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:
– Artificial atoms
– Single electron transistors
• Realizations:
– Semiconductor heterostructures
– Metallic grains
– Carbon buckyballs & nanotubes
– Single molecules
Level population
t 2L
L
L
1
t
(Spinless)
2
1
t2R
R
U
R
1
t
n1, n2
Vg
energy
2
1
2+U
2
2
2
1
F
1
F
F
F
1
1
Vg
Population switching
2
t 2L
energy
2
L
t
R
U
R
1
t
1
2
1
1
F
F
1
L
1
(Spinless)
t2R
F
2
2
F
1
[Weidenmüller et. al. `97, `99,
Silvestrov & Imry ’00 …]
n1, n2
Vg
2
2+U
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)
  V
2(   0 ) 1
n  arctan



2
1
0
2
Furusaki-Matveev prediction
Discontinuity in the
occupation of a level
coupled to a Luttinger liquid
with g<½
n0
1
PRL 88, 226404 (2002)
F
0
Model
• A single level quantum dot coupled to
– a Fermi Liquid (FL)
– a Luttinger Liquid (LL)
– a Charge Density Wave (CDW)
• Spinless electrons

Hˆ   0 aˆ  aˆ  H Lead ˆ  x  , ˆ x 
 t DL



1 


aˆ ˆ 0 ˆ 0aˆ  U DL  aˆ aˆ   :ˆ 0ˆ 0 :
2



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
H  t  (c j c j 1  c j 1c j )  U  (n j 1 / 2)(n j 1 1 / 2)
-1
XY
1
AFM
D
Phase separation -2
LL
2
CDW
U/t
FM
Half filling
Non interacting point
1
Filling
0
0.5
0.25  g  0.5
1  g  0 .5
2
0
U/t
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
DE 
vc
L
D2 
vc
gL
By fitting both curves to a polynomial in 1/L and
calculating the ratio of the linear coefficients
Results: Furusaki-Matveev jump
n0
L=300
1
L=100
F
0
 ≈ 0.13;
W
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
U DL
Dot-lead interaction
t DL
Dot-lead hopping
g
Vs
o
LL parameter
Velocity
Density at wires
edge
aFES
Fermi Edge Singularity
parameter
0
Renormalized level width
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):
n
1
–
0
+
–
N  2  0
 3  0
 
Z   0 0 
N 0   

S  i  
2N

0
0
  1
i  j 1
d 1
i j

0
+
–
d 2
 2 N  0
0
...

0
+
b
d 2 N 1 d 2 N

0

0 / b
a FES ln 
 sin   i   j / b

b
0
0

exp  S  i 
2N

i
   0   1  i
i 1


0: short time cutoff; 0: (renormalized) level width; aFES: Fermi edge singularity exponent
Coulomb gas parameters
Fermi liquid
aFES
0
 2
1  0U DL  
1

tan


 
2



0 teff
Bosonization
2
1
g
 U DL g 
1 


v
S 

General case
1  2d eff 
g
1 
g


2
2
teff  t DL cos 0U DL 2
 0 t DL
0 teff
2
2
2
teff  t DL cos d eff
0: 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
1 
d

tan
(XXZ) chain, from the Bethe Ansatz: eff

U DL
 2 1  U lead / 2tlead 
2




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, 0 and 0
• 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 0:
• For aFES<2, low energy physics is governed by a single
energy scale (“Kondo” temperature) and; Thus, for small
0, ndot  1 / 2 ~  0 / TK where: TK  00 1 2a FES  0
No power law behavior of the population in the dot!
Tk is reduced by repulsion in the lead or attractive dot-lead interaction, and viceversa
• When aFES>2,
population is
discontinuous
as a function
of 0 [Furusaki and
Matveev, PRL 2002]
Physical insight: Competition
of two effects
(I) 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
• Without interactions:
F
S ( ) ~  (  0 )
• Anderson orthogonality
catastrophe (’67):
S ( ) ~  (  00))(  0 )a
energy
Absorption spectrum:
0
e
orth
a orth  (d /  )
2
• Mahan exciton effect (’67): S()
aaorth
orth  a exciton
)( 
)
S ( ) ~S(() ~00()
)

