## Abstract

Observing quantum phase transitions in mesoscopic systems is a daunting task, thwarted by the difficulty of experimentally varying the magnetic interactions, the typical driving force behind these phase transitions. Here we demonstrate that in realistic coupled double-dot systems, the level energy difference between the two dots, which can be easily tuned experimentally, can drive the system through a phase transition, when its value crosses the difference between the intra- and inter-dot Coulomb repulsion. Using the numerical renormalization group and the semi-analytic slave-boson mean-field theory, we study the nature of this phase transition, and demonstrate, by mapping the Hamiltonian into an even-odd basis, that indeed the competition between the dot level energy difference and the difference in repulsion energies governs the sign and magnitude of the effective magnetic interaction. The observational consequences of this transition are discussed.

## Introduction

Quantum phase transitions (QPTs), where a system
changes its zero-temperature phase when a physical
parameter is continuously varied, is one of the focal
research areas in physics in general and condensed
matter physics in particular^{1,2}. While
usually QPTs are associated with bulk systems, they may
be observed even in mesoscopic systems. One simple
example is a single Kondo impurity - a magnetic moment
embedded in a Fermi sea^{3}. As was
pointed out by Anderson^{4}, the
system changes its character from a Kondo state, where
the electrons in the Fermi sea screen the impurity, to a
trivial state, where the magnetic interaction *J*
between the impurity and the electrons renormalizes to
zero, as the *z*-component of *J* changes
from being anti-ferromangetic (AFM) to ferromagnetic
(FM). Similarly, in a double-impurity system, one
expects, as the AFM interaction between the two
impurities is decreased, a transition from a state where
this interaction dominates, yielding a zero-spin entity
that is decoupled from the electrons, to a state where
each impurity is Kondo screened by the Fermi sea^{5}. In
addition, a phase transition has also been predicted as
the double dot system is driven from a triplet to a
singlet ground state as *J* changes sign,
similarly to the single-dot case^{6,7}. While
Kondo physics has been predicted^{8,9,10} and
observed^{11,12} in
quantum dots (QDs), observing such phase transitions in
these highly controllable systems proved a daunting
task, mainly due to the fact that it is very hard
experimentally to continuously modify the magnetic
interactions in the system. However, a transition
between singlet and triplet ground states of a single
dot can be induced experimentally by a magnetic field^{13,14} or by
changing the effective potential^{15}, leading
to a crossover from a non-Kondo to a Kondo regime.

In this paper we demonstrate that a realistic device,
consisting of two QDs with different energies and with
different inter-dot and intra-dot Coulomb interactions,
coupled in parallel to two leads, displays a QPT by
simply gate tuning the on-site energy difference between
the two QDs at the chemical potential corresponding to
the particle-hole (PH) symmetric point, where the
two-dot system is doubly occupied. (Similar setups have
already been studied, but with no energy difference
between the dots^{16} or in
the absence of inter-impurity repulsion^{17}, both of
which play an important role in our formulation.) After
demonstrating the transition numerically, employing the
numerical-renormalization-group (NRG) method, and
semi-analytically, using slave-boson mean field theory
(SBMFT), we show, by transforming the system Hamiltonian
to an even-odd basis, that the difference between the
dot energies, relative to the difference between the
inter- and intra-dot repulsions, plays the role of a
magnetic interaction, which changes its sign, from FM to
AFM, at the point where the QPT takes place. We also
discuss the nature of these two phases.

