Population Switching and Charge Sensing in Quantum Dots : A case for Quantum Phase Transitions
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Transcript of Population Switching and Charge Sensing in Quantum Dots : A case for Quantum Phase Transitions
Population Switching and Charge Sensing in Quantum
Dots: A case forQuantum Phase Transitions
Moshe Goldstein (Bar-Ilan Univ., Israel), Richard Berkovits (Bar-Ilan Univ., Israel),
Yuval Gefen (Weizmann Inst., Israel)
Support: Adams, BINA, GIF, ISF, Minerva, SPP 1285
PRL 104, 226805 (2010)
Outline
• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist: the effect of a charge sensor
• Extensions; spin effects
Quantum dots• “0D” systems:
– Semiconductor heterostructures – Metallic grains
– Carbon buckyballs & nanotubes – Single molecules
• Realizations:
– Artificial atoms– Single electron transistors
Quantum dots:A theorist’s view
H.c.ˆˆˆˆˆˆˆˆˆ;;,
,;,
,,,
ikRLiki
jiji
iiii
kRLkkk actnnUaaccH
R
1
2
Rt1
Rt2Lt2
ULt1
L
gii eV )0(Vg
• Traditional regimes:[Review: Alhassid, RMP ‘00]
– Open dots, – Closed dots,
• Last decade: intermediate dot-lead coupling, – Interference (e.g., Fano)
– Interactions (e.g., Kondo, population switching)
: level spacing; level width
1
2
energ
y
F
Level population
(spinless)
R1
2
Rt1
Rt2Lt2
ULt1
L
2
1
F1
2
F F2
1
Vg
Vg
n1, n2
+U
, g
Coulomb-blockade peak
Coulomb-blockade valley
1 2
Vg
n1, n2
+U
1
2F F
2
1
2
1
F
1
2
energ
y
F
Population switching(spinless)
R1
2
Rt1
Rt2Lt2
ULt1
L
[Baltin, Gefen, Hackenbroich & Weidenmüller‘97, ‘99; Silvestrov & Imry ’00; … Sindel et al. ‘05 …]
Related phenomena
• Charge sensing by QPC [widely used]
• Phase lapses[Heiblum group: Yacoby et al. ‘95; Shuster et al. ‘97; Avinun-Kalish et al. ‘05]
RL QD
QPCQPCU
– See also: MG, Berkovits, Gefen & Weidenmüller, PRB ‘09
Outline
• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist: the effect of a charge sensor
• Extensions; spin effects
Nature of the switching
Is the switching abrupt?
• Yes ? (1st order) quantum phase transition
• No ? continuous crossover
(at T=0)
A limiting case• Decoupled narrow level:
[Silvestrov & Imry ‘00]
– Switching is abrupt
– A single-particle problem:
not a QPT
[Marcus group: Johnson et al. ‘04] [Berkovits, von Oppen & Gefefn ‘05]
free energy
Vg
narrow level filled
narrow level empty
• Many levels:
Nature of the switching
Is the switching abrupt?
• Yes ? (1st order) quantum phase transition
• No ? continuous crossover
(at T=0, for a finite width narrow level)
Numerical results• Hartree-Fock: Two solutions, switching still abrupt
[Sindel et al. ’05, Golosov & Gefen `06, MG & Berkovits ‘07]
• FRG, NRG, DMRG: probably not [?] [Meden, von Delft, Oreg et al. ’07; MG & Berkovits, unpublished]
Outline
• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist: the effect of a charge sensor
• Extensions; spin effects
Basis transformation
[Kim & Lee ’07, Kashcheyevs et al. ’07, Silvestrov & Imry ‘07]
H.c.ˆˆˆˆˆˆˆˆˆ;,
,,;,
,,,
kRLkRL
RLkRLkkk actnnUaaccH
R1
2
Rt1
Rt2Lt2
ULt1
L
2
t
Electrostatic interaction
Level widths:
e.g,. RL tt 11 RL tt 22
RRt
ULt
L L RLRt
Coulomb gas expansion (I)
0
/1
0 0
1
0
2
0
122 expdd
...dd 232
N
T
nNN
N
SZN
N
nn
nN
mn mn
mni T
TS
2
1
2
1
1sin
ln1
T: temperature;
: short time cutoff;
=|t|2 level width
t
One level & lead:
Electron enters/exits
Coulomb gas (CG) of alternating positive/negative charges[Anderson & Yuval ’69; Wiegmann & Finkelstein ’78; Matveev ’91; Kamenev & Gefen ’97]
1/T
n1
0
– – –+ + +
Fugacity
Coulomb gas expansion (II)
L
L
R
R
RL
RL
N
n
N
n
Rn
Ln
nnnU
US
2
1
2
1
112
RRt
UL L R
Lt
Two levels & leads Two coupled CGs[Haldane ’78; Si & Kotliar ‘93]
1/T
n1, n2
1
0
– – –+ + ++ + + +– – – –
T
nURnR
LnL
RRRN
RN
NN
T LLLN
LN
N
R
N
L
SSS
Z
RL
RRRRN
RR
RL
LLLLN
LL
RL
/1
0 0
1
0
2
0
122
0,
/1
0 0
1
0
2
0
122
expdd
...dd
dd...
