Long-range interaction in a quantum model of interacting Fermi-particles
description
Transcript of Long-range interaction in a quantum model of interacting Fermi-particles
ASSISI, July 4, 2007
Long-range interaction in a quantum model of interacting Fermi-particles
Felix Izrailev
Instituto de Física, BUAPPuebla, México
in collaboration with:
in collaboration with
Fausto Borgonovi, Brescia, Italy
A. Smerzi – Trento, Italy
G. Berman – Los Alamos, USA
www.felix.izrailev.com
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• Introduction
• The model
• Mean field representation
• Delocalization border versus chaos border
• Short versus long-range interaction
• Conclusions
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M.Dyakonov, cond-mat/0110326
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p
pb p p ppp ttT 1
Solid state model of a quantum computer
xBtbtbB zppp
ppp ,sin- ,cos
1D spin-chain in - direction, in a rotating magnetic field :
- magnetic pulses
x
with
specified by
In what follows we analyze what is going on
during a single pulse
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1
0,2
~ L
k
zn
zk
knnk
zkk IIJIH
P
p
L
kk
itik
itipp IeIet pppp
1
1
02
1
nkJ ,
k
p
1tp
zyxk
zyxkI ,,,,
2
1 yk
xkk iIII
Total (time-dependent) Hamiltonian
1 pp ttt for
- Ising interaction between n-th and k-th qubit - Larmor frequency
- Rabi frequency
- Pauli matrices ;
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1
0,
)( 2L
k
zn
zk
knnk
zkpk
p IIJIH
py
kpxk
L
kp II sincos
1
0
kknkp J ,
kkk 1
2 p
Stationary Hamiltonian 1 pp ttt for
”Selective excitation” :
for simplicity :
with
”Non-selective excitation” : nkkp J ,
pp ;
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Hamiltonian under study
Nearest-neighbor interaction : 1,, kknk JJ
(dynamical model in - representation)
1
0
2
012
L
k
L
k
zk
zk
yk
zkk IIJIIH
where kk
z
matrix is diagonal for 0
off-diagonal matrix elements 2,
iH nk
nkkn HH ,, Other cases - !!
akxB kzk 0 and
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Mean field representation
1
00
L
k
yk
zkk IIH
2
010 2
L
k
zk
zk IIJV
in the basis where is diagonal :0H offbanddiag VVVV 0
zk
zkk
kkdiag IIbbV 112
00 JVHH
1
0
221
00
L
kk
L
kkHH
yk
ykk
kkband IIaaV 112
k
zk
zkkk
yk
ykkkoff IIbaIIbaV 11112
22
k
kkb
22
k
ka where
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22
1
2
1 2222 ka
kk
akk k for and
Magnetic field with a constant gradient
akxB kzk 0
then,
in this case, 1~ka 1kb and
Therefore, bandoffdiag VVV
Quasi-integrability !
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Integrability
if we neglect offdiag VV and
the model is :
1
0
2
01
L
k
L
k
yk
ykk
zkka IIJIH
where 22 kk and JJk 2
the model is integrable, independent on
kk J , !!
Young, Rieger (1996); Young (1997); Lieb, Schultz, Mattis (1961)
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Delocalization border
let us consider central band :
number of states - !2!2
!LL
LN cb
size of the band -
8
1 22 aLLE cb
Distance between directly coupled states -
22a
M
Ed
f
ff
should be compares with 20
JJVV
As a result,
24a
Jcr does not depends on L !!
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Chaos border
The mechanism of chaos is the band overlap
for non-selective regime the estimate is :
22216 La
LJch
- delocalization border
- chaos border
05.0crJ
100chJ ( for )
1 ; 100 a
do not coincide !!
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All-to-all interaction
Delocalization border coincide with the chaos border !
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Long-range interaction in a random model
nk
J nk
are random number in the interval
JJ ,
1101.0J
01.0J
short- range interaction
long- range interaction
under-critical region
over-critical region
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Conclusion / remarks
1. For the magnetic field with a constant gradient, one can avoid quantum chaos
2. The border of delocalization can be very different form the chaos border !
3. For short-range interaction the two borders different, however, for long-range interaction they are the same.
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