Ferromagnetism vs.
Antiferromagnetism in Double
Quantum Dots: Results from the
Bethe Ansatz
Robert Konik,
Brookhaven National Laboratory
Hubert Saleur,
Saclay/University of Southern California
Andreas Ludwig,
University of California, Santa Barbara
Galileo Galilei Institute for Theoretical Physics
Arcetri, Firenze
October 1, 2008
Overview
Themes:
1. Show classic Tsvelik-Wiegmann exact solution
of single level Anderson model can be extended
to more generalized dot systems
2: Show that from these exact solutions, interesting
(Kondo) physics arise that is non-perturbative in
nature
3. Discuss prospects of extending exact solvability
to non-equilibrium problems
Outline
1. Brief introduction to quantum dots
2. Kondo physics
3. Bethe ansatz solution of double quantum dots
4. Exact linear response conductance at T=0
-
Friedel sum rule
RKKY mediated Kondo effect
5. Beyond T=0 linear response
-
Finite temperature
Non-equilibrium
Quantum Dots: What are they?
Small groups (N~102-103) of electrons confined to
a small enough region such that
- Electronic levels are discrete
- Dot Coulombic effects are important
Dots interact with large electron reservoirs or “leads”
- Dot-lead coupling is origin of all the
physics I will discuss
Semiconductor Dots
quantum dot
Built from gated heterostructures
Fabrication of Semiconductor Quantum
Dots
quantum dot
Gates on top of GaAs/AlGaAs
heterostructure segregate a
portion of the two dimensional
electron gas
two dimensional
electron gas (2DEG)
Gates can arranged so that
a double dot structure is
obtained (here, in parallel)
in parallel: Chang et al. PRL 92, 176801 (2004)
in series: Petta et al. PRL 93, 186802 (2004)
Double Nanotube Quantum Dots
Gates allow chemical potentials
of dots to be tuned independently
N. Mason, M. Biercuk, C. Marcus, Science 303 (2004) 655
Regime of Interest: Dots with Single Active Level
low temperatures
EF At
and/or large level spacing,
δE
only level nearest Fermi
energy is relevant
lead L
lead R
V
V
dot
lead L
lead R
Energy Scales of a Dot with a Single Level
d+U
tunnel barriers
between dot-lead
Δ~V2
μL
R
εd
energy of singly
occupied dot
Vg
gate voltage making
dot tunable
energy of second
occupation
How Does Kondo Physics Arise in a Single Level
Dot
U/2
E
F
U/2
At symmetric point of
dot (U + 2εd = 0), one
EF electron sits on the dot
level.
Isolated electron acts
like magnetic impurity:
Kondo physics
What is Kondo Physics?
Two regimes separated by TK, the Kondo temperature:
T << TK: low temperature behaviour
lead L
lead R
Dot electron forms a singlet with electrons in leads
T >> TK: high temperature behaviour
Dot electron interacts only weakly with electrons in leads
Signature of Kondo Physics, T<<TK
impurity density of states, ρ 
Dynamically generated
spectral weight at the
Fermi level
(Abrikosov-Suhl resonance)
-TK
TK

Kondo Physics in Double Quantum Dots:
Possibilities
Suppose we two electrons on the two dots and effective channel of
electrons coupling to the dots:
i)
L
R
Ferromagnetic RKKY interaction binds
electrons into triplet – underscreened
spin-1 Kondo effect (standard picture)
R
Direct singlet formation between dot
electrons – no Kondo effect
R
Singlet formation as mediated by
electrons in leads – RKKY Kondo
effect
S=1
ii)
iii)
L
L
Double Quantum Dots in Parallel
Two dots are not capacitively or tunnel coupled
Anderson-like Hamiltonian of Double Dots
lead electrons
chemical potential
of dots
– Model
– Must
dot-lead tunneling
Coulomb
repulsion
is exactly solvable via Bethe ansatz
-RMK (PRL 99, 076602 (2007), New J. Phys. 9, 257 (2007))
explicitly check, however, that the multi-particle
wavefunctions are of Bethe ansatz form
Constraints on Parameter Space
i) Model is integrable if only one channel couples:
ce/o  (VL / R1cL / R  VL / R1cR / L ) / 2 VR12  VL12
for consistency need VL1/VR1 = VL2/VR2
ii) Energy of double occupancy is the same on the
two dots
U1 + 2εd1 = U2 + 2εd2
εd1 and εd2, cannot be adjusted independently
iii) U1(V1L2+V1R2) = U2 (V2L2+V2R2)
Small perturbations on these constraints do not affect the physics
Landauer-Büttiker Transport Theory
μL
RMK, H. Saleur, A. Ludwig
PRL 87 (2001) 236801
μR
PRB 66 (2002) 125304
left lead
right lead
- With μL>μR charge flows from left to right
- L-B requires knowledge of transmission probabilities, T(ε),
for electrons with energies, ε, in the range μL>ε>μR; current J is then
- Integrability allows one to both determine the correct scattering