dd
232
232
2
e
Coulomb gas expansion (III)CG can be rewritten as:[Cardy ’81; Si & Kotliar ‘93]
0
/1
0 0
1
0
2
0
1 ,expdd
...dd 23
1123221N
T
iiNN
i
N
NNNSyyyyZ
N
i
iiN
ji ji
ii ijjiih
T
TeeS
1
1
1111 sin
ln,
1/T0
00 00 0010 1001 0111 11
1,011,10 e
Ry 11,10
Uh RL 11
11
10
00
01
RG analysis (I)
2/2e
21
lnd
d hhhyyyy
hhhh yy ee
lnd
d 22
hhyh
he
lnd
d 2
• Generically (no symmetries):15 coupled RG equations [Cardy ’81; Si & Kotliar ‘93]
6 eqs.
6 eqs.
3 eqs.
11
10
00
01
• Solvable in Coulomb valley:
• Three stages of RG flow:
RG analysis (II)
,U11
10 01
00
(I) 1, U
U 1(II)
(III) ,1 U
Result: an effective Kondo model
zxy JJ ,
[Kim & Lee ’07, Kashcheyevs et al. ’07, ‘09, Silvestrov & Imry ‘07]
Digression: The Kondo problem• Realizations
– Magnetic impurity– QD with odd electron number
zzL ShxsSJHH ˆ)0(ˆˆˆˆ
tL
• Problem: divergences [Kondo ’64]
– susceptibility:
– Similarly: resistance, specific heat …
T
DJ
T
DJ
T22 lnln1
1~
,U
• Hamiltonian– J~t2/U>0: exchange
– hz: local magnetic field
D: bandwidth
(spinful)
Kondo: CG analysis
0
/1
0 0
1
0
2
0
122 expdd
...dd
2
232
N
T
nNN
N
xy SJ
ZN
N
nn
nz
N
mn mn
mnzi h
T
TJS
2
1
2
1
1sin
ln122
1/T
Sz
1/2
0
– – –+ + +–1/2
• Anderson & Yuval [’69]:– Anisotropic model (Jz≠Jxy)– expand in Jxy: Coulomb gas of spin-flips
Kondo: Phase diagram• RG equations:
22
lnd
dxy
z JJ
xyz
xy JJJ 2
lnd
d
• Ferromagnetic Kondo:– impurity decoupled– susceptibility: ~c(J)/T+…
• Anti-Ferromagnetic Kondo:– impurity strongly-coupled– susceptibility: ~1/TK+…
Kosterlitz-Thouless transition
JDTK 1exp TK: Kondo temperature
Back to our problem …
• Pseudo-spin (orbital) Kondo– Anisotropic
– Vg changes effective level separation switching
11
10 01
00
zxy JJ ,
UJ RLxy
00
112
UUJ
RR
R
LL
Lz
1111
R
RR
L
LLRLz
UUh
lnln
20 RL
RRtULt
L L R
R1
2
Rt1
Rt2Lt2
ULt1
L
Vg
nR, nL
LL+U
(spinless)
Implications
population switching is continuous (scale: TK)
No quantum phase transition[Kim & Lee ’07, Kashcheyevs et al. ’07, ‘09, Silvestrov & Imry ‘07]
• Anti-Ferromagetic Kondo model
• Gate voltage magnetic field hz
R
L
RL
RLK U
UUT ln
2exp 00
What was gained?
FDM Haldane on the Coulomb gas expansion:
“Though an expression such as [the Coulomb gas expansion] … could be taken as the starting point of a scaling theory …, the more direct ‘poor man’s’ approach … proves simpler and more complete in practice.”
