old ideas through new eyes: generalized density matrix revisited vladimir zelevinsky nscl / michigan...
TRANSCRIPT
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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
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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 …
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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, …
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GENERALIZED DENSITY MATRIX
SUPERSPACE – Hilbert space (many-body states) + single-particle space
Still an operator in many-body Hilbert space
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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
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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
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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
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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)
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TIME-DEPENDENT FORMULATION
Time – Dependent Mean Field:
Thouless – Valatin form
Self – consistent ground state energy
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BCS – HFB theory
Doubling single-particle space:
Effective self-consistent field
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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)
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INFORMATION ENTROPY of EIGENSTATES (a) function of energy; (b) function of ordinal number
ORDERING of EIGENSTATES of GIVEN SYMMETRY SHANNON ENTROPY AS THERMODYNAMIC VARIABLE
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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)
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COLLECTIVE MODES (RPA)
SOLUTION
First order(harmonic)
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A N H A R M O N I C I T Y
Next terms:
And so on …
Time-reversal invariance
Soft modes !
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J.F.C. Cocks et al. PRL 78 (1997) 2920.
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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
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3 3
2
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B(E3) values in Xe isotopes
Octupole energies
W. Mueller et al. 2006
M.P. Metlay et al. PRC 52 (1995) 1801
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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
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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
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EXAMPLE: CENTER-OF-MASS MOTION
“Band” – motion as a whole,
M - unknown inertial parameter
After transformation:
Pushing model
SOLUTION
M = m A
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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
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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
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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…
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HIGH – SPIN ROTATION
,
Calculate commutators in semiclassical approximation
Macroscopic Euler equation with fullyMicroscopic background
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TRIAXIAL ROTOR
CLASSICAL SOLUTION
CONSTANTSOF MOTION
TIME SCALE
SOLUTION(ELLIPTICAL FUNCTIONS)
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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
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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
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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)