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Gaussian processes and universal parametric decorrelations of wavefunctions
27 May 1996
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 215 (1996) 21-25
Gaussian processes and universal parametric decorrelations of wavefunctions D. Mitchell, Y. Alhassid, D. Kusnezov Center for Theoretical Physics. Sloane Physics Laboratory Yale Universit_v,New Haven, CT 06520. USA
Received 21
November 1995; accepted for publication 28 February 1996 Communicated
by P.R. Holland
Abstract Gaussian processes that depend on a parameter lead to universal parametric correlations of spectra and wavefunctions after an appropriate scaling of the parameter. The most general Gaussian process is described by the exponent 77 which characterizes the diffusive behavior of its energy levels. We discuss the scaling required for such processes and show that they lead to the same parametric correlators when considered as functions of (AX)“/* where X is the scaled parameter. Using simulations of Gaussian processes, we demonstrate this scaling and universality for the parametric decorrelation of the wavefunctions as measured by their overlap, and compute the respective distributions of that overlap. PACS: 05,4S.+b;
05.4O.Sj;
03.65.-w;
24.6O.L~
Random matrix theory (RMT) [ 1,2] has been successful in describing the statistical fluctuations of spectra and matrix elements in chaotic [3,4] or complex systems as well as in weakly disordered systems [ 51. For such systems that conserve time reversal symmetry, the local fluctuations are well described by the Gaussian orthogonal ensemble (GOE) while for systems without time reversal symmetry the relevant ensemble is the Gaussian unitary ensemble (GUE). In physical applications the system under consideration often depends on an external parameter x and it is of interest to study the correlation of the energy levels as a function of the parameter [6]. It was recently shown [7] that various spectra1 parametric correlations of a chaotic or disordered system are universal if the parameter is scaled appropriately. In particular the autocorrelator of the unfolded level velocities C(X - x’) = (aei/ax)(x) (aEi/aX)(X’) WaS stud-
0375.9601/96/$12.00
@ 1996 Elsevier Science
PII SO375-9601(96)00232-O
B.V. All rights reserved
ied as a function of x - x’. Using the supersymmetry method, it was found that under the scaling R = mx, the quantity C( x - x’)/C(O) is universal. Here C (0) 3 (&,/a~)~ is the generalized conductivity. The universality was shown for both a random matrix model H(x) = HI + X& where HO is a fixed matrix and HI is a GOE (or GUE) matrix, and for the motion of a particle in a weakly disordered system with an external parameter (potential or a magnetic field). It was also shown [ 81 that the density of states correlator (at different x’s) corresponds to time-dependent density correlations of the completely integrable Sutherland Hamiltonian [ 91, where the coordinates of the N particles play the role of the eigenvalues in the former problem. This connection was understood in terms of a Fokker-Planck equation that is satisfied by a quantity constructed from the wavefunction of the Sutherland mode1 [ 10,l 1 ]
22
D. Mitchell et al. /Physics
We define the Gaussian process (GP) as a natural extension of the random matrix ensembles of Wigner and Dyson in cases where the matrices depend on a parameter x. The most general GPs can be characterized by an exponent 7 through the diffusive behavior of their unfolded energy levels ALE?K Axv. In Ref. [ 121 we demonstrated the universality of wavefunction parametric cot-relators in chaotic and disordered systems using the framework of the GP (with 77= 2). In this paper we derive the universality of wavefunction correlators for a genera1 GP using Dyson’s Brownian motion model. For 77 f 2 the C (0) scaling has to be replaced by a different one, but the correlators (that can be defined for 7 # 2) are still universal and coincide with the v = 2 cot-relators if considered as a func-tion of (AX) VI*, where i is the appropriately scaled parameter. By studying the eigenfunctions’ overlap I(tii(X) I (cli(x’))12, we demonstrate that the wavefunctions decorrelate (before scaling) at x - x’ N N-‘/q, where N is the dimension of the random matrices. We also calculate the universal distributions of this eigenstates’ overlap at typical separations of the scaled parameter. We consider a system whose Hamiltonian H depends on a continuous parameter x and is chaotic or disordered for all values of x. We assume that its Hamiltonian is chosen as a representative of a stochastic process of random matrices H(x) . The process is assumed to be Gaussian with zero mean and translational-invariant so that its joint probability distribution is given by
X
dxdx’ Tr[H(x) J
,
K(x - x’) H(x’)]
(1)
>
Letters A 215 (1996) 21-25
Hij(X)
=O,
Hij(X)
H&1(X’)
a2
-f(X
-
w
X’)
,I?) rj,kl
(2)
’
where g$T ) = Si&aj[f 6$8j&,gbtT2)= 28i[6j&and % f(x) is normalized such that f( 0) = 1. The process defined by (2) is stationary since the correlation function f is invariant under translations in the parameter n. When viewed as a matrix with continuous indices x, x’, f (x - x’) is the inverse of K in ( 1). p = 1 is the case where the system has time reversal symmetry and H(x) are real symmetric matrices, while p = 2 is the case of broken time reversal symmetry where H(x) are complex Hermitian matrices. The parameter a determines the average level spacing A via a/A = m/r. A particular representation of a GP was introduced in Ref. [ 131to study the statistics of avoided crossings in chaotic systems. Various two-point correlation functions involving the spectra Ei(X), Ei(X’) and/or the wavefunctions fii (x), t,bi(x’), can be calculated from a two-point matrix ensemble obtained from Eq. (1) after integrating out H(x”) for all x” # x, x’. Since the process is Gaussian, it is straightforward to calculate this twomatrix distribution, P(KH’)
0: exp
-2a2(l
P _ ?)
