in Sec.2.4. Once $T=0$ limit is reached, no residual spin degrees are left in
the final state.
We now discuss the thermodynamics and the conductance as showed in
Figs.7.4(a),7.4(c) respectively.
At $T\sim\delta E$, the molecule entropy is determined by the three spin
degrees in the triplet state and $S_{mol}=\ln(3)$, this is the yellow region
in the entropy plot.
By lowering the temperature, at $T\leq T_{K}^{e}$ the first Kondo singlet is
formed from the screening of spin-$1/2$ of the molecule with the conduction
electron from the even channel. The molecule entropy counts only two spin
degrees left such that $S_{mol}=\ln(2)$ \- see green region in the plot. These
spin degrees of freedom are enough to allow finite transmission between leads
and molecule due to the Kondo effect. Thus, the conductance increases up to
$\mathcal{G}^{C}(T_{K}^{o}<T\leq T_{K}^{e})=\mathcal{G}_{0}$ as signature of
enhanced tunnelling due to the Kondo singlet formation, given by light yellow
region. However, at $T\ll T_{K}^{o}$ the second Kondo singlet is formed from
the previously unquenched molecule spin-$1/2$ with the conduction electron
from the odd channel. The molecule entropy counts only one spin degree left
and it drops to $S_{mol}=\ln(1)=0$, that is the dark blue region in the plot.
At this stage, there are no residual molecule spin degree available for
screening process and the conductance vanishes $\mathcal{G}^{C}(T\ll
T_{K}^{o})=0$, as we see from the dark region in the bottom.
The vanishing of the conductance from the unitary limit due to the two stage
Kondo effect [206] is more evident Fig.7.6. In particular, we clearly see the
lack of any conductive state for any $T\ll T_{K}^{o}$ and is persistent down
to $T=0$. In this case, the couplings are of the same nature indicating that
$\delta_{e}=\delta_{o}=\pi/2$. Hence, we have
$\mathcal{G}^{C}(T=0)\sim\sin^{2}(\delta_{e}-\delta_{o})\sim 0$ in the whole
zero temperature $N=8$ charge regime.

####  7.4.2 para-connected benzene transistor

In this section we focus on the para\- or $1,4$\- connected benzene
transistor, see the left schematic in Fig.7.1, with the right/drain lead
connected on the fourth site of the benzene molecule. We focus on the
different charge occupation values $N=6,7,8$ of the Coulomb staircase as seen
from the left plot in Fig.7.2 in terms of molecular entropy $S_{mol}$ in Fig.
7.4(b) and dc-regime electrical conductance $\mathcal{G}^{C}(T)$ in Fig.
7.4(d) as a function of temperature $T$ and gate voltage $V_{gate}$.
We note that the $N=6$ regime behaves similar to the $1,3$-geometry and the
same physics follows as before. Moreover, the similar conductive spikes in the
conductance plots appear in correspondence of the valence transitions as in
the $1,3$-geometry, compare Figs. 7.4(c),7.4(d). Hence, for the sake of the
more interesting $N=7$ and $N=8$ regimes, we omit those comments in this
section.

#### Spin doublet ground-state $S=\tfrac{1}{2}$ in $N=7$ charge sector: Kondo
effect with level crossing quantum phase transition

