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


• Introduction

• The model

• Mean field representation

• Delocalization border versus chaos border

• Short versus long-range interaction

• Conclusions


M.Dyakonov, cond-mat/0110326


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


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 ;


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 ;


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


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


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 !


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)


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 !!


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 !!


All-to-all interaction

Delocalization border coincide with the chaos border !


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


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.

long-range interaction in a quantum model of interacting fermi-particles

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