0 0
–––
–––
–––
noninteracting
Anderson
Mahan
a exciton  2 (d /  )
0
0

X-ray singularity physics (II)
Assume g=1 (Fermi Liquid)
t

 U DL 
d eff  tan 

 2 
1
e
U
Mahan
exciton
Scaling
dimension:
vs.
Anderson
orthogonality
2
2
1  2d eff 
1  2d eff  2d eff  
 
 1 
 
1 
2
  2 



 
For U>0 (repulsion)
<1  relevant
>
Mahan wins: Switching is continuous
X-ray singularity physics (III)
Assume g=1 (Fermi Liquid)
t
e

 U DL 
d eff  tan 

 2 
1
e
U
Mahan
exciton
Scaling
dimension:
vs.
Anderson
orthogonality
2
2
1  2d eff 
1  2d eff  2d eff  
 
 1 
 
1 
2
  2 



 
For U<0 (attraction)
>1  irrelevant
<
Anderson wins: Switching is discontinuous
Population: DMRG (A)
Density matrix renormalization group calculations on tight-binding
chains: L=100vs/vF and 0=10-4tlead [tlead – hopping matrix element]
Population: DMRG (B)
Density matrix renormalization group calculations on tight-binding
chains: L=100vs/vF and 0=10-4tlead [tlead – hopping matrix element]
Differential capacitance vs. a FES
Back to the original question
2
t 2L
L
1
L
1
t
L

ˆ

c
 ,k ,k cˆ,k 
  L , R;k
R
U
R
1
t
L U R
tL
Electrostatic
interaction
Hˆ 
t2R
tR
[Kim & Lee ’07,
Kashcheyevs et. al. ’07,
Silvestrov and Imry ‘07]

ˆ

a
   aˆ  UnˆL nˆR 
 L, R
R

ˆ
t
c
  ,k aˆ  H.c.
  L , R;k
Level
widths:
  2 t
2
Coulomb gas expansion

t
• One level & lead:
– Electron enters/exits
Coulomb gas (CG) of
positive/negative charges
[Anderson & Yuval ’69; Wiegmann &
Finkelstein ’78; Matveev ’91; Kamenev &
Gefen ’97]
L
tL
• Two levels & leads
L U R
tR
R
Two coupled CGs
[Haldane ’78; Si & Kotliar ‘93]
RG analysis
• Generically (no symmetries): ddlny   1 2  y   y y
d
  
  y e
15 coupled RG equations [Cardy
’81?]
d ln 
• Solvable in Coulomb valley: ddlnh   h   y e
ab
ab
ab
2
a

a
a
     U ,  
(II)
(III)
2
a
ab
b
a
e
ha  hb 2 h / 2
  b   a  b 
ha  h
11
   U ,     e1
10,11  0,1
    1    U  e
ab
ab
ha h
2


a
h11   L   R  U 
 R 
• Three stages of yRG
10,11 flow:
(I)
ab
J xy , J z
10
01
 1     U ,  
• Result: an effective Kondo model
00
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 …
L
tL
L U R
tR
R
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
g
3
4
Soft boundary
conditions
L W
x
Finite size scaling for LL leads
n
L
Vg
n
ln
 ln W
Vg
W
A different twist
L
• Adding a charge-sensor
tL
L U R
U QPC
tR
QPC
(Quantum Point Contact):
– 15 RG eqs. unchanged
– Three-component charge

e10,11  0,1  0,1, d QPC  
J z  J z  d QPC  2
population switching is discontinuous :
a quantum phase transition
R
X-ray singularity physics (I)
Electrons repelled/attracted to filled/empty dot:
tL
tR
 U 
L
eL
Mahan
exciton
Scaling
dimension:
vs.
e
R
R
Anderson
orthogonality
 d L d R  1  d L   d R  
1           
  2       

2
2
<1  relevant
>
Mahan wins: Switching is continuous
X-ray singularity physics (II)
L
e
tL
 L U eR
U QPC
e
Mahan
exciton
vs.
Anderson
orthogonality
tR
R
QPC
+
Extra
orthogonality
2
2
2


Scaling
d


d
d
1
d
d
 L
 L  R
QPC
R
dimension: 1        2             >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