co
dS/dt=SX(-8H/8S)=Sx(-2ASzez+Ji.Bex 1: 8(t-2;rn)).
n=-oo
(3·2)
The magnitude 8 2 is conserved in Eq. (3 · 2), and is now normalized to unity. Sis then
described in polar coordinates, i.e., S=(Sx, Sy, Sz)=(sin8cos¢, sin8sin¢, cosO). The
discrete map can be constructed for successive values {Sn}, where Sn is the value of
Sat t=2;rn+O, i.e., just after the n-th pulse. In the interval2;rn+O:::;; t:;:;;2;r(n+ 1)-0,
the magnetic field is not operative in Eq. (3·2) and thus Sn is transformed into r
=Rz(a)Sn at t=2;r(n+ 1)-0, where the operator Rz(a) denotes a rotation by angle a
=4;rASnz around the z axis. Then, the (n+1)-th pulse during 2;r(n+1)-0~t:::;;2;r
x(n+1)+0 rotates r by angle [3=-pB around the x axis, yielding Sn+I=Rx([J)r.
Eventually, the combined map Sn+I = Rx([J)Rz(a)Sn is obtained, which we have solved
with A=l.O and 0.0::5:pB~l.O. Investigation of our extensive data as a function of
pB indicates the presence of two characteristic fields pB1 ~ 0.1 and pB2 ~ 0.5, where
the fraction of chaotic trajectories increases strongly and the last KAM torus disappears, respectively. 12J
The corresponding quantum dynamics is governed by the time-dependent
Schrodinger equation for a wavefunction with the Hamiltonian being the quantum
version of Eq. (3 ·1). We solve this equation after rewriting it in matrix form at the
outset: A set of eigenstates of Sz is used as basis kets. A coefficient vector C for the
wavefunction satisfies the matrix equation
in dC/dt =fie .
(3·3)
fi is a (2S+l)X(2S+l) real-symmetric matrix ii=fio+V ~ 8(t-2;rn). Noting
n=-oo
Floquet's theorem, the solution of Eq. (3·3) just after the n-th pulse is
C(2;rn+O)= 1:[exp( -2;rinEa/n)][Xa t • C( +O)]Xa,
a
(3·4)
where the Ea's and Xa's are the quasienergies and eigenstate vectors, respectively,
obtained by solving the eigenvalue problem UXa=exp( -2;riEa/n)Xa. Here U is a
unitary matrix defined in terms of the time-ordering operator T as follows:
Quantum Chaos
(a)
<I>
(b)
389
ja)
(c
0
<I>
0
EJ
EJ
te)
(f)
Fig. 4. Contour maps of Pn(B, ¢) in the case JJ.B
=0.01 for very early stages of time evolution:
(a) n=O; (b) n=l; (c) n=2; (d) n=3; (e) n=4;
(f) n=8.
Fig. 5. The same as in Fig. 4, but in the case
JJ.B=l.O.
U = Texp[l:no(- i/n)H(t')dt'] =exp[(- i/n) V]exp[(- i/n)2n-ilo] .
(3·5)
The probability density function is given in terms of SU(2S+ 1) coherent state
representations as Pn(B, ¢)=1<8, ¢127Z'n+O>I 2 • In the following, we employ S=128
and choose n=(S(S+ 1))- 112 so that the observable spin magnitude maintains the
scaled value for the classical spin vector, i.e., S 2 =S(S+ 1)n2 =1.
In Figs. 4 and 5, very early stages of temporal evolution of initially (n=O)
localized wavepackets are shown. For a weak pulse (,uB=0.01), Pn(8,¢) shows a
unidirectional diffusion (see Fig. 4) corresponding to regular behaviors in classical
dynamics. However, for a strong pulse (.uB=l.O), remarkable isotropic and irregular diffusions begin after the period of "classical" stretching and folding-type diffusion
(see Fig. 5). Figures 4 and 5 also indicate the crossover time tc= 0(1) at which the
classical and quantum correspondence breaks down.
