# Linear response quantum transport through interacting multi-orbital
nanostructures

Emma L. Minarelli  School of Physics, University College Dublin, Belfield,
Dublin 4, Ireland  Centre for Quantum Engineering, Science, and Technology,
University College Dublin, Ireland  Jonas B. Rigo  School of Physics,
University College Dublin, Belfield, Dublin 4, Ireland  Centre for Quantum
Engineering, Science, and Technology, University College Dublin, Ireland
Andrew K. Mitchell  School of Physics, University College Dublin, Belfield,
Dublin 4, Ireland  Centre for Quantum Engineering, Science, and Technology,
University College Dublin, Ireland

###### Abstract

Nanoelectronics devices, such as quantum dot systems or single-molecule
transistors, consist of a quantum nanostructure coupled to a macroscopic
external electronic circuit. Thermoelectric transport between source and drain
leads is controlled by the quantum dynamics of the lead-coupled nanostructure,
through which a current must pass. Strong electron interactions due to quantum
confinement on the nanostructure produce nontrivial conductance signatures
such as Coulomb blockade and Kondo effects, which become especially pronounced
at low temperatures. In this work we first provide a modern review of standard
quantum transport techniques, focusing on the linear response regime, and
highlight the strengths and limitations of each. In the second part, we
develop an improved numerical scheme for calculation of the ac linear
electrical conductance through generic interacting nanostructures, based on
the numerical renormalization group (NRG) method, and explicitly demonstrate
its utility in terms of accuracy and efficiency. In the third part we derive
low-energy effective models valid in various commonly-encountered situations,
and from them we obtain simple analytical expressions for the low-temperature
conductance. This indirect route via effective models, although approximate,
allows certain limitations of conventional methodologies to be overcome, and
provides physical insights into transport mechanisms. Finally, we apply and
compare the various techniques, taking the two-terminal triple quantum dot and
the serial multi-level double dot devices as nontrivial benchmark systems.

##  I Introduction

The theory of mesoscopic quantum transport[1, 2, 3, 4] has gained importance
in recent years due to the development of advanced nanoelectronics devices,
such as semiconductor quantum dots[5] or single-molecule transistors.[6]
Cryogenic quantum devices can now even be fabricated in commercial CMOS
technology.[7] These systems typically consist of a nanostructure coupled to
source and drain leads, and are tunable in-situ by applying gate voltages. The
charge and/or heat current that flows between the macroscopic leads due to an
applied voltage bias and/or temperature gradient is controlled by the quantum
dynamics of the nanostructure. The setup is illustrated in Fig. 1. A deep
understanding of the underlying theory of quantum transport in such systems is
a pre-requisite for the development and design of devices underpinning
possible future quantum technologies.

To theoretically capture the diverse range of complex physics that can be
realized in quantum nanoelectronics devices, and to accurately predict the
resulting transport signatures, three steps must be undertaken. First, an
accurate microscopic model of the system must be formulated, taking into
account both the interacting, multi-orbital nanostructure and the macroscopic
metallic leads – as well as the coupling between them. Secondly, this model
must be solved. Typically, this must be done numerically using sophisticated
quantum many-body computational techniques. Often, the bare microscopic model
is too complex to be treated directly, and simplified effective models that
capture the physics of interest must be derived. Finally, the desired physical
observables (in this case quantum transport coefficients) must be obtained
from the solution of the model. Standard techniques for the latter require the
calculation of dynamical correlation functions for the interacting, lead-
coupled nanostructure – a formidable task in itself. Furthermore, these
standard techniques each have strengths and limitations and there is no all-
purpose black-box method.

In this paper we focus on quantum transport through interacting multi-orbital
nanostructures in the linear response regime, and discuss each of the above
three steps in detail.

Figure 1: Schematic of the basic quantum transport setup: a voltage bias
and/or temperature gradient between source and drain leads results in current
flow through the nanostructure.

The paper is organized as follows. In Sec. II we introduce and discuss in
detail the microscopic models describing nanoelectronics systems. We focus on
two-terminal setups, with non-interacting metallic leads connected in an
arbitrary way to an interacting, multi-orbital nanostructure (the
generalization to structured leads in a multi-terminal setup is discussed in
the Appendixes). We discuss two important and commonly-used limiting cases:
the proportionate coupling hybridization geometry (Sec. II.1) and the single-
orbital nanostructure (Sec. II.2). We emphasize that these limits are
typically not satisfied for realistic systems, and our main results do _not_
rely on either condition being met. Secs. II.3 and II.4 formalize the
dynamical quantities and quantum transport coefficients considered in this
paper, while Sec. II.5 briefly summarizes the NRG method employed for the
numerical calculations.

In Sec. III we give a modern review of various standard approaches to the
calculation of thermoelectric quantum transport coefficients. We relate the
different techniques in the appropriate limits where equivalences can be made,
and emphasize the benefits and limitations of each. We provide generalizations
of the conventional formulae where possible. We consider the Landauer approach
(Sec. III.1), the renormalized Landauer method (Sec. III.2), the Meir-Wingreen
formula (Sec. III.3), and the Kubo formula (Sec. III.4) as specific paradigms.

In Sec. IV we introduce an “improved” version of the Kubo formula for the ac
linear electrical conductance for use with NRG, demonstrating via explicit
benchmark calculations the benefit in terms of accuracy and efficiency over
the conventional Kubo approach. By contrast, in Sec. V we see that the Kubo
formula for heat conductance in NRG breaks down.

In Sec. VI we derive low-energy effective models valid for nanostructures in
the Coulomb-blockade regime (and consider separately the situations arising
for ground states with different spin). These effective models are analyzed in
terms of their RG flow, and the “emergent proportionate coupling” concept is
introduced, in which an effective single-channel description pertains at low
temperatures. This allows us to obtain analytic results for the low-
temperature conductance, directly in terms of the effective model parameters.
Analytical results are supported by numerical calculations. In Sec. VII we do
this instead near the charge degeneracy points separating two Coulomb blockade
diamonds. Depending on the spin states of the two degenerate sectors,
different effective models result (and low-$T$ transport is in consequence
found to be markedly affected).

In Sec. VIII we apply the recently-developed ‘auxiliary field’ mapping for
interacting quantum impurity-type models to derive a novel approximation for
conductance calculations. The approach exploits an equivalent non-interacting
formulation of an interacting model, so that the Landauer-Büttiker formula can
be employed.

Finally, in Sec. IX we apply the techniques discussed above to two nontrivial
quantum dot models (the triangular triple quantum dot system in both one- and
two-channel geometries; and the multi-level double quantum dot system in a
serial geometry). We compare and contrast the numerical methods and benchmark
the analytical approximations. A conclusion and outlook is provided in Sec. X,
while various derivations are given in the Appendixes.

##  II Models of nanodevices

We consider systems described by the generic Hamiltonian $H=H_{\rm
leads}+H_{\rm nano}+H_{\rm hyb}+H_{\rm gate}+H_{\rm field}$. Here,

$\displaystyle H_{\rm leads}=\sum_{\alpha}H_{\rm
leads}^{\alpha}=\sum_{\alpha,k,\sigma}\epsilon_{k}^{\phantom{\dagger}}c_{\alpha
k\sigma}^{\dagger}c_{\alpha k\sigma}^{\phantom{\dagger}}\;,$ (1)

describes source ($\alpha=s$) and drain ($\alpha=d$) leads as non-interacting
fermionic reservoirs at thermal equilibrium, with
$f_{\alpha}(\omega)=(1+\exp[(\omega-\mu_{\alpha})/k_{\rm B}T_{\alpha}])^{-1}$
the Fermi-Dirac distribution of lead $\alpha$ held at chemical potential
$\mu_{\alpha}$ and temperature $T_{\alpha}$. An electron with momentum $k$ and
spin $\sigma=\uparrow$ or $\downarrow$ is created in lead $\alpha$ by
$c_{\alpha k\sigma}^{\dagger}$, and $c_{\alpha k\sigma}$ is the corresponding
annihilation operator. We take the dispersion $\epsilon_{k}$ to be independent
of lead $\alpha$ and spin $\sigma$.

The isolated nanostructure is modelled by,

$\displaystyle H_{\rm
nano}=\sum_{i,j,\sigma}t_{ij}d^{\dagger}_{i\sigma}d^{\phantom{\dagger}}_{j\sigma}+H_{\rm
int}\;,$ (2)

where $t_{ij}=t_{ji}^{*}$ are elements of the single-particle matrix
$\textbf{H}_{\rm nano}^{(0)}$, and $H_{\rm int}$ describes electron-electron
interactions on the nanostructure. Here $d^{\dagger}_{i\sigma}$
($d_{i\sigma}$) creates (annihilates) a spin-$\sigma$ electron in orbital $i$
of the nanostructure, while
$\hat{n}_{i\sigma}=d^{\dagger}_{i\sigma}d^{\phantom{\dagger}}_{i\sigma}$ and
$\hat{n}_{i}=\sum_{\sigma}\hat{n}_{i\sigma}$ are number operators. In Sec.
IX.1 we study a triple quantum dot model of this form.

The tunnel-coupling between nanostructure and leads is described by the
hybridization Hamiltonian, $H_{\rm
hyb}=\sum_{\alpha,k,i,\sigma}(V_{\alpha,ik}d^{\dagger}_{i\sigma}c_{\alpha
k\sigma}^{\phantom{\dagger}}+\text{H.c.})$. Hereafter we shall take a common
simplification of this expression to the _two-channel_ limit in which the
tunneling matrix elements can be factorized as
$V_{\alpha,ik}=V_{\alpha,i}U_{\alpha,k}$ with $\sum_{k}|U_{\alpha,k}|^{2}=1$.
In this case, we define real-space lead orbitals localized at the
nanostructure position, given by
$c_{\alpha\sigma}=\sum_{k}U_{\alpha,k}c_{\alpha k\sigma}^{\phantom{\dagger}}$.
The hybridization Hamiltonian then becomes,

$\displaystyle H_{\rm
hyb}=\sum_{\alpha,i,\sigma}\left(V_{\alpha,i}d^{\dagger}_{i\sigma}c_{\alpha\sigma}^{\phantom{\dagger}}+\text{H.c.}\right)\;.$
(3)

This two-channel limit is a very good approximation when the dominant
nanostructure-lead hybridization is localized, and may be justified by the
contacting geometry of the device.

We can now identify the local retarded Green’s function of the _uncoupled_
leads at the nanostructure position. It is given by
$\mathcal{G}_{\alpha\beta}^{0}(\omega)\equiv\langle\langle
c_{\alpha\sigma}^{\phantom{\dagger}};c_{\beta\sigma}^{\dagger}\rangle\rangle^{0}_{\omega}=\delta_{\alpha\beta}\sum_{k}|U_{\alpha,k}|^{2}(\omega^{+}-\epsilon_{k})^{-1}$,
where $\omega^{+}=\omega+i0^{+}$. We take for simplicity a flat density of
states in equivalent leads,
$\rho_{\alpha}(\omega)=-\tfrac{1}{\pi}\text{Im}\leavevmode\nobreak\
\mathcal{G}_{\alpha\alpha}^{0}(\omega)=\rho_{0}\Theta(D-|\omega|)$ independent
$\alpha$, inside a band of half-width $D$ and with $\rho_{0}=1/(2D)$.

The ‘frontier orbitals’ of the nanostructure hybridizing with the leads can
also be identified as

$\bar{d}_{\alpha\sigma}^{\dagger}=\tfrac{1}{V_{\alpha}}\sum_{i}V_{\alpha,i}d_{i\sigma}^{\dagger}\;,$
(4)

with $V_{\alpha}^{2}=\sum_{i}|V_{\alpha,i}|^{2}$, which yields, $H_{\rm
hyb}=\sum_{\alpha,\sigma}(V_{\alpha}\bar{d}^{\dagger}_{\alpha\sigma}c_{\alpha\sigma}^{\phantom{\dagger}}+\text{H.c.})$.
However, note that in general
$\\{\bar{d}_{s\sigma}^{\dagger},\bar{d}^{\phantom{\dagger}}_{d\sigma}\\}\neq
0$ due to overlap between the frontier orbitals coupling to source and drain
leads, and hence these operators are not proper canonical fermions (for
example $\bar{d}_{s\sigma}=\bar{d}_{d\sigma}\equiv d_{\sigma}$ in the case of
the single-orbital nanostructure[8]).

The quantum dynamics of the lead-coupled nanostructure depend on the complex
hybridization matrices $\bm{\Delta}_{\alpha}(\omega)$, whose elements are the
individual hybridization functions
$\Delta_{\alpha,ij}(\omega)=V_{\alpha,i}(V_{\alpha,j})^{*}\mathcal{G}_{\alpha\alpha}^{0}(\omega)$.
The low-energy physics of such nanoelectronics devices,[9] including the low-
temperature transport properties, are controlled by the low-frequency form of
$\bm{\Delta}_{\alpha}(\omega)$. We therefore introduce the matrix
$\bm{\Gamma}_{\alpha}=-\text{Im}\leavevmode\nobreak\
\bm{\Delta}_{\alpha}(\omega=0)$, with elements
$\Gamma_{\alpha,ij}=\pi\rho_{0}V_{\alpha,i}(V_{\alpha,j})^{*}$, and we define
$\Gamma_{\alpha}=\pi\rho_{0}|V_{\alpha}|^{2}$.

Physical properties of the nanodevice so defined may be tuned in-situ by
application of suitable gate voltages. These may provide experimental control
[10, 5] over the tunneling matrix elements $V_{\alpha,i}$ entering in the
hybridization, or allow to manipulate the nanostructure single-particle level
energies through a term $H_{\rm gate}=eV_{\rm g}\sum_{i}\hat{n}_{i}$.
Transport properties are also strongly affected by the application of a local
magnetic field, modelled by the term $H_{\rm field}$.

###  II.1 Proportionate Coupling

A system is said to be under ‘proportionate coupling’ (PC) if
$\Gamma_{d,ij}=\lambda\Gamma_{s,ij}$ (or equivalently
$V_{d,i}=\sqrt{\lambda}V_{s,i}$), with $\lambda$ some real constant,
independent of the nanostructure level indices $i$ and $j$. [11,
*MeirWingreen_AMoutEquilib_PRL1993, 13] Evidently, this is not the generic
scenario for a multi-orbital nanostructure with spatially separated leads.

A system under PC simplifies to an effective _single-channel_ description,
with

$\displaystyle H_{\rm hyb}\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\
\sum_{i,\sigma}\left(V_{i}^{\phantom{\dagger}}d^{\dagger}_{i\sigma}c_{e\sigma}^{\phantom{\dagger}}+\text{H.c.}\right)$
(5)

where only the ‘even’ combination of source and drain leads couples to the
nanostructure, with
$c_{e\sigma}=\tfrac{1}{\sqrt{1+\lambda}}(c_{s\sigma}+\sqrt{\lambda}c_{d\sigma})$
and $V_{i}=V_{s,i}\sqrt{1+\lambda}$. The ‘odd’ combination of leads, defined
by
$c_{o\sigma}=\tfrac{1}{\sqrt{1+\lambda}}(\sqrt{\lambda}c_{s\sigma}-c_{d\sigma})$,
is formally decoupled on the level of the bare Hamiltonian (we contrast with
with the ‘emergent decoupling’ of a channel in Secs. VI and VII). However,
note that quantum transport still proceeds in the physical real-space basis of
source and drain leads.

In this case, one can also define a single linear combination of nanostructure
operators, the ‘PC frontier orbital’ $\bar{\bar{d}}_{\sigma}^{\dagger}$, to
which the even channel couples, given by

$\bar{d}_{\alpha\sigma}^{\dagger}\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\
\bar{\bar{d}}_{\sigma}^{\dagger}=\tfrac{1}{V}\sum_{i}V_{i}d_{i\sigma}^{\dagger}\;,$
(6)

with $V^{2}=\sum_{i}|V_{i}|^{2}$ (and $\Gamma=\pi\rho_{0}V^{2}$). The
hybridization Hamiltonian then takes the same form as that of a single-
channel, single-impurity model,

$H_{\rm hyb}\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\
V\sum_{\sigma}(\bar{\bar{d}}_{\sigma}^{\dagger}c_{e\sigma}^{\phantom{\dagger}}+\rm{H.c.})$
(7)

However, this is somewhat deceptive, since the other nanostructure operators
$d_{i\sigma}^{\dagger}$ are still coupled to
$\bar{\bar{d}}_{\sigma}^{\dagger}$ via $H_{\rm nano}$. Therefore one cannot in
general determine the dynamics or transport properties of a multi-orbital
system from a pure single impurity Anderson model, despite the form of Eq. 5.

The PC concept generalizes to systems with 2 or more leads with arbitrary
densities of states, for which the condition on the hybridization matrices
reads $\bm{\Delta}_{\alpha}(\omega)=\lambda_{\alpha}\bm{\Delta}(\omega)$, with
$\lambda_{\alpha}$ a real scalar and $\bm{\Delta}(\omega)$ equal for all
leads. Note that the PC condition can be satisfied only with equivalent leads,
where $\rho_{\alpha}(\omega)\equiv\rho(\omega)$. The case of inequivalent
leads is considered in Appendix C.

###  II.2 Anderson impurity model

The simplest nontrivial interacting nanostructure model in terms of which
quantum transport can be studied, comprises a single interacting site tunnel-
coupled to two leads – the Anderson impurity model (AIM),[8] [14,
*Anderson_LocalisedMoments1978]

$H_{\rm AIM}=H_{\rm
leads}+\epsilon_{d}(\hat{n}_{\uparrow}+\hat{n}_{\downarrow})+U_{d}\hat{n}_{\uparrow}\hat{n}_{\downarrow}+\sum_{\alpha,\sigma}\left(V_{\alpha}^{\phantom{\dagger}}d^{\dagger}_{\sigma}c_{\alpha\sigma}^{\phantom{\dagger}}+\rm{H.c.}\right)$
(8)

where $\hat{n}_{\sigma}=d_{\sigma}^{\dagger}d_{\sigma}$ and $\alpha=s,d$. The
model automatically satisfies the PC condition, and so reduces to a single-
channel model,

$H_{\rm AIM}=H_{\rm
leads}^{e}+\epsilon_{d}(\hat{n}_{\uparrow}+\hat{n}_{\downarrow})+U_{d}\hat{n}_{\uparrow}\hat{n}_{\downarrow}+V\sum_{\sigma}\left(d^{\dagger}_{\sigma}c_{e\sigma}^{\phantom{\dagger}}+\rm{H.c.}\right)$
(9)

where $H_{\rm leads}^{e}$ describes the even lead, coupled to the impurity
site at the local orbital $c_{e\sigma}$, and $V^{2}=V_{s}^{2}+V_{d}^{2}$, such
that $\Gamma=\pi\rho_{0}V^{2}$ is the total hybridization. The focus of this
paper is however multi-orbital two-channel systems.

