Eq. 5.24 is, in some sense, trivial in PC. Since the effective model is
noninteracting, we may use the conductance in dc-regime given by the
alternative Oguri formulation in Eq.4.105, here in the present context written
as,

$\mathcal{G}^{dc}(\omega=0,T=0)=\frac{e^{2}}{h}4\Gamma_{eff}^{s}\Gamma_{eff}^{d}\sum_{\sigma}|G_{sd,\sigma}^{R}(\omega=0,T=0)|^{2}\leavevmode\nobreak\
,$ (5.25)

where the required full Green’s function can be obtained from standard
equations of motion techniques as,

$G_{sd,\sigma}^{R}(\omega)=\frac{W_{sd\sigma}}{[\omega^{+}-W_{ss\sigma}-\Delta(\omega)][\omega^{+}-W_{dd\sigma}-\Delta(\omega)]-|W_{sd\sigma}|^{2}}\leavevmode\nobreak\
,$ (5.26)

where $\omega^{+}=\omega+i\eta$ for retarded correlators in the limit
$\eta\rightarrow 0^{+}$.
In the general case, one obtains the electrical conductance for $S=0$ system
reads,

$\boxed{\mathcal{G}^{dc}(0,0)=\left(\frac{e^{2}}{h}\right)\sum_{\sigma}\frac{4|\widetilde{W}_{sd\sigma}|^{2}}{(1+|\widetilde{W}_{sd\sigma}|^{2}-\widetilde{W}_{ss\sigma}\widetilde{W}_{dd\sigma})^{2}+(\widetilde{W}_{ss\sigma}+\widetilde{W}_{dd\sigma})^{2}}}\leavevmode\nobreak\
,$ (5.27)

where we have defined the dimensionless quantities
$\widetilde{W}_{\alpha\beta\sigma}=\pi\rho_{0}W_{\alpha\beta\sigma}$.
In the zero-field, particle-hole symmetric case where
$W_{ss\sigma}=W_{dd\sigma}=0$ and $W_{sd\sigma}\equiv W_{sd}$, the standard
result for the $T=0$ transmission of a tunnel-junction is recovered,

$\boxed{\mathcal{G}^{dc}(0,0)=\left(\frac{2e^{2}}{h}\right)\frac{4|\widetilde{W}_{sd}|^{2}}{(1+|\widetilde{W}_{sd}|^{2})^{2}}}\;,$
(5.28)

and the maximum conductance $\mathcal{G}^{dc}(0,0)=2e^{2}/h$ is attained when
$|\widetilde{W}_{sd}|^{2}=1$, which corresponds to intermediate values of
$W_{sd}$.
The conductance is therefore seen to depend strongly on particle-hole
asymmetry, generated and tuned in practice by application of gate voltages. In
particular, the maximum conductance in case of finite $W_{ss}$ and/or $W_{dd}$
is always reduced $\mathcal{G}^{dc}(0,0)<2e^{2}/h$ with respect to its
particle-hole symmetric counterpart.

####  5.1.2 Spin $S=1/2$

In this section we discuss the case of spin degenerate ground-state within CB
regime. The most interesting and subtle case arises when the isolated
nanostructure has an odd number of electrons $N$ and hosts a net spin
$S=\tfrac{1}{2}$ that means we consider the nanostructure ground-state
manifold as $\ket{\Psi^{N,S^{z}}_{0}}=\ket{Q=\text{odd};S^{z}=\text{half-
integer};j=0}$ belonging to $\mathcal{H}^{Q}_{0}$ Hilbert subspace only. In
this case, the odd $N$-electron charge sector satisfies the equation:

$\hat{H}\ket{\Psi^{N,S^{z}}_{0}}=\hat{H}\ket{N;S^{z}=\pm\tfrac{1}{2};j=0}=E^{N,S^{z}}_{0}\ket{N;S^{z}=\pm\tfrac{1}{2};j=0}\leavevmode\nobreak\
.$ (5.29)

The projector operator for the odd charge sector ground-state manifold using
Eq.5.10 reads as

$\widehat{P}_{0}\doteq\sum_{S^{z}=\pm\tfrac{1}{2}}\ket{N;S^{z};j=0}\bra{N;S^{z};j=0}\otimes\widehat{\mathbb{1}}_{leads}\leavevmode\nobreak\
.$ (5.30)

By applying this projector to Eq.5.15 yields to the effective model in sd-
basis as given in Eq.5.17. Later in this part we see manipulation of Eq.5.17
into different basis. In particular, the projection on
$\hat{H}_{nano}=\epsilon_{0}\widehat{P}_{0}$ because we recall the number
operator of the nanostructure is spin conserving, and $\sigma=\uparrow$ and
$\sigma=\downarrow$ nanostructure states have the same energy for $B=0$.
In the absence of a magnetic field or spin-orbit coupling terms in the bare
system, the effective model in Eq.5.17 therefore has full $SU(2)$ symmetry and
$W_{\alpha\beta}\equiv W_{\alpha\beta}$ independent of spin. We consider this
case explicitly below where the physics of $S=1/2$ ground-states is detailed
according to the particle-hole symmetry and sd-symmetry regimes. Extensions to
higher spin multiplet ground-states is discussed at the end of the chapter.

#### No potential scattering: $W_{ss}=W_{sd}=W_{ds}=W_{dd}\equiv 0$

In particle-hole symmetric regime, there is no contribution from the potential
scattering term $W_{\alpha\beta}$ in the system. We remark that, if
$W_{sd}=W_{ds}=0$ it is not guaranteed breaking of particle-hole symmetry, but
first we enforce vanishing contribution from all the potential scattering
terms, before relaxing this constraint in the next section.
Under such a condition, it proves useful to diagonalize the exchange coupling
matrix by a canonical rotation of the leads basis from $s,d$ to $e,o$ even/odd
basis [154, 42]. The transformation $\mathbb{U}^{\dagger}\mathbb{J}\mathbb{U}$
for the exchange scattering matrix $\mathbb{J}$ is defined by unitary matrix
$\mathbb{U}$ with elements $U_{i\alpha}$ for $i=e,o$ and $\alpha=s,d$:

$\mathbb{U}=\begin{pmatrix}U_{es}&U_{ed}\\\
U_{os}&U_{od}\end{pmatrix}\leavevmode\nobreak\ ,$ (5.31)

and it satisfies the properties $\mathbb{U}^{-1}=\mathbb{U}^{\dagger}$ such
that $\mathbb{U}^{\dagger}\mathbb{J}\mathbb{U}$ is diagonal. By performing
this transformation on the conduction electron operators we obtain

$c^{\dagger}_{\alpha\sigma}=\sum_{i=e,o}c^{\dagger}_{i\sigma}U_{i\alpha}\begin{cases}{c}^{\dagger}_{s\sigma}={c}^{\dagger}_{e\sigma}U_{es}+{c}^{\dagger}_{o\sigma}U_{od}\\\
{c}^{\dagger}_{d\sigma}={c}^{\dagger}_{e\sigma}U_{es}+{c}^{\dagger}_{o\sigma}U_{od}\end{cases}\leavevmode\nobreak\
,$ (5.32)

and similarly for $c_{\alpha\sigma}$. Hence, we apply the transformation in
Eq.5.31 to the exchange coupling Hamiltonian matrix form in Eq.5.17 and we
derive

$\displaystyle\hat{H}^{K}$
$\displaystyle=\sum_{\sigma\sigma^{\prime}}\hat{\mathbf{S}}\cdot\bm{\sigma}_{\sigma\sigma^{\prime}}\begin{matrix}\begin{pmatrix}c^{\dagger}_{s\sigma}&c^{\dagger}_{d\sigma}\end{pmatrix}\\\
\mbox{}\end{matrix}\begin{bmatrix}J_{ss}&J_{sd}\\\
J_{ds}&J_{dd}\end{bmatrix}\begin{pmatrix}c_{s\sigma^{\prime}}\\\
c_{d\sigma^{\prime}}\end{pmatrix}\leavevmode\nobreak\ ,$ (5.33)
$\displaystyle=\sum_{\sigma\sigma^{\prime}}\hat{\mathbf{S}}\cdot\bm{\sigma}_{\sigma\sigma^{\prime}}\begin{matrix}\begin{pmatrix}c^{\dagger}_{e\sigma}&c^{\dagger}_{o\sigma}\end{pmatrix}\\\
\mbox{}\end{matrix}\begin{bmatrix}J_{ee}&0\\\
0&J_{oo}\end{bmatrix}\begin{pmatrix}c_{e\sigma^{\prime}}\\\
c_{o\sigma^{\prime}}\end{pmatrix}=\sum_{i=e,o}J_{ii}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{ii}\leavevmode\nobreak\
,$

where $J_{eo}=J_{oe}\equiv 0$ by construction. As consequence, no direct spin
tunnelling between the even and odd leads arises any more. We can further
rewrite the diagonal elements in exchange coupling matrix as follows

$\displaystyle
J_{ee}=\frac{1}{2}\left(J_{+}+\sqrt{J^{2}_{-}+4|J_{sd}|^{2}}\right)\leavevmode\nobreak\
,$ (5.34a) $\displaystyle
J_{oo}=\frac{1}{2}\left(J_{+}-\sqrt{J^{2}_{-}+4|J_{sd}|^{2}}\right)\leavevmode\nobreak\
,$ (5.34b)

where $J_{\pm}=J_{ss}\pm J_{dd}$ and the potential scattering is zero. We also
introduce the notation $\delta=J_{ee}-J_{oo}$.
We define the even/odd basis version of the $\mathrm{T}$-matrix spectral
function in Eq.3.68 by applying rotation transformation in Eq.5.31 as follows

$t_{\alpha\beta}(\omega,T)=\sum_{ij=e,o}U^{\star}_{i\alpha}t_{ij}(\omega,T)U_{j\beta}\leavevmode\nobreak\
.$ (5.35)

The above expression for the inter-channel $\mathrm{T}$-matrix spectral
function $t_{sd}(\omega,T)$ reads

$\displaystyle t_{sd}(\omega,T)$
$\displaystyle\stackrel{{\scriptstyle(i)}}{{=}}U^{\star}_{es}t_{ee}(\omega,T)U_{ed}+U^{\star}_{os}t_{oo}(\omega,T)U_{od}\leavevmode\nobreak\
,$ (5.36)
$\displaystyle\stackrel{{\scriptstyle(ii)}}{{=}}-\pi\rho_{0}\left(U^{\star}_{es}\mathit{Im}T_{ee}(\omega,T)U_{ed}+U^{\star}_{os}\mathit{Im}T_{oo}(\omega,T)U_{od}\right)\equiv-\pi\rho_{0}\mathit{Im}T_{sd}(\omega,T)\leavevmode\nobreak\
,$

where in $(i)$ we use the fact that $J_{eo}=J_{oe}=W_{eo}=W_{oe}\equiv 0$ as
result of the canonical transformation in Eq.5.31 such that the off-diagonal
components $t_{eo}=t_{oe}\equiv 0$ and in $(ii)$ we give explicit expression
for even/odd basis inter-channel $\mathrm{T}_{sd}(\omega,T)$ matrix. We note
that Eq.5.36 can be further simplified under RG-flow of coupling parameters at
$\omega=T=0$, as discussed further below.
Before we continue with the discussion, we want to comment on the physical
implication of the canonical transformation. In the bare model, it is the
$T_{sd}(\omega,T)$ component describing the spin scattering in the coupled
lead-nanostructure-lead system to give a finite contribution to the
conductance. The application of canonical transformation makes $\hat{H}_{eff}$
diagonal such that the spin scattering is described only within even channel
and nanostructure and only within odd channel and nanostructure i.e.
$T_{ee}(\omega,T),T_{oo}(\omega,T)$ but no spin transfer occurs between even
and odd channel through the nanostructure tunnelling i.e.
$T_{eo}(\omega,T)=T_{oe}(\omega,T)=0$. The advantage of the diagonalised
$\hat{H}_{eff}$ with no mixed terms is the independent RG-flow of coupling
parameters for each even and odd channel separately. As consequence, once the
appropriate energy scale is reached, the complete channel decoupling is
possible by following the distinct channel evolution under RG - as we discuss
more in detail below. It is important to remark that, although the even/odd
basis is a convenient choice to study the renormalized system at low-energy
scale, only the bare basis is the actual physical basis and only it can be
used to define the conductance. In conclusion, once $\hat{H}_{eff}$ is found
in even/odd basis and the RG-flow analysed, it is necessary to apply the
inverse of the canonical transformation in order to restore the original
basis. This is a straightforward operation since the transformation is unitary
and therefore invertible.
We consider now two general scenarios happening in the particle-hole symmetric
system.
The first is the special case arises when $J_{oo}=0$: it happens when the odd
lead is decoupled on the level of $\hat{H}_{eff}$ and the effective model
reduces precisely to a single-channel Kondo model with a single
spin-$\tfrac{1}{2}$ impurity,

$\boxed{\hat{H}_{eff}=\hat{H}_{leads,e}+J_{ee}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{ee}}\;.$
(5.37)

From Eq. 5.34b, this occurs specifically when $|J_{sd}|^{2}=J_{ss}J_{dd}$. In
fact, this condition is always automatically satisfied when the bare model is
under PC condition and it exactly holds for single-orbital model or single-
impurity Anderson model. Indeed, this is physically natural since Eq. 5.37 is
the regular Schrieffer-Wolff transformation of the single-channel model using
$\hat{H}_{hyb}$ as defined in Eq.2.24 but now for even channel obtained under
PC, namely $\sum_{i\sigma}(V_{i}c^{\dagger}_{e\sigma}d_{i\sigma}+H.c.)$ for
given orbital $i$ of the nanostructure.
The conductance then follows from the MW formula under PC in Eq.3.66 but now
written in terms of the generalized transfer matrix under PC derived in
Eq.3.67, namely

$\displaystyle\widetilde{\mathbb{T}}_{PC}(\omega,T)\longrightarrow
4\frac{J_{ss}J_{dd}}{(J_{ss}+J_{dd})^{2}}\sum_{\sigma}t_{ee,\sigma}(\omega,T)\;,$
(5.38)

where we use the $\mathrm{T}$-matrix spectral function in Eq.3.68 evaluated
for the even channel. We remark the prefactor
$\tfrac{J_{ss}J_{dd}}{(J_{ss}+J_{dd})^{2}}\propto\tfrac{|V_{s}|^{2}|V_{d}|^{2}}{(|V_{s}|^{2}+|V_{d}|^{2})^{2}}$
is only valid within the Schrieffer-Wolff approximation for single-channel
model, with equivalent lead in wide flat band limit.
In case of no potential scattering and no magnetic field as considered here,
$t_{ee,\sigma}(\omega=0,T=0)=\sin^{2}(\pi/2)=1$ by FSR [38] and the resulting
electrical conductance expression reads

$\boxed{\mathcal{G}^{dc}(\omega=0,T=0)=\frac{2e^{2}}{h}4\frac{J_{ss}J_{dd}}{(J_{ss}+J_{dd})^{2}}}\leavevmode\nobreak\
,$ (5.39)

where $\mathcal{G}^{dc}$ is controlled by a pure geometric factor describing
the nanostructure-lead coupling. Indeed for $\omega,T\ll T_{K}$, with
$T_{K}\sim D\exp[-1/\rho_{0}J_{ee}]$ the Kondo temperature for the effective
model Eq. 5.37, we have $t_{ee,\sigma}(\omega,T)\simeq 1$ and so the
conductance approximately saturates its low-temperature bound,
$\mathcal{G}(T\ll T_{K})\simeq\mathcal{G}^{dc}(0,0)$.
Of course, the above scenario is special, and for most realistic systems the
exact PC condition $|J_{sd}|^{2}=J_{ss}J_{dd}$ is not expected to be
satisfied. Indeed, in general multi-orbital systems we find $|J_{sd}|^{2}\ll
J_{ss}J_{dd}$. In such cases, $J_{oo}\neq 0$ and the odd channel remains
formally coupled to the effective nanostructure spin-$\tfrac{1}{2}$. In the
even/odd basis with $W_{\alpha\beta}=0$, Eq. 5.17 reduces to the effective
model,

$\boxed{\hat{H}_{eff}=\hat{H}_{leads}+J_{ee}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{ee}+J_{oo}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{oo}}\;.$
(5.40)

The physics of this channel-anisotropic 2CK model are rich but well-known,
featuring a competition of Kondo screening between even and odd channels [79].
The model supports a non-Fermi liquid quantum critical point for
$J_{ee}=J_{oo}$, but from Eq. 5.34 we see that this only arises in these kinds
of systems when both $J_{ss}=J_{dd}$ and $J_{sd}=0$. Although one may be able
to tune to the sd-symmetric condition $J_{ee}=J_{oo}$, to realize $J_{sd}=0$
requires the suppression of through-nanostructure exchange-cotunnelling
processes. In principle, this could arise through gate-tunable many-body
quantum interference effects [154] that conspire to produce an exact
tunnelling node, but a real system exhibiting 2CK criticality driven by such
effects has not yet been reported. On the other hand, for complex multi-
orbital nanostructures, electron propagation across the entire structure
embodied by the exchange cotunnelling $J_{sd}$ is typically small in magnitude
(albeit finite) compared with the local terms $J_{ss}$ and $J_{dd}$. Therefore
it may still be possible to access the quasi-frustrated quantum critical
physics of the 2CK model at small $J_{sd}$ for intermediate temperatures or
energies [155, 156]. In the following we simply regard $J_{ee}$ and $J_{oo}$
as free independent parameters of the effective model, however obtained, and
consider the generic behaviour.
For concreteness, we now assume $J_{ss}$ is antiferromagnetic such that once
the canonical transformation is applied, the inequality $J_{ee}>J_{oo}>0$ is
fulfilled.
The perturbative scaling (poor man’s scaling [63]) starting from weak
coupling, indicates that both $J_{ee}$ and $J_{oo}$ get renormalized upwards
on reducing energy or temperature scale. The scaling invariant for the RG-flow
of each is its respective Kondo temperature, $T_{K}^{\alpha}\sim
D\exp[-1/\rho_{0}J_{\alpha\alpha}]$. However, the low-energy physics is
nonperturbative, and the 2CK strong coupling fixed point for a
spin-$\tfrac{1}{2}$ impurity is unstable [75]. Since $J_{ee}>J_{oo}$ we have
$T_{K}^{e}>T_{K}^{o}$ and the even channel ultimately flows under RG to strong
coupling while the odd channel flows back to weak coupling. According to the
couplings strength, two distinct regime appear, as we sketch in Fig.5.1.
The first case happens at strong coupling regime
$\delta=|J_{ee}-J_{oo}|>T^{e}_{K}\sim De^{-1/\rho_{0}J_{ee}}$ where
$T^{e}_{K}\equiv T^{1CK}_{K}$ single-channel Kondo temperature for the even
channel. From high to low temperature range, the system undergoes to

  1. 1.

$T\gg T^{1CK}_{K}$: at high-temperature scale, the system is effectively two-
channel model with both even, odd channel sectors are weakly coupled and
increase progressively at equal rate. The nanostructure is nearly free local
moment;

  2. 2.