## Model

The Hamiltonian that describes the two-QD system, depicted in the inset to Fig. 1, is given by

where *m* = 1, 2 denotes the QD index, ${\stackrel{\u02c6}{n}}_{m\sigma}={d}_{m\sigma}^{\u2020}{d}_{m\sigma}$
, ${\stackrel{\u02c6}{n}}_{m}={\sum}_{\sigma}{\stackrel{\u02c6}{n}}_{m\sigma}$
(${d}_{m\sigma}^{\u2020}$
creates an electron in QD *m*
with spin *σ*), and spin-degeneracy has been
assumed (i.e. no magnetic field). In order to reduce the
number of parameters, we assume the same intra-dot
interaction on both dots, *U*_{1} = *U*_{2}
= *U*, though this assumption is not necessary, and
our results also hold when these energies are different.
Without loss of generalization, we assume *ε*_{1}
≤ *ε*_{2}. We also assume that *U*_{12}
< *U*, as one would expect experimentally. Under
these assumptions, the six two-electron states have 3
distinct energies: the 4 states |*σ*, *σ′*>,
where each dot is occupied by a single electron (of
spins *σ* and *σ′*, respectively) are
degenerate with energy 2*ε*_{1} + Δ*ε*
+ *U*_{12}, where $\mathrm{\Delta}\epsilon \equiv {\epsilon}_{2}-{\epsilon}_{1}$
, while the state $|\uparrow \downarrow \mathrm{,0}\u3009$
where the two electrons occupy the dot with the lower
energy state, has energy 2*ε*_{1} + U. The
state $|\mathrm{0,}\uparrow \downarrow \u3009$
is always higher in energy since Δ*ε* ≥ 0 and $\mathrm{\Delta}U\equiv U-{U}_{12}>0$
.
Thus, as Δ*ε* increases from zero, the degeneracy
of the ground state changes from being 4 to 5, for Δ*ε*
= Δ*U*, and then to a non-degenerate ground state
for larger Δ*ε*. It is the transition around this
special point that we concentrate upon in this paper.
The full Hamiltonian of the double-QD system, connected
in parallel to a single channel in the leads, is then
given by

where ${c}_{k\sigma}^{\u2020}$
creates an electron with
spin *σ* in the left (L) or right (R) lead in
momentum state *k*. For simplicity, the tunneling
amplitude is chosen to be momentum, site and spin
independent, *V*_{mk} = *V*.

## Results

### Numerical Renormalization Group

We first describe density-matrix numerical
renormalization group (DM-NRG) results. We used the
open-access Budapest Flexible DM-NRG code^{18,19}. The
expectation values and the transmission spectral
function see below), required for the evaluation of the
conductance through the double dot device^{20}, were
calculated, assuming, for simplicity, equal couplings to
the left and right leads, Γ = *πρV*^{2},
and equal and constant density of states *ρ* in
the two leads, with a symmetric band of bandwidth *D*
around the Fermi energy. In the following we set *U*
to be the unit of energy. The bandwidth value in the
calculations is *D* = 3.33 *U*, the intra-
and inter- dot interaction difference is Δ*U* = *U*/6
and the coupling to the leads is Γ = *U*/15.

Defining the retarded Green functions ${G}_{ij,\sigma}^{r}(t-t\prime )=-\phantom{\rule{-.25em}{0ex}}i\theta (t)\u3008\{{d}_{i\sigma}(t),{d}_{j\sigma}(t\prime )\}\u3009$
and the transmission
spectral function ${t}_{\sigma}(\omega ;\mu ,T)=\frac{1}{\pi}Im[\mathrm{\Gamma}{\sum}_{ij}{\tilde{G}}_{ij,\sigma}^{r}(\omega )]$
, where ${\tilde{G}}_{ij,\sigma}^{r}(\omega )$
is the Fourier transform of ${G}_{ij,\sigma}^{r}(t)$
,
the current is given by^{20}

