Population Switching and Charge Sensing in Quantum Dots: A case for Quantum 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, (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 R L VgVg 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 energy Level population (spinless) R L VgVg VgVg n 1, n 2    +U  , g Coulomb- blockade peak Coulomb- blockade valley 1122

VgVg n 1, n 2    +U   energy Population switching (spinless) R 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] R L QD QPC –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 ?  (1 st 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 VgVg narrow level filled narrow level empty Many levels:

Nature of the switching Is the switching abrupt? Yes ?  (1 st 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 ’07; MG & Berkovits, unpublished]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] R L Electrostatic interaction Level widths: e.g., R L

Coulomb gas expansion (I) T: temperature;  : short time cutoff;  =  |t| 2  level width 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 n 1 0  – –– Fugacity

Coulomb gas expansion (II) R L Two levels & leads Two coupled CGs [Haldane ’78; Si & Kotliar ‘93] 1/T n 1, n  – – – – – – –

Coulomb gas expansion (III) CG can be rewritten as: [Cardy ’81; Si & Kotliar ‘93] 1/T 0 

RG analysis (I) Generically (no symmetries): 15 coupled RG equations [Cardy ’81; Si & Kotliar ‘93] 6 eqs. 3 eqs

Solvable in Coulomb valley: Three stages of RG flow: RG analysis (II) (I) (II) (III) Result: an effective Kondo model [Kim & Lee ’07, Kashcheyevs et al. ’07, ‘09, Silvestrov & Imry ‘07]

Digression: The Kondo problem Realizations – Magnetic impurity – QD with odd electron number L  Problem: divergences [Kondo ’64] – susceptibility: –Similarly: resistance, specific heat … Hamiltonian – J~t 2 /U>0: exchange – h z : local magnetic field D: bandwidth (spinful)

Kondo: CG analysis 1/T SzSz 1/2 0  – –– –1/2 Anderson & Yuval [’69] : – Anisotropic model (J z ≠J xy ) – expand in J xy : Coulomb gas of spin-flips

Kondo: Phase diagram RG equations: Ferromagnetic Kondo: –impurity decoupled –susceptibility:  ~c(J)/T+… Anti-Ferromagnetic Kondo: –impurity strongly-coupled –susceptibility:  ~1/T K +… Kosterlitz- Thouless transition T K : Kondo temperature

Back to our problem … Pseudo-spin (orbital) Kondo – Anisotropic – V g changes effective level separation  switching R L R L VgVg n R, n L LL  L +U   (spinless)

Implications population switching is continuous (scale: T K ) No quantum phase transition [Kim & Lee ’07, Kashcheyevs et al. ’07, ‘09, Silvestrov & Imry ‘07] Anti-Ferromagetic Kondo model Gate voltage  magnetic field h z

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 : 1 st order quantum phase transition Adding a charge-sensor (Quantum Point Contact): –15 RG eqs. unchanged – Three-component charge R L QPC Kosterlitz- Thouless transition

Reminder: X-ray edge singularity Without interactions: ––– noninteracting 0 S(  )   ––– Anderson ––– Mahan Anderson orthogonality catastrophe [’67] : Mahan exciton effect [’67] : energy e Absorption spectrum: 

R L e e X-ray singularity physics (I) Virtual fluctuations:

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

X-ray singularity physics (II) Mahan exciton Anderson orthogonality J xy Scaling dimension: R L QPC ee e Anderson wins: Switching is abrupt <+ vs. + Extra orthogonality >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 LLLL R QPC L1L1 L2L2 VgVg n R, n L, g L LL  L +U   VgVg n R, n L, g L LL  L +U   R QPC L1L1 L2L2  K K [CIR: Meden & Marquardt ’06]

Perturbations First order transition  switching smeared linearly in T, t LR 1.Finite T 2. Inter-dot hopping: R L QPC

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] R L   F  B

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

Extensions (II) Luttinger-liquid leads: –Repulsion  abrupt switching R L QPC 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]

R L 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) Finite size scaling:

Conclusions Population switching: –Usually: steep crossover, no quantum phase transition –Adding a charge sensor: 1 st order quantum phase transition Laboratory for various effects: – Anderson orthogonality, Mahan exciton, quantum Zeno effect, entanglement entropy; – Kondo