###  VII.2 Singlet-doublet transition

A more generic and interesting situation arises in spinful systems where
states of $H_{\rm nano}$ 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 non-degenerate spin-singlet, while with (odd)
$N+1$ electrons we have a spin-doublet. Both singlet and doublet must be taken
into account in the effective model near the charge degeneracy point between
$N$ and $N+1$ electrons.

The three accessible states in the low-energy nanostructure manifold can be
represented by a single spinful fermionic degree of freedom with operators
$f_{\sigma}^{\dagger}=|N+1;\sigma;1\rangle\langle N;0;1|$ and
$f_{\sigma}^{\phantom{\dagger}}=(f_{\sigma}^{\dagger})^{\dagger}$, together
with a constraint to exclude spurious double occupancy of this effective site.
Here we have additionally labelled states of $H_{\rm nano}$ by their total
$S^{z}$ quantum number $|n;S^{z};l\rangle$, where $\sigma=\uparrow,\downarrow$
denotes $S^{z}=+\tfrac{1}{2},-\tfrac{1}{2}$, and $l$ now labels the spin
multiplet. The corresponding energies are $E^{n}_{S^{z};l}$. Since we consider
only the $l=1$ ground state multiplets, we suppress the $l$ labels below. The
effective model Eq. 76 then reads,

$\begin{split}H_{\rm eff}=H_{\rm leads}+\epsilon\hat{n}_{f}&+B_{\rm
eff}\hat{S}^{z}_{f}+U_{c}\hat{n}_{f\uparrow}\hat{n}_{f\downarrow}\\\
&+\sum_{\begin{subarray}{c}\alpha=s,d\\\
\sigma=\uparrow,\downarrow\end{subarray}}\left(t_{\alpha\sigma}^{\phantom{\dagger}}f^{\dagger}_{\sigma}c^{\phantom{\dagger}}_{\alpha\sigma}+\rm{H.c.}\right)\;,\end{split}$
(80)

where $\hat{n}_{f\sigma}=f_{\sigma}^{\dagger}f_{\sigma}^{\phantom{\dagger}}$,
$\hat{n}_{f}=\hat{n}_{f\uparrow}+\hat{n}_{f\downarrow}$ and
$\hat{S}^{z}_{f}=\tfrac{1}{2}(\hat{n}_{f\uparrow}-\hat{n}_{f\downarrow})$.
Here $U_{c}\to\infty$ implements the hard-core constraint, while the effective
parameters are given by
$\epsilon=\tfrac{1}{2}[E^{N+1}_{\uparrow}+E^{N+1}_{\downarrow}]-E^{N}_{0}$,
$B_{\rm eff}=[E^{N+1}_{\uparrow}-E^{N+1}_{\downarrow}]$ and
$t_{\alpha\sigma}=V_{\alpha}\langle
N+1;\sigma|\bar{d}_{\alpha\sigma}^{\dagger}|N;0\rangle$. Eq. 80 is as such an
infinite-$U$ single impurity Anderson model in the mixed-valent regime,[8]
which possesses an emergent PC (independent of whether or not the bare model
is in PC). This implies that a single effective conduction electron channel
couples to the impurity $f$ level,
$c_{e\sigma}=\tfrac{1}{t_{\sigma}^{2}}[t_{s\sigma}c_{s\sigma}+t_{d\sigma}c_{d\sigma}]$,
where $t_{\sigma}^{2}=|t_{s\sigma}|^{2}+|t_{d\sigma}|^{2}$, and with effective
hybridization $\tilde{\Gamma}_{\sigma}=\pi\rho_{0}t_{\sigma}^{2}$.

Another consequence of the emergent PC in the effective model is that quantum
transport at low temperatures $T\ll\Delta E_{\rm min}$ may be obtained using
the MW formula Eq. 28 with transmission function,

$\displaystyle\mathcal{T}_{\rm
MW}(\omega,T)=\frac{4t_{s}^{2}t_{d}^{2}}{(t_{s}^{2}+t_{d}^{2})^{2}}\sum_{\sigma}t_{\sigma}(\omega,T)\;,\;$
(81)

where $t_{\sigma}(\omega,T)={\rm
Im}[-\tilde{\Gamma}_{\sigma}G_{\textit{ff};\sigma}(\omega,T)]$ is the t-matrix
spectrum and $G_{\textit{ff};\sigma}(\omega,T)\equiv\langle\langle
f_{\sigma}^{\phantom{\dagger}};f_{\sigma}^{\dagger}\rangle\rangle_{\omega,T}$
is the Green’s function of the effective $f$ level. Unlike Eq. 78 for the
spinless case, here the dynamics are nontrivial due to the constraint
$U_{c}\to\infty$. However, $G_{\textit{ff};\sigma}(\omega,T)$ can be
calculated numerically using e.g. 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.

In the absence of a magnetic field in the bare model, we have
$E^{N+1}_{\uparrow}=E^{N+1}_{\downarrow}$ such that the effective field
vanishes, $B_{\rm eff}=0$, and the effective tunneling amplitudes
$t_{\alpha\sigma}\equiv t_{\alpha}$ become independent of the spin label
$\sigma$ (hence $t_{\sigma}\equiv t$ and
$\tilde{\Gamma}_{\sigma}\equiv\tilde{\Gamma}$). We focus on this case below.

In Fig. 8 we consider the spectrum of the t-matrix $t_{\sigma}(\omega,T)\equiv
t(\omega,T)$ for the effective model Eq. 80 with $B_{\rm eff}=0$ and
$U_{c}\to\infty$, obtained by full NRG calculations. For given effective
parameters $\epsilon$ and $\tilde{\Gamma}$, the low-$T$ transport can then be
obtained via Eqs. 28, 81. Panel (b) shows the $T=0$ spectra at the charge
degeneracy point $\epsilon=0$ for different $\tilde{\Gamma}$. At small
$\tilde{\Gamma}/D\ll 1$ the spectrum is accurately approximated by that of an
equivalent non-interacting resonant level model[8] ($U_{c}=0$) with the same
$\tilde{\Gamma}$, but with an interaction-renormalized level energy
$\epsilon^{*}=\epsilon+{\rm Re}\Sigma_{\textit{ff}}(\omega=0)$, where
$\Sigma_{\textit{ff}}(\omega)$ is the $f$-level interaction self-energy,

$t(\omega,0)\simeq\frac{1}{1+(\omega-\epsilon^{*})^{2}/\tilde{\Gamma}^{2}}\;.$
(82)

This is shown for comparison as the dashed line in Fig. 8(b).

Figure 8: t-matrix for the effective model Eq. 80 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
$\epsilon=0$ (black line) and $|\epsilon|/\tilde{\Gamma}=0.5,1,1.5,2$ (red,
blue, green and magenta lines) with $\epsilon>0$ shown as solid lines and
$\epsilon<0$ as dashed lines. (b) $t(\omega,0)$ vs $\omega/\tilde{\Gamma}$ at
$\epsilon=0$ for $\tilde{\Gamma}/D=10^{-1},10^{-3},10^{-5},10^{-7}$ for the
green, blue, red and black lines. Eq. 82 is shown as the dashed line for
comparison to black line. (c) Behaviour of $\epsilon^{*}/\tilde{\Gamma}$ as a
function on $\tilde{\Gamma}$ for the same parameters as in panel (a), showing
small $\tilde{\Gamma}/D\ll 1$ asymptotes following Eq. 83.

At larger $\tilde{\Gamma}$, discrepancies from Eq. 82 at finite frequencies
become more pronounced, due mainly to the finite ${\rm
Im}\Sigma_{\textit{ff}}(\omega)$, leading to an enhanced peak height and
asymmetric shape. However, since ${\rm Im}\Sigma_{\textit{ff}}(0)=0$ by Fermi
liquid theory, $t(0,0)$ is always given exactly by Eq. 82, once $\epsilon^{*}$
is known.

Only $t(0,0)$ is needed for the $T\to 0$ conductance, and reference NRG
results for it as a function of $\tilde{\Gamma}$ are presented in Fig. 8(a)
for different $\epsilon/\tilde{\Gamma}$. The behaviour of $\epsilon^{*}$,
which yields $t(0,0)$ exactly from Eq. 82, is given for the same parameters in
Fig. 8(c). In the physically-relevant regime $\tilde{\Gamma}/D\ll 1$, the
behaviour is found to be simply,

$\epsilon^{*}/\tilde{\Gamma}\simeq
a\epsilon/\tilde{\Gamma}-b\log(\tilde{\Gamma}/D)+c\;,$ (83)

where numerically we find $a\simeq 0.93$, $b\simeq 0.29$ and $c\simeq 0.35$.
By contrast, at large $\tilde{\Gamma}/D\gg 1$, the self-energy correction can
be neglected for $\epsilon>0$, such that $\epsilon^{*}\to\epsilon$; while for
$\epsilon<0$ the Kondo effect gives $\epsilon^{*}\to 0$. This is confirmed
directly by the saturation values in Fig. 8(c).

Equipped with these results, the low-$T$ conductance of a system in the
vicinity of a singlet-doublet charge-degeneracy point can be straightforwardly
predicted, once the effective model parameters $\epsilon$ and $t_{s,d}$ have
been determined.

###  VII.3 Doublet-triplet transition

The charge degeneracy point between spin-doublet and spin-triplet 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, 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
(by contrast to an even-electron $S=0$ singlet, as considered in the previous
section).

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 the
effective impurity Hamiltonian comprises five states: two for the $N$ electron
doublet, and three for the $N+1$ electron triplet. For simplicity we consider
below the SU(2) spin-symmetric case arising when no external magnetic field
acts, such that states $|n;S^{z};l\rangle$ of a given spin multiplet $l$ are
degenerate ($E_{S^{z};l}^{n}\equiv E_{l}^{n}$ is independent of $S^{z}$). We
also set $l=1$ (ground state) and suppress $l$ labels in the following.

We construct an effective model using Eq. 76, casting the result in terms of
one effective fermionic site $f$, and one effective spin-$\tfrac{1}{2}$ degree
of freedom $\hat{\bm{S}}_{g}$, 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. Our effective model reads,

$\begin{split}H_{\rm eff}=H_{\rm
leads}&+\delta_{c}\hat{n}_{f}-J_{c}\hat{\bm{S}}_{f}\cdot\hat{\bm{S}}_{g}+\epsilon(\hat{n}_{f}-1)^{2}\\\
&+\sum_{\begin{subarray}{c}\alpha=s,d\\\
\sigma=\uparrow,\downarrow\end{subarray}}\left(t_{\alpha}^{\phantom{\dagger}}f^{\dagger}_{\sigma}c^{\phantom{\dagger}}_{\alpha\sigma}+\rm{H.c.}\right)\;,\end{split}$
(84)

where $\hat{\bm{S}}_{g}$ is a spin-$\tfrac{1}{2}$ operator for the effective
local moment,
$\hat{\bm{S}}_{f}=\tfrac{1}{2}\sum_{\nu,\nu^{\prime}}f_{\nu}^{\dagger}\bm{\sigma}_{\nu\nu^{\prime}}f_{\nu^{\prime}}^{\phantom{\dagger}}$
is the spin density and
$\hat{n}_{f}=\sum_{\sigma}f_{\sigma}^{\dagger}f_{\sigma}^{\phantom{\dagger}}$
is the number operator for the effective $f$ level, and

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

with $f_{\sigma}=(f_{\sigma}^{\dagger})^{\dagger}$ as usual. Here, the
$\tfrac{1}{\sqrt{2}}$ factors are Clebsch-Gordon coefficients implied by the
Wigner-Eckart theorem.

Of the 8 effective states of the $fg$ system in Eq. 84, we wish to retain only
the 5 that represent the $N$ electron doublet and the $N+1$ electron triplet
of the bare model:

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

The spurious states
$|\uparrow\downarrow\rangle_{f}\otimes|\uparrow\rangle_{g}$,
$|\uparrow\downarrow\rangle_{f}\otimes|\downarrow\rangle_{g}$ and
$\tfrac{1}{\sqrt{2}}(|\uparrow\rangle_{f}\otimes|\downarrow\rangle_{g}-|\downarrow\rangle_{f}\otimes|\uparrow\rangle_{g})$
are eliminated by the constraints in Eq. 84 by setting
$\delta_{c}=\tfrac{1}{4}J_{c}$ and sending $J_{c}\to\infty$.

With the above definitions, the effective tunneling amplitudes in Eq. 84 can
be obtained from the extremal weight matrix elements of the isolated
nanostructure,

$t_{\alpha}=V_{\alpha}\langle
N+1;S^{z}=1|\bar{d}_{\alpha\uparrow}^{\dagger}|N;S^{z}=\tfrac{1}{2}\rangle\;,$

although by SU(2) spin symmetry we may also write
$t_{\alpha}=\sqrt{2}V_{\alpha}\langle
N\text{+1;}S^{z}\text{=0}|\bar{d}_{\alpha\uparrow}^{\dagger}|N\text{;}S^{z}\text{=-}\tfrac{1}{2}\rangle$.
Deviations from the precise charge-degeneracy point are captured by the
parameter $\epsilon=E^{N+1}-E^{N}$. The effective model is therefore a mixed-
valent single impurity Anderson model, side-coupled to an additional
spin-$\tfrac{1}{2}$ local moment by a ferromagnetic interaction.

Figure 9: t-matrix $t(0,0)$ at $T=0$ as a function of the hybridization
$\tilde{\Gamma}$ near a doublet-triplet charge-degeneracy point, described by
the effective model Eq. 84, obtained by NRG. Plotted for $\epsilon=0$ (black
line) and $|\epsilon|/\tilde{\Gamma}=0.5,1,1.5,2$ (red, blue, green and
magenta lines) with $\epsilon>0$ shown as solid lines and $\epsilon<0$ as
dashed lines.