###  II.3 Green’s functions and t-matrix

The retarded Green’s functions for the interacting, lead-coupled nanostructure
at equilibrium in Zubarev [16] notation are specified by
$G_{ij,\sigma}(\omega,T)\equiv\langle\langle
d_{i\sigma}^{\phantom{\dagger}};d_{j\sigma}^{\dagger}\rangle\rangle_{\omega,T}$.
These are the Fourier transforms of the time-domain retarded correlators
$G_{ij,\sigma}(t,T)=-i\theta(t)\langle\\{d_{i\sigma}^{\phantom{\dagger}},d_{j\sigma}^{\dagger}(t)\\}\rangle_{T}$.
The corresponding non-interacting Green’s functions,
$G_{ij,\sigma}^{(0)}(\omega)$, are defined for $H_{\rm int}=0$ and are
temperature independent. These are related through the Dyson equation by the
interaction self-energy,

$\displaystyle[\bm{G}_{\sigma}(\omega,T)]^{-1}=[\bm{G}_{\sigma}^{(0)}(\omega)]^{-1}-\bm{\Sigma}_{\sigma}(\omega,T)\;,$
(10)

where
$[\bm{G}_{\sigma}^{(0)}(\omega)]^{-1}=(\omega^{+}-eV_{g}+B\sigma)\textbf{I}-\textbf{H}_{\rm
nano}^{(0)}-\sum_{\alpha}\bm{\Delta}_{\alpha}(\omega)$ with
$\sigma=\pm\tfrac{1}{2}$ for $\uparrow/\downarrow$.

The local t-matrix equation for the system is [8],

$\displaystyle\mathcal{G}_{\alpha\beta,\sigma}(\omega,T)=\mathcal{G}_{\alpha\beta}^{0}(\omega)+\mathcal{G}_{\alpha\alpha}^{0}(\omega)T_{\alpha\beta,\sigma}(\omega,T)\mathcal{G}_{\beta\beta}^{0}(\omega)\;,\;\;\;$
(11)

with the t-matrix itself defined as
$T_{\alpha\beta,\sigma}(\omega,T)=\sum_{i,j}(V_{\alpha,i})^{*}V_{\beta,j}G_{ij,\sigma}(\omega,T)\equiv
V_{\alpha}^{*}V_{\beta}\overline{G}_{\alpha\beta,\sigma}(\omega,T)$, where the
latter is given in terms of the frontier orbital Green’s function
$\overline{G}_{\alpha\beta,\sigma}=\langle\langle\bar{d}_{\alpha\sigma}^{\phantom{\dagger}};\bar{d}_{\beta\sigma}^{\dagger}\rangle\rangle$.

In proportionate coupling, this simplifies further to
$T_{\alpha\beta,\sigma}(\omega,T)=V_{\alpha}^{*}V_{\beta}\overline{\overline{G}}_{\sigma}(\omega,T)$
with
$\overline{\overline{G}}_{\sigma}=\langle\langle\bar{\bar{d}}_{\sigma}^{\phantom{\dagger}};\bar{\bar{d}}_{\sigma}^{\dagger}\rangle\rangle$.
We may also write a t-matrix equation in the even/odd channel basis,
$\mathcal{G}_{ee,\sigma}(\omega,T)=\mathcal{G}_{ee}^{0}(\omega)+[\mathcal{G}_{ee}^{0}(\omega)]^{2}T_{ee,\sigma}(\omega,T)$,
where
$T_{ee,\sigma}(\omega,T)=V^{2}\overline{\overline{G}}_{\sigma}(\omega,T)$ and
$T_{oo,\sigma}(\omega,T)=0$ in PC.

###  II.4 Observables

We consider the electrical and heat conductances due to a voltage bias $\Delta
V_{b}$ and/or temperature gradient $\Delta T$ between leads.

The electrical current $I_{C}^{\alpha}=-eI_{N}^{\alpha}$ into lead
$\alpha=s,d$ is given in terms of the particle current
$I_{N}^{\alpha}=\langle\dot{N}^{\alpha}\rangle$. Here,
$\dot{\Omega}=\tfrac{d}{dt}\hat{\Omega}$ denotes the time derivative of an
operator, and $\hat{N}^{\alpha}=\sum_{k,\sigma}c_{\alpha
k\sigma}^{\dagger}c_{\alpha k\sigma}^{\phantom{\dagger}}$ is the total number
operator for lead $\alpha$.

Similarly, the energy current is defined as
$I_{E}^{\alpha}=\langle\dot{H}_{\rm leads}^{\alpha}\rangle$, where the
Hamiltonian for lead $\alpha$ is given in Eq. 1. From the first law of
thermodynamics, $dE=dQ+\mu dN$, the heat current operator follows as
$I^{\alpha}_{Q}=I^{\alpha}_{E}-\mu_{\alpha}I^{\alpha}_{N}$, where
$\mu_{\alpha}$ is the chemical potential of lead $\alpha$. Note that all
thermal expectation values $\langle\dots\rangle$ must be calculated in the
full lead-coupled system.

Due to current conservation in the two-channel setup, $I^{d}=-I^{s}\equiv I$,
and so we drop the lead index $\alpha$ hereafter.

The bias voltage is related to a chemical potential difference between leads,
$e\Delta V_{b}=\mu_{s}-\mu_{d}$. For $\Delta V_{b}>0$, electrical current
flows from source to drain. In particular, an ac bias can be incorporated on
the level of the Hamiltonian by adding a time-dependent source term, $H\to
H+H_{\rm bias}(t)$ with,

$\displaystyle H_{\rm bias}(t)=-e\Delta V_{b}\cos(\omega t)\hat{N}_{s}\;$ (12)

where $\omega$ is the ac driving frequency. The dc limit is obtained as
$\omega\to 0$.

In contrast, the effect of a temperature gradient $\Delta T=T_{s}-T_{d}$
cannot be incorporated by adding a source term to the Hamiltonian. The
temperature of the leads enters through their respective distribution
functions $f_{\alpha}(\omega)$.

In this paper we will focus on the linear response regime, $\Delta V_{b}\to 0$
and $\Delta T\to 0$, in which case the nanostructure remains close to
equilibrium, and quantum transport coefficients can be calculated from
equilibrium correlation functions. We expand[17] the electrical and heat
currents to first order in $\Delta V_{b}$ and $\Delta T$ to obtain,

$\begin{pmatrix}I_{C}\\\
I_{Q}\end{pmatrix}=\begin{pmatrix}\chi_{CC}&\chi_{CQ}\\\
\chi_{QC}&\chi_{QQ}\end{pmatrix}\begin{pmatrix}\Delta V_{b}\\\ \Delta
T/T\end{pmatrix}\;,$ (13)

where the susceptibilities $\chi$ are related to the desired transport
coefficients. With $I_{C}=G_{C}\Delta V_{b}$ defined at $\Delta T=0$, and
$I_{Q}=-\kappa\Delta T$ defined at $I_{C}=0$, we identify the electrical
conductance $G_{C}=\chi_{CC}$ and the heat conductance
$\kappa=(\chi_{CQ}\chi_{QC}/\chi_{CC}-\chi_{QQ})/T$. We also define the heat
conductance at $\Delta V_{b}=0$ as $K_{Q}=-\chi_{QQ}/T$. Similarly, $\Delta
V_{b}=-S\Delta T$ measured at $I_{C}=0$ defines the thermopower (Seebeck
coefficient) $S=\chi_{CQ}/T\chi_{CC}$. In the following we denote the charge
and heat conductance quanta as $G_{0}=e^{2}/h$ and $K_{0}=1/h$.

The Lorenz ratio $L=\kappa/TG_{C}$ plays an important role in understanding
the nature of thermoelectric transport in nanoelectronic devices. In
particular, quantum transport in the Fermi liquid regime typically satisfies
the Wiedemann-Franz (WF) law,[18, 19] in which the Lorenz ratio as $T\to 0$
takes the special value $L_{0}=\pi^{2}k_{\rm B}^{2}/3e^{2}$. This is expected
when the carriers of both charge and heat are the same fermionic
quasiparticles (or simply the bare electrons themselves). Although violations
of the WF law are often associated with non-Fermi liquid physics in bulk
systems,[20] in the context of mesoscopic quantum devices, violations are
known in Fermi liquid systems[21, 22, 23] as well as in non-Fermi liquids.[24]
Counter-intuitively the WF law can also be satisfied in certain non-Fermi
liquid systems.[25, *van2020electric, 27] This highlights the richer
thermoelectric capabilities of complex nanostructures.

Another important quantity is the Johnson-Nyquist noise, which describes the
equilibrium fluctuations of the current in time.[28, *nyquist1928thermal] Its
frequency-resolved spectrum is defined as,

$C(\omega)=e^{2}\int_{-\infty}^{\infty}dt\leavevmode\nobreak\ e^{i\omega
t}\left[\langle\dot{N}_{d}(0)\dot{N}_{d}(t)\rangle-\langle\dot{N}_{d}\rangle^{2}\right]\;.$
(14)

The equilibrium current fluctuations[30] are given by $\Delta I_{C}\equiv
e^{2}\tfrac{d}{dt}\langle\hat{N}_{d}-\langle\hat{N}_{d}\rangle\rangle^{2}=2C(0)$.

Finally, we note that the average entropy production rate
$\langle\hat{\sigma}\rangle$ can also be obtained from the particle and energy
currents from the second law of thermodynamics,[31]

$\langle\hat{\sigma}\rangle=\left(\frac{\mu_{s}}{k_{\rm
B}T_{s}}-\frac{\mu_{d}}{k_{\rm B}T_{d}}\right)I_{N}-\left(\frac{1}{k_{\rm
B}T_{s}}-\frac{1}{k_{\rm B}T_{d}}\right)I_{E}\;\;$ (15)

The signal-to-noise ratio of the currents in nanoelectronic devices is bounded
by the entropy production rate, according to the thermodynamic uncertainty
relations (TURs).[32]

###  II.5 NRG

The gold-standard method of choice for solving equilibrium generalized quantum
impurity models of the type discussed above is often considered to be Wilson’s
Numerical Renormalization Group (NRG).[33, *bulla2008numerical]
Thermodynamical quantities and can be accurately obtained within NRG at
essentially any energy scale or temperature, down to $T=0$. The numerical
results for quantum transport coefficients presented in this work are obtained
from dynamical correlation functions calculated using the full-density-matrix
NRG approach.[35] An advantage of this method is that real-frequency dynamics
with excellent resolution at low energies and temperatures can be obtained for
strongly interacting models. A disadvantage is that the impurity part of the
Hamiltonian must be diagonalized exactly, which limits the ability of NRG to
treat more complex nanostructures; and two-lead calculations are
computationally expensive. This motivates the development of new techniques,
simplified effective models, and reusable approximate results for these
models. We undertake this in Secs. IV-VII.

However, we emphasize that our results are not specific to NRG and that any
suitable quantum impurity solver can be used.

##  III Exact techniques for
linear response

In this section we review some commonly used techniques for the calculation of
quantum transport. We focus on the linear response regime for 2-lead models of
the type introduced in Sec. I, and highlight the limitations of each. Here we
assume that source and drain leads are metallic with a constant density of
states $\rho_{0}$. The generalization to equivalent leads with arbitrary
density of states $\rho(\omega)$ is given in Appendix A. The multi-terminal
case is discussed briefly in Appendix B, while the situation for inequivalent
leads is considered in Appendix C.

###  III.1 Landauer

For non-interacting systems (Eq. 2 with $H_{\rm int}=0$), the Landauer formula
for the dc currents may be used [36, *Landauer1970, *LandauerButtiker1985,
*Buttiker1986],

$\displaystyle I_{C}(T)$ $\displaystyle=\frac{e}{h}\int
d\omega\leavevmode\nobreak\ [f_{d}(\omega)-f_{s}(\omega)]\mathcal{T}_{\rm
L}(\omega)\;,$ (16a) $\displaystyle I_{Q}(T)$ $\displaystyle=\frac{1}{h}\int
d\omega\leavevmode\nobreak\
(\mu_{d}-\omega)[f_{d}(\omega)-f_{s}(\omega)]\mathcal{T}_{\rm L}(\omega)\;.$
(16b)

Within linear response, the transmission function $\mathcal{T}_{\rm
L}(\omega)$ assumes its equilibrium form, and the Fermi functions
$f_{\alpha}(\omega)$ can be expanded to first order in $\Delta V_{b}$ and
$\Delta T$. This yields the susceptibilities $\chi_{CC}=e^{2}\Phi_{0}$,
$\chi_{CQ}=\chi_{QC}=e\Phi_{1}$ and $\chi_{QQ}=\Phi_{2}$, in terms of the
transport integrals, $\Phi_{n}(T)=h^{-1}\int d\omega\leavevmode\nobreak\
\omega^{n}[-\partial_{\omega}f_{\rm eq}(\omega)]\mathcal{T}_{\rm L}(\omega)$,
where $f_{\rm eq}(\omega)=(1+\exp[\omega/k_{\rm B}T])^{-1}$ is the equilibrium
Fermi function (and its derivative is $-\partial_{\omega}f_{\rm
eq}(\omega)=\text{sech}[\omega/(2T)]^{2}/(4T)$). In particular, the electrical
and heat conductances are,

$\displaystyle G_{C}(T)$ $\displaystyle=\frac{e^{2}}{h}\int
d\omega\leavevmode\nobreak\ [-\partial_{\omega}f_{\rm
eq}(\omega)]\mathcal{T}_{\rm L}(\omega)\;,$ (17a) $\displaystyle K_{Q}(T)$
$\displaystyle=\frac{1}{hT}\int d\omega\leavevmode\nobreak\
\omega^{2}[-\partial_{\omega}f_{\rm eq}(\omega)]\mathcal{T}_{\rm
L}(\omega)\;,$ (17b)

such that $G_{C}(T\to 0)=(e^{2}/h)\mathcal{T}_{\rm L}(0)$ and $K_{Q}(T\to
0)=(\pi^{2}k_{\rm B}^{2}T/3h)\mathcal{T}_{\rm L}(0)$ provided the transmission
function remains finite as $\omega\to 0$. Note also that $\Phi_{1}(T)=0$ under
particle-hole symmetry $\mathcal{T}_{\rm L}(\omega)=\mathcal{T}_{\rm
L}(-\omega)$, such that $\kappa=K_{Q}$. Under these rather typical conditions,
$\kappa/TG_{C}=L_{0}$ and the Wiedemann-Franz law is automatically satisfied,
independently of microscopic details. In the marginal case $\mathcal{T}_{\rm
L}(\omega)\sim 1/\ln^{2}|\omega|$ characteristic of spin-flip scattering, the
Lorenz ratio picks up logarithmic temperature corrections to the Wiedemann-
Franz limit, $L-L_{0}\sim a/\ln|bT|$. The Lorenz ratio is typically strongly
enhanced at a transmission node; for example $L=9\zeta(3)k_{\rm
B}^{2}/(\ln(4)e^{2})$ for $\mathcal{T}_{\rm L}(\omega)\sim|\omega|$ and
$L=7\pi^{2}k_{\rm B}^{2}/(5e^{2})$ for $\mathcal{T}_{\rm
L}(\omega)\sim\omega^{2}$.

The transmission function can be expressed in the Caroli [40] form in terms of
Green’s functions for the lead-coupled (but non-interacting) nanostructure and
the hybridization matrices,[41, *Beenakker_RandomMatrix1997]

$\displaystyle\mathcal{T}_{\rm L}(\omega)$ $\displaystyle=4\sum_{\sigma}{\rm
Tr}[\bm{G}_{\sigma}^{(0)}(\omega)\bm{\Gamma}^{d}\bm{G}_{\sigma}^{(0)}(\omega)^{*}\bm{\Gamma}^{s}]$
(18a) $\displaystyle\equiv
4\Gamma_{d}\Gamma_{s}\sum_{\sigma}\overline{G}^{(0)}_{sd,\sigma}(\omega)\overline{G}^{(0)}_{ds,\sigma}(\omega)^{*}\;,$
(18b)

where the simplified form of Eq. 18b follows from the definition of the non-
interacting frontier orbital Green’s functions
$\overline{G}_{\alpha\beta,\sigma}^{(0)}=\langle\langle\bar{d}_{\alpha\sigma}^{\phantom{\dagger}};\bar{d}_{\beta\sigma}^{\dagger}\rangle\rangle^{(0)}$
and the corresponding hybridizations
$\Gamma_{\alpha}=\pi\rho_{0}|V_{\alpha}|^{2}$. Note that the transmission
function in the non-interacting limit $H_{\rm int}=0$ is seen from this
expression to be temperature-independent (since non-interacting retarded
Green’s functions are temperature-independent). Furthermore, in the absence of
a magnetic field (whence we have time-reversal symmetry), the expression
simplifies further to $\mathcal{T}_{\rm
L}(\omega)=8\Gamma_{d}\Gamma_{s}|\overline{G}_{ds,\uparrow}^{(0)}(\omega)|^{2}$.
This has a physically intuitive interpretation: combined with Eqs. 17, the
expressions relate the conductance to the probability amplitude for electrons
to propagate across the nanostructure.

Finally, we note that under proportionate coupling (see Sec. II.1), Eq. 18b
reduces to,

$\displaystyle\mathcal{T}_{\rm L}(\omega)\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\
4\Gamma_{d}\Gamma_{s}\sum_{\sigma}|\overline{\overline{G}}_{\sigma}^{(0)}(\omega)|^{2}$
(19)

where
$\overline{\overline{G}}_{\sigma}^{(0)}=\langle\langle\bar{\bar{d}}_{\sigma}^{\phantom{\dagger}};\bar{\bar{d}}_{\sigma}^{\dagger}\rangle\rangle^{(0)}$
is the local non-interacting nanostructure Green’s function. In the special
case of a non-interacting single-orbital nanostructure (resonant level
model),[8]
$|\overline{\overline{G}}_{\sigma}^{(0)}(\omega)|^{2}\simeq[(\omega-\epsilon_{d})^{2}+(\Gamma_{s}+\Gamma_{d})^{2}]^{-1}$
takes the form of a Lorentzian of width $\Gamma_{s}+\Gamma_{d}$, centered on
the level position $\epsilon_{d}$.

These results can be generalized to two or more inequivalent leads with
arbitrary densities of states (Appendices B and C). The key constraint for the
standard Landauer approach, however, is the non-interacting limit.

###  III.2 Renormalized Landauer

As first discussed by Oguri,[43, *Oguri_TransmissionProb_2001,
*Oguri_QuasiParticlePRB2001] all the results of the previous section can be
immediately applied to interacting systems in the Fermi liquid regime at low
temperatures $T\to 0$. One simply replaces [46] the non-interacting
nanostructure Green’s functions by their interacting counterparts,
$G_{ij,\sigma}^{(0)}\to G_{ij,\sigma}$. The Green’s functions are evaluated at
the Fermi energy $\omega=0$ because the $T\to 0$ limit of Eqs. 17 picks out
the zero frequency value of the transmission function, $\mathcal{T}_{\rm
L}(0)$.

