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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 MAT I C S
(1)
[P,Q] = trace ( [p,q] R)
if P = trace (pR), Q = trace (qR)
(2)
[Q+q, R]=0
Saturation condition
DYNAMICS
Two-body Hamiltonian:
Exact GDM equation of motion:
Generalized mean field:
R, S, W – operators in Hilbert space
STRATEGY
• 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
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)
ANHARMONICITY
Next terms:
Time-reversal
invariance
And so on …
Soft modes !
J.F.C. Cocks et al. PRL 78 (1997) 2920.
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
Octupole energies
B(E3) values in Xe isotopes
W. Mueller et al. 2006
M.P. Metlay et al. PRC 52 (1995) 1801
CONSERVATION LAWS
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 = mA
ROTATION
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,…
ISOLATED ROTATIONAL BAND
Collective Hamiltonian
- transformation
GDM equation
Adiabatic (slow)
rotation
Linear term: cranking model
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
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
PROBLEMS
•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
(partly
solved)
THANKS
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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)