Figure 1a
depicts a two-dimensional plot of transmission spectral
function at the particle-hole symmetry chemical
potential, ${\mu}_{PH}\equiv {\epsilon}_{1}+(U+2{U}_{12}+\mathrm{\Delta}\epsilon )/2$
,
as a function of energy *ω* and energy difference
Δ*ε* between the two dots, for a fixed Δ*U*
and low temperature *T* = 3 · 10^{−6} *U*.
The most striking feature of the data is the sharp
change of behavior near $\mathrm{\Delta}\epsilon \simeq \mathrm{\Delta}U$
.
For $\mathrm{\Delta}\epsilon \lesssim \mathrm{\Delta}U$
(lower part of Fig. 1a)
the transmission spectral function displays a sharp peak
at the Fermi level (Fig. 1b),
while for $\mathrm{\Delta}\epsilon \gtrsim \mathrm{\Delta}U$
there is a sharp dip at the Fermi level inside a wider
peak (Fig. 1c).
The transition between these two regimes is very sharp:
Fig. 1d
displays the linear response conductance at that
chemical potential (which is proportional to the
transmission spectral function at *ω* = 0), as a
function of Δ*ε*/Δ*U*. We see that the
conductance *G* drops sharply from *G* = 2*e*^{2}/*h*
to nearly zero, indicating a quantum phase transition.
In agreement with this interpretation, the critical
regime becomes wider with increasing temperature^{1,2}. We will
demonstrate below why this transition is indeed a QPT.
The phase for $\mathrm{\Delta}\epsilon \ll \mathrm{\Delta}U$
is relatively well understood: there are 4 degenerate
states, as each dot is singly occupied with either spin.
The states $|\uparrow ,\uparrow \u3009$
,
$\mathrm{1/}\sqrt{2}(|\uparrow ,\downarrow \u3009+|\downarrow ,\uparrow \u3009)$
and $|\downarrow ,\downarrow \u3009$
form an *S* = 1 entity (see also^{21}), while
$1/\sqrt{2}(|\uparrow ,\downarrow \u3009-|\downarrow ,\uparrow \u3009)$
is an *S* = 0 entity,
that for equal coupling of the two dots to the leads,
is, in fact, decoupled from the triplet (no tunneling
through the leads). Thus this phase corresponds to the
underscreened Kondo impurity^{7,22}. On the
other hand, for $\mathrm{\Delta}\epsilon \gg \mathrm{\Delta}U$
the first dot is doubly occupied, while the second one
is empty, and thus there is no net magnetic moment on
the double-dot system. Accordingly one would expect a
small, finite contribution to the spectral function at *ω*
= 0, and to the conductance, from the tails of the
standard Coulomb blockade peaks. However, we find that
the conductance there is exactly zero, within numerical
accuracy. This is due to a sharp dip at the transmission
spectral function in the middle of a wider peak, that
reaches all the way to zero (Fig. 1c).
As we will show below, this is due to a two-stage
screening mechanism.

In order to further demonstrate the peculiar role
played by the chemical potential *μ*_{PH},
we plot in Fig. 2a
the conductance as a function of chemical potential, for
different values of Δ*ε*, at the same temperature,
*T* = 3 · 10^{−6} *U*. Indeed, while
for Δ*ε* < Δ*U* the conductance exhibits a
sharp Kondo peak in the middle of the double-occupation
valley (A conductance peak at the PH symmetric point,
similar in appearance to our case, also appears near the
singlet-triplet transition, tuned by a magnetic field as
studied in^{23}), the
conductance *G*(*μ*_{PH})
vanishes there around Δ*ε* = Δ*U*, in
accordance with the vanishing of the transmission
spectral function at the chemical potential.
Intriguingly, as shown in Fig. 2b,
as temperature is increased, *G*(*μ*_{PH})
for that value of Δ*ε* increases, developing a
sharp resonance, which is suppressed for yet higher
temperatures. This might be expected from the form of
the transmission spectral function: as the temperature
increases beyond the width of the dip, the conductance
should increase, and then decrease when the temperature
becomes larger than the wider peak. The dependence of *G*(*μ*_{PH})
on temperature, for various Δ*ε*, is plotted in
Fig. 2c.
Such a non-monotonic dependence of the conductance on
temperature, as seen for $\mathrm{\Delta}\epsilon /\mathrm{\Delta}U\gtrsim 1$
,
is usually expected in the context of two-stage Kondo
screening, with *T*_{K1} and *T*_{K2}
the temperatures for the first and second stages. As the
temperature *T* is reduced below *T*_{K1}
the conductance starts to rise, but then falls towards
zero for *T* < *T*_{K2}. As
seen in Fig. 2,
*T*_{K2} becomes non-zero at $\mathrm{\Delta}\epsilon /\mathrm{\Delta}U\simeq 1$
,
and increases sharply with increasing Δ*ε*/Δ*U*.
Interestingly, for large enough values of Δ*ε*/Δ*U*,
*T*_{K2} becomes larger than *T*_{K1}
and the first stage does not fully form - the
conductance decreases monotonically to zero with
decreasing temperature due to local singlet formation.