( x Tr(H2 + H’2 - 2fHH’)
(3)
, >
where H and H’ denote the Hamiltonian at x and x’, respectively. With an appropriate (N-dependent) scaling of x, all two-point correlation functions become universal in the limit of large N. Here we derive this by using a particular GP - the Brownian motion mode1 of Dyson r 14,10,111, fiij = -yHij + fij (t) .
and its measure D [ H(x) ] E n, dH(x) is a product over the continuous variable x of the corresponding Gaussian ensemble measure dH( x) . An example of such a process is the continuous matrix mode1 of Ref. [8], used to show the connection to the Sutherland model. Since cumulants of order higher than second vanish, a GP is alternatively defined by its first two moments
In (4) y is a “friction” dom forces fij(t) =O,
(4) coefficient and Fij ( t) are ran-
Fij(t) F;(P)
= rgL<{,S(t-t’)
.
(5)
Eqs. (4) and (5) describe a GP as a function of r with f ( t - t’) = exp( -~]t - t’l ) and are equivalent to a Fokker-Planck equation for the distribution P( H, I)
D. Mitchell er al. /Physics
(6) Since the process H(t)is to be stationary, we choose the initial distribution to be the equilibrium one, P(H) lr;exp(-/3TrH2/2a’), so that P(H,t) = P (H) is just the corresponding GE at all times t. For the equilibrium distribution to be a solution of (6)) it is necessary that r satisfies the fluctuation-dissipation theorem s)t.) r/2y = m, or I’/y =
u?/p.
(7)
The process therefore depends on two parameters, a* and I’. We would like to determine their N dependence such that correlation functions of (appropriately scaled) eigenvalues and/or eigenfunctions would be independent of N in the limit of large N. Dyson [ 141 showed that the eigenvalues of H(t)describe a diffusive Coulomb gas, that, when disturbed from equilibrium, will come into local equilibrium within a microscopic time scale I N (yN>-‘. This time scale is distinguished from the macroscopic time scale t - y-’ which is needed for global equilibrium. If we choose the microscopic time scale to be independent of N, i.e. y x 1/N, we expect to find N-independent correlators. A simple way to realize this condition is to make local properties such as the average level spacing and the amplitude r of the random forces independent of N. The first condition of N-independent average level spacing is equivalent to working with the unfolded energies and is satisfied by the scaling a2 0; N. Both conditions guarantee that for an N-independent mean level spacing, the mean squared variations of the matrix elements (A Hij) 2 = giJ$i f A t and of the unfolded energies invoking conclude follows
(Aci)2 = 2TAt are independent of N. By the fluctuation-dissipation theorem (7) we again the scaling y c( 1/N. From (4) it then that
This GP leads to N-independent correlations in the limit of large N. To show that any GP leads to universal correlations we note from (3) that the two-point correlations are determined by f E f(x - x’) (and N) . To obtain the same correlations as from the above Brownian process, we need to identify
Letters A 215 (1996) 21-25
23
f(x - x’) ++ exp (-$I
- i’)
From (9) it is clear that for large N we need to know only the leading-order behavior of f, which we assume to be f(X - X’) =
1 -
KIX
-
X’I’
,
( IO)
for x close to x’. The interpretation of the parameter 7 will be clarified later. Eq. (9) is then satisfied in the limit of large N, providing we scale x to .? = N]/~.x and identify K12 -
.?‘I”
*
#It
-
f’l
.
