Rings, boxes and spins with dissipative environments

• Motivation

• Rings – particle + environment & conductance [1]

• Coulomb Boxes -- Relaxation resistance [1]

→ Non-equilibrium quantum critical point

• Spin dephasing on a ring – mapping to a spinless problem [2]

[1] Y. Etzioni, B. Horovitz and P. Le Doussal, Phys. Rev. Lett. 106, 166803 (2011)

[2] B. Horovitz, P. Le Doussal and G. Zarand, Euro. Phys. Lett. 95, 57004 (2011)

Baruch Horovitz Department of Physics, Ben-Gurion University, Beer-Sheva, Israel

Workshop on Quantum Filed Theory aspects of Condensed Matter Physics, Frascati 9/2011

Coulomb box motivation

J. Gabelli, G. Fève, J.-M. Berroir, B. Plaçais, A. Cavanna, B. Etienne, Y. Jin, D. C. Glattli, Science 313, 499 (2006)

0

20 0

0

1 (1 ) ( )(1/ )

g q

qg q

QV i QRC

Q C i C R OV C i R

2

2

2

B ttiker, Pr tre and Thomas 1993

non-interacting: / (2 )

Mora & Le Hur (2010) -- interacting=1: small dots / 2

Large dots /

q c

c q

q

ü ê

R h N e

N R h e

R h e

2

2

int 2

Expand in particle coordinate R( )

1{ ( ) ( ) [ ( )]}2

Integrate bath coordinates, dissipation is obtained if

[ ( ) ( ')]| | ( ) ( ) '( ')

Since

i i bath ii

dRS d M R Q L Qd

R RS d R R d d

22

2

0

( ) [cos ( ),sin ( )]

1 1 cos[ ( ) ( ')]' 2 ( ')

is an external flux (in units of / ) Long range interaction

x

x

R R

dS d MR d d dd

hc e

Caldeira-Legget environment

N̂

0N

20

2 20

2Ambegaokar, Eckern & Schon (82)

/ 2 2 single particle energies:ˆcharging energy: [ ] (m- ) / 2 m is winding

=|t| (0) (

c g g c

c x

c dot lead

E e C V E N

E N N M

N

0 0

, ' ' , ' '

2

0), 0, ˆ 2 [ ] 1 / 2

ˆ ˆ( ') [ , ] ( ') [ , ]

( ) 2 4 ( ) (4

c

t c c x

t t t t t t t t

c c

t N

E N N M E N

K i t t N N K i t t

K E E K E

2 20 0/ )C (1 ) c qe i C R

Mapping

02 2 3

( 0)

( 1)

without noise

( ) ( ) (0) ( ) , ( ) | |

( ) ( ) cos ( ) sin ( )

0 / ( ), v1 1 1 1ln

R

x

i j ij

x y

t R

R c

T

R

M t Et B t B

M t t t t E

E i

x x ξ

2 0 00

0

00 0

00 0

[Hofstetter & Zwerger 1997.]

v v[ln ln ] ...

v / , /Equilibrium: lim lim b 1

Non-equilibrium: lim lim b 0

c c

c

E

E

b

E M

Langevin dynamics - nonequilibrium

10−1

100

101

0.88

0.9

0.92

0.94

0.96

0.98

1

E/ηwc

Eηv

10−0.9

10−0.2

0

5

10x 10

−3

E/ηwc

E(2)

ηv

?2

0 1

0

0

Linear response to (coupling ) is ( ) 1/

Linear response to (coupling ) is ( ) ( )constant term ( ) is missing?Claim: can be eliminated in total flux , or

(

R

x x

x

x x

E E R i

K K i K RK

Et

K

1

0 00

1

0 0 10

2 20 0 002

) ( ) is periodic, for dc response ( ) 0

lim lim ( ) / ( ) 1/

v v v1 1 2 1 4 1 1sin ln sin sin [ln ln ]( ) 2 2

unexpected small parameter sin(1

Keldysh

x x x

E x x R

R c c c

K Et K d

K i K d

bE

/ 2 ) fixed point at 1/ 2R

Equilibrium vs non-equilibrium

Thouless charge pump

slow change of by 1 unit with / 2

1 / 2

i.e. the particle comes back to the same position on the ring and a unit charge has been transported.

x x R t t

x tdt dt

20

20

sin(1/ 2 ) 0 has =1/(2 n)but cos cos ~ with n>1 is consistent

with Spohn-Zwerger "theorem" cos cos ~ for > |----- ----------------

1/ 2

nt

t

R

R

t

t

2 /ringG e h

Box Experiment

8

8

2 22

' 2 2

sample with many (e.g. Al), 1meV

sweep gate voltage at a rate / 10 Hz<<

need: level spacing << , <<10 Hzcharge fluctuations -- quantized noise

ˆ ˆ( ) 24 4

c c

c

Q t tc R c

N

E

T

e eS e N NE E

21 0 00 02 20

0 0

2

( ) ( )

For large C , expect independent of

[1 O( )]

qg

g q

q

C N hR N dNC e

C R N

hR ee

Spins2 2

0 02

2

0 02

22

0 0

2

1[ ] ( )2

' ( ) ( )2

1 [ ( ) ]2

( ) ( cos sin , sin cos ) , =mr=0: rotation invariance, is conserved

1 [2

x y y x x x y y

ring

z z

ring z

H V rm r r r

pH S p S p S p S pmr

H pmr

mrJ p S

H Jmr

h S

h

2 2 2( ) 1 ] ( ) [ ( ), ( ),1] / 1

For < 3 the ground state is a spin coherent state | ( )

x yh h

n S n

n

Adding environment:

2 2

( , ) are coordinates of a dissipative environment.

( ) 1 ( , )

Dynamics of are independent of the spin-orbit coupling.

Spin dynamics: ( )

ringH H V

p Vmr Mr

d ddt d

h S

S Sh S

0

0 0

2

( )

The solution is a linear mapping ( ) ( , )

In particular for = 2 the rotation has a unit vector ( )as axis of rotation and the rotation angle.

2 ( 1 1) incommensurate

i ijS R

h S

N

Spin 1/2

0

0

( )0 0

2

,

2 4 41, 1, 2 , 2 2 ,

( , ) ( ) e ( )

1 (1 1 ) incommensurate2

spin correlations involve ( ) e e of spinless problem1S ( ) (0) sin [ ( ) ( )] cos ( ) sin (4 2 2

t

iGspin

ia iaa

x x G G G G G G

U

G

P t

t S P t P t P t P

2 2

21 2 ,

/ 2,

4

)

S ( ) (0) cos ( )sin does not dephase

Large expect ( ) ~ ?

small perturbation cos has a finite correction, no dephasing.2

z z G G

z

a aa

t

t S P tS

P t t t

Conclusions & messages

1

0 0 10

2

1. Non-equilibrium limit

lim lim ( ) / ( ) 1/

/

E x x R

ring

K i K d

G e h

2

2

2. Quantized noise experiment

( ) 24Q

c

eSE

03. Spin dephasing via e etia ia