$T^{1CK}_{K}<T<\delta$: at intermediate temperature, channels start splitting
with similar slope of increment/decrement for the even/odd channel
respectively. The even coupling exceeds the odd one because of
$J_{ee}>J_{oo}$;

  3. 3.

$T=T^{1CK}_{K}$: channels decoupling is completed and there is no more spin
scattering between the odd channel sector and the nanostructure;

  4. 4.

$0<T<T^{1CK}_{K}$: the system undergoes to Kondo screening of the
nanostructure spin due to the even electrons sector and it becomes effectively
a single-channel model. At this energy scale we understand the system having
an emergent proportionate coupling condition;

  5. 5.

$T=0$: at strictly zero temperature, the system is exactly in single-channel
characterised by $J_{oo}\equiv 0,\leavevmode\nobreak\ J_{ee}\neq 0$ and
$t_{ee}(0,0)=1$ is calculated exactly using the FRS.

The second case happens at the opposite regime, namely at weak channel
anisotropy $\delta=|J_{ee}-J_{oo}|<T^{e}_{K}\sim De^{-1/\rho_{0}J_{ee}}$ where
$T^{e}_{K}\equiv T^{2CK}_{K}$ two-channel Kondo temperature. Again, going from
high to low temperature range, we find the system undergoing

  1. 1.

$T\gg T^{2CK}_{K}$: at high-temperature scale, the system is in two-channel
configuration with both leads weakly coupled;

  2. 2.

$T^{\star}<T<T^{2CK}_{K}$: the system reaches a plateau where even and odd
channels compete for screening the spin of the nanostructure and the couplings
have similar strength i.e. $J_{ee}\simeq J_{oo}$. The system shows a
frustrated state which lasts until the even channel overcomes the odd one with
consequent channels decoupling and $T^{\star}\sim\delta^{2}/T_{K}^{2CK}$ sets
the crossover;

  3. 3.

$0<T\leq T^{\star}$: the channel frustration is relived at $T^{\star}$ and the
model is effectively in single-channel i.e. $J_{ee}>J_{oo}\to 0$. The Kondo
singlet is formed from the screened nanostructure spin by the even electrons
channel. At this energy scale the system presents an emergent proportionate
coupling property;

  4. 4.

$T=0$: as in the previous case, at strictly zero temperature the system is
exactly in single-channel with $J_{oo}\equiv 0,\leavevmode\nobreak\ J_{ee}\neq
0$ and the $t_{ee}(0,0)=1$ value is calculated exactly using the FRS.

(a)

(b)

Figure 5.1: Sketch of $\mathrm{T}$-matrix spectral in even/odd basis at $T=0$,
as function of energy scales $\omega$, for the particle-hole symmetric $S=1/2$
system. $(a)$ Strong coupling: $\delta=|J_{ee}-J_{oo}|>T^{e}_{K}\equiv
T^{1CK}_{K}$ $(b)$ Weak coupling: $\delta=|J_{ee}-J_{oo}|<T^{e}_{K}\equiv
T^{2CK}_{K}$, $T^{\star}\sim\delta^{2}/T_{K}^{2CK}$.

For a given $J_{ee}$, the system couplings flow in RG sense through both
coupling regimes without undergoing to any phase transition. The continuous
shifts from one regime to another of the RG-flow implies the crossover at
$T^{1CK}_{K}$ for the strong coupling regime merges with the one at
$T^{2CK}_{K}$ for the weak coupling. As result, we have
$T^{1CK}_{K}=T^{2CK}_{K}\equiv T^{e}_{K}\sim De^{-1/\rho_{0}J_{ee}}$. Hence,
in order to incorporate both couplings regime, channel decoupling occurs at
the minimum between $T^{\star}$ and $T^{e}_{K}$ temperature. The outcome is a
single-channel model [9] but now only in even basis. We observe that
unconstrained channel configuration and the PC found at $\omega=T=0$ regime
are incipient properties of $\hat{H}_{eff}$ in even/odd basis: both the
independent spin sectors and PC are not feature of the original bare model.
The properties of the 2CK model $\mathrm{T}$-matrix were discussed in [81].
This physical picture is confirmed by numerically-exact NRG calculations for
the effective 2CK model Eq. 5.40, in Fig. 5.2. The RG-flow is vividly
demonstrated in the $T=0$ spectrum of the $\mathrm{T}$-matrix for the even
channel (solid lines) and odd channel (dashed lines), plotted for systems with
fixed $J_{ee}$ but decreasing $\delta=J_{ee}-J_{oo}>0$ (and
$W_{\alpha\beta}=0$). We highlight three features: $(i)$ in all cases, the
$\mathrm{T}$-matrix spectrum of the even channel is always exactly pinned to 1
at zero frequency and zero temperature, $t_{ee}(0,0)=1$; $(ii)$ by contrast,
the odd channel $\mathrm{T}$-matrix spectrum vanishes, $t_{oo}(0,0)=0$;
$(iii)$ relief of incipient frustration is characterized by a new scale
$T^{\rm FL}$, where the effective model can be described by Fermi liquid
theory.

Figure 5.2: NRG results for the $T=0$ $\mathrm{T}$-matrix spectrum
$t_{\alpha\alpha}(\omega,0)$ of the effective 2CK model, Eq. 5.40. Even (odd)
channel spectra shown as the solid (dashed) lines, for fixed $J_{ee}/D=0.3$
and $\mathbb{W}=\mathbb{0}$, with different $J_{oo}$. In all cases,
$t_{ee}(0,0)=1$ and $t_{oo}(0,0)=0$, despite incipient frustration of Kondo
screening for small $\delta=J_{ee}-J_{oo}$.

Importantly, we see an emergent decoupling of the odd lead at low energies
$|\omega|\ll T^{\rm FL}$ in all cases, where
$T^{FL}=\min(T_{K}^{e},T^{\star})$ is a dynamically-generated Fermi liquid
scale. Here $T^{\star}\sim\delta^{2}/T_{K}^{e}$ is an energy scale embodying
the frustration of Kondo screening from the two channels. For large
$\delta>T_{K}^{e}$ we have $T^{\star}>T_{K}^{e}$ and the RG-flow of the odd
channel is cut off on the scale of $T_{K}^{e}$ as the even channel flows to
strong coupling. On the other hand, for small $\delta<T_{K}^{e}$ we have
$T^{\star}<T_{K}^{e}$ and the system is described by the frustrated (non-Fermi
liquid) 2CK fixed point over an intermediate temperature range
$T^{\star}<T<T_{K}^{e}$. The even channel flows to strong coupling and the odd
channel flows to weak coupling only for $T\ll T^{\star}$ in this case.
Without potential scattering, it is then guaranteed that $t_{ee}(0,0)=1$ and
$t_{oo}(0,0)=0$, as dictated by phase-shift arguments: due to the decoupling
at $\omega=T=0$, we may write
$t_{\alpha\alpha}(0,0)=\sin^{2}(\delta_{\alpha})$, see discussion at Eq.3.68
for vanishing frequency energy and temperature. The Kondo effect in the even
channel confers a phase shift $\delta_{e}=\pi/2$ whereas in the decoupled odd
channel $\delta_{o}=0$. We note that although the non-Fermi liquid 2CK regime
has been accessed experimentally in quantum dot devices, this is the result of
fine-tuning and ingenious quantum engineering [85]. The more standard
situation is for the odd channel to decouple at low temperatures, as described
above.
Since such systems flow under RG to an effective single-channel description
involving only the even lead combination, we have an emergent PC condition.
This greatly simplifies the calculation and interpretation of the low-
temperature quantum transport. On the lowest temperature and energy scales
$\omega,T\ll T^{\rm FL}$, the alternative formulation of the Oguri argument in
dc-regime derived in Eq.4.105 yields the electrical conductance. We may now
relate the inter-channel $\mathrm{T}$-matrix $T_{sd,\sigma}$ to the even
channel $\mathrm{T}$-matrix $T_{ee,\sigma}$ using the rotation $\mathbb{U}$ in
Eq.5.31 diagonalizing the exchange matrix as we shown in Eq.5.36. Hence, we
write the alternative Oguri formula now for the sd-model with equivalent
metallic leads, namely

$\displaystyle\mathcal{G}^{dc}(\omega=0,T=0)$
$\displaystyle=\frac{2e^{2}}{h}4|\pi\rho_{0}\mathit{Im}T_{sd,\sigma}(0,0)|^{2}\leavevmode\nobreak\
,$ (5.41)
$\displaystyle=\frac{2e^{2}}{h}4|\pi\rho_{0}\left(U^{\star}_{es}\mathit{Im}T_{ee,\sigma}(\omega,T)U_{ed}+U^{\star}_{os}\mathit{Im}T_{oo,\sigma}(\omega,T)U_{od}\right)|^{2}\leavevmode\nobreak\
,$

here equivalently written in sd and even/odd basis, where the latter
expression is because $J_{eo}=W_{eo}=0$ so that
$T_{eo,\sigma}(\omega,T)=T_{oe,\sigma}(\omega,T)=0$ from the canonical
transformation in Eq.5.31. As we mentioned previously, this equation further
simplifies according to the RG-flow of coupling parameters: in the current
case $W_{\alpha\beta}=0$, at $\omega=T=0$ we find $J_{oo}=0$ so that
$T_{oo}(0,0)=0$ whereas $J_{ee}\neq 0$ so that $T_{ee}(0,0)\neq 0$ and due to
the emergent decoupling it contributes in

$\pi\rho_{0}\mathit{Im}T_{ee,\sigma}(0,0)=t_{ee,\sigma}(0,0)\equiv
1\leavevmode\nobreak\ ,$ (5.42)

where we used the definition of the $\mathrm{T}$-matrix spectral function in
Eq.3.68 and the FSR in Eq.2.39 [47, 52]. We can indeed think of
$\hat{H}_{eff}$ as now being an effective renormalized model. Therefore the
$T=0$ electrical conductance follows as,

$\boxed{\mathcal{G}^{dc}(\omega=0,T=0)=\left(\frac{2e^{2}}{h}\right)4|U^{\star}_{es}U_{ed}|^{2}=\left(\frac{2e^{2}}{h}\right)\frac{4|J_{sd}|^{2}}{4|J_{sd}|^{2}+J_{-}^{2}}}\;.$
(5.43)

This result also applies accurately for all $T\ll T^{FL}$. The low-temperature
conductance can therefore be determined directly from the effective exchange
couplings $J_{ss}$, $J_{dd}$, and $J_{sd}$.
Indeed, the factor $|J_{sd}|^{2}/(|J_{sd}|^{2}+J_{-}^{2})$ appearing in Eq.
5.43 reduces as it must to $|V_{s}V_{d}|^{2}/(|V_{s}|^{2}+|V_{d}|^{2})^{2}$ in
the case of exact PC in the bare model, for which we have
$|J_{sd}|^{2}=J_{ss}J_{dd}$ and $J_{ss}/J_{dd}=|V_{s}/V_{d}|^{2}$ by means of
the Schrieffer-Wolff transformation [154]. In some sense, Eq. 5.43 contains
the generalization of Eq.3.67 but now for emergent PC condition. This is
equivalent to the factor $\Gamma_{s}\Gamma_{d}/(\Gamma_{s}+\Gamma_{d})^{2}$ we
used in the derivation of $\mathrm{T}$-matrix spectral function in Eq.3.68.
As conclusion on this part: the even/odd basis transformation allows for
independent channel sectors. Under RG-flow, the $\mathrm{T}$-matrix spectral
function flows along different paths according to channel symmetry and takes
specific values at $\omega=T=0$ in Fermi liquid regime, namely $t_{ee}(0,0)=1$
and $t_{oo}(0,0)=0$. We stress that, in case we had studied the RG-flow of
coupling parameters in original basis, we would have dealt with a proper two-
channel model with entangled contributions from lead-nanostructure, lead-lead
electronic flow and non-diagonal
$\mathbb{J}_{\alpha\beta},\mathbb{W}_{\alpha\beta}$ matrices. Hence, under RG
we would not have found the energy scale with incipient PC properties. On the
contrary, the canonical transformation defined in Eq.5.31 combined with RG-
flow of couplings in even/odd basis offers an emergent physics at low-energy
scale.

#### Finite potential scattering $W_{\alpha\beta}\neq 0$ in sd-symmetry

The results of the previous section hold when
$W_{\alpha\beta}=W_{\beta\alpha}=W_{\alpha\alpha}=W_{\beta\beta}=0$. Although
an important simple limit, the resulting behaviour is not generic since even
at particle-hole symmetry ($W_{ss}=W_{dd}=0$), the through-nanostructure
cotunnelling term $W_{sd}$ is typically finite. In this section we consider
finite potential scattering $W_{\alpha\beta}\neq 0$ in the sd-symmetric limit
- meaning $J_{ss}=J_{dd}\equiv J$ and $W_{ss}=W_{dd}\equiv W$ (or equivalently
$J_{-}=W_{-}=0$). In this case, the full model has an overall mirror symmetry
with respect to exchanging $s$ and $d$ leads.
At high temperatures $T\gg T_{K}$, renormalization due to the Kondo effect is
weak, and the contribution to conductance is dominated by finite source-drain
cotunnelling, $W_{sd}$. Conductance in this limit is therefore given
approximately by Eq. 5.27, obtained for $J_{\alpha\beta}=0$. However, richer
and more complex behaviour results at lower temperatures, where there is a
subtle interplay between $\mathbb{W}$ and $\mathbb{J}$ matrices. We therefore
focus on the low-$T$ physics in the following.
First, we note that in the sd-symmetric limit, the even/odd channels are
simply the symmetric/antisymmetric combinations of source and drain leads,

$\displaystyle
c_{e\sigma}=\frac{1}{\sqrt{2}}\left(c_{s\sigma}+c_{d\sigma}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
c_{o\sigma}=\frac{1}{\sqrt{2}}\left(c_{s\sigma}-c_{d\sigma}\right)\leavevmode\nobreak\
,$ (5.44)

which yields,

$\displaystyle J_{ee}=J+J_{sd}\quad$ $\displaystyle;\quad
J_{oo}=J-J_{sd}\leavevmode\nobreak\ ,$ (5.45a) $\displaystyle
W_{ee}=W+W_{sd}\quad$ $\displaystyle;\quad W_{oo}=W-W_{sd}\leavevmode\nobreak\
,$ (5.45b)

and $J_{eo}=J_{oe}=W_{eo}=W_{oe}=0$. Note that with sd-symmetry, the lead
transformation diagonalizing the exchange matrix $\mathbb{J}$ given in Eq.5.31
also diagonalizes the potential scattering matrix $\mathbb{W}$. We write the
effective model as,

$\boxed{\hat{H}_{eff}=\widetilde{H}_{leads}+J_{ee}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{ee}+J_{oo}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{oo}}\;,$
(5.46)

where we have incorporated the local potentials $W_{ee}$ and $W_{oo}$ into the
definition of the lead Hamiltonian,
$\widetilde{H}_{leads}=\sum_{\alpha=e,o}[\hat{H}_{leads,\alpha}+W_{\alpha\alpha}^{\phantom{\dagger}}\sum_{\sigma}c_{\alpha\sigma}^{\dagger}c_{\alpha\sigma}^{\phantom{\dagger}}]$
where $\hat{H}_{leads,\alpha}$ is the free $\alpha$-lead Hamiltonian as
defined in Eq.5.17.
Analysis of Eq. 5.46 proceeds similarly to that of Eq. 5.40, except now we
account for the potential scattering by rediagonalizing the lead Hamiltonian
$\widetilde{H}_{leads}$. This can be done implicitly using Green’s function
techniques, which allow us to relate the local lead Green’s functions
including the potential to those of the bare leads. We find
$[\widetilde{G}^{0}_{\alpha\alpha,\sigma}(\omega)]^{-1}=[G^{0}_{\alpha\alpha,\sigma}(\omega)]^{-1}-W_{\alpha\alpha}$
for $\alpha=e,o$. This implies that the renormalized Fermi level density of
states is $\widetilde{\rho}_{\alpha}=-\tfrac{1}{\pi}{\rm
Im}\widetilde{G}^{0}_{\alpha\alpha,\sigma}(0)=\rho_{0}/(1+\widetilde{W}_{\alpha\alpha}^{2})$,
where we again use the notation for dimensionless potential scattering
$\widetilde{W}_{\alpha\beta}=\pi\rho_{0}W_{\alpha\beta}$.
In fact, the quantities controlling the Kondo physics and RG-flow in such
problems are the dimensionless parameters
$j_{\alpha}=\widetilde{\rho}_{\alpha}J_{\alpha\alpha}$, rather than the bare
exchange couplings $J_{\alpha\alpha}$. For $j_{e}>j_{o}$, the even lead flows
to strong coupling, while the odd lead decouples; the opposite applies for
$j_{e}<j_{o}$. Indeed, even with $J_{ee}>J_{oo}$ and $W_{ee}>W_{oo}$, one may
be able to realize $j_{e}=j_{o}$, as required for 2CK criticality. It may
therefore be possible to tune across the 2CK quantum phase transition in a
given system by tuning gate voltages.
We now explore the effect of the local potentials $W_{ee}$ and $W_{oo}$ on the
dynamics and conductance. For $W_{\alpha\beta}=0$ considered in the previous
section, we argued that the emergent decoupling of the odd lead at low
energies and temperatures $\omega,T\ll T^{FL}$ produced an effective PC
condition, with $t_{ee,\sigma}(0,0)$ and hence $G_{C}(0)$ then being
controlled by an effective _single_ channel Kondo (1CK) model, Eq. 5.37. With
finite $W_{ee}$ and $W_{oo}$, the lead with the smaller $j_{\alpha}$ still
decouples asymptotically. However, $t_{ee,\sigma}(0,0)<1$ and
$t_{oo,\sigma}(0,0)>0$ due to the potential scattering.
This physical picture is confirmed in Fig. 5.3 where we plot the $T=0$ spectra
of the even and odd $\mathrm{T}$-matrix for the 2CK model Eq. 5.46 in the
centre and right panels, for fixed exchange couplings $J_{ee}$ and $J_{oo}$,
but with different potential scattering strengths chosen for illustration to
be $W=3W_{sd}$. Below an emergent scale $T^{FL}\sim T_{K}^{e}$ (here $\sim
10^{-3}D$), the $\mathrm{T}$-matrix of the even channel exhibits a Kondo
resonance, while the incipient RG-flow of the odd channel is arrested.

Figure 5.3: NRG results for the $T=0$ $\mathrm{T}$-matrix spectrum of the
effective 1CK model Eq. 5.37 (left panel), and 2CK model Eq. 5.46 (even/odd
channels in the centre/right panels). For the 2CK model, we take the sd-
symmetric case $J_{-}=W_{-}=0$, with fixed $J/D=0.25$ and $J_{sd}/D=0.05$,
varying $W=3W_{sd}$. For the 1CK model we use the same parameters as for the
even channel of the corresponding 2CK model ($J_{ee}/D=0.3$ and
$W_{ee}=\tfrac{4}{3}W$). Dotted lines in the left/centre panels are given by
Eq. 5.49, while Eq. 5.47 is used in the right panel.