Similar to the meta-connected configuration, decreasing $V_{gate}$ smoothly
increases the molecule charge when leads are connected. However, we can still
identify the Coulomb blockade regimes and mixed-valence crossovers. In the
voltage range $-43\leavevmode\nobreak\ eV<V_{gate}\lesssim
36\leavevmode\nobreak\ eV$, the ground-state of the isolated molecule is
essentially identified by a doublet spin $S=1/2$ state. However, the different
lead-molecule-lead configuration exhibits new physics from the $1,3$-connected
model. This is further evidence of the major role of symmetries in the low-
energy behaviour of the system.
We start with analysing the molecule entropy in Fig.7.4(b) in the $N=7$
sector.
In analogy to the $1,3$ configuration, at high temperature $T\sim\delta E$ for
the entire voltage range $-43\leavevmode\nobreak\
eV<V_{gate}\lesssim-36\leavevmode\nobreak\ eV$, the doublet ground-state
comprises both spin and pseudospin degrees of freedom. This is the red region
in the plot characterised by $S_{mol}=\ln(2+2)$. The pseudospin freezing out
of the weakest parity doublet occurs only at lower temperature $T=T^{\star}$.
Only the lowest pseudospin $p^{\star}$ is accessible, such that the entropy is
reduced to $S_{mol}=\ln(2)$, as seen in the green region. At $T\leq
T^{\star}$, the system is described as a generalized two-channel spin-$1/2$
Kondo model. By lowering even further the temperature, the thermodynamics
before and after $V_{gate}\sim-40\leavevmode\nobreak\ eV$ is evidently of
different nature. At $-40\leavevmode\nobreak\
eV<V_{gate}\lesssim-36\leavevmode\nobreak\ eV$ in the temperature scale
$T_{K}\ll T^{\star}$, under RG-flow of coupling parameters we find that the
lowest energy doublet with parity $p^{\star}$ is screened by the conduction
electrons from the strongest channel, while the weakest channel has to detach
from the molecule. The Kondo singlet forms $T\ll T_{K}$ and the signature of
this process is the entropy value $S_{mol}=\ln(1)=0$ which persists down to
$T=0$ \- that is the dark blue region developing from the green area to the
bottom of the diagram in Fig.7.4(b). Opposed to this strong coupling phase, at
$-43\leavevmode\nobreak\ eV<V_{gate}<-40\leavevmode\nobreak\ eV$ we find a
persistence of the local moment phase for any $T\leq T^{\star}$, as indicated
by the large green region sharply confined on the right at
$V_{gate}=-40\leavevmode\nobreak\ eV$ and developing till $T=0$.
We note that $SU(2)$ symmetry holds in the system for the whole voltage range
$-43\leavevmode\nobreak\ eV<V_{gate}\lesssim-36\leavevmode\nobreak\ eV$.
However, the sudden jump in thermodynamics values at
$V_{gate}\sim-40\leavevmode\nobreak\ eV$ suggests a first order level crossing
transition from Kondo screened to local moment phase should take place in the
coupled system. This observation is indeed corroborated by the eigenspectrum
analysis from NRG. As consequence of the pseudospin freezing process happening
at $T\leq T^{\star}$, the lowest ground-state doublet is
$\ket{\Psi^{7}_{0}}=\ket{7;\pm 1/2;-}$ before
$V_{gate}\sim-40\leavevmode\nobreak\ eV$. However, at
$V_{gate}\sim-40\leavevmode\nobreak\ eV$ we find an inversion of the ground-
state parity eigenvalue and the previously frozen doublet
$\ket{\Psi^{7}_{0}}=\ket{7;\pm 1/2;+}$ becomes the ground-state doublet. In
conclusion at $V_{gate}\sim-40\leavevmode\nobreak\ eV$ the system undergoes to
a switch of the ground-state doublet state: an unequivocal signature of level
crossing quantum phase transition. Now, we further investigate the special
point at $V_{gate}\sim-40\leavevmode\nobreak\ eV$ and in its boundary.
As we analyse for the $1,3$ system, we compute the average of
$\widehat{P}_{mol}$ using NRG data. Opposed to the previous result, we never
find a gate value where both parity doublet multiplets equate. Moreover,
$\langle\widehat{P}_{mol}\rangle$ runs from negative values with limit at
$\sim-1$ to positive ones with limit at $\sim+1$ with a discontinuity at
$V_{gate}\sim-40\leavevmode\nobreak\ eV$. This discontinuity evidently
confirms the level crossing transition already seen in the switching of
ground-state energy at the special $V_{gate}$ value.

Figure 7.7: Molecular entropy in the $N=7$ regime, as a function of
temperature $T$ for two different gate voltages
$V_{gate}=-39\leavevmode\nobreak\ eV$ (black curve) and
$V_{gate}=-41\leavevmode\nobreak\ eV$ (green curve) on either side of
$V_{gate}\sim-40\leavevmode\nobreak\ eV$ value where the level crossing
quantum phase transition occurs in the $1,4$-connected benzene junction. At
$T\ll T_{K}$, for $V_{gate}=-39\leavevmode\nobreak\ eV$ the doublet spin-$1/2$
state with parity $p=+1$ is Kondo quenched whereas for
$V_{gate}=-41\leavevmode\nobreak\ eV$ the doublet spin-$1/2$ state with parity
$p=-1$ remains unscreened down to $T=0$.