We proceed to examine Pn(B, ¢)in large n(~1) regions beyond tc. Figures 6(a)
"-'(c), (a')""'(c') and (a")""'(c"), while they have no exact classical counterparts in these
time regimes, retain some signatures of periodic orbits at ,uB=0.01, coexisting KAM
orbits and localized chaos at ,uB=0.2 and global chaos at ,uB=l.O, respectively. We
have attempted to characterize these structures in terms of the singularity spectra
/(a)/ 4 > which have proven effective recently in quantifying multifractal aspects of
chaotic systems. While details will be described elsewhere, 13 > we summarize here the
major issue: Widths of /(a), i.e., distribution of local dimensions of Pn(B, ¢), for ,uB
=0.01 and 1.0 fall into a narrow range during the time evolution. On the other hand,
390
K. Nakamura
(C)
Fig. 6. Time evolution of three-dimensional pictures of Pn(B, ¢)for n:>l. (a)-(c) JlB=O.Ol. From
the left, n=70, 90 and 110; (a')-(c') tlB=0.2. From the left, n=60, 110 and 120; (a")-(c") tlB=l.O.
From the left, n=90, 110 and 120.
those for J.tB=0.2 extend over a much broader range. The large fluctuation of local
dimensions for J.tB=0.2 signifies the inhomogeneous distribution of measures
Pn(B, ¢),as seen in Figs. 6(a'}·"(c'). It clearly reflects the coexistence of KAM orbits
and chaos characteristic of a transitional region leading to a global chaos. The small
fluctuations for J.tB=O.Ol and 1.0 signify the uniform distribution of measures in Figs.
6(a)--.....(c) and 6(a")--.....(c"). The small fluctuation for J.tB=l.O reflects highly organized
ergodicity in the corresponding classical dynamics.
Thus semiclassical wavefunctions exhibit quite new phenomena which can be
seen in neither the classical nor quantum limit. The outcome of this section is
summarized as: (1) Despite the complete absence of classical and quantum correspondence, the long-time .behavior of wavefunctions maintains the ergodic or nonergodic
features possessed by the underlying classical dynamics; (2) the enhanced fluctuation
of local dimensions of wavefunctions, which is found at a transitional region leading
to a global chaos, persists throughout the time evolution.
§4. Universal dynamical system behind the quantum chaos
--a single parameter case
We have so far observed salient aspects of quantum chaos by taking its typical
examples experimentally-accessible in magnetic phenomena. We have solved
eigenvalue problems for Hermitian operator H(t) and Unitary one U(t) where t
represents hereafter a nonintegrability parameter (e.g., strength of magnetic fields).
In the analyses of quantum chaos, in general, we encounter the eigenvalue problems
Quantum Chaos
391
of the forms H(t)ln>=xnln> and U(t)ln>=e-.:rnln> corresponding to autonomous and
driven nonautonomous systems, respectively. Here we can naturally ask a question:
If tis regarded as quasi-time and both of eigenfunctions {In>} and eigenvalues {xn} as
dynamical variables, what will be the resultant dynamical system? While Pechukas
derived some interesting equations along this line, 4 > his major concern was lying in the
derivation of level-spacing distributions such as Wigner-Dyson2 > or Poisson distributions. Therefore, he chose, besides the variables of {xn}, matrix elements for a
nonintegrable part of H(t) as dynamical variables. But, this methodology inevitably
suppresses the essential part of information possessed by eigenfunctions: In the case
of the finite (N-) dimensional Hilbert space, for instance, a set of eigenfunctions of
H(t) has N 2 complex variables, while nondiagonal matrix elements employed by
Pechukas have only N(N -1)/2 complex variables. Since the ingredients of quantum
chaos are involved not only in energy spectra but also in wavefunctions, it is highly
desirable to take {In>} and {xn} on the same footing.
We now concentrate on autonomous nonintegrable systems. Let H(t)=Ho+ tV
be the Hamiltonian for a quantum bound system which is classically nonintegrable.