Turning now to the low-temperature quantum transport, the PC form of the
effective model implies that we may use the MW formula Eq. 28, with
transmission coefficient Eq. 81 given in terms of the t-matrix
$t_{\sigma}(\omega,T)\equiv t(0,0)={\rm
Im}[-\tilde{\Gamma}G_{\textit{ff};\sigma}(\omega,T)]$, where
$G_{\textit{ff};\sigma}(\omega,T)\equiv\langle\langle
f_{\sigma}^{\phantom{\dagger}};f_{\sigma}^{\dagger}\rangle\rangle_{\omega,T}$
is the (spin-independent) Green’s function of the effective $f$ level and
$\tilde{\Gamma}=\pi\rho_{0}(|t_{s}|^{2}+|t_{d}|^{2})$ is the effective
hybridization as before.

The $T\to 0$ conductances of the physical system therefore depend on the
dynamics of the effective model Eq. 84 as $T\to 0$ and $\omega\to 0$, as
encoded in $t(0,0)$. In Fig. 9 we plot $t(0,0)$ as a function of
$\tilde{\Gamma}$ for different $\epsilon/\tilde{\Gamma}$, obtained from NRG.
This is the analogous plot for the doublet-triplet transition as Fig. 8(a) for
the singlet-doublet transition. Although we find qualitatively similar
behaviour to that of the singlet-doublet case, the details are different.
Indeed, here the full spectrum $t(\omega,0)$ shows much more pronounced
asymmetries than the simple Lorentzian Eq. 82, even at very small
$\tilde{\Gamma}$. However, in the physically-relevant regime
$\tilde{\Gamma}\ll 1$ we find $t(0,0)$ for the effective model is still
accurately approximated by Eqs. 82 and 83 but with modified $a\simeq 1.15$,
$b\simeq 0.18$ and $c\simeq 0.7$.

With these NRG results the low-$T$ conductances for real systems near a
doublet-triplet charge degeneracy point can be obtained simply from the
effective model parameters $\epsilon$ and $t_{s,d}$. Comparison of Fig. 8(a)
and Fig. 9 shows that for a given $\tilde{\Gamma}$ and $\epsilon$, the
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. 84,
obtained perturbatively to 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. Although irrelevant for the singlet-doublet case
considered in the previous section (whose many-body ground state is always a
non-degenerate 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. 84. 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 neglectable at experimentally-relevant temperatures. We therefore
argue that the effective model Eq. 84, and results of this section, remain
valid close to the charge degeneracy point where there is a clear separation
of scales.

###  VII.4 General spinful case

We briefly touch upon the general spinful situation near a charge degeneracy
point. Since $N$ and $N+1$ electron sectors of the nanostructure are connected
to first order in $H_{\rm hyb}$ by tunneling of a single electron, nontrivial
spin-charge entanglement arises near the charge-degeneracy point between
spin-$S$ and $S\pm\tfrac{1}{2}$ ground states. The two most common such
scenarios (doublet to singlet and doublet to triplet) were considered
explicitly above.

Generalizing now to the charge-degeneracy point between an $N$ electron
spin-$S$ multiplet ground state, and an $N+1$ electron spin-$S+\tfrac{1}{2}$
multiplet ground state, it is easily seen that an effective model with the
same form as Eq. 84 must arise – but with $\bm{\hat{S}}_{g}$ now a spin-$S$
operator, and with $t_{\alpha}=V_{\alpha}\langle
N+1;S^{z}=S+\tfrac{1}{2}|\bar{d}_{\alpha\uparrow}^{\dagger}|N;S^{z}=S\rangle$.
However $\epsilon=E^{N+1}-E^{N}$ as before.

$4S+3$ physical states of the spin $S$ and $S+\tfrac{1}{2}$ multiplets are
retained in the ground state manifold of the effective model, meaning that
$4S+1$ spurious states must be eliminated in Eq. 84 using the constraints
$\delta_{c}$ and $J_{c}$. This implies setting $\delta_{c}=\tfrac{1}{2}SJ_{c}$
and sending $J_{c}\to\infty$.

Low-$T$ quantum transport can be obtained in the same way as above from a
knowledge of $t(0,0)$, which can be calculated for the effective model using
e.g. NRG.

##  VIII Auxiliary field formulation

Recently in Ref. 114, topological properties of the Mott metal-insulator
transition in the Hubbard model were uncovered by mapping the interaction
self-energy of the effective impurity problem within dynamical mean field
theory, to auxiliary non-interacting degrees of freedom.

Here we explore the consequences of the same mapping for quantum transport. We
start with an application to the paradigmatic single-impurity Anderson model,
Eq. 8.

The auxiliary field mapping is an exact representation of the single-particle
dynamics for an interacting system in terms of a completely non-interacting
one.[114] The Dyson equation, Eq. 10, for the Anderson impurity reads,

$\displaystyle G_{\sigma}(\omega)\equiv\langle\langle
d_{\sigma}^{\phantom{\dagger}};d_{\sigma}^{\dagger}\rangle\rangle=\frac{1}{\omega^{+}-\epsilon_{d}-\sum_{\alpha}\Delta_{\alpha}(\omega)-\Sigma_{\sigma}(\omega)}\;,\qquad$
(85)

where
$\Delta_{\alpha}(\omega)=V_{\alpha}^{2}\mathcal{G}^{0}_{\alpha\alpha}(\omega)$
is the hybridization between the impurity and the physical lead $\alpha=s,d$,
and $\Sigma_{\sigma}(\omega)$ is the interaction self-energy. In the absence
of a magnetic field, we have SU(2) spin symmetry, and so we write
$\Sigma_{\sigma}(\omega)\equiv\Sigma(\omega)$. For convenience we absorb the
static contribution to the self-energy into the definition of the renormalized
level $\epsilon_{d}^{*}=\epsilon_{d}+{\rm Re}\leavevmode\nobreak\ \Sigma(0)$,
and work with the dynamical part of the self-energy
$\tilde{\Sigma}(\omega)=\Sigma(\omega)-{\rm Re}\leavevmode\nobreak\
\Sigma(0)$.

Figure 10: Schematic of the auxiliary field mapping. The interaction self-
energy $\tilde{\Sigma}(\omega)$ of the single-impurity Anderson model (left)
is mapped to an auxiliary non-interacting tight-binding chain (right). The
current between physical source and drain leads due to a bias voltage in the
interacting system is reproduced in the mapped non-interacting system through
a zero-current constraint for the auxiliary ‘lead’ in the 3-terminal Landauer-
Büttiker formula.

Following Ref. 114 we now interpret $\tilde{\Sigma}(\omega)\to\Delta^{\rm
aux}(\omega)$ as a hybridization to a bath of auxiliary non-interacting
degrees of freedom. The effect of the electronic scattering due to the Coulomb
interaction on the impurity is reproduced exactly by proper choice of the
auxiliary bath. Here we describe the auxiliary system as a semi-infinite
linear chain,

$\displaystyle H_{\rm
aux}=\sum_{n=0}^{\infty}\sum_{\sigma}\left(t_{n}^{\phantom{\dagger}}f_{n\sigma}^{\dagger}f_{(n+1)\sigma}+{\rm
H.c.}\right)\;,$ (86)

where for simplicity we have now assumed particle-hole symmetry,
$\epsilon_{d}=-U_{d}/2$ (such that $\epsilon_{d}^{*}=0$ and Eq. 86 does not
require inclusion of onsite potentials). The auxiliary chain is coupled at one
end to the impurity,

$\displaystyle H_{\rm imp-aux}=\sum_{\sigma}\left(V^{\phantom{\dagger}}_{\rm
aux}d_{\sigma}^{\dagger}f_{0\sigma}^{\phantom{\dagger}}+{\rm H.c.}\right)\;.$
(87)

The mapping is illustrated schematically in Fig. 10.

The impurity-auxiliary chain hybridization function is thus given by
$\Delta^{\rm aux}(\omega)=V_{\rm aux}^{2}\mathcal{G}^{0}_{\rm aux}(\omega)$,
where $\mathcal{G}^{0}_{\rm aux}(\omega)=\langle\langle
f_{0\sigma}^{\phantom{\dagger}};f_{0\sigma}^{\dagger}\rangle\rangle^{0}$ is
the boundary Green’s function of the isolated auxiliary system. The latter can
be expressed simply as a continued fraction[115] in terms of the tight-binding
parameters $\\{t_{n}\\}$ in Eq. 86 as $\mathcal{G}^{0}_{\rm
aux}(\omega)=1/[\omega^{+}-\Delta^{\rm aux}_{0}(\omega)]$ with $\Delta^{\rm
aux}_{n}(\omega)=t_{n}^{2}/[\omega^{+}-\Delta^{\rm aux}_{n+1}(\omega)]$.

The auxiliary parameters $V_{\rm aux}$ and $\\{t_{n}\\}$ are uniquely
determined[114] by setting $\Sigma(\omega)\to\Delta^{\rm aux}(\omega)=V_{\rm
aux}^{2}\mathcal{G}^{0}_{\rm aux}(\omega)$. Specifically, $V_{\rm
aux}^{2}=-\tfrac{1}{\pi}{\rm Im}\int d\omega\Sigma(\omega)$ initializes an
iterative scheme in which successive $t_{n}$ are obtained by
$t_{n}^{2}=-\tfrac{1}{\pi}{\rm Im}\int d\omega\Delta_{n}^{\rm aux}(\omega)$.
At half-filling, $V_{\rm aux}=U_{d}/2$. The other auxiliary parameters are
regular and well-behaved, with the recursion being efficient and numerically
stable. Since the original self-energy $\Sigma(\omega)$ is a continuous
function, the recursion does not terminate (the auxiliary chain is semi-
infinite); however the $t_{n}$ settle down to a regular pattern after a finite
number of steps.

The key point for the present discussion is not the specific form of these
parameters for a given model, but rather the fact that this mapping exists and
is unique.

For the mapped non-interacting system (Fig. 10, right) we have _three_
effective leads, with a resonant level impurity Green’s function,

$\displaystyle
G_{\sigma}(\omega)=\frac{1}{\omega^{+}-\epsilon_{d}^{*}-\sum_{\gamma}\Delta^{\gamma}(\omega)}\;,\qquad$
(88)

where $\gamma=s,d$ or ${\rm aux}$.

We now wish to calculate the current $I^{d}$ flowing from source lead to drain
lead due to a source-drain bias voltage $\Delta V_{b}$. In the physical
interacting system (Fig. 10, left), the linear response conductance follows
from the MW formula, Eq. 28a, with the interacting impurity Green’s function,
Eq. 85. For the mapped non-interacting system (Fig. 10, right), we may use the
3-terminal linearlized Landauer-Büttiker (LB) formula for the current into
lead $\gamma$,

$I^{\gamma}(T)=\frac{e}{h}\sum_{\beta\neq\gamma}\int
d\omega\leavevmode\nobreak\ [\partial_{\omega}f_{\rm
eq}(\omega)](\mu_{\gamma}-\mu_{\beta})\mathcal{T}_{\text{L}\>\gamma\beta}(\omega)\;,$
(89)

where $\mu_{\gamma}$ is the chemical potential of lead $\gamma$ and
$\mathcal{T}_{\text{L}\;\gamma\beta}(\omega)=4\Gamma_{\gamma}(\omega)\Gamma_{\beta}(\omega)\sum_{\sigma}|G_{\sigma}(\omega)|^{2}$
is the analogue of Eq. 19, with $G_{\sigma}(\omega)$ the effective non-
interacting Green’s function given in Eq. 88, and
$\Gamma_{\gamma}(\omega)=-{\rm Im}\leavevmode\nobreak\
\Delta^{\gamma}(\omega)$. Eq. 89 is a generalization of the usual LB formula
to the case with inequivalent leads with arbitrary density of states (Appendix
C). This is important because the auxiliary ‘lead’ has a specific form that
must be accounted for. We will however assume for simplicity that the source
and drain leads are equivalent: $\Gamma_{s}(\omega)=\pi V_{s}^{2}\rho(\omega)$
and $\Gamma_{d}(\omega)=\pi V_{d}^{2}\rho(\omega)$ with the same (but
otherwise arbitrary) density of states $\rho(\omega)$.

Of course, the auxiliary lead is not a physical lead and so we do not apply a
voltage to it ($\mu_{\rm aux}=0$), and no current flows into or out of it
($I^{\rm aux}=0$). The latter property is also required by current
conservation in the physical system, $I^{s}=-I^{d}$. From Eq. 89, these
constraints imply that $V^{2}_{s}\mu_{s}+V^{2}_{d}\mu_{d}=0$. The voltage bias
$e\Delta V_{b}\equiv\mu_{s}-\mu_{d}$ must be split across source and drain
leads in a specific way to satisfy this constraint, with $\mu_{s}=e\Delta
V_{b}(1+V^{2}_{s}/V^{2}_{d})^{-1}$ and $\mu_{d}=-e\Delta
V_{b}(1+V^{2}_{d}/V^{2}_{s})^{-1}$. We note however that the $\Delta V_{b}\to
0$ linear response conductance does not depend on the details of this
splitting.

Substituting in Eq.89, we obtain,

$\begin{split}I^{d}(T)=\Delta V_{b}\frac{e^{2}}{h}\leavevmode\nobreak\
\frac{4\pi V^{2}_{d}V^{2}_{s}}{V^{2}_{d}+V^{2}_{s}}\int
d\omega&\leavevmode\nobreak\ [-\partial_{\omega}f_{\rm
eq}(\omega)]\rho(\omega)\\\
&\times\sum_{\gamma,\sigma}\Gamma_{\gamma}(\omega)|G_{\sigma}(\omega)|^{2}\;.\end{split}$
(90)

This reduces correctly to Eq. 105 since
$\sum_{\gamma}\Gamma_{\gamma}(\omega)|G_{\sigma}(\omega)|^{2}=-{\rm
Im}\leavevmode\nobreak\ G_{\sigma}(\omega)$ from Eq. 88. Furthermore, in the
wide flat band limit of the leads (when $\rho(\omega)\to\rho_{0}$ and
$\Gamma_{\alpha}(\omega)\to\Gamma_{\alpha}=\pi\rho_{0}V_{\alpha}^{2}$ for
$\alpha=s,d$), Eq. 90 recovers the standard PC form of the MW formula, Eq.
28a, as expected.

