symmetries of the local densities

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Symmetries of the local densities

S.G.Rohoziński, J. Dobaczewski, W. NazarewiczUniversity 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:

Properties:

Time and charge reversed matrices:

(the „breve” representation of the original antisymmetric pairing tensor)

r, r’ – position vectors, s, s’=+1/2,-1/2 – spin indices,

t, t’ =+1/2,-1/2 – isospin indices

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Spin-isospin structure of density matrices

Nonlocal densities

p-h, scalar and vector:

p-p, scalar and vector:

k=0 (isoscalar), k=1, 2, 3 (isovector)Properties:

t0=0, t1,2,3=1

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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 matricesHermitian one-body operator in the Fock space:

Unitary transformation generated by G:

Transformation of the nucleon field operators under U:

Transformation of the density matrices:(Black circle stands for integral and sum )

ts,3d r

g=g+ - a single-particle operator

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Transformation of the generalized density matrix:

Generalized transformation matrix

Transformed density matrix

Transformed mean field Hamiltonian

Two observations1. A symmetry U of H (HU=UHU+=H ) can be broken

in the mean field approximation:

2. 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(π), 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