1

Elements of Random Matrix Theory and chaos-order

transition in finite quantum systems

1. MotivationWhen we pass from hadron-light nucleus, light nucleus-light nucleus collisions at low, middle and high energies to relativistic and

ultrarelativistic heavy ion collisions we get the new and unequal possibility: to create the high density and high temperature hadronic matter and to get the information on the properties of the matter under extreme conditions. In such new situation the volume of

information increases sharply as well as the background information. The Figure illustrates how the volume of the information increase with energy and the mass of beams. Some time the background information

,p +p,A

Pb+Pb √sNN = 17.3GeV NA 49 SPS

CERN

Au+Au √sNN = 200 GeV STAR RHIC BNL

PbPb √sNN= 5.5 TeV

Alice LHC CERN

From hadron-light nucleus, light nucleus-light nucleus collisions at low, medium and high energies to relativistic and ultrarelativistic heavy ion collisions we create a hadronic matter at high extreme conditions. In such new situation the volume of information increases drastically. A natural question arises as to how identify the useful signal that would be unambiguously associated with a certain physical process ?

3

• Statistical approach: recent theoretical analyses of the multiplicity fluctuations demonstrate that corresponding results are different in different ensembles and it is sensitive to conservation laws obeyed by a statistical system.

• Transport theory: the underlying concept of transport models is based on numerical solution of the Boltzmann equation. However, exact numerical solutions, for example, of the BUU equation are very difficult to obtain and, therefore, various approximate treatments have been developed. The only proof is that numerical solutions provide the asymptotic limit at equilibrium, which can be deduced from the Boltzmann equation (the BUU equation without potential and Pauli-blocking factor)

• Equation of motion of the Brownian particle

Brownian motion model for the matrix H

• Suppose that

(Dyson, 1962)

• We require that each be a

random variable

Standard statistical

mechanics

One considers an ensemble of identical physical systems, all governed by the same Hamiltonian but differing in initial conditions, and calculates thermo-dynamic functions by averaging over this ensemble.

Wigner proposed to consider ensembles of dynamic systems governed by different Hamiltonians with some common symmetry property.

A New Statistical Mechanics

This novel statistical approach focuses attention on the generic properties which are common to all members of the ensemble and determined by the

underlying fundamental symmetries.

Each of the elements is a random variable.

They are statistically independent of each other.

There are several useful statistical measures of spectral fluctuations.Among the most popular are nearest-neighbor spacing distributions P(s). Their asymptotic forms for large N cannot be written in a simple form, but they are surprisingly well approximated by the simple expressions obtained for N=2.

Joint probability of the independent matrix elements

Joint probability ofthe eigen-values

HTHTJHHH 1,0,,

)()()()( 222212121111 HPHPHPHP

*2112 , ijij HHHH

For A=0

2

2

DC

Wigner distribution

N

iiEEES

1

)()(

We define a staircase function

),'(')'()(1

EEdEESENE N

i

Giving the number of points on the energy axis Which are below or equal to E. Here

1

0)(x

0x

1xfor

for

NEEEE .....0 321

We separate N(E) in a smooth part and the reminder that will define the fluctuating part )(EN fl

i1iis XX

; N1i ,E ii XX);()()( ENEXEN fl

)(EX

1E 2E 5E … .

N

.1;2

)(2

2

qessPs

w

.0;)( qesP s;0s

;)1()(1

qqsq

qBr esqsP ;1

21

q

q q

q

Quantum chaos: the matrix is the Hamiltonian. Though the Hamiltonian is not random (it is given), it is observed and conjectured (but not understood) that (when the corresponding classical mechanical problem is chaotic), its spectrum has the same distribution as the spectrum of a random matrix. Random Hamiltonians belong to Gaussian ensembles, like GUE, GOE or GSE depending on their symmetries

Disordered systems: the matrix is the Hamiltonian. It is naturally a random matrix due to disorder. We average over samples or over variations of an external parameter (a magnetic field). Again, the Hamiltonian can belong to Gaussian ensembles, like GUE, GOE or GSE depending on their symmetries

order chaos

chaos order

totoRMT successfully describes the spectral RMT successfully describes the spectral fluctuation properties of complex atomic nuclei, fluctuation properties of complex atomic nuclei, complex atoms, complex molecular, quantum complex atoms, complex molecular, quantum dots and biological systems. dots and biological systems.

11. M. L. Mehta, Random Matrices, 2004 (Amsterdam: Elsevier).. M. L. Mehta, Random Matrices, 2004 (Amsterdam: Elsevier).

22. T. A. Brody et al., Rev. Mod. Phys. . T. A. Brody et al., Rev. Mod. Phys. 5353 (1981) 385. (1981) 385.

33.H.A. Weidenmuller, G.E.Mitchell, Rev. Mod. Rhys. .H.A. Weidenmuller, G.E.Mitchell, Rev. Mod. Rhys. 8181 (2009)539.(2009)539.