The above arguments generalize trivially to any multi-orbital two-lead system
in PC. The Green’s function for the PC frontier orbital
$\bar{\bar{d}}_{\sigma}$ coupling to the leads can always be expressed as,

$\displaystyle\overline{\overline{G}}_{\sigma}(\omega)\equiv\langle\langle\bar{\bar{d}}_{\sigma}^{\phantom{\dagger}};\bar{\bar{d}}_{\sigma}^{\dagger}\rangle\rangle=\frac{1}{\omega^{+}-\epsilon_{d}-\sum_{\alpha}\Delta_{\alpha}(\omega)-\Sigma^{\prime}_{\sigma}(\omega)}\;,\qquad$
(91)

where $\Sigma^{\prime}_{\sigma}(\omega)$ includes the effect of scattering
from coupling of $\bar{\bar{d}}_{\sigma}$ to the other nanostructure degrees
of freedom, as well as accounting for electronic interactions. Following the
same steps as before, $\Sigma^{\prime}_{\sigma}(\omega)$ can be mapped to a
single non-interacting auxiliary chain (Eq. 86) coupled at one end to a single
resonant level (Eq. 87), which is also coupled to the physical source and
drain leads. Schematically, the mapped system is identical to that depicted in
Fig. 10(b).

The non-PC case is more subtle, since the equivalent non-interacting form of
$\overline{G}_{sd,\sigma}(\omega)$ for use in Eq. 16 must be determined. In
general this requires mapping the effective self-energies to _two_ auxiliary
chains, attached to both frontier orbitals $\bar{d}_{s\sigma}$ and
$\bar{d}_{d\sigma}$. We leave this for future investigation.

In summary, quantum transport for interacting systems can be understood in
terms of the non-interacting Landauer-Büttiker formula, in which the self-
energy plays the role of an additional fictitious lead, subject to a zero-
current constraint. This formulation provides a simple way of viewing the
correction to quantum transport due to interactions.

Furthermore, the auxiliary chain representation may provide a route to simple
approximations, given its convenient structure and well-defined asymptotic
form. For example, at $T=0$ in the metallic Kondo screened case,[114]

$t_{n}\sim\frac{D}{2}\sqrt{1-\frac{2(-1)^{n}}{n+d}}\;,$ (92)

for large $n$, where $D$ is the effective bandwidth and $d\sim 1/Z$ is related
to the quasiparticle weight $Z$. The conductance formulae can be expressed in
terms of the auxiliary chain parameters.

An application of the auxiliary field method is given for the TQD system in
the next section.

##  IX Applications and Comparison of techniques

The goal of this section is to demonstrate the results of the previous
sections by application to specific multi-orbital strongly-correlated systems.
Our chosen test systems are the two-lead triangular triple quantum dot (TQD)
and the serial multilevel double quantum dot (MLDQD).

###  IX.1 Triple quantum dot

Figure 11: Illustration of the triple quantum dot devices studied. (a) PC
coupling geometry; (b) non-PC geometry.

Here we consider the TQD model $H=H_{\rm leads}+H_{\rm TQD}+H_{\rm hyb}$, with
$H_{\rm leads}$ given in Eq. 1 and,

$\begin{split}&H_{\rm TQD}=\\\
&\sum_{j=1,2,3}\left(\epsilon_{j}\hat{n}_{j}+U_{j}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow}\right)+\sum_{i\neq
j}\left(U_{ij}^{\prime}\hat{n}_{i}\hat{n}_{j}+\sum_{\sigma}t_{ij}^{\phantom{\dagger}}d^{\dagger}_{i\sigma}d^{\phantom{\dagger}}_{j\sigma}\right)\;,\end{split}$
(93)

where $\hat{n}_{j}=\sum_{\sigma}\hat{n}_{j\sigma}$, and the inter-dot
tunneling matrix elements satisfy $t_{ij}=t_{ji}^{*}$. We take for simplicity
equivalent dots with $\epsilon_{j}\equiv\epsilon$, $U_{j}\equiv U$,
$U^{\prime}_{ij}\equiv U^{\prime}$, and consider the mirror symmetric case
$t_{12}=t_{13}\equiv t$ but $t_{23}\equiv t^{\prime}$. We consider two
geometries for the TQD-lead hybridization,

$\displaystyle H_{\rm hyb}^{I}$
$\displaystyle=\sum_{\alpha=s,d}\sum_{\sigma}\left(V_{\alpha}^{\phantom{\dagger}}d^{\dagger}_{1\sigma}c_{\alpha\sigma}^{\phantom{\dagger}}+{\rm
H.c.}\right)\;,$ (94a) $\displaystyle H_{\rm hyb}^{II}$
$\displaystyle=\sum_{\sigma}\left(V_{s}^{\phantom{\dagger}}d^{\dagger}_{2\sigma}c_{s\sigma}^{\phantom{\dagger}}+V_{d}^{\phantom{\dagger}}d^{\dagger}_{3\sigma}c_{d\sigma}^{\phantom{\dagger}}+{\rm
H.c.}\right)\;.$ (94b)

For simplicity we also take $V_{s}=V_{d}$. Importantly, note that $H_{\rm
hyb}^{I}$ in Eq. 94a (illustrated in Fig. 11a) satisfies the PC geometry
condition while $H_{\rm hyb}^{II}$ in Eq. 94b (Fig. 11b) is non-PC and
therefore irreducibly 2-channel.

The behavior of TQD systems is notoriously rich, with aspects of their complex
physics having been uncovered in both experiments (see e.g. Refs. 116, 117,
118, 119, 120, 121) and theory (see e.g. Refs. 122, 123, 124, 125, 126, 127,
*oguri2011kondo, 129, 130, 131, 132, 133). Our purpose here is to examine the
quantum transport properties in a systematic and consistent fashion, applying
and comparing the techniques discussed above. We solve the underlying quantum
impurity problems using NRG.

Figure 12: AC conductance $G_{C}(\omega,T)$ as a function of driving frequency
$\omega$ at different temperatures $T$ as indicated in the color scale, for
the TQD model in the PC geometry (see Fig. 11(a) and Eq. 94a). Results shown
for the Kondo screened phase $t^{\prime}>t$ (left) and local moment phase
$t^{\prime}<t$ (right). NRG results are obtained using the ac-generalized MW
formula, Eq. 39, within the effective single-channel model. Model parameters:
$U=0.4D$, $\epsilon=-U/2$, $U^{\prime}=0$, $V_{s}=V_{d}=0.071D$, $t=5\times
10^{-4}D$ and $t^{\prime}=10^{-2}D$ (left) or $t^{\prime}=0$ (right). NRG
parameters: $\Lambda=2.5$, $M_{K}=3000$.

####  IX.1.1 Proportionate Coupling

First we consider the PC case, Eq. 94a (Fig. 11a) with $U^{\prime}=0$. The
behavior of the linear dc electrical conductance $G_{C}(T)$ for this system
was reported in Ref. 122. In Fig. 12 we consider instead the dynamical
conductance $G_{C}(\omega,T)$, plotted vs ac driving frequency $\omega$ at
different temperatures $T$. The TQD in this geometry supports a quantum phase
transition driven by geometric frustration in the triangular arrangement of
dots.[122] The transition, tuned by the ratio $t/t^{\prime}$, embodies the
competition between an antiferromagnetic-Kondo Fermi liquid with a screened
spin-singlet ground state, and a ferromagnetic-Kondo singular Fermi liquid
with a degenerate local moment ground state. [52] We show how this plays out
in the ac conductance in Fig. 12, which we calculate via Eq. 39. As such, the
conductance is controlled by the equilibrium retarded Green’s function for dot
1 (interacting, lead-coupled), since in this geometry
$\overline{\overline{G}}_{\sigma}(\omega,T)=G_{11,\sigma}(\omega,T)\equiv\langle\langle
d_{1\sigma}^{\phantom{\dagger}};d_{1\sigma}^{\dagger}\rangle\rangle$. This is
in turn obtained from the Dyson equation Eq. 10, with the TQD self-energy
matrix calculated in NRG via Eq. 31.

For $t^{\prime}>t$ (left panel) we see a Kondo resonance, with strongly
enhanced ac conductance for $|\omega|,T\ll T_{\rm K}$. As the temperature is
increased, the resonance is lost. To understand this dynamical transport
behavior in the TQD, the screening mechanism[122] must be understood. At
temperatures $T\lesssim U$, the dots become essentially singly-occupied
(assuming $t,t^{\prime}\ll U$ as here). On the scale $J\sim t^{2}/U\ll U$,
effective antiferromagnetic exchange interactions then bind the three spins of
the TQD into a combined doublet ground state,

$\displaystyle|+;S_{\rm
TQD}^{z}=\tfrac{1}{2}\rangle=\tfrac{1}{\sqrt{2}}[|\uparrow\uparrow\downarrow\rangle-|\uparrow\downarrow\uparrow\rangle]\;,$
(95)

where the basis states $|\sigma_{1},\sigma_{2},\sigma_{3}\rangle$ are labelled
by the spins on the three dots, and the other component of the doublet with
$S_{\rm TQD}^{z}=-\tfrac{1}{2}$ is obtained by replacing
$\uparrow\leftrightarrow\downarrow$. This doublet then couples to the leads
with an effective exchange interaction $J_{\rm eff}^{+}\sim+V^{2}/U$, which
generates spin-flip scattering of conduction electrons, and results in the
Kondo effect below the Kondo temperature[8] $T^{+}_{\rm K}\sim
D\exp(-1/\rho_{0}J^{+}_{\rm eff})$. For the parameters used, we have
$T^{+}_{\rm K}\approx 10^{-5}D$, with $D$ the conduction electron bandwidth.

For $T\ll T^{+}_{\rm K}$ the conduction electron scattering rate is on the
order of $T^{+}_{\rm K}$. Therefore, when a small dc bias voltage is applied,
the enhanced electronic scattering boosts the net current flowing from source
to drain.[134, 135] The conductance in this case can reach its maximum value,
$G_{C}=2e^{2}/h$. In the ac case, conductance is still strongly enhanced,
provided that the electronic scattering time is much shorter than the period
of the voltage bias oscillations, or $|\omega|\ll T^{+}_{\rm K}$. As the
frequency increases, the current does not have time to build up fully before
the bias voltage changes sign. At large frequencies, the TQD sees an
‘averaged’ bias and the conductance is low. The resonance condition is
therefore $|\omega|\sim T^{+}_{\rm K}$ when $T\ll T^{+}_{\rm K}$. This is seen
in the low-$T$ limit (turquoise line) in the left panel of Fig. 12, which is
found to have asymptotic behavior in the Kondo-screened phase following,

$\displaystyle G_{C}(\omega,0)/G_{0}$
$\displaystyle=2-\bar{a}(\omega/T^{+}_{\rm K})^{2}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ |\omega|\ll T^{+}_{\rm
K}\leavevmode\nobreak\ $ (96a)
$\displaystyle=\bar{b}/\ln^{2}(|\omega|/T^{+}_{\rm K})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle:\leavevmode\nobreak\ |\omega|\gg T^{+}_{\rm
K}\leavevmode\nobreak\ $ (96b)

with $\bar{a},\bar{b}$ constants of order 1.

As the temperature is increased, the Kondo effect is destroyed, the resonance
condition is lost, and the low-frequency ac conductance decreases. For $T\gg
T^{+}_{\rm K}$, the conductance is low, being the result of incoherent
sequential tunneling only (it is not boosted by spin-flip scattering from the
Kondo effect). This is seen in the left panel of Fig. 12 by the sequence of
lines from turquoise to purple on increasing temperature.

Upon tuning $t^{\prime}$ from $>t$ to $<t$, the isolated TQD ground state
changes, and this results in a quantum phase transition in the lead-coupled
system.[122] For $t^{\prime}<t$ (right panel, Fig. 12), the collective TQD
doublet ground state (in the singly-occupied limit) is,

$\displaystyle|-;S_{\rm
TQD}^{z}=\tfrac{1}{2}\rangle=\tfrac{1}{\sqrt{6}}[2|\downarrow\uparrow\uparrow\rangle-|\uparrow\uparrow\downarrow\rangle-|\uparrow\downarrow\uparrow\rangle]\;.$
(97)

This doublet state again forms on the scale of $J\sim t^{2}/U$, but its
effective coupling to the leads is now _ferromagnetic_ , $J^{-}_{\rm eff}<0$.
This results in the suppression of spin-flip scattering on reducing the
temperature[8] below $T\sim J$, and hence a suppression of the conductance. In
the dc limit, we have $G_{C}(0,0)=0$, embodying the emergent decoupling of the
TQD from the leads. However, singular Fermi liquid corrections to the local
moment fixed point at finite energy [136] lead to logarithmic corrections to
the low-frequency ac conductance [109],

$\displaystyle
G_{C}(\omega,0)/G_{0}=\bar{\alpha}_{1}/\ln^{2}(\bar{\alpha}_{2}|\omega|/J)\qquad$
$\displaystyle:\leavevmode\nobreak\ |\omega|\ll J\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $ (98)

At energies $\gg J$, the collective TQD doublet $|-;S_{\rm TQD}^{z}\rangle$
has not developed, and lead conduction electron scattering is dominated by the
direct coupling between the leads and dot 1. In this regime the physics is
controlled by an effective single-impurity Kondo temperature $T^{1}_{\rm
K}\simeq T_{\rm K}^{+}$. If $J\gg T_{\rm K}^{1}$, then scattering is
incoherent and weak at energies $\gg J$. Correspondingly, the ac conductance
for $|\omega|\gg J$ is small. However, if $J\ll T_{\rm K}^{1}$ then Kondo-
enhanced scattering with a rate $\sim T_{\rm K}^{1}$ controls conductance at
frequencies $|\omega|\gg J$, following Eq. 96. In particular, we expect in
this regime Kondo resonant ac conductance for $|\omega|\sim T^{1}_{\rm K}\gg
J$, but suppressed conductance for $|\omega|\ll J$.