Ho and V are a classically integrable part and a nonintegrable perturbation, respectively. Both of them are Hermitian, i.e., Hot=Ho and Vt= V. A single parameter
t denotes the strength of nonintegrability. A manifold with a definite symmetry is
considered. Then we can assume discrete eigenvalues {xn(t)} to be nondegenerate,
suppressing a negligible possibility of accidental degeneracies. (When the
Hamiltonian includes several nonintegrability parameters, this assumption is not
valid which will be described in § 5.) Eigenfunctions {ln(t)>} are complex orthonormal and form a complete set. If we take t as quasi-time, equations of motion for
Xn(t), ln(t)> and <n(t)l can be obtained from the time (real time!)-independent
Schrodinger equation H(t)ln(t)>=xnln(t)> as 15 >
dxn/dt=Pn[ = Vnn=<n(t)l Vln(t)>],
(4·1a)
(4 ·1b)
(4·1c)
(4·1d)
Equations (4·1c) and (4·1d) have been derived as follows: Taking the t derivative of
the SchrOdinger equation, we have Vln>+Hdln>/dt=Pnln>+xndln>/dt. Its lefthand side equals
~lm> Vmn+ ~lm>xm<ml(dln>/dt)
m
m
where <ml(dln>/dt)= Vmn(xn-xm)- 1 for m=l=n and <nl(dln>/dt)=(d<nl/dt)ln>=O are
used. Noting xn(t)=l=const (=0), we arrive at Eqs. (4·1c) and (4·1d). Here, the
equality <nl(dln>/dt)=O and its Hermitian conjugate prove to be valid so long as a
single parameter is changed (see § 5). For the study below, it is convenient to rewrite
392
K. Nakamura
Eqs. (4 ·1) in a perfectly canonical formalism.
Let us define Anm as
(4·2)
which reduces to Anm=<nli- 1 [Ho, V]lm>=<niAim>. The operator A is /-independent
and proves to be Hermitian. A can thereby be decomposed as A= L t L. Here L is
an appropriate /-independent operator which has its unique inverse L - 1• Anm thus
becomes
(4·3)
Anm=<niLt Lim>.
By using Eqs. (4·2) and (4·3), Eqs. (4·1b~d) become
dPn/dt=2 m(*n)
~ AnmAmn(Xn-Xm)- 3 ,
(4·1b')
(4 ·1c')
(4·1d')
Equation (4·1c') and (4·1d'), though substantially the same as Eqs. (4·1c) and (4·1d),
take advantageous forms for our study on the complete integrability below. Lin>
and <niL t are now read as complex dual vectors. In contrast with previous results, 4'
Eqs. (4 ·1a) and (4 ·1b' ~ d') describe the dynamics for both eigenfunctions and
eigenvalues and suggest a perfectly canonical formalism.
We now show that these equations can also be attained from an effective 'classical' Hamiltonian. Let us introduce a classical N-particle Hamiltonian with internal complex dual space in N dimensions for each particle:
(4·4)
with Anm defined as a scalar product of complex vectors <niLt and Lim>, consistent
with Eqs. (4·2) and (4·3). Applying Poisson brackets given by
aA
aB )
aiLin> a<niLt '
(4·5)
one gets a set of canonical equations from Eq. (4·4) as 15 '
dxn/dt={fi, Xn}=ailjapn,
(4·6a)
dPn/dt={fi, Pn}=-afi/axn,
(4·6b)
d(iLin>)/dt={il, iLin>}=ail;a<niLt,
(4·6c)
d(<niLt)/dt={il, <niLt}=-ail/aiLin>.
(4·6d)
Equations (4·6) together with Eq. (4·4) are exactly the same with Eqs. (4·1a) and
(4 ·1b' ~d'), if the number N in the latter is finite. Clearly, this equivalence holds even
in the limit N=oo. Equations (4·6) with (4·4) prove to be a Calogero-Moser
N-particle system in 1 + 1 dimensions with internal complex vector space, which
Quantum Chaos
393
happened to be proposed in a completely different context of nonlinear dynamics,I 6 >
though in the present case Amn, Lin> and <niLt are defined in a much more intricate
way. The number of degree of freedom of the system, Eqs. (4·6), is N(N+1),
consisting of N for positions and momenta of N particles (levels) and N 2 for internal
freedom (complex eigenstates) possessed by particles (levels).