(11)
Our analysis shows that the correlations are determined by ~I.52 - i’/T = N( 1 - f) alone. The nonuniversal constant K can be absorbed by an additional resealing of P. For that purpose, we calculate the quantity (Aei)2 z [EL(a) - ~i(?‘)]~ for small AB = .Z’- .?. From At-i M ($i(x)lH(x’) -H(x)l~/~i(x))/A, and using
(3),
(4//h-2)~lA.f17.
(2) and (ll), we find (h~i)~ z Therefore, we can eliminate K by
scaling P with (A~~/lA~i-J’r)~l~ (calculated in the limit Ax 4 0). One can combine it with the N’/q scaling of the original x, to obtain a single scaling
The leading-order
behavior of f. t 13)
is now independent A+ c( 1AxlV ,
of
K.
Since ( 14)
we observe that (for small Ax) the energy levels execute a diffusion as a function of the parameter x with an exponent 0. For r] = 1 (Dyson process) we obtain the ordinary diffusion. For q = 2, ( 12) reduces to the scaling introduced in Ref. [ 71, X = [ ( a6i/d.x)2] “‘x. As argued earlier, the parametric correlators depend only on N( 1 - f) = ifl&\X - X’I” (see Eq. (13)); they are therefore universal as a function of 1%- X’\‘1/2 and in particular independent of 7. To calculate the
24
D. Mitchell
Letters A 215 (19%)
21-25
r~.~~l....1...~l..~~1~.~““‘.1
N -
A
et al. /Physics
1.0
g
z G z x
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X m -
I....“““““‘.“““‘1
?_
ool .o.o
0.0 0.5
1.0
1.5
’
2.0
= cos xHi + sin xH2 ,
J
1
Fig. 2. The parametric overlap as a function (circles) and 7 = 2 (solid line) GPs.
Fig. I. Top: the parametric overlap 1(@i(X) 1 @i(O))/* as a function of x of the GOP ( 13) for (widest to narrowest) N = 50.100,200 and 300. Bottom: the same quantity as a function of the scaled parameter X (see ( 11) ). Notice that the overlap becomes universal.
H(x)
0.5
p/z
x
universal correlations, we can therefore such as the q = 2 process [ 151
’
’
’
0
I
.
where HI, Hz are uncorrelated GE matrices belonging to the same symmetry class. This process is translational-invariant with f(x - x’) = cos (x - x’). According to (12) and (13) the scaling x ---f X = d-fix would lead to universal correlations. To demonstrate the necessity of an Ndependent scaling we compute the parametric overlap I(rcli(x) I @i(x’))12 th a t measures the decorrelation of wavefunctions. Notice that since this quantity is independent of the particular eigenstate chosen, we can average over i to obtain better statistics. Fig. 1 (top) shows this quantity as a function of the unscaled variable x - x’ computed from simulations of the GOP ( 15) for N = 50 (widest), 100, 200 and 300 (narrowest). In the simulations we used about 30 samples and averaged over the middle third of the spectrum. We see that the decorrelation is not universal and is more rapid for larger N. In Fig. 1 (bottom) the same quantity is shown but now as a function of the scaled variable X = max. The decorrelation is now universal and independent of N. By solving the Brownian motion Eq. (4) (for 77= 1) or by using
for 11 = 1
#““I”“1
” x=0.5
x=0.18
1 5-
use any GP
(15)
“.I”“1
of .@
a i
J
-1
l-l
0 Ll
=
0
1
<~k,(x)J~,(o)>
Fig. 3. The distribution of wavefunction overlaps, P(u), with u = (Jli( x) 1k( I’)), calculated for several values of the scaled parameter 1 from simulations of the GOP ( 13). These distributions are universal and independent of N.
a construction of discrete GPs (for 0 < v < 2), it is possible to verify that this correlator is the same for all q’s if plotted against Ix - x’/V/~ [ 161. Fig. 2 shows that the overlap correlator for Dyson’s GP (v = 1) coincides with the v = 2 correlator. The distribution of wavefunction amplitudes also behaves in a universal manner. In Fig. 3, we show the universal distributions P((+k(x) 1 &(x’>)) for typical values off - X’. At X - R’ = 0, the distribution
D. Mitchell et ~1. /Physics
Letters A 215 (19Y6,6)21-25
25
We acknowledge H. Attias for useful discussions. This work was supported in part by the Department of Energy Grant DE-FG02-9 1ER40608.