Since the odd channel is decoupled at $\omega=T=0$, the $\mathrm{T}$-matrix
spectral function $t_{oo,\sigma}(0,0)$ is determined completely by $W_{oo}$.
From the $\mathrm{T}$-matrix equation, Eq. 2.36, together with the bare and
renormalized Green’s functions $G^{0}_{oo,\sigma}(0)=-i\pi\rho_{0}$ and
$\widetilde{G}^{0}_{oo,\sigma}(0,0)=-i\pi\widetilde{\rho}_{o}$, we immediately
obtain,

$t_{oo,\sigma}(0,0)=\frac{\widetilde{W}_{oo}^{2}}{1+\widetilde{W}_{oo}^{2}}\;.$
(5.47)

This result is equivalent to a phase shift
$\delta_{o}=\tan^{-1}[\widetilde{W}_{oo}]$ in the odd channel, since
$t_{oo,\sigma}(0,0)=\sin^{2}(\delta_{o})$ from Eq.3.68 evaluated at zero
energy and zero frequency according to FSR. Thus, Eq.5.47 is an exact
expression. The full complex odd-channel $\mathrm{T}$-matrix follows as

$\pi\rho_{0}T_{oo,\sigma}(0,0)=(\widetilde{W}^{-1}_{oo}-i)\times
t_{oo,\sigma}(0,0)\leavevmode\nobreak\ .$ (5.48)

For the systems considered in Fig. 5.3, we plot the corresponding values of
$t_{oo,\sigma}(0,0)$ obtained from Eq. 5.47 in the right panel as the dotted
horizontal lines. The full odd-channel $\mathrm{T}$-matrix obtained by NRG is
seen to saturate to these values for $|\omega|\ll T^{\rm FL}$, as anticipated.
More interestingly, comparison between the left and centre panels of Fig. 5.3
shows that the low-energy dynamics of the _even_ channel can be accurately
understood in terms of an effective single-channel Kondo model with the same
$J_{ee}$ and $W_{ee}$. This is again a consequence of the emergent decoupling
of the odd channel. Indeed, provided $\delta\equiv J_{ee}-J_{oo}\gg T_{K}^{e}$
\- as is fairly standard condition for bare systems with at least one
antiferromagnetic coupling - the correct Kondo scales are also well
reproduced.
For the bare Andersonian nanostructure model, calculating the low-energy
behaviour of the scattering $\mathrm{T}$-matrix is deeply nontrivial. Even for
the single-impurity Anderson model, $t_{ee,\sigma}(0,0)=\sin^{2}(\delta_{e})$
requires knowledge of the even-channel phase shift $\delta_{e}$. In this case,
the Friedel sum rule dictates that $\delta_{e}=\tfrac{\pi}{2}n_{imp}$ in terms
of the excess charge due to the impurity, $n_{imp}$. But the latter is
renormalized by interactions and must itself be calculated using many-body
techniques away from particle-hole symmetry. On the other hand, the above
physical arguments and NRG results in Fig. 5.3 establish that
$t_{ee,\sigma}(0,0)$ can be obtained from an effective 1CK model, Eq. 5.17.
Treating the exchange coupling $J_{ee}$ as a perturbation to the even-channel
charge, we may then write
$n_{imp}\simeq\tfrac{2}{\pi}\tan^{-1}[\widetilde{W}_{ee}]+1$, where the first
term is the excess charge induced in a free bath due to a boundary potential
$W_{ee}$, and we add $1$ for the singly-occupied impurity local moment in the
Kondo model. Here, we use $\simeq$ symbol to indicate the independence of
contribution in the phase-shift from the Kondo singlet formation and
$\widetilde{W}_{ee}$ is taken into account. This approximation gives
$\delta_{e}\simeq\tan^{-1}[\widetilde{W}_{ee}]+\tfrac{\pi}{2}$ where deviation
from half-filling regime is given by the correction term
$\tan^{-1}[\widetilde{W}_{ee}]$. The two contributions in $\delta_{e}$ can be
either constructive or destructive, as we discuss in Fig.5.5.
By combining all the elements, the $\mathrm{T}$-matrix spectral function from
the Friedel sum rule reads,

$t_{ee,\sigma}(0,0)\simeq\frac{1}{1+\widetilde{W}_{ee}^{2}}\;,$ (5.49)

and the full even-channel $\mathrm{T}$-matrix is therefore given by

$\pi\rho_{0}T_{ee,\sigma}(0,0)=-(\widetilde{W}_{ee}+i)\times
t_{ee,\sigma}(0,0)\leavevmode\nobreak\ .$ (5.50)

These predictions are quantitatively substantiated in Fig. 5.4, which shows
NRG results for a pure 1CK model with exchange coupling $J_{ee}$ and potential
scattering $W_{ee}$. The $\mathrm{T}$-matrix spectral function $t_{ee,\sigma}$
at $\omega=T=0$ is plotted in panel $(a)$ as a function of $W_{ee}$ for
different $J_{ee}$. Even for rather large $J_{ee}/D=0.4$, the exact results
agree accurately with Eq. 5.49 (dashed line) over the entire range of
$W_{ee}$.
The full energy dependence of the $\mathrm{T}$-matrix spectral function is
shown in panel $(b)$ for fixed $J_{ee}$ with increasing $W_{ee}$. The
asymptotic $\omega=0$ value reported in $(a)$ can be extracted for
$|\omega|\ll T^{e}_{K}$, with the Kondo scale $T^{e}_{K}$ decreasing as the
potential scattering strength $W_{ee}$ increases. Under RG-flow we have
$T^{e}_{K}(J_{ee},W_{ee})=D\exp[-1/\widetilde{\rho}_{e}J_{ee}]$ and the Fermi
level density of states incorporating $W_{ee}$ is
$\widetilde{\rho}_{e}=\rho_{0}/(1+\widetilde{W}_{ee}^{2})$ as before. Hence,
for the 1CK model we may write,

$T^{e}_{K}(J_{ee},W_{ee})\simeq
T^{e}_{K}(J_{ee},0)\times\exp[-\widetilde{W}^{2}_{ee}/\rho_{0}J_{ee}]\;\,$
(5.51)

and this is confirmed in Fig. 5.4$(c)$ for the same systems as in panel $(a)$.
Eq. 5.49 can therefore be used to find the low-energy behaviour of the even-
channel $\mathrm{T}$-matrix spectral function in the generalized 2CK model,
while Eq. 5.51 provides an estimate of the regime of applicability of this
result. Remarkably, the low-energy scattering behaviour can therefore be
extracted purely from a knowledge of the effective model parameters $J_{ee}$
and $W_{ee}$, and full solution of the generalized 2CK model is not required.

Figure 5.4: NRG results for the effective 1CK model, Eq. 5.17. $(a)$ $T=0$
spectrum of the $\mathrm{T}$-matrix spectral function at the Fermi energy
$t_{ee}(0,0)$, as a function of the local potential $W_{ee}$, for different
$J_{ee}$. Dashed line is Eq. 5.49, showing only small deviations even at
rather large $J_{ee}$. $(b)$ The full energy-dependence of the
$\mathrm{T}$-matrix spectral function $t_{ee}(\omega,0)$ is shown for
$J_{ee}/D=0.3$ with $W_{ee}/D=0,0.2,0.4,0.6,0.8$ increasing in the direction
of the arrow. $(c)$ Dependence of the Kondo temperature $T_{K}^{e}$ on
$W_{ee}$ for the same $J_{ee}$ used in $(a)$, confirming Eq. 5.51.

Finally, we use this information on the dynamics to extract the generic low-
temperature conductance properties. We use the alternative formulation of the
Oguri argument in Eq. 4.105 but now in the even/odd basis of Eq. 5.44 to write

$\mathcal{G}^{dc}(\omega=0,T=0)=\frac{2e^{2}}{h}4|\pi\rho_{0}T_{ee,\sigma}(0,0)-\pi\rho_{0}T_{oo,\sigma}(0,0)|^{2}$
(5.52)

and using the form of the even/odd $\mathrm{T}$-matrix spectral function
derived in this section, we obtain the electrical conductance,

$\boxed{\mathcal{G}^{dc}(\omega=0,T=0)=\left(\frac{2e^{2}}{h}\right)\frac{4(1+\widetilde{W}_{ee}\widetilde{W}_{oo})^{2}}{(1+\widetilde{W}_{ee}^{2})(1+\widetilde{W}_{oo}^{2})}}\;.$
(5.53)

This is the main result of this section: it allows the Kondo-renormalized low-
temperature conductance to be calculated purely from a knowledge of the
effective model parameters. Note that the result holds approximately for all
$T\ll T^{FL}$.
In the case of sd-symmetry as considered here, only $W_{ee}=W+W_{sd}$ and
$W_{oo}=W-W_{sd}$ are required. Furthermore, Eq. 5.53 implies the existence of
an exact quantum-interference effect conductance node,

$\mathcal{G}^{dc}(\omega=0,T=0)=0\qquad\text{when}\qquad
W_{sd}^{2}=W^{2}+(1/\pi\rho_{o})^{2}\leavevmode\nobreak\ .$ (5.54)

In principle, this condition can be satisfied on tuning gate voltages in
multi-orbital nanostructures. Although conductance nodes due to single-
particle quantum interference in effective noninteracting systems are well-
known [16], their many-body counterparts are comparatively poorly understood -
even though strong electron interactions are ubiquitous in e.g. semiconductor
quantum dot devices and single molecule transistors. We emphasize that in such
systems, electron interactions are crucial to understand the low-temperature
conductance properties. Indeed, the $T=0$ conductance node, Eq. 5.54, is
driven by the Kondo effect and a finite conductance is expected at higher
temperatures, $T\gg T_{K}$, see Fig. 5.5$(a)$ the orange curve opposed to the
black one. This Kondo blockade has been proposed as a mechanism for efficient
quantum interference effect transistors [154]. The above analysis puts such
phenomena in the context of a more general framework.

Figure 5.5: Conductance of the generalized 2CK model Eq. 5.17 at
$sd$-symmetry, obtained by NRG using the improved Kubo method. $(a)$
$\mathcal{G}^{dc}(T)$ vs $T$ for fixed $J/D=0.25$, $J_{sd}/D=0.05$ and $W=0$
with different $W_{sd}/D=0,0.1,0.2,0.3,0.4,0.6$ for the solid black, red,
blue, green, violet and orange lines. Dotted lines are the low-$T$ predictions
of Eq. 5.53. $(b)$ $\mathcal{G}^{dc}(0)$ vs $W_{sd}$, for systems with the
same parameters as in $(a)$, showing the conductance node in dimensionless
quantity near $\widetilde{W}_{sd}=1$, compared with Eq. 5.53 (dashed line).
(c) $\mathcal{G}^{dc}(T)$ vs $T$ for fixed $J/D=0.15$, $J_{sd}/D=0.05$,
$W/D=0.2$, varying $W_{sd}/D=0,0.1,0.2,0.3$ for the black, red, blue and green
solid lines (again compared with Eq. 5.53, dotted lines).

Another way to present the quantum-interference effect in the system it is via
phase-shift. At $T=0$, $\delta_{e}\simeq+\pi/2+\pi/2\simeq+\pi$: the
correction due finite potential scattering interferes constructively with the
Kondo singlet formation. As result, the conductance reaches its unitary limit
- see black curve in Fig. 5.5$(a)$ and the point from effective model where
curve $\mathcal{G}^{dc}(T=0)/\mathcal{G}_{0}=2$ in $(b)$. At $T\gg T_{K}$,
$\delta_{e}\simeq-\pi/2+\pi/2\simeq 0$: now the conductance curve presents an
inverted behaviour to the standard Kondo enhanced curve. The unitary limit is
found again but in high-energy scale - see orange curve Fig. 5.5$(a)$. At
$T=0$, $\delta_{e}\simeq 0$ indicates the potential scattering interferes
destructively and the Kondo effect is completely destroyed by the correction
from $\widetilde{W}_{ee}$ term. See point from effective model where the curve
vanishes in Fig. 5.5$(b)$.
The quantitative accuracy of Eq. 5.53 is established by NRG results in Fig.
5.5. The dc-conductance $\mathcal{G}^{dc}(\omega=0,T)$ of the full 2CK model,
Eq. 5.17, under sd-symmetry is obtained within NRG using the improved Kubo
approach, Eq. 4.81. In Fig. 5.5$(a)$ we plot $\mathcal{G}^{dc}$ as a function
of $T$ for fixed $J$, $J_{sd}$ and $W=0$, with different $W_{sd}$. Although
the high-$T$ conductance is rather accurately reproduced by 5.27, Kondo
correlations strongly affect the behaviour at low temperatures $T\ll T_{K}$,
with $T_{K}$ the emergent Kondo temperature. The horizontal dotted lines are
the result of Eq. 5.53, and are seen to agree very well with the low-
temperature conductance from NRG. In particular, the Kondo resonance at
$W_{sd}=0$ (black line) is inverted to a Kondo blockade for $W_{sd}=0.6D$
(orange line). $\mathcal{G}^{dc}(0)$ is plotted for the same parameters in
Fig. 5.5$(b)$ as a function of $W_{sd}$, with NRG results (points) compared
with Eq. 5.53 (dashed line), showing the conductance node predicted by Eq.
5.54. Fig. 5.5$(c)$ confirms the more general form of Eq. 5.53 for finite $W$
and $W_{sd}$.

#### Finite potential scattering $W_{\alpha\beta}\neq 0$ in broken sd-symmetry

Finally, we consider the most general case, in which both $\mathbb{J}$ and
$\mathbb{W}$ are finite, and sd-symmetry is broken i.e. $J_{-}\neq 0$ and
$W_{-}\neq 0$ for $J_{\pm}=J_{ss}\pm J_{dd}$ and $W_{\pm}=W_{ss}\pm W_{dd}$.
First, we note that for arbitrary $J_{\pm}$, $J_{sd}$ and $W_{\pm}$, $W_{sd}$
the lead basis rotation matrix $\mathbb{U}$ that diagonalizes the exchange
matrix $\mathbb{J}$ will not in general diagonalize the potential scattering
matrix $\mathbb{W}$. That is, although
$\mathbb{U}^{\dagger}\mathbb{J}\mathbb{U}$ is constructed such that
$J_{eo}=J_{oe}^{\star}=0$, $\mathbb{U}^{\dagger}\mathbb{W}\mathbb{U}$ still
yields finite $W_{eo}=W^{\star}_{oe}\neq 0$. This means that the even/odd
channels defined in this way are not distinct; and in particular the odd
channel does not completely decouple at low energies, as required for the
previous analysis. In the remaining of this part we develop additional basis
transformations to bring the couplings into a basis form suitable for channel
decoupling under RG.
The first step is therefore to define a new basis, in terms of which we can
identify _distinct_ conduction electron channels, here labelled as a and b. We
introduce the transformation matrix $\mathbb{P}$ from sd into ab basis and we
define for the potential scattering in Eq.5.24 the equation
$\mathbb{\mathcal{W}}=\mathbb{P}^{\dagger}\mathbb{W}\mathbb{P}$ to be
diagonal, such that,

$\displaystyle\mathcal{W}_{aa}$
$\displaystyle=\tfrac{1}{2}(W_{+}+\sqrt{W_{-}^{2}+4|W_{sd}|^{2}})\leavevmode\nobreak\
,$ (5.55a) $\displaystyle\mathcal{W}_{bb}$
$\displaystyle=\tfrac{1}{2}(W_{+}-\sqrt{W_{-}^{2}+4|W_{sd}|^{2}})\leavevmode\nobreak\
,$ (5.55b)

and where $\mathcal{W}_{ab}=\mathcal{W}^{\star}_{ba}=0$ by construction. The
Eq. 5.55 is as such the potential scattering analogue of Eq. 5.34 for the
exchange couplings. Incorporating these local potentials into the definition
of the free lead Hamiltonian in the $a/b$ basis of $\mathcal{H}_{leads}$ with
$\alpha=a,b$, we have renormalized Fermi level densities of states, namely
$\widetilde{\rho}_{\alpha}=\rho_{0}/[1+(\pi\rho_{0}\mathcal{W}_{\alpha\alpha})^{2}]$.
We remark that the same transformation $\mathbb{P}$ is applied to the exchange
matrix $\mathbb{J}$, noting that
$\mathbb{\mathcal{J}}=\mathbb{P}^{\dagger}\mathbb{J}\mathbb{P}$ is not in
general diagonal. We may now identify the relevant dimensionless quantities to
analyse the RG-flow as
$j_{\alpha\beta}=\sqrt{\widetilde{\rho}_{\alpha}\widetilde{\rho}_{\beta}}\mathcal{J}_{\alpha\beta}$
and the $2\times 2$ matrix of these couplings is denoted $\mathbb{j}$.
As second step, we introduce the transformation matrix $\mathbb{Q}$ from ab
into eo basis. Instead of applying this transformation to
$\mathbb{\mathcal{J}}$ that does not have a diagonal form generally, we
perform the final rotation of the lead basis to diagonalize $\mathbb{j}$
matrix and the expression
$\mathbb{j}^{\prime}=\mathbb{Q}^{\dagger}\mathbb{j}\mathbb{Q}$ yields the
desired even/odd exchange couplings $j^{\prime}_{ee}$ and $j^{\prime}_{oo}$.
It is in _this_ basis that we can say that the more strongly coupled even
channel flows to strong coupling and the more weakly coupled odd channel
decouples, analogous to the analysis of the previous section. This assumes
that $j^{\prime}_{ee}>j^{\prime}_{oo}$, which is the conventional situation.
However, according to strongest antiferromagnetic coupling in the bare model,
it is also possible that $j^{\prime}_{oo}>j^{\prime}_{ee}$, in which case the
roles of the even and odd channels in the following are reversed.
The same transformation $\mathbb{Q}$ is applied to the potential scattering
and it is given by
$\mathbb{W}^{\prime}=\mathbb{Q}^{\dagger}\mathbb{\mathcal{W}}\mathbb{Q}$ where
we define the dimensionless quantity
$\widetilde{W}^{\prime}_{\alpha\beta}=\pi\rho_{0}W^{\prime}_{\alpha\beta}$. As
outcome, $\mathbb{W}^{\prime}$ is in the desired even/odd but it is not
diagonal, hence
$\widetilde{W}^{\prime}_{ee}=\widetilde{W}^{\prime}_{eo}=\widetilde{W}^{\prime}_{oe}=\widetilde{W}^{\prime}_{oo}\neq
0$.
The emergent decoupling of the odd channel in this basis at low energies and
temperatures $\omega,T\ll T_{K}$ yields the nontrivial result
$t^{\prime}_{eo}(0,0)=t^{\prime}_{oe}(0,0)=0$. This is a consequence of Kondo
renormalization, and scattering between even and odd channels is finite at
higher energies. In the fully asymmetric case, there is no lead rotation that
yields $t^{\prime}_{eo}(\omega,T)=0$ at all energies and temperatures. The
above prescription, although somewhat complicated, is needed in order to
identify the specific lead combination where inter-channel scattering vanishes
at low energies. We have confirmed this result explicitly with NRG.
In this basis, the analogue of Eqs. 5.47 and 5.49 for the $\mathrm{T}$-matrix
spectra are
$t^{\prime}_{oo,\sigma}(0,0)=(\widetilde{W}^{\prime}_{oo})^{2}/(1+(\widetilde{W}^{\prime}_{oo})^{2})$
and $t^{\prime}_{ee,\sigma}(0,0)=1/(1+(\widetilde{W}^{\prime}_{ee})^{2})$,
with the corresponding complex $\mathrm{T}$-matrices given by

