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