Population Switching and Charge Sensing in

Quantum Dots A case forQuantum Phase Transitions

PRL 104, 226805 (2010)

- 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

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

- Artificial atoms
- Single electron transistors

- Realizations

- Semiconductor heterostructures

- Metallic grains

- Carbon buckyballs nanotubes

- Single molecules

Quantum dotsA theorists view

Vg

- Traditional regimes
- Review Alhassid, RMP 00
- Open dots, GgtgtD
- Closed dots, GltltD

- Last decade intermediate dot-lead coupling, GltD
- Interference (e.g., Fano)
- Interactions (e.g., Kondo, population switching)

D level spacing G level width

Level population

Coulomb-blockade peak

, g

(spinless)

Vg

?G1?

?G2?

Coulomb-blockade valley

1

Population switching

(spinless)

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

- 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?

(at T0)

- Yes ? ?(1st order) quantum phase transition
- No ? ? continuous crossover

A limiting case

narrow level empty

- Decoupled narrow level
- Silvestrov Imry 00
- Switching is abrupt
- A single-particle problem
- not a QPT

narrow level filled

- Many levels

Marcus group Johnson et al. 04

Berkovits, von Oppen Gefefn 05

Nature of the switching

- Is the switching abrupt?

(at T0, for a finite width narrow level)

- Yes ? ?(1st order) quantum phase transition
- No ? ? continuous crossover

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Berkovits, unpublished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

e.g.,

- Kim Lee 07, Kashcheyevs et al. 07,

Silvestrov Imry 07

Electrostatic interaction

Level widths

Coulomb gas expansion (I)

- Coulomb gas (CG) of alternating positive/negative

charges - Anderson Yuval 69 Wiegmann Finkelstein

78 Matveev 91 Kamenev Gefen 97

One level lead Electron enters/exits

Fugacity

T temperature x short time cutoff Gprt2

level width

Coulomb gas expansion (II)

- Two coupled CGs
- Haldane 78 Si Kotliar 93

Two levels leads

Coulomb gas expansion (III)

- CG can be rewritten as
- Cardy 81 Si Kotliar 93

RG analysis (I)

- Generically (no symmetries)
- 15 coupled RG equations Cardy 81 Si Kotliar

93

6 eqs.

6 eqs.

3 eqs.

RG analysis (II)

- Solvable in Coulomb valley
- Three stages of RG flow

11

10

01

(I)

00

(II)

(III)

Result an effective Kondo model

Kim Lee 07, Kashcheyevs et al. 07, 09,

Silvestrov Imry 07

Digression The Kondo problem

(spinful)

- Realizations
- Magnetic impurity
- QD with odd electron number

- Hamiltonian
- Jt2/Ugt0 exchange
- hz local magnetic field

- Problem divergences Kondo 64
- susceptibility
- Similarly resistance, specific heat

D bandwidth

Kondo CG analysis

- Anderson Yuval 69
- Anisotropic model (Jz?Jxy)
- expand in Jxy Coulomb gas of spin-flips

Kondo Phase diagram

- RG equations

- Ferromagnetic Kondo
- impurity decoupled
- susceptibility cc(J)/T

- Anti-Ferromagnetic Kondo
- impurity strongly-coupled
- susceptibility c1/TK

Kosterlitz-Thouless transition

TK Kondo temperature

Back to our problem

11

(spinless)

10

01

- Pseudo-spin (orbital) Kondo
- Anisotropic
- Vg changes effective level separation ? switching

00

Implications

- Anti-Ferromagetic Kondo model
- Gate voltage ? magnetic field hz

- population switching is continuous (scale TK)
- No quantum phase transition
- Kim Lee 07, Kashcheyevs et al. 07, 09,

Silvestrov Imry 07

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

mans 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

- Adding a charge-sensor (Quantum Point Contact)
- 15 RG eqs. unchanged
- Three-component charge

Kosterlitz-Thouless transition

- population switching is discontinuous
- 1st order quantum phase transition

Reminder X-ray edge singularity

Absorption spectrum

energy

- Without interactions

w0

- Anderson orthogonality catastrophe 67

e

noninteracting

- Mahan exciton effect 67

Anderson

Mahan

X-ray singularity physics (I)

Virtual fluctuations

e

e

X-ray singularity physics (I)

Electrons repelled/attracted to filled/empty dot

(Jz)

e

e

Mahan exciton

Anderson orthogonality

vs.

Jxy Scaling dimension

lt1 ? relevant

gt

Mahan wins Switching is continuous

X-ray singularity physics (II)

e

e

e

Mahan exciton

Anderson orthogonality

Extra orthogonality

vs.

Jxy Scaling dimension

gt1 ? irrelevant

lt

Anderson wins Switching is abrupt

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?

Use Friedels sum rule!

- continuous switching

abrupt switching

? ? TK

?GL?

?GL?

?GL?

?GL?

CIR Meden Marquardt 06

Perturbations

- Finite T
- Inter-dot hopping

- First order transition ?
- switching smeared linearly in T, tLR

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

Extensions (I)

- Dot-lead interactions

- Mahan Anderson
- Repulsion ? continuous switching

Extensions (II)

- Luttinger-liquid leads
- Repulsion ? abrupt switching

- 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

Switching in a Luttinger liquid (I)

- Density Matrix RG calculations

- Luttinger liquid parameter g3/4
- Soft boundary conditions

Switching in a Luttinger liquid (II)

- Finite size scaling

W

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