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

Population Switching and Charge Sensing in Quantum Dots : A case for Quantum Phase Transitions. PRL 104, 226805 (2010). Moshe Goldstein (Bar-Ilan Univ., Israel) , Richard Berkovits (Bar-Ilan Univ., Israel) , Yuval Gefen (Weizmann Inst., Israel).

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

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  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 • Artificial atoms • Single electron transistors • “0D” systems: • Realizations: • Semiconductor heterostructures • Metallic grains • Carbon buckyballs & nanotubes • Single molecules

  4. R L Quantum dots:A theorist’s view Vg • Traditionalregimes: [Review: Alhassid, RMP ‘00] • Opendots, G>>D • Closeddots,G<<D • Last decade: intermediatedot-lead coupling,G<D • Interference (e.g., Fano) • Interactions (e.g., Kondo, populationswitching) D: level spacing; G: level width

  5. Level population n1, n2 1 2 Vg 2 0 e1 e2+U 1 energy R 2 L 2 1 1 Coulomb-blockade peak , g (spinless) Vg G1 G2 Coulomb-blockade valley 1

  6. Population switching energy 2 1 2 1 2 1 R L 2 1 n1, n2 1 Vg 0 e2+U e2 (spinless) [Baltin, Gefen, Hackenbroich & Weidenmüller ‘97, ‘99; Silvestrov & Imry ’00; … Sindel et al. ‘05 …]

  7. QD R QPC L 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 T=0) • Yes ?(1st order) quantum phase transition • No ? continuous crossover

  10. free energy Vg 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 T=0, for afinite 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. R L L R Basis transformation [Kim & Lee ’07, Kashcheyevs et al. ’07, Silvestrov & Imry ‘07] e.g., Electrostatic interaction Level widths:

  15. n 1 t – – – 0 + + + 1/T Coulomb gas expansion (I) Coulomb gas (CG) of alternatingpositive/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; G=pr|t|2: level width

  16. R n1, n2 1 L t – – – – – – – + 0 + + + + + + 1/T Coulomb gas expansion (II) Two coupled CGs [Haldane ’78; Si & Kotliar ‘93] Two levels& leads

  17. 11 01 10 00 11 01 11 01 10 00 00 10 00 t 0 1/T Coulomb gas expansion (III) CG can be rewritten as: [Cardy ’81; Si & Kotliar ‘93]

  18. 11 01 10 00 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 Realizations Magnetic impurity QD with odd electron number L  (spinful) • Hamiltonian • J~t2/U>0: exchange • hz: local magnetic field • Problem: divergences [Kondo ’64] • susceptibility: • Similarly: resistance, specific heat … D: bandwidth

  21. Sz 1/2 0 1/T t –1/2 – – – + + + 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: c~c(J)/T+… • Anti-Ferromagnetic Kondo: • impurity strongly-coupled • susceptibility: c~1/TK+… Kosterlitz-Thouless transition TK: Kondo temperature

  23. Back to our problem … R R L L nR, nL 1 Vg 0 eL+U eL 11 (spinless) 10 01 • Pseudo-spin (orbital) Kondo • Anisotropic • Vgchanges effective level separation  switching 00

  24. Implications • Anti-Ferromagetic Kondo model • Gate voltagemagnetic field hz population switching is continuous (scale: TK) Noquantum 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 man’s’ 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. R L QPC But … population switching is discontinuous : 1st order quantum phase transition • Adding a charge-sensor(Quantum Point Contact): • 15 RG eqs. unchanged • Three-component charge Kosterlitz-Thouless transition

  28. S(w) w0 w 0 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) R L Virtual fluctuations: e e

  30. X-ray singularity physics (I) R L Electrons repelled/attracted to filled/empty dot (Jz): e e Mahan exciton Anderson orthogonality vs. Jxy Scaling dimension: <1  relevant > Mahan wins: Switching is continuous

  31. R L QPC X-ray singularity physics (II) e e e Mahan exciton Anderson orthogonality Extra orthogonality + vs. Jxy Scaling dimension: >1  irrelevant < + Anderson wins: Switching is abrupt

  32. A different perspective • Detector constantlymeasures the level population • Population dynamics suppressed: Quantum Zeno effect • A sensormay induce a phase transition

  33. Noninvasive charge sensing? continuous switching L1 L1 R R QPC L2 QPC L2 nR, nL, gL nR, nL, gL 1 1 Vg Vg 0 0 eL+U eL eL+U eL Use Friedel’s sum rule! abrupt switching   TK GL GL [CIR: Meden & Marquardt ’06] GL GL

  34. L R QPC Perturbations • Finite T • 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, …]  B F L   R • 2-impurity Kondo with zexchange [Andrei et al. ’99, Garst et al. ‘94]

  37. L R QPC Extensions (I) • Dot-lead interactions: • Mahan&Anderson • Repulsion continuous switching

  38. R L QPC 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. R L Switching in a Luttinger liquid (I) • Density Matrix RG calculations: • Luttinger liquid parameter: g=3/4 • Soft boundary conditions:

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

  41. Conclusions • Population switching: • Usually: steep crossover, noquantum phase transition • Adding a charge sensor: 1st orderquantum phase transition • Laboratory for various effects: • Anderson orthogonality, Mahan exciton, quantum Zeno effect, entanglement entropy; • Kondo

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