$\displaystyle\pi\rho_{0}T^{\prime}_{oo,\sigma}(0,0)=[(\widetilde{W}^{\prime})^{-1}_{oo}-i]\times
t^{\prime}_{oo,\sigma}(0,0)\leavevmode\nobreak\ ,$ (5.56)
$\displaystyle\pi\rho_{0}T^{\prime}_{ee,\sigma}(0,0)=-[\widetilde{W}^{\prime}_{ee}+i]\times
t^{\prime}_{ee,\sigma}(0,0)\leavevmode\nobreak\ .$

The low-temperature conductance then follows from alternative version of the
Oguri formulation in Eq. 4.105,

$\boxed{\mathcal{G}^{C}(0,0)=\left(\frac{2e^{2}}{h}\right)|\Omega_{e}^{\phantom{{}^{\prime}}}T^{\prime}_{ee,\sigma}(0,0)+\Omega_{o}^{\phantom{{}^{\prime}}}T^{\prime}_{oo,\sigma}(0,0)|^{2}}\;,$
(5.57)

where the inter-channel $\mathrm{T}$-matrix $T_{ds,\sigma}$ in the physical
basis of d and s leads is brought first into the basis of a and b channels
using the transformation $\mathbb{P}$, and then into the even/odd basis via
$\mathbb{Q}$. This yields the coefficients,

$\Omega_{\gamma}=2\pi\rho_{0}\sum_{\alpha,\beta\in
a,b}P_{d\alpha}Q_{\alpha\gamma}P^{\star}_{s\beta}Q^{\star}_{\beta\gamma}\;.$
(5.58)

The low-temperature conductance obtained from NRG is found to agree well with
these predictions.
We emphasize that the benefit of the above framework is that the low-
temperature conductance can be accurately estimated, purely from a knowledge
of the effective 2CK model parameters $\mathbb{J}$ and $\mathbb{W}$. The 2CK
model itself does not need to be solved. Furthermore, since any microscopic
multi-orbital system hosting a net spin-$\tfrac{1}{2}$ coupled to source and
drain leads can in principle be mapped to such a model, the problem of
calculating low-temperature transport becomes one of determining effective
model parameters. These can be estimated perturbatively using Eq. 5.15, or
nonperturbatively using more sophisticated techniques, such as model machine
learning [152].

####  5.1.3 Spin $S=1$

When the nanostructure hosts an even number of electrons, a high-spin state
with $S=1$ may result. This is especially prevalent in single molecule
junctions with Hund’s coupling. The effective model in this case is again Eq.
5.17, but now with $\hat{\mathbf{S}}$ a spin-$1$ operator whereas
$\hat{\mathbf{s}}_{\alpha\beta}$ is still spin-$\tfrac{1}{2}$ operator for the
leads.
In the case with exact PC on the level of the bare model, an effective single-
channel spin-$1$ Kondo model results namely Eq. 5.37 with $\hat{\mathbf{S}}$ a
spin-$1$ operator. The low-energy physics is that of the underscreened Kondo
effect - a singular Fermi liquid with a residual free spin-$\tfrac{1}{2}$
local moment surviving down to $T=0$ [17]. Conductance is again Kondo-enhanced
in this case, but with logarithmic corrections to the low-temperature approach
to the fixed point conductance [157]. This situation can arise in parallel
double quantum dots; although the exact PC condition on the hybridization
matrix is challenging to realize in practice.
The more generic scenario is when the effective nanostructure spin-$1$ is
delocalized and the PC condition is not satisfied. In this case, both $J_{ee}$
and $J_{oo}$ remain finite in the even/odd lead basis. The nanostructure
spin-$1$ is exactly screened on the lowest temperature scales - where we
assume both exchange couplings are antiferromagnetic. However, since
$J_{ee}>J_{oo}$ we have $T_{K}^{e}>T_{K}^{o}$ and a two-stage Kondo effect
results similar to what we discuss in Sec.2.4. In this case, the odd lead
participates in screening of the nanostructure spin-$1$, and remains coupled
down to $T=0$, in contrast to the spin-$\tfrac{1}{2}$ case discussed above.
The nanostructure spin-$1$ is underscreened by the even lead to an effective
spin-$\tfrac{1}{2}$ on the scale of $T_{K}^{e}$, and then fully screened down
to $S=0$ by the odd lead on the scale of $T_{K}^{o}$. Correspondingly, the
conductance is Kondo enhanced for $T_{K}^{o}<T<T_{K}^{e}$ but suppressed on
the lowest temperature scales $T\ll T_{K}^{o}$. These scenarios have been
explored experimentally in [158, 159, 160, 161].

As conclusion, we summarize our findings and method developed across this
section. We start with a general non-proportionate coupling model with a net
ground-state spin $S=0,1/2,1$ at energy scale $T\ll E_{C}$. The original
system is characterized by spin degeneracy. Besides the $S=0$ system which
satisfies the PC condition at the level of the effective model
$\hat{H}_{eff}$, the other two cases require the application of basis
transformation from the bare system basis into suitable even/odd basis which
allow independent RG-flow of coupling in even and odd channel sector. Then, we
study the resulting effective model under RG and we find which energy scale
the channel decoupling occurs. At characteristic temperature depending on the
ground-state spin, the model reduces to single-channel state: it acquires an
incipient PC property and it has renormalised coupling in correspondence to
the strongest channel. In order to evaluate the electrical conductance, we
need to reverse engineer the canonical transformation to bring back the
renormalized effective model in even/odd basis into original basis. We
calculate the $\mathrm{T}$-matrix spectral function and study the RG-flow of
couplings using it. The conductance is studied by means of the alternative
Oguri formula; it is the crucial quantity in the analysis of emergent PC in
Coulomb blockade systems.

###  5.2 Emergent proportionate coupling in mixed-valence regime

This section studies systems with degeneracy both in spin $\hat{S}^{z}$ and in
charge $Q$.
We consider the generic electrical conductance properties of systems near a
Coulomb peak, corresponding to the charge degeneracy point between $N$ and
$N+1$ electrons on the nanostructure i.e. states separated by unitary charge
variation. Typically, one can tune to the Coulomb peaks in experiment by
applying suitable plunger gate voltages [12]. We expect enhanced cotunnelling
conductance at such points due to strong nanostructure charge fluctuations
arising in the mixed-valence regime (MV). In the stability diagram, see sketch
in Fig.3.8, the MV regime is found at the crossover between two well-defined
charge states where we find simultaneously $V_{bias}\to 0$ and finite
$V_{gate}$, here denoted as $V^{\star}_{gate}$.
By contrast with the Coulomb blockade situation of the previous Sec.5.1 where
electrons visit intermediate virtual excited states and spin-flip processes
occur, here electrons can tunnel from source to drain leads without leaving
the nanostructure ground-state manifold; although electron interactions still
typically renormalize the conductance. The fact that at charge degeneracy
points the electrons do not tunnel to excited states - on the contrary to the
Coulomb blockade system - stems into the different order of perturbation
expansion in $\hat{H}_{hyb}$.
Effective models near charge-degeneracy points can be obtained perturbatively
to _first_ order in the hybridization $\hat{H}_{hyb}$ using BWPT. Here,
$\hat{H}_{hyb}$ connects two adjacent electron states of the isolated
nanostructure i.e. $N$ and $N+1$ electrons state. There are no tunnelling
events to other charge states to first-order in $\hat{H}_{hyb}$. The suitable
energy scale where the derived effective model is a solid description of the
system physics is identified as follows. The temperature has to be $(i)$
smaller than the energy difference of states having the same charge $Q$ but
belonging to different eigenstates $l$ i.e. $E^{Q}_{l+1}-E^{Q}_{l}$ and $(ii)$
smaller than the energy difference of states belonging to the same eigenstates
but in two different charge sectors i.e. $E^{Q+1}_{l}-E^{Q}_{l}$. To sum up,
we assume $T\ll min(E^{Q}_{l+1}-E^{Q}_{l},E^{Q+1}_{l}-E^{Q}_{l})\doteq\Delta
E_{min}$, where this minimum energy is the lowest excited state not included
in $\widehat{P}_{0}$ projector on the ground-state manifold defined in Eq.5.10
\- note that in these expressions we omit $S^{z}$ index for conciseness.
Hence, $Q=N-1$ or $Q=N+2$ states are not energetically reachable from the
ground-state energy levels in the chosen temperature range.
To place our discussion on a general ground, in the ground-state manifold
characterised by $\left\\{\ket{\Psi^{Q,S^{z}}_{0}}\right\\}$ state and
$\left\\{E^{Q,S^{z}}_{0}\right\\}$ energy, we consider first a nondegenerate
model where there is a finite energy gap $\Delta E_{0}$ between the ground-
state energy levels, namely

$\displaystyle E^{N,S^{z}}_{0}(V_{gate})=\epsilon\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ E^{N+1,S^{z}\pm
1/2}_{0}(V_{gate})=\epsilon+\epsilon^{\prime}\leavevmode\nobreak\ ,$ (5.59)
$\displaystyle\Rightarrow\Delta E_{0}(V_{gate})\stackrel{{\scriptstyle non-
deg}}{{=}}E^{N+1,S^{z}\pm
1/2}_{0}-E^{N,S^{z}}_{0}=\epsilon^{\prime}\leavevmode\nobreak\ ,$

where $\epsilon^{\prime}$ is a gate voltage-dependent energy splitting to
differentiate the ground-state levels in the two charge sectors. Next, we take
a system in MV regime: the gate voltage is tuned so that all the ground-state
energy levels are not degenerate, namely

$\Delta E_{0}(V^{\star}_{gate})\stackrel{{\scriptstyle
MV}}{{=}}0\quad\Leftrightarrow\quad\epsilon^{\prime}=0\leavevmode\nobreak\ .$
(5.60)

This brief consideration on the energy difference between ground-state energy
levels is relevant because although the derivation developed through this
section stems into the MV regime, the method is also valid for nondegenerate
systems in the close vicinity at the MV degenerate point - as we discuss more
later.
We now continue with the general structure of the BWPT calculation for models
at the valence transition.

#### Analytical method: generalities

The model Hamiltonian of reference in this work is defined by the two-channel
system in Eq.5.1. We now present the general expressions used to calculate the
effective model in MV regime. Then, we detail those according to the spinless
and spinful electrons systems and on the latter for two different charge
sector transitions.
At the crossover between $\ket{N}$ and $\ket{N+1}$ charge sectors, we define
the projector operator for a general MV transition as follows

$\widehat{P}_{0}=\sum_{S^{z}\in
G.S.}\ket{N;S^{z};j=0}\bra{N;S^{z};j=0}\otimes\widehat{\mathbb{1}}_{leads}+\sum_{S^{z}\in
G.S.}\ket{N+1;S^{z};j=0}\bra{N+1;S^{z};j=0}\otimes\widehat{\mathbb{1}}_{leads}\leavevmode\nobreak\
,$ (5.61)

where we use the definition in Eq.5.10, with $j=0$ is the ground-state in
given charge sector and $\sum_{S^{z}\in G.S.}$ is the summation over ground-
state $S^{z}$ components for a given $Q$. The Eq.5.61 is the projector
operator onto the manifold of the retained states in the $\ket{N}$ and
$\ket{N+1}$ charge sector of $\hat{H}_{nano}$. We keep only the ground-state
manifold characterised by $\ket{\Psi^{N,S^{z}}_{0}}$ state, $E^{N,S^{z}}_{0}$
energy and $\ket{\Psi^{N+1,S^{z\prime}}_{0}}$ state, $E^{N+1,S^{z\prime}}_{0}$
energy.
By applying the projector defined in the MV given in Eq.5.61 to the bare model
in Eq.5.1, we calculate the effective model: according to the energy range
$T\ll\Delta E_{min}$, it is given by the $0^{th}$ and $1^{st}$ orders of
Eq.5.13. In particular, the $0^{th}$-order is trivially obtained by:

$\widehat{P}_{0}\hat{H}_{leads}\widehat{P}_{0}\equiv\hat{H}_{leads}\leavevmode\nobreak\
,$ (5.62)

because the lead operators commute with the nanostructure states and

$\widehat{P}_{0}\hat{H}_{nano}\widehat{P}_{0}=\epsilon^{\prime}\big{(}\ket{N+1}\bra{N}\ket{N}\bra{N+1}\big{)}\equiv\epsilon^{\prime}\ket{N+1}\bra{N+1}\leavevmode\nobreak\
,$ (5.63)

with $\epsilon^{\prime}=0$ precisely at the MV charge degeneracy point.
From Eqs.5.62,5.63, we see that the actual form of $\hat{H}_{nano}$ does not
enter in the projection calculation; only the relative splitting between
states in the ground-state manifold. Furthermore, as we mentioned before, the
perturbative scheme is not only applicable to MV regime: the whole calculation
is valid also for nondegenerate systems near the MV point. When we present the
various results for spinless and spinful models under MV regime, the
generalization to nondegenerate model is given by simply adding the energy
constant term $\epsilon^{\prime}$ to the effective MV expression.
The actual demanding calculation is the $1^{st}$-order of Eq.5.13:
$\widehat{P}_{0}\hat{H}_{hyb}\widehat{P}_{0}$. Recalling that we can only
sandwich states that are separated by a unitary charge variation - hence
tunnelling between states with same charge is zero - the projection for the
hybridization term takes the general form:

$\displaystyle\widehat{P}_{0}\hat{H}_{hyb}\widehat{P}_{0}=\sum_{S^{z},S^{z\prime}\in
G.S.}\Big{(}\ket{N;S^{z}}\bra{N+1;S^{z\prime}}$
$\displaystyle\big{(}\bra{N,S^{z}}\hat{H}_{hyb}\ket{N+1;S^{z\prime}}\big{)}+$
(5.64)
$\displaystyle+\ket{N+1;S^{z\prime}}\bra{N;S^{z}}\big{(}\bra{N+1;S^{z\prime}}\hat{H}_{hyb}\ket{N;S^{z}}\big{)}\Big{)}\leavevmode\nobreak\
,$

where all states are taken to be the $j=0$ ground-state for each $Q,S^{z}$
sector.
Hence, the complete effective Hamiltonian in the MV regime we aim to calculate
reads:

$\displaystyle\hat{H}_{eff}=\hat{H}_{leads}$
$\displaystyle+\epsilon^{\prime}\sum_{S^{z}\in
G.S.}\ket{N+1;S^{z}}\bra{N+1;S^{z}}+$ (5.65)
$\displaystyle+\sum_{S^{z},S^{z\prime}\in
G.S.}\Big{(}\bra{N;S^{z}}\hat{H}_{hyb}\ket{N+1;S^{z\prime}}\ket{N,S^{z}}\bra{N+1;S^{z\prime}}\big{)}+H.c.\Big{)}+\mathcal{O}(\hat{H}_{hyb})^{2}\leavevmode\nobreak\
.$

with $\epsilon^{\prime}=0$ precisely at the MV charge degeneracy point.
The remaining of this part is devoted to the calculation of Eq.5.64, according
to the prominent systems in case of spinless and spinful models

####  5.2.1 Spinless model

When the nanostructure and the leads are effectively spinless - which may
arise physically in the presence of a strong magnetic field as we have in the
general two-channel model in Eq.5.1 \- the effective model Hamiltonian in
Eq.5.13 takes a simple form as we discuss in this section.
We use the same model Hamiltonian in Eq.5.1 but we introduce some
simplification for the case of spinless system in MV regime. We consider
$\hat{H}_{hyb}$ to consist of tunnelling processes between the $0^{th}$-sites
on the semi-infinite chain baths and the frontier orbitals of the
nanostructure. For a nanostructure of $m=1,\dots,M$ orbitals, we adopt the
convention in which the lead $\alpha=s$ couples to $m=1$ and $\alpha=d$
couples to $m=M$. The corresponding hybridization term reads as
$\hat{H}_{hyb}=V_{s1}c^{\dagger}_{s0}d_{1}+V^{\star}_{s1}d^{\dagger}_{1}c_{s0}+V_{dM}c^{\dagger}_{d0}d_{M}+V^{\star}_{dM}d^{\dagger}_{M}c_{d0}$.
Using the general definition of the charge-basis state given in Eq.5.8, the
spinless charge basis states now read:

$\ket{\phi^{Q}_{l}}=\ket{n_{1}}_{1}\bigotimes\ket{n_{2}}_{2}\bigotimes\ket{n_{3}}_{3}\bigotimes\dots\bigotimes\ket{n_{M}}_{M}\leavevmode\nobreak\
,$ (5.66)

with the unique combination of occupations $\left\\{n_{i}\right\\}$ consistent
with $Q=\sum_{j}n_{j}$ indexed by $j$, see Eq.5.5. The eigenstates of
$\hat{H}$ are then constructed as linear combination of these states:

$\ket{\Psi^{Q}_{l}}=\sum_{j}c_{Q,l}(j)\ket{\phi^{Q}_{l}}\leavevmode\nobreak\
,\quad\text{with}\leavevmode\nobreak\ \leavevmode\nobreak\ c_{Q,l}(j)\equiv
c_{Q}(n_{1},n_{2},\dots,n_{M})\leavevmode\nobreak\ ,$ (5.67)

where summation $\sum_{j}$ involves all combinations of
$\left\\{n_{i}\right\\}$ consistent with the property $Q=\sum_{j}n_{j}$ in
Eq.5.5 and $l=0$ is taken to be the ground-state in a given charge sector $Q$.
The two energy-degenerate ground-state with $N$ and $N+1$ electrons are
$\ket{\Psi^{Q=N}_{0}}\equiv\ket{N},\ket{\Psi^{Q=N+1}_{0}}\equiv\ket{N+1}$.
Those states satisfy the equations:

$\displaystyle\hat{H}\ket{\Psi^{Q=N}_{0}}\equiv\hat{H}\ket{N}=E^{N}_{0}\ket{N}\leavevmode\nobreak\
,$ (5.68)
$\displaystyle\hat{H}\ket{\Psi^{Q=N+1}_{0}}\equiv\hat{H}\ket{N+1}=E^{N+1}_{0}\ket{N+1}\leavevmode\nobreak\
.$

The spinless projector operator $\widehat{P}_{0}$ under MV regime, see
definition in Eq.5.61, now reads:

$\widehat{P}_{0}\doteq\ket{N}\bra{N}\otimes\widehat{\mathbb{1}}_{leads}+\ket{N+1}\bra{N+1}\otimes\widehat{\mathbb{1}}_{leads}\leavevmode\nobreak\
.$ (5.69)

It is useful to introduce the operators in the reduced ground-state manifold
of the nanostructure:

$f^{\dagger}=\ket{N+1}\bra{N}\quad;\quad
f=\ket{N}\bra{N+1}\leavevmode\nobreak\ ,$ (5.70)

which are Fermionic operators satisfying the anti-commutation relations:

$\\{f,f\\}=0\quad,\quad\\{f^{\dagger},f^{\dagger}\\}=0\quad,\quad\\{f,f^{\dagger}\\}=\mathbb{1}\leavevmode\nobreak\
,$ (5.71)

and those commute with the lead operators i.e. $[f,c_{\alpha 0}]=0$. We note
that the Fermionic operators in Eq.5.70 are constructed over the nanostructure
ground-state basis. Hence, those describe the physical process of unitary
charge variation on a given $\ket{\Psi^{Q}_{0}}$ state and we can refer to
them as charge operators. This interpretation is relevant when it comes to
recognise the derived effective derived models under PC condition - as we see
later in this chapter. We can use the $f^{\dagger},f$ operators in Eq.5.70 to
rewrite the spinless projector in Eq.5.69 as

$\widehat{P}_{0}=(ff^{\dagger})\otimes\widehat{\mathbb{1}}_{leads}+(f^{\dagger}f)\otimes\widehat{\mathbb{1}}_{leads}.$
(5.72)

We observe that the physical process described by the $f^{\dagger},f$
operators is a single-transmission between the leads and the nanostructure
ground-state of a spinless electron. Hence, $f^{\dagger},f$ operators
represent only two possible transmission events separated by a unitary charge
variation i.e. either empty or one electron state.
At this point, we have ready the analytical structure to implement degenerate
perturbation calculations for a spinless model and we proceed with presenting
its results.

