rings, boxes and spins with dissipative environmentsrings, boxes and spins with dissipative...
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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