[J. Phys. C 11, 5015 (1978)]
Outline
• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist: the effect of a charge sensor
• Extensions; spin effects
But …
population switching is discontinuous :1st order quantum phase transition
2/2 QPCzz JJ
• Adding a charge-sensor (Quantum Point Contact):
– 15 RG eqs. unchanged
– Three-component charge
RRtULt
L L R
QPCQPCU
QPCe ,1,01,011,10
2tan2 1QPCQPCQPC U
Kosterlitz-Thouless transition
Reminder: X-ray edge singularity
• Without interactions:
)(~)( 0 S
––– noninteracting
0
S()
––– Anderson
––– Mahan
orth))((~)( 00 S
excitonorth)(~)( 0 S
• Anderson orthogonality catastrophe [’67]:
• Mahan exciton effect [’67]:
energ
y
Fe
Absorption spectrum:
)(~)( 0 S 2
orth)/(
)/(2exciton
orth))((~)( 00 S )(~)( 0 S
RRtULt
L L Re e
X-ray singularity physics (I)Virtual fluctuations:
X-ray singularity physics (I)
Mahan exciton
Anderson orthogonality
Jxy Scaling dimension:
Mahan wins: Switching is continuous
>
vs.
<1 relevant
RRtULt
L L Re e
Electrons repelled/attracted to filled/empty dot (Jz):
22
2
11
RLRL
U
11
2tan2 1
X-ray singularity physics (II)
Mahan exciton
Anderson orthogonality
Jxy Scaling dimension:
RRtULt
L L R
QPCQPCU
e e
e
Anderson wins: Switching is abrupt
< +
vs. + Extra orthogonality
222
2
11
QPCRLRL >1 irrelevant
A different perspective• Detector constantly measures the level
population
• Population dynamics suppressed: Quantum Zeno effect
! A sensor may induce a phase transition
Noninvasive charge sensing?
continuous switching
Use Friedel’s sum rule!
abrupt switching
L L L L
RRtU1Lt
L R
QPCQPCU
L1
2Lt
L2
L
LL
LLL n
h
eG 22
21
212
sin4
Vg
nR, nL, gL
L L+U
Vg
nR, nL, gL
L L+U
RRtU1L
t
L R
QPCQPCU
L1
2Lt
L2
K [CIR: Meden &Marquardt ’06]
Perturbations
First order transition switching smeared linearly in T, tLR
1. Finite T
2. Inter-dot hopping:
RRt
ULt
L L R
QPCQPCU
LRt
Outline
• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist: the effect of a charge sensor
• Extensions; spin effects
Related models• Bose-Fermi Kondo
[Kamenev & Gefen ’97, Le Hur ’04, Borda et al. ’05, Florens et al. ’07, ‘08, …]
• 2-impurity Kondo with z exchange[Andrei et al. ’99, Garst et al. ‘94]
RRJzzILJ
L
RL
Rz
LzzzRL xsSJSSIHHH
,
)0(ˆˆˆˆˆˆˆ
)0(ˆˆ)0(ˆˆˆˆˆ xSJxsSJHHH BxzFFFBF
BJFJF B
Extensions (I)
– Mahan & Anderson– Repulsion continuous switching
QPCURe ,1,011,10
RRtULt
L L R
QPCQPCULU RU
22
2
1
UR
UL
UR
UL
zz JJ
• Dot-lead interactions:
Extensions (II)
• Luttinger-liquid leads:
– Repulsion abrupt switching
RRtULt
L L R
QPCQPCU
11 gJJ zz
• Luttinger-liquid & dot-lead interaction:– Edge singularity given by CFT & Bethe ansatz [Ludwig & Affleck ’94; MG, Weiss & Berkovits, EPL ‘09]
– Many novel effects even for single level, single lead[MG, Weiss & Berkovits, PRB ’05, ’07, ’08; J. Phys. Conden. Matt. ‘07; Physica E ’10; PRL ‘10]
2/WL
RRtULt
L L R
• Luttinger liquid parameter: g=3/4• Soft boundary conditions:
Switching in a Luttinger liquid (I)
• Density Matrix RG calculations:
W
Switching in a Luttinger liquid (II)
WV
n
g
ln
• Finite size scaling:
LV
n
g
Conclusions• Population switching:
– Usually: steep crossover, no quantum phase transition
– Adding a charge sensor: 1st order quantum phase transition
• Laboratory for various effects:– Anderson orthogonality, Mahan exciton,
quantum Zeno effect, entanglement entropy;– Kondo