The complete integrability of Eqs. (4·6) will be elucidated, merely by rewriting
the novel findings of Gibbons and Hermsen 16> in our notations. Let us define N X N
matrices:
Pnm = OnmPn + (1- Onm)iAnm(Xn- Xm)-l ,
Xnm=OnmXn,
L=(LID, LIZ>,···, Lin>,···, LIN>),
where Ll n> are N-component column vectors. Then, Lax representation of Eqs. (4 · 6)
can be obtained. Here we show its complete algebraic solution:
X(t)= U(t)[X(O)+ P(O)t] u- 1(t),
P(t)= U(t)P(O)U- 1(t),
L(t)=L(O)U- 1(t),
Lt(t)=U(t)Lt(o).
(4·7)
U(t) is a unitary matrix defined in terms of the time-ordering operator T, i.e.,
Thus, if we would have a knowledge of eigenfunctions and eigenvalues at t = + 0 (i.e.,
in the integrable or weak-coupling limit), those at t >O (i.e., in the nonintegrable or
strong-coupling regime) will be provided by the solution in Eqs. (4·7).
While we have been confining ourselves to the autonomous system, the same
argument as above proves to hold even for nonintegrable driven systems in the case
when the driving field consists of periodic pulse trains.17l The problem of the time
(real time!)-dependent Schrodinger equation is now reduced to an eigenvalue problem
for the one-period propagator:
exp(- itV)·exp( -2m"Ho)ln(t))=exp(- ixn(t))in(t)>,
(4·8)
where t is a nonintegrability parameter, representing the pulse strength. By taking
the t derivative of Eq. (4·8), we obtain: 17>
dxn/dt=Pn,
dPn/dt= 41 m-¢n
~ <niLt Llm><miLt Lln>cos[(xn-xm)/2]sin- 3[(xn-xm)/2],
d(Lin>)/dt=( -i/4)
~
m*n
Llm><miLt Lln>sin- 2 [(xn-xm)/2],
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K. Nakamura
(4 ·9)
L and L t are !-independent operators, though their definition here is a bit different
from Eq. (4·3). We find that Eqs. (4·9) constitute a generalized Sutherland system in
1 + 1 dimensions whose complete integrability was proved by Nakamura and
Mikeska. 17>
In this way, we arrive at the issue: There exists a universal 'classical' dynamical
system which underlies the quantum chaos. It is a completely-integrable manyparticle system with internal complex vector space, whose effective Hamiltonian is
given by
N 1
1 N
H= "'£12Pi+---{~1 ~1 <niLt ·Lim><miLt ·Lin>.P(xn-xmlw, w').
(m*n)
(4·10)
Poisson brackets and equations of motion also take universal forms described by Eqs.
(4·5) and (4·6), respectively. In Eq. (4·10), .P(zlw, w') is the doubly-periodic Weierstrass function. .P(zloo, oo)=z- 2 and .P(ziO, i7r/2)= fsin- 2 (z/2) correspond to the
nonintegrable autonomous and driven (pulsed) nonautonomous systems, respectively.
§ 5.
Reduction to a field-theoretical complex-Grassmannian sigma model
-several parameters case
So long as a single parameter is to be changed in the Hamiltonian, the desymmetrized spectra exhibit no degeneracy expect for accidental ones. We can see only
avoided crossings. The presence of a multitude of avoided crossings is a quantummechanical manifestation of the classical chaos. In§ 4, this situation was examined
from a viewpoint of nonlinear dynamics. When several parameters are to be changed,
however, we see degeneracies in general. 18 > For example, crossings of eigenvalues at
degenerate points have codimensions 2, 3 and 5 depending on the real, complex and
quaternionic Hermitian nature of Hamiltonians, respectively. The codimension here
implies the number of independent parameters. Let us consider the eigenvalue
problem:
d
H(t)in(t)>=(Ho+ L! tp. Vp.)in(t)>=xn(t)in(t)>,
P.=l
(5·1)
where t=(ft, t2, ···, td) denotes a set of nonintegrability parameters. Here the
adiabatic ansatz is imposed: Throughout the change of coordinates in the highdimensional parameter space (t), no transition between eigenstates is assumed to
occur. Just as in the previous section, the Hamiltonian H(t) in Eq. (5·1) is assumed
to have bound spectra. Hot =Ho and VP. t = VP. for ,u=1, 2, ···,d. Let us consider the
desymmetrized spectra of Eq. (5 ·1). The spectra are now constructed in the space t,
in which each eigenvalue constitutes a hyper-energy surface. In general, we can see
the contacts of adjacent energy surfaces at degenerate points. In the neighbourhood
of each degeneracy, a pair of energy surfaces constitute two sheets of a double-cone
(diabolo). 19 > Figure 7 illustrates a diabolical degenerate point and an avoided crossing in the parameter space. The diabolical point is the source of topological singularity: Let us consider a closed circuit around a degenerate point in t space (see Fig. 7).
