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

K I N E M A T I C S

[P,Q] = trace ( [p,q] R)

if P = trace (pR), Q = trace (qR)

(1)

(2) [Q+q, R]=0

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>

Single-particle density matrix of the ground state

[R, H]=0

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

From occupation numbers in the shell model solution (dots)From thermodynamic entropy defined by level density (lines)

Gaussian level density

839 states (28 Si)

COLLECTIVE MODES (RPA)

SOLUTION

First order(harmonic)

A N H A R M O N I C I T Y

Next terms:

And so on …

Time-reversal invariance

Soft modes !

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

Effect formally exists (in the limit of small frequencies)

but we need the condensate of phonons, therefore

consideration beyond RPA is needed.

Single-particle strength is strongly fragmented. This leads to the suppression of the enhancement effect.

Monopole phonons – Poisson distributionMultipole phonons (L>0) – no exact solution

Looking for collective enhancement of the atomic EDM

3 3

2

B(E3) values in Xe isotopes

Octupole energies

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

Constant of motion

[p , W{R}] = W{[p , R]}

Rotated field = field of rotated density Self-consistency

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

R O T A T I O NAngular momentum conservation,

Non – Abelian group, X – Euler angles

Transformation to the body-fixed frame

But they can depend on I=(Je)

Scalars

Transformed EQUATION:

Phonons, quasiparticles,…

Coriolis andcentrifugal effects

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

Deformed mean field

Rotational part

Angular momentumself-consistency

Even system:

Tensor of inertia

Pairing:

Nonaxiality, “centrifugal” corrections

Coriolis attenuation problem, wobbling, vibrational bands …

Transition rates, Alaga rules…

HIGH – SPIN ROTATION

,

Calculate commutators in semiclassical approximation

Macroscopic Euler equation with fullyMicroscopic background

TRIAXIAL ROTOR

CLASSICAL SOLUTION

CONSTANTSOF 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 WFind self-consistently W for given microscopic Hamiltonian (analytically for anisotropic harmonic oscillator withresidual 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)