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

• 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

Conclusions