References
0.0
0.5
1.0
1.5
X Fig. 4. The level velocity cot-relator C(x)/C(O) versus X from simulations of ( 13). The results are identical to those of Ref. 17 1.
is obviously a a-function centered at 1. As R - 2 increases, the distribution widens gradually, with its centroid moving towards zero. Finally, in Fig. 4, we show for reference the level velocity correlator that we calculate from the q = 2 GP (14). The curve obtained is identical to that of Ref. [ 71. This particular correlator is meaningful only for 7 = 2 since it requires the existence of the level derivative. In conclusion, we have characterized the most general translational-invariant Gaussian process that respects a given symmetry according to the diffusive behavior of its energy levels AE? c( AC?. The parametric wavefunction correlators remain universal upon an appropriate scaling of the parameter if considered as functions of IX - _x’I”/~. In particular we have calculated the universal distributions of the wavefunctions’ overlap at typical values of the scaled parameter.
[I] E.P. Wigner, Ann. Math. 53 (1951) 36; 62 (1955) 548. [2] M.L. Mehta, Random matrices, 2nd Ed. (Academic Press. New York, 1991). [3] 0. Bohigas, M.J. Giannoni and C. Schmit, Phys. Rev. Lctt. 52 (1984) 1. [41 M.C. Gutzwiller, Chaos in classical and quantum mechanics (Springer, New York, 1990) (Academic Press, New York, 1965). [51 B.L. Altshuler and B.1. Shklovskii, Sov. Phys. JETP 91 (1986) 220. 161 J. Goldberg, U. Smilansky, M.V. Berry, W. Schweizer, G. Wunner and G. Zeller, Nonlinearity 4 ( 199 I ) I. [71 B.D. Simons and B.L. Altshuler, Phys. Rev. Len. 70 ( 1993) 4063; Phys. Rev. B 48 (1993) 5422. [f31 B.D. Simons, PA. Lee and B.L. Altshuler, Phys. Rev. Lea. 70 ( 1993) 4122; 72 ( 1994) 64; Nucl. Phys. B 409 (1993) 487. 191 B. Sutherland, J. Math. Phys. 12 (1971) 246; 12 (1971) 251; Phys. Rev. A 4 (1971) 2019; 5 (1972) 1372; F.J. Dyson, J. Math. Phys. 13 (1972) 90. 1101 C.W.J. Beenakker, Phys. Rev. Lett. 70 (1993) 4126. [Ill 0. Narayan and B.S. Shastry, Phys. Rev. Len 71 (1993) 2106. 1121 Y. Alhassid and H. Attias, Phys. Rev. Lett. 74 ( 1995) 4635. [131 M. Wilkinson, J. Phys. A 22 (1989) 2795. [I41 F.J. Dyson, J. Math. Phys. 3 (1962) 1191. 1151 E.A. Austin and M. Wilkinson, Nonlinearity 5 t 1992) 1137. I161 H. Attias, Ph.D. Thesis (Yale, 1995).