### Slave-Boson Mean-Field Theory

In order to gain more insight into the physics behind
the transition and in order to look beyond the linear
response regime, we have employed the slave-boson
mean-field approach in the Kotliar-Ruckenstein (KR)
formulation^{24}. In
this method (see *Methods*) one ends up with an
effective non-interacting Hamiltonian, with renormalized
parameters ${\tilde{\epsilon}}_{m}$
and ${\tilde{V}}_{m}$
, which, on average, obey the
same constraints as the full interacting Hamiltonian.
The transmission spectral function for the effective
non-interacting model can then be easily expressed in
terms of the renormalized parameters,

where ${\tilde{\mathrm{\Gamma}}}_{m}=\pi \rho {\tilde{V}}_{m}^{2}$
(note that even if *V*_{1}
= *V*_{2} in the original Hamiltonian, the
renormalized parameters ${\tilde{V}}_{m}$
do not have to be the same
when ${\epsilon}_{1}\ne {\epsilon}_{2}$
). The temperature and
chemical potential dependence of *t*(*ω*; *μ*,
*T*) arise from the dependence of the renormalized
parameters ${\tilde{\epsilon}}_{m}$
and ${\tilde{\mathrm{\Gamma}}}_{m}$
on these parameters. The
resulting conductance, as a function of chemical
potential, for the special point Δ*ε* = Δ*U*
is depicted in Fig. 3a
for $\mathrm{D}=3.33\phantom{\rule{.25em}{0ex}}U,\mathrm{\Delta}U=U/6$
and $\mathrm{\Gamma}=U\mathrm{/30}$
.
The results of the SBMFT approximation closely resemble
those of the accurate NRG calculation (except for an
overestimated width of the middle region), with the
conductance going to zero at the symmetry point, only to
increase with increasing temperature, giving rise again
to a finite-temperature Kondo effect. The temperature
dependence of the conductance at Δ*ε* = Δ*U*
in the SBMFT treatment is plotted along with the NRG
results in Fig. 2c.
The similarity between NRG and SBMFT results gives
additional credence to this approximation, at least in
this parameter regime. The above comparison was done for
${\mathrm{\Gamma}}_{NRG}=2{\mathrm{\Gamma}}_{SBMF}$
, which is a known
discrepancy between NRG and SBMFT^{25}.

In addition to the features in the linear-response
conductance, the predicted dip in the spectral function
can be probed by measuring the voltage-dependent
differential conductance *G*(*V*) through the
double-dot system for Δ*ε* = Δ*U*.
Figure 3b
depicts *G*(*V*) for several temperatures,
using the SBMFT approximation. As one might expect, *G*(*V*)
exhibits a dip at zero bias, corresponding to the shift
of the peaks in the spectral function from the Fermi
energy. At high enough voltage the Kondo effect is
suppressed, though in the SBMFT approach, this appears
as an unphysical abrupt transition^{26}.