This is precisely what is observed in the right panel of Fig. 12 at low-$T$
(turquoise line), where we have chosen $J\sim T_{\rm K}^{1}$. This gives rise
to a non-monotonic behavior as the temperature is increased, with the Kondo
effect suppressed at $T\ll J$ by ferromagnetic correlations, activated around
$T\sim T_{\rm K}^{1}$ due to incipient screening of dot 1, and then again
suppressed thermally for $T\gg T_{\rm K}^{1}$.

The dynamical ac conductance $G_{C}(\omega,T)$ therefore contains much richer
information on the scattering and screening processes than the steady-state dc
conductance $G_{C}(T)$.

Figure 13: DC electrical conductance $G_{C}(T)$ (top), heat conductance
$K_{Q}(T)$ (middle), and Lorenz ratio $L(T)$ (bottom) as a function of
temperature, for the same TQD system as in Fig. 12. NRG results obtained using
the MW formulae Eq. 28 (solid lines) for both the Kondo screened phase
($t^{\prime}>t$, blue) and the unscreened local moment phase ($t^{\prime}<t$,
red). Dashed lines show the result of the effective single-impurity, single-
channel Kondo model, with effective exchange coupling $J_{\rm eff}$ obtained
from model machine learning methods, Eq. 102.

In Fig. 13 we investigate the temperature-dependence of dc thermoelectric
properties in the same system. The solid lines in the top panel for the dc
electrical conductance $G_{C}(T)$ are obtained by taking the $\omega\to 0$
limit of the $G_{C}(\omega,T)$ shown in Fig. 12. In practice, the dc limit is
realized for $|\omega|\ll\max(T_{\rm K},T)$ and the NRG data are well-
converged in this regime. This is formally and numerically equivalent to using
the MW formula Eq. 28a. We again show the representative behavior in each
phase (blue line for $t^{\prime}>t$, red line for $t^{\prime}<t$). The middle
panel shows the corresponding curves for the heat conductance $K_{Q}(T)$,
obtained from the MW formula Eq. 28b, with transmission function Eq. 27b, and
t-matrix $t_{ee,\sigma}(\omega,T)=-\Gamma\leavevmode\nobreak\ {\rm
Im}G_{11,\sigma}(\omega,T)$. The lower panel shows the Lorenz ratio
$L=\kappa/TG_{C}$ in units of the limiting WF value,[18] $L_{0}=\pi^{2}k_{\rm
B}^{2}/3e^{2}$.

We consider first the behavior in the Kondo-screened phase for $t^{\prime}>t$
(blue lines). At low temperatures $T\ll T_{\rm K}^{+}$, the physics of the TQD
is typical of a Fermi liquid,[19] with quadratic corrections to the strong
coupling fixed point. Here we find,

$\displaystyle G_{C}(T)/G_{0}$ $\displaystyle=2-a(T/T^{+}_{\rm K})^{2}$ (99a)
$\displaystyle K_{Q}(T)/TK_{0}$
$\displaystyle=\frac{2\pi^{2}}{3}-b(T/T^{+}_{\rm K})^{2}$ (99b) $\displaystyle
L(T)/L_{0}$ $\displaystyle=1-c(T/T^{+}_{\rm K})^{2}$ (99c)

where $a,b,c$ are constants of order 1 depending on the specific definition of
$T_{\rm K}^{+}$ used. Fitting to the NRG data yields $b/a\approx 9$, while
$c=3b/2\pi^{2}-a/2$. At the lowest temperatures, the TQD behaves like a
renormalized non-interacting system with $G_{c}\sim e^{2}/h$ and $K_{Q}\sim
k_{\rm B}T/h$ while $L\to L_{0}$. Note that the geometric factor
$4\Gamma_{s}\Gamma_{d}/(\Gamma_{s}+\Gamma_{d})^{2}$ in Eq. 29 describing the
relative coupling strength of the TQD to source and drain leads drops out here
for $V_{s}=V_{d}$. In general for $V_{s}\neq V_{d}$, both $G_{C}$ and $K_{Q}$
are multiplied by this overall constant (independent of temperature), such
that the Lorenz ratio $L$ is always unaffected by the coupling geometry in PC.
This is not necessarily the case for non-PC systems.

At higher temperatures $T\gg T_{\rm K}^{+}$ in the Kondo phase, spin-flip
scattering of conduction electrons from the TQD show up in thermoelectric
properties as,

$\displaystyle G_{C}(T)/G_{0}$
$\displaystyle=\frac{\alpha_{1}}{\ln^{2}(\alpha_{2}T/T_{\rm K}^{+})}$ (100a)
$\displaystyle K_{Q}(T)/TK_{0}$
$\displaystyle=\frac{\gamma_{1}}{\ln^{2}(\gamma_{2}T/T_{\rm K}^{+})}$ (100b)
$\displaystyle L(T)/L_{0}$ $\displaystyle=1-\frac{\xi}{\ln(\gamma_{2}T/T_{\rm
K}^{+})}$ (100c)

where we find $\gamma_{1}/\alpha_{1}=\pi^{2}/3$ and
$\gamma_{2}/\alpha_{2}\approx 1.7$. It then follows that
$\xi=\ln(\gamma_{2}/\alpha_{2})>0$.

The numerical results in Fig. 13 show the full crossover behavior. Note also
that although $G_{C}(\omega,0)$ and $G_{C}(0,T)\equiv G_{C}(T)$ appear
similar, the details of their functional forms are different. The Lorenz ratio
$L$ is found to approach the WF value $L_{0}$ from below for both $T\ll T_{\rm
K}^{+}$ and $T\gg T_{\rm K}^{+}$ in the Kondo phase, although in the latter
case the limit is saturated logarithmically slowly. At intermediate
temperatures, especially $T\sim T_{\rm K}^{+}$, we see strong violations of
the WF law, driven by electronic scattering from interactions.[132] This
phenomenology is also known experimentally in single-electron transistors.[22]
Such violations are in general expected for interacting nanostructures. Note
however that at intermediate temperatures, $L\neq L_{0}$ does not imply non-
Fermi liquid physics: the WF law strictly applies[18] only in the limit $T\to
0$, where most conventional systems exhibit Fermi liquid behavior and satisfy
the WF law.

Turning now to the ferromagnetic local moment phase realized in the TQD for
$t^{\prime}<t$ (red lines in Fig. 13), singular Fermi liquid corrections
appear in the thermoelectric properties at low temperatures $T\ll J$,

$\displaystyle G_{C}(T)/G_{0}$
$\displaystyle=\frac{\alpha^{\prime}_{1}}{\ln^{2}(\alpha^{\prime}_{2}T/J)}$
(101a) $\displaystyle K_{Q}(T)/TK_{0}$
$\displaystyle=\frac{\gamma^{\prime}_{1}}{\ln^{2}(\gamma^{\prime}_{2}T/J)}$
(101b) $\displaystyle L(T)/L_{0}$
$\displaystyle=1-\frac{\xi}{\ln(\gamma^{\prime}_{2}T/J)}$ (101c)

where the ratios $\gamma^{\prime}_{1}/\alpha^{\prime}_{1}=\pi^{2}/3$ and
$\gamma^{\prime}_{2}/\alpha^{\prime}_{2}\approx 1.7$ as well as
$\xi=\ln(\gamma^{\prime}_{2}/\alpha^{\prime}_{2})$ are the same as for Eq.
100. In this local moment phase, the electrical and heat conductances are
suppressed relative to the Kondo phase; in particular for $T\to 0$ we have
$G_{C}\to 0$, and $K_{Q}/T\to 0$ corresponding to a transmission node.
However, their ratio remains finite and $L\to L_{0}$ as $T\to 0$. The fact
that the effective ferromagnetic Kondo effect in the TQD asymptotically
satisfies the WF law is a signature of its low-energy singular Fermi liquid
correlations. By contrast, conventional transmission nodes typically yield
strongly enhanced Lorenz ratios.[23]

For $T\gg J$ the incipient Kondo screening of dot 1 results in behavior
described by Eq. 100. The crossover between Eqs. 100c and 101c upon lowering
the temperature from $T\gg J\sim T_{\rm K}^{+}$ to $T\ll J$ is therefore
predicted to give a sign-change in $L(T)-L_{0}$, and this fingerprint of the
ferromagnetic Kondo effect is observed in the lower panel of Fig. 13.

The above results support the perturbative prediction[122] that the physics of
the half-filled TQD in the PC coupling geometry is always described by an
effective single-channel, single spin-$\tfrac{1}{2}$ Kondo model (Eq. 61) at
the lowest temperatures. This is to be expected from RG theory, since the
Kondo model is the minimal effective model, comprising only the most RG-
relevant terms.[8] The resulting universality implies that the low-energy
behavior of dynamic observables in the TQD is the same as that of the Kondo
model when rescaled.

Since the observables of interest here are the thermoelectric transport
coefficients, we expect that $G_{C}(T)$ and $K_{Q}(T)$ calculated via Eqs.
27b, 29 for the effective Kondo model will precisely match those of the full
TQD at low temperatures. This is because the electrical and heat conductances
are controlled by the spectrum of the scattering t-matrix
$t_{ee,\sigma}(\omega,T)$, which should agree in bare and effective models at
low temperatures and energies – even though the single-channel Kondo model
itself has no such capacity for quantum transport measurements. This
illustrates a broader point that low-energy effective models with greatly
reduced complexity can still be used to obtain the correct transport
properties at low temperatures – provided the effective model parameters have
been accurately determined.

However, the perturbative prediction of the coupling constant $J_{\rm
eff}^{\pm}$ via Eq. 51 is naturally approximate, and is in fact rather
inaccurate for $U<D$ (this is seen already in the Schrieffer-Wolff mapping of
the AIM to the Kondo model[137]).

A better estimation of $J_{\rm eff}^{\pm}$ can be obtained by techniques
borrowed from machine learning,[99] where the ‘distance’ between the bare TQD
model and the low-energy effective Kondo model is encoded in a loss function,
which is then minimized by gradient descent. Since our target observables are
determined by the t-matrix $t_{ee,\sigma}(\omega,T)$, we define a loss
function based on the symmetrized Kullback-Leibler divergence,[138, 139]

$\displaystyle L(J_{\rm eff})$ $\displaystyle=\tilde{D}_{\rm KL}(t^{\rm
TQD}_{ee,\sigma}\|t^{\rm K}_{ee,\sigma})+\tilde{D}_{\rm KL}(t^{\rm
K}_{ee,\sigma}\|t^{\rm TQD}_{ee,\sigma})\;,$ (102a)
$\displaystyle\tilde{D}_{\rm KL}(P\|Q)$
$\displaystyle=\int^{\Lambda_{U}}_{\Lambda_{L}}d\omega\leavevmode\nobreak\
\frac{P(\omega)}{\omega}\ln\left(\frac{P(\omega)}{Q(\omega)}\right)\;,$ (102b)

where the roles of un-normalized distributions $P(\omega)$ and $Q(\omega)$ in
Eq. 102b are here played by the zero-temperature energy-resolved electronic
scattering amplitudes in the bare TQD model and the effective Kondo model,
given by $t_{ee,\sigma}(\omega,0)$. The Kullback-Leibler divergence therefore
compares the t-matrices of the TQD and Kondo models over the desired energy
range $\Lambda_{L}\leq\omega\leq\Lambda_{U}$. The loss function $L(J_{\rm
eff})$ is minimized by tuning the coupling in the Kondo model. The best-fit
effective model is obtained for $\partial_{J_{\rm eff}}L(J_{\rm eff})=0$ by
gradient descent. We implemented a simple numerical finite-difference
approximation to the gradient of $L(J_{\rm eff})$ for the optimization
procedure, since finding the derivative of dynamical correlation functions is
in general a hard problem; but we note that the exact gradient can in
principle be calculated using the recently-developed ‘differentiable NRG’
technique.[89]

Using the above systematic approach, we extracted the numerically-exact
effective Kondo couplings $J_{\rm eff}$ for the TQD systems considered in Fig.
13. For the Kondo screened case $t^{\prime}>t$, the effective model pertains
for all $T\ll U$, and so we chose an upper cutoff for the fitting
$\Lambda_{U}=0.1\times U$. To capture the Kondo resonance we choose a low-
energy cutoff $\Lambda_{L}=10^{-10}D\ll T_{\rm K}^{+}$. The model machine
learning technique yielded $J^{+}_{\rm eff}=0.1624D$. This should be compared
with the perturbative result[122] for these bare parameters, $J^{+}_{\rm
eff}=0.2D$ (but recall the exponential sensitivity of the Kondo temperature on
$J_{\rm eff}$). The t-matrix of the resulting single-channel Kondo model is
then used to compute the full temperature-dependence of the conductances
$G_{C}(T)$ and $K_{Q}(T)$. These are plotted as the blue dashed lines in Fig.
13, and are seen to agree perfectly over the full range of applicability for
the effective model, $T\ll U$.

More interesting is the behavior in the local moment phase obtained in the TQD
for $t^{\prime}<t$. The perturbative prediction[122] $J^{-}_{\rm
eff}\simeq-0.07D$ fails to capture the first-stage renormalization from the
incipient screening of dot 1 at temperatures $T\gg J$, and does not yield even
qualitatively correct results for the low-$T$ physics. Using the model machine
learning approach however, we find a surprisingly strong ferromagnetic
coupling $J^{-}_{\rm eff}=-19.826D$. Since the effective Kondo model is here
valid only for $T\ll J$ we set the upper cutoff to $\Lambda_{U}=10^{-6}D$
(while $\Lambda_{L}=10^{-10}D$ again). The t-matrix for the optimized
effective model was used to calculate $G_{C}(T)$ and $K_{Q}(T)$ – see red
dashed lines in Fig. 13. As expected, it agrees quantitatively with the exact
results for the TQD in the low-temperature regime $T\ll J$.