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 215 (1996) 21-25
Gaussian processes and universal parametric decorrelations of wavefunctions D. Mitchell, Y. Alhassid, D. Kusnezov Center for Theoretical Physics. Sloane Physics Laboratory Yale Universit_v,New Haven, CT 06520. USA
Received 21
November 1995; accepted for publication 28 February 1996 Communicated
by P.R. Holland
Abstract Gaussian processes that depend on a parameter lead to universal parametric correlations of spectra and wavefunctions after an appropriate scaling of the parameter. The most general Gaussian process is described by the exponent 77 which characterizes the diffusive behavior of its energy levels. We discuss the scaling required for such processes and show that they lead to the same parametric correlators when considered as functions of (AX)“/* where X is the scaled parameter. Using simulations of Gaussian processes, we demonstrate this scaling and universality for the parametric decorrelation of the wavefunctions as measured by their overlap, and compute the respective distributions of that overlap. PACS: 05,4S.+b;
05.4O.Sj;
03.65.-w;
24.6O.L~
Random matrix theory (RMT) [ 1,2] has been successful in describing the statistical fluctuations of spectra and matrix elements in chaotic [3,4] or complex systems as well as in weakly disordered systems [ 51. For such systems that conserve time reversal symmetry, the local fluctuations are well described by the Gaussian orthogonal ensemble (GOE) while for systems without time reversal symmetry the relevant ensemble is the Gaussian unitary ensemble (GUE). In physical applications the system under consideration often depends on an external parameter x and it is of interest to study the correlation of the energy levels as a function of the parameter [6]. It was recently shown [7] that various spectra1 parametric correlations of a chaotic or disordered system are universal if the parameter is scaled appropriately. In particular the autocorrelator of the unfolded level velocities C(X - x’) = (aei/ax)(x) (aEi/aX)(X’) WaS stud-
0375.9601/96/$12.00
@ 1996 Elsevier Science
PII SO375-9601(96)00232-O
B.V. All rights reserved
ied as a function of x - x’. Using the supersymmetry method, it was found that under the scaling R = mx, the quantity C( x - x’)/C(O) is universal. Here C (0) 3 (&,/a~)~ is the generalized conductivity. The universality was shown for both a random matrix model H(x) = HI + X& where HO is a fixed matrix and HI is a GOE (or GUE) matrix, and for the motion of a particle in a weakly disordered system with an external parameter (potential or a magnetic field). It was also shown [ 81 that the density of states correlator (at different x’s) corresponds to time-dependent density correlations of the completely integrable Sutherland Hamiltonian [ 91, where the coordinates of the N particles play the role of the eigenvalues in the former problem. This connection was understood in terms of a Fokker-Planck equation that is satisfied by a quantity constructed from the wavefunction of the Sutherland mode1 [ 10,l 1 ]
22
D. Mitchell et al. /Physics
We define the Gaussian process (GP) as a natural extension of the random matrix ensembles of Wigner and Dyson in cases where the matrices depend on a parameter x. The most general GPs can be characterized by an exponent 7 through the diffusive behavior of their unfolded energy levels ALE?K Axv. In Ref. [ 121 we demonstrated the universality of wavefunction parametric cot-relators in chaotic and disordered systems using the framework of the GP (with 77= 2). In this paper we derive the universality of wavefunction correlators for a genera1 GP using Dyson’s Brownian motion model. For 77 f 2 the C (0) scaling has to be replaced by a different one, but the correlators (that can be defined for 7 # 2) are still universal and coincide with the v = 2 cot-relators if considered as a func-tion of (AX) VI*, where i is the appropriately scaled parameter. By studying the eigenfunctions’ overlap I(tii(X) I (cli(x’))12, we demonstrate that the wavefunctions decorrelate (before scaling) at x - x’ N N-‘/q, where N is the dimension of the random matrices. We also calculate the universal distributions of this eigenstates’ overlap at typical separations of the scaled parameter. We consider a system whose Hamiltonian H depends on a continuous parameter x and is chaotic or disordered for all values of x. We assume that its Hamiltonian is chosen as a representative of a stochastic process of random matrices H(x) . The process is assumed to be Gaussian with zero mean and translational-invariant so that its joint probability distribution is given by
X
dxdx’ Tr[H(x) J
,
K(x - x’) H(x’)]
(1)
>
Letters A 215 (1996) 21-25
Hij(X)
=O,
Hij(X)
H&1(X’)
a2
-f(X
-
w
X’)
,I?) rj,kl
(2)
’
where g$T ) = Si&aj[f 6$8j&,gbtT2)= 28i[6j&and % f(x) is normalized such that f( 0) = 1. The process defined by (2) is stationary since the correlation function f is invariant under translations in the parameter n. When viewed as a matrix with continuous indices x, x’, f (x - x’) is the inverse of K in ( 1). p = 1 is the case where the system has time reversal symmetry and H(x) are real symmetric matrices, while p = 2 is the case of broken time reversal symmetry where H(x) are complex Hermitian matrices. The parameter a determines the average level spacing A via a/A = m/r. A particular representation of a GP was introduced in Ref. [ 131to study the statistics of avoided crossings in chaotic systems. Various two-point correlation functions involving the spectra Ei(X), Ei(X’) and/or the wavefunctions fii (x), t,bi(x’), can be calculated from a two-point matrix ensemble obtained from Eq. (1) after integrating out H(x”) for all x” # x, x’. Since the process is Gaussian, it is straightforward to calculate this twomatrix distribution, P(KH’)
0: exp
-2a2(l
P _ ?)
