- Email: [email protected]
Numerical simulation of level statistics
17 August 1998 PHYSICS
LETTERS
A
Physics Letters A 245 ( 1998) 183-188
ELSBVIER
~~rn~~cal simulation of level statistics T. Yukawa a*b.l a Coordination Center for Research and Education. The Graduate ~aiversi~ forAdvanced Studies, ShonanVillage, Hayarna, Kanagawa 240-01, Japan h lnstitate of Particle and Nuclear Studies, KEK, High Energy Acceierator I-l Oho, Tsukuba-City. Ibaraki 30$, Japan
Research organization,
Received 14 October 1997; revised manuscript received 7 April 1998; accepted for publication 29 April 1998 Communicated by J.I? Vigier
Abstract The statistics of energy eigenvalues for a class of Hamiltonians written as H = FIe + tV is studied through the dynamics of levels regarding the perturbation parameter t as time. The statistical properties of eigenvalues are investigated by measuring quantities such as the nearest-neigh~r level spacing dist~bution and the Dyson-Metha & statistics for two types of ensembles known as orthogonal and unitary, and for three values of the system parameter controlling chaos. @ 1998 Elsevier Science B.V.
1. Introduction
Historically, the statistical properties of energy levels of an isolated system were initially studied by neutron resonance spectra of heavy atomic nuclei [ I]. They have drawn much attention recently in the context of quantum chaos, namely, the quantum mechanical property of a dynamical system which exhibits chaos in its classical trajectories [ 2,3]. The spectral property, which was previously regarded to be typical in a system with moderate complexity such as heavy atomic nuclei, has turned out to be rather universal in most isolated systems whose phase space of classical motion is dominated by chaotic trajectories. For example, the nearest-neighbor level spacing distribution Pa(S) as well as the Dyson-Meh~ statistics &(L) for any non-integrable system show the universal behav-
’ E-mail: yukawa~t~o~.kek.jp,
ior characteristic in neutron resonance spectra when the system is fully chaotic [4]. As the study of statistical property of energy spectra is extended to various isolated quantum systems like molecules, micro-clusters, or simple model systems such as billiards, types of statistics other than that of nuclei have been found. Various studies indicate the existence of distributions having properties ranging continuously from the one seen in nuclei to the uncorrelated distribution depending on the chaotic nature of the system. For example, for the nearest-neighbor level spacing distribution two extreme cases are known experimentally as well as theoretically, namely, the Wigner dis~ibution in the chaotic limit and the Poisson distribution for integrable systems. For distributions in between these two limits several dis~ibutions [S-9] have been proposed phenomenologically interpolating these two extreme cases. However, they fail to reproduce the level repulsion phenomenon which is known to persist in the intermediate cases. To carry out stud-
0375-9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00357-0
184
7: Yukawa/Physics
Letters A 245 (1998) 183-188
ies beyond phenomenological levels we ask ourselves, (i) is there any relation between the transition from the Poisson to the Wigner distribution and chaos of the system, and (ii) what is the statistical law in the level statistics which can explain the intermediate distributions seen in numerical simulations. For answering these questions we need a fundamental understanding of the level statistics based on standard statistical mechanical principles. There have been attempts in this direction which treat energy levels dynamically by regarding them as particles in one dimension, and interpreting a parameter in the Hamiltonian as time [ 10-121. In this way it becomes possible to understand the random matrix theory in terms of the universal language of physics, instead of giving an ensemble a priori. Also, the theory can predict distributions intermediate between the Wigner and the Poisson distribution by varying a control parameter [ 131. In the next section we briefly review the theory, which is followed by numerical simulations. The last section is devoted to examining the results and their significance.
