Date post: | 18-Dec-2015 |
Category: |
Documents |
Upload: | lindsey-singleton |
View: | 215 times |
Download: | 0 times |
Luigi AmicoMATIS – INFM & DMFCI Università di Catania
Collaboration with:
K. Hikami (Tokyo)
A. Osterloh
H. Frahm
Integrable spin boson models
(Hannover)
Superconductivity
Mesoscopics
Theory group
Materials and Technologies
for Information and communication Sciences
OUTLINE
The models & their physical origins.
Rotating wave approx.: integrable models of the Tavis-Cummings type.
Integrable models beyond the rotating wave approximation.
Conclusions.
)()(2
1)
2
1( SaSahaSSagBSGaaH j
zBc
Spin-orbit coupling in semiconducting heterostructures
ShSzgm
H ˆ2
1 2
Ac
ep
0
)(
)(
22
22
z
xyzy
yxzx
kOkk
kOkk
FM SC FM
In the Landau gauge: Ay=Bx
zy
x
Rashba, 1960
Bulk-IA:
Dresselhaus, 1955
Zutic, Fabian, das Sarma 2004;Shliemann, Egues, Loss 2003
)()( kuku
Superconducting nanocircuits
ppext
ppext
ILEE
IL~
The two states are given from the clockwise-anticlockwise currents of the secondary. (Nanocircuits for quantum computation: Maklhlin, Schoen, Shnirman 2001; Murali et. al. 2002; Paternostro et al. 2003).
xJJ
zCC
xJJ
EE
ENE
EE
)(cos)( : SQUIDa forenergy Josephson
:energy ticElectrosta
cos :junction singlea ofenergy Josephson
Chiorescu et al. 2004
Amico, Hikami 2005
Two SQUID’s
algebras. e(2) or h(4) either span,h where
2)()(~
223
3
h
ShhVShhiMShhLhSEH zC
yxpp
xextJ
N
jjjjjjj
zyx
N
jjjjCT
SaSahSaSag
SwaaH
1
,,1
)()(
Structure of the models
“Rotating terms” “Counter-rotating terms”
(no number cons) Traditionally emploied in:
Dissipative quantum mechanics (Caldeira-Leggett. Ref. U. Weiss )
Quantum optics (single mode: Jaynes/Tavis-Cummings. Ref. Scully, Zubairy)
Less traditionally: semiconducting heterostructure
nanocircuits (a lot of work by: G. Falci & coworkers 1993-2005)
(Zutic, Fabian, das Sarma 2004)
Rotating Vs Counter-rotating terms
• the corresponding coupling constant h is not small; • the frequency of the bosonic fields cannot be adjusted to a “resonance ”.
2
0 :..g
EEchaS R
Energy shifts due to Rot. or CR terms in perturbation theory:
0,
0,
1,
1,
The counter-rotating terms important if:
These regimes are going to be the working point for many applications; the dynamics is very complicated and “new” physics might emerge.
CR
R
2
0 :..h
EEchaS CR
It is easy to handle with models with only rotating OR counter rotat. terms.
The problem to deal with the terms at the SAME time is unsolved.
Simple example: Tavis-Cummings
)( aSSagSwaaH zCT
How to insert CR terms to keep the exact solvability?
Tavis-Cummings with Counter-Rotating terms:
Tavis-Cummings is solved exactly (T-C 1969; Hepp-Lieb 1973).
)( aSSahHH CT
Is not solvable.
)1(2
ssS
aaSM zConstants of the
motion: EH CT
Integrability: QIS method
)()()()()()( RTTTTR
Existence of a pair of matrices R(), Tsatisfying the Yang-Baxter eq.
Ctt
Ttrt
,0)(),(
)()( 0
Transfer matrix.
t() is taken as generating functional for the Hamiltonian.
And for the integrals of the motion. Ex.: H=d/dlog t()] 0.
St. Petersbourg group 1980; Korepin et al. book 1993
Tavis-Cummings model from the XXX R-matrix
)()()( SB LLT
000
00
00
000
)(R
1
1
)(
a
aaaLB
z
z
SSS
SSL
)(
Comment: the tr0 In the auxiliary space.
R-matrix
Monodromy matrix Lax matrices g
)()( 0 Ttrt
Bogolubov, Bullough, Timonen 1996
0
11
0)()(lim
ft
tgH CT
Beyond the RWA. I: Boundary Twist to the XXX Tavis Cummings
)()()( SSBB LKLKtrt
In the present case K can be general 2X2 C-number matrices
and .
Previous literature: KB=KS=K diagonal: various type of
non-linearities (like ) aaS z Rybin, Kastelewicz, Timonen, Bogolubov 1998
Important remark:
Fixed in such a way the final model is “interesting”.
