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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
Con
cept
ofth
e
Uni
tary
Cor
rela
tion
Ope
rato
rM
etho
d#3
unita
ryco
rrel
atio
nop
erat
or
�� ���
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�� �
��
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% corr
elat
ion
func
tion
)+*�-,�
.0/1324
2
576 89 6:;< =>?A@9�B:DCB<89 B:
corr
elat
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ates
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elat
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ave
func
tion
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ysy
stem
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ors
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tent
ial
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spac
e GTS�VU�'S
W )YX�VU�Z
Cor
rela
ted
Ham
ilton
ian
in2b
ody
App
roxi
mat
ion
#4
Ham
ilton
ian
in2b
ody
appr
oxim
atio
n
G �'
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G �� ���
G �� ���
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ectio
ns
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1= �?))) 9�B: ?) 9 B:2
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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������ �� ���� � ���� � � �� �� �� � � �� � ��������� �� � �� �
� many-body trial state��"!# is a correlated Slater determi-
nant of all one-body states with $ �� � $&% �('��)!# � * �� # � * + , �� �.- /0/0/1 �� �32 �4
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...
Equa
tion
ofSt
ate
for
anIn
tera
ctin
gFe
rmiG
asin
2bod
yA
ppro
xim
atio
n#6
corr
.Ham
ilton
ian
ener
gype
rpa
rtic
leas
func
tion
of
� �
% dire
ctte
rms
ofor
ders
�� � ,
�� � and
�� � ,res
p.
��� �
,
� and
��� �(
�'
� ����� � )
% exch
ange
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sgr
oww
ith
smal
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ower
sof
� �
than
the
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ctte
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% high
dens
ity
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vior
isde
term
ined
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eef
fect
ive
mas
ste
rms
(
�� � ���� � )
G � �'�*
G � �'
� � �
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�G S�VU��G ��VU�
�
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h.
�G ����
G � �
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� � ��
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� G��N� �� ,��� �G �N� �� ,��� xc
h.
Homework Problem
Potential & Optimal Correlator#7
Bethe’s homework problem (1973)� neutron matter ( � � �
) interacting via the repulsive coreof the
-����component of the Reid potential� ��� � � � ���� � MeV fm ����������� � fm � - � � 8 �
parametrization & matter-optimal correlator� choose a parametrization for the correlation function ���
and minimize the energy expectation value !5���� ��� � atsome given density ( � � � fm ��� )
� � ��� � � ��� ! �" # $ ��� �&%'� ��������� 8 " �)(
➼
� � � � � fm" � � ��* � fm+ � * *-,
0 1 2 3 4. [fm]
0
0.05
0.1
0.15
0.2
0.25
0.3
[fm]
/10 2 .43657.98 : 2 .43
Hom
ewor
kPr
oble
m
Loca
lPot
entia
ls&
Eff.
Mas
sC
orre
ctio
ns#8
loca
lpot
entia
lsef
fect
ive
mas
sco
rrec
tions
01
23
4
B [fm
]
-250255075100
[MeV
fm
� ]
01
23
4
B [fm
]
0
0.2
0.4
0.6
0.81
1.2
B�39 B:
B�� 39 B:
B�� +9�B:
�' � ��
9�B:1=
�' � �29�B:
% core
ofth
epo
tent
ial
isdr
amat
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duce
d
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tion
alm
ediu
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nge
repu
l-si
onby
the
kine
tic
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ntia
l
% mas
sre
duct
ion
wit
hin
the
rang
eof
the
inte
ract
ion
% enha
ncem
ento
fthe
kine
tic
ener
gyby
addi
tion
alra
dial
mas
s
Hom
ewor
kPr
oble
m
Equa
tion
ofSt
ate
in2b
ody
App
rox.
#9
00.
51
1.5
2� [f
m
�� ]
02
2.5
33.
5
���
[fm
�� ]
0
500
1000
1500
2000
[MeV
]
00.
10.
20.
30.
40.
5
01
1.5
2
� �
[fm
�� ]
050100
150
200
250
300
[MeV
]
�� ��
noco
rrel
ator
���� w
ith
mat
ter-
opti
mal
corr
elat
or
MC
Var
iatio
n[P
RD
16(7
7)30
81]
� /
�
FHN
C[P
L61
B(76
)393
]/[P
RA
16(7
7)12
58]
✔re
duct
ion
ofth
een
ergy
to
� �
ofth
eun
corr
elat
edva
lue
✔go
odag
reem
ent
wit
hre
fere
nce
for
low
dens
itie
s
�����
fm
N�
✘”o
verb
indi
ng”
ofup
to20
%at
high
dens
itie
s
� fm
N�
➼2b
ody
appr
oxim
atio
ngi
ves
res-
onab
lede
scri
ptio
nfo
rlo
wde
n-si
ties
but
need
sco
rrec
tion
sfo
rhi
ghde
nsit
ies
➼3b
ody
cont
ribu
tion
sre
leva
nt!!!
Hom
ewor
kPr
oble
m
Ener
gyC
ontr
ibut
ions
inD
etai
l#1
0
ener
gyco
ntri
butio
nsdi
rect
&ex
chan
gete
rms
00.
