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