2. Statistical mechanics
of levels
We consider an isolated system with time-reversal invariance for simplicity. Applications to a system without time-reversal invariance are straightforward, and they will be mentioned later without proof. Suppose a class of Hamiltonians written as
(1)
H=Ho+ty
where the perturbation strength r runs along the real axis. We assume that there are no degenerate levels, or if there are operators which commute with the Hamiltonian, we choose a subspace of the Hilbert space belonging to an irreducible representation of the operators. Representing the Hamiltonian by a basis set which is independent of the parameter t, for example, a complete set of eigenfunctions of Ha, the matrix X = X0 + tPa can be regarded as a solution of the classical N( N + 1) /2 free particle equations dX dt=
P,
dP dt=
0
with initial conditions,
(2)
X(0)
= x0,
P(0)= PO,
(3)
where X and P are matrices corresponding to H and V, respectively. For systems with time-reversal invariance Xu and PO are given by real symmetric matrices. Changing the representation basis by the orthogonal transformation 0, X= O-‘EC?,
P= CTIVO,
(4)
so that E is diagonal, the equations (Eq. (2) ) now become dE . ~+i[M,El
=V,
z+i[M,V]
where M E i(dQ/dr)O-‘. matrix G by
of the motion
=O,
Introducing
G=i[E,V]
(5)
a Hermitian
(6)
the equation of motion for G follows as dG dt +i[M,G]
=O.
A class of equations such as those for V and G is known as the Lax form. In this case the trace of matrices constructed by products of any powers and orderings of the matrices V and G are constants of the motion provided [V, G] # 0. By explicitly writing these equations (5)-(7), the equations of the motion are shown to obey the socalled generalized Calogero-Moser (GCM) equation, which has been studied extensively [ 14,151 as an example of a completely integrable system. It turns out to be obvious from the starting linear equation, i.e. the Schrodinger equation, that the GCM equation is completely integrable. To study the statistical property of energy spectra, especially the fluctuation of the level distribution, let us briefly describe the procedure employed in Ref. [ 131. We split a set of energy levels into subsets consisting of N sequential levels. Each subset is put in a box of size N by fixing the positions of the two end levels at 0 and N, so that the average level density is constant (= 1) . By these boundary conditions the parity and the translational invariance are broken explicitly, and most constants of the motion are not constant anymore, which implies the system loses complete integrability.
7: YukawdPhysics
Letters A 245 (1998)
From among the remaining constants of the motion we pick two quantities which have the lowest powers of V and G, and are expected to be important: Tr V* and Tr G*. The statistical equilibrium distribution is then given by the Boltzmann distribution, dw = exp
- $TrV’-
with the Lagrange multipliers 0 and y. The integral measure dT should be proportional to the phase space volume with the symmetry constraints on matrices X and P. For the time-reversal invariant system they are real symmetric and the measure is given by
n
( dJ&,,
dP,,n
1
(9)
l
nr,n
while for the time-reversal non-invariant case the matrices are complex and Hermitian constraints should be imposed. By changing the variables from {Xnm} and {P,l,} to {Ent,&} and {V&G&!, where {&}(a = are rotation angles of the orthogl,...,N(N-1)/2) onal transformation 0, and carrying out integrations over all non-observed variables except {En}, the joint probability distribution for {En) is given by P( (E,)) X
= const
(Em 1 i- (y/P)(E,,,
&I2 - En)*
4
>
’
(10)
where cy should be chosen to be either 1 for the orthogonal ensemble or 2 for the unitary ensemble depending on the time-reversal property of the system being either invariant or non-invariant, respectively. For the two limiting cases of parameter y/P we obtain the known distributions; the Poisson distribution, P( (E,)) when y/P
-+ const , + co, and the Gaussian
185
when y//3 --f 0 with NT//~ being fixed, where EC = ( l/N) 23, En. In these two cases various statistical observables are known analytically. We try to obtain the intermediate distribution numerically in the following section.
dT
gTrG2 >
X
183-188
(11) distribution,
3. Numerical simulation Although it is expected from the consideration on the two extreme limits that inte~edia~ dis~ibutions will be obtained by varying the parameter y/P in the range [ 0, co), here we check it for the orthogonal case (cu = I) and the unitary case (cu = 2) numerically. Energy spectra are simulated by the Metropolis Monte Carlo method so that the joint distribution function reaches Eq. (10) in statistical equilibrium. Initially N+l levelsareplacedatn (=O,l,...,N),andtwo end levels EO and EN are fixed at 0 and N, respectively, throughout the simulation to keep the average level density 1. We choose the parameter r - y/P = 100.0, 1.0, and 0.01 representing the state of system being regular, intermediate, and chaotic, respectively, for the orthogonal ensemble (1~ = 1) and the unitary ensemble ((Y = 2). The total number of levels N is taken to be 400. In Figs. l-3 three types of spectra are compared for the orthogonal ensemble. For the chaotic system (Fig. 3) the level repulsion is so strong that the level densities near the end points are larger than in the central region. Then the average level density deviates below from 1 systematically reflecting the higher density at both ends. Our numerical simulation for varying r indicates the levef repulsion effect becomes signi~cant when r gets smaller than 1. We also observed similar results for the unitary case (a = 2). In order to avoid the boundary effect we chose 200 levels between the level number 100 and 300 for measurements. In this region the level densities for all cases are almost constant, and levels are resealed as E;=c,E;+ioo+c2
(12)
(i=O,l,...,
200).