For a Fermi liquid, the imaginary part of the interaction self-energy vanishes
at low temperatures and energies [47],

$\displaystyle\text{Im}\leavevmode\nobreak\
\bm{\Sigma}_{\sigma}(\omega=0,T=0)\leavevmode\nobreak\
\overset{\text{FL}}{\longrightarrow}\leavevmode\nobreak\ 0$ (20)

while the real part can be viewed as renormalizing the single particle
nanostructure Hamiltonian,

$\displaystyle\textbf{H}_{\rm nano,\sigma}=\textbf{H}_{\rm
nano}^{(0)}+\text{Re}\leavevmode\nobreak\ \bm{\Sigma}_{\sigma}(0,0)\;.$ (21)

This follows from the form of the Dyson equation, Eq. 10. At $T=0$ and
$\omega=0$, the Green’s functions of the interacting system are the same as
those of a system with $H_{\rm int}=0$ but with a renormalized single-particle
Hamiltonian $\textbf{H}_{\rm nano,\sigma}$.[46] Quantum transport properties
of the renormalized non-interacting system can then be obtained within the
Landauer formalism. We emphasize that the form of the renormalized non-
interacting system need not be found explicitly: only the Green’s functions
$G_{ij,\sigma}(0,0)$ of the interacting system are required.

The electrical conductance through a generic nanostructure in the Fermi liquid
regime as $T\to 0$ therefore follows as,

$\displaystyle
G_{C}(0)=\left(\frac{e^{2}}{h}\right)4\Gamma_{d}\Gamma_{s}\sum_{\sigma}\overline{G}_{sd,\sigma}(0,0)\overline{G}_{ds,\sigma}(0,0)^{*}\;,\qquad$
(22)

and in the case of proportionate coupling,

$\displaystyle G_{C}(0)\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\
\left(\frac{e^{2}}{h}\right)4\Gamma_{d}\Gamma_{s}\sum_{\sigma}|\overline{\overline{G}}_{\sigma}(0,0)|^{2}\;.$
(23)

The above equations can be reformulated in terms of the scattering t-matrix.
Using current conservation, one may write $\mathcal{T}_{\rm
L}(0,0)=(2/\pi\rho_{0})^{2}\sum_{\sigma}\mathcal{G}_{sd,\sigma}(0,0)\mathcal{G}_{ds,\sigma}(0,0)^{*}$,
and thence from Eq. 11 we find,

$\displaystyle
G_{C}(0)=\left(\frac{e^{2}}{h}\right)(2\pi\rho_{0})^{2}\sum_{\sigma}T_{sd,\sigma}(0,0)T_{ds,\sigma}^{*}(0,0)\;,\qquad$
(24)

or in the case of time-reversal symmetry ($B=0$),
$G_{C}(0)=\tfrac{2e^{2}}{h}\times 4|\pi\rho_{0}T_{ds,\uparrow}(0,0)|^{2}$.
These expressions allow the conductance to be obtained from standard
diagrammatic techniques for the scattering t-matrix [48,
*Yamada316ImSelfDiag], or from Fermi liquid theory. [47, 50]

Similar expressions can be obtained for the heat conductance. Alternatively,
since the approach is only valid for Fermi liquids in the low-temperature
limit, the heat conductance may be obtained from the electrical conductance
via the Wiedemann-Franz law, $\kappa/TG_{C}=L_{0}$ (assuming $\mathcal{T}_{\rm
L}(0,0)\neq 0$ as above).

We note that the renormalized Landauer formulation only holds for systems
described by Fermi liquid theory at low temperatures and energies. Genuine
non-Fermi liquid states[51] (characterized by non-vanishing
$\text{Im}\leavevmode\nobreak\ \Sigma(0,0)$ and fractional residual entropy),
arising due to a frustration of Kondo screening for example, can also be
engineered in quantum dot devices, and alternative approaches must then be
used. However, we emphasize here that the Fermi liquid scenario is very much
the standard one. Indeed, even in cases where the nanostructure hosts free
local moments as $T\to 0$ (so-called ‘singular Fermi liquids’), arising due to
ferromagnetic interactions or Kondo underscreening for example, Eq. 20 still
holds asymptotically.[52]

Rather, the key limitation of the renormalized Landauer approach is the $T\to
0$ limit. This may be particularly problematical when comparing with results
of experiments performed at base temperatures larger than emergent energy
scales (such as the Kondo temperature). For example, the general
considerations of Sec. III.1 suggest that the thermoelectric Lorenz ratio for
standard quantum dot devices should saturate to $L_{0}=\pi^{2}k_{\rm
B}^{2}T/3h$ as $T\to 0$. However, experiments often appear to show non-Fermi
liquid signatures [53, *ralph19942] because the low-temperature limit is not
reached in practice, and electron-electron interactions give a finite
$\text{Im}\leavevmode\nobreak\ \Sigma(\omega,T)$.

###  III.3 Meir-Wingreen

A general framework for calculating electrical conductance through generic
interacting nanostructures was devised by Meir and Wingreen (MW).[11,
*MeirWingreen_AMoutEquilib_PRL1993, 13] The approach was subsequently extended
to thermoelectric transport by Costi and Zlatić.[19] The dc electrical and
energy currents into the drain lead are given by,

$\displaystyle I_{C}(T)$ $\displaystyle=\frac{e}{h}\int
d\omega\leavevmode\nobreak\ X(\omega,T)\;,$ (25a) $\displaystyle I_{Q}(T)$
$\displaystyle=\frac{1}{h}\int d\omega\leavevmode\nobreak\
(\mu_{d}-\omega)X(\omega,T)\;,$ (25b)

where the kernel

$X(\omega,T)=-\text{Im}\leavevmode\nobreak\
\text{Tr}\sum_{\sigma}\bm{\Gamma}^{d}\left(\bm{G}_{\sigma}^{<}(\omega,T)+2f_{d}(\omega)\bm{G}_{\sigma}(\omega,T)\right)\;,$
(26)

now involves the nanostructure lesser Green’s functions,
$\bm{G}_{\sigma}^{<}(\omega,T)$. The above equations hold in the general non-
equilibrium situation at finite bias $\Delta V_{b}$ and temperature gradient
$\Delta T$. However, in this case the full non-equilibrium retarded and lesser
Green’s functions must be obtained for the interacting problem. Since this is
typically deeply nontrivial,[55] we again focus on the linear response regime
where the nanostructure remains close to equilibrium. But even in this limit,
the explicit first-order variation of the Green’s functions with respect to
$\Delta V_{b}$ and $\Delta T$ is required to calculate the conductances
$G_{C}$ and $K_{Q}$. In particular, $\partial_{\Delta
V_{b}}\bm{G}_{\sigma}^{<}|_{\Delta V_{b}=0}$ and $\partial_{\Delta
T}\bm{G}_{\sigma}^{<}|_{\Delta T=0}$ are not accessible using standard
equilibrium techniques.

An important simplification arises in the case of proportionate coupling (Sec.
II.1), since then the lesser Green’s functions in Eq. 26 can be eliminated to
yield the Landauer form,

$\displaystyle X(\omega,T)$ $\displaystyle\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\
\left[f_{d}(\omega)-f_{s}(\omega)\right]\mathcal{T}_{\rm MW}(\omega,T)$ (27a)
$\displaystyle\mathcal{T}_{\rm MW}(\omega,T)$
$\displaystyle=\frac{4\Gamma_{d}\Gamma_{s}}{\Gamma_{d}+\Gamma_{s}}\sum_{\sigma}\left[-\text{Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega,T)\right]$ (27b)

Importantly, the Landaurer form of Eq. 27a in proportionate coupling implies
that the conductances in linear response can be obtained using Eqs. 17a and
17b, but with $\mathcal{T}_{\rm L}(\omega)\to\mathcal{T}_{\rm MW}(\omega,T)$
given now by Eqs. 27b. This is because the leading linear variations in
$\Delta V_{b}$ and $\Delta T$ of the currents $I_{C}$ and $I_{Q}$ are now
carried solely by the lead Fermi functions, and not the nanostructure
transmission function (first order variations in $\Delta V_{b}$ and $\Delta T$
of $\mathcal{T}_{\rm MW}(\omega,T)$ appear only to quadratic order in the
currents). Linear response conductances in proportionate coupling can
therefore be obtained for interacting nanostructures at finite temperatures
using only _equilibrium_ retarded nanostructure Green’s functions,

$\displaystyle G_{C}(T)$ $\displaystyle\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\ \frac{e^{2}}{h}\int
d\omega\leavevmode\nobreak\ [-\partial_{\omega}f_{\rm
eq}(\omega)]\mathcal{T}_{\rm MW}(\omega,T)\;,$ (28a) $\displaystyle K_{Q}(T)$
$\displaystyle\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\ \frac{1}{hT}\int
d\omega\leavevmode\nobreak\ \omega^{2}[-\partial_{\omega}f_{\rm
eq}(\omega)]\mathcal{T}_{\rm MW}(\omega,T)\;,$ (28b)

with the full energy- and temperature-dependent transmission $\mathcal{T}_{\rm
MW}(\omega,T)$ here given by Eq. 27b. Analogous expressions can be obtained
for the cross susceptibility $\chi_{CQ}=\chi_{QC}$ in Eq. 13.

Further insight is gained by expressing the transmission function in terms of
the t-matrix,

$\displaystyle\mathcal{T}_{\rm
MW}(\omega,T)=\frac{4\Gamma_{d}\Gamma_{s}}{(\Gamma_{d}+\Gamma_{s})^{2}}\sum_{\sigma}t_{ee,\sigma}(\omega,T)$
(29)

where
$t_{ee,\sigma}(\omega,T)=-\pi\rho_{0}\text{Im}T_{ee,\sigma}(\omega,T)\equiv(\Gamma_{d}+\Gamma_{s})[-\text{Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega,T)]$ is the spectrum of the even
channel t-matrix. The dimensionless geometric factor
$4\Gamma_{d}\Gamma_{s}/(\Gamma_{d}+\Gamma_{s})^{2}$ in Eq. 29 describes the
relative coupling to source and drain leads, and takes its maximal value of 1
in the symmetric limit $\Gamma_{s}=\Gamma_{d}$. The spectrum of the t-matrix
can take values $0\leq t_{ee,\sigma}\leq 1$ only. The lower bound arises
because $t_{ee,\sigma}$ is proportional to the local nanostructure density of
states $A_{\sigma}=-\tfrac{1}{\pi}\text{Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}\geq 0$. The upper bound can be seen from the
t-matrix equation (and similarly noting that
$\mathcal{A}_{ee,\sigma}=-\tfrac{1}{\pi}\text{Im}\leavevmode\nobreak\
\mathcal{G}_{ee,\sigma}\geq 0$). The transmission function itself therefore
satisfies $0\leq\mathcal{T}_{\rm MW}\leq 2$, independently of $\omega$ and
$T$. The low temperature and low energy behavior of the t-matrix can be
obtained from the Friedel sum rule for Fermi liquid systems,
$t_{ee,\sigma}(0,0)=\sin^{2}(\tfrac{\pi}{2}n_{\rm imp})$ [56]. Here $n_{\rm
imp}$ is the thermodynamic ‘excess charge’ of the system due to the
nanostructure at $T=0$. Note however that in a local moment phase (a ‘singular
Fermi liquid’ with a residual unscreened spin on the nanostructure), this
expression becomes $t_{ee,\sigma}(0,0)=\cos^{2}(\tfrac{\pi}{2}n_{\rm imp})$
[51]. For genuine non-Fermi liquid situations, the Friedel sum rule cannot be
used.[51]

The electrical conductance in the low temperature limit follows as,

$\displaystyle G_{C}(0)\leavevmode\nobreak\
\overset{\text{PC}}{\longrightarrow}\leavevmode\nobreak\
\left(\frac{e^{2}}{h}\right)\frac{4\Gamma_{d}\Gamma_{s}}{(\Gamma_{d}+\Gamma_{s})^{2}}\sum_{\sigma}t_{ee,\sigma}(0,0)\;,$
(30)

which thereby can take a maximum value of $2e^{2}/h$. We note that Eq. 30 is a
generalization of Eq. 23. The expressions coincide for a Fermi liquid system
since
$t_{ee,\sigma}(\omega,T)=(\Gamma_{d}+\Gamma_{s})[-\text{Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega,T)]$ with
$\text{Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(0,0)=[\text{Im}\leavevmode\nobreak\
\overline{\overline{\Sigma}}_{\sigma}(0,0)-\Gamma_{d}-\Gamma_{s}]\times|\overline{\overline{G}}_{\sigma}(0,0)|^{2}$,
and where the local self-energy $\text{Im}\leavevmode\nobreak\
\overline{\overline{\Sigma}}_{\sigma}(0,0)=0$ vanishes in a Fermi liquid by
definition[47].

The above formulation in terms of the t-matrix is particularly important
because it allows the low-$T$ transport to be described using simplified low-
energy effective models, whose t-matrix $t_{ee,\sigma}(\omega,T)$ still
provides a faithful representation of the electronic scattering in the leads
induced by the nanostructure. In many cases, an effective Kondo model may be
used, which significantly reduces the nanostructure complexity. This is
discussed further in Sec. VI and demonstrated explicitly for the TQD in Sec.
IX.1.

Finally, we emphasize that quantum transport calculations using the MW
approach require the accurate determination of nanostructure Green’s
functions. These follow immediately from the Dyson equation, Eq. 10, once the
self-energies are known. In NRG the self-energy matrix can be obtained via,

$\bm{\Sigma}_{\sigma}(\omega,T)=[\bm{G}_{\sigma}(\omega,T)]^{-1}\bm{F}_{\sigma}(\omega,T)\;,$
(31)

with elements $F_{ij,\sigma}=\langle\langle
d_{i\sigma}^{\phantom{\dagger}};[H_{\rm
int},d_{j\sigma}^{\dagger}]\rangle\rangle$. This generalizes the result for
the AIM in Ref. 57, and is found in practice to yield highly accurate results
within NRG. Conductance calculated in this way for interacting PC systems
reproduces exact analytical results when known,[58, *mitchell2012universal]
and has been shown to agree with experimental results for a wide range of
nanoelectronics devices (see e.g. Refs. 60, 61, 62, 63).

For interacting systems in linear response, the main limitation of the MW
formulation in terms of equilibrium retarded Green’s functions is the PC
requirement [55]: this places constraints on the form of the hybridization for
nanostructures with multiple degrees of freedom, and requires equivalent
source and drain leads.

####  III.3.1 Ng Ansatz

For systems not in PC, the currents can be obtained from Eq. 25, using the
general form of Eq. 26. This requires a knowledge of the non-equilibrium
lesser Green’s function matrix, $\bm{G}^{<}_{\sigma}(\omega,T)$. From the
Keldysh equation [64, *LangrethLinearRespNEq1976]
$\bm{G}^{<}_{\sigma}=\bm{G}_{\sigma}^{\phantom{<}}\bm{\Sigma}_{\sigma}^{<}\bm{G}_{\sigma}^{*}$
this is related to the usual retarded nanostructure Green’s functions
$\bm{G}_{\sigma}$ through the lesser self-energy $\bm{\Sigma}_{\sigma}^{<}$.
Here
$\bm{\Sigma}_{\sigma}^{<}=\bm{\Sigma}_{\sigma;0}^{<}+\bm{\Sigma}_{\sigma;{\rm
int}}^{<}$ where
$\bm{\Sigma}_{\sigma;0}^{<}=2i(f_{s}\bm{\Gamma}^{s}+f_{d}\bm{\Gamma}^{d})$
involves the non-interacting hybridization , while $\bm{\Sigma}_{\sigma;{\rm
int}}^{<}$ accounts for electronic interactions. Similarly,
$\bm{\Sigma}_{\sigma}^{>}=\bm{\Sigma}_{\sigma;0}^{>}+\bm{\Sigma}_{\sigma;{\rm
int}}^{>}$ with
$\bm{\Sigma}_{\sigma;0}^{>}=2i([f_{s}-1]\bm{\Gamma}^{s}+[f_{d}-1]\bm{\Gamma}^{d})$.
Unfortunately, $\bm{\Sigma}_{\sigma;{\rm int}}^{</>}$ are notoriously
difficult to calculate out of equilibrium.

A simple Ansatz was proposed by Ng in Ref. 66, *Dong2002 for the non-
equilibrium single-impurity Anderson model to simplify such calculations.
Generalizing to the multi-orbital situation,[68, 69, 70] we suppose that the
lesser and greater self-energies take a factorized form,

$\begin{rcases*}\bm{\Sigma}_{\sigma}^{<}=\bm{\Sigma}_{\sigma;0}^{<}\leavevmode\nobreak\
\bm{\Lambda}_{\sigma}\\\
\bm{\Sigma}_{\sigma}^{>}=\bm{\Sigma}_{\sigma;0}^{>}\leavevmode\nobreak\
\bm{\Lambda}_{\sigma}\qquad\end{rcases*}\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm Ng\leavevmode\nobreak\ Ansatz}$ (32)

where $\bm{\Lambda}_{\sigma}\equiv\bm{\Lambda}_{\sigma}(\omega,T)$ is a common
function to be determined. From the identity
$\bm{\Sigma}_{\sigma}^{>}-\bm{\Sigma}_{\sigma}^{<}=2i{\rm
Im}\bm{\Sigma}_{\sigma}$, where $\bm{\Sigma}_{\sigma}$ is the usual retarded
interaction self-energy defined in Eq. 10, it then follows that,

$\bm{\Lambda}_{\sigma}=\bm{{\rm
I}}-[\bm{\Gamma}^{s}+\bm{\Gamma}^{d}+2\delta\bm{{\rm I}}]^{-1}[{\rm
Im}\bm{\Sigma}_{\sigma}]\;,$ (33)

with $\delta\to 0^{+}$ for regularization. These expressions imply a specific
form for the non-equilibrium distribution function matrix $\bm{f}^{NE}_{\rm
int}$ which connects the lesser and retarded interaction self-energies through
$\bm{\Sigma}^{<}_{\sigma;{\rm int}}=-2i\bm{f}_{\rm int}^{\rm
NE}\leavevmode\nobreak\ {\rm Im}\bm{\Sigma}_{\sigma}$. Specifically, the Ng
Ansatz may be stated as $\bm{f}^{\rm NE}_{\rm
int}\to[f_{s}\bm{\Gamma}^{s}+f_{d}\bm{\Gamma}^{d}][\bm{\Gamma}^{s}+\bm{\Gamma}^{d}+2\delta\bm{{\rm
I}}]^{-1}$, which is the well-known pseudo-equilibrium distribution[13]
obtained for non-interacting systems out of equilibrium. Of course, the true
distribution function is in general modified for interacting systems out of
equilibrium.