To get a complete intuition about the pseudospin freezing process with related
system phases, we plot $S_{mol}$ as function of temperature in the phases
either side of $V_{gate}\sim-40\leavevmode\nobreak\ eV$, namely at
$V_{gate}=-39\leavevmode\nobreak\ eV$ and $V_{gate}=-41\leavevmode\nobreak\
eV$ in black and green curve respectively in Fig.7.7. We distinguish again two
plateaux at $S_{mol}=\ln(2+2)$ and $S_{mol}=\ln(2)$, marked with blue and red
dashed lines respectively. At high temperature, the entropy value is
determined by entangled spin and pseudospin degrees both before and after the
special $V_{gate}$ point - at this scale the two curves approximately overlap.
At $V_{gate}=-39\leavevmode\nobreak\ eV$ once pseudospin freezing temperature
$T=T^{\star}$ is reached, the molecule ground-state is represented solely by
$\ket{\Psi^{7}_{0}}=\ket{7;\pm 1/2;+}$ doublet. This spin doublet state is
associated to $S_{mol}=\ln(2)$ value, that is the local moment phase as
indicated by intermediate plateau in the black curve. This residual spin-$1/2$
degree is Kondo quenched at lower temperature $T_{K}\ll T^{\star}$, reaching
$S_{mol}=\ln(1)$ value as seen in the black tail approaching zero entropy
value as $T\to 0$ at the left corner of the plot. On the contrary, at
$V_{gate}=-41\leavevmode\nobreak\ eV$ below the pseudospin freezing
temperature $T=T^{\star}$, the molecule ground-state is now represented solely
by $\ket{\Psi^{7}_{0}}=\ket{7;\pm 1/2;-}$ state. This is now the prevailing
spin doublet state to which corresponds $S_{mol}=\ln(2)$ entropy value. The
system preserves this local moment phase till $T=0$ with residual molecule
spin-$1/2$ degrees of freedom, as showed by the long plateau in green curve.
To sum up: the additional orbital degeneracies encoded in the pseudospin
degrees $(i)$ are separately distinct from the spin degrees at $T\ll
T^{\star}$ $(ii)$ bring $1,4$-connected system at
$V_{gate}=-40\leavevmode\nobreak\ eV$ to an inversion of parity in the
molecule ground-state leading to a level crossing phase transition.
Considering the interplay between parity and Kondo singlet formation, we
complete our understanding by extracting information on the couplings nature.
We find that at $V_{gate}=-39\leavevmode\nobreak\ eV$, the exchange coupling
with parity $p=+1$ is antiferromagnetic in the even channel and ferromagnetic
in the odd channel whereas the exchange coupling with parity $p=-1$ is
ferromagnetic in both channels. At $V_{gate}=-41\leavevmode\nobreak\ eV$, the
coupling nature is the same with the exception for the exchange coupling with
parity $p=+1$ turns antiferromagnetic also in the odd channel. Across the
transition, the sign of the coupling of the ground-state doublet therefore
changes.
We study now the electrical conductance in Fig.7.4(d).
As before, at $T^{\star}\ll T\sim\delta E$, temperature is beyond the regime
for observing any many-body quantum effect. Hence, the conductance is almost
negligible, that is the black top region in the plot. We analyse separately
the transport for $V_{gate}>-40\leavevmode\nobreak\ eV$ and
$V_{gate}<-40\leavevmode\nobreak\ eV$. In the range $-40\leavevmode\nobreak\
eV<V_{gate}\lesssim-36\leavevmode\nobreak\ eV$, at temperature $T^{\star}$ the
freezing out the doublet state with parity $p=-1$ occurs and the conductance
slightly increases as indicated by the violet sharp region. However, it is
only at $T\ll T_{K}$ that under RG-flow the Kondo singlet is formed from the
residual spin-$1/2$ doublet with $p=+1$ screened by the conduction electrons
from the even channel (coupling $J^{+}_{e}>0$) while the odd channel (coupling
$J^{+}_{o}<0$) separates at $T\to 0$. The system reaches strong coupling phase
which determines a Kondo enhanced conductance $\mathcal{G}^{C}(T\ll
T_{K};-40\leavevmode\nobreak\ eV<V_{gate}<-36\leavevmode\nobreak\
eV)=\mathcal{G}_{0}$, showed by the large bright yellow region in the plot. At
$V_{gate}=-40\leavevmode\nobreak\ eV$ the conductance abruptly changes,
confirming the level crossing occurring at this point. At
$-43\leavevmode\nobreak\ eV<V_{gate}<-40\leavevmode\nobreak\ eV$, the system
undergoes initially at $T=T^{\star}$ to pseudospin freezing of spin doublet
with $p=-1$ followed then by a ground-state inversion that turns back the
lowest energy doublet with $p=-1$ parity. The resulting two-channel spin-$1/2$
is a ferromagnetic Kondo model. Under RG-flow, residual spin-$1/2$ doublet
with $p=-1$ remains unquenched till $T=0$ since both even and odd channels
with couplings $J^{-}_{e},J^{-}_{o}<0$. Within this voltage range, the coupled
system remains in a zero conductive state at any temperature, as indicated by
the black column occupying half of the $N=7$ regime. Thus, we conclude the
benzene junction at $V_{gate}=-40\leavevmode\nobreak\ eV$ behaves like a
switch: the conductance changes from unitary limit $\mathcal{G}_{0}$ to zero
just by tuning the gate voltage value in the vicinity of
$V_{gate}=-40\leavevmode\nobreak\ eV$.
The switching behaviour is more evident in the conductance curve at $T=0$, see
plot in Fig.7.8. Only at the crossover from $N=6$ to $N=7$ sector there is
finite electron tunnelling to lift the system from the Colomb blockade regime.
Within the doublet multiplet ground-state, the conductance takes maximum value
at $V_{gate}>-40\leavevmode\nobreak\ eV$ then vanishes at
$V_{gate}=-40\leavevmode\nobreak\ eV$ in discontinuous fashion. Ultimately,
the conductance increases again at the valence transition between
$7\leftrightarrow 8$ sectors. We confirm this result also analytically, namely
before $V_{gate}=-40\leavevmode\nobreak\ eV$, the antiferromagnetic
$J^{+}_{e}>0$ contributes to finite $\delta_{e}\sim\pi/2$ whereas the
ferromagnetic $J^{+}_{o}<0$ has vanishing phase shift $\delta_{o}=0$ so we
obtain maximum conductance $\mathcal{G}^{C}(T=0)\sim\sin^{2}(\pi/2-0)\sim 1$.
After $V_{gate}=-40\leavevmode\nobreak\ eV$, both couplings are ferromagnetic
giving $\delta_{e}=\delta_{o}\sim 0$ and so we have zero conductance.