Quantum Chaos
Then, the eigenfunction ln(t)> joining
the degeneracy acquires the topological
phase6 > given by (we take Einstein's contraction for Greek letters)
395
energy
(5·2)
This phase is a manifestation of the fact
that eigenfunctions are multi-valued in a
parameter space, though of course
I
single-valued in the configuration space.
I
I
-i<nlatpln> in Eq. (5·2) plays the role of
a gauge potential, which will be shown
below. We may expect that the presc
ence of many degenerate points and
avoided crossings in energy spectra Fig. 7. Diabolical degenerate point and avoided
crossing in 2D parameter space.
would be an indicator of the quantum
chaos characterized by several nonintegrability parameters.
We take {t"}, {xn(t)} and {ln(t)>} as d-dimensional 'space-time coordinates',
components of a vector field and elements of a matrix field, respectively. By making
the t" derivative of Eq. (5 ·1), we obtain (hereafter, atp is abbreviated as a")
Q:j
(5·3)
with
A;:'n=- i<mla"ln>.
(5·4)
We have also
a"xn=<nl V"ln>.
(5·5)
Equations (5·3) and (5·4) form a closed system, which governs {In>}. On the other
hand, {xn} can be solved, separately by using Eq. (5·5) whose right-hand side is
completely determined by the solution {In>} in Eqs. (5·3) and (5·4). So, we focus our
attention on Eqs. (5·3) and (5·4).
Let us define the rna trix field
X=(ID, 12>, ·--,IN>).
(5·6)
Equations (5·3) and (5·4) are then rewritten as
a~=iXA",
(5. 3')
Ap=-iXta~.
(5 ·4')
These forms remind us of a complex-Grassmannian sigma model. 20> In fact, for a
transformation
396
K. Nakamura
X'(t)=X(t) U(t)
with
(5·7)
ut U=1, (5·4') transforms as
(5·8)
which is characteristic of a non-Abelian gauge field. Under the transformation in
Eqs. (5·7) and (5·8), Eq. (5·3') remains invariant. Equations like (5·3') prove to be
obtained from the gauge-invariant action:
(5·9)
The topological invariants are given by
(5·10)
Diagonal elements Qnn in Eq. (5·10), which were recently introduced by Mead in the
case d=2 (for an infinitesimal rectangle), 21 > are generalizations of the adiabatic phase
in nondegenerate systems. 6> (So long as one treats a single level, the 3rd term in the
integrand in Eq. (5·10) does not appear except for in the degenerate case. 2 ll' 22 l) The
system in Eqs. (5·3') and (5·4') is completely integrable and admits instanton solutions
in the case d=2. Further details will be described elsewhere.23>
Thus, the problem of quantum chaos characterized by several nonintegrability
parameters proves to be reduced to a simplified version of the d -dimensional
complex-Grassmannian sigma model. Singularities at degenerate points are described by gauge fields. We can extend the present theory to a driven system as well.
§ 6.
Summary and discussion
In the initial half of the present paper, we have shown two realistic examples of
quantum chaos in condensed matter physics. The first one is concerned with the
chaotic cyclotron motion of electrons in a conducting disk of mesoscopic size. The
irregularity in a single-electron energy spectrum is found to be induced by a multitude
of avoided level-crossings. Resultant homogeneous level structures lead to a reduction of diamagnetism in 2D dilute electron-gas systems. We find also the critical
magnetic field below which the anomalous feature of diamagnetism can be observed.