Ultimately, the quantum phase transition in the TQD on tuning $t^{\prime}/t$
can be traced to the quantum phase transition in the effective Kondo model on
changing the sign of the coupling $J_{\rm eff}$. Specifically, we have
antiferromagnetic coupling $J_{\rm eff}^{+}>0$ for $t^{\prime}>t$ and
ferromagnetic coupling $J_{\rm eff}^{-}<0$ for $t^{\prime}<t$. However, to
obtain quantitatively accurate results from the effective model requires a
more sophisticated non-perturbative estimation of the effective parameters,
such as the method described above.

Figure 14: DC electrical conductance $G_{C}(0,0)$ at $T=0$ for the TQD model
in PC geometry, as a function of applied magnetic field $B$. Blue line for
Kondo screened phase $t^{\prime}>t$, red line for local moment phase
$t^{\prime}<t$. Inset shows magnified view around the field-induced ground-
state transition in the isolated TQD, which results in a level-crossing QPT in
the lead-coupled system for $t^{\prime}>t$ but re-entrant Kondo for
$t^{\prime}<t$. Model parameters: $U=0.4D$, $U^{\prime}=0$, $\epsilon=-U/2$,
$V_{s}=V_{d}=0.1D$ and $(t,t^{\prime})=(0.094,0.04)$ and $(0.08,0.1)$ for red
and blue lines, respectively. NRG parameters: $\Lambda=2.5$, $M_{K}=2000$.

We consider now transport through the TQD in the presence of a finite local
magnetic field $B$. To do this we add a term to the TQD Hamiltonian $H_{\rm
field}=-B\sum_{j}\hat{S}^{z}_{j}$. The t-matrix controlling quantum transport
in this coupling geometry becomes spin-dependent
$t_{ee,\uparrow}(\omega,T)\neq t_{ee,\downarrow}(\omega,T)$, and therefore the
current for up and down spin electrons is unequal. Within the MW formalism,
the total (spin-summed) linear conductances follows from Eqs. 27, 29.

Focusing on the $T=0$ dc linear electrical conductance $G_{C}(0)$ as a
function of magnetic field strength $B$ in Fig. 14, we again compare the
behavior in the Kondo phase ($t^{\prime}>t$, blue line) and the local moment
phase ($t^{\prime}<t$, red line). For $B=0$, we see from Fig. 13 that
$G_{C}(0)=0$ for $t^{\prime}<t$ but $G_{C}(0)\simeq 2e^{2}/h$ for
$t^{\prime}>t$ (deviations from the ideal unitarity limit for a single-
electron transistor arise if particle-hole symmetry is broken). For
$t^{\prime}>t$, the degenerate TQD doublet ground state $|+;S^{z}_{\rm
TQD}=\pm\tfrac{1}{2}\rangle$ at $B=0$ gets split at finite $B$. The Kondo
effect is still operative, providing the Zeeman splitting $\sim|B|$ is much
less than the Kondo binding energy $T_{\rm K}^{+}$. However, the Kondo effect
is completely destroyed[8] and the low-$T$ conductance collapses[134] for
$B\gg T_{\rm K}^{+}$, since then the unique TQD ground state
$|+;+\tfrac{1}{2}\rangle$ is singled out and spin-flip excitations involving
$|+;-\tfrac{1}{2}\rangle$ become suppressed. The blue line in Fig. 14 shows
the expected[131, 140] rapid decrease in the conductance as $B$ increases.

As $B$ increases further to $B\sim J$, the ground state of the isolated TQD
with $t^{\prime}>t$ changes from $|+;+\tfrac{1}{2}\rangle$ to the extremal
weight state of the spin-quartet $|\uparrow\uparrow\uparrow\rangle$. Exactly
at the transition $B=B^{*}$, the $|+;+\tfrac{1}{2}\rangle$ and
$|\uparrow\uparrow\uparrow\rangle$ TQD states are precisely degenerate.
However, they cannot be interconverted by tunneling to the leads because they
have different _parity_ under the mirror symmetry operation[122] that swaps
the labels on dots 2 and 3 ($|+;S^{z}_{\rm TQD}\rangle$ is symmetric while
$|\uparrow\uparrow\uparrow\rangle$ is antisymmetric). This means that
$\langle\uparrow\uparrow\uparrow|\hat{S}^{+}_{1}|+;+\tfrac{1}{2}\rangle=0$, as
can be seen directly from the form of Eq. 95 (although the symmetry argument
holds to all orders and does not require the half-filled limit approximation).
As a consequence, there can be no field-induced Kondo effect[96, 141, 142,
143, 144, 145] in the TQD system for $t^{\prime}>t$, and we see a first-order
(level-crossing) quantum phase transition in the full lead-coupled system,
with a change in many-body ground states at the critical field strength
$B^{*}$. This shows up in the $T=0$ conductance as a small discontinuity – see
inset, Fig. 14 (blue line).

The $t^{\prime}<t$ phase is more interesting because, although no Kondo effect
occurs at $B=0$, we now have the possibility of a field-induced Kondo effect
at $B=B^{*}$. This is because the ground state of the isolated TQD at finite
$B$ crosses over from $|-;S^{z}_{\rm TQD}=+\tfrac{1}{2}\rangle$ to
$|\uparrow\uparrow\uparrow\rangle$ as a function of $B$ for $t^{\prime}<t$,
and both states have the same parity (they are both antisymmetric under
$2\leftrightarrow 3$ exchange). At $B^{*}$, the two states are degenerate and
can therefore be interconverted by coupling to the leads. The matrix element
$\langle\uparrow\uparrow\uparrow|\hat{S}^{+}_{1}|-;+\tfrac{1}{2}\rangle\neq
0$, as can be seen from Eq. 97, and therefore an effective Kondo model of the
form Eq. 61 arises at $B=B^{*}$, where the two components of the effective
impurity spin-$\tfrac{1}{2}$ are played here by the degenerate TQD states
$|-;+\tfrac{1}{2}\rangle$ and $|\uparrow\uparrow\uparrow\rangle$, similar to
the finite-field singlet-triplet Kondo effect in quantum dots.[96, 141] This
novel prediction for a field-induced Kondo effect involving the crossing of
spin doublet and spin quartet states in the TQD is borne out by the NRG
results shown as the red lines in Fig. 14, where a dramatic conductance
enhancement arises for $t^{\prime}<t$ at $B\sim J$. There is no quantum phase
transition as a function of $B$ for $t^{\prime}<t$.

The TQD therefore hosts Kondo-boosted conductance at $B=0$ for $t^{\prime}>t$,
but at $B\sim J$ for $t^{\prime}<t$. This provides sensitive control over the
current through the TQD, with a high on-off ratio, by tuning magnetic fields.

Figure 15: Mapping of the TQD self-energy in PC geometry to an auxiliary non-
interacting chain Eq. 86, with effective tunneling matrix elements $\delta
t_{n}=t_{n}-t_{\infty}$ plotted against chain site number $n$ close to the
physical TQD (left) and far from it (right). Inset shows the input self-
energy. Blue or red symbols or lines for Kondo screened or local moment
phases, respectively. Parameters same as Fig. 12.

Finally, we turn to the auxiliary field representation of the interacting TQD
in Fig. 15. As discussed above, transport in the PC coupling geometry is
controlled by the Green’s function of dot 1,
$\overline{\overline{G}}_{\sigma}(\omega)=G_{11,\sigma}(\omega)$. From Eq. 91
we extract the effective single-impurity self-energy
$\Sigma^{\prime}_{\sigma}(\omega)$, which incorporates both the effect of dots
2 and 3 as well as the electronic interactions. Defining the dynamical part
$\tilde{\Sigma}^{\prime}_{\sigma}(\omega)=\Sigma^{\prime}_{\sigma}(\omega)-{\rm
Re}\Sigma^{\prime}_{\sigma}(0)$, we map this to an effective non-interacting
auxiliary chain Eq. 86 by setting
$\tilde{\Sigma}^{\prime}_{\sigma}(\omega)\to\Delta_{\sigma}^{\rm
aux}(\omega)$, as described in Sec. VIII. For simplicity we consider the case
$B=0$ where
$\tilde{\Sigma}^{\prime}_{\sigma}(\omega)\equiv\tilde{\Sigma}^{\prime}(\omega)$,
and hence $\Delta_{\sigma}^{\rm aux}(\omega)\equiv\Delta^{\rm aux}(\omega)$,
and half-filling. For the same TQD model parameters as in Fig. 13, we extract
the effective self-energy at $T=0$, perform the auxiliary chain mapping, and
plot the chain hopping parameters $\delta t_{n}=t_{n}-t_{\infty}$ as a
function of chain position $n$ in Fig. 15. As before, blue lines/points
correspond to the Kondo phase $t^{\prime}>t$ while red lines/points are for
the local moment phase $t^{\prime}<t$. Inset shows the behavior of the self-
energy $\tilde{\Sigma}^{\prime}(\omega)$ from which the $\delta t_{n}$ are
extracted in each case.

The overall behavior of the $\delta t_{n}$ is rather regular; the early part
of the chain (small $n$) roughly corresponds to high-energy features in
$\tilde{\Sigma}^{\prime}(\omega)$, while low-energy information is encoded at
large $n$. In particular, since the self-energy for both $t^{\prime}>t$ and
$t^{\prime}<t$ is essentially equivalent down to $|\omega|\sim J$ (see inset),
the resulting $\delta t_{n}$ at small $n$ are essentially identical for both
phases – see left panel of Fig. 15. On the other hand, the low-energy
$|\omega|\to 0$ form of the self-energy is very different in the two phases,
with $-{\rm Im}\tilde{\Sigma}^{\prime}(\omega)\sim\omega^{2}$ characteristic
of the Fermi liquid correlations in the $t^{\prime}>t$ Kondo phase,[8] but
with $-{\rm Im}\tilde{\Sigma}^{\prime}(\omega)\sim\ln^{2}(|\omega|/J)$ in the
$t^{\prime}<t$ local moment phase, resulting from singular Fermi liquid
corrections.[136] The large $n$ behavior of $\delta t_{n}$ in the right panel
of Fig. 15 is therefore different in the two phases. In the Kondo phase we
have asymptotic behavior $\delta t_{n}\sim(-1)^{n}/n$, satisfying Eq. 92 as
expected, while in the local moment phase, we see a crossover in the envelope
function of $\delta t_{n}$ at $n\sim 10^{5}$ (red points), corresponding to
the minimum in $-{\rm Im}\tilde{\Sigma}^{\prime}(\omega)$ on the scale of
$|\omega/D|\sim 10^{-5}$ (red line, inset). The asymptotic behavior is found
to be $\delta t_{n}\sim(-1)^{n}/\ln^{2}(n)$.

In the auxiliary field representation of the TQD, the system is completely
non-interacting, and transport properties can be determined using the LB
formula, Eq. 90. In general, one must of course determine the auxiliary chain
parameters before applying the LB formula. We obtained them numerically
exactly here from a knowledge of the self-energy to demonstrate the procedure;
but the rather simple form of the coefficients $t_{n}$ might also justify the
use of simple approximations to capture the essence of interaction effects on
transport in a heuristic way.

Figure 16: Serial dc electrical conductance $G_{C}(T)$ through the TQD in non-
PC coupling geometry (see Fig. 11(b) and Eq. 94b), comparing the numerically-
exact improved Kubo formula (Eqs. 36, 37, 44, blue line) with common
approximations based on Green’s functions: Ng ansatz (Eqs. 28a, 34, green
line), Landauer (Eqs. 17a, 18 but with interacting Green’s functions, orange
line), and Oguri renormalized Landauer (Eq. 22, red dashed). Model parameters:
$U=0.4D$, $U^{\prime}=0$, $\epsilon=-U/2$, $V_{s}=V_{d}=0.12D$, $t=0.02D$,
$t^{\prime}=0.005D$. NRG parameters: $\Lambda=3$, $M_{K}=10000$.

####  IX.1.2 Non-Proportionate Coupling

We now turn to the more complex case in which leads are attached to dots 2 and
3 (see Fig. 11b and Eq. 94b). Since the coupling geometry does not satisfy the
PC condition, the MW formulation involving only retarded TQD Green’s functions
used in the previous section cannot be applied here. Instead, we obtain the
linear electrical conductance by NRG from the Kubo formula Eq. 36, using the
‘improved’ version of the current-current correlator, Eq. 44.

In Fig. 16 we consider $G_{C}(T)$ for the TQD at half-filling with typical
parameters, comparing the result of the improved Kubo formula (blue line, Eqs.
36, 37, 44) with approximations involving only retarded equilibrium single-
particle Green’s functions. The dc conductance shows a strong Kondo
enhancement at low-temperatures $T\ll T_{\rm K}\sim 10^{-5}D$, signifying the
involvement of collective TQD spin states in series transport.

The red dashed line is the Oguri result Eq. 22, for the asymptotic $T\to 0$ dc
conductance $G_{C}(0)$. The agreement with the result from the Kubo formula
for $T\ll T_{\rm K}$ is as expected for a system with a Fermi liquid ground
state. The green line is obtained using the Ng Ansatz, in which the
approximate transmission function Eq. 34 is employed in Eq. 28a. It reduces to
the Oguri result, as expected, for $T\ll T_{\rm K}$, and captures the
qualitative features of the true result, but significant deviations are
observed for $T\sim T_{\rm K}$. Ironically, simply using the interacting
retarded Green’s functions in the non-interacting Landauer formula, Eq. 17a
(orange line) does a better job.

For linear response electrical transport in non-PC geometries, we conclude
that the improved Kubo formula is the method-of-choice. Within NRG, the
accuracy of the method was validated in Fig. 2; it is in fact also simpler and
less computationally expensive than the standard approximate alternatives.

In the lower panel of Fig. 17 we consider the series dc electrical conductance
through the TQD as a function of gate voltage $\delta
V_{G}=\epsilon-\epsilon_{0}$, defined such that $\delta V_{G}=0$ corresponds
to half-filling. The conductance $G_{C}$ is calculated using NRG via the
improved Kubo formula at different temperatures, corresponding to the
different gate voltage trace lines. We use this to test the analytic
predictions of Secs. VI and VII, which are shown as the points in Fig. 17.

