SU(2) and SU(4) Kondo effect in double quantum dots
SU(2) and SU(4) Kondo effect in double quantum
Jernej Mravlje∗, Anton Ramšak†,∗ and Tomaž Rejec†,∗
∗Jožef Stefan Institute, Jamova 39, Ljubljana, Slovenia
†Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, Slovenia
Abstract. We investigate serial double quantum dot systems with on-site and inter-site interaction
by means of Schönhammer-Gunnarsson projection-operator method. The ground state is established
by the competition between extended Kondo phases and localized singlet phases in spin and
charge degrees of freedom. We present and discuss different phases, as discerned by characteristic
correlation functions. We discuss also how different phases would be seen in linear transport
measurements.
Keywords: Kondo effect, double quantum dots, conductance
PACS: 73.23.-b, 73.63.Kv, 72.15.Qm
INTRODUCTION
In the last decade the advances in experimental techniques enabled the exploration
of intriguing many-body effects occurring in solid-state systems such as the Kondo
effect [1] by means of measuring the conductance of nanoscale electrical circuits. Tiny
pools of electrons defined by electrodes – quantum dots (QDs) – constitute artificial
atoms/molecules. Additional gates enable tuning of the orbital levels as well as the
tunneling rates, which makes systematic exploration of various effects experimentally
accessible. The Kondo effect is essentially the increased scattering rate (with phase shifts
near π/2) at low temperatures due to magnetic impurities in host metals. In transport
experiments through quantum dots it is seen in another disguise: it is discerned as
the amplification of the conductance towards unitary limit. Interesting way to proceed
further is to analyze the consequences of inter-impurity interaction by looking at the
transport through double quantum dot (DQD) systems.
The characteristic feature of the two-impurity Kondo physics is that the two impurities
either form an inter-impurity singlet, which is virtually decoupled from conduction elec-
trons or they form a double Kondo state SU(2)×SU(2), in which each spin characterized
by the SU(2) symmetry group is screened by the conduction electrons [2] depending
on the scales of the energies of the inter-impurity singlet formation J and Kondo state
formation TK . When the symmetry of the Hamiltonian is larger the Kondo temperature
is enhanced. For double quantum dots, which have the capacitative interaction V tuned
near the value of the on-dot interaction U , the SU(4) Kondo effect occurs [3].
Here we report our results on the competition between extended Kondo and localized
singlet phases in serial DQD systems with inter-dot interaction in the point of particle-
hole symmetry [4] and discuss also the phases which occur outside this point. The SU(4)
Kondo phase cannot be explored directly by transport experiment through a DQD as the
http://arxiv.org/abs/0704.1979v1
conductance is small irrespective of whether the system is in the SU(4) Kondo state
or not. Nevertheless, the scale of the SU(4) condensation energy can be estimated by
tuning the system away from the point of SU(4) symmetry until the SU(4) Kondo state
collapses. The boundary is easy to discern from the conductance data as the conductance
is unity whenever the crossover between the phases takes place.
MODEL AND METHOD
We model DQDs by the two-impurity Anderson Hamiltonian H = Hd +Hl, where Hd
corresponds to the isolated dots
Hd = ∑
i=1,2
(εni +Uni↑ni↓)+Vn1n2 − t ∑
(c†1σ c2σ +h.c.),
with ni = ni↑+ni↓, niσ = c
iσ ciσ . The dots are coupled by a tunneling matrix element t
and a capacitive V term. The on-site energies ε and the Hubbard repulsion U are taken
equal for both dots. Hl describes the noninteracting left and right tight-binding leads
with hopping parameter t0 and the coupling of the leads to the DQD. We denote the
characteristic tunneling rate of an isolated electron from the dot to the lead by Γ = t ′2/t0,
where t ′ is the parameter characterizing the dot-lead hopping.