#### Analytical result: the effective spinless Hamiltonian

In this part we want to calculate the effective low-energy spinless
Hamiltonian $\hat{H}_{eff}$ as expressed in Eq.5.65. As discussed at the
beginning of this section, the projection operation over $\hat{H}_{leads}$ and
$\hat{H}_{nano}$ is directly given in Eq.5.62 and Eq.5.63, respectively.
Moreover, the expression for $\widehat{P}_{0}\hat{H}_{hyb}\widehat{P}_{0}$
requires the calculation of the matrix elements appearing in Eq.5.64, where
the nanostructure operators act on the spinless charge basis states defined in
Eq.5.67. The outcome of this projection calculation reads:

$\widehat{P}_{0}\hat{H}_{hyb}\widehat{P}_{0}=\left(c^{\dagger}_{s0}\widetilde{V}_{s}+c^{\dagger}_{d0}\widetilde{V}_{d}\right)\ket{N}\bra{N+1}+\ket{N+1}\bra{N}\left(\widetilde{V}^{\star}_{s}c_{s0}+\widetilde{V}^{\star}_{d}c_{d0}\right)\leavevmode\nobreak\
,$ (5.73)

where the effective low-energy hybridization parameters of the model
$\widetilde{V}_{\alpha}$ now appear as consequence of the projection operation
with expressions

$\widetilde{V}_{s}\doteq
V_{s1}\bra{N}d_{1}\ket{N+1}\quad,\quad\widetilde{V}_{d}\doteq
V_{dM}\bra{N}d_{M}\ket{N+1}\leavevmode\nobreak\ ,$ (5.74)

where these expressions can be expressed in terms of matrix elements of the
operators $d_{1},d_{M}$ in the product basis $\left\\{\phi_{l}^{Q}\right\\}$.
In particular, $\widetilde{V}_{\alpha}\leavevmode\nobreak\
\propto\leavevmode\nobreak\ V_{\alpha m}c_{Q}(n_{1},n_{2},\dots,n_{M})n_{m}$,
$\widetilde{V}^{\star}_{\alpha}\leavevmode\nobreak\
\propto\leavevmode\nobreak\ V^{\star}_{\alpha
m}c_{Q}(n_{1},n_{2},\dots,n_{M})(1-n_{m})$ and
$\widetilde{V}_{d},\widetilde{V}^{\star}_{d}$ include an extra $(-1)^{Q}$
factor due to anticommuting $M$ sites. These effective couplings are obtained
from the physical hybridization strength $V_{\alpha m}$ defined in the bare
model in Eq.5.1 multiplied by matrix element in the eigenbasis for the ground-
states connected by $d_{1},d_{M}$, that is $(i)$ the expansion constants
$c_{Q}(\dots)$ defined in Eq.5.67 and $(ii)$ a combination of occupation
number $n_{m}$ to show the anticommuting property of operators on the states.
Before proceeding with the derivation of $\hat{H}_{eff}$, we observe that the
$\widetilde{V}_{\alpha}$ factors can be readily used to calculate the total
levelwidth function defined in Eq.3.13 but now dressed by the effective
hybridization. Assuming the standard scenario of equivalent metallic leads
with constant density of states within the wide-flat band approximation, we
have $\rho_{0,s}=\rho_{0,d}\equiv\rho_{0}$ and the spinless effective
levelwidth function reads:

$\Gamma^{\alpha}_{eff}=\pi|\widetilde{V}_{\alpha}|^{2}\rho_{0}\leavevmode\nobreak\
.$ (5.75)

Ultimately, the electrical conductance is only governed by the effective
coupling $\widetilde{V}_{\alpha}$ terms and those emerge as straightforward
result from the applied perturbative scheme.
We return now to the derivation of the effective model: by inserting all these
results in Eq.5.65, we obtain the effective low-energy spinless Hamiltonian in
MV regime:

$\displaystyle\hat{H}_{eff}$ $\displaystyle\stackrel{{\scriptstyle
MV}}{{=}}\hat{H}_{leads}+\left(c^{\dagger}_{s0}\widetilde{V}_{s}+c^{\dagger}_{d0}\widetilde{V}_{d}\right)f+f^{\dagger}\left(\widetilde{V}^{\star}_{s}c_{s0}+\widetilde{V}^{\star}_{d}c_{d0}\right)\leavevmode\nobreak\
,$ (5.76) $\displaystyle\stackrel{{\scriptstyle non-
deg}}{{=}}\hat{H}_{leads}+\left(c^{\dagger}_{s0}\widetilde{V}_{s}+c^{\dagger}_{d0}\widetilde{V}_{d}\right)f+f^{\dagger}\left(\widetilde{V}^{\star}_{s}c_{s0}+\widetilde{V}^{\star}_{d}c_{d0}\right)+\epsilon^{\prime}f^{\dagger}f\leavevmode\nobreak\
,$

where we give both the charge degenerate MV regime and the nondegenerate case,
using the $f$ operators definition in Eq.5.70 and the effective
hybridizations.
Before proceeding any further, we compare the derived model in Eq.5.76 with
the original model presented in Eq.5.1. The nanostructure region is initially
a finite, interacting chain representing a multi-orbital configuration. As
outcome of the projection, the whole nanostructure region is identified as an
effective, noninteracting single-site by the $f,f^{\dagger}$ operators. As
consequence, the physical tunnelling matrices assume their effective version
$\widetilde{V}_{\alpha}$ as we discuss above. And the spinless $f,f^{\dagger}$
operators correctly describe a single-transmission between the semi-infinite
lead chains and the single-effective nanostructure.
We also note that Hamiltonian expression as written in Eq.5.76 reflects the
fundamental requirement for proportionate coupling condition as discussed in
Sec.3.3.3. The effective nanostructure is coupled directly to both lead chains
and the system satisfies the proportionality equation as given in Eq.3.63 but
now written for the effective levelwidth function $\Gamma^{\alpha}_{eff}$
defined in Eq.5.75.
By means of these observations, the Hamiltonian in Eq.5.76 can be factorised
and we can combine both physical $s,d$ leads into a unified effective one.
This is achieved by the canonical lead transformation
$c_{o}=[\widetilde{V}_{s}c_{so}+\widetilde{V}_{d}c_{do}]/\widetilde{V}$ with
$\widetilde{V}^{2}=\widetilde{V}^{2}_{s}+\widetilde{V}^{2}_{s}$. This single
lead couples with a unique effective hybridization $\widetilde{V}$ that is the
total spinless effective hybridization for the PC model in MV regime. These
manipulation brings the effective model in Eq.5.76 into the effective
Hamiltonian for a spinless PC model in MV regime:

$\boxed{\hat{H}_{eff}\stackrel{{\scriptstyle
MV}}{{=}}\hat{H}_{leads}+c^{\dagger}_{0}\widetilde{V}f+f^{\dagger}\widetilde{V}^{\star}c_{0}}\leavevmode\nobreak\
,$ (5.77)

where this low-energy Hamiltonian is fully determined by the $0^{th}$-site of
the combined lead chain and the $f^{\dagger},f$ operators on the effective
nanostructure, mediated by an effective PC hybridization $\widetilde{V}$. The
Eq.5.77 is most important result of the spinless section: we have shown that
performing BWPT calculation on a spinless interacting model under its MV
condition, we obtain an effective Hamiltonian that has an emergent PC
property. We conclude by saying that it is straightforward to generalize
$\hat{H}_{eff}$ for the nondegenerate charge system by adding to Eq.5.77 the
term $\epsilon^{\prime}f^{\dagger}f\equiv(E^{N+1}_{0}-E^{N}_{0})\hat{n}_{f}$.

#### Spinless effective model as resonant-level model

So far, we have developed an operative scheme to derive an effective model
satisfying PC condition at low-energy scale while starting from a microscopic
model that does not satisfy such a property. Now, we continue our discussion
on the type of model the Eq.5.77 can be mapped to.
Recalling the definition on the noninteracting resonant level model given in
Eq.2.42, we recognize the analogies between our derived effective model in
Eq.5.77 and the NI-RLM. If we release the spin index in the NI-RLM Hamiltonian
in Eq.2.42, the expression constitutes a two-level-charge system characterised
by tunnelling processes with unitary charge variation between a unique bath
chain and an nanostructure. Hence, the spinless NI-RLM is a proper two-level
system: the tunnelled electron can either be on one bath or on the
nanostructure
Thus, the derived Hamiltonian for a spinless mixed-valence model in
proportionate coupling in Eq.5.77 is recognised as noninteracting resonant
level model as defined in Eq.2.42, [162].
As last observation, we notice that the mapped effective model shows specific
features - for instance the PC property and the lack of Coulomb interaction -
arising from the perturbative BWPT scheme.
In conclusion, Eq.5.77 shows PC property - opposed to the bare model. This
incipient PC condition greatly facilitates quantum transport calculation in
the low energy and temperature scales where the effective model
$\hat{H}_{eff}$ provides a solid description. Furthermore, Eq.5.77 \- and its
mapped system - is noninteracting despite of the strong electron correlations
typically present on the physical bare nanostructure. We make use of these
emergent properties in the next part where we calculate the electrical
conductance.

#### Analytical result: electrical conductance for spinless mixed-valence
model

Due to the incipient PC property appearing in the spinless effective model in
Eq.5.77 and to the mapping $\hat{H}_{eff}\equiv\hat{H}^{NI-RLM}$, the
effective model is exactly solvable. In particular, we can write a single-
particle description for the electrical conductance formula.
Considering $\hat{H}_{eff}$, the noninteracting retarded correlator of the
effective $f$ single-level reads as
$G^{R}_{0,ff}(\omega,T)=\lim_{\eta\rightarrow
0^{+}}1/(\omega+i\eta-\epsilon^{\prime}-\Gamma^{\alpha}_{eff})$. Hence, we can
apply the MW formula under PC in Eq.3.66 to calculate the electrical
conductance for the spinless $\hat{H}_{eff}$ by means of its effective
parameters. The current expression then reads:

$I=\frac{e}{h}\int
d\omega\big{(}f^{\mu_{L}}(\omega)-f^{\mu_{R}}(\omega\big{)}Tr[\widetilde{\mathbb{T}}_{PC}(\omega,T)]\leavevmode\nobreak\
,$ (5.78)

where we use $\widetilde{\mathbb{T}}_{PC}(\omega,T)$ generalised transfer
matrix under PC in Eq.3.67 for standard equivalent metallic leads in wide-flat
band approximation - the explicit expression for the noninteracting RLM reads
as:

$\displaystyle\widetilde{\mathbb{T}}_{PC}(\omega,T)=$ $\displaystyle
4\frac{\Gamma^{s}_{eff}\Gamma_{eff}^{d}}{(\Gamma_{eff}^{s}+\Gamma_{eff}^{d})}\frac{\Gamma^{s}_{eff}+\Gamma_{eff}^{d}}{\Gamma_{eff}^{s}+\Gamma_{eff}^{d}}(-\mathit{Im}G^{R}_{0,ff}(\omega,T))\leavevmode\nobreak\
,$ (5.79)
$\displaystyle=4\frac{\Gamma_{eff}^{s}\Gamma_{eff}^{d}}{(\Gamma_{eff}^{s}+\Gamma_{eff}^{d})^{2}}(\Gamma_{eff}^{s}+\Gamma_{eff}^{d})\left(-\lim_{\eta\rightarrow
0^{+}}\frac{-(\eta+\Gamma_{eff}^{s}+\Gamma_{eff}^{d})}{(\omega-\epsilon^{\prime})^{2}+(\eta+\Gamma_{eff}^{s}+\Gamma_{eff}^{d})^{2}}\right)\leavevmode\nobreak\
,$
$\displaystyle=4\frac{\Gamma_{eff}^{s}\Gamma_{eff}^{d}}{(\omega-\epsilon^{\prime})^{2}+(\Gamma_{eff}^{s}+\Gamma_{eff}^{d})^{2}}\leavevmode\nobreak\
,$

where we use the effective gamma function definition in Eq.5.75 and
$\epsilon^{\prime}=E^{N+1}_{0}-E^{N+1}_{0}$ to account for nondegenerate
charge states. To conclude, the MW formula under PC $\hat{H}_{eff}$ spinless
model is equal to:

$\boxed{I=\frac{e}{h}\int
d\omega\big{(}f^{\mu_{L}}(\omega)-f^{\mu_{R}}(\omega\big{)}Tr\left(4\frac{\Gamma_{eff}^{s}\Gamma_{eff}^{d}}{(\omega-\epsilon^{\prime})^{2}+(\Gamma_{eff}^{s}+\Gamma_{eff}^{d})^{2}}\right)}\leavevmode\nobreak\
,$ (5.80)

whose spectral function - the term inside round parenthesis - is described by
a Lorentzian function centred at $\omega=\epsilon^{\prime}$ with full-width at
half-max equal to $\Gamma_{eff}^{s}+\Gamma_{eff}^{d}$. Moreover, the
conductance is generally decaying as
$\Gamma_{eff}^{\alpha}\leavevmode\nobreak\ \propto\leavevmode\nobreak\
|\widetilde{V}_{\alpha}|^{2}$. We now discuss the possible scenarios occurring
for Eq.5.80.
At strictly zero energy and temperature scales i.e. $\omega=T=0$ and in full
MV regime i.e. $\epsilon^{\prime}\equiv 0$, we can use the MW current formula
in Eq.5.80 under linear response to calculate the differential conductance:

$\mathcal{G}^{dc}(0,0)=\frac{e^{2}}{h}\left(4\frac{|\widetilde{V}_{s}|^{2}|\widetilde{V}_{d}|^{2}}{(|\widetilde{V}_{s}|^{2}+|\widetilde{V}_{d}|^{2})^{2}}\right)\equiv\mathcal{G}_{0}\widetilde{\Gamma}\leavevmode\nobreak\
,$ (5.81)

where we use the definition in Eq.5.75. From this expression, the conductance
is fully determined by the effective hybridization parameters in the
definition of the geometric factor $\widetilde{\Gamma}$ arising from BWPT -
which ultimately are related to the bare hybridization of the bare model, see
Eq.3.70. Furthermore, we obtain the maximum conductance value in the spinless
system. The value is enhanced to $e^{2}/h$ in case of symmetric effective
hybridizations $\widetilde{V}_{s}\equiv\widetilde{V}_{d}$ which does not
necessarily imply the symmetry in the bare hybridization $V_{s1}\equiv
V_{dM}$. By means of the current expression, we can also identify the MV
region extending around the crossover point between $N$ and $N+1$ charge
sector for a radius equal to
$\Gamma_{eff}^{s}+\Gamma_{eff}^{d}\leavevmode\nobreak\
\propto\leavevmode\nobreak\ |\widetilde{V}_{s}|^{2}+|\widetilde{V}_{d}|^{2}$.
At $\omega$, $T\simeq 0$ and $\epsilon^{\prime}\simeq 0$, the small energy and
temperature regimes allow to study the system in the vicinity of the charge
degenerate crossover with radius again equal to
$\Gamma_{eff}^{s}+\Gamma_{eff}^{d}$ \- hence, we regard the system as in
pseudo-MV regime. The corresponding spectral function has the same half-width
at half-max of the previous case but now it is centred at $\epsilon^{\prime}$.
In the general situation for finite $\omega,T$ and nondegenerate charge system
$\epsilon^{\prime}\neq 0$, the system is outside the MV area identified by the
radius $\Gamma_{eff}^{s}+\Gamma_{eff}^{d}$. The conductance is derived from
the MW formula in Eq.5.80 where we need to solve the integration over the
whole energy spectrum. The electrical conductance is now function of:

$\mathcal{G}^{C}(\omega,T)=\mathit{f}\Big{\\{}\epsilon^{\prime},\widetilde{V}_{s},\widetilde{V}_{d}\Big{\\}}\leavevmode\nobreak\
,$ (5.82)

which follows directly from Eq.5.80.