Using Eqs. 32, 33 it can be shown that the linear response electrical and heat
conductances can be obtained from Eq. 28, with an _effective_ transmission
function of Landauer-like form,

$\mathcal{T}_{\rm Ng}(\omega,T)=4\sum_{\sigma}{\rm
Tr}[\bm{G}_{\sigma}(\omega,T)\bm{\Gamma}^{d}\bm{\Lambda}_{\sigma}(\omega,T)\bm{G}_{\sigma}(\omega,T)^{*}\bm{\Gamma}^{s}]\;,$
(34)

where the retarded nanostructure Green’s functions $\bm{G}_{\sigma}(\omega,T)$
are evaluated at equilibrium in the presence of interactions and coupling to
the leads. Note that Eq. 34 is as such the analogue of Eq. 18a.

By construction, the above Ansatz is exact in the non-interacting limit, as
well as in equilibrium (no voltage bias or temperature gradient), and
automatically satisfies the continuity equation implying current conservation.
Indeed, since ${\rm Im}\bm{\Sigma}_{\sigma}(0,0)=0$ in a Fermi liquid system,
$\bm{\Lambda}_{\sigma}(0,0)=\bm{{\rm I}}$ from Eq. 33, such that Eqs. 28, 34
reduce correctly to the Oguri result, Eq. 22.

In the case of a single-orbital nanostructure, the Ng Ansatz yields $f_{\rm
int}^{\rm NE}\to f^{\rm
NE}_{0}\equiv[f_{s}\Gamma_{s}+f_{d}\Gamma_{d}]/(\Gamma_{s}+\Gamma_{d})$. To
check the validity of the approximation, we computed the non-equilibrium
lesser and retarded self-energies of the Anderson impurity model in
perturbation theory to second order in the interaction $U$. We found[71] a
non-zero correction to the above expression for $f_{\rm int}^{\rm NE}$
proportional to the voltage bias $\Delta V_{b}$, which implies a finite
correction to the linear-response conductance obtained via Eq. 34 already at
order $U^{2}$. This correction is given in Appendix D.

Therefore despite the simplicity and utility of the Ng Ansatz approach, it is
clearly a somewhat uncontrolled approximation for interacting systems, and
cannot be relied upon to yield accurate results in the general multi-orbital
case. For further discussion and rigorous extensions, see Ref. 72, 73. In Sec.
IX.1 we illustrate the use of the Ng Ansatz approach for the TQD, comparing to
exact results from the Kubo formula.[74, *Kubo1957, *IzumidaSakai1997] In
Secs. VI, VII we devise alternative routes to the low-$T$ conductance of non-
PC systems.

###  III.4 Kubo

The first-order correction to a thermodynamical observable from its
equilibrium value due to a source term in the Hamiltonian can be computed
using standard perturbation theory.[8] In the context of linear-response
transport coefficients, the result is known as the Kubo formula.[74,
*Kubo1957, *IzumidaSakai1997] The derivation for electrical conductance due to
a bias voltage is straightforward, because a difference in chemical potential
between source and drain leads can be incorporated as a perturbation on the
level of the Hamiltonian (even though strictly speaking the chemical potential
is a statistical property of leads treated as infinite thermal reservoirs at
equilibrium).

We treat the ac bias $H_{\rm bias}$ in Eq. 12 as the perturbation, switched on
adiabatically in the infinitely distant past, and calculate the resulting
electrical current $I_{C}(\omega,T)=-e\langle\dot{N}^{d}\rangle$ into the
drain lead at time $t=0$, to first order in the bias voltage $\Delta V_{b}$.
Here, $\omega$ is the frequency of the ac bias voltage. The result is,

$I_{C}(\omega,T)=-ie^{2}\Delta V_{b}{\rm Tr}\int_{0}^{\infty}e^{-\eta
t^{\prime}}[\hat{N}^{s},\hat{\rho}_{\rm
eq}(T)]\dot{N}^{d}(t^{\prime})\cos(\omega t^{\prime})\leavevmode\nobreak\
dt^{\prime}\;,$ (35)

where $\hat{\rho}_{\rm eq}(T)$ is the equilibrium density matrix operator
evaluated at temperature $T$,
$\dot{N}^{d}(t^{\prime})=e^{i\hat{H}t^{\prime}}\dot{N}^{d}e^{-i\hat{H}t^{\prime}}$
with $\hat{H}$ the equilibrium Hamiltonian of the full system, and $\eta\to
0^{+}$ is required for convergence of the integral.

With some manipulation, this expression can be recast in terms of the current-
current correlation function
$K(\omega,T)=\langle\langle\dot{N}^{s};\dot{N}^{d}\rangle\rangle_{\omega,T}$
which is itself the Fourier transform to the frequency domain of the retarded
function $K(t,T)=-i\theta(t)\langle[\dot{N}^{s},\dot{N}^{d}(t)]\rangle_{T}$.
The dynamical conductance follows as,

$G_{C}(\omega,T)=\left(\frac{e^{2}}{h}\right)\frac{-2\pi\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\ K(\omega,T)}{\omega}\;.$ (36)

In the dc limit $\omega\to 0$ we obtain the steady state conductance,

$G_{C}(T)\equiv\lim_{\omega\to 0}\leavevmode\nobreak\ G_{C}(\omega,T)\;.$ (37)

With $\dot{N}^{\alpha}=i[\hat{H},\hat{N}^{\alpha}]$, one can express
$K(\omega,T)$ in Eq. 36 in terms of correlation functions involving the
nanostructure frontier orbitals and the local lead orbitals,

$K(\omega,T)=V_{s}V_{d}\sum_{\sigma,\sigma^{\prime}}\langle\langle\bar{d}_{s\sigma}^{\dagger}c_{s\sigma}^{\phantom{\dagger}}-c_{s\sigma}^{\dagger}\bar{d}_{s\sigma}^{\phantom{\dagger}}\leavevmode\nobreak\
;\leavevmode\nobreak\
\bar{d}_{d\sigma^{\prime}}^{\dagger}c_{d\sigma^{\prime}}^{\phantom{\dagger}}-c_{d\sigma^{\prime}}^{\dagger}\bar{d}_{d\sigma^{\prime}}^{\phantom{\dagger}}\rangle\rangle_{\omega,T}$
(38)

Eqs. 36–38 have the advantage in linear response that they do not require the
special proportionate coupling geometry. They are also applicable in the
general dynamical regime of an ac driving bias voltage. NRG results for the
linear conductance of non-PC systems via the Kubo formula have been shown to
reproduce exact results in the rare cases where analytic solutions are
known,[77] and quantitative agreement with experimental results have been
demonstrated.[78, 79, 80]

We now remark on the equivalence of Eqs. 36–38 to the MW result Eqs. 25a and
27 in the PC limit. In this case, we may write Eq. 38 in terms of the even/odd
lead basis and the PC frontier orbital $\bar{\bar{d}}_{\sigma}$. The even lead
orbital can then be eliminated by exploiting current conservation
$\dot{N}^{s}=-\dot{N}^{d}$. This yields,

$\displaystyle
K(\omega,T)\overset{\text{PC}}{\longrightarrow}\tfrac{V_{d}^{2}V_{s}^{2}}{V_{d}^{2}+V_{s}^{2}}\sum_{\sigma,\sigma^{\prime}}\langle\langle\bar{\bar{d}}_{\sigma}^{\dagger}c_{o\sigma}^{\phantom{\dagger}}{\rm-}c_{o\sigma}^{\dagger}\bar{\bar{d}}_{\sigma}^{\phantom{\dagger}};c_{o\sigma^{\prime}}^{\dagger}\bar{\bar{d}}_{\sigma^{\prime}}^{\phantom{\dagger}}{\rm-}\bar{\bar{d}}_{\sigma^{\prime}}^{\dagger}c_{o\sigma^{\prime}}^{\phantom{\dagger}}\rangle\rangle_{\omega,T}$

Provided the original source and drain leads are equivalent, the transformed
odd lead is then formally decoupled from the rest of the system (cf. emergent
decoupling in Secs. VI and VII). This means that exact eigenstates of the full
system are product states of the form
$|\Psi\rangle=|\psi\rangle_{n-e}\otimes|\phi\rangle_{o}$, where
$|\psi\rangle_{n-e}$ has support in the nanostructure-even-lead Fock space,
while $|\phi\rangle_{o}$ lives in the decoupled odd-lead Fock space. This
allows a factorization into single-particle Green’s functions. After some
manipulation, one obtains

$\begin{split}G_{C}(\omega,T)\overset{\text{PC}}{\longrightarrow}\left(\frac{e^{2}}{h}\right)\frac{4\Gamma_{d}\Gamma_{s}}{\Gamma_{d}+\Gamma_{s}}\sum_{\sigma}\int_{-\infty}^{\infty}d\omega^{\prime}\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega^{\prime},T)&\\\
\times\left[\frac{f_{\rm eq}(\omega^{\prime}+\omega)-f_{\rm
eq}(\omega^{\prime}-\omega)}{2\omega}\right]&\end{split}$ (39)

where the integral over Fermi functions arises here because of the original
trace over decoupled odd lead states. Eq. 39 is as such the ac generalization
of Eq. 28a, and reduces to it in the dc limit $\omega\to 0$, wherein the
expression in the square brackets becomes $\partial_{\omega^{\prime}}f_{\rm
eq}(\omega^{\prime})$. In the low-temperature limit $T\to 0$, we have

$G_{C}(\omega,0)\overset{\text{PC}}{\longrightarrow}\left(\frac{e^{2}}{h}\right)\frac{4\Gamma_{d}\Gamma_{s}}{\Gamma_{d}+\Gamma_{s}}\sum_{\sigma}\left[\frac{-\int_{-\omega}^{\omega}d\omega^{\prime}\leavevmode\nobreak\
{\rm Im}\leavevmode\nobreak\
\overline{\overline{G}}_{\sigma}(\omega^{\prime},0)}{2\omega}\right]$ (40)

which is determined by the average of the nanostructure spectral function over
a window set by the ac frequency. Ref. 81 proposed to use this as a means of
measuring the nanostructure spectral function from dynamical conductance
measurements.

Finally, we turn to the Kubo formula for heat transport.[17] This is more
subtle, since there is no source term in which to do perturbation theory, that
can be added to the Hamiltonian that mimics a temperature gradient. An
indirect solution utilizing the Tolman-Ehrenfest effect[82] was proposed by
Luttinger.[17] The energy current couples to both a gravitational potential
gradient as well as a temperature gradient (“heat has weight”). A
gravitational potential difference between leads can be incorporated into the
Hamiltonian as a source term, leading to an expression for the linear-response
energy current analogous to Eq. 35. Importantly, Luttinger showed[17] that the
linear response heat conductance due to a gravitational potential difference
with zero temperature difference, is identical to that due to a temperature
difference with zero gravitational potential difference. Exploiting this
correspondence, one may work the previous steps of this section to obtain,

$K_{Q}(T)=\left(\frac{1}{hT}\right)\lim_{\omega\to
0}\frac{-2\pi\leavevmode\nobreak\ {\rm Im}\leavevmode\nobreak\
K^{\prime}(\omega,T)}{\omega}\;,$ (41)

with $K^{\prime}(\omega,T)=\langle\langle\dot{H}_{\rm leads}^{s};\dot{H}_{\rm
leads}^{d}\rangle\rangle$. Similarly, the cross susceptibility
$\chi_{CQ}=\chi_{QC}$ appearing in Eq. 13, which controls thermopower, is
given by $\chi_{CQ}(T)=\tfrac{e}{h}\lim_{\omega\to 0}[\tfrac{2\pi}{\omega}{\rm
Im}\leavevmode\nobreak\ K^{\prime\prime}(\omega,T)]$, with
$K^{\prime\prime}(\omega,T)=\langle\langle\dot{N}^{s};\dot{H}_{\rm
leads}^{d}\rangle\rangle$.

Finally, note that in linear response, the Kubo formula provides a connection
between the ac conductance and the spectrum of the Johnson-Nyquist noise Eq.
14, via the fluctuation-dissipation theorem,[81, 30]

$C(\omega,T)=-2e^{2}f_{\rm BE}(\omega)[{\rm Im}K(\omega,T)]=2\hbar\omega
f_{\rm BE}(\omega)G_{C}(\omega,T)\;,$ (42)

where $f_{\rm BE}(\omega)=(\exp[\omega/k_{\rm B}T]-1)^{-1}$ is the Bose-
Einstein distribution.

##  IV “Improved” charge transport Kubo formula for NRG

A well-defined and finite conductance in the dc limit $\omega\to 0$ implies
from Eqs. 36 and 37 that the current-current correlator ${\rm
Im}\leavevmode\nobreak\ K(\omega,T)\sim\omega$ at low energies. The physics of
lead-coupled interacting nanostructures is often controlled by emergent low
energy scales such as the Kondo temperature $T_{\rm K}$. The RG structure of
such problems means that the limiting $\omega\to 0$ behavior can in practice
be extracted for finite $|\omega|\ll T,T_{\rm K}$. However in this regime
${\rm Im}\leavevmode\nobreak\ K(\omega,T)$ is itself very small. Any numerical
estimation of $K(\omega,T)$ must therefore be very accurate over a sufficient
range of $|\omega|\ll T,T_{\rm K}$ so that the gradient can be identified and
the conductance deduced.

Even with highly accurate methods such as NRG,[33, *bulla2008numerical, 35,
83] this can be a challenge. Good results are only obtained when the number of
kept states in NRG is pushed very high. A further issue with the NRG
implementation is that, even with the full density matrix approach,[35, 83]
dynamical information is not reliably obtained for $|\omega|\ll T$. This can
make it difficult to extract the dc conductance numerically using Eqs. 36 and
37.

Finally, we note that the explicit form of $K(\omega,T)$ in Eq. 38 depends on
the nanostructure frontier orbitals and the local lead orbitals, and is model-
specific.

Below we propose a conceptually simple but practically useful reformulation
that substantially overcomes these limitations and is found to greatly improve
accuracy of the Kubo formula for charge transport when implemented in NRG.

We utilize the Zubarev equations of motion[16] for retarded Green’s functions
of bosonic operators $\hat{A}$ and $\hat{B}$,

$\displaystyle\omega^{+}\langle\langle\hat{A};\hat{B}\rangle\rangle-\langle[\hat{A},\hat{B}]\rangle=\langle\langle\hat{A};[\hat{H},\hat{B}]\rangle\rangle=\langle\langle[\hat{A},\hat{H}];\hat{B}\rangle\rangle$

With $\dot{\Omega}\equiv\tfrac{d}{dt}\hat{\Omega}=i[\hat{H},\hat{\Omega}]$,
and choosing $\hat{A}=\dot{N}^{s}$ and $\hat{B}=\hat{N}^{d}$, we may therefore
write,
$\langle\langle\dot{N}^{s};\dot{N}^{d}\rangle\rangle=i\omega^{+}\langle\langle\dot{N}^{s};\hat{N}^{d}\rangle\rangle+\langle[[\hat{H},\hat{N}^{s}],\hat{N}^{d}]\rangle$.
Since the expectation value
$\langle[[\hat{H},\hat{N}^{s}],\hat{N}^{d}]\rangle$ is real, we obtain the
current-current correlator appearing in Eq. 36 as,

${\rm Im}\leavevmode\nobreak\ K(\omega,T)=\omega{\rm Re}\leavevmode\nobreak\
\langle\langle\dot{N}^{s};\hat{N}^{d}\rangle\rangle_{\omega,T}\;.$ (43)

Conveniently, the factor of $\omega$ in Eq. 43 cancels with that in the
denominator of Eq. 36, leading to greater stability in the numerical
evaluation of the dc conductance via the Kubo formula. In fact, one can apply
the equations of motion a second time to obtain,

${\rm Im}\leavevmode\nobreak\ K(\omega,T)=-\omega^{2}{\rm
Im}\leavevmode\nobreak\
\langle\langle\hat{N}^{s};\hat{N}^{d}\rangle\rangle_{\omega,T}\;,$ (44)

which involves the total number operators for the leads only. Within NRG, this
is a convenient reformulation from a technical point of view because the Kubo
formula is now independent of the details of the nanostructure or
hybridization Hamiltonians: once the NRG code is set up to calculate the
correlator $\langle\langle\hat{N}^{s};\hat{N}^{d}\rangle\rangle_{\omega,T}$,
the same code can be used to calculate the conductance of any system without
modification. Importantly, we find that Eq. 44 yields much more accurate
results in practice, and that a greatly reduced number of states $M_{K}$ need
to be kept in the NRG calculations to obtain converged results than when using
the standard Kubo formula. This is in part because nontrivial dynamical
contributions to the conductance arise at every step of the NRG within this
formulation, since $\hat{N}^{\alpha}$ has weight on all Wilson chain orbitals.

###  IV.1 Calculation of
$\langle\langle\hat{N}^{s};\hat{N}^{d}\rangle\rangle_{\omega,T}$

The Lehmann representation of generic spectral functions of the type
$\mathcal{A^{BC}}(\omega,T)=\int\frac{dt}{2\pi}e^{i\omega
t}\langle\hat{\mathcal{B}}(t)\hat{\mathcal{C}}\rangle_{T}$ is given by,

$\mathcal{A^{BC}}(\omega,T)=\sum_{a,b}\langle
b|\hat{\mathcal{C}}|a\rangle\frac{e^{-E_{a}/k_{\rm B}T}}{Z}\langle
a|\hat{\mathcal{B}}|b\rangle\delta(\omega-E_{b}+E_{a})\;,$ (45)

where $\\{|a\rangle\\}$ constitutes the _complete_ set of eigenstates of the
Hamiltonian satisfying the Schrödinger equation $H|a\rangle=E_{a}|a\rangle$,
and $Z$ is the partition function. In principle, the desired retarded
correlator
$\tilde{K}(\omega,T)=\langle\langle\hat{N}^{s};\hat{N}^{d}\rangle\rangle_{\omega,T}$
can be obtained from Eq. 45 as $\frac{1}{\pi}{\rm Im}\leavevmode\nobreak\
\tilde{K}(\omega,T)=\mathcal{A}^{N^{d}N^{s}}(\omega,T)-\mathcal{A}^{N^{s}N^{d}}(-\omega,T)$.

However, calculating such an object within NRG is somewhat unconventional
because the operators $\hat{N}^{s}$ and $\hat{N}^{d}$ are not local impurity
operators, but rather live on the Wilson chains for the leads.

Within NRG, $H_{\rm leads}$ in Eq. 1 is discretized logarithmically and mapped
to Wilson chains of the form,[33, *bulla2008numerical]

$H_{\rm leads}\to H_{\rm leads}^{\rm
WC}=\sum_{\alpha,\sigma}\sum_{n=0}^{\infty}\left(t_{n}^{\phantom{\dagger}}f_{\alpha
n\sigma}^{\dagger}f_{\alpha(n+1)\sigma}+{\rm H.c.}\right)\;,$ (46)

where $t_{n}\sim D\Lambda^{-n/2}$ at large $n$, $D$ is the bare conduction
electron bandwidth as before, and $\Lambda>1$ is the discretization parameter.
We have assumed for simplicity leads with particle-hole symmetry, although
this is immaterial. The mapping is defined such that the nanostructure
hybridizes only with the Wilson ‘zero’ orbitals $f_{\alpha 0\sigma}$,

$\displaystyle H_{\rm hyb}\to H_{\rm hyb}^{\rm
WC}=\sum_{\alpha,i,\sigma}\left(V_{\alpha,i}d^{\dagger}_{i\sigma}f_{\alpha
0\sigma}^{\phantom{\dagger}}+\text{H.c.}\right)\;.$ (47)

The lead number operators are $\hat{N}^{\alpha}=\sum_{n}\hat{n}^{f}_{\alpha
n}$ where $\hat{n}^{f}_{\alpha n}=\sum_{\sigma}f_{\alpha
n\sigma}^{\dagger}f_{\alpha n\sigma}^{\phantom{\dagger}}$.

