Integrable spin boson models

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Integrable spin boson models
Luigi AmicoMATIS INFM DMFCI Università di
Catania
Superconductivity Mesoscopics Theory
group
Collaboration with K. Hikami (Tokyo)
A. Osterloh H. Frahm
Materials and Technologies for Information and
communication Sciences
(Hannover)
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OUTLINE

• The models their physical origins.
• Rotating wave approx. integrable models of the
  Tavis-Cummings type.
• Integrable models beyond the rotating wave
  approximation.
• Conclusions.

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Spin-orbit coupling in semiconducting
heterostructures
x
Bulk-IA Dresselhaus, 1955
FM
SC
FM
z
Rashba, 1960
y
In the Landau gauge AyBx
Zutic, Fabian, das Sarma 2004Shliemann, Egues,
Loss 2003
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Superconducting nanocircuits
Chiorescu et al. 2004
Two SQUIDs
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).
Amico, Hikami 2005
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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-Cummin
gs. Ref. Scully, Zubairy) Less traditionally
semiconducting heterostructure
nanocircuits (a lot of work by G.
Falci coworkers 1993-2005)
(Zutic, Fabian, das Sarma 2004)
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Rotating Vs Counter-rotating terms
Energy shifts due to Rot. or CR terms in
perturbation theory
CR
R
W
w
The counter-rotating terms important if

• the corresponding coupling constant h is not
  small
• the frequency of the bosonic fields cannot be
  adjusted to a resonance .

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.
These regimes are going to be the working point
for many applications the dynamics is very
complicated and new physics might emerge.
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Simple example Tavis-Cummings
Tavis-Cummings is solved exactly (T-C 1969
Hepp-Lieb 1973).
Constants of the motion
Tavis-Cummings with Counter-Rotating terms
Is not solvable.
How to insert CR terms to keep the exact
solvability?
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Integrability QIS method
Existence of a pair of matrices R(l), T(l)
satisfying the Yang-Baxter eq.
Transfer matrix.
t(l) is taken as generating functional for the
Hamiltonian. And for the integrals of the motion.
Ex. Hd/dllog t(l) l0.
St. Petersbourg group 1980 Korepin et al. book
1993
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Tavis-Cummings model from the XXX R-matrix
R-matrix
Monodromy matrix
Lax matrices
Comment the tr0 In the auxiliary space.
Bogolubov, Bullough, Timonen 1996
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XXX-Tavis-Cummings model
The Transfer matrix t(l) generates the
hamiltonian
and the costants of motion

Jurco 1989 Bogolubov, Bullough, Timonen 1996
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Beyond the RWA. I Boundary Twist to the XXX
Tavis Cummings
Previous literature KBKSK diagonal various
type of non-linearities (like )
Rybin, Kastelewicz, Timonen, Bogolubov 1998
In the present case K can be general 2X2
C-number matrices and .
Quantum boundary
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(h).
K(h) non-diagonal no number conserv.
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Tavis-Cummings type counter rotating terms
Two different bondary twist for the bosonic and
spin degrees of freedom
Constants of the motion
Restrictions on the parameters u and v have the
same sign
l, D, x,z free.
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Exact solution
)


)
P

)
The Energy is
obtained in
terms of roots of a set of algebraic eqs Bethe
eqs
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The problem with the twisted XXX chains

• Because 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

General property Any non singular twist for
XXX chain can be put in a diagonal form
unitarily (Amico, Hikami 2003 Ribeiro,
Martins,2004).
The optimal working point is reached by tuning
the capacitance to
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Exact solution
The 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)
(Dyson-Maleev)
Then
The impurity is
With
The auxiliary monodromy matrix represents 2 sites
with 2 different representations
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The solution of the auxiliary problem
The transfer matrix fulfills the Baxter eq.
Where Q(l) are (2j2s2)x(2j2s2) matrices
satisfying
and
The eigenvalue of ta is
The Bethe eqs. are
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The solution of the spin-boson problem
The bosonic limit 1) e infty 2) h
0 in the energy the BE.
Energy
Bethe eqs
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Model of Rabi type
Assuming KBKS restricts the coefficients
uv and 2wy
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More general coupling constants XXX with open
boundaries.

• Idea non diagonal boundary Hxxx a S bS- c
  Sz.

Sklyanin 1989 De Vega, Ruiz 1993 Goshal,
Zamolodchikov 1994.
For spin chain, Algebraic BA by Melo, Ribeiro,
Martins 2004 . The eigenvalue of t(l) is
obtained by contractions.
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Possible Hamiltonian

• Quasi-classical limit of

Amico, Osterloh, Frham 2005.
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Beyond the XXX models spin-boson from the XYZ
R-matrix
a,b,c,d parametrized in terms of theta
functions
R-matrix
Baxter 1972
spin S Sklyanin, Takebe 1996 Takebe 1992.
Lax matrices
Sklyanin 1989 De Vega, Ruiz 1993 Inami, Konno
1994.
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The XYZ spin-phase model
Work by Felder Varchenko 1996 Gould, Zhang,
Zhao 2002 Fan, Hou, Shi 1997
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Conclusions

• XXX Counter rotating terms can be included by
  applying general boundary twist (restriction on
  the coefficients).
• XYZ spin boson with CR terms can be obtained
  with a diagonal boundary

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Relation with Gaudin models
Wijx Wijy g/sin(ui-uj) Wijy g cot(ui-uj)
Exact solution
Gaudin 1976
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Multi mode.

• The lattice L1...N L1 U L2
• Holstein-Primakov transf. in one of the
  sublattices. In the large sj limit

i
Exact solution