####  5.2.2 Spin model: singlet-doublet transition

A more generic and interesting situation arises in spinful systems where
states of $\hat{H}_{nano}$ form a spin multiplet structure. Here we focus on
the most common case where the ground state of the nanostructure with an even
number of electrons $N$ is a degenerate spin-singlet state, while with odd
$N+1$ electrons we have a degenerate spin-doublet state. Both singlet and
doublet must be taken into account in the effective model near the charge
degeneracy point between $N$ and $N+1$ electrons.
In the derivation of the effective model for singlet-doublet valence
transition, we use the general two-channel model defined in Eq.5.1 with spin
charge basis state introduced in Eq.5.8. As it will be clear from the
derivation, we insert in the bare hybridization also the spin label i.e.
$V_{\alpha m\sigma}$ such that we can define the coupling for each spinful
channel of the system. The bare model presents $3$-fold degeneracy including
only $\ket{\Psi^{N,S^{z}=0}_{0}},\ket{\Psi^{N+1,S^{z}=\pm 1/2}_{0}}$ \- as
consistent with the energy regime $T\ll\Delta E_{min}$ considered here and
with the order of perturbation expansion $\mathcal{O}(\hat{H}_{hyb})^{2}$. The
energy-degenerate charge sectors $N$ and $N+1$ defined on the nanostructure
ground-state manifold satisfy the equations:

$\displaystyle\hat{H}\ket{\Psi^{Q=N,S^{z}=0}_{0}}\equiv\hat{H}\ket{N;S^{z}=0}=E^{N,S^{z}=0}_{0}\ket{N;S^{z}=0}\leavevmode\nobreak\
,$ (5.83) $\displaystyle\hat{H}\ket{\Psi^{Q=N+1,S^{z}=\pm
1/2}_{0}}\equiv\hat{H}\ket{N+1;S^{z}=\pm 1/2}=E^{N+1,S^{z}=\pm
1/2}_{0}\ket{N+1;S^{z}=\pm 1/2}\leavevmode\nobreak\ .$

and extensions to higher spin multiplet ground-states is discussed at the end
of the chapter although, in practice, the $T\to 0$ transport of our interest
is typically controlled by the lowest spin multiplet in $N$ and $N+1$ electron
sectors, and we now focus only on these situations.
The spin projector operator $\widehat{P}_{0}$ in the MV regime, see definition
in Eq.5.10, now reads:

$\widehat{P}_{0}\doteq\ket{N;0}\bra{N;0}\otimes\widehat{\mathbb{1}}_{leads}+\sum_{S^{z}=\pm
1/2}\ket{N+1;S^{z}}\bra{N+1;S^{z}}\otimes\widehat{\mathbb{1}}_{leads}\leavevmode\nobreak\
.$ (5.84)

In analogy to the spinless case, it is helpful to introduce the spinful
operators:

$f^{\dagger}_{\sigma}=\ket{N+1;\sigma}\bra{N;0}\quad;\quad
f_{\sigma}=\ket{N;0}\bra{N+1;\sigma}\leavevmode\nobreak\ ,$ (5.85)

with $\sigma=\uparrow$ for $S^{z}=+1/2$ and $\sigma=\downarrow$ for
$S^{z}=-1/2$. These charge operators are again Fermionic operators satisfying
the anti-commutation relations
$\\{f_{\sigma},f_{\sigma^{\prime}}\\}=0\leavevmode\nobreak\
,\leavevmode\nobreak\
\\{f^{\dagger}_{\sigma},f^{\dagger}_{\sigma^{\prime}}\\}=0\leavevmode\nobreak\
,\leavevmode\nobreak\
\\{f_{\sigma},f^{\dagger}_{\sigma^{\prime}}\\}=\mathbb{1}$ and the commutation
relation with lead operators i.e. $\\{f_{\sigma},c_{0\sigma}\\}=0$. We also
observe that the $f^{\dagger}_{\sigma},f_{\sigma}$ conserve the spin. With
this respect, these operators are understood as like the $\hat{S}^{z}$-spin
operator. This demonstrates the fact that $\hat{H}_{hyb}$ returns non-
vanishing matrix elements only if it is calculated between two different
charge states separated by $\Delta Q=\pm 1$ but with spin conservation during
the whole transmission process. Analogously to the spinless case, we use
$f^{\dagger}_{\sigma},f_{\sigma}$ operators to rewrite the spin projector in
Eq.5.84 as:

$\widehat{P}_{0}=\sum_{\sigma\in\\{\uparrow,\downarrow\\}}(f_{\sigma}f^{\dagger}_{\sigma})\otimes\widehat{\mathbb{1}}_{leads}+(f^{\dagger}_{\sigma}f_{\sigma})\otimes\widehat{\mathbb{1}}_{leads}\leavevmode\nobreak\
.$ (5.86)

We continue in the next part to implement the BWPT calculation on the spinful
MV model.

#### Analytical result: the effective spin Hamiltonian

We aim now to calculate the effective low-energy spin Hamiltonian
$\hat{H}_{eff}$ as expressed in Eq.5.65. The projection on $\hat{H}_{leads}$
is trivially given by Eq.5.62 and on $\hat{H}_{nano}$ by Eq.5.63 \- just now
introducing also the spin label. We focus on
$\widehat{P}_{0}\hat{H}_{hyb}\widehat{P}_{0}$ where we need to compute the
matrix elements appearing in Eq.5.64, where the nanostructure operators act on
the spin charge basis states defined in Eq.5.8. The outcome of the projection
calculation reads as:

$\widehat{P}_{0}\hat{H}_{hyb}\widehat{P}_{0}=\\!\\!\\!\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{\alpha=s,d}\left(c^{\dagger}_{\alpha
0\sigma}\widetilde{V}_{\alpha\sigma}\ket{N;0}\bra{N+1;\sigma}+\ket{N+1;\sigma}\bra{N;0}\widetilde{V}^{\star}_{\alpha\sigma}c_{\alpha
0\sigma}\right)\leavevmode\nobreak\ ,$ (5.87)

where the effective low-energy hybridization parameters of the model
$\widetilde{V}_{\alpha\sigma}$ per $\alpha$ channel with $\sigma$ spin are
emergent parameters of arising from the perturbation calculation. These
effective elements are defined by the physical hybridization strength
$V_{\alpha m\sigma}$ from the bare spinful model - here accounting all the
degree of freedom - multiplied by matrix elements, namely

$\widetilde{V}_{s\sigma}\doteq
V_{s1\sigma}\bra{N;0}d_{1\sigma}\ket{N+1;\sigma}\quad,\quad\widetilde{V}_{d\sigma}\doteq
V_{dM\sigma}\bra{N;0}d_{M\sigma}\ket{N+1;\sigma}\leavevmode\nobreak\ ,$ (5.88)

where these matrix elements can again be expressed in terms of the original
product state basis for the actual evaluation. In particular,
$\widetilde{V}_{\alpha\sigma}\leavevmode\nobreak\ \propto\leavevmode\nobreak\
V_{\alpha
m\sigma}c_{Q}(n_{1},n_{2},\dots,n_{M})n_{m}(\Pi_{y=1}^{m-1}(-1)^{n_{y\uparrow}+n_{y\downarrow}})$
and $\widetilde{V}^{\star}_{\alpha\sigma}\leavevmode\nobreak\
\propto\leavevmode\nobreak\ V^{\star}_{\alpha
m\sigma}c_{Q}(n_{1},n_{2},\dots,n_{M})(1-n_{m})(\Pi_{y=1}^{m-1}(-1)^{n_{y\uparrow}+n_{y\downarrow}})$
where the matrix elements are expressed by $(i)$ the expansion constants
$c_{Q}(\dots)$ given in Eq.5.8 and $(ii)$ a string of occupation numbers
$n_{m\sigma}$ according to the anticommuting property of the Fermionic spinful
operators on the charge state - here according to the spin indexing order in
Eq.5.8 where each $m^{th}$-site is filled first by a spin up electron and then
by a spin down electron.
As for the spinless case, it is straightforward to implement the spinful
effective levelwidth function using the effective hybridization
$\widetilde{V}_{\alpha\sigma}$ in Eq.5.88, namely:

$\Gamma^{\alpha}_{eff,\sigma}=\pi|\widetilde{V}_{\alpha\sigma}|^{2}\rho_{0}\leavevmode\nobreak\
,$ (5.89)

and the electrical conductance is controlled only by these effective
parameters arising from the BWPT calculation.
We continue now with the derivation of the effective model: by inserting all
the above results in Eq.5.65, we calculate the effective low-energy spin
Hamiltonian in MV regime:

$\displaystyle\hat{H}_{eff}$ $\displaystyle\stackrel{{\scriptstyle
MV}}{{=}}\hat{H}_{leads}+\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{\alpha=s,d}\left(c^{\dagger}_{\alpha
0\sigma}\widetilde{V}_{\alpha\sigma}f_{\sigma}+f^{\dagger}_{\sigma}\widetilde{V}^{\star}_{\alpha\sigma}c_{\alpha
0\sigma}\right)+B_{eff}\hat{S}^{z}_{f}\leavevmode\nobreak\ ,$ (5.90)
$\displaystyle\stackrel{{\scriptstyle non-
deg}}{{=}}\hat{H}_{leads}+\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\sum_{\alpha=s,d}\left(c^{\dagger}_{\alpha
0\sigma}\widetilde{V}_{\alpha\sigma}f_{\sigma}+f^{\dagger}_{\sigma}\widetilde{V}^{\star}_{\alpha\sigma}c_{\alpha
0\sigma}\right)+B_{eff}\hat{S}^{z}_{f}+\epsilon^{\prime}\hat{n}_{f}\leavevmode\nobreak\
,$

where we use Eqs.5.85,5.88, the $z$-component of the total nanostructure spin
$\hat{S}^{z}_{f}=\tfrac{1}{2}(\hat{n}_{f\uparrow}-\hat{n}_{f\downarrow})$ for
$\hat{n}_{f_{\sigma}}=f_{\sigma}^{\dagger}f_{\sigma}^{\phantom{\dagger}}$ and
$B_{eff}=(E^{N+1;\uparrow}_{0}-E^{N+1;\downarrow}_{0})$. In the second line
for nondegenerate charge system we use
$\hat{n}_{f}=\hat{n}_{f\uparrow}+\hat{n}_{f\downarrow}$ and
$\epsilon^{\prime}=\tfrac{1}{2}(E^{N+1;\uparrow}_{0}-E^{N+1;\downarrow}_{0})-E^{N;0}_{0}$.
We remark that the Eq.5.90 ascribes a noninteracting model. Before we proceed
with the derivation, in analogy to the discussion of the spinless model, the
interacting multi-orbital nanostructure is condensed to a unique effective
nanostructure with $f^{\dagger}_{\sigma},f_{\sigma}$ operators. The effective
hybridizations $\widetilde{V}_{\alpha\sigma}$ in Eq.5.88 couple this unique
site, that is the lowest energy single-orbital, with the $0^{th}$-site of both
chain leads.
By following the same steps we did for the spinless system, we would attempt
to directly factorise the terms in Eq.5.90. However, we cannot apply this yet
because the spin states set is now enlarged with respect to the bare system
states. We need to remove the extra spurious spin states under appropriate
energy constraints - as we discuss now.
According to the energy regime $T\ll\Delta E_{min}$ as we assume for the MV
modelling, states $Q=N-1$ or $Q=N+2$ are neither energetically favourable, nor
are appearing within $\mathcal{O}(\hat{H}_{hyb})^{2}$ order of perturbation
expansion from the ground-states to first order. We denote then the three
states $\ket{N;0}$, $\ket{N+1;\pm\tfrac{1}{2}}$ described by the bare model in
Eq.5.1 as admissible states of system. However, we have four effective states
of the spinful f-orbital, namely:

$\ket{0}_{f}\quad,\quad\ket{\uparrow}_{f}\leavevmode\nobreak\
,\leavevmode\nobreak\
\ket{\downarrow}_{f}\quad,\quad\ket{\uparrow\downarrow}_{f}\leavevmode\nobreak\
.$ (5.91)

We must eliminate the spurious state $\ket{\uparrow\downarrow}_{f}$.
The $\hat{H}_{eff}$ in Eq.5.90 has a $4$-fold degeneracy and that is not
consistent with single-electron transmission within energy scale $T\ll\Delta
E_{min}$, which gives $3$-fold degeneracy MV states. In order to eliminate the
extra unwanted states that lead to energetically unfavoured tunnelling events,
we introduce a fictitious on-site Coulomb potential as a constraint, $U_{c}$.
We add this to the derived effective model in Eq.5.90 that is:

$\hat{H}^{\prime}_{eff}=\hat{H}_{eff}+U_{c}\hat{n}_{f_{\uparrow}}\hat{n}_{f_{\downarrow}}\leavevmode\nobreak\
.$ (5.92)

The Eq.5.92 describes the effective single Fermionic degree of freedom with
operators $f^{\dagger}_{\sigma},f_{\sigma}$ defining the three accessible
states $\\{\ket{0},\ket{\uparrow},\ket{\downarrow}\\}$ under the energy
constraint $U_{c}\rightarrow+\infty$ to ensure
$\braket{\hat{n}_{f_{\uparrow}}\hat{n}_{f_{\downarrow}}}=0$. Thus, the
effective interacting model in Eq.5.92 returns the correct $3$-fold degeneracy
consistent with the bare model spin degeneracy.
We note that after implementing the Coulomb potential constraint,
$\hat{H}^{\prime}_{eff}$ is subjected to an additional energy scale. Next to
$T\ll\Delta E_{min}$ required for applying the perturbative scheme up to order
$\mathcal{O}(\hat{H}_{hyb})^{2}$, we now have also $k_{B}T\ll U_{c}$ to
project out spurious states in Eq.5.90 such as the doubly occupied one. This
condition is always satisfied for $U_{c}\to\infty$ in the effective model in
Eq.5.92, or implemented in practice. It is indeed known that the Coulomb
interaction and the energy level spacing between states at same $Q$ charge
sector control the conductance curve for a given temperature [46].
Having restored the correct ground-state degeneracy in the effective model by
means of introducing the fictitious constraint $U_{c}$ and taking its limit to
infinity, we proceed with the factorization of $\hat{H}^{\prime}_{eff}$ in
Eq.5.92 to formulate an effective single-channel model as previously,

$c_{0\sigma}=\frac{1}{\widetilde{V}_{\sigma}}\left(\widetilde{V}_{s\sigma}c_{s0\sigma}+\widetilde{V}_{d\sigma}c_{d0\sigma}\right)\leavevmode\nobreak\
,$ (5.93)

with

$\widetilde{V}_{\sigma}^{2}=\widetilde{V}_{s\sigma}^{2}+\widetilde{V}_{d\sigma}^{2}\leavevmode\nobreak\
.$ (5.94)

As before, we define
$\Gamma_{eff,\sigma}=\pi|\widetilde{V}_{\sigma}|^{2}\rho_{0}$ that is the
spinful effective hybridization for the PC model in the singlet-doublet MV
transition.
At the end of these manipulation on Eq.5.92 we obtain the effective
Hamiltonian for a spinful PC model in singlet-doublet transition:

$\boxed{\hat{H}^{\prime}_{eff}=\hat{H}_{leads}+\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\left(c^{\dagger}_{0\sigma}\widetilde{V}_{\sigma}f_{\sigma}+f^{\dagger}_{\sigma}\widetilde{V}^{\star}_{\sigma}c_{0\sigma}\right)+U_{c}\hat{n}_{f_{\uparrow}}\hat{n}_{f_{\downarrow}}}\leavevmode\nobreak\
,$ (5.95)

with the constraint $U_{c}\rightarrow+\infty$. The low-energy Hamiltonian in
Eq.5.95 is fully determined by the $0^{th}$-site of the combined lead chain
and the $f^{\dagger}_{\sigma},f_{\sigma}$ operators on the effective
nanostructure, mediated by the effective PC hybridization
$\widetilde{V}_{\sigma}$. This is the most important result of this part:
under the appropriate energy condition, the effective model calculated from
BWPT presents an emergent PC property. It is straightforward to generalize
$\hat{H}^{\prime}_{eff}$ model to nondegenerate charge states by adding a term
$+\epsilon^{\prime}\sum_{\sigma}f^{\dagger}_{\sigma}f_{\sigma}$ with
$\epsilon^{\prime}=\tfrac{1}{2}(E^{N+1;\uparrow}_{0}+E^{N+1;\downarrow}_{0})-E^{N;0}_{0}$.

#### Spinful effective model as Anderson impurity model

In the previous part we calculated the effective model satisfying
proportionate coupling condition starting from a microscopic model that does
not satisfy such a property. To complete the analysis, it remains to identify
a mapping for Eq.5.95.
According to the discussion in Sec.2.2, the infinitely strong interacting
effective model $\hat{H}^{\prime}_{eff}$ in Eq.5.95 is equivalent to spinful
infinite-U Anderson impurity model in the MV transition between singlet and
doublet charge sector. By means of the large $U$ limit, tunnelling to any
multi-occupied states are inhibited and the effective model can be manipulated
into a form that satisfies PC condition. This property is of great advantage
for conductance calculation, as we discuss in the next part.

#### Analytical and NRG results: electrical conductance for spinful singlet-
doublet mixed-valence model

An important consequence of the emergent PC in $\hat{H}^{\prime}_{eff}$ is
that the quantum transport at low energy $T\ll\Delta E_{min}$ can be obtained
using the MW formula under PC - as we calculate below.
In the derived effective model, the interacting retarded correlator of the
effective single Fermionic degree of freedom is, in analogy to Eq.2.43, given
by $G^{R}_{ff\sigma}(\omega,T)=\lim_{\eta\rightarrow
0^{+}}1/(\omega+i\eta-\epsilon^{\prime}-\Gamma_{eff,\sigma}-\Sigma_{ff\sigma}(\omega,T))$
where $\Sigma_{ff\sigma}(\omega,T)$ is the $f$-level interaction self-energy
according to the standard equivalent single metallic lead model under wide-
flat conduction band limit. Unlike for the previous spinless case, here the
dynamics are nontrivial due to the constraint $U_{c}\to\infty$. However,
$G^{R}_{ff\sigma}(\omega,T)$ can be calculated numerically using NRG and
hence, the quantum transport properties can be obtained. This requires only
the solution of the effective model, rather than the full bare model.
Reference results for the effective model can therefore be calculated and
reused for any microscopic system.
We start the derivation of MW current formula under PC for the singlet to
doublet valence transition by writing its expression in Eq.3.66 as

$I=\frac{2e}{h}\int
d\omega\big{(}f^{\mu_{L}}(\omega)-f^{\mu_{R}}(\omega\big{)}Tr[\widetilde{\mathbb{T}}_{PC}(\omega,T)]\leavevmode\nobreak\
,$ (5.96)

for $\widetilde{\mathbb{T}}_{PC}(\omega,T)$ the generalised transfer matrix in
PC, see Eq.3.67, now for the model Hamiltonian in Eq.5.95 is given by the
expression

$\widetilde{\mathbb{T}}_{PC}(\omega,T)=4\sum_{\sigma}\frac{|\widetilde{V}_{s\sigma}|^{2}|\widetilde{V}_{d\sigma}|^{2}}{(|\widetilde{V}_{s\sigma}|^{2}+|\widetilde{V}_{d\sigma}|^{2})^{2}}t_{ff,\sigma}(\omega,T)\leavevmode\nobreak\
,$ (5.97)

where the $\mathrm{T}$-matrix spectral function defined in Eq.3.68 now for the
effective single Fermionic degree of freedom reads as

$\displaystyle
t_{ff,\sigma}(\omega,T)=\Gamma_{eff}\left(-\mathit{Im}G^{R}_{ff\sigma}(\omega)\right)\leavevmode\nobreak\
,$ (5.98) $\displaystyle t_{ff,\sigma}(\omega=0,T=0)\stackrel{{\scriptstyle
FL}}{{=}}\frac{\Gamma_{eff,\sigma}^{2}}{(\epsilon^{\star})^{2}+\Gamma^{2}_{eff,\sigma}}\leavevmode\nobreak\
,$

where in the second line we use $G^{R}_{ff\sigma}(\omega,T)$ correlator for
Fermi liquid system characterised by interaction-renormalised single Fermionic
level energy
$\epsilon^{\star}=\epsilon^{\prime}+{\mathit{Re}}\Sigma_{ff}(\omega\to 0,T\to
0)$, and vanishing $\mathit{Im}\Sigma_{ff}(\omega\to 0,T\to 0)\to 0$ \- see
discussion for Eq.2.44. We remark that the Fermi liquid results are exact only
at $\omega=T=0$ where $t_{ff,\sigma}(0,0)$ spectra is described by a
Lorentzian curve centred at $\epsilon^{\star}$ and having half-width at half-
max as $\Gamma_{eff,\sigma}$. Hence, any deviation from zero temperature and
energy scales presents $t_{ff,\sigma}(\omega,T)$ spectra with deformed
Lorentzian shape due to non-negligible $\mathit{Im}\Sigma_{ff}(\omega,T)$.

