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 [Weidenmüller 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 g<½ PRL 88, 226404 (2002) 1 n0 e0 eF

Model A single level quantum dot coupled to Spinless electrons a Fermi Liquid (FL) a Luttinger Liquid (LL) a Charge Density Wave (CDW) Spinless electrons

Numerical method: Density Matrix Renormalization Group (DMRG)

Infinite size DMRG

Finite size DMRG Iteration improve dramatically the accuracy

Model and phase diagram for the wire -1 U/t CDW LL 1 Phase separation -2 2 FM XY AFM D Non interacting point Half filling 0.5 Filling 1 U / t 2 Haldane (1981)

Evaluating the Luttinger Liquid parameter g g can be evaluated by calculating the addition spectrum and the energy of the first excitation, since By fitting both curves to a polynomial in 1/L and calculating the ratio of the linear coefficients

Results: Furusaki-Matveev jump n0 1 L=300 W L=100 eF e0 G ≈ 0.13; g=0.42 Slope is linear in L suggesting a first order transition in the thermodynamic limit Y. Weiss, M. Goldstein and R. Berkovits PRB 77, 205128 (2008).

Parameter space for a level coupled to a Luttinger Liquid Coupling Parameters Wire parameters LL parameter Velocity Density at wires edge Dot-lead interaction Dot-lead hopping Fermi Edge Singularity parameter aFES Renormalized level width G0

Yuval-Anderson approach The system can be mapped onto a classical model of alternating charges (Coulomb gas) on a circle of circumference b (inverse temperature): b n 1 t – + x0: short time cutoff; G0: (renormalized) level width; aFES: Fermi edge singularity exponent

Coulomb gas parameters General case Bosonization Fermi liquid aFES G0 n0: density of states at the lead edge; g, vs: LL parameters In general, deff can be found using boundary conformal field theory results [Affleck and Ludwig, J.Phys.A 1994] In particular, for the Nearest-Neighbor (XXZ) chain, from the Bethe Ansatz:

Conclusions from this mapping: Thermodynamic properties, such as population, dynamic capacitance, entropy and heat capacity: Are universal, i.e., depend on the microscopic model only through aFES, G0 and e0 Are identical to their counterparts in the anisotropic Kondo model M. Goldstein, Y. Weiss, and R. Berkovits, Europhys. Lett. 86, 67012 (2009)

Lessons from the Kondo problem For small enough G0: For aFES<2, low energy physics is governed by a single energy scale (“Kondo” temperature) and; Thus, for small e0, where: No power law behavior of the population in the dot! Tk is reduced by repulsion in the lead or attractive dot-lead interaction, and vice-versa When aFES>2, population is discontinuous as a function of e0 [Furusaki and Matveev, PRL 2002]

Physical insight: Competition of two effects Anderson Orthogonality Catastrophe, which leads to suppression of the tunneling – zero level width (II) Quasi-resonance between the tunneling electron and the hole left behind (Mahan exciton), which leads to an enhancement of the tunneling – finite level width For a Fermi liquid and no dot-lead interaction (II) wins – finite level width Attractive dot-lead interaction or suppression of LDOS in the lead (LL) suppresses (II) and may lead to (I) gaining the upper hand zero level width

Reminder: X-ray edge singularity Absorption spectrum: energy Without interactions: w0 Anderson orthogonality catastrophe (’67): e ––– noninteracting Mahan exciton effect (’67): S(w) w w0 ––– Anderson ––– Mahan

X-ray singularity physics (II) Assume g=1 (Fermi Liquid) e Mahan exciton Anderson orthogonality vs. For U>0 (repulsion) Scaling dimension: <1  relevant > Mahan wins: Switching is continuous

X-ray singularity physics (III) Assume g=1 (Fermi Liquid) e e Mahan exciton Anderson orthogonality vs. For U<0 (attraction) Scaling dimension: >1  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 (g<1) leads or attractive dot-lead Interactions will change the picture. population switching is discontinuous : a quantum phase transition

Abrupt population switching Soft boundary conditions

Finite size scaling for LL leads W

A different twist Adding a charge-sensor (Quantum Point Contact): L QPC Adding a charge-sensor (Quantum Point Contact): 15 RG eqs. unchanged Three-component charge population switching is discontinuous : a quantum phase transition

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

X-ray singularity physics (II) QPC e e e Mahan exciton Anderson orthogonality vs. + Extra orthogonality Scaling dimension: >1  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