Generalized Density Matrix Revisited Old Ideas Through New Eyes: Generalized Density Matrix Revisited Vladimir Zelevinsky NSCL / Michigan State University Baton Rouge, LSU, Mardi Gras Nuclear Physics Workshop February 19, 2009

THEORETICAL PROBLEMS Hamiltonian 2-body, 3-body… Space truncation Continuum Method of solution Symmetries Approximations Mean field (HF, HFB) RPA, TDHF Generator coordinate Cranking model Projection methods …

How to solve the quantum many-body problem Full Schroedinger equation Shell-model (configuration interaction) Variational methods ab-initio Mean field (HF, DFT) BCS, HFB RPA, QRPA, … Monte-Carlo, …

GENERALIZED DENSITY MATRIX SUPERSPACE – Hilbert space (many-body states) + single-particle space Still an operator in many-body Hilbert space

[P,Q] = trace ( [p,q] R) [Q+q, R]=0 Saturation condition K I N E M A T I C S [P,Q] = trace ( [p,q] R) (1) if P = trace (pR), Q = trace (qR) [Q+q, R]=0 (2) Saturation condition

D Y N A M I C S Two-body Hamiltonian: Exact GDM equation of motion: Generalized mean field: R, S, W – operators in Hilbert space

S T R A T E G Y Microscopic Hamiltonian Collective band Nonlinear set of equations saturated by intermediate states inside the band Symmetry properties and conservation laws to extract dependence of matrix elements inside the band on quantum numbers

Hartree – Fock approximation Collective space Ground state |0> [R, H]=0 Single-particle density matrix of the ground state Self-consistent field Single-particle basis |1) Single-particle energies e(1) Occupation numbers n(1)

TIME-DEPENDENT FORMULATION Time – Dependent Mean Field: Thouless – Valatin form Self – consistent ground state energy

BCS – HFB theory Doubling single-particle space: Effective self-consistent field

MEAN FIELD OUT OF CHAOS Between Slater determinants |k> Complicated = chaotic states (look the same) Result: Averaging with Mean field as the most regular component of many-body dynamics Fluctuations, chaos, thermalization (through complexity of individual wave functions)

INFORMATION ENTROPY of EIGENSTATES (a) function of energy; (b) function of ordinal number ORDERING of EIGENSTATES of GIVEN SYMMETRY SHANNON ENTROPY AS THERMODYNAMIC VARIABLE

EFFECTIVE TEMPERATURE of INDIVIDUAL STATES Gaussian level density 839 states (28 Si) EFFECTIVE TEMPERATURE of INDIVIDUAL STATES From occupation numbers in the shell model solution (dots) From thermodynamic entropy defined by level density (lines)

COLLECTIVE MODES (RPA) SOLUTION First order (harmonic)

A N H A R M O N I C I T Y Soft modes ! Next terms: Time-reversal invariance And so on … Soft modes !

J.F.C. Cocks et al. PRL 78 (1997) 2920.

Single-particle strength is strongly fragmented. Looking for collective enhancement of the atomic EDM Single-particle strength is strongly fragmented. This leads to the suppression of the enhancement effect. Effect formally exists (in the limit of small frequencies) but we need the condensate of phonons, therefore consideration beyond RPA is needed. Monopole phonons – Poisson distribution Multipole phonons (L>0) – no exact solution

2 3 3

B(E3) values in Xe isotopes Octupole energies B(E3) values in Xe isotopes W. Mueller et al. 2006 M.P. Metlay et al. PRC 52 (1995) 1801

C O N S E R V A T I O N L A W S [p , W{R}] = W{[p , R]} Constant of motion [p , W{R}] = W{[p , R]} Self-consistency Rotated field = field of rotated density

RESTORATION of SYMMETRY If the exact continuous symmetry is violated by the mean field, there appears a Goldstone mode, zero frequency RPA solution and a band; entire band has to be included in external space X – collective coordinate(s) conjugate to violated P Transformation of the intrinsic space , [ s , P ] = 0 New equation: [ s + H (P – p), r ] =0

EXAMPLE: CENTER-OF-MASS MOTION “Band” – motion as a whole, M - unknown inertial parameter After transformation: Pushing model SOLUTION M = m A

Angular momentum conservation, R O T A T I O N Angular momentum conservation, Non – Abelian group, X – Euler angles Transformation to the body-fixed frame But they can depend on I=(Je) Scalars Transformed EQUATION: Coriolis and centrifugal effects Phonons, quasiparticles,…

I S O L A T E D R O T A T I O N A L B A N D Collective Hamiltonian - transformation GDM equation Adiabatic (slow) rotation Linear term: cranking model

Nonaxiality, “centrifugal” corrections Deformed mean field Rotational part Angular momentum self-consistency Even system: Tensor of inertia Pairing: Transition rates, Alaga rules… Nonaxiality, “centrifugal” corrections Coriolis attenuation problem, wobbling, vibrational bands …

HIGH – SPIN ROTATION , Calculate commutators in semiclassical approximation Macroscopic Euler equation with fully Microscopic background

TRIAXIAL ROTOR TIME SCALE SOLUTION (ELLIPTICAL FUNCTIONS) CLASSICAL SOLUTION CONSTANTS OF MOTION TIME SCALE SOLUTION (ELLIPTICAL FUNCTIONS)

MICROSCOPIC SOLUTION The same for W Separate harmonics using elliptical trigonometry Solve equations for matrix elements of r in terms of W Find self-consistently W for given microscopic Hamiltonian (analytically for anisotropic harmonic oscillator with residual quadrupole-quadrupole forces) Find moments of inertia

P R O B L E M S (partly solved) Interacting collective modes - spherical case - deformed case LARGE AMPLITUDE COLLECTIVE MOTION SHAPE COEXISTENCE TWO- and MANY-CENTER GEOMETRY Group dynamics - interacting bosons - SU(3)… EXACT PAIRING; BOSE - systems * CHAOS AND KINETICS

THANKS Spartak Belyaev (Kurchatov Center) Abraham Klein (University of Pennsylvania) Dietmar Janssen (Rossendorf) Mark Stockman (Georgia State University) Eugene Marshalek (Notre Dame) Vladimir Mazepus (Novosibirsk) Pavel Isaev (Novosibirsk) Vladimir Dmitriev (Novosibirsk) Alexander Volya (Florida State University)