In the absence of a magnetic field in the bare model, we have
$E^{N+1;\uparrow}_{0}=E^{N+1;\downarrow}_{0}$ such that the effective field
vanishes, $B_{eff}=0$, and the effective tunnelling amplitudes become
independent of the spin label $\sigma$ hence
$\widetilde{V}_{\alpha\sigma}\equiv\widetilde{V}_{\alpha}$ and
$\Gamma_{eff,\sigma}\equiv\Gamma_{eff}$.
The $\mathrm{T}$-matrix spectral function is identified by the following
expression

$t_{ff}(\omega=T=0)=\lim_{\omega\rightarrow
0}\frac{(\Gamma_{eff})^{2}}{(\omega-\epsilon^{\star})^{2}+(\Gamma_{eff})^{2}}\leavevmode\nobreak\
,$ (5.99)

We notice that $t_{ff}(\omega=0,T=0)$ is the core quantity of interest for the
electrical conduction calculation and we can evaluate it easily once we know
both the effective hybridization $\Gamma_{eff}$ \- as we derive already in
Eq.5.88 \- and the renormalised $\epsilon^{\star}$ value. Once the effective
model is fully derived, $\epsilon^{\star}$ still remains an unknown quantity
we are left to calculate and about it we discuss more below. However, for the
infinite-$U$ metallic AIM, $\Sigma_{ff}(0,0)$ and hence $\epsilon^{\star}$ are
completely determined by $\Gamma_{eff}$ and $\epsilon^{\prime}$. Therefore,
the conductance in the MV spinful model is function of:

$\mathcal{G}^{C}(\omega,T\to
0)=\mathit{f}\Big{\\{}\epsilon^{\prime};\Gamma_{eff}\Big{\\}}\leavevmode\nobreak\
.$ (5.100)

In conclusion, we can use the MW current formula for $\hat{H}_{eff}$ spinful
model to calculate the conductance at $T=0$ and vanishing $\omega$, namely

$\boxed{\begin{aligned}
&\mathcal{G}^{dc}(0,0)=\frac{2e^{2}}{h}\left(4\frac{|\widetilde{V}_{s}|^{2}|\widetilde{V}_{d}|^{2}}{(|\widetilde{V}_{s}|^{2}+|\widetilde{V}_{d}|^{2})^{2}}t_{ff}(\omega=T=0)\right)\leavevmode\nobreak\
,\\\
&t_{ff}(\omega=T=0)=\frac{1}{1+(\epsilon^{\star})^{2}/(\Gamma^{s}_{eff}+\Gamma^{d}_{eff})^{2}}\end{aligned}}\leavevmode\nobreak\
.$ (5.101)

In particular, we can use the Friedel sum rule result, see Eq.2.39, to write
the spectra $t_{ff}(\omega=T=0)$ in term of the excess charge of the single
Fermionic degree of freedom, namely

$t_{ff}(\omega=T=0)=\sin^{2}\left(\frac{\pi}{2}\hat{n}_{f}(T=0)\right)\leavevmode\nobreak\
.$ (5.102)

The $\hat{n}_{f}$ quantity undergoes to renormalization under RG-flow and it
presents a highly non-trivial behaviour, hence the above equation is not ideal
for practical implementations without a full solution of $\hat{H}_{eff}$. For
this reason, it is more convenient to compute using NRG the spectra
$t_{ff}(\omega=T=0)$ defined by the effective parameters as derived from the
BWPT. We comment that a numerical demonstration of the accuracy of Eq.5.101 in
a complex impurity model - the benzene molecule - is given in Chapter 7. What
we find is the analytical prediction is perfectly verified by NRG data.
In case we need to compute the under linear response at $T=0$ but finite
$\omega$, the $\mathrm{T}$-matrix spectral function expression to use is as
follows,

$t_{ff}(\omega,T\to
0)=\sum_{\alpha}\frac{(\Gamma^{\alpha}_{eff})^{2}}{(\omega-\epsilon^{\star})^{2}+(\Gamma^{\alpha}_{eff})^{2}}\leavevmode\nobreak\
.$ (5.103)

Figure 5.6: $\mathrm{T}$-matrix for the effective model Eq.5.95 at $T=0$,
describing the dynamics in the vicinity of a singlet-doublet charge degeneracy
point, calculated from NRG. $(a)$ $t(0,0)$ as a function of $\tilde{\Gamma}$
for $\epsilon^{\prime}=0$ at charge degeneracy (black line) and
$|\epsilon^{\prime}|/\tilde{\Gamma}=0.5,1,1.5,2$ (red, blue, green and magenta
lines) with $\epsilon^{\prime}>0$ shown as solid lines and
$\epsilon^{\prime}<0$ as dashed lines. $(b)$ $t(\omega,0)$ vs
$\omega/\tilde{\Gamma}$ at $\epsilon^{\prime}=0$ charge degeneracy point for
$\tilde{\Gamma}/D=10^{-1},10^{-3},10^{-5},10^{-7}$ for the green, blue, red
and black lines. $t(\omega,0)$ in Eq. 5.103 is shown as the dashed line for
comparison to black line. $(c)$ Behaviour of $\epsilon^{\star}/\tilde{\Gamma}$
as a function on $\tilde{\Gamma}$ for the same parameters as in panel $(a)$,
showing small $\tilde{\Gamma}/D\ll 1$ asymptotes following Eq. 5.104.

In Fig. 5.6, we now consider the spectrum of $\mathrm{T}$-matrix
$t_{ff}(\omega,T\to 0)$ for the effective model in Eq.5.95 with $B_{eff}=0$
and $U_{c}\to\infty$ obtained by full NRG calculation. For given effective
parameters $\epsilon^{\prime}$ and $\Gamma_{eff}$, the low-$T$ transport can
then be obtained via Eqs.5.101,5.103.
We start with the charge degenerate system for $\epsilon^{\prime}=0$ in the
valence transition between singlet and doublet spin multiplet, as we show in
panel $(b)$ of Fig. 5.6 for different effective $\Gamma_{eff}$ \- in the plot
published in the paper [1] this is indicated as $\tilde{\Gamma}$ so we adopt
this notation for the sake of clarity from now on. According to the ratio
$\tilde{\Gamma}/D$, we distinguish two opposite regimes. At
$\tilde{\Gamma}/D\ll 1$, $t_{ff}(\omega,T\to 0)$ spectrum is accurately
approximated by that of an equivalent NI-RLM with $U_{c}=0$ under Fermi liquid
description. The results of $t_{ff}(\omega,T\to 0)$ in Eq.5.103 are shown for
comparison as dashed lined in Fig. 5.6$(b)$. On the contrary, at
$\tilde{\Gamma}/D\gg 1$, discrepancies from the Lorentzian form of
$t_{ff}(\omega,T\to 0)$ in Eq.5.103 becomes more pronounced at finite
frequency, due mainly to $\mathit{Im}\Sigma_{ff}(\omega,T\to 0)\neq 0$ leading
to an enhanced peak height and asymmetric shape.
As we mentioned previously, once the effective parameters are derived, the
only remaining quantity we need to calculate in order to determine the
electrical conductance is the interaction-renormalised energy
$\epsilon^{\star}$. Hence, it is useful to find its asymptotic behaviour as
function of the effective $\tilde{\Gamma}$ as we show in Fig. 5.6$(c)$. In the
physically-relevant regime $\tilde{\Gamma}/D\ll 1$, the behaviour is found to
be simply,

$\epsilon^{\star}/\tilde{\Gamma}\simeq
a\epsilon^{\prime}/\tilde{\Gamma}-b\log(\tilde{\Gamma}/D)+c\;,$ (5.104)

where numerically we find $a\simeq 0.93$, $b\simeq 0.29$ and $c\simeq 0.35$.
By contrast, at large $\tilde{\Gamma}/D\gg 1$, we have two scenarios according
to the sign of $\epsilon^{\prime}$. If $\epsilon^{\prime}>0$, the self-energy
correction can be neglected such that $\epsilon^{\star}\to\epsilon^{\prime}$
and the Kondo effect is suppressed. If $\epsilon^{\prime}<0$, the Kondo effect
renormalises the effective energy giving $\epsilon^{\star}\to 0$. This is
indeed confirmed directly by the saturation values and the low/high
$\tilde{\Gamma}$ regime behaviour of $\epsilon^{\star}$, which yields
$t_{ff}(\omega=T=0)$ exactly from Eq. 5.101, as presented in Fig. 5.6$(c)$.
The quantity $t_{ff}(\omega=T=0)$ is required to compute the zero energy and
temperature scales conductance and reference NRG results for it as a function
of $\tilde{\Gamma}$ are presented in Fig. 5.6$(a)$ for different
$\epsilon/\tilde{\Gamma}$. At $\tilde{\Gamma}/D\ll 1$ for any
$\epsilon^{\prime}$, the spectra $t_{ff}(\omega=T=0)\simeq 0$. At
$\tilde{\Gamma}/D\gg 1$, for $\epsilon^{\prime}\leq 0$,
$t_{ff}(\omega=T=0)\simeq 1$ whereas at $\epsilon^{\prime}>0$, for increasing
$\epsilon^{\prime}$ value we have decreasing $t_{ff}(\omega=T=0)$. At
intermediate $\tilde{\Gamma}$ regime, the spectra of $t_{ff}(\omega=T=0)$
representing states away from the valence singlet-doublet multiplet
transition, spread around the charge degeneracy curve - in black line - where
maximum conductance is measured. In panel $(b)$ we verify an important
observation: there is a one-to-one correspondence between the effective
hybridizations and the spectra $t_{ff}(\omega=T=0)$. This is precisely the
crucial advantage of computing the conductance by means of effective
quantities rather than solving the microscopic model per se.
Equipped with these results, the low-temperature conductance of a system in
the vicinity of a singlet-doublet charge-degeneracy point can be
straightforwardly predicted, once the effective model parameters
$\epsilon^{\prime}$ and $\widetilde{V}_{\alpha}$ have been determined.

####  5.2.3 Numerical study for serial configuration of two-impurity with
single-orbital model: example of singlet-double mixed-valence case

So far, in the modelling of systems under MV regime, we always implicitly
consider a single-site on each lead coupled to the $m^{th}$ degree of freedom
on the nanostructure via $V_{\alpha m\sigma}$ hybridization. Prior to any form
of diagonalization, this bare hybridization matrix is composed by finite
entries in each matrix element. We remark that the $m$-label stands for either
an impurity site or an orbital in more complex structures.
To complete our examination of emergent proportionate coupling MV systems, we
consider now a nanostructure with two-orbital and we couple it with leads in
non-PC set-up. Under this condition, a unique correspondence between effective
and bare couplings can be established. In this configuration, we intend to
have one unique $V_{s\sigma}$ coupling between the source lead and
nanostructure characterised by a single-orbital and similarly for
$V_{d\sigma}$ on the drain lead side.
In this section, starting from a single-orbital system whose microscopic model
does not show PC property, we aim to numerically compute its effective model
showing emergent new properties and using those to compute the electrical
conductance. We pin down the discussion considering a serial configuration of
two-impurity which is by construction lacking of PC property, see more
discussion on PC vs no-PC in Sec.3.3.3. We chose this impurity system to be in
MV regime. In the following, we give the derivation with explicit numerical
values. Extension to multiple-impurity set-up follows as direct
generalization.
We start with the two-impurity model Hamiltonian, namely:

$\displaystyle\hat{H}=\hat{H}_{leads}$
$\displaystyle+\sum_{\sigma,\sigma^{\prime}\in\\{\uparrow,\downarrow\\}}\sum_{\beta=\\{1,2\\}}\big{(}\epsilon_{d_{\beta}}d^{\dagger}_{\beta\sigma}d_{\beta\sigma}+U_{\beta}\hat{n}_{d\beta\sigma}\hat{n}_{d\beta\sigma\prime}+t(d^{\dagger}_{1\sigma}d_{2\sigma}+d^{\dagger}_{2\sigma}d_{1\sigma})+V_{gate}\hat{n}_{d_{\beta}\sigma}\big{)}+$
(5.105)
$\displaystyle+\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\Big{(}V_{s\sigma}c^{\dagger}_{s\sigma}d_{1\sigma}+V_{d\sigma}c^{\dagger}_{d\sigma}d_{2\sigma}+H.c.\Big{)}\leavevmode\nobreak\
,$

where we explicitly take $\epsilon_{d1}\neq\epsilon_{d2}$ to avoid further
charge degeneracy in the model and particle-hole symmetry regime meaning that
$\epsilon_{d_{\beta}}=-U_{\beta}/2$. The intrinsic two-impurity asymmetry
factor is given only by $\nu=V_{d\sigma}/V_{s\sigma}$.
We follow now the protocol detailed in Sec.5.2 to derive the effective model
with an emergent proportionate coupling property from the above two-impurity
single-orbital model Hamiltonian in MV regime. At $V_{gate}=0$, the lowest
state is a collective two-impurity spin singlet state. We tune $V_{gate}$ to
the MV transition point between this singlet state and a state with a single
electron on the two-impurity structure giving a doublet state.
The two-impurity structure has basis states in $N=2$ electrons sector,

$\ket{\uparrow,\uparrow}\quad,\quad\ket{\uparrow,\downarrow}\quad,\quad\ket{\downarrow,\uparrow}\quad,\quad\ket{\downarrow,\downarrow}\leavevmode\nobreak\
,$ (5.106)

and in $N=1$ electrons sector,

$\ket{\uparrow,-}\quad,\quad\ket{\downarrow,-}\quad,\quad\ket{-,\uparrow}\quad,\quad\ket{-,\downarrow}\leavevmode\nobreak\
.$ (5.107)

This many-body basis describes states in the isolated nanostructure, which is
then used to construct the matrix Hamiltonian of $\hat{H}_{nano}$. Upon
diagonalization, we obtain the normalized eigenvectors numerically.
Furthermore, we can identify the lowest spin multiplet and at which $V_{gate}$
value it crossed another spin multiplet state - this is the MV point.
In our numerical analysis we use as bare parameters
$\epsilon_{d1}=-10\leavevmode\nobreak\ eV,\leavevmode\nobreak\
U_{1}=20\leavevmode\nobreak\ eV$, $\epsilon_{d2}=-6\leavevmode\nobreak\
eV,\leavevmode\nobreak\ U_{2}=12\leavevmode\nobreak\ eV$, hopping
$t=1.5\leavevmode\nobreak\ eV$. We find the two lowest energy-degenerate two-
impurity ground-state are the charge $N=1,\leavevmode\nobreak\ N+1=2$ sectors
and their crossover occurs at $V^{\star}_{gate}=6.06\leavevmode\nobreak\ eV$.
To sum up, the energy degenerate eigenbasis states defining the two-impurity
ground-state read as:

$\displaystyle\ket{\Psi^{N=1,S^{z}=\sigma}_{0}}=-0.95\ket{\sigma,-}+0.32\ket{-,\sigma}\leavevmode\nobreak\
,$ (5.108)
$\displaystyle\ket{\Psi^{N=2,S^{z}=0}_{0}}=-0.70\big{(}\ket{\uparrow,\downarrow}-\ket{\downarrow,\uparrow}\big{)}+0.13\big{(}\ket{\uparrow\downarrow,-}-\ket{-,\uparrow\downarrow}\big{)}\leavevmode\nobreak\
,$

for $\sigma=\uparrow,\downarrow$ .
By means of these degenerate ground-state, in analogy to Eqs.5.10,5.61, we
define the projector operator of the two-impurity model, namely:

$\widehat{P}_{0}=\sum_{\sigma}\left(\ket{\Psi^{N=1,S^{z}=\sigma}_{0}}\bra{\Psi^{N=1,S^{z}=\sigma}_{0}}+\ket{\Psi^{N=2,S^{z}=0}_{0}}\bra{\Psi^{N=2,S^{z}=0}_{0}}\right)\leavevmode\nobreak\
,$ (5.109)

and from it we can define $f^{\dagger}_{\sigma},f_{\sigma}$ Fermionic
operators analogously to Eq.5.85.
Once we complete the projection operation up to
$\mathcal{O}(\hat{H}_{hyb})^{2}$, the resulting effective model is the same as
Eq.5.90, just with different spin label for $f^{\dagger}_{\sigma},f_{\sigma}$
and leads operators,

$\hat{H}_{eff}=\hat{H}_{leads}+\sum_{\sigma,\sigma^{\prime}\in\\{\uparrow,\downarrow\\}}\sum_{\alpha=s,d}\left(c^{\dagger}_{\alpha
0\sigma}\widetilde{V}_{\alpha}f_{\sigma^{\prime}}+f^{\dagger}_{\sigma^{\prime}}\widetilde{V}^{\star}_{\alpha}c_{\alpha
0\sigma}\right)\leavevmode\nobreak\ ,$ (5.110)

with effective hybridizations

$\displaystyle\widetilde{V}_{s}=-0.3475V_{s\sigma}\quad;\quad\widetilde{V}_{d}=0.7066V_{d\sigma}\leavevmode\nobreak\
,$ (5.111)

where these numerical pre-factors result form matrix element tunnelling
amplitude from the projection calculation - see for comparison Eq.5.88. Due to
the single-orbital coupling, there is a unique effective 2-impurity asymmetry
factor, namely: $\tilde{\nu}=\widetilde{V}_{d}/\widetilde{V}_{s}$. After
introducing fictitious on-site Coulomb potential as we previously, see
Eq.5.92, to render the effective model only $3$-fold spin degenerate, the
final effective model is equivalent to Eq.5.95 \- here again with opposite
spin label for $f_{\sigma}$ and lead operators. At this point, we can identify
the direct proportionality among couplings, namely:

$\widetilde{V}_{s}\doteq\widetilde{V}\quad;\quad\widetilde{V}_{d}=\tilde{\nu}\widetilde{V}_{s}\doteq\tilde{\nu}\widetilde{V}_{\sigma}\leavevmode\nobreak\
,$ (5.112)

where the single-lead effective model with
$\widetilde{V}^{\prime}=\widetilde{V}\sqrt{1+\tilde{\nu}^{2}}$ and
$c_{0\sigma}=\widetilde{V}/\widetilde{V}^{\prime}\left(c_{s0\sigma}+\tilde{\nu}c_{d0\sigma}\right)$
such that the effective asymmetric factor $\tilde{\nu}$ can be written as
ratio of the numerical values resulting from the exact diagonalization the
isolated nanostructure multiplied for the intrinsic asymmetric factor from the
bare model. Hence, due to the emergent proportionality, the ratio of effective
couplings corresponds to the ratio of the bare ones.
In conclusion, the effective model of spinful serial two-impurity model in MV
transition regime reads

$\hat{H}^{\prime}_{eff}=\hat{H}_{leads}+\sum_{\sigma\in\\{\uparrow,\downarrow\\}}\left(c^{\dagger}_{0\sigma}\widetilde{V}^{\prime}f_{\sigma}+f^{\dagger}_{\sigma}\widetilde{V}^{\star\prime}c_{0\sigma}\right)+U_{c}\hat{n}_{f_{\uparrow}}\hat{n}_{f_{\downarrow}}\leavevmode\nobreak\
,$ (5.113)

that is analogous to Eq.5.95 with
$\widetilde{V}^{\prime}=\sqrt{0.121V^{2}_{s}+0.5V^{2}_{d}}$ and
$U_{c}\to+\infty$.
We have completed the derivation of the effective model giving a numerical
sense to the various effective parameters. Now, we make use of the incipient
PC property of $\hat{H}^{\prime}_{eff}$ to derive the electrical conductance
using the MW formula under PC as we show in Eqs.5.101,5.103. With this
effective model and the results of Fig.5.6, the conductance can now be readily
calculated.
In conclusion, in this section we show from the exact diagonalization of the
isolated $\hat{H}_{nano}$ with double-orbital, we derived the essential
quantity not only to define the effective model with emergent PC property but
also to evaluate the electrical conductance.