51
1.5
2
� [fm
�� ]
02
2.5
33.
5
� �
[fm
�� ]
0
100
200
300
400
500
[MeV
]
00.
51
1.5
2
� [fm
�� ]
02
2.5
33.
5� �
[fm
�� ]
-1000
100
200
300
400
500
[MeV
].
���
� � �1� � 2
�,3
� +
����
dire
ctex
chan
ge
% loca
lte
rms
(
�� � �� )
give
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nco
ntri
buti
on,
effe
ctiv
em
ass
term
(
�� � ���� � )w
ins
athi
ghde
nsit
ies
% exch
ange
cont
ribu
tion
isin
the
or-
der
of10
%of
the
dire
ctte
rms
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
!5 � � � � !5 ��� � ! � � � � � !� � � �! � � � � � � ��
����� ��' � � - � � � - � ! � � � � � �� - ��� �� - � �� xch. � xxch.
� contains all terms up to order� �' , especially all local
3body contributions� 3body effective mass contributions of order
�� ' are ne-glected
✘ high-dimensional numerical integrations over “rough”functions necessary (Monte Carlo methods)
Hom
ewor
kPr
oble
m
3bod
yC
ontr
ibut
ions
toth
eEo
S#1
2
00.
51
1.5
2� [f
m
�� ]
02
2.5
33.
5
���
[fm
�� ]
0
500
1000
1500
2000
[MeV
]
00.
10.
20.
30.
40.
5
01
1.5
2
� �
[fm
�� ]
050100
150
200
250
300
[MeV
]
�� ��
���� w
ith
mat
ter-
opti
mal
corr
.
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��
����1�,3�
��1� +�
��;���
��
MC
Var
iatio
n[P
RD
16(7
7)30
81]
� /
�
FHN
C[P
L61
B(76
)393
]/[P
RA
16(7
7)12
58]
✔lo
calc
ontr
ibut
ions
ofth
e3b
ody
orde
rra
ise
the
ener
gy
✔va
riat
iona
lup
per-
boun
dpr
op-
erty
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ored
✔ve
rygo
odde
scri
ptio
nof
the
low
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ity
regi
me
✔po
tent
ialc
ontr
ibut
ion
prod
uces
dom
inan
teff
ect
➼3b
ody
orde
rse
ems
tobe
suffi
-ci
ent
for
are
ason
able
appr
oxi-
mat
ion
Form
ulat
ion
of
Effe
ctiv
eH
ighe
rO
rder
Con
trib
utio
ns#1
3
dens
ity-d
epen
dent
corr
elat
ors
% corr
elat
orpa
ram
eter
sw
ith
dim
en-
sion
ofa
leng
th(i
.e.
� and
� )ar
esc
aled
asa
func
tion
ofde
nsit
y
�������
��
� �� ��
�) X� , ���
�� ��' ����� �
✔ef
fect
ive
pote
ntia
lsan
dm
asse
sbe
-co
me
dens
ity-
depe
nden
t
✔co
nsis
tent
sim
ulat
ion
ofhi
gher
or-
der
effe
cts
for
all
corr
elat
edob
-se
rvab
les
(ene
rgie
s,de
nsit
ies.
..)
3bod
yco
ntac
tint
erac
tion
% loca
l3b
ody
cont
ribu
tion
sap
prox
-im
ated
bya
sing
le3b
ody
cont
act
inte
ract
ion
S� �� �'�
�� , ��
� �� ,��
�
✔ex
pect
atio
nva
lue
give
san
addi
-ti
onal
term
prop
orti
onal
to
�� � fo
rth
eEo
S G � � X� 'G � ������ �
✘ap
plic
able
only
for
the
ener
gyex
-pe
ctat
ion
valu
e,no
pred
icti
onfo
rot
her
obse
rvab
les
% the
free
para
met
er� or
� has
tobe
fixed
byfit
ting
the
ener
gyto
high
eror
der
resu
lts
orto
som
ein
depe
nden
tcal
cula
tion
s
Hom
ewor
kPr
oble
m
Effe
ctiv
eH
ighe
rO
rder
Con
trib
utio
nsto
the
EoS
#14
00.
51
1.5
2� [f
m
�� ]
02
2.5
33.
5
���
[fm
�� ]
0
500
1000
1500
2000
[MeV
]
00.
10.
20.
30.
40.
5
01
1.5
2
� �
[fm
�� ]
050100
150
200
250
300
[MeV
]
�� ��
���� w
ith
mat
ter-
opti
mal
corr
.
