Coupling quantum dots to leads:Universality and QPT Richard Berkovits Bar-Ilan University Moshe Goldstein (BIU), Yuval Weiss (BIU) and Yuval Gefen (Weizmann)
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Transcript Coupling quantum dots to leads:Universality and QPT Richard Berkovits Bar-Ilan University Moshe Goldstein (BIU), Yuval Weiss (BIU) and Yuval Gefen (Weizmann)
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 0aˆ aˆ H Lead ˆ x , ˆ x
1
t DL aˆ ˆ 0 ˆ 0aˆ U DL aˆ aˆ :ˆ 0ˆ 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 (cj c j 1 cj 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
0 0
Z
N 0
S i
–
+
N 2 0
2N
0
0
1
i j 1
d 1
i j
3 0
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 tDL cos 0U DL 2
0 t DL
0 teff
2
2
2
teff tDL 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]
2
2
1
U
/
2
t
lead
lead
• In particular, for the Nearest-Neighbor
1
d
tan
(XXZ) chain, from the Bethe Ansatz: eff
U DL
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 00 1 2a 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:
energy
Absorption spectrum:
F
S () ~ ( 0 )
• Anderson orthogonality
catastrophe (’67):
S () ~ ( 00))( 0 )aorth
e
a orth (d / )
2
• Mahan exciton effect (’67): S()
S () ~S(()~00())(
0 )0)aaorth
orth
0
aexciton
–––
–––
–––
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
L
1
t
L
tL
L
Electrostatic
interaction
Hˆ
ˆ
c
,k ,k cˆ,k
L , R;k
1
t2R
R
U
R
1
t
U
R
tR
[Kim & Lee ’07,
Kashcheyevs et. al. ’07,
Silvestrov and Imry ‘07]
ˆ
a
aˆ UnˆLnˆR
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 , e1
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, dQPC
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:
L
tL
Mahan
exciton
Scaling
dimension:
eL
U
R
vs.
Anderson
orthogonality
tR
d L d R 1 d L d R
1
2
2
2
e
R
<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