In NRG, one starts from the ‘impurity’ Hamiltonian $H_{\rm nano}$, and then
defines a sequence of Hamiltonians $H_{n}$, obtained iteratively by addition
of a successive Wilson chain orbitals,[33, *bulla2008numerical, 84]

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle H_{\rm nano}+H_{\rm
hyb}^{\rm WC}\;,$ (48) $\displaystyle H_{n}$ $\displaystyle=$ $\displaystyle
H_{n-1}+t_{n-1}\sum_{\alpha,\sigma}\left(f_{\alpha(n-1)\sigma}^{\dagger}f_{\alpha
n\sigma}^{\phantom{\dagger}}+{\rm H.c.}\right)\;,\qquad$ (49)

with Eq. 49 defined for $n>0$. The exact (discretized) Hamiltonian $H^{\rm
disc}=H_{\rm nano}+H_{\rm leads}^{\rm WC}+H_{\rm hyb}^{\rm WC}$ is recovered
as $H^{\rm disc}=\lim_{n\to\infty}H_{n}$. The traditional rescaling of $H_{n}$
is omitted here for pedagogical simplicity.

Clearly the Fock space dimension of $H_{n}$ grows exponentially with $n$, and
the exact solution of even the discretized model is intractable. Instead, at
each step, $H_{n}$ is diagonalized and only the low-energy manifold of
eigenstates is retained for the next step. This constitutes an RG
transformation;[33, *bulla2008numerical] the physics on successively lower
energy scales is revealed on increasing $n$.

The iterative scheme is initialized in some convenient basis
$\\{|\phi_{a}\rangle_{0}\\}$ spanning $H_{0}$ by forming the corresponding
Hamiltonian matrix $\bm{H_{0}}$, with elements
$[\bm{H_{0}}]_{ab}={{}_{0}}\langle\phi_{a}|H_{0}|\phi_{b}\rangle_{0}$. The
unitary matrix $\bm{U_{0}}$ is constructed such that
$\bm{U_{0}}^{\dagger}\bm{H_{0}}\bm{U_{0}}=\bm{D_{0}}$ is diagonal. Eigenstates
$|\psi_{k}\rangle_{0}=\sum_{a}[\bm{U_{0}}]_{ak}|\phi_{a}\rangle_{0}$ are used
to construct $\bm{H_{1}}$, with matrix elements
${{}_{1}}\langle\phi_{j;i}|\hat{H}_{1}|\phi_{j^{\prime};i^{\prime}}\rangle_{1}$
defined in terms of basis states
$|\phi_{j;i}\rangle_{1}=|\psi_{j}\rangle_{0}\otimes|\gamma_{i}\rangle_{1}$
spanning the Fock space of $H_{1}$. Here $\\{|\gamma_{i}\rangle_{n}\\}$ are
the 16 states defined on the $n^{\rm th}$ Wilson shell corresponding to
$f_{\alpha n\sigma}$. We now diagonalize the matrix $\bm{H_{1}}$ to find
$\bm{U_{1}}$.

Truncation of the $M$-dimensional Fock space of $H_{n}$ is accomplished in NRG
by discarding high energy states at that iteration.[33, *bulla2008numerical]
$M_{K}$ eigenstates of $H_{n}$ are retained up to an energy cutoff $E_{\rm
cut}$. The $M\times M$ matrix $\bm{U_{n}}$ is therefore reshaped to an
$M\times M_{K}$ matrix $\bm{\tilde{U}_{n}}$. This implies that
$\bm{\tilde{U}_{n}}^{\dagger}\bm{H_{n}}\bm{\tilde{U}_{n}}=\bm{\tilde{D}_{n}}$
is a diagonal matrix of dimension $M_{K}\times M_{K}$ containing only the
lowest eigenvalues of $H_{n}$. The restricted set of $M_{K}$ retained
eigenvectors at iteration $n$ are obtained as
$|\psi_{k}\rangle_{n}=\sum_{j,i}[\bm{\tilde{U}_{n}}]_{(j;i)k}|\phi_{j;i}\rangle_{n}$.
Only these retained states at iteration $n$ are used to construct the
Hamiltonian matrix at iteration $n+1$. The approximate Hamiltonian in the
basis of
$|\phi_{j;i}\rangle_{n+1}=|\psi_{j}\rangle_{n}\otimes|\gamma_{i}\rangle_{n+1}$
is denoted $\bm{\tilde{H}_{n+1}}$. The truncation at each step means that the
Fock space dimension is roughly independent of $n$.

Crucially, the structure of the Wilson chain implies that discarded states at
a given iteration are unimportant for constructing the low-energy states at a
later iteration.[33, *bulla2008numerical] Furthermore, useful information can
be extracted at each step[33, *bulla2008numerical] since $\bm{\tilde{H}_{n}}$
may be regarded as a renormalized version of the full Hamiltonian at an
effective temperature $T_{n}\sim D\Lambda^{-n/2}$.

With this formalism established, we return to the calculation of the
correlator $\tilde{K}(\omega,T)$. The exact result for the discretized model
could in principle be obtained from Eq. 45 given the complete set of exact
eigenstates of $H^{\rm disc}$. In NRG however, the iterative diagonalization
and truncation procedure means that only the approximate (renormalized)
eigenstates of $\bm{\tilde{H}_{n}}$ at each step are known. The Anders-
Schiller (AS) basis[85, *anders2005real] comprises the _discarded_ states at
each step and is a complete, albeit approximate, basis with which to compute
spectral functions via the Lehmann sum, Eq. 45. Refs. 35, 83 reformulated the
problem in terms of the full density matrix (FDM) established on the AS basis,
wherein the spectrum
$\mathcal{A^{BC}}(\omega,T)=\sum_{n}w_{n}\mathcal{A}^{\mathcal{BC}}_{n}(\omega,T)$
consists of a weighted sum of contributions from each NRG iteration. This
requires matrix representations of the operators $\hat{\mathcal{B}}$ and
$\hat{\mathcal{C}}$ at iteration $n$ in the eigenbasis of
$\bm{\tilde{H}_{n}}$. For the correlator $\tilde{K}(\omega,T)$ we require[87]
specifically
$\bm{\tilde{N}^{\alpha}_{n}}=\sum_{m=0}^{n}\bm{\tilde{N}^{\alpha,m}_{n}}$,
where
$[\bm{\tilde{N}^{\alpha,m}_{n}}]_{kk^{\prime}}={{}_{n}}\langle\psi_{k}|\hat{n}^{f}_{\alpha
m}|\psi_{k^{\prime}}\rangle_{n}$. For $n=0$,
$\bm{\tilde{N}^{\alpha}_{0}}\equiv\bm{\tilde{N}^{\alpha,0}_{0}}$ can be
explicitly evaluated from the known exact eigenstates of $H_{0}$. For $n>0$ we
split
$\bm{\tilde{N}^{\alpha}_{n}}=\bm{\tilde{N}^{\alpha,<}_{n}}+\bm{\tilde{N}^{\alpha,n}_{n}}$
into two contributions. The on-shell term can be directly evaluated at each
step,
$[\bm{\tilde{N}^{\alpha,n}_{n}}]_{kk^{\prime}}=\sum_{j,i}[\bm{\tilde{U}_{n}}]^{*}_{(j;i)k}[\bm{\tilde{U}_{n}}]_{(j;i)k^{\prime}}\times{{}_{n}}\langle\gamma_{i}|\hat{n}^{f}_{\alpha
n}|\gamma_{i}\rangle_{n}$ since the matrix elements involved are trivial;
while
$[\bm{\tilde{N}^{\alpha,m}_{n}}]_{kk^{\prime}}=\sum_{j,j^{\prime},i}[\bm{\tilde{U}_{n}}]^{*}_{(j;i)k}[\bm{\tilde{U}_{n}}]_{(j^{\prime};i)k^{\prime}}\times[\bm{\tilde{N}^{\alpha,m}_{n-1}}]_{jj^{\prime}}$
for $0\leq m<n$. The latter property implies the recursion relation
$[\bm{\tilde{N}^{\alpha,<}_{n}}]_{kk^{\prime}}=\sum_{j,j^{\prime},i}[\bm{\tilde{U}_{n}}]^{*}_{(j;i)k}[\bm{\tilde{U}_{n}}]_{(j^{\prime};i)k^{\prime}}\times[\bm{\tilde{N}^{\alpha}_{n-1}}]_{jj^{\prime}}$.
The required operator matrices $\bm{\tilde{N}^{\alpha}_{n}}$ can then all be
obtained iteratively, starting from $\bm{\tilde{N}^{\alpha}_{0}}$.

With this, the correlator $\tilde{K}(\omega,T)$ can be obtained using standard
FDM-NRG as described in Refs. 35.

We emphasize that the technical implementation of the above is rather
straightforward within the usual NRG framework, and is compatible with abelian
and non-abelian quantum numbers in general symmetry settings,[88] and with
differentiable programming within $\partial$NRG.[89] The number operator
matrix elements can also be calculated on the generalized Wilson chain used in
the interleaved NRG (iNRG) method.[90, *stadler2016interleaved]

###  IV.2 Numerical Results for the AIM

As a proof-of-principle demonstration, we calculate the ac electrical
conductance $G_{C}(\omega,T)$ vs driving frequency $\omega$ at $T=0$ with NRG
for the two-lead AIM (Eq. 8) using the Kubo formula Eq. 36. In Figs. 2(a-f) we
compare the standard implementation of the Kubo formula Eq. 38 (left panels)
with the “improved” Kubo formula Eq. 44 (right panels), for different numbers
$M_{K}$ of NRG kept states as specified in the legend. (a-b) are calculated
for NRG discretization parameter $\Lambda=2$; (c,d) for $\Lambda=2.5$; and
(e,f) for $\Lambda=3$. As a reference, we provide the MW result (dashed line)
calculated within NRG within the equivalent single-channel AIM (Eq. 9)
obtained via Eq. 28a, which should be considered the numerically-exact result
for this system.

Figure 2: (a)-(f) Comparison of NRG results for the Kubo (left) and “improved”
Kubo (right) formulae for the ac conductance $G_{C}(\omega,0)$ at $T=0$ for
the two-lead AIM, with different numbers $M_{K}$ of NRG kept states as given
in the legend, and different NRG discretization parameters $\Lambda$. Results
benchmarked against the MW formula, evaluated within the equivalent single-
channel AIM (red dashed line). (g) Low-frequency behaviour, comparing Kubo and
improved-Kubo with a Fermi liquid fit
$2-G_{C}(\omega,0)/G_{0}=c_{2}(\omega/T_{\rm K})^{2}+c_{3}|\omega/T_{\rm
K}|^{3}$. AIM parameters: $U_{d}=0.4D$, $\epsilon_{d}=-\tfrac{1}{2}U_{d}$,
$V_{s}=V_{d}=0.07D$.

Fig. 2(a) shows that the standard Kubo formula yields rather poor results,
even at large $M_{K}=10000$ for $\Lambda=2$. The situation improves with
increasing $\Lambda$ [see panels (c,e)] due to the tradeoff[92] between
$M_{K}$ and $\Lambda$. The best performance, with an error of $\sim 5\%$, was
obtained with $\Lambda=3$ and $M_{K}=10000$ [blue line, panel (e)], although
even here the results are not fully converged with respect to $M_{K}$. By
contrast, the improved Kubo results [panels (b,d,f)] are highly stable, and
essentially fully converged even for remarkably low $M_{K}=4000$ for all
$\Lambda$ considered. This shows that accurate results can be obtained within
NRG using the improved Kubo formulation at relatively low computational cost.

Fig. 2(g) shows the low-frequency behavior, comparing Kubo and improved Kubo
with a Fermi liquid fit. The improved Kubo formula is found to capture the
expected asymptotic form very accurately. Indeed, for $\Lambda=2$ and
$M_{K}=10000$, the improved Kubo method satisfies the exact dc limit result
$G_{C}(0,0)/G_{0}=2$ to within $0.01\%$.

Although in this benchmark comparison, accurate results can be alternatively
obtained at low computational cost from the MW formula using the equivalent
single-channel model, we emphasize that for general non-PC systems this is not
possible; and genuine two-channel NRG calculations are notoriously demanding.

Note that the Hubbard satellite feature at $\omega\sim U_{d}$ seen in the MW
result is not captured by either Kubo approach. The lack of resolution at high
energies appears to be due to the logarithmic discretization of the bath
Hamiltonian in NRG. All other features are well-described; and the improved-
Kubo approach confers a significant accuracy and efficiency gain.

##  V Failure of heat transport
Kubo formula in NRG

Although the Kubo formula for charge transport as implemented in NRG can yield
highly accurate results, the same is unfortunately not true for heat
transport. Perhaps surprisingly, we find that the Kubo formula for heat
transport, Eq. 41, does not yield even qualitatively correct results when
implemented in NRG.

In NRG the required current-current correlation function
$K^{\prime}(\omega,T)$ takes the form,

$\displaystyle\langle\langle\dot{H}_{\rm leads}^{s};\dot{H}_{\rm
leads}^{d}\rangle\rangle$
$\displaystyle=-V_{s}V_{d}t_{0}^{2}\sum_{\sigma,\sigma^{\prime}}\langle\langle\hat{\mathcal{O}}_{s\sigma};\hat{\mathcal{O}}_{d\sigma^{\prime}}\rangle\rangle\;,$
(50a) $\displaystyle{\rm with,}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \hat{\mathcal{O}}_{\alpha\nu}$
$\displaystyle=\bar{d}^{\dagger}_{\alpha\nu}f_{\alpha
1\nu}-f^{\dagger}_{\alpha 1\nu}\bar{d}_{\alpha\nu}\;,$ (50b)

where $f_{\alpha 1\nu}$ is the $n=1$ Wilson orbital defined in Eq. 46.
Although the $n=0$ Wilson orbital $f_{\alpha 0\nu}$ may be interpreted as a
discretized version of the local lead orbital $c_{\alpha\nu}$, the $n=1$
Wilson orbital $f_{\alpha 1\nu}$ has no direct physical meaning in the bare
model.

Figure 3: (a) Heat conductance $|K_{Q}(T)|$ vs $T$ for the two-lead AIM,
calculated in NRG via the Kubo formula (Eq. 41; red line), compared with the
MW result (Eq. 28b; blue line). Solid lines for $K_{Q}>0$, dashed for
$K_{Q}<0$. Parameters used: $U_{d}=0.4D$, $\epsilon_{d}=-U_{d}/2$,
$V_{s}=V_{d}=0.07D$, $\Lambda=3$, $M_{K}=11000$ (2-channel Kubo) or $4000$
(1-channel MW). (b) Same as panel (a) except for dc charge conductance
$G_{C}(T)$, obtained with NRG via the standard Kubo formula (Eq. 36; red line)
or MW (Eq. 28a; blue line).

We use Eq. 50 to calculate $K_{Q}(T)$ via Eq. 41 for the two-lead AIM using
NRG, and present numerical results in Fig. 3(a) as the red line. For
comparison, we show the MW result computed via Eq. 28b for the effective one-
channel system as the blue line. The latter should be regarded as the
numerically-exact result. We see that $K_{Q}(T)$ from the Kubo formula fails
to capture the correct physics, with apparently noisy data, erratic sign-
changes, and diverging behavior at low-$T$ rather than vanishing as $\sim T$.
However, other physical quantities computed in the same NRG run, such as the
electrical conductance $G_{c}(T)$ obtained by the Kubo formula Eq. 36,
correctly recover the expected results, see Fig. 3(b).

The high-quality two-channel NRG calculations here are fully converged (Fig.
3(a) is not improved by increasing the number of kept states $M_{K}$). We have
also implemented an “improved” version of the heat Kubo formula along the
lines of Sec. IV, but the underlying failure of NRG demonstrated above is not
resolved by this.

We note that the heat Kubo formula for the exactly-solvable $U_{d}=0$ resonant
level model coupled to Wilson chains also fails to capture the correct
behavior, including the low-$T$ asymptote $K_{Q}(T)\sim T$. This indicates
that the problem lies with the representation of the lead Hamiltonian as a
discretized Wilson chain, rather than with the iterative diagonalization
procedure in NRG. We therefore believe the failure of NRG in this context to
be because Wilson chains are not true thermal reservoirs,[93] as argued in
Ref. 94 – arbitrary amounts of energy $\Delta E$ cannot be dissipated by
Wilson chains without changing their temperature. We hypothesize that this
issue may afflict thermal transport calculations using the Kubo formula for
_any_ discretized system.

We emphasize that the above breakdown is not a problem with the heat Kubo
formula itself, but rather a failure of NRG to capture the proper behavior of
the required current-current correlation function defined on the discretized
Wilson chain, Eq. 50. Note that for systems in the PC geometry, the MW formula
Eq. 28b can be used instead. This relates the heat conductance in linear
response to nanostructure equilibrium retarded Green’s functions, which are
very accurately calculated within NRG.[35] This approach within NRG yields the
correct behavior of $K_{Q}(T)$, as demonstrated in Ref. 19. The problem of how
to obtain $K_{Q}(T)$ via NRG for non-PC systems therefore remains an open
question.

##  VI Emergent Proportionate Coupling: Coulomb Blockade

In most realistic systems (with the notable exception of single-dot devices)
the orbital complexity and spatial structure of the nanostructure prevents a
PC description. However, in certain situations the low-energy effective model
for the system may be in PC even though the bare model is not. This is
particularly useful for the MW formulation of quantum transport involving only
retarded single-particle nanostructure Green’s functions. We explore these
scenarios below for the case of systems in the Coulomb blockade (CB) regime.