Within SBMFT, the emergence of a Kondo peak in the
spectral function, and the resulting zero-bias anomaly
in the conductance, are due to the renormalization of
the effective energies ${\tilde{\epsilon}}_{i}$
toward the chemical
potential. For a single dot, in the Coulomb-blockade
valley corresponding to total unit occupation, each spin
state is half-occupied, on average. Thus, in order to
obtain the same occupation by an effective
non-interacting model, the energy of each spin state
lies exactly at the Fermi energy, leading to a resonance
at that energy which is interpreted as the
Abrikosov-Suhl resonance associated with the Kondo
effect (for a review, see^{3}). On the
other hand, in the single-dot Coulomb-blockade valleys
which correspond to either zero or double occupation,
the renormalized energy levels are shifted to well above
or below the Fermi level, leading to suppression of the
spectral function at the Fermi energy. Thus, in order to
understand the features in the spectral function in the
double-dot system, one needs to determine the
corresponding energy shifts. For Δ*ε* < Δ*U*
the two dots are singly occupied, and thus their
energies are degenerate, and, as in the single-dot case,
are shifted to the chemical potential, leading to a peak
at the spectral function at that energy. On the other
hand, for Δ*ε* ≥ Δ*U*, the occupation of dot 1
is larger than that of dot 2, so the energies ${\tilde{\epsilon}}_{1}$
and ${\tilde{\epsilon}}_{2}$
straddle the Fermi energy
symmetrically (since the occupation has to add up to *n*
= 2). Moreover, the two symmetric Abrikosov-Suhl
resonances on the two sides of the Fermi energy give
rise to the exact same transmission amplitude, which
interfere destructively due to a phase difference of *π*
between sub-resonance and sup-resonance transmission
through the individual dots. This interference is the
origin of the central dip and the finite-temperature
effect. Interestingly, exactly at the transition point,
where Δ*ε* = Δ*U*, the occupations of the two
dots in the SBMFT treatment are *n*_{1} =
6/5 and *n*_{2} = 4/5, so the renormalized
energies assume specific values in the mixed-valence
regime, but the conductance is still exactly zero during
the above-mentioned interference effect.

The SBMFT formulation offers an alternative point of
view of the physics of the transition. Concentrating on
the PH symmetry point, *μ* = *μ*_{PH},
the constraints of the SBMFT equations require ${\tilde{\epsilon}}_{1}=-\phantom{\rule{-.25em}{0ex}}{\tilde{\epsilon}}_{2}$
and ${\tilde{V}}_{1}={\tilde{V}}_{2}$
. Transforming into the
even-odd basis, ${d}_{oe\sigma}=\frac{1}{\sqrt{2}}({d}_{1\sigma}\pm {d}_{2\sigma})$
,
the effective SBMFT Hamiltonian takes the following
form:

So in the even-odd language, the leads are only coupled
to the even state, with possible hopping between the
even and the odd state, which is proportional to ${\tilde{\epsilon}}_{1}$
. As was shown above, for Δ*ε*
< Δ*U* the SBMFT equations lead to ${\tilde{\epsilon}}_{1}=0$
,
resulting in a single state coupled to the leads,
exactly on resonance, which is the standard SBMFT Kondo
solution. On the other hand, for Δ*ε* ≥ Δ*U* ${\tilde{\epsilon}}_{1}$
becomes finite and grows,
thus allowing for tunneling between the even and odd
states, resulting in splitting of the energies
symmetrically around the Fermi energy. This is identical
to the results of the SBMFT calculation for side-coupled
quantum dot (depicted in the inset to Fig. 4b)^{27}, where
Δ*ε* − Δ*U* plays the role of the effective
magnetic interaction between the dots. Thus in this
language, the QPT discussed above is expressed in terms
of the standard Kondo transition from a FM to AFM
interaction^{4}.