The parameters cl and cs are fixed with conditions E: = 0.0, E&, = 200.0 so that the level densities are 1. Then, we measure the nearest-neighbor level spacing distribution Po( S) defined by
Z Yukawa/Physics
186
Letters A 245 (1998) 183-188
Energy Spectra olthogad, r=o.ot
Energy Spectra olttloQalal, 1;100.0 400.0
400.0
f
300.0
300.0
%
200.0
W
loo.0
loo.0
Fig. I. Energy spectra of a regular time-reversal invariance (a = 1)
system
(f
= 100.0)
with
Fig. 3. Energy spectra time-reversal invariance
Energy Spectra or(hoowl, r-i.0 400.0
system
(r
= 0.01)
with
Nearest-NeighborSpacing
I-
300.0
of a chaotic (a = 1).
1.0
r
0.8 -
0.6 200.0
8 z
f
0.4 loo.0 02
0.0
L 0
100
200 LevalNlJmeW
Fig. 2. Energy spectra of an intermediate time-reversal
invariance
PO(S) 0: (number
300
system
I 400
(f = 1.0) with
(cy = 1)
of pairs of neighboring
with spacing between S and S + AS).
0
Fig. 4. Nearest-neighbor level spacing distributions for the orthogonal ensemble. The three lines correspond to three values of f: 100.0 (dot-dashed), 1.0 (dotted), and 0.01 (solid).
levels (13)
In Fig. 4 we show three observed distributions with r = 0.01 (solid), 1.0 (dotted) and 100.0 (dotdashed) for the orthogonal ( LY= 1) case with AS =
0.1. The transition from the Poisson to the Wigner distribution is clearly observed, although the distribution for r = 1.0 is already rather close to that of r = 0.01, which is almost same as the Wigner distribution. Fig. 5 shows PO(S) for the unitary ensemble
T. Yukawa/Physics
Letters A 245 (1998)
,
I
187
Delta3 Statiiics
Nearest-Neighbor Spacing l.Of
183-188
cm-lI
1
Fig. 5. Nearest-neighbor level spacing distributions for the unitary ensemble. The three lines correspond to three values of I‘: 100.0 (dot-dashed), 1.0 (dotted), and 0.01 (solid).
where we can clearly find stronger level repulsion compared to the orthogonal case as the theory predicts a quadratic repulsion for cy = 2, while a linear one for the cz = 1 case. The last measurements are for the Dyson-Mehta & statistics, which has been defined originally by
Fig. 6. Dyson-Mehta delta3 statistics for the orthogonal ensembles. The three lines correspond to three values of I‘: 100.0 (dot-dashed), I.0 (dotted), and 0.01 (solid).
Delta3 Statistii
Eo+L
QL)=(~!;;
-AEJ [N(E)
Bl*dE
EO
, > (14)
where N(E) = C, t?( E - E,,), and (. . .) indicates averaging over the lower end of the integration, Eo. Parameters A and B are fixed at the values giving the minimum &. For numerical simplicity we interchange the E, N axes and define N+M
a,~,=(~,& J
NO
[E(N)
-A’N-B’I*dN
, >
(15) where E(N) is the inverse function of N( E). Here, N is considered to be a real variable, while NO and M are integers. L&(M) obviously has the same property as the original Dyson-Mehta statistics. In this definition
Fig. 7. Dyson-Mehta delta3 statistics for the unitary ensembles. Three lines correspond to three values of f: 100.0 (dot-dashed), I.0 (dot), and 0.01 (solid).
(. . .) should be understood as averaging over No. The parameters A’ and B’ are determined so as to give the minimum &. In Figs. 6 and 7 we show the orthogonal case and the unitary case, respectively. A smooth
188
transition between in both ensembles.
