Unitary Correlation Operator MethodThe Unitary Correlation Operator Method for Interacting Fermi...
Transcript of Unitary Correlation Operator MethodThe Unitary Correlation Operator Method for Interacting Fermi...
The Uni
tary
Cor
rela
tion
Ope
rato
rM
etho
d
for
Inte
ract
ing
Ferm
iGas
es
Rob
ertR
oth
(GSI
)
Rau
isch
holz
haus
enX
I@
Ros
tock
,Jul
y1-
4,19
99
Problem & Overview#2
Q: How to treat short-range correlations in-duced by the strong repulsive core of the2body interaction within a many-bodysystem?
A: Unitary Correlation Operator Method!
...some keywords
★ elements: correlation operator/function,correlated states, correlated opera-tors/Hamiltonian, equation of state,correlated density matrix
★ approximations: 2body approximation,3body contributions, effective higher or-der contributions
★ application I: homework problem forneutron matter, equation of state, 2bodydensity & occupation numbers
★ application II: phenomenological GCT-potential for nuclear matter, equation ofstate
The
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tary
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Interacting Fermi GasAssumptions and Trial State
#5
many-body trial state� free fermi gas & unitary correlation operator provides
trial state for the interacting fermi gas� free one-body states are momentum eigenstates with
momentum����
and � -fold spin-isospin degeneracy������ �� ���� � ���� � � �� �� �� � � �� � ��������� �� � �� �
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next steps...� calculate correlated energy expectation value
!5 � � !# ��76 �� !# 98;: � � # �� !6 �� # <8;:� minimize the energy as a function(al) of the correlation
function ➼ optimal correlator� calculate any observable you like, e.g. equation of state,
densities...
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Homework Problem
Potential & Optimal Correlator#7
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Contributions of the
3rd Cluster Order#11
3body correlations
✘ no closed analytic expression for 3body correlated oper-ators available
➼ product approximation: partial summation of theBaker-Hausdorff expansion of the correlated operator
➼ generalized coordinate transformation: correlations for-mulated by a norm conserving 3body coordinate trans-formation
3body approximation of the EoS
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Correlated
Densities & Occupation Numbers#15
correlated 2body density matrix
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Fourier transform of correlated 1body density
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Summary & Outlook#20
what was shown here...
✔ UCOM is a new and promising approach totame short-range correlations in the context ofquantum many-body problems
✔ unitary coordinate transformation of states oroperators to introduce correlations
✔ in 2body approximations the correlated inter-action has a simple analytic form
✔ higher order contributions can be calculatedexplicitly or included effectively
✔ convincing results for the homework problemand nuclear matter with the GCT potential
coming up next...
★ application to the zoo of atomic many-bodysystem: liquid � He, liquid
�He and droplets,
trapped Bose-Einstein condensates...
★ nuclear many-body systems with realistic in-teractions: tensor correlations!!!
★ use of the correlated realistic interactions as in-put for simulation of dynamics: nuclear colli-sions (FMD) and BEC dynamics.