���� +
dens
ity-
dep.
corr
elat
or
���� +
3bod
yco
ntac
tint
erac
tion
MC
Var
iatio
n[P
RD
16(7
7)30
81]
� /
�
FHN
C[P
L61
B(76
)393
]/[P
RA
16(7
7)12
58]
% para
met
er
� or� is
choo
sen
tofit
the
MC
Vre
sult
at
�'
fm
N�
✔ve
rygo
odag
reem
ent
wit
hre
f-er
ence
over
the
who
lera
nge
for
the
dens
ity-
dep.
corr
elat
or
✔pe
rfec
tre
sult
sfo
rlo
wan
din
-te
rmed
iate
dens
itie
sw
ith
3bod
yco
ntac
tint
erac
tion
➼ea
syan
dsu
cces
sful
way
tosi
m-
ulat
eth
eef
fect
sbe
yond
2bod
yap
prox
imat
ion
Correlated
Densities & Occupation Numbers#15
correlated 2body density matrix
� � ��� � �� - � �� ��� �� � � �� � ��� � ��� � � � � � � ��� � ��� !� � ��� � �� - � �� � � �� � � �� � ��� � � * � � � � � � � � � � � � *
➼ diagonal elements of the correlated 2body density in2body approximation
!� � ��� ��� ��� - � � � � � � ��� �� � ��� � ��� �! � � � ��
� � � - � � � � ��� - � ���� � #
correlated occupation numbers
� � - � � �� - � �� � ��� � � � � � � � � !� � - � � �� - � �� � ��� � � * � � � � � � *
� off-diagonal elements of the correlated 1body density in2body approximation
!� � - � ��� ��� - � � � � � - � ��� - � � � !� � - � � � � ��� - � �➼ correlated momentum space occupation numbers by
Fourier transform of correlated 1body density
! � ��� � � � � � �� � ������ � � � � !� � - � ��� ��� �
Hom
ewor
kPr
oble
m
Cor
rela
ted
2bod
yD
ensi
tyM
atri
x#1
6
0
0.2
0.4
0.6
0.81
[
� � ]
�;���� fm
��
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;= ��= fm
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02
46
8
B [� ���
]
0
0.2
0.4
0.6
0.81
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� �
;�����
fm��
�� ���-,�
�1 �49 B:
� �1 �4 �9 B:
� �1 �4 �9 B:
+de
nsit
y-de
pend
entc
orr.
(den
sity
-dep
ende
nce
fixed
byen
ergy
fit)
✔de
nsit
yis
redu
ced
toal
mos
tze
row
ithi
nth
eco
rere
gion
(,����� fm
)
✔en
hanc
emen
tof
the
dens
ity
outs
ide
the
core
such
that
the
inte
gral
ispr
e-se
rved
(uni
tari
ty)
➼co
rrel
atio
nho
le
✘si
zeof
the
corr
elat
ion
hole
grow
sco
mpa
red
toth
eex
chan
geho
lew
ith
incr
easi
ngde
nsit
y
✔de
nsit
y-de
pend
ent
corr
elat
orre
-du
ces
this
effe
ct
Hom
ewor
kPr
oble
m
Cor
rela
ted
Occ
upat
ion
Num
bers
#17
0
0.250.5
0.751
�;���� fm
��
� �
;= ��= fm
��
00.
51
1.5
22.
5
� [� �
]
-0.2
50
0.250.5
0.751
�;= �� fm
��
� �
;�����
fm
��
�� ��� �
�9 �:' �
���� 9 �:' �
���� 9 �:' �
+de
nsit
y-de
pend
entc
orr.
✔co
rrel
atio
nsle
adto
popu
lati
onof
stat
esab
ove
the
Ferm
iedg
e
✔st
ates
wit
hin
the
Ferm
isp
here
are
depo
pula
ted
acco
rdin
gly
✘ne
gati
v–
unph
ysic
al–
occu
pati
onnu
mbe
rsfo
rhi
ghde
nsit
ies
wit
hde
nsit
y-in
depe
nden
tcor
rela
tor
✘2b
ody
appr
oxim
atio
nbr
eaks
dow
n!
✔ef
fect
ive
high
eror
der
incl
usio
nvi
ath
ede
nsit
y-de
pend
ent
corr
elat
orcu
res
the
prob
lem
Gam
mel
-Chr
istia
n-Th
aler
-Pot
entia
la
Sem
i-R
ealis
ticIn
tera
ctio
n#1
8
GC
T-po
tent
ial
optim
alco
rrel
ator
s
00.
51
1.5
22.
5
B [fm
]
-400
-300
-200
-1000
100
[MeV
]
00.
51
1.5
22.
5
B [fm
]
0
0.1
0.2
0.3
0.4
[fm
]
S � ��-,�
) X� �� ,��,
� �
;��
� �
;�=� �
;=�
� �
;==
% spin
-iso
spin
-dep
ende
ntha
rd-c
ore
pote
ntia
l
% tens
orco
mpo
nent
sno
tinc
lude
d
% para
met
ers
fixed
byen
ergy
min
i-m
izat
ion
for
nucl
ear
mat
ter
% full
spin
-iso
spin
depe
nden
tco
rre-
lato
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mpa
red
toov
er-a
llco
rr.
Gam
mel
-Chr
istia
n-Th
aler
-Pot
entia
l
Equa
tion
ofSt
ate
in2b
ody
App
rox.
#19
0.2
0.4
0.6
0.8
1
� [fm
�� ]
1.4
1.6
1.8
22.
22.
4
� �
[fm
�� ]
-35
-30
-25
-20
-15
-10-5
[MeV
]
�� ��
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ith
ST-d
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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.
Recommended