The genereting function for the integrals is the first
order coefficient of t().
SB KK “Quantum boundary”
K() non-diagonal: no “number” conserv.
Tavis-Cummings type + counter rotating terms
)()()(2
)(),,(2)(),(
aSSavSaaSuSaauv
aazSvuyaavuSuvwHz
xz
SB KK
Two different bondary twist for the bosonic and spin
degrees of freedom:
Restrictions on the parameters:
u and v have the same sign; x,z free.
aavuaazSuvSvuC
SaSavSaaSuSaauvSuvySuvwCxz
zxz
)()(2
)()()(2),0(2),0(
2
1
])/([uv/)(y
);( u)(v-xw
vuxvuvuz
Constants of the motion:
1
1
2
U
VU
V
K
V
YXUYXUV
YUVUK
S
B
The problem with the twisted XXX chainsBecause of the relation between the coupling constants the
CR terms can be rotated away:
• Possible application for nanocircuit: an hidden working point is revealed where the interaction is “effectively weak”:
CR)without (
;2
)tan(2
expexp
1 HHGG
VU
Z
VU
UV
aaSSG
The optimal working point is reached by tuning the capacitance to
General property: Any non singular twist for
XXX chain can be put in a diagonal form unitarily (Amico, Hikami 2003; Ribeiro, Martins,2004).
Exact solutionThe idea is to obtain the bosonic problem starting
from a suitable “auxiliary” spin problem.We exploit that the bosonic algebra can be obtained via a
singular limit (contraction) of su(2):
Then:
With:
The auxiliary monodromy matrix represents 2 sites with 2
different representations:
The “impurity” is
(Dyson-Maleev)
The solution of the auxiliary problemThe transfer matrix fulfills the Baxter eq.
Where Q() are (2j+2s+2)x(2j+2s+2) matrices satisfying
and
The eigenvalue of ta is
The Bethe eqs. are
The solution of the spin-boson problem
The bosonic limit: 1) infty; 2) in the energy & the BE
Bethe eqs:
Energy:
2/32
11
11
0000
12
3212 ;
12
22 ;
312
; ; 4 ;4
V
UXVUYx
VUV
VUUYXVUUVUVUVy
V
UYXUVUVx
V
YVUX
V
Uy
VUV
VUUYUx
yUVyVxUx
More general coupling constants: XXX with open boundaries.
Idea: non diagonal boundary: Hxxx+ a S+ +bS- +c Sz.
)()()( SB LLT )()()()()( 10 TKTKtrt
Sklyanin 1989; De Vega, Ruiz 1993; Goshal, Zamolodchikov 1994.
/11/
1//1
//
//
d
cK
d
cK
For spin chain, Algebraic BA by: Melo, Ribeiro, Martins 2004 .
The eigenvalue of t() is obtained by contractions.
dc
S
Sdcdct
M
i ii
iij
M
i ii
ii
1
~2~
2/~
2/~
2
1
8
3
1
21
1
2122
M
ji ij
ij
ij
ij
jj
jj
jj
jj
S
Sdcdc
2/~
2/~
21
83
21
2122
Beyond the XXX models: spin-boson from the XYZ R-matrix
)2()()2()(
)2()()2()(
)2()()2()(
)2()()2()(
11111121
01011121
01110130
11010130
wwd
wwc
wwb
wwa
)()()( SJ LLT
)(00)(
0)()(0
0)()(0
)(00)(
)(
ad
bc
cb
da
R
3
30
02
21
1
22
11
33
00)(
SwSwSiwSw
SiwSwSwSwLS
R-matrix
Lax matrices )()()()()( 10 TKTKtrt
Sklyanin 1989; De Vega, Ruiz 1993; Inami, Konno 1994.
a,b,c,d parametrized in terms of theta
functions:
Baxter 1972
spin S: Sklyanin, Takebe 1996; Takebe 1992.
Znab z
bn
ain
aiz
222
2exp),(
The XYZ spin-phase model
)()( SeSehSeSegNSNSH iiiizz
BSB
SB
SB
SBSB
SB
SBSB
S
SBSBSB
SBS
SBSBSB
SB
zzzsnk
zzsng
zzsn
zzdnzzcn
zzsn
zzdnzzcn
zsn
zzsnzzdnzzcn
zzsnzsn
zzsnzzdnzzcn
zzsn
e wher /h
; /1
/2
2
1
2
1
matricesboundary diagonal uK
Work by: Felder Varchenko 1996; Gould, Zhang, Zhao 2002; Fan, Hou, Shi 1997