( x Tr(H2 + H’2 - 2fHH’)
(3)
, >
where H and H’ denote the Hamiltonian at x and x’, respectively. With an appropriate (N-dependent) scaling of x, all two-point correlation functions become universal in the limit of large N. Here we derive this by using a particular GP - the Brownian motion mode1 of Dyson r 14,10,111, fiij = -yHij + fij (t) .
and its measure D [ H(x) ] E n, dH(x) is a product over the continuous variable x of the corresponding Gaussian ensemble measure dH( x) . An example of such a process is the continuous matrix mode1 of Ref. [8], used to show the connection to the Sutherland model. Since cumulants of order higher than second vanish, a GP is alternatively defined by its first two moments
In (4) y is a “friction” dom forces fij(t) =O,
(4) coefficient and Fij ( t) are ran-
Fij(t) F;(P)
= rgL<{,S(t-t’)
.
(5)
Eqs. (4) and (5) describe a GP as a function of r with f ( t - t’) = exp( -~]t - t’l ) and are equivalent to a Fokker-Planck equation for the distribution P( H, I)
D. Mitchell er al. /Physics
(6) Since the process H(t)is to be stationary, we choose the initial distribution to be the equilibrium one, P(H) lr;exp(-/3TrH2/2a’), so that P(H,t) = P (H) is just the corresponding GE at all times t. For the equilibrium distribution to be a solution of (6)) it is necessary that r satisfies the fluctuation-dissipation theorem s)t.) r/2y = m, or I’/y =
u?/p.
(7)
The process therefore depends on two parameters, a* and I’. We would like to determine their N dependence such that correlation functions of (appropriately scaled) eigenvalues and/or eigenfunctions would be independent of N in the limit of large N. Dyson [ 141 showed that the eigenvalues of H(t)describe a diffusive Coulomb gas, that, when disturbed from equilibrium, will come into local equilibrium within a microscopic time scale I N (yN>-‘. This time scale is distinguished from the macroscopic time scale t - y-’ which is needed for global equilibrium. If we choose the microscopic time scale to be independent of N, i.e. y x 1/N, we expect to find N-independent correlators. A simple way to realize this condition is to make local properties such as the average level spacing and the amplitude r of the random forces independent of N. The first condition of N-independent average level spacing is equivalent to working with the unfolded energies and is satisfied by the scaling a2 0; N. Both conditions guarantee that for an N-independent mean level spacing, the mean squared variations of the matrix elements (A Hij) 2 = giJ$i f A t and of the unfolded energies invoking conclude follows
(Aci)2 = 2TAt are independent of N. By the fluctuation-dissipation theorem (7) we again the scaling y c( 1/N. From (4) it then that
This GP leads to N-independent correlations in the limit of large N. To show that any GP leads to universal correlations we note from (3) that the two-point correlations are determined by f E f(x - x’) (and N) . To obtain the same correlations as from the above Brownian process, we need to identify
Letters A 215 (1996) 21-25
23
f(x - x’) ++ exp (-$I
- i’)
From (9) it is clear that for large N we need to know only the leading-order behavior of f, which we assume to be f(X - X’) =
1 -
KIX
-
X’I’
,
( IO)
for x close to x’. The interpretation of the parameter 7 will be clarified later. Eq. (9) is then satisfied in the limit of large N, providing we scale x to .? = N]/~.x and identify K12 -
.?‘I”
*
#It
-
f’l
.
(11)
Our analysis shows that the correlations are determined by ~I.52 - i’/T = N( 1 - f) alone. The nonuniversal constant K can be absorbed by an additional resealing of P. For that purpose, we calculate the quantity (Aei)2 z [EL(a) - ~i(?‘)]~ for small AB = .Z’- .?. From At-i M ($i(x)lH(x’) -H(x)l~/~i(x))/A, and using
(3),
(4//h-2)~lA.f17.
(2) and (ll), we find (h~i)~ z Therefore, we can eliminate K by
scaling P with (A~~/lA~i-J’r)~l~ (calculated in the limit Ax 4 0). One can combine it with the N’/q scaling of the original x, to obtain a single scaling
The leading-order
behavior of f. t 13)
is now independent A+ c( 1AxlV ,
of
K.