22220

2

22 c

z

b

y

a

xm

m

pH

Heiss,Nazmitdinov,Radu, Phys.Rev.C52(1995) R1179, 3032

32

20

2Yr

m

3321 )2/1()1(321

nnnnnn

321 )( pnnnqN shell

0321 )2/)([ pqpnnnq

0)2/[ pqN shell

3pnqn

Heiss, Nazmitdinov, Radu, PRL 72 (1994) 2351

E

λ

Heiss, Nazmitdinov, Radu, PRL 72, 2351 (1994)

E

λ

gap

gapshell

shell

shell

Pronouncedshell structure

(quantum numbers)

Pronouncedshell structure

(quantum numbers)

Shell structureabsent

Shell structureabsent

closed trajectory(regular motion)

trajectory does not close

]eV/[

nm22.1

3 kineff EkTm

h

p

h

If the carrier motion in a solid is limited in a layer of a thickness of the order of the carrier de Broglie wavelength (λ), one will observe effects of size quantizationsize quantization.

Quantum dots (QD)Quantum dots (QD) are small boxes (2 – 10 nm on a side, corresponding to 10 to 50 atoms in diameter), contained in semiconductor, and holding a number of electrons.

One can consider the QD as a tiny laboratories in which quantum and classical effects of electron-electron interaction can be studied.

Mesoscopic system: quantum fluctuations are very important !

Constant interaction model:

i

iC

NeNU

2)(

22

CeCeE NNadd // 221

macroscopic energy

The Hamiltonian for the axially symmetric (x = y 0) two-electron quantum dot in magnetic field reads

Dineykhan & Nazmitdinov, Phys.Rev. B55 (1997) 13707

Introducing the relative r = r1 - r2 and center-of-mass R = (r1+ r2)/2 coordinates;

the conjugated momenta P = M* dR/dt, p = dr/dt, where M* = 2m* and = m*/2,

the Hamiltonian separates into the center-of-mass and relative-motion terms

For a perpendicular magnetic field B|| z we choose the gauge

and obtain

where is the Larmor frequency.

Radionov, Aberg, Guhr, Phys.Rev. E70 (2004) 036207

stateszyx 400;0),1(),1( 20

220

2

Simonović and Nazmitdinov, PRB 67 (2003) 041305(R)

Poincaré surfaces of sections z = 0, pz > 0 of the classical relative motion of two electrons in an axially symmetric QD with: (a) ωz/Ω = 5/2, (b) ωz/Ω = 2, (c) ωz/Ω = 3/2, (d) ωz/Ω = 1, (e) ωz/Ω = 2/3 and (f) ωz/Ω = 1/2. The sections (b), (c) and (f)

indicate that for the corresponding ratios ωz/Ω the system is integrable.

;0, zyx

Poincaré surfaces of sections z = 0, pz > 0 at ωz/Ω : a)=5/2, (b) = 2, (c) = 3/2, (d) = 1, (e) = 2/3,(f) = 1/2.

m=0

Coulomb problem:

V.A. Fock, Zs.f.Physik 98, 145 (1935)

Simonović and Nazmitdinov, PRB 67 (2003) 041305(R) - For specific values of the ratio ωz/Ω the Hamiltonian Hrel (in the 3D approach) becomes integrable !!!

20

2220 2

zLL mc

eB;1/)1 z

Beside the energy and , the additional integral of motion is and the problem has a spherical symmetry, O(3).

zl2zl

;/)2 Kz

,220

22 zAc z

New integrals of motion similar to Runge-Lentz vector are found at thesespecific values of magnetic field !

;2/1,2K 20

2/ KzL

zLB 022

00

1/20

2 zzL !!!

,00 l

kRW

0*0 m

l

,2 *cm

BeL

2The potential surface of the effective potential and

50,9607.2/,5.2/,0 00 WLz Rm

The potential barrier with the ridge alongthe line z=0 is high enough to separate themotions in vicinities of the minima.

Birman, Nazmitdinov, Yukalov, Phys.Rep.526(2013)1

• QDs, as functioning elements for electronics or spintronics, should be able to carry through electrical and, possibly, spin current. Therefore, QD should be opened !

• Electronics, as well as future quantum computers are based on transmission and transformation (correction) of the shape of signals. This means that one has to be able to create the devices with predictable ability to transform

non-linear characteristics.

• The sensitivity of the dots to external magnetic field, can be made very different by making use of different shape (depth and width) of the confining potential.

Classical (Hamiltonian) chaos

Regular Chaotic

diffusive dot

ballistic dot

2D Schrodinger equation

2

R

RR

i

2

'

* )2

'()2

'()(kmnmn

kkt

kktkC )exp()( q

qqmn ikLakt

)exp()()( ikLkdkCLP 2

)()( mn

mn LtLP

222

)exp()(2

)exp()()( LEiEtE

dEikLkdktLt mnmnmn

2mn

mntG

Nazmitdinov, Pichugin, Rotter, and Seba, Phys.Rev.B66 (2002) 085322

Study of quantum correlations in finite quantum systems under external fields provide answers on fundamental aspects: symmetry breaking phenomena,

stability of these systems, link between quantum and clasical description and nature of quantum coherence in transport.

The RMT approach is free from various assumptions

concerning the background of the measurements and it provides reliable information about correlations induced by external or internal perturbations