1. Coupling quantum dots to leads:Universality and QPT
Richard Berkovits
Bar-Ilan University
Moshe Goldstein (BIU),
Yuval Weiss (BIU)
and Yuval Gefen (Weizmann)

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

4. Level population Vg energy (Spinless) n1, n2 Vg e1 e2+U 2 2 2 2 1 1 1

5. 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’]

6. 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.]

7. 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)

9. Furusaki-Matveev prediction
Discontinuity in the occupation of a level coupled to a Luttinger liquid with g<½
PRL 88, (2002) 1 n0 e0 eF

10. 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

11. Numerical method:
Density Matrix Renormalization Group (DMRG)

12. Infinite size DMRG

13. Finite size DMRG
Iteration improve dramatically the accuracy

14. 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)

15. 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

16. Results: Furusaki-Matveev jumpn0 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, (2008).

17. 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

18. 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

19. 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:

20. 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, (2009)

21. 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]

22. 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

23. 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

24. 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

25. 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

26. Population: DMRG (A)
Density matrix renormalization group calculations on tight-binding chains: L=100vs/vF and G0=10-4tlead [tlead – hopping matrix element]

27. Population: DMRG (B)
Density matrix renormalization group calculations on tight-binding chains: L=100vs/vF and G0=10-4tlead [tlead – hopping matrix element]

28. Differential capacitance vs. a
FES

29. 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:

30. 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

31. 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

32. 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]

33. 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

34. Abrupt population switching
Soft boundary
conditions

35. Finite size scaling for LL leadsW

36. 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

37. 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

38. 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

39. A different perspective
Detector constantly measures the level population
Population dynamics suppressed: Quantum Zeno effect
Sensor may induce a phase transition

40. 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