Figure 17: Gate-controlled TQD in the non-PC coupling geometry. Top panel: TQD
occupancy $\langle\hat{N}_{\rm TQD}\rangle$ in the full lead-coupled,
interacting system, showing Coulomb blockade staircase. Middle panel: Low-
energy Fermi liquid scale $T_{\rm FL}$, defined as the temperature below which
TQD degrees of freedom are quenched (in practice via the TQD entropy $S_{\rm
TQD}(T_{\rm FL})=0.1$). Bottom panel: dc series electrical conductance,
obtained from NRG using the improved Kubo formula, Eqs. 36, 37, 44 (lines) at
different temperatures given in the legend. Symbols show the $T=0$ effective
model predictions: $S=1/2$ CB regime (cross points via Eq. 71), $S=0$ CB
regime (circle point via Eq. 54), singlet-doublet MV crossover (square points
via Eq. 80). Vertical dashed lines show MV transitions in the isolated TQD for
reference. Model parameters: $U=3D$, $U^{\prime}=D$, $t=0.3D$,
$t^{\prime}=0.1D$ and $V_{s}=V_{d}=0.3D$. Gate voltage defined such that
$\delta V_{G}=0$ is at half-filling. NRG parameters: $\Lambda=3$,
$M_{K}=6000$.

Before discussing transport, we comment on the underlying Kondo physics of
this system.[125, 146] The number of electrons on the isolated TQD is
conserved and takes integer values. We define $\langle\hat{N}_{\rm
TQD}\rangle$ in terms of the TQD number operator $\hat{N}_{\rm
TQD}=\sum_{j,\sigma}d_{j\sigma}^{\dagger}d_{j\sigma}^{\phantom{\dagger}}$.
With finite interdot repulsion $U^{\prime}>0$, the half-filled $N_{\rm TQD}=3$
electron sector is well-separated in gate voltage from the $N_{\rm TQD}=2,4$
sectors. The transitions between these filling sectors are marked as the
vertical grey dashed lines in Fig. 17, and correspond to the charge-degeneracy
points in the isolated TQD (for the parameters used, they arise at $\delta
V_{G}/D\approx\pm 1.2$). Analysis of the TQD shows that for $N_{\rm TQD}=3$ we
have a spin-doublet ground state (effective impurity $S=\tfrac{1}{2}$), while
for $N_{\rm TQD}=2$ we have an effective $S=1$ spin-triplet ground state, and
for $N_{\rm TQD}=4$ the ground state is an effective $S=0$ spin-singlet. This
results in a rich range of gate-tunable many-body physics on attaching the
leads. In the top panel of Fig. 17 we see how the sharp Coulomb blockade (CB)
steps are now smoothed into a continuous evolution of $\langle\hat{N}_{\rm
TQD}\rangle$. However, one can still identify clear CB regimes and mixed-
valence (MV) crossovers. In particular, we have defined $\langle\hat{N}_{\rm
TQD}\rangle=3$ at $\delta V_{G}=0$, while, for the parameters used, the center
of the triplet and singlet CB regimes with $\langle\hat{N}_{\rm
TQD}\rangle=2,4$ occur at $\delta V_{G}/D\approx\pm 2$.

Deep in the spin-doublet CB regime, one can derive an effective
$S=\tfrac{1}{2}$ two-channel Kondo model of the type given in Eq. 56. Note
that in standard situations, the cross terms $J_{sd}$ and $W_{sd}$ are finite;
these are the processes in the effective model responsible for mediating a
source-drain current. We consider explicitly the $sd$ symmetric case with
$V_{s}=V_{d}$, and so $J_{ss}=J_{dd}\equiv J$ and $W_{ss}=W_{dd}\equiv W$.
Deviations from $\langle\hat{N}_{\rm TQD}\rangle=3$ at finite $\delta V_{G}$
lead to finite potential scattering terms $W_{ss}$ and $W_{dd}$.

From the results of Sec. VI.2.2, we expect an effective spin-$\tfrac{1}{2}$
single-channel Kondo effect on the lowest energy scales, forming with the even
lead combination. The resulting Kondo temperature $T_{\rm K}$ in this regime
is shown in the middle panel of Fig. 17, and exhibits the typical behavior of
Eq. 70. At $T=0$ we therefore expect Kondo-boosted conductance. This is shown
in the lower panel of Fig. 17 as the blue line, where the conductance is seen
to reach up to $G_{C}(0)\simeq 2e^{2}/h$ at the particle-hole symmetric point
$\delta V_{G}=0$. In fact, the conductance remains close to this maximum value
throughout most of the doublet regime, despite significant deviations in
$\langle\hat{N}_{\rm TQD}\rangle$ away from half-filling. The $T=0$
conductance only begins to attenuate when close to the mixed-valence points.

This behavior can be understood from the effective low-energy model. With the
chosen bare TQD parameters $U=3D$, $U^{\prime}=D$ and $V_{s}=V_{d}=U/10$, the
perturbative estimation of the effective model parameters via Eq. 51 is quite
accurate. The low-$T$ conductance can then be estimated from Eq. 71. For
strong interactions, we find that the effective potential scattering
parameters $W_{ee}$ and $W_{oo}$ remain small until close to the MV points
(where the applicability of Eq. 56 and hence Eq. 71 breaks down anyway). The
numerical values of $G_{C}(0)$ predicted in the doublet CB regime are shown in
the lower panel of Fig. 17 as the cross points, and agree rather well with the
full NRG calculations.

The red lines in Fig. 17 show NRG results for the conductance at higher
temperatures. Since the conductance remains Kondo-enhanced for $T\ll T_{\rm
K}$ but is strongly reduced as the Kondo effect is destroyed for $T\gg T_{\rm
K}$, we expect a non-trivial behaviour of $G_{C}(T)$ at a fixed $T$ on
changing the gate voltage $\delta V_{G}$, because $T_{\rm K}$ itself depends
on $\delta V_{G}$. We plot the conductance at $T/D=10^{-5},10^{-4},10^{-3}$ in
the lower panel (these same temperatures are marked as the horizontal lines in
the middle panel). As expected, we see reduced conductance as the ratio
$T/T_{\rm K}$ increases.

Deep in the CB spin-singlet regime around $\delta V_{G}/D=-2$, the low-energy
effective model is described by Eq. 52, with conductance controlled simply by
the effective cotunneling amplitude $W_{sd}$ via Eq. 54. There is no low-
energy scale in this regime; the effective model is valid for all $T\ll U$ (we
denote the scale where the spin singlet ground state forms as $T_{\rm K}$, but
this should be understood as a Fermi liquid scale rather than a Kondo scale
per se). The perturbative estimation of $W_{sd}$ and hence $G_{C}(0)$ is shown
as the circle point in the lower panel of Fig. 17 and agrees well with NRG
calculations in this regime. As anticipated, the conductance is small since
electrons must tunnel through the interacting TQD unaided by the Kondo effect.

In the CB spin-triplet regime, both even and odd conduction electron channels
are involved in screening the collective TQD spin states (see Sec. VI.3). The
delicate interplay between the effective couplings $J_{ee}$ and $J_{oo}$ (and
hence the Kondo scales $T_{\rm K}^{e}$ and $T_{\rm K}^{o}$) produces more
complex behavior in the conductance. The Fermi liquid scale $T_{\rm FL}$
corresponds here to the temperature below which the TQD spin is exactly
screened, $T_{\rm FL}\equiv\min(T_{\rm K}^{e},T_{\rm K}^{o})=T_{\rm K}^{o}$.
From NRG we see a Kondo peak reaching up to the maximum $G_{C}=2e^{2}/h$ at
around $\delta V_{G}\approx 1.7$ (that is, away from the center of the CB
window at $\delta V_{G}\simeq 2$ where $\langle\hat{N}_{\rm TQD}\rangle=2$).
This can be viewed as a many-body quantum interference effect resulting from
different conductance pathways around the TQD ring.[103, 147] The
insensitivity of the conductance in this regime to the ratio $T/T_{\rm K}$
implies a two-stage screening with $T_{\rm K}^{e}\gg T_{\rm K}^{o}$. The
first-stage partial screening arises at relatively high temperatures $T_{\rm
K}^{e}>T$ and is chiefly responsible for the enhanced conductance.

We now consider the MV crossovers between the above regimes, where the TQD
charge is strongly fluctuating and the CB approximation breaks down. For
$\delta V_{g}/D\approx-1.2$ the isolated TQD has degenerate ground states with
$N_{\rm TQD}=3$ and $4$ electrons, corresponding to a singlet-doublet MV
transition. This is smoothed into a crossover on attaching the leads and the
TQD occupancy varies smoothly as a function of gate; the many-body ground
states in the Kondo-screened doublet CB regime and the CB singlet regime are
continuously connected. In the vicinity of the singlet-doublet MV point we use
the effective model Eq. 80, calculating the effective parameters
$\tilde{\Gamma}$ and $\epsilon$ from the isolated TQD eigenstates and
energies. Since this effective model is in PC, the transmission function
within the MW formalism then follows from t-matrix as plotted in Fig. 8. The
low-$T$ conductance obtained using this approximation are shown as the solid
square points in Fig. 17, and capture the behaviour of the exact $T=0$ result
(blue line).

At $\delta V_{g}/D=+1.2$ the system is at the MV point between the $N_{\rm
TQD}=3$ doublet and the $N_{\rm TQD}=2$ triplet. An effective model describing
the coupling of these 5 TQD states to the leads to first order in the
hybridization was derived in Eq. 84. However, diagonalization of the isolated
TQD shows multiple low-lying excited states in this regime. The analysis of
Sec. VII.3 requires that the retained 5-fold-degenerate ground state TQD
manifold be well-separated from excited states, and hence breaks down in this
case. In addition, the coupling to the odd conduction channel is not captured
by first-order perturbation theory, but 2-stage Kondo screening involving both
even and odd channels is observed in NRG calculations for the full lead-
coupled system. This becomes important when $T\lesssim T_{\rm K}^{o}$.
Predictions of the conductance using Eq. 84 in this regime (not shown)
considerably underestimate the exact results. This demonstrates that care must
be taken when using results of the effective models to check that the
conditions used to derive them are satisfied for a given system. Of course, in
such cases the effective models could be generalized to include more TQD
states and/or go beyond first-order perturbation theory.

Figure 18: (a) Schematic illustration of the two-lead serial MLDQD model. (b)
Charge stability diagram of the isolated DQD as a function of gate voltages
$\delta V_{gL}$ and $\delta V_{gR}$. Charge configurations $(\delta
N_{L},\delta N_{R})$ correspond to the left and right dot charges, relative to
half-filling. (c) NRG calculations of the series dc linear conductance through
the MLDQD with $M=3$ levels per dot, as a function of gate voltages $\delta
V_{gL}$ and $\delta V_{gR}$, obtained using the improved Kubo formula at
finite temperature $T/D=10^{-3}$. Model parameters: $\delta\epsilon=0.2D$,
$U=0.2D$, $V=0.2D$, $t=0.1D$. NRG parameters: $\Lambda=5$, $M_{K}=4096$. We
exploit left/right symmetry and a particle-hole transformation to relate the
quadrant $|\delta V_{gR}|\leq\delta V_{gL}\geq 0$ to the other three
quadrants. The periodic structure _within_ the calculated quadrant is a result
of the multi-level nature of the dots (not enforced symmetrization).

###  IX.2 Serial multilevel double quantum dot

In this section we consider the serial MLDQD model illustrated in Fig. 18(a),
consisting of two equivalent multilevel dots, tunnel-coupled together and each
coupled to its own lead. The Hamiltonian reads $H=H_{\rm leads}+H_{\rm
DQD}+H_{\rm hyb}$, where $H_{\rm leads}$ is given in Eq. 1 as usual, and,

$\begin{split}H_{\rm
DQD}=&\sum_{\alpha=L,R}\left(\sum_{j=1}^{M}\leavevmode\nobreak\
\sum_{\sigma=\uparrow,\downarrow}\epsilon_{j}\hat{n}_{\alpha
j\sigma}+U\hat{N}_{\alpha}^{2}\right)+U^{\prime}\hat{N}_{L}\hat{N}_{R}\\\
&+\sum_{j,k,\sigma}\frac{t}{M}\left(d_{Lj\sigma}^{\dagger}d_{Rk\sigma}^{\phantom{\dagger}}+{\rm
H.c.}\right)+\sum_{\alpha}V_{g\alpha}\hat{N}_{\alpha}\;,\\\ H_{\rm
hyb}=&\sum_{\alpha,j,\sigma}\frac{V_{j}^{\alpha}}{\sqrt{M}}\left(d_{\alpha
j\sigma}^{\dagger}c_{\alpha\sigma}^{\phantom{\dagger}}+{\rm
H.c.}\right)\;,\end{split}$ (103)

where $M$ is the number of levels per dot and $d_{\alpha j\sigma}^{\dagger}$
($d_{\alpha j\sigma}$) is the creation (annihilation) operator for level
$j=1,2,...,M$ of dot $\alpha=L,R$ with spin $\sigma=\uparrow,\downarrow$. We
define number operators $\hat{n}_{\alpha j\sigma}=d_{\alpha
j\sigma}^{\dagger}d_{\alpha j\sigma}^{\phantom{\dagger}}$ and
$\hat{N}_{\alpha}=\sum_{j,\sigma}\hat{n}_{\alpha j\sigma}$. In this work we
use $M=3$, and a constant level spacing
$\delta\epsilon=\epsilon_{j+1}-\epsilon_{j}$ with $\epsilon_{2}=0$, set
$U^{\prime}=U/2$ and $V_{j}^{\alpha}\equiv V$, and define $\delta
V_{g\alpha}=V_{g}^{0}+V_{g\alpha}$ such that $\delta V_{gL}=\delta V_{gR}=0$
is at half-filling.