Deep in the CB regime, the nanostructure has a well-defined number of
electrons, with charge fluctuations suppressed at low temperatures $k_{\rm
B}T\ll E_{C}$ by the effective nanostructure charging energy $E_{C}$ (the
microscopic origin of which is the Coulomb repulsion described by $H_{\rm
int}$ in Eq. 2). Incoherent transport, involving sequential tunneling events
between the nanostructure and leads, is therefore also suppressed.[95, 96, 97]
This is the dominant transport mechanism for nanostructures with a net spin
$S=0$. By contrast, Kondo-enhanced spin-flip scattering in spinful
nanostructures can give a substantial boost to low-temperature coherent
transport, with the unitarity limit of a perfect single-electron transistor
attainable experimentally.[98]

In both cases, the structure of the low-energy effective model can be obtained
perturbatively to second order in $H_{\rm hyb}$, by projecting onto the ground
state manifold of $N$-electron states for the isolated nanostructure $|N;{\rm
gs},j\rangle$, and eliminating virtual excitations to nanostructure states
$|N\pm 1;{\rm ex},k\rangle$ with $N\pm 1$ electrons. Using Brillouin-Wigner
perturbation theory, we may write,

$H_{\rm eff}^{N}\simeq H_{\rm leads}+\hat{1}_{\rm gs}^{N}H_{\rm
nano}\hat{1}_{\rm gs}^{N}+\hat{1}_{\rm gs}^{N}H_{\rm hyb}(E_{\rm
gs}^{N}-H_{\rm nano})^{-1}H_{\rm hyb}\hat{1}_{\rm gs}^{N}\;,$ (51)

where $\hat{1}_{\rm gs}^{N}=\sum_{j}|N;{\rm gs},j\rangle\langle N;{\rm gs},j|$
and $H_{\rm nano}|N;{\rm gs},j\rangle=E_{{\rm gs}}^{N}|N;{\rm gs},j\rangle$.
By construction, $H_{\rm hyb}|N;{\rm gs},j\rangle$ generates (a superposition
of) excited eigenstates of $H_{\rm nano}$ satisfying $H_{\rm nano}|N\pm 1;{\rm
ex},k\rangle=E_{{\rm ex},k}^{N\pm 1}|N\pm 1;{\rm ex},k\rangle$.

Parameters of the effective model are then related to matrix elements of the
type $\langle N+1;{\rm ex},k|\bar{d}_{\alpha\sigma}^{\dagger}|N;{\rm
gs},j\rangle$ and $\langle N-1;{\rm
ex},k|\bar{d}_{\alpha\sigma}^{\phantom{\dagger}}|N;{\rm gs},j\rangle$, which
may be computed straightforwardly for the isolated nanostructure for a given
model.

We note that although the structure of the effective model for generalized
quantum impurity systems can be obtained from Eq. 51, the perturbative
estimation of its coupling constants is accurate only in the limit of strong
nanostructure interactions and weak hybridization. Recently in Ref. 99,
machine learning techniques have been employed to determine the numerical
value of the effective model parameters non-perturbatively. We illustrate the
quantitative accuracy of such a model machine learning approach to describe
the low-temperature physics of nanostructures by explicit calculations in Sec.
IX.1.

In the following we focus on the generic behavior of the underlying effective
models. Ultimately, the accuracy of predictions using these models depends on
the accurate determination of the effective parameters.

###  VI.1 $S=0$

For isolated nanostructures with an even number of electrons $N$, the ground
state is often (but not always) a unique spin singlet state with total $S=0$.
Applying Eq. 51 yields an effective model in the form of a renormalized tunnel
junction,

$H_{\rm eff}=H_{\rm
leads}+\sum_{\alpha,\beta}\sum_{\sigma}W_{\alpha\beta}^{\sigma}c_{\alpha\sigma}^{\dagger}c_{\beta\sigma}^{\phantom{\dagger}}\;,$
(52)

where $\alpha,\beta\in s,d$ and
$W_{\alpha\beta}^{\sigma}=(W_{\beta\alpha}^{\sigma})^{*}$ are effective
potential scattering parameters obtained after integrating out the
nanostructure. Note that in the presence of a magnetic field on the
nanostructure, the excited states $|N\pm 1;{\rm ex},k\rangle$ and excitation
energies $(E_{\rm gs}^{N}-E_{{\rm ex},k}^{N\pm 1})$ depend on spin $\sigma$,
which in general endow $W_{\alpha\beta}^{\sigma}$ with a spin dependence,
which we retain here for generality.

Eq. 52 is in some sense trivially in PC. Since the effective model is non-
interacting, we may use the Landauer formula for conductance, Eqs. 17, 18b. In
the low-temperature limit $T\to 0$, the electrical conductance is given by
$G_{C}(0)=(e^{2}/h)4\tilde{\Gamma}_{s}\tilde{\Gamma}_{d}\sum_{\sigma}|\mathcal{G}_{sd,\sigma}(0)|^{2}$.
Here $\mathcal{G}_{sd,\sigma}(\omega)\equiv\langle\langle
c_{s\sigma}^{\phantom{\dagger}};c_{d\sigma}^{\dagger}\rangle\rangle$ is the
full Green’s function connecting local orbitals in source and drain leads in
the presence of the $W_{\alpha\beta}^{\sigma}$ terms in Eq. 52. It can be
expressed in terms of the free lead Green’s functions of the isolated $H_{\rm
leads}$, which we write as
$\mathcal{G}_{\alpha\beta,\sigma}^{0}(\omega)=\delta_{\alpha\beta}/[\omega^{+}-\Delta(\omega)]$,
with $\Delta(\omega)$ so defined. The effective hybridizations are then
$\tilde{\Gamma}_{\alpha}=-{\rm Im}\Delta(0)=1/(\pi\rho_{0})$, equal for both
leads. The required full Green’s function can be obtained from standard
equations of motion techniques as,

$\mathcal{G}_{sd,\sigma}(\omega)=\frac{W_{sd}^{\sigma}}{[\omega^{+}-W_{ss}^{\sigma}-\Delta(\omega)][\omega^{+}-W_{dd}^{\sigma}-\Delta(\omega)]-|W_{sd}^{\sigma}|^{2}}$
(53)

In the zero-field, particle-hole symmetric case where
$W_{ss}^{\sigma}=W_{dd}^{\sigma}=0$ and $W_{sd}^{\sigma}\equiv W_{sd}$, the
standard result for the $T=0$ transmission of a tunnel-junction is recovered,

$G_{C}(0)=\left(\frac{2e^{2}}{h}\right)\frac{4|\tilde{W}_{sd}|^{2}}{(1+|\tilde{W}_{sd}|^{2})^{2}}\;.$
(54)

where we have defined the dimensionless quantities
$\tilde{W}^{\sigma}_{\alpha\beta}=\pi\rho_{0}W_{\alpha\beta}^{\sigma}$. The
maximum conductance $G_{C}=2e^{2}/h$ is attained when
$|\tilde{W}_{sd}|^{2}=1$, which corresponds to intermediate values of
$W_{sd}$.

In the general case, one obtains,

$G_{C}(0)=\left(\frac{e^{2}}{h}\right)\sum_{\sigma}\frac{4|\tilde{W}_{sd}^{\sigma}|^{2}}{(1+|\tilde{W}_{sd}^{\sigma}|^{2}-\tilde{W}_{ss}^{\sigma}\tilde{W}_{dd}^{\sigma})^{2}+(\tilde{W}_{ss}^{\sigma}+\tilde{W}_{dd}^{\sigma})^{2}}$
(55)

The conductance is therefore seen to depend strongly on particle-hole
asymmetry, generated and tuned in practice by application of gate voltages. In
particular, the maximum conductance $G_{C}<2e^{2}/h$ for finite $W_{ss}$
and/or $W_{dd}$ is always reduced with respect to its particle-hole symmetric
counterpart.

###  VI.2 $S=\tfrac{1}{2}$

The more interesting and subtle case arises when the isolated nanostructure
has an odd number of electrons $N$ and hosts a net spin $S=\tfrac{1}{2}$. Eq.
51 then becomes a generalized Schrieffer-Wolff transformation.[100, 101] In
the absence of other ground state degeneracies, the effective model takes the
form of a generalized two-channel Kondo (2CK) model,[102]

$H_{\rm eff}=H_{\rm
leads}+\sum_{\alpha,\beta}\sum_{\sigma}W_{\alpha\beta}^{\sigma}c_{\alpha\sigma}^{\dagger}c_{\beta\sigma}^{\phantom{\dagger}}+\sum_{\alpha,\beta}J_{\alpha\beta}\hat{\bm{S}}\cdot\hat{\bm{s}}_{\alpha\beta}+\tilde{B}\hat{S}^{z}\;,$
(56)

where $\hat{\bm{S}}$ is a spin-$\tfrac{1}{2}$ operator for the nanostructure,
$\hat{S}^{z}$ is its $z$-component, and
$\hat{\bm{s}}_{\alpha\beta}=\tfrac{1}{2}\sum_{\nu,\nu^{\prime}}c_{\alpha\nu}^{\dagger}\bm{\sigma}_{\nu\nu^{\prime}}c_{\beta\nu^{\prime}}^{\phantom{\dagger}}$
is the spin density of the leads ($\bm{\sigma}$ is a vector of Pauli
matrices). The effective model parameters $J_{\alpha\beta}$ (exchange
coupling) and $W_{\alpha\beta}^{\sigma}$ (potential scattering) can be
straightforwardly calculated from matrix elements of the isolated
nanostructure states, as described in Ref. 103. Hermiticity requires that
$W_{\alpha\beta}^{\sigma}=(W_{\alpha\beta}^{\sigma})^{*}$ and
$J_{\alpha\beta}=(J_{\beta\alpha})^{*}$. In the absence of a magnetic field or
spin-orbit coupling terms in the bare system, the effective model has full
SU(2) symmetry and $W_{\alpha\beta}^{\sigma}\equiv W_{\alpha\beta}$
independent of spin, and $\tilde{B}=0$. We consider this case explicitly
below. In the following we use the notation $J_{\pm}=J_{ss}\pm J_{dd}$ and
$W_{\pm}=W_{ss}\pm W_{dd}$.

The dynamics of the effective model, discussed further below, are
characterized by the spectrum of the scattering t-matrix,
$t_{\alpha\alpha,\sigma}(\omega,T)=-\pi\rho_{0}{\rm
Im}T_{\alpha\alpha,\sigma}(\omega,T)$, where
$T_{\alpha\beta,\sigma}(\omega,T)$ is defined by the t-matrix equation Eq. 11.
For Eq. 56, the t-matrix can be expressed as
$T_{\alpha\beta,\sigma}(\omega,T)=W_{\alpha\beta}+\langle\langle\hat{a}_{\alpha\sigma}^{\phantom{\dagger}};\hat{a}_{\beta\sigma}^{\dagger}\rangle\rangle$
where,

$\hat{a}_{\alpha\uparrow}=\sum_{\gamma}\left[W_{\alpha\gamma}c_{\gamma\uparrow}+\tfrac{1}{2}J_{\alpha\gamma}(c_{\gamma\uparrow}\hat{S}^{z}+c_{\gamma\downarrow}\hat{S}^{-})\right]\;,$
(57)

and similarly for $\hat{a}_{\alpha\downarrow}$. The retarded correlator
$\langle\langle\hat{a}_{\alpha\sigma}^{\phantom{\dagger}};\hat{a}_{\beta\sigma}^{\dagger}\rangle\rangle$
can be computed directly in NRG. However, an ‘improved’ version of the
t-matrix can be obtained in a similar fashion to the self-energy method for
the retarded nanostructure Green’s functions, Eq. 31. We define a 2x2 matrix
Dyson equation for the full (impurity coupled) lead Green’s functions,
$[\bm{\mathcal{G}}_{\sigma}(\omega,T)]^{-1}=[\bm{\mathcal{G}}_{\sigma}^{0}(\omega)]^{-1}-\bm{\tilde{\Sigma}}_{\sigma}(\omega,T)$,
where the self-energy matrix $\bm{\tilde{\Sigma}}_{\sigma}(\omega,T)$
incorporates all the effects of the nanostructure. Rearranging Eq. 11 gives,

$\displaystyle\bm{\tilde{\Sigma}}_{\sigma}(\omega,T)=[\bm{\mathcal{G}}_{\sigma}(\omega,T)]^{-1}\bm{\mathcal{G}}_{\sigma}^{0}(\omega)\bm{T}_{\sigma}(\omega,T)\;.$
(58)

In practice we therefore compute both $\bm{\mathcal{G}}_{\sigma}(\omega,T)$
and $\bm{T}_{\sigma}(\omega,T)$ in NRG to obtain
$\bm{\tilde{\Sigma}}_{\sigma}(\omega,T)$ by Eq. 58. This gives an improved
estimation of $\bm{\mathcal{G}}_{\sigma}(\omega,T)$ from the Dyson equation,
and hence an improved version of $\bm{T}_{\sigma}(\omega,T)$ through the
t-matrix equation.

In terms of quantum transport, the Kubo formula can be used to obtain the dc
linear response electrical conductance (Eqs. 36, 37) for the effective model.
The retarded current-current correlator
$K(\omega,T)=\langle\langle\dot{N}^{s};\dot{N}^{d}\rangle\rangle\equiv\langle\langle\hat{\Omega};\hat{\Omega}\rangle\rangle$
can be expressed in terms of the composite operator,

$\displaystyle\hat{\Omega}=-i\left(J_{sd}\hat{\bm{S}}\cdot\hat{\bm{s}}_{sd}+\sum_{\sigma}W_{sd}^{\phantom{\dagger}}c_{s\sigma}^{\dagger}c_{d\sigma}^{\phantom{\dagger}}\right)-{\rm
H.c.}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (59)

where we have used $\dot{N}^{\alpha}=i[\hat{H},\hat{N}^{\alpha}]$ and current
conservation $\dot{N}^{s}=-\dot{N}^{d}$. The form of the ‘improved’ Kubo
formula (Sec. IV) is model-independent and so Eq. 44 remains the same. The
corresponding Kubo formula for heat conductance (Eq. 41) follows similarly.

To gain further insight into the expected behaviour and to make progress
analytically, we consider various limiting cases below.

####  VI.2.1 No potential scattering: $W_{\alpha\beta}=0$

For $W_{\alpha\beta}=0$, it proves useful to diagonalize the matrix of
exchange couplings, $\bm{U}^{\dagger}\bm{J}\bm{U}$, by a canonical rotation of
the lead basis $\bm{c_{\sigma}}=\bm{U}\bm{\tilde{c}_{\sigma}}$, where
$\bm{c_{\sigma}}=(c_{s\sigma},c_{d\sigma})^{\rm T}$ and
$\bm{\tilde{c}_{\sigma}}=(c_{e\sigma},c_{o\sigma})^{\rm T}$. In the new
‘even/odd’ basis, the exchange couplings are,

$\displaystyle J_{ee}$
$\displaystyle=\tfrac{1}{2}(J_{+}+\sqrt{J_{-}^{2}+4|J_{sd}|^{2}})$ (60a)
$\displaystyle J_{oo}$
$\displaystyle=\tfrac{1}{2}(J_{+}-\sqrt{J_{-}^{2}+4|J_{sd}|^{2}})$ (60b)

and where $J_{eo}=J_{oe}=0$ by construction.

A special case arises when $J_{oo}=0$, since then the odd lead is decoupled
and the effective model reduces precisely to a single-channel Kondo model with
a single spin-$\tfrac{1}{2}$ impurity,

$H_{\rm eff}=H_{\rm leads}^{e}+J_{ee}\hat{\bm{S}}\cdot\hat{\bm{s}}_{ee}\;.$
(61)

From Eq. 60b, this occurs specifically when $|J_{sd}|^{2}=J_{ss}J_{dd}$. In
fact, this condition is always automatically satisfied when the bare model is
in PC. Indeed, this is physically natural since Eq. 61 is the regular
Schrieffer-Wolff transformation[100] of the single-channel model using Eq. 5
obtained under PC.

The conductance then follows from Eq. 28 with

$\displaystyle\mathcal{T}_{\rm MW}(\omega,T)\overset{{\rm
PC}}{\longrightarrow}\frac{4J_{ss}J_{dd}}{(J_{ss}+J_{dd})^{2}}\sum_{\sigma}t_{ee,\sigma}(\omega,T)\;.$
(62)

With no potential scattering and no magnetic field as considered here,
$t_{ee,\sigma}(0,0)=1$ by the Friedel sum rule,[8] and so
$G_{C}(0)=(2e^{2}/h)\times\tfrac{4J_{ss}J_{dd}}{(J_{ss}+J_{dd})^{2}}$ is
controlled by a pure geometric factor describing the nanostructure-lead
coupling. Indeed for $\omega,T\ll T_{K}$, with $T_{K}\sim
D\exp[-1/\rho_{0}J_{ee}]$ the Kondo temperature for the effective model Eq.
61, we have $t_{ee,\sigma}(\omega,T)\simeq 1$ and so the conductance
approximately saturates its low-temperature bound, $G_{C}(T\ll T_{K})\simeq
G_{C}(0)$.

Figure 4: NRG results for the $T=0$ t-matrix spectrum
$t_{\alpha\alpha}(\omega,0)$ of the effective 2CK model, Eq. 63. Even (odd)
channel spectra shown as the solid (dashed) lines, for fixed $J_{ee}/D=0.3$
and $\bm{W}=\bm{0}$, with different $J_{oo}$. In all cases, $t_{ee}(0,0)=1$
and $t_{oo}(0,0)=0$, despite incipient frustration of Kondo screening for
small $\delta=J_{ee}-J_{oo}$.

Of course, the above scenario is special, and for most realistic systems the
exact PC condition $|J_{sd}|^{2}=J_{ss}J_{dd}$ is not expected to be
satisfied. In such cases, $J_{oo}\neq 0$ and the odd channel remains formally
coupled to the effective nanostructure spin-$\tfrac{1}{2}$. In the even/odd
basis with $W_{\alpha\beta}=0$, Eq. 56 reduces to,

$H_{\rm eff}=H_{\rm
leads}+J_{ee}\hat{\bm{S}}\cdot\hat{\bm{s}}_{ee}+J_{oo}\hat{\bm{S}}\cdot\hat{\bm{s}}_{oo}\;.$
(63)

The physics of this channel-anisotropic 2CK model are rich but well-known,
featuring a competition of Kondo screening between the even and odd
channels.[104] The model supports a non-Fermi liquid quantum critical point
for $J_{ee}=J_{oo}$, but from Eq. 60 we see that this only arises in these
kinds of systems when both $J_{ss}=J_{dd}$ and $J_{sd}=0$. Although one may be
able to tune to the $sd$-symmetric condition $J_{ee}=J_{oo}$, to realize
$J_{sd}=0$ requires the suppression of through-nanostructure exchange-
cotunneling processes. In principle, this could arise through gate-tunable
many-body quantum interference effects[103] that conspire to produce an exact
tunneling node, but a real system exhibiting 2CK criticality driven by such
effects has not yet been reported. On the other hand, for complex multiorbital
nanostructures, electron propagation across the entire structure embodied by
the exchange cotunneling $J_{sd}$ is typically small in magnitude (albeit
finite) compared with the local terms $J_{ss}$ and $J_{dd}$. Therefore it may
still be possible to access the quasi-frustrated quantum critical physics of
the 2CK model at small $J_{sd}$ for intermediate temperatures or
energies.[105, *vzitko2006kondo] In the following we simply regard $J_{ee}$
and $J_{oo}$ as free independent parameters of the effective model, however
obtained, and consider the generic behaviour.

