Coupling quantum dots to leads:Universality and QPT
Richard Berkovits Bar-Ilan University Moshe Goldstein (BIU), Yuval Weiss (BIU) and Yuval Gefen (Weizmann) Quantum dots 0D systems: Realizations: Artificial atoms
Single electron transistors Realizations: Semiconductor heterostructures Metallic grains Carbon buckyballs & nanotubes Single molecules Level population Vg energy (Spinless) n1, n2 Vg e1 e2+U 2 2 2 2 1 1 1 Population switching energy Also relevant for:
(Spinless) 1 2 energy 2 1 2 1 1 2 Vg n1, n2 e2 e2+U [Weidenmller et. al. `97, `99, Silvestrov & Imry 00 ] Also relevant for: Charge sensing by QPC [widely used] Phase lapses [Heiblum group 97,05] Is the switching abrupt?
Yes ? (1st order) quantum phase transition No ? continuous crossover Numerical data (FRG, NRG, DMRG) indicate: No [see also: Meden, von Delft, Oreg et al.] Lets simplify the question:
Could a single state coupled to a lead exhibit an abrupt population change as function of an applied gate voltage? (i.e. a quantum phase transition) Furusaki-Matveev prediction
Discontinuity in the occupation of a level coupled to a Luttinger liquid with g0 (repulsion) Scaling dimension: Mahan wins: Switching is continuous X-ray singularity physics (III)
Assume g=1 (Fermi Liquid) e e Mahan exciton Anderson orthogonality vs. For U1 irrelevant < Anderson wins: Switching is discontinuous Population: DMRG (A) Density matrix renormalization group calculations on tight-binding chains: L=100vs/vF and G0=10-4tlead [tlead hopping matrix element] Population: DMRG (B) Density matrix renormalization group calculations on tight-binding chains: L=100vs/vF and G0=10-4tlead [tlead hopping matrix element] Differential capacitance vs. a
FES Electrostatic interaction
Back to the original question R L R L [Kim & Lee 07, Kashcheyevs et. al. 07, Silvestrov and Imry 07] Electrostatic interaction Level widths: Coulomb gas expansion One level & lead:
Electron enters/exits Coulomb gas (CG) of positive/negative charges [Anderson & Yuval 69; Wiegmann & Finkelstein 78; Matveev 91; Kamenev & Gefen 97] R L Two coupled CGs [Haldane 78; Si & Kotliar 93] Two levels & leads RG analysis Generically (no symmetries):
15 coupled RG equations [Cardy 81?] Solvable in Coulomb valley: Three stages of RG flow: 11 (I) (II) 10 01 (III) 00 Result: an effective Kondo model Arriving at 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, Silvestrov and Imry 07] Nevertheless population switching is discontinuous :
Considering Luttinger liquid (g1 irrelevant < + Anderson wins: Switching is abrupt A different perspective
Detector constantly measures the level population Population dynamics suppressed:Quantum Zeno effect Sensor may induce a phase transition Conclusions Population switching: a steep crossover,
No quantum phase transition Adding a third terminal (or LL leads): 1st order quantum phase transition Laboratory: Anderson orthogonality, Mahan exciton & Quantum Zeno effect