The charge stability diagram of the isolated DQD as a function of $\delta
V_{gL}$ and $\delta V_{gR}$ for representative parameters is shown in Fig.
18(b). The charge configurations $(\delta N_{L},\delta N_{R})$ correspond to
the left and right dot charges, relative to half-filling, which for $M=3$
means 3 electrons per dot. Lines denote degeneracies between two isolated DQD
charge configurations; filled circles correspond to the triple points. The
deformed hexagonal structure of the diagram is a ubiquitous feature of
experimental double dot systems with both local interactions $U$ on each dot,
and capacitive interactions $U^{\prime}$ between dots.[148, 149, 62] Since
typically $U^{\prime}<U$, not all hexagons are equivalent, and pronounced
even-odd (parity) effects are observed. For DQDs described by Eq. 103 with
many levels $M$, configurations $(n,m)$ are equivalent to $(n+2x,m+2y)$ with
integer $x$ and $y$. For finite $M$ the periodic structure is truncated.

When the leads are attached, Kondo effects can arise at low temperatures –
provided either one or both of the dots host a net spin-$\tfrac{1}{2}$ local
moment. The left dot may therefore participate in a Kondo effect when it is in
the CB regime with an odd number of electrons (that is, deep inside a CB
hexagon $(N_{L},N_{R})$ with odd $N_{L}$, independent of $N_{R}$). Similarly
Kondo effects with the right dot require odd $N_{R}$. However, the inter-dot
coupling competes with these individual dot-lead Kondo effects when both dots
host local moments.[150] Therefore the Kondo effects arising in sectors
$(N_{L},N_{R})$, with $N_{L}$ and $N_{R}$ both odd, are only partially
developed (note that realistic Anderson-type models of double quantum dots do
not support a quantum phase transition between Kondo and RKKY phases,[151] as
would be predicted from the oversimplified two-impurity Kondo model[152]).
Strong DQD charge fluctuations arise in the vicinity of the lines separating
two hexagons (MV lines). The lines connecting configurations $(N_{L}+1,N_{R})$
and $(N_{L},N_{R}+1)$ are particularly important: such configurations are
separated in gate-voltage space from other configurations by the finite inter-
dot capacitance, and are interconverted by the inter-dot tunneling. Including
dot-lead coupling, this can produce an orbital Kondo effect, which has an
emergent SU(4) symmetry in the limit of strong capacitive interactions.[153,
62, 154]

In terms of electronic transport through the device, we expect the highest
series conductance at the triple points between three charge configurations
[filled circles in Fig. 18(b)], since here an electron can pass between leads
by traversing the DQD without leaving the ground state charge configurations.
Conductance at the triple points can reach up to $G_{C}=2e^{2}/h$, even in the
incoherent (sequential tunneling) regime. Enhanced conductance is also
expected in CB regimes when both dots host a net spin-$\tfrac{1}{2}$ local
moment, since enhanced dot-lead scattering processes due to the Kondo effect
generated by finite $V_{j}^{\alpha}$ become correlated by the interdot
coupling $t$. However, the finite interdot coupling necessary for a series
current to flow favours interdot singlet formation and therefore weakens the
dot-lead Kondo effects as discussed above (the phase shift of the leads must
be less than $\pi/2$ for $|t|>0$). Thus the series conductance in this regime
must be less than the maximum for a single electron transistor,
$G_{c}<2e^{2}/h$. By contrast, in the CB regimes inside hexagons where one or
both dots host an even number of electrons and therefore have a net spin
$S=0$, the Kondo effect is inoperative and conductance is strongly suppressed
by the CB effect, $G_{C}\sim 0$. Finally, we distinguish between MV lines
where spin or orbital Kondo effects can occur [black lines in Fig. 18(b)], and
those where DQD charge fluctuations are not accompanied by Kondo [blue lines
in Fig. 18(b)]. Conductance is expected to be enhanced when electronic
scattering is Kondo-boosted. Conductance at triple points at the intersection
of three such lines (black circles) will be high relative to triple points
with only one such Kondo-boosted MV line (red circles).

In Fig. 18(c) we present full NRG results for the series dc linear conductance
through the device at finite temperature, as a function of the dot gate
voltages $\delta V_{gL}$ and $\delta V_{gR}$. Conductance maps of this type
are commonly obtained from experimental transport measurements;[155] our aim
here is to simulate such results. NRG calculations for the irreducible two-
channel MLDQD model utilize the improved Kubo formula (Eqs. 36 and 44), which
yields highly accurate transport properties at relatively low computational
cost. This is important, because to generate a high-resolution conductance map
as a function of gate voltages requires many NRG runs (in the case of Fig.
18(c), 2601 separate runs were performed). We verified that conductance
results were fully converged at temperature $T=10^{-3}D$ with the NRG
parameters used ($\Lambda=5$ and $M_{K}=4096$), which required about 30
minutes of computing time per NRG run on a standard desktop machine. To
achieve similar quality results using other methods would be significantly
more costly. Note that an equivalent single-channel formulation is not
possible for such a model, and the standard PC form of the MW formula is
inapplicable in non-PC coupling geometries. The Kubo formula is therefore
required to make contact with experimental results, which are naturally at
finite temperature.

The quantitative numerical results shown in Fig. 18(c) confirm the qualitative
physical picture discussed above in relation to the charge stability diagram,
Fig. 18(b). In particular, we see maximal conductance $G_{C}\sim 2e^{2}/h$ at
triple points which lie at the intersection between three Kondo-boosted MV
lines. Conductance is also relatively enhanced in CB hexagons where both dots
host a net spin-$\tfrac{1}{2}$, although the competition between interdot
singlet and Kondo singlet formation reduces the series conductance to $\sim
0.4e^{2}/h$. CB hexagons where one or both dots are $S=0$ give extremely low
conductance, $G_{C}\sim 0$. The periodic structure of the diagram owes to the
multilevel nature of the dots; distortions to the pattern at high $|\delta
V_{g\alpha}|$ are a physical feature of systems with a finite number of dot
levels (in this case, $M=3$ per dot) since particle and hole excitations are
asymmetrical.

##  X Conclusion

In this paper we reviewed and developed the theory of mesoscopic quantum
transport through interacting nanostructures in the linear response regime. We
presented new numerical techniques as well as analytical results which help
overcome the limitations of conventional methodologies. This opens the door to
a more systematic and rigorous theoretical treatment of quantum transport,
bringing more complex systems within reach. In particular, our approach –
which treats orbital complexity, strong electron interactions, and nontrivial
lead hybridization on an equal footing – is essential for the realistic
simulation of generic nanoelectronics devices and to make quantitative contact
with experiments.

In linear response, the Kubo formula[74, 75] is arguably the most general and
powerful formulation, but it can be challenging to extract accurate numerical
results from it in practice, and it offers little analytical insight into
underlying transport mechanisms. In Sec. IV we derive an improved formulation
of the Kubo formula for electrical conductance for use with NRG, demonstrating
its practical advantages over the conventional implementation.

The Meir-Wingreen approach[11] is also rather general, but only takes a
convenient form in terms of equilibrium correlation functions in the linear-
response limit when the nanostructure-lead coupling geometry satisfies the
‘proportionate coupling’ condition. This turns out to be highly restrictive
for realistic multi-orbital systems where source and drain leads are spatially
separated (and the basic Landauer formula[36, *Landauer1970,
*LandauerButtiker1985, *Buttiker1986] is inapplicable due to the presence of
electron interactions).

Instead, in Secs. VI and VII we introduce and exploit the concept of ‘emergent
proportionate coupling’, in which an effective combination of lead states
decouples low temperatures, due to the renormalization from electron
interactions. Despite the irreducible two-channel form of the bare model,
under RG the system flows to an effective single-channel strong coupling
description. This enables us to express low-temperature quantum transport
coefficients directly in terms of effective model parameters. In addition to
facilitating conductance calculations, the approach also provides physical
insights into the underlying transport mechanisms at low temperatures.

We also derive connections between the various standard quantum transport
formulations, in cases where equivalences can be made. In particular, we
analyze the Kubo formula in the proportionate coupling limit, to obtain a
generalized Landauer form valid in the ac frequency regime, and for structured
(energy-dependent) leads in a multi-terminal setup (see Appendixes).

Finally, in Sec. IX, we compare and contrast the various techniques discussed
above, benchmarking against known results where possible. We do this for two
nontrivial systems: the two-terminal double and triple quantum dot models. Our
results and analysis indicate that numerical calculations for such systems are
under control, and that simple analytical predictions, although approximate,
are nevertheless sufficiently accurate to capture the essential physics of
interest in appropriate limits.

Finally, we emphasize that the results presented in this paper pertain to the
linear-response regime. The full non-equilibrium problem at finite voltage
bias, and/or with a finite temperature gradient between leads, is highly
challenging. In many interacting, multi-orbital systems, non-linear transport
calculations remain an open problem.

###### Acknowledgements.

We are grateful to S. Sen, P. Wong, D. Goldhaber-Gordon, W. Pouse, for helpful
discussions. We acknowledge funding from the Irish Research Council through
the Laureate Award 2017/2018 grant IRCLA/2017/169 (AKM/JBR) and the Enterprise
Partnership Scheme grant EPSPG/2017/343 (ELM).

##  Appendix A Quantum transport with
structured leads

The formalism presented in Sec. III assumed that the leads were metallic, with
a constant density of states $\rho_{0}$ inside a wide band of half-width
$D$.[81] A more general formulation takes into account an arbitrary lead
density of states $\rho(\omega)$. There are circumstances where this can
become important; for example when $\rho(\omega)\sim|\omega|^{r}$ has a low-
energy pseudogap [156, 157] as with graphene leads, whence
$\rho_{0}\equiv\rho(\omega=0)$ is nominally zero.

The Kubo formulae Eqs. 36 and 41 remain unchanged, but the correlation
functions $K(\omega,T)$ and $K^{\prime}(\omega,T)$ must now be calculated in
the presence of the structured leads.

For equivalent leads with proportionate coupling, the MW ac electrical
conductance formula Eq. 39 becomes,

$\begin{split}G_{C}(\omega,T)=\left(\frac{e^{2}}{h}\right)\frac{4\pi
V_{d}^{2}V_{s}^{2}}{V_{d}^{2}+V_{s}^{2}}\sum_{\sigma}\int_{-\infty}^{\infty}d\omega^{\prime}\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega^{\prime},T)&\\\
\times\frac{1}{2\omega}\Big{[}\rho(\omega^{\prime}+\omega)\\{f_{\rm
eq}(\omega^{\prime}+\omega)-f_{\rm eq}(\omega^{\prime})&\\}\\\
-\rho(\omega^{\prime}-\omega)\\{f_{\rm eq}(\omega^{\prime}-\omega)-f_{\rm
eq}(\omega^{\prime})&\\}\Big{]}\end{split}$ (104)

which in the dc limit $\omega\to 0$ reduces to,

$\begin{split}G_{C}(T)=\left(\frac{e^{2}}{h}\right)\frac{4\pi
V_{d}^{2}V_{s}^{2}}{V_{d}^{2}+V_{s}^{2}}\sum_{\sigma}\int_{-\infty}^{\infty}d\omega^{\prime}\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega^{\prime},T)&\\\
\times\rho(\omega^{\prime})\partial_{\omega^{\prime}}f_{\rm
eq}(\omega^{\prime})&\;.\end{split}$ (105)

For the heat conductance, we similarly obtain,

$\begin{split}K_{Q}(T)=\left(\frac{1}{hT}\right)\frac{4\pi
V_{d}^{2}V_{s}^{2}}{V_{d}^{2}+V_{s}^{2}}\sum_{\sigma}\int_{-\infty}^{\infty}d\omega^{\prime}\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega^{\prime},T)&\\\ \times\omega^{\prime
2}\rho(\omega^{\prime})\partial_{\omega^{\prime}}f_{\rm
eq}(\omega^{\prime})&\;.\end{split}$ (106)

Note in particular that Eqs. 105 and 106 are required in the pseudogap case,
since the quantity $\rho(\omega){\rm Im}G_{\sigma}(\omega)$ appearing under
the integral is subject to a generalized Friedel sum rule.[51] In particular,
$\rho(\omega){\rm Im}\leavevmode\nobreak\ G_{\sigma}(\omega)$ may remain
finite as $\omega\to 0$ in Kondo phases, even though $\rho(\omega\to 0)\to 0$.

##  Appendix B Multi-terminal transport

We consider here briefly the situation for interacting nanostructures coupled
to more than two leads. The system Hamiltonian is still given by Eqs. 1–3, but
with the lead index now $\alpha=1,2,3,...$ rather than just $s$ and $d$.

Focusing on charge transport in linear response, we are interested in elements
of the conductance tensor,

$\displaystyle
G_{C}^{\alpha\beta}=\frac{dI_{C}^{\alpha}}{dV_{b}^{\beta}}\bigg{|}_{V_{b}^{\beta}\to
0}$ (107)

where, as before, $I^{\alpha}=-e\langle\dot{N}^{\alpha}\rangle$ is the
electrical current into lead $\alpha$ and $V_{b}^{\beta}=\mu_{\beta}/e$ is the
voltage bias applied to lead $\beta$. The corresponding dc Kubo formula reads,

$G_{C}^{\alpha\beta}(T)=\frac{e^{2}}{h}\lim_{\omega\to
0}\left[\frac{-2\pi\leavevmode\nobreak\ {\rm
Im}K_{\alpha\beta}(\omega,T)}{\omega}\right]\;,$ (108)

where the current-current correlator
$K_{\alpha\beta}(\omega)=\langle\langle\dot{N}^{\alpha};\dot{N}^{\beta}\rangle\rangle$
can be evaluated at any temperature $T$. This formulation is totally general
and holds for any interacting or non-interacting system with any number of
leads (which can also be structured and inequivalent).

