Symmetries of the local densities

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Symmetries of the local densities

• S.G.Rohozinski, J. Dobaczewski, W. Nazarewicz
• University of Warsaw, University of Jyväskylä
• The University of Tennessee, Oak Ridge National
  Laboratory
• XVI Nuclear Physics Workshop Pierre Marie
  Curie
• Superheavy and exotic nuclei
• Kazimierz Dolny, Poland, 23. 27. September 2009

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The matter is

• A contemporary standard approach to the theory of
  nuclear structure The density functional theory
• Starting point H nuclear effective Hamiltonian
• Original approach HFB LDA
• 
  d3rd3r (r,r)
  
• (HFB)
• 
  
• 
  (LDA)

• Generalization (a new starting point)
• Construction of the Hamiltonian density
• (archetype The Skyrme Hamiltonian density)

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Outline

• What is the matter?
• Density matrices and densities
• Generalized matrices and HFB equation
• Transformations of the density matrices
• Symmetries of the densities
• General forms of the local densities with a given
  symmetry
• Final remarks

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Density p-h and p-p matrices
(the breve representation of the original
antisymmetric pairing tensor)
r, r position vectors, s, s1/2,-1/2 spin
indices, t,
t 1/2,-1/2 isospin indices
Time and charge reversed matrices
Properties
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Spin-isospin structure of density matrices
Nonlocal densities p-h, scalar and vector p-p,
scalar and vector k0 (isoscalar), k1, 2, 3
(isovector) Properties
t00, t1,2,31
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Local densities
(Tensor is decomosed into the trace Jk
(scalar), antisymmetric part Jk (vector) and
symmetric traceless tensor )

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Generalized density matrix
Generalized mean field Hamiltonian
Lagrange multiplier matrix
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where the p-h and p-p mean field Hamiltonians are
HFB equation
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Transformations of density matrices
Hermitian one-body operator in the Fock space
gg - a single-particle operator
Unitary transformation generated by G
Transformation of the nucleon field operators
under U
(Black circle stands for integral and sum
)
Transformation of the density matrices
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• Transformation of the generalized density matrix
• Generalized transformation matrix
• Transformed density matrix
• Transformed mean field Hamiltonian

• Two observations
• A symmetry U of H (HUUHUH ) can be broken in
  the mean field approximation
• The symmetry of the density matrix (and the mean
  field Hamiltonian) is robust in the iteration
  process

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Symmetries of the densities

• The symmetry of the mean field, if appears, is,
  in general, only a sub-symmetry of the
  Hamiltonian H
• When solving the HFB equation the symmetry of the
  density matrix should be assumed in advance
• There are physical and technical reasons for the
  choice of a particular symmetry of the density
  matrix
• Considered symmetries
• 1. Spin-space
  symmetries
• - Orthogonal and rotational
  symmetries, O(3) and SO(3)
• - Axial symmetry SO(2), axial
  and mirror symmetry SO(2)xSz
• - Point symmetries D2h,
  inversion, signatures Rx,y,z(p), simplexes
  Sx,y,z
• (in the all above cases
  , which
  means that
• p-h and p-p densities are
  transformed in the same way)
• 2. Time reversal T
• 3. Isospin symmetries
• - p-n symmetry (no
  proton-neutron mixing)
• - p-n exchange symmetry
• 
  

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General forms of the densities with a given
symmetry

• The key construction of an arbitrary isotropic
  tensor field as a function of the position
  vector(s) r, (r )
• (Generalized Cayley-Hamilton Theorem)
• A simple example
• The O(3) symmetry (rotations and inversion)
• Independent scalars
• Scalar nonlocal densities
• (Pseudo)vector nonlocal densities

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• Local densities
• Real p-h
• Complex isovector p-p
• Vanishing pseudovector p-h and p-p
• Gradients of scalar functions

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• Differential local densities
• Scalar
• Vector
• (er is the unit vector in radial direction,
• Jk stands for the antisymmetric part of the
  (pseudo)tensor densities)
• All other differential densities vanish.

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• The SO(3) symmetry (rotations alone)
• (There is no difference between scalars and
  pseudoscalars,
• vectors and pseudovectors, and tensors and
  pseudotensors)
• Nonlocal densities
• Scalar (without any change)
• Vector (pseudovector)

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• Local densities
• Scalar
• Vector

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• Traceless symmetric tensor
• Axial symmetry (symmetry axis z)

SO(2) vector (in the xy plane) SO(2) scalar, S3
pseudoscalar (perpendicular to the xy plane)
Tensor fields are functions of and
separately
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Final remarks

• The nuclear energy density functional theory is
  the basis of investigations of the nuclear
  structure at the present time (like the
  phenomenological mean field in the second half of
  the last century)
• Knowledge of properties of the building blocks
  of the functional densities with a given
  symmetry is of the great practical importance