### The Hamiltonian in the language of even-odd states

Motivated by the insight provided by the SBMFT results, we rewrite the Hamiltonian (Eq. 1) in the even-odd basis,

where the parameters of the new Hamiltonian are ${\epsilon}_{av}=\frac{1}{2}({\epsilon}_{1}+{\epsilon}_{2}),t=\frac{1}{2}\mathrm{\Delta}\epsilon $
,
$\tilde{U}=U-\phantom{\rule{-.25em}{0ex}}\mathrm{\Delta}U\mathrm{/2,}{\tilde{U}}_{12}=U-\phantom{\rule{-.25em}{0ex}}3\mathrm{\Delta}U\mathrm{/4,}$
$\phantom{\rule{.25em}{0ex}}{J}_{F}=-\phantom{\rule{-.25em}{0ex}}\mathrm{\Delta}U$
,
and *A* = Δ*U*/2. The parity ladder operators
are defined ${P}_{\sigma}^{+(-)}={c}_{e(o),\sigma}^{\u2020}{c}_{o(e),\sigma}$
and will contribute very
little to the dynamics in the given range. The full
Hamiltonian, including tunneling and leads becomes

which shows, as before, that the odd level is entirely
decoupled from the leads. While most of the specific
values are of little qualitative importance, there are
two new terms which draw the most interest. The hopping
amplitude *t* gives rise to an effective AFM
interaction *J*_{AF} = 2*t* =
Δ*ε*(unlike the usual side coupled effective
interaction ${J}_{AF}\sim \frac{4{t}^{2}}{\mathrm{\Delta}U}$
, here the extra terms of *J*_{F}
= −Δ*U* and *A* = Δ*U*/2 change the
spectrum so that ${J}_{AF}=2t=\mathrm{\Delta}\epsilon $
)
0.4 in the *n*_{e} = *n*_{0}
= 1 subspace, while the interaction *J*_{F}
is a FM one. As a result, the effective Kondo model
would represent a conduction spin coupled AFM to a
localized spin which is further coupled to another spin
via ${J}_{tot}={J}_{AF}+{J}_{F}=\mathrm{\Delta}\epsilon -\mathrm{\Delta}U$
(There is an additional n = 2 level which is degenerate
with the spin-triplet state. This level does not incur
spin-flip processes through the leads and does not
contribute to the Kondo screening.) The competition
between *J*_{F} and *J*_{AF}
governs the quantum phase transition between the two
phases mentioned. This analysis is supported by
Fig. 4,
which shows the spin-spin correlation between the even
and odd states, an indicator for the sign of the
magnetic interaction and its magnitude. The transition
from FM to AFM occurs a little below $\mathrm{\Delta}\epsilon /\mathrm{\Delta}U\simeq 1$
due to level renormalization through the continuum, a
temperature dependent effect. For FM ${J}_{tot}<\phantom{\rule{-.25em}{0ex}}0(\mathrm{\Delta}\epsilon <\mathrm{\Delta}U)$
the two localized spins form an *S* = 1 triplet,
leading to an under-screened Kondo impurity^{6} (with an
additional, uncoupled level with *S* = 0, which
gives rise to $\u3008{\overrightarrow{S}}_{e}\cdot {\overrightarrow{S}}_{o}\u3009=0.25\cdot \frac{3}{4}$
). As Δ*ε* increases
further $\u3008{\overrightarrow{S}}_{e}\cdot {\overrightarrow{S}}_{o}\u3009$
become negative, eventually saturating at $0.75\cdot \frac{1}{2}$
, as the ground state for $\mathrm{\Delta}\epsilon \gg \mathrm{\Delta}U$
is an equal superposition of the two singlets, $\frac{1}{\sqrt{2}}\phantom{\rule{0.25em}{0ex}}(|\uparrow \downarrow \mathrm{,0}\u3009-|\mathrm{0,}\uparrow \downarrow \u3009)$
and $\frac{1}{\sqrt{2}}\phantom{\rule{0.25em}{0ex}}(|\uparrow ,\downarrow \u3009-|\downarrow ,\uparrow \u3009)$
. The transition from
positive to negative spin-spin correlation function
demonstrates that indeed Δ*ε* − Δ*U* plays the
role of the magnetic interaction which drives the system
between the two phases.

