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

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... Anderson orthogonality, Mahan exciton, quantum Zeno effect, entanglement ... Artificial atoms Single electron transistors R L Vg Traditional regimes ... – PowerPoint PPT presentation

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Title: 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
2
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

8
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
11
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

13
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
15
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
16
Coulomb gas expansion (II)
  • Two coupled CGs
  • Haldane 78 Si Kotliar 93

Two levels leads
17
Coulomb gas expansion (III)
  • CG can be rewritten as
  • Cardy 81 Si Kotliar 93

18
RG analysis (I)
  • Generically (no symmetries)
  • 15 coupled RG equations Cardy 81 Si Kotliar
    93

6 eqs.
6 eqs.
3 eqs.
19
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
20
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
21
Kondo CG analysis
  • Anderson Yuval 69
  • Anisotropic model (Jz?Jxy)
  • expand in Jxy Coulomb gas of spin-flips

22
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
23
Back to our problem
11
(spinless)
10
01
  • Pseudo-spin (orbital) Kondo
  • Anisotropic
  • Vg changes effective level separation ? switching

00
24
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

25
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)

26
Outline
  • Introduction
  • Is population switching a QPT?
  • Coulomb gas analysis
  • A surprising twist the effect of a charge sensor
  • Extensions spin effects

27
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

28
Reminder X-ray edge singularity
Absorption spectrum
energy
  • Without interactions

w0
  • Anderson orthogonality catastrophe 67

e
noninteracting
  • Mahan exciton effect 67

Anderson
Mahan
29
X-ray singularity physics (I)
Virtual fluctuations
e
e
30
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
31
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
32
A different perspective
  • Detector constantly measures the level population
  • Population dynamics suppressed Quantum Zeno
    effect
  • A sensor may induce a phase transition

33
Noninvasive charge sensing?
Use Friedels sum rule!
  • continuous switching

abrupt switching
? ? TK
?GL?
?GL?
?GL?
?GL?
CIR Meden Marquardt 06
34
Perturbations
  1. Finite T
  2. Inter-dot hopping
  • First order transition ?
  • switching smeared linearly in T, tLR

35
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

37
Extensions (I)
  • Dot-lead interactions
  • Mahan Anderson
  • Repulsion ? continuous switching

38
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

39
Switching in a Luttinger liquid (I)
  • Density Matrix RG calculations
  • Luttinger liquid parameter g3/4
  • Soft boundary conditions

40
Switching in a Luttinger liquid (II)
  • Finite size scaling

W
41
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
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