7: Yukawa/Physics L.&em A 245 (1998) 183-188
the two extreme cases is apparent
4. Discussion Regarding the Hamiltonian H = Ho + tV as a solution of the classical motion of N( N + 1) /2 free particles when it is represented in matrix form, X(t) = X(0) + tP(O), we have used the standard technique of statistical mechanics to study the statistics of energy spectra. We have obtained the intermediate level statistics between the two known limits, for example, the Poisson and the Wigner distribution for the nearest-neighbor spacing distribution. Numerical simulations were performed by the Metropolis Monte Carlo method. The energy spectra show the boundary effect significantly in the chaotic cases, while other observables such as Po( S) and & (L) show a gradual transition as the system changes from regular to chaotic. The level repulsion phenomenon is found to persist in the intermediate distribution, which is in agreement with experiments. Since we have the joint distribution function expressed in simple analytic forms, it is challenging to calculate the nearest-neighbor spacing distributions and the Dyson-Mehta & statistics analytically. In order to examine the analytic properties of those observables, the techniques developed for fluctuation analysis in Ref. [ 161 will be equally powerful in the present case.
Acknowledgement I thank H. Kawai of the theory division at KEK for many stimulating discussions, and N. Tsuda for his help in the computation.
References [II I21
[31 [41 [51 [61 (71 [81 [91 IlO1 [I11 [I21 [I31 1141 [I51 [ 161
C.E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965). 0. Bohigas, M.-J. Giannoni, Chaotic Motion and Random Matrix Theories, in Lecture Note in Physics Vol. 209 (Springer, Berlin, 1984) pp. l-99. MC. Gutzwiller. Chaos in Classical and Quantum Mechanics (Springer, Berlin, 1990). SW. McDonald, A.N. Kaufman, Phys. Rev. Lea. 42 (1979) 1189. T.A. Brody, Lea. Nuovo Cim. 7 ( 1973) 482. M.V. Berry, M. Robnik, J. Phys. A 17 (1984) 2413. T. lshikawa, T. Yukawa, Phys. Rev. Lett. 54 (1985) 1617. E. Hailer, H. Kiippel, L.S. Cederbaum, Phys. Rev. Lett. 52 (1984) 1665. T.H. Seligman, J.M. Verbaarschot, M.R. Zimbauer, Phys. Rev. Lett. 53 (1984) 215. F.J. Dyson, J. Math. Phys. 3 (1962) 157. P. Pechukas, Phys. Rev. Lett. 51 (1983) 943. T. Yukawa, Phys. Rev. Lett. 54 (1985) 1883; Phys. Lett. A 116 (1986) 227. T. Yukawa, T. lshikawa, Prog. Theor. Phys. Suppl. 98 ( 1989) 157. J. Gibbons, T. Hermsen, Physica D 11 ( 1984) 337. S. Wojciechowski, Phys. Lett. A 111 (1985) 101. T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandy, S.S.M. Wong, Rev. Mod. Phys. 53 (1981) 385; J.B. French, V.K.B. Kota, A. Pandy, S. Tomsovic, Ann. Phys. 181 (1988) 198, and references therein.
LETTERS
A
Physics Letters A 245 ( 1998) 183-188
ELSBVIER
~~rn~~cal simulation of level statistics T. Yukawa a*b.l a Coordination Center for Research and Education. The Graduate ~aiversi~ forAdvanced Studies, ShonanVillage, Hayarna, Kanagawa 240-01, Japan h lnstitate of Particle and Nuclear Studies, KEK, High Energy Acceierator I-l Oho, Tsukuba-City. Ibaraki 30$, Japan
Research organization,
Received 14 October 1997; revised manuscript received 7 April 1998; accepted for publication 29 April 1998 Communicated by J.I? Vigier
Abstract The statistics of energy eigenvalues for a class of Hamiltonians written as H = FIe + tV is studied through the dynamics of levels regarding the perturbation parameter t as time. The statistical properties of eigenvalues are investigated by measuring quantities such as the nearest-neigh~r level spacing dist~bution and the Dyson-Metha & statistics for two types of ensembles known as orthogonal and unitary, and for three values of the system parameter controlling chaos. @ 1998 Elsevier Science B.V.