To calculate the ground state of the system we use the Schönhammer and Gunnarsson
projection-operator basis [5, 6] |Ψλλ ′〉 = Pλ1Pλ ′2
, which consists of projectors Pλ i;
P0i =
1−ni↑
1−ni↓
, P1i = ∑σ niσ (1−niσ̄ ), P2i = ni↑ni↓ and additional operators
involving the operators in leads. We used up to ∼ 100 additional combinations of
operators consisting of, for example, P3i =P0iv̂P1i, where v̂ denotes the tunneling to/from
dot i. These operators are applied to the state
, which is the ground state of the
auxiliary noninteracting DQD Hamiltonian of the same form as H, but with U,V = 0,
renormalized parameters ε, t, t ′ → ε̃, t̃, t̃ ′ and additional parameter t̃ ′′ which corresponds
to hopping from left dot to right lead and vice versa which although absent in the original
Hamiltonian is present in the effective Hamiltonian in some parameter regimes.
The conductance is calculated using the sine formula [7], G = G0 sin
2[(E+ −
E−)/4t0L], where G0 = 2e
2/h and E± are the ground state energies of a large auxiliary
ring consisting of L non-interacting sites and an embedded DQD, with periodic and
anti-periodic boundary conditions, respectively.
GROUND STATE AND CONDUCTANCE OF DQD WITH
INTER-DOT INTERACTION
Detached DQDs
The starting point towards the understanding of the ground state of DQDs are the
filling properties of isolated DQDs (i.e. of the Heitler-London or the two-site Hubbard
model). The first electron is added when ε = t, and the second when ε =−t +J+[(U +
V )−|U −V |]/2, where J = [−|U −V |+
(U −V )2 +16t2]/2 is the difference between
singlet and triplet energies. When n = 2 the ground state is [α(|↑↓〉− |↓↑〉)+β (|20〉−
|02〉)]/
2, where α/β = 4t/(V −U +
(U −V )2 +16t2). The range of ε where single
occupation is favorable is progressively diminished when V 6=U . For large t or at (and
near) V =U the molecular bonding and anti-bonding orbitals are formed as is seen here
from α ∼ β .
Attached DQDs and conductance
As we attach DQDs to the leads the ground state either is or is not reminiscent of the
ground state of the isolated system. Here the latter possibility is always due to some kind
of the Kondo effect. In the top panels of Fig. 1 the ground state of DQDs are presented
with pictograms for V = 0,U on the left and right, respectively. The near vertical dividing
lines correspond to values of parameters where the ground state of the isolated system is
degenerate due to matching energies of states with different occupancies, for example,
the rightmost line corresponds to E(0) = 0= E(1) = ε− t. The horizontal U-shaped line
is given by J = 2.2TK, where the scale of the Kondo condensation energy is estimated
by TK =
UΓ/2exp(−πε(ε +U)/2Γ) for U/Γ = 15.
FIGURE 1. – Top panels: phases of serial DQDs for V = 0 (left) and V =U (right). The occupancy of
the DQD falls from left to right. Extended Kondo phases (with leads in pictograms) and localized singlet
phases (without leads in pictograms) occur. – Bottom panels: Conductance and spin-spin correlation of the
DQD for t above (full and dotted lines for V = 0; full, dotted and dashed lines for V =U ) and below the
localized singlet formation threshold. Note the approaching of S1 ·S2 towards −3/8 for large t indicating
the formation of the orbital singlet.
For n= 0,4 interaction between electrons (or holes) is not important, hence the ground
state is not interesting. For n = 1,3 the ground-state of the isolated DQD is a free spin
in (anti-)bonding orbital, which is, when the leads are attached, at low-temperatures
screened by conduction electrons as in ’ordinary’ single impurity Anderson model. The
most interesting part of the diagrams corresponds to n ∼ 2. Here the ground state of
the isolated system is a non-degenerate singlet but the tunneling to the leads breaks
this singlet whenever roughly twice the Kondo condensation energy exceeds the triplet
excitation energy J. For V ∼ U the J is enhanced hence the area corresponding to
SU(2)×SU(2) Kondo is diminished. Near the symmetric point, however, another kind
of the Kondo effect arises for V ∼U as a consequence of larger symmetry of the V =U
Hamiltonian, which partially restores the occurrence of the Kondo phase.