For concreteness we now assume antiferromagnetic couplings $J_{ee}>J_{oo}>0$.
Perturbative scaling (poor man’s scaling) starting from weak coupling,
indicates that both $J_{ee}$ and $J_{oo}$ get renormalized upwards on reducing
the energy or temperature scale. The scaling invariant for the RG flow of each
is its respective Kondo temperature, $T_{\rm K}^{\alpha}\sim
D\exp[-1/\rho_{0}J_{\alpha\alpha}]$. However, the low-energy physics is non-
perturbative, and the 2CK strong coupling fixed point for a
spin-$\tfrac{1}{2}$ impurity is unstable.[50] Since $J_{ee}>J_{oo}$ we have
$T_{\rm K}^{e}>T_{\rm K}^{o}$ and the even channel ultimately flows under RG
to strong coupling while the odd channel flows back to weak coupling.

This physical picture is confirmed by numerically-exact NRG calculations for
the effective 2CK model Eq. 63, in Fig. 4. The RG flow is vividly demonstrated
in the $T=0$ spectrum of the t-matrix for the even channel (solid lines) and
odd channel (dashed lines), plotted for systems with fixed $J_{ee}$ but
decreasing $\delta=J_{ee}-J_{oo}>0$ (and $W_{\alpha\beta}=0$). We highlight
three features[58]: (i) in all cases, the t-matrix spectrum of the even
channel is always exactly pinned to 1 at zero frequency and zero temperature,
$t_{ee}(0,0)=1$; (ii) by contrast, the odd channel t-matrix vanishes,
$t_{oo}(0,0)=0$; (iii) relief of incipient frustration is characterized by a
new scale $T^{\rm FL}$.

Importantly, we see an emergent decoupling of the odd lead at low energies
$|\omega|\ll T^{\rm FL}$ in all cases, where $T^{\rm FL}=\min(T_{\rm
K}^{e},T^{*})$ is a dynamically-generated Fermi liquid scale. Here
$T^{*}\sim\delta^{2}/T_{\rm K}^{e}$ is an energy scale embodying the
frustration of Kondo screening from the two channels. For large $\delta>T_{\rm
K}^{e}$ we have $T^{*}>T_{\rm K}^{e}$ and the RG flow of the odd channel is
cut off on the scale of $T_{\rm K}^{e}$ as the even channel flows to strong
coupling. On the other hand, for small $\delta<T_{\rm K}^{e}$ we have
$T^{*}<T_{\rm K}^{e}$ and the system is described by the frustrated (non-Fermi
liquid) 2CK fixed point over an intermediate temperature range $T^{*}<T<T_{\rm
K}^{e}$. The even channel flows to strong coupling and the odd channel flows
to weak coupling only for $T\ll T^{*}$ in this case. Without potential
scattering, it is then guaranteed that $t_{ee}(0,0)=1$ and $t_{oo}(0,0)=0$, as
dictated by phase-shift arguments: due to the decoupling at $\omega=T=0$, we
may write $t_{\alpha\alpha}(0,0)=\sin^{2}(\delta_{\alpha})$. The Kondo effect
in the even channel confers a phase shift $\delta_{e}=\pi/2$ whereas in the
decoupled odd channel $\delta_{o}=0$. We note that although the non-Fermi
liquid 2CK regime has been accessed experimentally in quantum dot devices,[53,
*ralph19942] this is the result of fine-tuning and ingenious quantum
engineering. The more standard situation is for the odd channel to decouple at
low temperatures, as described above.

Since such systems flow under RG to an effective single-channel description
involving only the even lead combination, we have an emergent PC condition.
This greatly simplifies the calculation and interpretation of the low-
temperature quantum transport. On the lowest temperature and energy scales
$\omega,T\ll T^{\rm FL}$, Eq. 24 yields the electrical conductance. We may now
relate the inter-channel t-matrix $T_{sd,\sigma}$ to the even channel t-matrix
$T_{ee,\sigma}$ using the rotation $\bm{U}$ diagonalizing the exchange matrix,
$T_{ds,\sigma}(0,0)=\sum_{(\alpha,\beta)\in(e,o)}U_{d\alpha}U_{s\beta}^{*}T_{\alpha\beta,\sigma}(0,0)$.
Note that $T_{eo,\sigma}(\omega,T)=T_{oe,\sigma}(\omega,T)=0$ since
$J_{eo}=W_{eo}=0$, while for $W_{\alpha\beta}=0$ we have $T_{oo}(0,0)=0$ and
$|\pi\rho_{0}T_{ee,\sigma}(0,0)|^{2}=[t_{ee,\sigma}(0,0)]^{2}=1$ due to the
emergent decoupling. Therefore the $T=0$ conductance follows as,

$G_{C}(0)=\left(\frac{2e^{2}}{h}\right)\frac{4|J_{sd}|^{2}}{4|J_{sd}|^{2}+J_{-}^{2}}\;.$
(64)

This result also applies accurately for all $T\ll T^{\rm FL}$. The low-
temperature conductance can therefore be determined directly from the
effective exchange couplings $J_{ss}$, $J_{dd}$, and $J_{sd}$. Eq. 64 is as
such the generalization of Eq. 30 to the emergent PC case.

Indeed, the factor $|J_{sd}|^{2}/(|J_{sd}|^{2}+J_{-}^{2})$ appearing in Eq. 64
reduces, as it must to $|V_{s}V_{d}|^{2}/(|V_{s}|^{2}+|V_{d}|^{2})^{2}$ in the
case of exact PC in the bare model, for which we have
$|J_{sd}|^{2}=J_{ss}J_{dd}$ and $J_{ss}/J_{dd}=|V_{s}/V_{d}|^{2}$. This is
equivalent to the well-known geometric factor
$\Gamma_{s}\Gamma_{d}/(\Gamma_{s}+\Gamma_{d})^{2}$ appearing in Eq. 30 after
cancelling factors of $\pi$ and $\rho_{0}$.

Figure 5: NRG results for the $T=0$ t-matrix spectrum of the effective 1CK
model Eq. 61 (left panel), and 2CK model Eq. 67 (even/odd channels in the
centre/right panels). For the 2CK model, we take the $sd$-symmetric case
$J_{-}=W_{-}=0$, with fixed $J/D=0.25$ and $J_{sd}/D=0.05$, varying
$W=3W_{sd}$. For the 1CK model we use the same parameters as for the even
channel of the corresponding 2CK model ($J_{ee}/D=0.3$ and
$W_{ee}=\tfrac{4}{3}W$). Dotted lines in the left/centre panels are given by
Eq. 69, while Eq. 68 is used in the right panel.

####  VI.2.2 $sd$-symmetry and finite potential scattering

The results of the previous section hold when $W_{\alpha\beta}=0$. Although an
important simple limit, the resulting behavior is not generic since even at
particle-hole symmetry ($W_{ss}=W_{dd}=0$), the through-nanostructure
cotunneling term $W_{sd}$ is typically finite. In this section we consider
finite potential scattering $W_{\alpha\beta}\neq 0$ in the $sd$-symmetric
limit – meaning $J_{ss}=J_{dd}\equiv J$ and $W_{ss}=W_{dd}\equiv W$ (or
equivalently $J_{-}=W_{-}=0$). In this case, the full model has an overall
mirror symmetry with respect to exchanging $s$ and $d$ leads

At high temperatures $T\gg T_{\rm K}$, renormalization due to the Kondo effect
is weak, and the contribution to conductance is dominated by finite source-
drain cotunneling, $W_{sd}$. Conductance in this limit is therefore given
approximately by Eq. 55, obtained for $J_{\alpha\beta}=0$. However, richer and
more complex behavior results at lower temperatures, where there is a subtle
interplay between $\bm{W}$ and $\bm{J}$. We therefore focus on the low-$T$
physics in the following.

First, we note that in the $sd$-symmetric limit, the even/odd channels are
simply the symmetric/antisymmetric combinations of source and drain leads,

$\displaystyle
c_{e\sigma}=\tfrac{1}{\sqrt{2}}[c_{s\sigma}+c_{d\sigma}]\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ ;\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
c_{o\sigma}=\tfrac{1}{\sqrt{2}}[c_{s\sigma}-c_{d\sigma}]$ (65)

which yields,

$\displaystyle J_{ee}$ $\displaystyle=J+J_{sd}$ $\displaystyle;\qquad J_{oo}$
$\displaystyle=J-J_{sd}$ (66a) $\displaystyle W_{ee}$ $\displaystyle=W+W_{sd}$
$\displaystyle;\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ W_{oo}$ $\displaystyle=W-W_{sd}$
(66b)

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

$H_{\rm eff}=\tilde{H}_{\rm
leads}+J_{ee}\hat{\bm{S}}\cdot\hat{\bm{s}}_{ee}+J_{oo}\hat{\bm{S}}\cdot\hat{\bm{s}}_{oo}\;,$
(67)

where we have incorporated the local potentials $W_{ee}$ and $W_{oo}$ into the
definition of the lead Hamiltonian, $\tilde{H}_{\rm
leads}=\sum_{\alpha=e,o}[H_{\rm
leads}^{\alpha}+W_{\alpha\alpha}^{\phantom{\dagger}}\sum_{\sigma}c_{\alpha\sigma}^{\dagger}c_{\alpha\sigma}^{\phantom{\dagger}}]$.

Analysis of Eq. 67 proceeds similarly to that of Eq. 63, except now we account
for the potential scattering by rediagonalizing the lead Hamiltonian
$\tilde{H}_{\rm leads}$. This can be done implicitly using Green’s function
techniques, which allow us to relate the local lead Green’s functions
including the potential to those of the bare leads. We find
$[\tilde{\mathcal{G}}^{0}_{\alpha\alpha,\sigma}(\omega)]^{-1}=[\mathcal{G}^{0}_{\alpha\alpha,\sigma}(\omega)]^{-1}-W_{\alpha\alpha}$
for $\alpha=e,o$. This implies that the renormalized Fermi level density of
states is $\tilde{\rho}_{\alpha}=-\tfrac{1}{\pi}{\rm
Im}\tilde{\mathcal{G}}^{0}_{\alpha\alpha,\sigma}(0)=\rho_{0}/(1+\tilde{W}_{\alpha\alpha}^{2})$,
where we again use the notation
$\tilde{W}_{\alpha\beta}=\pi\rho_{0}W_{\alpha\beta}$.

In fact, the quantities controlling the Kondo physics and RG flow in such
problems are the dimensionless parameters
$j_{\alpha}=\tilde{\rho}_{\alpha}J_{\alpha\alpha}$, rather than the bare
exchange couplings $J_{\alpha\alpha}$. For $j_{e}>j_{o}$, the even lead flows
to strong coupling, while the odd lead decouples; the opposite applies for
$j_{e}<j_{o}$. Indeed, even with $J_{ee}>J_{oo}$ and $W_{ee}>W_{oo}$, one may
be able to realize $j_{e}=j_{o}$, as required for 2CK criticality. It may
therefore be possible to tune across the 2CK quantum phase transition in a
given system by tuning gate voltages.

We now explore the effect of the local potentials $W_{ee}$ and $W_{oo}$ on the
dynamics and conductance. For $W_{\alpha\beta}=0$ considered in the previous
section, we argued that the emergent decoupling of the odd lead at low
energies and temperatures $\omega,T\ll T^{\rm FL}$ produced an effective PC
condition, with $t_{ee,\sigma}(0,0)$ and hence $G_{C}(0)$ then being
controlled by an effective _single_ channel Kondo (1CK) model, Eq. 61. With
finite $W_{ee}$ and $W_{oo}$, the lead with the smaller $j_{\alpha}$ still
decouples asymptotically. However, $t_{ee,\sigma}(0,0)<1$ and
$t_{oo,\sigma}(0,0)>0$ due to the potential scattering.

This physical picture is confirmed in Fig. 5 where we plot the $T=0$ spectra
of the even and odd t-matrix for the 2CK model Eq. 67 in the centre and right
panels, for fixed exchange couplings $J_{ee}$ and $J_{oo}$, but with different
potential scattering strengths $W=3W_{sd}$. Below an emergent scale $T^{\rm
FL}\sim T_{\rm K}^{e}$ (here $\sim 10^{-3}D$), the t-matrix of the even
channel exhibits a Kondo resonance, while the incipient RG flow of the odd
channel is arrested.

Since the odd channel is decoupled at $\omega=T=0$, the t-matrix
$t_{oo,\sigma}(0,0)$ is determined completely by $W_{oo}$. From the t-matrix
equation, Eq. 11, together with the bare and renormalized Green’s functions
$\mathcal{G}^{0}_{oo,\sigma}(0)=-i\pi\rho_{0}$ and
$\tilde{\mathcal{G}}^{0}_{oo,\sigma}(0,0)=-i\pi\tilde{\rho}_{o}$, we
immediately obtain,

$t_{oo,\sigma}(0,0)=\frac{\tilde{W}_{oo}^{2}}{1+\tilde{W}_{oo}^{2}}\;.$ (68)

This result is equivalent to a phase shift
$\delta_{o}=\tan^{-1}[\tilde{W}_{oo}]$ in the odd channel, since
$t_{oo,\sigma}(0,0)=\sin^{2}(\delta_{o})$ [56]. The full odd-channel t-matrix
is $\pi\rho_{0}T_{oo,\sigma}(0,0)=(\tilde{W}^{-1}_{oo}-i)\times
t_{oo,\sigma}(0,0)$.

For the systems considered in Fig. 5, we plot the corresponding values of
$t_{oo,\sigma}(0,0)$ obtained from Eq. 68 in the right panel as the dotted
horizontal lines. The full odd-channel t-matrix obtained by NRG is seen to
saturate to these values for $|\omega|\ll T^{\rm FL}$, as anticipated.

More interestingly, comparison between the left and centre panels of Fig. 5
shows that the low-energy dynamics of the _even_ channel can be accurately
understood in terms of an effective single-channel Kondo model with the same
$J_{ee}$ and $W_{ee}$. This is again a consequence of the emergent decoupling
of the odd channel. Indeed, provided $\delta\equiv J_{ee}-J_{oo}\gg T_{\rm
K}^{e}$ (as is fairly standard), the correct Kondo scales are also well
reproduced.

For the bare Andersonian nanostructure model, calculating the low-energy
behaviour of the scattering t-matrix is deeply nontrivial. Even for the
single-impurity Anderson model, $t_{ee,\sigma}(0,0)=\sin^{2}(\delta_{e})$
requires knowledge of the even-channel phase shift $\delta_{e}$. In this case,
the Friedel sum rule dictates that $\delta_{e}=\tfrac{\pi}{2}n_{\rm imp}$ in
terms of the excess charge due to the impurity, $n_{\rm imp}$.[51] But the
latter is renormalized by interactions and must itself be calculated using
many-body techniques away from particle-hole symmetry. On the other hand, the
above physical arguments and NRG results in Fig. 5 establish that
$t_{ee,\sigma}(0,0)$ can be obtained from an effective 1CK model, Eq. 61.
Treating the exchange coupling $J_{ee}$ as a perturbation to the even-channel
charge, we may then write $n_{\rm
imp}\simeq\tfrac{2}{\pi}\tan^{-1}[\tilde{W}_{ee}]+1$ (it was shown in Ref. 107
that the phase shifts due to $J_{ee}$ and $\tilde{W}_{ee}$ are additive when
the Hamiltonian is transformed in the potential scattering eigenbasis), where
the first term is the excess charge induced in a free bath due to a boundary
potential $W_{ee}$, and we add $1$ for the singly-occupied impurity local
moment in the Kondo model. This approximation gives
$\delta_{e}\simeq\tan^{-1}[\tilde{W}_{ee}]+\tfrac{\pi}{2}$ and hence from the
Friedel sum rule,

$t_{ee,\sigma}(0,0)\simeq\frac{1}{1+\tilde{W}_{ee}^{2}}\;.$ (69)

Using the relation between the phase shift and the argument of the t-matrix
[8] $\delta_{e}=\arg[T_{ee,\sigma}(0,0)]$ the full even-channel t-matrix can
be determined to be $\pi\rho_{0}T_{ee,\sigma}(0,0)=-(\tilde{W}_{ee}+i)\times
t_{ee,\sigma}(0,0)$.

Figure 6: NRG results for the effective 1CK model, Eq. 61. (a) $T=0$ spectrum
of the t-matrix at the Fermi energy $t_{ee}(0,0)$, as a function of the local
potential $W_{ee}$, for different $J_{ee}$. Dashed line is Eq. 69, showing
only small deviations even at rather large $J_{ee}$. (b) The full energy-
dependence of the t-matrix $t_{ee}(\omega,0)$ is shown for $J_{ee}/D=0.3$ with
$W_{ee}/D=0,0.2,0.4,0.6,0.8$ increasing in the direction of the arrow. (c)
Dependence of the Kondo temperature $T_{\rm K}^{e}$ on $W_{ee}$ for the same
$J_{ee}$ used in (a), confirming Eq. 70.

These predictions are quantitatively substantiated in Fig. 6, which shows NRG
results for a pure 1CK model with exchange coupling $J_{ee}$ and potential
scattering $W_{ee}$. The t-matrix $t_{ee,\sigma}$ at $\omega=T=0$ is plotted
in panel (a) as a function of $W_{ee}$ for different $J_{ee}$. Even for rather
large $J_{ee}/D=0.4$, the exact results agree accurately with Eq. 69 (dashed
line) over the entire range of $W_{ee}$.

The full energy dependence of the t-matrix is shown in panel (b) for fixed
$J_{ee}$ with increasing $W_{ee}$. The asymptotic $\omega=0$ value reported in
(a) can be extracted for $|\omega|\ll T^{e}_{\rm K}$, with the Kondo scale
$T^{e}_{\rm K}$ decreasing as the potential scattering strength $W_{ee}$
increases. Since $T^{e}_{\rm K}\sim D\exp[-1/\tilde{\rho}_{e}J_{ee}]$ and the
Fermi level density of states incorporating $W_{ee}$ is
$\tilde{\rho}_{e}=\rho_{0}/(1+\tilde{W}_{ee}^{2})$ as before, for the 1CK
model we may write,

$T^{e}_{\rm K}(W_{ee})\simeq T^{e}_{\rm
K}(0)\times\exp[-\tilde{W}^{2}_{ee}/\rho_{0}J_{ee}]\;.$ (70)

This is confirmed in Fig. 6(c) for the same systems as in panel (a).

Eq. 69 can therefore be used to find the low-energy behaviour of the even-
channel t-matrix in the generalized 2CK model,[102] while Eq. 70 provides an
estimate of the regime of applicability of this result. Remarkably, the low-
energy scattering behaviour can therefore be extracted purely from a knowledge
of the effective model parameters $J_{ee}$ and $W_{ee}$, and full solution of
the generalized 2CK model is not required.