1. Introduction
Historically, the statistical properties of energy levels of an isolated system were initially studied by neutron resonance spectra of heavy atomic nuclei [ I]. They have drawn much attention recently in the context of quantum chaos, namely, the quantum mechanical property of a dynamical system which exhibits chaos in its classical trajectories [ 2,3]. The spectral property, which was previously regarded to be typical in a system with moderate complexity such as heavy atomic nuclei, has turned out to be rather universal in most isolated systems whose phase space of classical motion is dominated by chaotic trajectories. For example, the nearest-neighbor level spacing distribution Pa(S) as well as the Dyson-Meh~ statistics &(L) for any non-integrable system show the universal behav-
’ E-mail: yukawa~t~o~.kek.jp,
ior characteristic in neutron resonance spectra when the system is fully chaotic [4]. As the study of statistical property of energy spectra is extended to various isolated quantum systems like molecules, micro-clusters, or simple model systems such as billiards, types of statistics other than that of nuclei have been found. Various studies indicate the existence of distributions having properties ranging continuously from the one seen in nuclei to the uncorrelated distribution depending on the chaotic nature of the system. For example, for the nearest-neighbor level spacing distribution two extreme cases are known experimentally as well as theoretically, namely, the Wigner dis~ibution in the chaotic limit and the Poisson distribution for integrable systems. For distributions in between these two limits several dis~ibutions [S-9] have been proposed phenomenologically interpolating these two extreme cases. However, they fail to reproduce the level repulsion phenomenon which is known to persist in the intermediate cases. To carry out stud-
0375-9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00357-0
184
7: Yukawa/Physics
Letters A 245 (1998) 183-188
ies beyond phenomenological levels we ask ourselves, (i) is there any relation between the transition from the Poisson to the Wigner distribution and chaos of the system, and (ii) what is the statistical law in the level statistics which can explain the intermediate distributions seen in numerical simulations. For answering these questions we need a fundamental understanding of the level statistics based on standard statistical mechanical principles. There have been attempts in this direction which treat energy levels dynamically by regarding them as particles in one dimension, and interpreting a parameter in the Hamiltonian as time [ 10-121. In this way it becomes possible to understand the random matrix theory in terms of the universal language of physics, instead of giving an ensemble a priori. Also, the theory can predict distributions intermediate between the Wigner and the Poisson distribution by varying a control parameter [ 131. In the next section we briefly review the theory, which is followed by numerical simulations. The last section is devoted to examining the results and their significance.
2. Statistical mechanics
of levels
We consider an isolated system with time-reversal invariance for simplicity. Applications to a system without time-reversal invariance are straightforward, and they will be mentioned later without proof. Suppose a class of Hamiltonians written as
(1)
H=Ho+ty
where the perturbation strength r runs along the real axis. We assume that there are no degenerate levels, or if there are operators which commute with the Hamiltonian, we choose a subspace of the Hilbert space belonging to an irreducible representation of the operators. Representing the Hamiltonian by a basis set which is independent of the parameter t, for example, a complete set of eigenfunctions of Ha, the matrix X = X0 + tPa can be regarded as a solution of the classical N( N + 1) /2 free particle equations dX dt=
P,
dP dt=
0
with initial conditions,
(2)
X(0)
= x0,
P(0)= PO,
(3)
where X and P are matrices corresponding to H and V, respectively. For systems with time-reversal invariance Xu and PO are given by real symmetric matrices. Changing the representation basis by the orthogonal transformation 0, X= O-‘EC?,
P= CTIVO,
(4)
so that E is diagonal, the equations (Eq. (2) ) now become dE . ~+i[M,El
=V,
z+i[M,V]
where M E i(dQ/dr)O-‘. matrix G by
of the motion
=O,
Introducing
G=i[E,V]
(5)
a Hermitian
(6)
the equation of motion for G follows as dG dt +i[M,G]
=O.
A class of equations such as those for V and G is known as the Lax form. In this case the trace of matrices constructed by products of any powers and orderings of the matrices V and G are constants of the motion provided [V, G] # 0. By explicitly writing these equations (5)-(7), the equations of the motion are shown to obey the socalled generalized Calogero-Moser (GCM) equation, which has been studied extensively [ 14,151 as an example of a completely integrable system. It turns out to be obvious from the starting linear equation, i.e. the Schrodinger equation, that the GCM equation is completely integrable. To study the statistical property of energy spectra, especially the fluctuation of the level distribution, let us briefly describe the procedure employed in Ref. [ 131. We split a set of energy levels into subsets consisting of N sequential levels. Each subset is put in a box of size N by fixing the positions of the two end levels at 0 and N, so that the average level density is constant (= 1) . By these boundary conditions the parity and the translational invariance are broken explicitly, and most constants of the motion are not constant anymore, which implies the system loses complete integrability.