states to use as well as their exact scattering matrices (i.e. T(ε))
Computation of Scattering Amplitudes
Exact solvability means one can construct exact many-body eigenfunctions:
2
2
V
V
 ( p)  -2 tan-1 ( 1  2 )
p -  d1 p -  d 2
The momenta are quantized according to the Bethe ansatz equations:
gives scattering amplitude
N. Andrei, Phys. Lett. 87A, 299 (1982)
These quantization conditions can be recast as Tp  sin2((δimpp))
Linear response conductance
T=0 Linear Response Conductance in Double
Quantum Dots
two electrons
on dots forming
L
singlet; system
is p-h symmetric
one electron on dots forming a
singlet with conduction
electrons; standard Kondo effect
L
R
Rapid variation due to
interference between
dots
εd2
L
εd1
R
decreasing dots’ chemical
potential with εd1-εd2
kept fixed
R
RKKY Kondo effect
Evidence for Kondo Physics at the Particle-Hole
Symmetric Point
Impurity Density of States
Spin-Spin Correlation Function
S1 gS2
Slave boson mft
RMK + M. Kulkarni
An Abrikosov-Suhl resonance exists
Ferromagnetic like correlations exist
between dots although over all
ground state is a singlet
S1 gS2
= ¼ for ferro
S1 gS2
= -¾ for antiferro
The Friedel Sum Rule in Double Dots
Friedel sum rule states:
Scattering phase of electrons
at Fermi energy is proportional
to the number of electrons
displaced by impurity, Ndisplaced
δ = πNdisplaced
G = 2e2/h sin2(δ)
Ndisplaced has two contributions
1) Electrons on dots
2) deviations in electron
density in leads
Langreth, Phys. Rev. 150, 516 (1966)
Beyond Linear Response at T=0: Finite Temperature
and Out-of-Equilibrium
Challenge: Compute transmission probability, T(ε), at finite energies
Two difficulties in doing so:
1) Non-unique construction of electron
charge excitations
one
parameter
family
used;
reproduces
Kondo
physics
2) Does this construction of the electron
behave well under map from original
(two leads 1,2) to integrable basis
(even, odd)
Yes, maybe.
spin excitations
Finite Temperature Linear Response Conductance
in Single Dots: Universal Scaling Curve
no parameter fit
Kaminski et al.
numerical renormalization
group - Costi et al.
Finite Temperature Linear Response Conductance
in Double Quantum Dots
Dot levels well separated
L
Dot levels close together
εd1
εd2
R
Out-of-Equilibrium Conductance in a Single Dot
In zero magnetic field, ansatz fails:
1) At large voltages, the computed
conductance does not behave as
G ~ 1/log2(V/TK)
2) At small voltages, produces incorrect
coefficient, α, of second order term in voltage
2e2
G
(1  V 2 )
h
Likely culprit: Wrong choice of one parameter family
of scattering states
Why? Ansatz does work at qualitative level in large
magnetic fields where ambiguity of choice is lifted
Out-of-Equilibrium Magnetoconductance
Observations
Bethe Ansatz Computations
T=0 peak value occurs at Δμ<H; agrees
with experiments on carbon nanotube dots
Kogan et al.
Phys. Rev. Lett. 93, 166602 (2004)
 R-μL)/H
We can make similar computations for the noise at large fields and bias
(Gogolin, RMK, Ludwig, and Saleur, Ann. Phys. (Leipzig) 16, 678 (2007))
Other Strategies for Out-of-Equilibrium Transport:
Open Bethe Ansatz
Our approach: Equilibrium scattering data
out of equilibrium
results
Open Bethe ansatz: attempt to construct wavefunctions of Bethe type
with non-equilibrium boundary conditions
Mehta and Andrei,
PRL 96, 216802 (2006)
μL
μR
left lead
right lead
Wavefunctions encode
the different number
of electrons in the two
leads
Nature of Wavefunctions in Open Bethe Ansatz
In the interacting resonant level model, to construct such
wavefunctions, single particle wavefunctions with removable
singularities must be used:
|Ψ(x)|2
|Ψ(0)|2
|Ψ(0)|2 ≠ |Ψ(0+)|2≠ |Ψ(0-)|2
|Ψ(x)|2
Probabilitistic interpretation of wavefunction?
Can physics be changed by changing wavefunction on a set of
measure zero?
Conclusions
Exactly solvable variants of the Anderson model
applicable to multi-dot systems exist
- double dots in series
(RMK, PRL 99, 076602 (2007))
- dots exhibiting Fano resonances
(RMK, J. Stat. Mech L11001 (2004))
These solutions can exhibit unexpected physics:
- singlet formation in a double dot where
a triplet might be expected
Out-of-equilibrium no generic approach employing
exact solvability seems to be available
Finite Temperature Conductance Through a Single
Dot
Conductance, G
For the dot, it is the conductance
(not the resistance) that increases
as T is lowered
Kondo
singlet
free spin
TK
T