Since ( 14)
we observe that (for small Ax) the energy levels execute a diffusion as a function of the parameter x with an exponent 0. For r] = 1 (Dyson process) we obtain the ordinary diffusion. For q = 2, ( 12) reduces to the scaling introduced in Ref. [ 71, X = [ ( a6i/d.x)2] “‘x. As argued earlier, the parametric correlators depend only on N( 1 - f) = ifl&\X - X’I” (see Eq. (13)); they are therefore universal as a function of 1%- X’\‘1/2 and in particular independent of 7. To calculate the
24
D. Mitchell
Letters A 215 (19%)
21-25
r~.~~l....1...~l..~~1~.~““‘.1
N -
A
et al. /Physics
1.0
g
z G z x
1.0
0.5
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
X m -
I....“““““‘.“““‘1
?_
ool .o.o
0.0 0.5
1.0
1.5
’
2.0
= cos xHi + sin xH2 ,
J
1
Fig. 2. The parametric overlap as a function (circles) and 7 = 2 (solid line) GPs.
Fig. I. Top: the parametric overlap 1(@i(X) 1 @i(O))/* as a function of x of the GOP ( 13) for (widest to narrowest) N = 50.100,200 and 300. Bottom: the same quantity as a function of the scaled parameter X (see ( 11) ). Notice that the overlap becomes universal.
H(x)
0.5
p/z
x
universal correlations, we can therefore such as the q = 2 process [ 151
’
’
’
0
I
.
where HI, Hz are uncorrelated GE matrices belonging to the same symmetry class. This process is translational-invariant with f(x - x’) = cos (x - x’). According to (12) and (13) the scaling x ---f X = d-fix would lead to universal correlations. To demonstrate the necessity of an Ndependent scaling we compute the parametric overlap I(rcli(x) I @i(x’))12 th a t measures the decorrelation of wavefunctions. Notice that since this quantity is independent of the particular eigenstate chosen, we can average over i to obtain better statistics. Fig. 1 (top) shows this quantity as a function of the unscaled variable x - x’ computed from simulations of the GOP ( 15) for N = 50 (widest), 100, 200 and 300 (narrowest). In the simulations we used about 30 samples and averaged over the middle third of the spectrum. We see that the decorrelation is not universal and is more rapid for larger N. In Fig. 1 (bottom) the same quantity is shown but now as a function of the scaled variable X = max. The decorrelation is now universal and independent of N. By solving the Brownian motion Eq. (4) (for 77= 1) or by using
for 11 = 1
#““I”“1
” x=0.5
x=0.18
1 5-
use any GP
(15)
“.I”“1
of .@
a i
J
-1
l-l
0 Ll
=
0
1
<~k,(x)J~,(o)>
Fig. 3. The distribution of wavefunction overlaps, P(u), with u = (Jli( x) 1k( I’)), calculated for several values of the scaled parameter 1 from simulations of the GOP ( 13). These distributions are universal and independent of N.
a construction of discrete GPs (for 0 < v < 2), it is possible to verify that this correlator is the same for all q’s if plotted against Ix - x’/V/~ [ 161. Fig. 2 shows that the overlap correlator for Dyson’s GP (v = 1) coincides with the v = 2 correlator. The distribution of wavefunction amplitudes also behaves in a universal manner. In Fig. 3, we show the universal distributions P((+k(x) 1 &(x’>)) for typical values off - X’. At X - R’ = 0, the distribution
D. Mitchell et ~1. /Physics
Letters A 215 (19Y6,6)21-25
25
We acknowledge H. Attias for useful discussions. This work was supported in part by the Department of Energy Grant DE-FG02-9 1ER40608.
References
0.0
0.5
1.0
1.5
X Fig. 4. The level velocity cot-relator C(x)/C(O) versus X from simulations of ( 13). The results are identical to those of Ref. 17 1.
is obviously a a-function centered at 1. As R - 2 increases, the distribution widens gradually, with its centroid moving towards zero. Finally, in Fig. 4, we show for reference the level velocity correlator that we calculate from the q = 2 GP (14). The curve obtained is identical to that of Ref. [ 71. This particular correlator is meaningful only for 7 = 2 since it requires the existence of the level derivative. In conclusion, we have characterized the most general translational-invariant Gaussian process that respects a given symmetry according to the diffusive behavior of its energy levels AE? c( AC?. The parametric wavefunction correlators remain universal upon an appropriate scaling of the parameter if considered as functions of IX - _x’I”/~. In particular we have calculated the universal distributions of the wavefunctions’ overlap at typical values of the scaled parameter.
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