7: YukawdPhysics
Letters A 245 (1998)
From among the remaining constants of the motion we pick two quantities which have the lowest powers of V and G, and are expected to be important: Tr V* and Tr G*. The statistical equilibrium distribution is then given by the Boltzmann distribution, dw = exp
- $TrV’-
with the Lagrange multipliers 0 and y. The integral measure dT should be proportional to the phase space volume with the symmetry constraints on matrices X and P. For the time-reversal invariant system they are real symmetric and the measure is given by
n
( dJ&,,
dP,,n
1
(9)
l
nr,n
while for the time-reversal non-invariant case the matrices are complex and Hermitian constraints should be imposed. By changing the variables from {Xnm} and {P,l,} to {Ent,&} and {V&G&!, where {&}(a = are rotation angles of the orthogl,...,N(N-1)/2) onal transformation 0, and carrying out integrations over all non-observed variables except {En}, the joint probability distribution for {En) is given by P( (E,)) X
= const
(Em 1 i- (y/P)(E,,,
&I2 - En)*
4
>
’
(10)
where cy should be chosen to be either 1 for the orthogonal ensemble or 2 for the unitary ensemble depending on the time-reversal property of the system being either invariant or non-invariant, respectively. For the two limiting cases of parameter y/P we obtain the known distributions; the Poisson distribution, P( (E,)) when y/P
-+ const , + co, and the Gaussian
185
when y//3 --f 0 with NT//~ being fixed, where EC = ( l/N) 23, En. In these two cases various statistical observables are known analytically. We try to obtain the intermediate distribution numerically in the following section.
dT
gTrG2 >
X
183-188
(11) distribution,
3. Numerical simulation Although it is expected from the consideration on the two extreme limits that inte~edia~ dis~ibutions will be obtained by varying the parameter y/P in the range [ 0, co), here we check it for the orthogonal case (cu = I) and the unitary case (cu = 2) numerically. Energy spectra are simulated by the Metropolis Monte Carlo method so that the joint distribution function reaches Eq. (10) in statistical equilibrium. Initially N+l levelsareplacedatn (=O,l,...,N),andtwo end levels EO and EN are fixed at 0 and N, respectively, throughout the simulation to keep the average level density 1. We choose the parameter r - y/P = 100.0, 1.0, and 0.01 representing the state of system being regular, intermediate, and chaotic, respectively, for the orthogonal ensemble (1~ = 1) and the unitary ensemble ((Y = 2). The total number of levels N is taken to be 400. In Figs. l-3 three types of spectra are compared for the orthogonal ensemble. For the chaotic system (Fig. 3) the level repulsion is so strong that the level densities near the end points are larger than in the central region. Then the average level density deviates below from 1 systematically reflecting the higher density at both ends. Our numerical simulation for varying r indicates the levef repulsion effect becomes signi~cant when r gets smaller than 1. We also observed similar results for the unitary case (a = 2). In order to avoid the boundary effect we chose 200 levels between the level number 100 and 300 for measurements. In this region the level densities for all cases are almost constant, and levels are resealed as E;=c,E;+ioo+c2
(12)
(i=O,l,...,
200).
The parameters cl and cs are fixed with conditions E: = 0.0, E&, = 200.0 so that the level densities are 1. Then, we measure the nearest-neighbor level spacing distribution Po( S) defined by
Z Yukawa/Physics
186
Letters A 245 (1998) 183-188
Energy Spectra olthogad, r=o.ot
Energy Spectra olttloQalal, 1;100.0 400.0
400.0
f
300.0
300.0
%
200.0
W
loo.0
loo.0
Fig. I. Energy spectra of a regular time-reversal invariance (a = 1)
system
(f
= 100.0)
with
Fig. 3. Energy spectra time-reversal invariance
Energy Spectra or(hoowl, r-i.0 400.0
system
(r
= 0.01)
with
Nearest-NeighborSpacing
I-
300.0
of a chaotic (a = 1).