####  5.2.4 Spin model: doublet-triplet transition

The charge degeneracy point between spin-doublet and spin-triplet multiplet
states also yields nontrivial strongly-correlated electron behaviour - yet it
can again be captured in a simple effective model in PC. The doublet-triplet
transition scenario is more common in the context of single molecule
junctions, see Chapter 7, where Hund’s Rule or other complex multi-orbital
interactions can yield a high spin $S=1$ ground-state in the even electron
sector of the isolated nanostructure. This is opposed to an even-electron
$S=0$ singlet, as we considered in the singlet to doublet valence transition.
Here we shall take $N$ to be odd, with the $N$-electron nanostructure hosting
a net $S=\tfrac{1}{2}$ doublet state; while in the (even) $N+1$ electron
sector, the nanostructure ground state is an $S=1$ triplet. Focusing again on
the low-$T$ transport, we construct an effective model in which only the
lowest energy spin multiplet of each sector is retained. In this case, by
means of the BWPT, the derived effective nanostructure Hamiltonian comprises
five states: two for the $N$ electron doublet, and three for the $N+1$
electron triplet. We refer to those spin states described by the bare model as
admissible. For simplicity we consider below the $SU(2)$ spin-symmetric case
arising when no external magnetic field acts i.e. $B=0$ in the bare model in
Eq.5.1, such that states $\ket{N;S^{z}=\pm 1/2}$ for the ground-state spin
doublet are degenerate and eigenstate $E_{0}^{N;S^{z}=\pm 1/2}\equiv
E_{0}^{N}$ is independent of $S^{z}$.
We construct an effective model using Eq. 5.65, casting the result in terms of
one effective Fermionic site $f_{\sigma}$, and one effective
spin-$\tfrac{1}{2}$ degree of freedom $\hat{\mathbf{S}}_{g}$, together with
carefully chosen constraints - which are implemented through additional hard-
core terms in the Hamiltonian. This allows us to capture the various spin-
multiplet transitions from doublet to triplet by adding a single electron. By
applying the projector operator in Eq.5.61 up to first order in
$\hat{H}_{hyb}$, the resulting effective model at doublet to triplet valence
transition reads

$\boxed{\begin{aligned} \hat{H}_{eff}&\stackrel{{\scriptstyle
MV}}{{=}}\hat{H}_{leads}-J_{c}\hat{\mathbf{S}}_{f}\cdot\hat{\mathbf{S}}_{g}+\sum_{\begin{subarray}{c}\alpha=s,d\\\
\sigma=\uparrow,\downarrow\end{subarray}}\left(\widetilde{V}_{\alpha\sigma}^{\phantom{\dagger}}f^{\dagger}_{\sigma}c^{\phantom{\dagger}}_{\alpha\sigma}+\rm{H.c.}\right)+\delta_{c}\hat{n}_{f}\leavevmode\nobreak\
,\\\ &\stackrel{{\scriptstyle no-
deg}}{{=}}\hat{H}_{leads}-J_{c}\hat{\mathbf{S}}_{f}\cdot\hat{\mathbf{S}}_{g}+\sum_{\begin{subarray}{c}\alpha=s,d\\\
\sigma=\uparrow,\downarrow\end{subarray}}\left(\widetilde{V}_{\alpha\sigma}^{\phantom{\dagger}}f^{\dagger}_{\sigma}c^{\phantom{\dagger}}_{\alpha\sigma}+\rm{H.c.}\right)+\delta_{c}\hat{n}_{f}+\epsilon^{\prime}(\hat{n}_{f}-1)^{2}\leavevmode\nobreak\
,\end{aligned}}$ (5.114)

and we refer to it as an $fg$-system whose elements are defined as follows.
The $\hat{\mathbf{S}}_{g}$ is a spin-$\tfrac{1}{2}$ operator for the effective
local moment,
$\hat{\mathbf{S}}_{f}=\tfrac{1}{2}\sum_{\nu,\nu^{\prime}}f_{\nu}^{\dagger}\bm{\sigma}_{\nu\nu^{\prime}}f_{\nu^{\prime}}^{\phantom{\dagger}}$
is the spin density,
$\hat{n}_{f}=\sum_{\sigma}f_{\sigma}^{\dagger}f_{\sigma}^{\phantom{\dagger}}$
is as before the number operator for the effective $f$ level, and

$\displaystyle f^{\dagger}_{\uparrow}$
$\displaystyle=|N\text{+1;}S^{z}\text{=1}\rangle\langle
N\text{;}S^{z}\text{=}\tfrac{1}{2}|+\tfrac{1}{\sqrt{2}}|N\text{+1;}S^{z}\text{=0}\rangle\langle
N\text{;}S^{z}\text{=-}\tfrac{1}{2}|\leavevmode\nobreak\ ,$ (5.115)
$\displaystyle f^{\dagger}_{\downarrow}$
$\displaystyle=|N\text{+1;}S^{z}\text{=-1}\rangle\langle
N\text{;}S^{z}\text{=-}\tfrac{1}{2}|+\tfrac{1}{\sqrt{2}}|N\text{+1;}S^{z}\text{=0}\rangle\langle
N\text{;}S^{z}\text{=}\tfrac{1}{2}|\leavevmode\nobreak\ ,$

with $f_{\sigma}=(f_{\sigma}^{\dagger})^{\dagger}$ as usual. Here, the
$\tfrac{1}{\sqrt{2}}$ factor is Clebsch-Gordon coefficient implied by the
Wigner-Eckart theorem. The $\widetilde{V}_{\alpha\sigma}$ is effective
tunnelling amplitude between allowed valence transitions and it is equal to
the matrix elements derived from diagonal representation of the isolated
nanostructure, namely

$\widetilde{V}_{\alpha\sigma}=\begin{cases}V_{\alpha\sigma}\langle
N+1;\hat{S}^{z}=1|d_{\alpha\uparrow}^{\dagger}|N;\hat{S}^{z}=\tfrac{1}{2}\rangle\leavevmode\nobreak\
,\\\ V_{\alpha\sigma}\langle
N+1;\hat{S}^{z}=-1|d_{\alpha\downarrow}^{\dagger}|N;\hat{S}^{z}=-\tfrac{1}{2}\rangle,\end{cases}$
(5.116)

here we recall $d_{\alpha\sigma}^{\dagger},d_{\alpha\sigma}$ is nanostructure
operators in the bare model Eq.5.1 where we use the standard
$d_{s\sigma}=d_{1\sigma}$ and $d_{d\sigma}=d_{M\sigma}$ and by $SU(2)$ spin
symmetry and Wigner-Eckart theorem it is straightforward to obtain the other
transitions, that is

$\widetilde{V}_{\alpha\sigma}=\begin{cases}\sqrt{2}V_{\alpha\sigma}\langle
N+1;\hat{S}^{z}=0|d_{\alpha\uparrow}^{\dagger}|N;\hat{S}^{z}=-\tfrac{1}{2}\rangle\leavevmode\nobreak\
,\\\ \sqrt{2}V_{\alpha\sigma}\langle
N+1;\hat{S}^{z}=0|d_{\alpha\downarrow}^{\dagger}|N;\hat{S}^{z}=\tfrac{1}{2}\rangle\leavevmode\nobreak\
.\end{cases}$ (5.117)

In the second line of Eq.5.114 we present the expression for the effective
model away from the doublet triplet valence transition regime. Those
deviations from the precise charge-degeneracy point are captured by the
parameter $\epsilon^{\prime}=E^{N+1}_{0}-E^{N}_{0}$.
We note the $fg$-system in Eq. 5.114 includes eight effective states: we wish
to retain only the five admissible states that represent the $N$ electron
doublet and the $N+1$ electron triplet of the bare model, namely:

$\displaystyle|N;S^{z}=\tfrac{1}{2}\rangle$
$\displaystyle\equiv|-\rangle_{f}\otimes|\uparrow\rangle_{g}\;,$ (5.118)
$\displaystyle|N;S^{z}=-\tfrac{1}{2}\rangle$
$\displaystyle\equiv|-\rangle_{f}\otimes|\downarrow\rangle_{g}\;,$
$\displaystyle|N+1;S^{z}=1\rangle$
$\displaystyle\equiv|\uparrow\rangle_{f}\otimes|\uparrow\rangle_{g}\;,$
$\displaystyle|N+1;S^{z}=0\rangle$
$\displaystyle\equiv\tfrac{1}{\sqrt{2}}(|\uparrow\rangle_{f}\otimes|\downarrow\rangle_{g}+|\downarrow\rangle_{f}\otimes|\uparrow\rangle_{g})\;,$
$\displaystyle|N+1;S^{z}=-1\rangle$
$\displaystyle\equiv|\downarrow\rangle_{f}\otimes|\downarrow\rangle_{g}\;.$

We identify as spurious states
$|\uparrow\downarrow\rangle_{f}\otimes|\uparrow\rangle_{g}$,
$|\uparrow\downarrow\rangle_{f}\otimes|\downarrow\rangle_{g}$ and
$\tfrac{1}{\sqrt{2}}(|\uparrow\rangle_{f}\otimes|\downarrow\rangle_{g}-|\downarrow\rangle_{f}\otimes|\uparrow\rangle_{g})$
and we can eliminate them by the energy constraints implemented in Eq. 5.114
by setting $\delta_{c}=\tfrac{1}{4}J_{c}$ and sending then $J_{c}\to\infty$.
By means of the energy constraint, the effective model in Eq. 5.114 is
therefore a mixed-valence single impurity Anderson model, side-coupled to an
additional spin-$\tfrac{1}{2}$ local moment by a ferromagnetic interaction.
Because the AM is by construction fulfilling the PC property, $\hat{H}_{eff}$
presents such an emergent property that facilitates the evaluation of the
electrical conductance - as we see in the next part. This can be seen from the
fact that in Eq.5.114 is an effective one-channel model, with
$c_{0\sigma}=[\widetilde{V}_{s\sigma}+\widetilde{V}_{d\sigma}]/\widetilde{V}_{\sigma}$
and
$\widetilde{V}_{\sigma}^{2}=\widetilde{V}_{s\sigma}^{2}+\widetilde{V}_{d\sigma}^{2}$,
describing the first site of a combined lead.

#### NRG results: electrical conductance for spinful doublet-triplet mixed-
valence model

Turning now to the low temperature and low energy scales quantum transport,
the PC form of the $\hat{H}_{eff}$ in Eq. 5.114 \- once we implement the
energy constraints to discard the additional unwanted states - implies that we
may use the MW formula in PC version, see Eq.3.66. Therefore, the electrical
conductance can be successfully evaluated from the current equations in
Eqs.5.103,5.101.
As before, we consider the conductance at $T\to 0$. Moreover, we adopt the
notation of effective hybridization labelled
$\tilde{\Gamma}=\pi|\widetilde{V}_{\sigma}|^{2}\rho_{0}$ as in Fig. 5.7
published in the paper [1], so we use it from now on for the sake of clarity.
The $T\to 0$ electrical conductance of the physical system depends on the
dynamics of the effective model Eq. 5.114 as $T\to 0$ and $\omega\to 0$, as
encoded in $\mathrm{T}$-matrix spectral function $t_{ff}(\omega=0,T=0)$. In
Fig. 5.7 we plot $t_{ff}(\omega=0,T=0)$ as a function of the effective
$\tilde{\Gamma}$ for different $\epsilon^{\prime}/\tilde{\Gamma}$, obtained
from NRG. This is the analogous plot for the doublet-triplet transition as
Fig. 5.6$(a)$ for the singlet-doublet transition. Although we find
qualitatively similar behaviour to that of the singlet-doublet case according
to $\epsilon^{\prime}$ parity, the details are different due to the different
valence transition. Indeed, here the full spectrum $t_{ff}(\omega,T=0)$ shows
much more pronounced asymmetries than the simple Lorentzian function in the
spectra of Eq. 5.103, even at very small $\tilde{\Gamma}$. However, in the
physically-relevant regime $\tilde{\Gamma}\ll 1$ we find
$t_{ff}(\omega=0,T=0)$ for the effective model is still accurately
approximated by the spectra defined in Eq. 5.101 with asymptotic behaviour
given by Eq. 5.104 but now with modified $a\simeq 1.15$, $b\simeq 0.18$ and
$c\simeq 0.7$ parameters numerically computed.

With these NRG results, the low-temperature conductances for real systems near
a doublet-triplet charge degeneracy point can be obtained simply from the
effective model parameters $\epsilon^{\prime}$ and effective hybridization
defining the tunnelling amplitudes $\widetilde{V}_{\alpha\sigma}$. Comparison
of Fig. 5.6$(a)$ and Fig. 5.7 shows that for a given effective
$\tilde{\Gamma}$ and $\epsilon^{\prime}$, the electrical conductance is larger
in the vicinity of the doublet-triplet charge degeneracy point that than near
the singlet-doublet charge degeneracy point, which we attribute to enhanced
spin fluctuations in the former.
Finally, we remark that the above analysis of the effective model Eq. 5.114,
obtained perturbatively using BWPT up to $\mathcal{O}(\hat{H}_{hyb})^{2}$
leading order in the nanostructure-lead hybridization, accounts for the
dominant contributions to conductance. However, higher-order corrections will
in general result in an effective coupling to the neglected odd conduction
electron channel, taking the effective model out of PC condition. Although
irrelevant for the singlet-doublet case considered in the previous section -
whose many-body ground-state is always a nondegenerate singlet due to the
Kondo effect - in the doublet-triplet case the involvement of the odd channel
is expected to induce a second-stage Kondo screening, quenching the $\ln(2)$
residual entropy found for Eq. 5.114. Since such terms arise only at higher
order in perturbation theory, the associated Kondo scale for the second-stage
screening is expected to be very small, and hence may be neglected at
experimentally-relevant temperatures. We therefore argue that the effective
model Eq. 5.114, and results of this section, remain valid close to the charge
degeneracy point where there is a clear separation of scales.

Figure 5.7: $\mathrm{T}$-matrix $t_{ff}(\omega=0,T=0)$ at $T=0$ as a function
of the hybridization $\tilde{\Gamma}$ near a doublet-triplet charge degeneracy
point, described by the effective model Eq. 5.114, obtained by NRG. Plotted
for $\epsilon^{\prime}=0$ (black line) charge degenerate and
$|\epsilon^{\prime}|/\tilde{\Gamma}=0.5,1,1.5,2$ (red, blue, green and magenta
lines) with $\epsilon^{\prime}>0$ shown as solid lines and
$\epsilon^{\prime}<0$ as dashed lines.

###  5.3 Generalized spinful model

As common feature of this chapter, the BWPT requires the projection of the
whole system states onto the lowest spin-multiplet ground-state manifold of
the nanostructure. Considering only single-electron tunnelling events and
appropriate energy and temperature scales, real quantum impurity systems may
present higher spin multiplet ground-states. In those situations, the BWPT is
again a valid method to successfully derive effective models exhibiting an
emergent PC condition - confirming the general applicability of degenerate
perturbation theory.
In this section, we briefly mention the generalization of the main equations
used in BWPT for general spin models. The resulting $\hat{H}_{eff}$ takes
analogue form to the one derived for the lowest spin multiplet case - hence,
we find again emergent PC property that was lacking in the microscopic bare
model. At the end, we give some short quantitative analysis for the general
spin model under CB and MV regime.
We consider $N$ electron spin-$S$ multiplet ground-state and $N+1$ electron
spin-$S+\tfrac{1}{2}$ multiplet ground-state. The generalized charge spin
multiplet ground-state $\ket{\Psi_{0}^{Q}}$, based on the definition of charge
state in Eq.5.8, is defined as:

$\ket{Q=N;S^{z}}=\begin{cases}\ket{N;\left\\{-S,-S+1,\dots,S+1,S\right\\}}\leavevmode\nobreak\
,\\\
\ket{N+1;\left\\{-S-\dfrac{1}{2},-S+\dfrac{1}{2},\dots,S-\dfrac{1}{2},S+\dfrac{1}{2}\right\\}}\leavevmode\nobreak\
,\end{cases}$ (5.119)

with corresponding generalized projector operator defined as:

$\displaystyle\widehat{P}_{0}$
$\displaystyle=\left(\ket{N;-S}\bra{N;-S}+\ket{N;-S+1}\bra{N;-S+1}+\dots+\ket{N;S}\bra{N;S}\right)\otimes\widehat{\mathbb{1}}_{leads}\leavevmode\nobreak\
,$ (5.120) $\displaystyle\widehat{P}_{0}$
$\displaystyle=\Big{(}\ket{N+1;-S-\tfrac{1}{2}}\bra{N+1;-S-\tfrac{1}{2}}+\ket{N+1;-S+\tfrac{1}{2}}\bra{N+1;-S+\tfrac{1}{2}}+\dots+$
$\displaystyle\hskip
28.45274pt+\ket{N+1;+S+\tfrac{1}{2}}\bra{N+1;+S+\tfrac{1}{2}}\Big{)}\otimes\widehat{\mathbb{1}}_{leads}\leavevmode\nobreak\
,$

as straightforward generalization of Eq.5.10. Using these generalized formulas
we can perform the projection operation to any spin multiplet ground-state
manifold.
In the case of the MV regime in Sec.5.2, we presented the two most common
scenarios for singlet-doublet and doublet-triplet valence transition. However,
within the energy window $T\ll\Delta E_{min}$, other valence transitions are
compatible with unitary charge and spin variation.
In the context of singlet-doublet crossover, for example $N$ charge sector
state comprises next to $\ket{N;S^{z}=0}$ also $\ket{N;S^{z}=-1,0,+1}$ that is
consistent with single-electron tunnelling events with
$\ket{N+1;S^{z}=\pm\tfrac{1}{2}}$ state. Hence, this is the complete spin