1. 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

2. 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 …

3. 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, …

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

5. [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

6. 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

7. 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

8. 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)

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

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

11. 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)

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

13. 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)

14. COLLECTIVE MODES (RPA)
SOLUTION
First order
(harmonic)

15. 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 !

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

17. 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

19. 2 3 3

20. 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

21. 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

22. 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

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

24. 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,…

25. 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

26. 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 …

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

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

29. 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

30. 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

31. 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)