1.0
r
0.8 -
0.6 200.0
8 z
f
0.4 loo.0 02
0.0
L 0
100
200 LevalNlJmeW
Fig. 2. Energy spectra of an intermediate time-reversal
invariance
PO(S) 0: (number
300
system
I 400
(f = 1.0) with
(cy = 1)
of pairs of neighboring
with spacing between S and S + AS).
0
Fig. 4. Nearest-neighbor level spacing distributions for the orthogonal ensemble. The three lines correspond to three values of f: 100.0 (dot-dashed), 1.0 (dotted), and 0.01 (solid).
levels (13)
In Fig. 4 we show three observed distributions with r = 0.01 (solid), 1.0 (dotted) and 100.0 (dotdashed) for the orthogonal ( LY= 1) case with AS =
0.1. The transition from the Poisson to the Wigner distribution is clearly observed, although the distribution for r = 1.0 is already rather close to that of r = 0.01, which is almost same as the Wigner distribution. Fig. 5 shows PO(S) for the unitary ensemble
T. Yukawa/Physics
Letters A 245 (1998)
,
I
187
Delta3 Statiiics
Nearest-Neighbor Spacing l.Of
183-188
cm-lI
1
Fig. 5. Nearest-neighbor level spacing distributions for the unitary ensemble. The three lines correspond to three values of I‘: 100.0 (dot-dashed), 1.0 (dotted), and 0.01 (solid).
where we can clearly find stronger level repulsion compared to the orthogonal case as the theory predicts a quadratic repulsion for cy = 2, while a linear one for the cz = 1 case. The last measurements are for the Dyson-Mehta & statistics, which has been defined originally by
Fig. 6. Dyson-Mehta delta3 statistics for the orthogonal ensembles. The three lines correspond to three values of I‘: 100.0 (dot-dashed), I.0 (dotted), and 0.01 (solid).
Delta3 Statistii
Eo+L
QL)=(~!;;
-AEJ [N(E)
Bl*dE
EO
, > (14)
where N(E) = C, t?( E - E,,), and (. . .) indicates averaging over the lower end of the integration, Eo. Parameters A and B are fixed at the values giving the minimum &. For numerical simplicity we interchange the E, N axes and define N+M
a,~,=(~,& J
NO
[E(N)
-A’N-B’I*dN
, >
(15) where E(N) is the inverse function of N( E). Here, N is considered to be a real variable, while NO and M are integers. L&(M) obviously has the same property as the original Dyson-Mehta statistics. In this definition
Fig. 7. Dyson-Mehta delta3 statistics for the unitary ensembles. Three lines correspond to three values of f: 100.0 (dot-dashed), I.0 (dot), and 0.01 (solid).
(. . .) should be understood as averaging over No. The parameters A’ and B’ are determined so as to give the minimum &. In Figs. 6 and 7 we show the orthogonal case and the unitary case, respectively. A smooth
188
transition between in both ensembles.
7: Yukawa/Physics L.&em A 245 (1998) 183-188
the two extreme cases is apparent
4. Discussion Regarding the Hamiltonian H = Ho + tV as a solution of the classical motion of N( N + 1) /2 free particles when it is represented in matrix form, X(t) = X(0) + tP(O), we have used the standard technique of statistical mechanics to study the statistics of energy spectra. We have obtained the intermediate level statistics between the two known limits, for example, the Poisson and the Wigner distribution for the nearest-neighbor spacing distribution. Numerical simulations were performed by the Metropolis Monte Carlo method. The energy spectra show the boundary effect significantly in the chaotic cases, while other observables such as Po( S) and & (L) show a gradual transition as the system changes from regular to chaotic. The level repulsion phenomenon is found to persist in the intermediate distribution, which is in agreement with experiments. Since we have the joint distribution function expressed in simple analytic forms, it is challenging to calculate the nearest-neighbor spacing distributions and the Dyson-Mehta & statistics analytically. In order to examine the analytic properties of those observables, the techniques developed for fluctuation analysis in Ref. [ 161 will be equally powerful in the present case.
Acknowledgement I thank H. Kawai of the theory division at KEK for many stimulating discussions, and N. Tsuda for his help in the computation.
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