Population Switching and Charge Sensing in Quantum Dots: A case for Quantum Phase Transitions

1
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
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Outline

• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist the effect of a charge sensor
• Extensions spin effects

3
Quantum dots

• 0D systems

• Artificial atoms
• Single electron transistors

• Realizations

• Semiconductor heterostructures

• Metallic grains

• Carbon buckyballs nanotubes

• Single molecules

4
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
5
Level population
Coulomb-blockade peak
, g
(spinless)
Vg
?G1?
?G2?
Coulomb-blockade valley
1
6
Population switching
(spinless)
Baltin, Gefen, Hackenbroich Weidenmüller 97,
99 Silvestrov Imry 00 Sindel et al. 05

7
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

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Outline

• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist the effect of a charge sensor
• Extensions spin effects

9
Nature of the switching

• Is the switching abrupt?

(at T0)

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

10
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
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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

12
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

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Outline

• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist the effect of a charge sensor
• Extensions spin effects

14
Basis transformation
e.g.,

• Kim Lee 07, Kashcheyevs et al. 07,
  Silvestrov Imry 07

Electrostatic interaction
Level widths
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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
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Coulomb gas expansion (II)

• Two coupled CGs
• Haldane 78 Si Kotliar 93

Two levels leads
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Coulomb gas expansion (III)

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

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RG analysis (I)

• Generically (no symmetries)
• 15 coupled RG equations Cardy 81 Si Kotliar
  93

6 eqs.
6 eqs.
3 eqs.
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RG analysis (II)

• Solvable in Coulomb valley
• Three stages of RG flow

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10
01
(I)
00
(II)
(III)
Result an effective Kondo model
Kim Lee 07, Kashcheyevs et al. 07, 09,
Silvestrov Imry 07
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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
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Kondo CG analysis

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

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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
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Back to our problem
11
(spinless)
10
01

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

00
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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

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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)

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Outline

• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist the effect of a charge sensor
• Extensions spin effects

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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

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Reminder X-ray edge singularity
Absorption spectrum
energy

• Without interactions

w0

• Anderson orthogonality catastrophe 67

e
noninteracting

• Mahan exciton effect 67

Anderson
Mahan
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X-ray singularity physics (I)
Virtual fluctuations
e
e
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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
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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
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A different perspective

• Detector constantly measures the level population
• Population dynamics suppressed Quantum Zeno
  effect

• A sensor may induce a phase transition

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Noninvasive charge sensing?
Use Friedels sum rule!

• continuous switching

abrupt switching
? ? TK
?GL?
?GL?
?GL?
?GL?
CIR Meden Marquardt 06
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Perturbations

1. Finite T
2. Inter-dot hopping

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

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Outline

• Introduction
• Is population switching a QPT?
• Coulomb gas analysis
• A surprising twist the effect of a charge sensor
• Extensions spin effects

36
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

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Extensions (I)

• Dot-lead interactions

• Mahan Anderson
• Repulsion ? continuous switching

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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

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Switching in a Luttinger liquid (I)

• Density Matrix RG calculations

• Luttinger liquid parameter g3/4
• Soft boundary conditions

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Switching in a Luttinger liquid (II)

• Finite size scaling

W
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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