These findings will be a vehicle for our future study on open billiards: If we make
small holes on the billiard boundary and connect them with current-flowing thin
channels, we have a new possibility of observing the quantum chaos by means of
quantum transports such as magneto-conductance or Hall conductance. We may see
there an interesting interplay between a deterministic quantum chaos and disorder
(impurity)-induced quantum Hall effect in 2D.
The other example is a semiclassical spin system radiated by periodic pulse
trains. Such a spin system is conceivable, e.g., in an assembly of spin 1/2 systems in
electron spin resonances. We find: Despite the complete absence of the classical and
quantum correspondence, semiclassical wavefunctions exhibit turbulent diffusions in
Quantum Chaos
397
the long-time regime. The wavefunctions show also a quantum analog of the coexistence of KAM tori and chaos in a transitional region leading to a global chaos. Its
invariant probability measure is characterized by anomalous fluctuations of singularities.14> Our spin system, when suitable dissipations will be incorporated, will yield
remarkable phenomena such as chaotic trains of spin-echo bursts. This system will
also be a nice model to simulate Bloch oscillations in driven Josephson junctions24 >
because Cooper-pair operators in a small junction can be described by quantum
spins. 25 >
In the last half of the present paper, we have found a universal dynamical system
lying behind the quantum chaos. In case when a single nonintegrability parameter
characterizes the Hamiltonian, the eigenvalue problem is reduced to a (1 +I)dimensional many-particle system, where the parameter and both of eigenfunctions
and eigenvalues are regarded as quasi-time and dynamical variables, respectively.
The above system has inter-particle interactions of Weierstrass Sf function type and
proves to be completely integrable. On the basis of our findings, many conjectures
become possible: An avoided crossing and its motion for a change of the quasi-time
may be interpreted as a single soliton motion. 26 > Excitation of small or large number
of solitons will correspond to the near-integrable or chaotic nature of the underlying
classical dynamical system, respectively. The statistical mechanics approach
attempted by Dyson5>and Pechukas4>is of course interesting, though it is not a unique
way of analysing the quantum chaos. Regarding this approach, however, the proof
for the ergodicity hypothesis on our completely-integrable system remains unprovided.
Extention of our analyses has been made to more generic quantum systems
characterized by several nonintegrability parameters. Since we see many degeneracies in this case, a drastic change in the methodology is required. A set of nonintegrability parameters and both of eigenfunctions and eigenvalues are taken as 'spacetime coordinates' and field variables, respectively. Matrix gauge fields in Eq. (5·4)
are introduced, whose diagonal elements give rise to the adiabatic phase of current
interest. 6> The resultant field theory proves to be described by a simplified version of
a d-dimensional complex-Grassmannian sigma model. The gauge fields capture the
topological singularities at degenerate points. In a particular case d=2, this model
admits instanton solutions with various topological charges. Excitation of small or
large number of instantons will be related to the near-integrability and chaos of
underlying classical systems, respectively.
The mixing and ergodic features of classical chaos have helped to establish its
relationship with the formalism of equilibrium statistical mechanics. 27 > In the field of
quantum chaos, however, neither the nonzero Kolmogorov-Sinai entropy nor positive
Lyapunov exponent can be expected, except for in systems with infinite degrees of
freedom and/ or in some of open quantum systems. Nonetheless, we see many
complicated issues in the quantum-mechanical treatment of chaotic systems. Both
the universal dynamical system and complex-Grassmannian sigma model that we
have obtained will be a guiding vehicle for understanding the above complexities.
398
K. Nakamura
Acknowledgements
The main part of the present paper is based on a series of our latest works. The
author would like to thank collaborators, in particular, C. Jeffries, M. Lakshmanan,
H. J. Mikeska, A. Shuda and H. Thomas. He acknowledges valuable conversations
with P. Gaspard and S. A. Rice and their nice hospitalities during his sabbatical leave
at the University of Chicago.
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