Symmetries
The Kondo effect occurs as the consequence of the degeneracy of states of iso-
lated impurities. If one looks at the ground state of two isolated impurities coupled
by a capacitative (but not tunneling) term V = U , one sees that the 6 states |σ1σ2〉,
|20〉 and |02〉 are degenerate. Indeed, by introducing the pseudospin operator [8] T̃ i =
1/2∑ll′=1,2 ∑σ c
ll′cl′σ , where τ
i are the Pauli matrices, and the combined spin-
pseudospin operators W i j = SiT̃ j, one sees that the Hamiltonian is SU(4) symmetric.
As long as the SU(4) symmetry breaking terms are small enough V −U, t . TK[SU(4)],
the ground state is an SU(4) ’spin’ screened by the electrons in the leads.
Orbital representation
A complementary way is to rewrite the Hamiltonian in the basis of orbital operators
cb,a = (c1 ± c2)/
Hd = ∑
α=a,b
εαnα +
nα↑nα↓+nα↑nᾱ↓
naσ nbσ +
Cflip−Sflip
where notation ā = b, b̄ = a is used. The last term of Hd consists of isospin-flip Cflip =
T+a T
b +h.c. and spin-flip Sflip = S
b +h.c. operators, where S
λ = c
λ↓cλ↑ = (S
† are
spin and T−λ = cλ↑cλ↓ = (T
† isospin lowering and raising operators for the orbitals
λ = b,a (or sites λ = 1,2). The full spin (isospin) algebra is closed with operators
Szλ = (nλ↑−nλ↓)/2 and T
λ = (nλ −1)/2, respectively.
When V = U , the spin- and isospin-flip terms in Hd are absent: the Hamiltonian is
mapped exactly to the two-level Hamiltonian with intra- and inter-level interaction U
with the bonding and anti-bonding levels coupled to even and odd transmission channels,
respectively. When V 6= U this mapping is no longer strictly valid: the electrons try to
avoid the inter-level repulsion by occupying aligned spin-states in different orbitals,
and the isospin-flip terms induce the fluctuations of charge between orbitals. Both
mechanisms prohibit electrons from occupying the well-defined orbital states.
FIGURE 2. Phases of DQD in the point of particle-hole symmetry. The boundaries between Kondo and
localized singlet phases are given by peaks in conductance and abrupt changes in correlation functions.
The boundaries of the orbital spin singlet state are given by S1 ·S2 = −3/16 and ∆n21 = ∆n2b on the upper
and lower side, respectively. Note the extension of the Kondo phase behind the line J = 2.2TK (dashed) at
V ∼U .
Numerical results
In the lower panels of Fig. 1 the conductance and inter-dot spin-spin correlations are
plotted. Note that the orbital picture is indeed more robust for the V =U case as indicated
by the broad plateaus in conductance corresponding to the SU(2) Kondo effect of a spin
residing in the (anti-)bonding orbitals. Moreover, J is enhanced when compared to the
V = 0 case: absence of singlet phase signalled by no peak with unitary conductance and
minor spin-spin correlation for all ε occurs only for smaller t. Note also that conductance
is small whenever the ground state is practically geometrically separable into parts. In
that case the flux can be transported out of the auxiliary ring through the boundary
between the parts, yielding zero conductance in our approach [7]. In Fig. 2 we indicate
the phases in the (J/TK,V/U) plane. Details are given in Ref. [4].
ACKNOWLEDGMENTS
We acknowledge the support of SRA under grant Pl-0044.
REFERENCES
1. A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press, 1993.
2. B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett. 61, 125–128 (1988).
3. M. R. Galpin, D. E. Logan, and H. R. Krishnamurthy, Phys. Rev. Lett. 94, 186406 (2005).
4. J. Mravlje, A. Ramšak, and T. Rejec, Phys. Rev. B 73, 241305(R) (2006).
5. K. Schönhammer, Phys. Rev. B 13, 4336 (1976).
6. O. Gunnarsson, and K. Schönhammer, Phys. Rev. B 31, 4815 (1985).
7. T. Rejec, and A. Ramšak, Phys. Rev. B 68, 035342 (2003).
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	INTRODUCTION
	MODEL AND METHOD
	GROUND STATE AND CONDUCTANCE OF DQD WITH INTER-DOT INTERACTION
	Detached DQDs
	Attached DQDs and conductance
	Symmetries
	Orbital representation
	Numerical results