For equivalent leads and a nanostructure in PC, further simplifications can be
made. Eq. 108 can then be cast in the Landauer-Büttiker form. The first step
to show this is a rotation of the lead basis,
$\vec{f}_{\sigma}=\bm{U}\vec{c}_{\sigma}$, where
$\vec{c}_{\sigma}^{\leavevmode\nobreak\
\dagger}=(c_{1\sigma}^{\dagger},c_{2\sigma}^{\dagger},c_{3\sigma}^{\dagger},...)$
and similarly for $\vec{f}_{\sigma}^{\leavevmode\nobreak\ \dagger}$, and
$\bm{U}$ is unitary. We define
$f_{1\sigma}=\sum_{\gamma}U_{1\gamma}c_{\gamma\sigma}$ specifically such that
$U_{1\gamma}=V_{\gamma}/V$ where $V_{\gamma}$ are the nanostructure-lead
couplings in Eq. 5 and $V^{2}=\sum_{\gamma}|V_{\gamma}|^{2}$. The
nanostructure therefore couples only to the single transformed lead orbital
$f_{1\sigma}$.

Note $\tilde{\mathcal{G}}_{\alpha\beta}^{0}(\omega)\equiv\langle\langle
f_{\alpha\sigma}^{\phantom{\dagger}};f_{\beta\sigma}^{\dagger}\rangle\rangle^{0}=\sum_{\gamma}U_{\alpha\gamma}^{\phantom{*}}U_{\beta\gamma}^{*}\mathcal{G}_{\gamma\gamma}^{0}(\omega)$
is only diagonal if the original leads are equivalent ($-\tfrac{1}{\pi}{\rm
Im}\mathcal{G}_{\gamma\gamma}^{0}(\omega)=\rho(\omega)$ independent of
$\gamma$). In this case, $-\tfrac{1}{\pi}{\rm
Im}\tilde{\mathcal{G}}_{\alpha\beta}^{0}(\omega)=\delta_{\alpha\beta}\rho(\omega)$,
meaning that the subsystem containing the nanostructure and the lead $f_{1}$
is strictly decoupled from the other orthogonal lead combinations $f_{2}$,
$f_{3}$, and so on.

For concreteness, consider now $G_{C}^{12}$, for which we require
$K_{12}(\omega)=\langle\langle\dot{N}^{1};\dot{N}^{2}\rangle\rangle$. Current
conservation implies $\sum_{\gamma}\dot{N}^{\gamma}=0$. Furthermore, from Eq.
5, we may write $\dot{N}^{\gamma}\equiv
i[\hat{H},\hat{N}^{\gamma}]=i\sum_{\sigma}(V_{\gamma}^{\phantom{\dagger}}\bar{\bar{d}}_{\sigma}^{\dagger}c_{\gamma\sigma}^{\phantom{\dagger}}-{\rm
H.c.})$. This allows us to express
$\dot{N}^{1}\equiv\dot{N}^{1}+a\sum_{\gamma}\dot{N}^{\gamma}=i\sum_{\sigma}(\tilde{V}_{12}^{\phantom{\dagger}}\bar{\bar{d}}_{\sigma}^{\dagger}f_{2\sigma}^{\phantom{\dagger}}-{\rm
H.c.})$, with $f_{2\sigma}=\sum_{\gamma}U_{2\gamma}c_{\gamma\sigma}$
constructed to be orthogonal to $f_{1\sigma}$ via judicious choice of the
constant $a$. With $f_{1\sigma}$ and $f_{2\sigma}$ so defined, $\dot{N}^{2}$
can now be expanded in the basis of the $f$ orbitals. We again exploit current
conservation to eliminate the $f_{1\sigma}$ orbital, which yields
$\dot{N}^{2}\equiv\dot{N}^{2}+b\sum_{\gamma}\dot{N}^{\gamma}=i\sum_{\eta}^{\prime}\sum_{\sigma}(\tilde{V}_{2\eta}^{\phantom{\dagger}}\bar{\bar{d}}_{\sigma}^{\dagger}f_{\eta\sigma}^{\phantom{\dagger}}-{\rm
H.c.})$ where the primed sum excludes $\eta=1$.

Since $f_{2\sigma}$, $f_{3\sigma}$, … are decoupled from each other as well as
from $\bar{\bar{d}}_{\sigma}$, we may factorize the correlator $K_{12}(t)$
into two pieces corresponding to single-particle Green’s functions for the
nanostructure and the $f_{2}$ lead respectively. In the frequency domain this
results in a convolution. After some manipulation, one finds,

$\begin{split}{\rm
Im}K_{12}(\omega)=\tilde{V}_{12}\tilde{V}_{22}\sum_{\sigma}\int_{-\infty}^{\infty}d\omega^{\prime}\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega^{\prime},T)&\\\
\times\Big{[}\rho(\omega^{\prime}+\omega)\\{f_{\rm
eq}(\omega^{\prime}+\omega)-f_{\rm eq}(\omega^{\prime})&\\}\\\
-\rho(\omega^{\prime}-\omega)\\{f_{\rm eq}(\omega^{\prime}-\omega)-f_{\rm
eq}(\omega^{\prime})&\\}\Big{]}\end{split}$ (109)

where $\tilde{V}_{12}\tilde{V}_{22}=-V_{1}^{2}V_{2}^{2}/V^{2}$ follows
directly from the above prescription together with the orthonormality
condition $\bm{U}\bm{U}^{\dagger}=\bm{I}$.

Substituting Eq. 109 into Eq. 108 and taking the $\omega\to 0$ limit yields
the dc electrical conductance for the multi-terminal setup,

$\begin{split}G_{C}^{\alpha\neq\beta}(T)=\left(\frac{e^{2}}{h}\right)\frac{4\pi
V_{\alpha}^{2}V_{\beta}^{2}}{V^{2}}\sum_{\sigma}\int_{-\infty}^{\infty}&d\omega^{\prime}\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega^{\prime},T)\\\
&\times\rho(\omega^{\prime})\partial_{\omega^{\prime}}f_{\rm
eq}(\omega^{\prime})\;.\end{split}$ (110)

which holds for interacting nanostructures coupled to an arbitrary number of
equivalent structured leads. By current conservation, the diagonal components
of the conductance tensor follow as
$G_{C}^{\alpha\alpha}(T)=-\sum_{\beta\neq\alpha}G_{C}^{\alpha\beta}(T)$.

##  Appendix C Inequivalent leads

For systems in which the leads are inequivalent (meaning that the density of
states $\rho_{\alpha}(\omega)$ is not the same for all leads $\alpha$), the PC
condition cannot be satisfied. The strategy employed in the previous
appendices to decouple orthogonal lead combinations cannot therefore be used
in such cases. The Kubo formula Eq. 36, with the current-current correlator
$K(\omega,T)$ given by Eq. 38, can still be used – but for generic interacting
nanostructures, it cannot be cast in the form of local retarded Green’s
functions for the nanostructure.

However, for non-interacting systems, an approximate generalization of the
Landauer formula can be used. We write Eqs. 16, 17 but with a transmission
function that now involves the full energy-dependent hybridizations,

$\displaystyle\mathcal{T}_{\gamma\beta}(\omega)$
$\displaystyle=4\sum_{\sigma}{\rm
Tr}[\bm{G}_{\sigma}^{(0)}(\omega)\bm{\Gamma}^{\gamma}(\omega)\bm{G}_{\sigma}^{(0)}(\omega)^{*}\bm{\Gamma}^{\beta}(\omega)]$
(111a) $\displaystyle\equiv
4\Gamma_{\gamma}(\omega)\Gamma_{\beta}(\omega)\sum_{\sigma}\overline{G}^{(0)}_{\beta\gamma,\sigma}(\omega)\overline{G}^{(0)}_{\gamma\beta,\sigma}(\omega)^{*}\;,$
(111b)

where $\bm{\Gamma}_{\alpha}(\omega)=-{\rm Im}\bm{\Delta}_{\alpha}(\omega)$ and
$\Gamma_{\alpha}(\omega)=\pi|V_{\alpha}|^{2}\rho_{\alpha}(\omega)$. In the
standard two-terminal setup,
$\mathcal{T}(\omega)\equiv\mathcal{T}_{sd}(\omega)$.

We speculate that this result is in fact exact, and is derivable from the Kubo
formula by utilizing Wick’s theorem to factorize the two-particle
susceptibility $K(t)$ into single-particle retarded Green’s functions for the
nanostructure and for the leads.[71] We emphasize that this can only be done
for non-interacting systems since the decoupling step relies here on Wick’s
theorem rather than a canonical transformation of the lead basis which is only
possible for PC systems.

The generalization of Landauer-Büttiker to multi-terminal transport with
inequivalent leads is given by Eq. 90, with
$\mathcal{T}_{\gamma\beta}(\omega)$ given above in Eq. 111.

##  Appendix D Perturbative correction to non-equilibrium distribution for the
AIM

The general form of the MW formula (Eqs. 25, 26), applicable also for
interacting non-PC systems, requires the lesser Green’s function
$\bm{G}_{\sigma}^{<}$. The latter is related to the lesser self-energy via the
Keldysh equation
$\bm{G}^{<}_{\sigma}=\bm{G}_{\sigma}^{\phantom{<}}\bm{\Sigma}_{\sigma}^{<}\bm{G}_{\sigma}^{*}$.[64,
*LangrethLinearRespNEq1976] One may then express the lesser interaction self-
energy in terms of its retarded counterpart through the nonequilibrium
distribution function, $\bm{\Sigma}^{<}_{\sigma;{\rm
int}}=-2i\bm{f}_{\sigma;{\rm int}}^{\rm NE}\leavevmode\nobreak\ {\rm
Im}\bm{\Sigma}_{\sigma}$. Of course, $\bm{f}_{\sigma;{\rm int}}^{\rm NE}$ is
in general not known.

The Ng Ansatz[66, *Dong2002] amounts to a simple approximation to this unknown
function, $\bm{f}^{\rm NE}_{\sigma;{\rm int}}\to\bm{f}^{\rm
NE}_{0}=[f_{s}\bm{\Gamma}^{s}+f_{d}\bm{\Gamma}^{d}][\bm{\Gamma}^{s}+\bm{\Gamma}^{d}+2\bm{{\rm
I}}0^{+}]^{-1}$. The latter is the pseudo-equilibrium distribution[13] for a
non-interacting system out of equilibrium. For the AIM Eq. 8, the Ng Ansatz
corresponds to $f_{\sigma;{\rm int}}^{\rm
NE}\to[f_{s}\Gamma_{s}+f_{d}\Gamma_{d}]/(\Gamma_{s}+\Gamma_{d})$. Assuming a
symmetric voltage bias split $\pm\tfrac{1}{2}\Delta V_{b}$ across source and
drain leads, this becomes,

$f_{\rm Ng}(\omega)=f_{\rm eq}(\omega)+\tfrac{1}{2}e\Delta
V_{b}[-\partial_{\omega}f_{\rm
eq}(\omega)]\frac{\Gamma_{s}-\Gamma_{d}}{\Gamma_{s}+\Gamma_{d}}+\mathcal{O}(\Delta
V_{b}^{2})\;,$ (112)

where we have expanded to linear-order in the bias, as required to determine
the linear-response conductance via the MW formula.

Since the above is exact for $U_{d}=0$, a natural question pertains to
perturbative corrections in $U_{d}$. To that end, we compute $f_{\sigma;{\rm
int}}^{\rm NE}=i\Sigma^{<}_{\sigma;{\rm int}}/2{\rm Im}\Sigma_{\sigma}$ for
the AIM up to second-order in the interaction $U_{d}$ using nonequilibrium
Kadanoff-Keldysh perturbation theory.[64, *LangrethLinearRespNEq1976] We then
expand the result to linear order in the bias $\Delta V_{b}$ and compare to
Eq. 112. After some tedious calculations we obtain,[71]

$\begin{split}f_{\sigma;{\rm int}}^{\rm NE}(\omega)=f_{\rm
eq}(\omega)+\tfrac{1}{2}e\Delta V_{b}[-\partial_{\omega}f_{\rm
eq}(\omega)]\frac{\Gamma_{s}-\Gamma_{d}}{\Gamma_{s}+\Gamma_{d}}&\Big{[}1+U_{d}^{2}h_{\sigma}(\omega)\Big{]}\\\
+&\mathcal{O}(\Delta V_{b}^{2},U_{d}^{3})\;,\end{split}$ (113)

where the correction is controlled by the function,

$h_{\sigma}(\omega)=\frac{\pi}{\Gamma_{s}+\Gamma_{d}}\int d\omega_{1}\int
d\omega_{2}\leavevmode\nobreak\
\bar{f}(\omega,\omega_{1},\omega_{2})\times\bar{A}_{\sigma}(\omega,\omega_{1},\omega_{2})\;,$
(114)

with $\bar{f}(\omega,\omega_{1},\omega_{2})=f_{\rm
eq}(\omega_{1}+\omega_{2}-\omega)[f_{\rm eq}(\omega_{1})+f_{\rm
eq}(\omega_{2})-1]-f_{\rm eq}(\omega_{1})f_{\rm eq}(\omega_{2})$ and
$\bar{A}_{\sigma}(\omega,\omega_{1},\omega_{2})=\mathcal{A}^{0}_{\sigma}(\omega_{1})\mathcal{A}^{0}_{\bar{\sigma}}(\omega_{2})\mathcal{A}^{0}_{\bar{\sigma}}(\omega_{1}+\omega_{2}-\omega)$
with the latter given in terms of the non-interacting impurity spectral
function,
$\mathcal{A}^{0}_{\sigma}(\omega)=\tfrac{1}{\pi\Gamma}\times[1+(\omega-\epsilon_{d\sigma})^{2}/\Gamma^{2}]^{-1}$.

We therefore find a non-zero correction at order $U_{d}^{2}\Delta
V_{b}^{\phantom{2}}$, which contributes to the linear-response conductance. Of
course, this analysis is not really needed for the AIM itself, since the PC
formulation Eq. 28 here applies. However, we provide the above discussion to
illustrate a general strategy for systematically improving upon the Ng Ansatz
for non-PC systems, by including the effect of interactions perturbatively.

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