## Discussion

We have demonstrated in this paper that for a realistic
double quantum-dot device, one can tune the system
through a quantum phase transition, leading to a sharp
change in the conductance and in the shape of the
zero-bias anomaly. The transition between the reported
phases and their respective conductance signatures have
been studied here as two facets of the same effect. On
one hand it is described by the destructive vs
constructive interference between the two dot branches.
On the other side it is described by the transition
between ferro- or anti-ferromagnetic interaction between
two impurities in a side coupled setup, which are
relatively well understood. The predicted features in
the conductance, either as a function of gate voltage
(Fig. 2a,b)
or as a function of temperature (Fig. 2c),
can be easily checked in the double-dot setup, depicted
in the inset of Fig. 1,
where each dot is controlled by a different gate
voltage. The intra- and inter-dot interactions, *U*
and *U*_{12}, and consequently Δ*U* =
*U* − *U*_{12}, are usually
determined by the geometry of the system and cannot be
easily modified. However the energy difference between
the two dots, $\mathrm{\Delta}\epsilon =\left|{\epsilon}_{1}-{\epsilon}_{2}\right|$
, can be readily tuned. Thus,
for a given setup, one can change this relative voltage
until the Kondo peak at the valley midpoint is
suppressed and the conductance vanishes. As discussed
above, this should happen around the point where Δ*ε*
reaches the value Δ*U*. While in this paper we have
concentrated on the single-channel case, where both dots
are connected to the same channel in the leads, one can
easily extend the calculation to the general case of
mixed channels, where the lead states the two dots
couple to are not identical. Even in this case one
expects that the transition studies here will also give
rise to an observable effect as long as these lead
wave-functions are not orthogonal. Thus, we hope that
the results presented here will stimulate experiments in
this direction.

## Methods

Within the KR slave-boson framework^{24}, the
Hamiltonian is replaced by an effective non-ineracting
Hamiltonian where slave bosons are added as an
accounting tool for the different many-body states of
the system. In order to accommodate the larger Hilbert
space of the two-dot Hamiltonian, an enlarged boson
space is introduced^{25}, where
each boson accounts for one of the impurity dimer
states: *e* for empty dimer, *p*_{σm}
for single occupation of an electron with spin *σ*
on dot *m*, *x*_{m} for two
electrons on dot *m*, ${y}_{s\ell}$
for two electrons on different
dots with total spin *s* and z-component of total
spin $\ell $
,
*h*_{σm} for triple occupation, with
a missing electron on dot *m* with spin *σ*,
and *b* for four electrons in the dimer. Since the
system can be in only one of these states (or a linear
combination with total weight equal to unity), the
bosons are constrained to have unit total occupation $I={e}^{\u2020}e+{\sum}_{\sigma m}{p}_{\sigma m}^{\u2020}{p}_{\sigma m}+{\sum}_{i}{x}_{i}^{\u2020}{x}_{i}+{\sum}_{sl}{y}_{sl}^{\u2020}{y}_{sl}+{\sum}_{\sigma m}{h}_{\sigma m}^{\u2020}{h}_{\sigma m}+{b}^{\u2020}b$
. Given the
boson operators, the electron number operators are
constrained to be ${\stackrel{\u02c6}{n}}_{\sigma m}={\stackrel{\u02c6}{Q}}_{\sigma m}$
, where ${\stackrel{\u02c6}{Q}}_{\sigma m}={e}^{\u2020}e+{p}_{\sigma m}^{\u2020}{p}_{\sigma m}+{\sum}_{m}{x}_{m}^{\u2020}{x}_{m}+{y}_{12\sigma}^{\u2020}{y}_{12\sigma}+$
$\frac{1}{2}{\sum}_{s}{y}_{s0}^{\u2020}{y}_{s0}+{h}_{-\sigma m}^{\u2020}{h}_{-\sigma m}+{\sum}_{\sigma}{h}_{\sigma -m}^{\u2020}{h}_{\sigma -m}+{b}^{\u2020}b$
. Including
these constraints, the Hamiltonian (2)
can be exactly mapped into the slave-boson Hamiltonian,
${H}_{SB}={H}_{2QD}+{H}_{lead}+{H}_{cons}$
, where: