R.G.Nazmitdinov

1. R.G.Nazmitdinov Elements of Random Matrix Theory and chaos-order transition in finite quantum systems 1

2. 1. Motivation When 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 Motivation 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 ? ,p +p,A • PbPb √sNN= 5.5 TeV • Alice LHC CERN Au+Au √sNN = 200 GeV STAR RHIC BNL Pb+Pb √sNN = 17.3GeV NA 49 SPS CERN

3. 3

4. Macroscopic approach

5. How to detect signal ? • 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)

6. Coulomb gas Molecular dynamics: • Equation of motion of the Brownian particle

7. Brownian motion model for the matrix H • Suppose that (Dyson, 1962) • We require that each H be a random variable

8. 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.

9. Joint Probability Joint probability of the independent matrix elements Joint probability of the eigen-values 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.

10. Random distribution of matrix elements

11. Random distribution of matrix elements For A=0

12. Random distribution of eigenvalues

13. Random distribution of eigenvalues Wigner distribution

14. We define a staircase function N Giving the number of points on the energy axis Which are below or equal to E. Here … . for for We separate N(E) in a smooth part and the reminder that will define the fluctuating part

15. Brody distribution

16. Some conclusions

17. Some conclusions 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

18. RMT successfully describes the spectral fluctuation properties of complex atomic nuclei, complex atoms, complex molecular, quantum dots and biological systems. to to order chaos chaos order 1. M. L. Mehta, Random Matrices, 2004 (Amsterdam: Elsevier). 2. T. A. Brody et al., Rev. Mod. Phys. 53 (1981) 385. 3.H.A. Weidenmuller, G.E.Mitchell, Rev. Mod. Rhys. 81 (2009)539.

19. Octupole deformation and Shell effects Heiss,Nazmitdinov,Radu, Phys.Rev.C52(1995) R1179, 3032

20. Single-particle spectrum for AHO

21. Chaotic vs Regular Heiss, Nazmitdinov, Radu, PRL 72 (1994)2351 E λ

22. Spectra and NND Heiss, Nazmitdinov, Radu, PRL 72, 2351 (1994) E λ

23. Pronounced shell structure (quantum numbers) Shell structure absent shell gap shell gap shell closed trajectory (regular motion) trajectory does not close

24. 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. Basics 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 ofsize quantization. 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 !

25. Conductance

26. Coulomb blockade: Constant interaction model: macroscopic energy

27. Two-electron quantum dot Dineykhan & Nazmitdinov, Phys.Rev. B55 (1997) 13707 The Hamiltonian for the axially symmetric (x = y0) two-electron quantum dot in magnetic field reads

28. Two-electron quantum dot Introducing the relative r = r1 -r2and 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.

29. Spectrum 2e QD in magnetic field

30. Two-electron quantum dot Radionov, Aberg, Guhr, Phys.Rev. E70 (2004) 036207

31. Integrable cases 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.

32. Quantum vs Classical m=0 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.

33. Runge-Lenz vector Coulomb problem: V.A. Fock, Zs.f.Physik 98, 145 (1935)

34. Integrable cases Simonović and Nazmitdinov, PRB 67 (2003) 041305(R) - For specific values of theratio ωz/Ωthe HamiltonianHrel(in the 3D approach) becomes integrable !!! Beside the energy and,the additional integral of motion is and the problem has a spherical symmetry, O(3). New integrals of motionsimilar to Runge-Lentz vector are found at these specific values of magnetic field !

35. !!!

36. The potential surface of the effective potential and The potential barrier with the ridge along the line z=0 is high enough to separate the motions in vicinities of the minima.

37. a precursor of a phase transition !!! Geometrical crossover - Birman, Nazmitdinov, Yukalov, Phys.Rep.526(2013)1

38. Transport through quantum dots • 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.

39. Classical (Hamiltonian) chaos Regular Chaotic

40. Quantum Billiards diffusive dot 2D Schrodinger equation ballistic dot

41. the poles of S matrix and

42. Quantum Billiards correlation function and power spectrum

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

44. Summary 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