Figure 7: Conductance of the generalized 2CK model Eq. 56 at $sd$-symmetry,
obtained by NRG using the improved Kubo method. (a) $G_{C}(T)$ vs $T$ for
fixed $J/D=0.25$, $J_{sd}/D=0.05$ and $W=0$ with different
$W_{sd}/D=0,0.1,0.2,0.3,0.4,0.6$ for the solid black, red, blue, green, violet
and orange lines. Dotted lines are the low-$T$ predictions of Eq. 71. (b)
$G_{C}(0)$ vs $W_{sd}$, for systems with the same parameters as in (a),
showing the conductance node near $\tilde{W}_{sd}=1$, compared with Eq. 71
(dashed line). (c) $G_{C}(T)$ vs $T$ for fixed $J/D=0.15$, $J_{sd}/D=0.05$,
$W/D=0.2$, varying $W_{sd}/D=0,0.1,0.2,0.3$ for the black, red, blue and green
solid lines (again compared with Eq. 71, dotted lines).

Finally, we use this information on the dynamics to extract the generic low-
temperature conductance properties. We use Eq. 24 in the even/odd basis of Eq.
65 to write
$G_{C}(0)=(2e^{2}/h)\times|\pi\rho_{0}T_{ee,\sigma}(0,0)-\pi\rho_{0}T_{oo,\sigma}(0,0)|^{2}$.
Using the form of the even/odd t-matrices derived in this section, we obtain,

$G_{C}(0)=\left(\frac{2e^{2}}{h}\right)\frac{(1+\tilde{W}_{ee}\tilde{W}_{oo})^{2}}{(1+\tilde{W}_{ee}^{2})(1+\tilde{W}_{oo}^{2})}\;.$
(71)

This is the main result of this section: it allows the Kondo-renormalized
low-$T$ conductance to be calculated purely from a knowledge of the effective
model parameters. Note that the result holds approximately for all $T\ll
T^{\rm FL}$.

In the case of $sd$-symmetry as considered here, only $W_{ee}=W+W_{sd}$ and
$W_{oo}=W-W_{sd}$ are required. Furthermore, Eq. 71 implies the existence of
an exact quantum-interference effect conductance node,

$G_{C}(0)=0\qquad\text{when}\qquad W_{sd}^{2}=W^{2}+(1/\pi\rho_{o})^{2}$ (72)

This condition can in principle be satisfied on tuning gate voltages in multi-
orbital nanostructures. Although conductance nodes due to single-particle
quantum interference in effective non-interacting systems are well-known,
their many-body counterparts are comparatively poorly understood – even though
strong electron interactions are ubiquitous in e.g. semiconductor quantum dot
devices and single molecule transistors. We emphasize that in such systems,
electron interactions are crucial to understand the low-temperature
conductance properties. Indeed, the $T=0$ conductance node, Eq. 72, is driven
by the Kondo effect (a finite conductance is expected at higher temperatures,
$T\gg T_{\rm K}$). This “Kondo Blockade” has been proposed as a mechanism for
efficient quantum interference effect transistors.[103] The above analysis
puts such phenomena in the context of a more general framework.

The quantitative accuracy of Eq. 71 is established by NRG results in Fig. 7.
The dc conductance $G_{C}(T)$ of the full 2CK model, Eq. 56, under $sd$
symmetry is obtained within NRG using the improved Kubo approach, Eqs. 36 and
44. In Fig. 7(a) we plot $G_{C}$ as a function of $T$ for fixed $J$, $J_{sd}$
and $W=0$, with different $W_{sd}$. Although the high-$T$ conductance is
rather accurately reproduced by 55, Kondo correlations strongly affect the
behaviour at low temperatures $T\ll T_{\rm K}$, with $T_{\rm K}$ the emergent
Kondo temperature. The horizontal dotted lines are the result of Eq. 71, and
are seen to agree very well with the low-temperature conductance from NRG. In
particular, the Kondo resonance at $W_{sd}=0$ (black line) is inverted to a
Kondo blockade for $W_{sd}=0.6D$ (orange line). $G_{C}(0)$ is plotted for the
same parameters in Fig. 7(b) as a function of $W_{sd}$, with NRG results
(points) compared with Eq. 71 (dashed line), showing the conductance node
predicted by Eq. 72. Fig. 7(c) confirms the more general form of Eq. 71 for
finite $W$ and $W_{sd}$.

####  VI.2.3 finite potential scattering and broken $sd$-symmetry

Finally, we consider the most general case, in which both $\bm{J}$ and
$\bm{W}$ are finite, and $sd$-symmetry is broken ($J_{-}\neq 0$ and $W_{-}\neq
0$).

First, we note that for arbitrary $J_{\pm}$, $J_{sd}$ and $W_{\pm}$, $W_{sd}$
the lead basis rotation $\bm{U}$ that diagonalizes the exchange matrix
$\bm{J}$ will not in general diagonalize the potential scattering matrix
$\bm{W}$ (that is, although $\bm{U}^{\dagger}\bm{J}\bm{U}$ is constructed such
that $J_{eo}=J_{oe}^{*}=0$, $\bm{U}^{\dagger}\bm{W}\bm{U}$ still yields finite
$W_{eo}=W^{*}_{oe}\neq 0$). This means that the even/odd channels defined in
this way are not distinct; and in particular the odd channel does not
completely decouple at low energies, as required for the previous analysis.

The first step is therefore to define a new basis, in terms of which we can
identify _distinct_ conduction electron channels, labelled $a$ and $b$. We
define $\bm{\mathcal{W}}=\bm{P}^{\dagger}\bm{W}\bm{P}$ to be diagonal, such
that,

$\displaystyle\mathcal{W}_{aa}$
$\displaystyle=\tfrac{1}{2}(W_{+}+\sqrt{W_{-}^{2}+4|W_{sd}|^{2}})$ (73a)
$\displaystyle\mathcal{W}_{bb}$
$\displaystyle=\tfrac{1}{2}(W_{+}-\sqrt{W_{-}^{2}+4|W_{sd}|^{2}})$ (73b)

and where $\mathcal{W}_{ab}=\mathcal{W}^{*}_{ba}=0$ by construction. Eq. 73 is
as such the potential scattering analog of Eq. 60 for the exchange couplings.
Incorporating these local potentials into the definition of the free lead
Hamiltonian in the even/odd basis $\mathcal{H}_{\rm leads}$, we have
renormalized Fermi level densities of states
$\tilde{\rho}_{\alpha}=\rho_{0}/[1+(\pi\rho_{0}\mathcal{W}_{\alpha\alpha})^{2}]$.
The same transformation $\bm{P}$ is applied to the exchange matrix $\bm{J}$,
noting that $\bm{\mathcal{J}}=\bm{P}^{\dagger}\bm{J}\bm{P}$ is not in general
diagonal.

We may now identify the relevant dimensionless quantities to analyze the RG
flow as
$j_{\alpha\beta}=\sqrt{\tilde{\rho}_{\alpha}\tilde{\rho}_{\beta}}\mathcal{J}_{\alpha\beta}$.
The $2\times 2$ matrix of these couplings is denoted $\bm{j}$. We perform a
final rotation of the lead basis to diagonalize this matrix,
$\bm{j}^{\prime}=\bm{Q}^{\dagger}\bm{j}\bm{Q}$, to yield the desired even/odd
exchange couplings $j^{\prime}_{ee}$ and $j^{\prime}_{oo}$. It is in _this_
basis that we can say that the more strongly coupled even channel flows to
strong coupling and the more weakly coupled odd channel decouples,[108]
analogous to the analysis of the previous section. Potential scattering in
this basis is given by
$\bm{W}^{\prime}=\bm{Q}^{\dagger}\bm{\mathcal{W}}\bm{Q}$, and we define
$\tilde{W}^{\prime}_{\alpha\beta}=\pi\rho_{0}W^{\prime}_{\alpha\beta}$.

The emergent decoupling of the odd channel in this basis at low energies and
temperatures $\omega,T\ll T_{\rm K}$ yields the non-trivial result
$t^{\prime}_{eo}(0,0)=t^{\prime}_{oe}(0,0)=0$. This is a consequence of Kondo
renormalization, and scattering between even and odd channels is finite at
higher energies. In the fully asymmetric case, there is no lead rotation that
yields $t_{eo}(\omega,T)=0$ at all energies and temperatures. The above
prescription, although somewhat complicated, is needed in order to identify
the specific lead combination where inter-channel scattering vanishes at low
energies. We have confirmed this result explicitly with NRG.

In this basis, the analog of Eqs. 68 and 69 for the t-matrix spectra are
$t^{\prime}_{oo,\sigma}(0,0)=(\tilde{W}^{\prime}_{oo})^{2}/(1+(\tilde{W}^{\prime}_{oo})^{2})$
and $t^{\prime}_{ee,\sigma}(0,0)=1/(1+(\tilde{W}^{\prime}_{ee})^{2})$, with
the corresponding complex t-matrices given by
$\pi\rho_{0}T^{\prime}_{oo,\sigma}(0,0)=[(\tilde{W}^{\prime})^{-1}_{oo}-i]\times
t^{\prime}_{oo,\sigma}(0,0)$ and
$\pi\rho_{0}T^{\prime}_{ee,\sigma}(0,0)=-[\tilde{W}^{\prime}_{ee}+i]\times
t^{\prime}_{ee,\sigma}(0,0)$. The low-$T$ conductance then follows from Eq.
24,

$G_{C}(0)=\left(\frac{2e^{2}}{h}\right)|\Omega_{e}^{\phantom{{}^{\prime}}}T^{\prime}_{ee,\sigma}(0,0)+\Omega_{o}^{\phantom{{}^{\prime}}}T^{\prime}_{oo,\sigma}(0,0)|^{2}\;,$
(74)

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

$\Omega_{\gamma}=2\pi\rho_{0}\sum_{\alpha,\beta\in
a,b}P_{d\alpha}Q_{\alpha\gamma}P^{*}_{s\beta}Q^{*}_{\beta\gamma}\;.$ (75)

The low-temperature conductance obtained from NRG is found to agree well with
these predictions.

We emphasize that the benefit of the above framework is that the low-
temperature conductance can be accurately estimated, purely from a knowledge
of the effective 2CK model parameters $\bm{J}$ and $\bm{W}$. The 2CK model
itself does not need to be solved. Furthermore, since any microscopic multi-
orbital system hosting a net spin-$\tfrac{1}{2}$ coupled to source and drain
leads can in principle be mapped to such a model, the problem of calculating
low-$T$ transport becomes one of determining effective model parameters. These
can be estimated perturbatively using Eq. 51, or non-perturbatively using more
sophisticated techniques, such as model machine learning.[99]

###  VI.3 $S=1$

When the nanostructure hosts an even number of electrons, a high-spin state
with $S=1$ may result. This is especially prevalent in single molecule
junctions. The effective model in this case is again Eq. 56, but now with
$\hat{\bm{S}}$ a spin-1 operator ($\hat{\bm{s}}_{\alpha\beta}$ are still
spin-$\tfrac{1}{2}$ operators for the leads).

In the case with exact PC on the level of the bare model, an effective single-
channel spin-1 Kondo model results (Eq. 61 with $\hat{\bm{S}}$ a spin-1
operator). The low-energy physics is that of the underscreened Kondo effect
[50] – a singular Fermi liquid with a residual free spin-$\tfrac{1}{2}$ local
moment surviving down to $T=0$. Conductance is again Kondo-enhanced in this
case, but with logarithmic corrections to the low-temperature approach to the
fixed point conductance.[109] This situation can arise in parallel double
quantum dots; although the exact PC condition on the hybridization matrix is
challenging to realize in practice.

The more generic scenario is when the effective nanostructure spin-1 is
delocalized and the PC condition is not satisfied. In this case, both $J_{ee}$
and $J_{oo}$ remain finite in the even/odd lead basis. The nanostructure
spin-1 is exactly screened on the lowest temperature scales[110] (assuming
both exchange couplings are antiferromagnetic). However, since $J_{ee}>J_{oo}$
we have $T_{\rm K}^{e}>T_{\rm K}^{o}$ and a ‘two-stage’ Kondo effect results:
the odd lead participates in screening of the nanostructure spin-1, and
remains coupled down to $T=0$, in contrast to the spin-$\tfrac{1}{2}$ case
discussed above. The nanostructure spin-1 is underscreened by the even lead to
an effective spin-$\tfrac{1}{2}$ on the scale of $T_{\rm K}^{e}$, and then
fully screened down to $S=0$ by the odd lead on the scale of $T_{\rm K}^{o}$.
Correspondingly, the conductance is Kondo enhanced for $T_{\rm K}^{o}<T<T_{\rm
K}^{e}$ but suppressed on the lowest temperature scales $T\ll T_{\rm K}^{o}$.
These scenarios have been explored experimentally in e.g. Refs. 111, 61, 112,
113.

##  VII Emergent Proportionate Coupling: Charge Degeneracy

In this section we consider the generic electrical conductance properties of
systems near a Coulomb peak, corresponding to the charge degeneracy point
between $N$ and $N+1$ electrons on the nanostructure. Typically, one can tune
to the Coulomb peaks in experiment by applying suitable plunger gate voltages.
We expect enhanced cotunneling conductance at such points due to strong
nanostructure charge fluctuations (mixed-valence).[97] By contrast with the
Coulomb blockade situation of the previous section,[95] here electrons can
tunnel from source to drain leads without leaving the nanostructure ground
state manifold; although electron interactions still typically renormalize the
conductance.

Our aim in this section is to understand the low-temperature conductance
properties of arbitrary systems in this regime, in terms of simple effective
models. By studying the generic behaviour of such models, we again ultimately
relate the conductance directly to the effective model parameters, avoiding
the need to solve the effective models each time for every new system
considered.

Effective models near charge-degeneracy points can be obtained perturbatively
to _first_ order in the hybridization $H_{\rm hyb}$, which connects $N$ and
$N+1$ electron states of the isolated nanostructure. We define $\hat{1}_{\rm
gs}=\sum_{j=1}^{m_{N}}|N;j\rangle\langle
N;j|+\sum_{k=1}^{m_{N+1}}|N+1;k\rangle\langle N+1;k|$ as a projector onto the
manifold of _retained_ states in the $N$ and $N+1$ electron sectors of $H_{\rm
nano}$. We only keep the $m_{N}$ lowest energy states of the $N$ electron
sector and the $m_{N+1}$ lowest states of the $N+1$ electron sector. Here
$H_{\rm nano}|n;l\rangle=E^{n}_{l}|n;l\rangle$, where the index $l=1,2,3,...$
labels eigenstates of the isolated $n$-electron nanostructure in order of
increasing energy $E_{l}^{n}$. The truncated set of states included in
$\hat{1}_{\rm gs}$ will be sufficient to describe the transport at low
temperatures $T\ll\Delta E_{\rm min}$, where $\Delta E_{\rm min}$ is the
energy of the lowest lying state not included in $\hat{1}_{\rm gs}$. The
typically large charging energy in small nanostructures means that removing an
electron from an $N$ electron ground state, or adding an electron to an $N+1$
electron ground state, yields a highly excited state that can be safely
neglected. However, one should also check that the gap to excitations within
the $N$ and $N+1$ electron sectors is large enough to justify the specific
truncation scheme used. Therefore we take $\Delta E_{\rm
min}=\min[E_{1}^{N-1},E_{1}^{N+2},E_{m_{N}+1}^{N},E_{m_{N+1}+1}^{N+1}]$. In
practice, the $T\to 0$ transport is typically controlled by the lowest spin
multiplet in $N$ and $N+1$ electron sectors, and we now focus on these
situations. The effective model is then defined as,

$H_{\rm eff}=H_{\rm
leads}+\sum_{n=N}^{N+1}\sum_{l=1}^{m_{n}}E_{l}^{n}|n;l\rangle\langle
n;l|+\hat{1}_{\rm gs}H_{\rm hyb}\hat{1}_{\rm gs}$ (76)

We consider specific prominent cases below, and give example applications for
the TQD in Sec. IX.1.

###  VII.1 Spinless case

When the nanostructure and leads are effectively spinless (which may arise
physically in the presence of a strong magnetic field, for example) Eq. 76
takes a simple form. In particular, the standard non-degenerate case where a
single state can be retained in each of the $N$ and $N+1$ electron sectors
yields an effective resonant level model,[8]

$H_{\rm eff}=H_{\rm leads}+\epsilon
f^{\dagger}f+\sum_{\alpha=s,d}\left(t_{\alpha}^{\phantom{\dagger}}f^{\dagger}c^{\phantom{\dagger}}_{\alpha}+t_{\alpha}^{*}c_{\alpha}^{\dagger}f^{\phantom{\dagger}}\right)\;,$
(77)

where $f^{\dagger}=|N+1;1\rangle\langle N;1|$ and $f=(f^{\dagger})^{\dagger}$
are canonical fermionic operators with the proper anticommutation relations.
Here the effective model parameters are $\epsilon=E_{1}^{N+1}-E_{1}^{N}$ and
$t_{\alpha}=V_{\alpha}\langle N+1;1|\bar{d}_{\alpha}^{\dagger}|N;1\rangle$,
which can be obtained straightforwardly from the diagonal representation of
the isolated $H_{\rm nano}$.

Importantly, note that the effective model Eq. 77 is in PC, even though the
bare model is in general not. This emergent PC property greatly facilitates
quantum transport calculations at low energy and temperature scales where the
effective model provides a valid description. Furthermore, the effective model
is non-interacting, despite the strong electron correlations typically present
on the physical nanostructure. This means we can write down the effective
$f$-level Green’s function exactly,

$G_{\textit{ff}}(\omega)=\frac{1}{\omega^{+}-\epsilon-\sum_{\alpha}|t_{\alpha}|^{2}\mathcal{G}^{0}_{\alpha\alpha}(\omega)}\;.$
(78)

Electrical and heat conductances then follow immediately from the Landauer
formula, Eq. 17, with the transmission function being the analog of Eq. 19,

$\mathcal{T}_{\rm
L}(\omega)=4\tilde{\Gamma}_{s}\tilde{\Gamma}_{d}|G_{\textit{ff}}(\omega)|^{2}\simeq\frac{4\tilde{\Gamma}_{s}\tilde{\Gamma}_{d}}{(\omega-\epsilon)^{2}+(\tilde{\Gamma}_{s}+\tilde{\Gamma}_{d})^{2}}\;,$
(79)

where $\tilde{\Gamma}_{\alpha}=\pi\rho_{0}|t_{\alpha}|^{2}$ and we have used
the wide flat band approximation
$|t_{\alpha}|^{2}\mathcal{G}^{0}_{\alpha\alpha}(\omega)\simeq-i\tilde{\Gamma}_{\alpha}$.
Eq. 79 is as such a Lorentzian of width
$(\tilde{\Gamma}_{s}+\tilde{\Gamma}_{d})$ and centred on $\omega=\epsilon$.
The effective parameters $\epsilon$, $t_{s}$ and $t_{d}$ are therefore
sufficient to calculate the transport properties for $T\ll\Delta E_{\rm min}$.
In particular, note that the maximum low-$T$ conductance is obtained at the
charge degeneracy point (where $\epsilon=0$) and when $t_{s}=t_{d}$ (which
does not necessarily imply the symmetry $V_{s}=V_{d}$ in the bare model).