Figure 7.8: Electrical conductance $\mathcal{G}^{C}(T=0)$ as a function of
$V_{gate}$ encompassing $N=6,\,7,\,8$ sector in the $1,4$-connected molecule-
lead coupled benzene junction. In the $N=7$ charge sector, the benzene
junction behaves as a switch where the Kondo enhanced conductance prior to the
level crossing phase transition switches to zero at
$V_{gate}\sim-40\leavevmode\nobreak\ eV$. As $V_{gate}$ is further tuned to
more negative values, the molecule enters the $N=8$ regime and the conductance
eventually recovers the unitarity limit due to the underlying partial Kondo
screening of the $S=1$ spin triplet state.

#### Spin triplet ground-state $S=1$ in $N=8$ charge sector: effective single-
channel underscreened Kondo model

As the gate voltage is tuned further down, in the $1,4$-meta configuration the
molecule accommodates another electron to give $N=8$ and the molecular ground-
state crosses from doublet $S=1/2$ to $S=1$ spin triplet regime. This is
observed in the gate voltage range $-48\leavevmode\nobreak\ eV\leq
V_{gate}\leq-43\leavevmode\nobreak\ eV$, see Fig.7.2. As in the meta-
configuration, the triplet ground-state offers enough spin degrees for
realising Kondo screened phase. However, the different couplings nature
determines half screened state from $1,3$ case - as we discuss now.
From second order perturbation expansion results in $\hat{H}_{hyb}$, we find
the exchange coupling in the even channel is antiferromagnetic $J_{e}>0$
whereas in the odd channel is ferromagnetic $J_{o}<0$. As consequence, $J_{e}$
is RG relevant parameter while $J_{o}$ is RG irrelevant. Under RG-flow, deep
in the $N=8$ sector the benzene junction behaves as an effective single-
channel spin-$1$ Kondo model undergoing to partial Kondo screening \- see more
comments in Sec.2.4. In the limit $T\to 0$, there are residual unquenched spin
degrees of freedom.
We now discuss the thermodynamics and the conductance as showed in
Figs.7.4(b),7.4(d) respectively.
As before, at $T\sim\delta E$, the molecule entropy is determined by the three
spin degrees in the triplet state as $S_{mol}=\ln(3)$, this is the yellow
region in the entropy plot. As seen previously, in this regime we have weak
electron tunnelling - low conductance means no Kondo effect, see black area in
the conductance plot.
By lowering the temperature, at $T\ll T_{K}^{e}$ the Kondo singlet is formed
from the screening of spin-$1/2$ of the molecule with the conduction electron
tunnelling from the even channel with positive $J_{e}$. The molecule entropy
is reduced by a unit as $S_{mol}=\ln(2)$ as represented by the green region of
the diagram. Thus, the conductance increases up to $\mathcal{G}^{C}(T\ll
T_{K}^{e})=\mathcal{G}_{0}$ as signature of enhanced tunnelling due to the
Kondo singlet formation, given by light yellow region in the plot. By
decreasing even further the temperature down to $T\to 0$, under RG-flow the
Kondo singlet state persists while the odd channel with ferromagnetic $J_{o}$
decouples from the molecule. Although there is still a spin-$1/2$ degree of
freedom on the molecule now, there is no second stage Kondo because the odd
channel couples ferromagnetically and decouples under RG-flow. This model can
be regarded as single-channel underscreened Kondo model. The system reaches a
partial Kondo screened state with a local moment ground-state accompanied by
unitary limit of conductance $\mathcal{G}^{C}(T\ll T_{K}^{e})=\mathcal{G}_{0}$
which persists down to $T=0$.
The enhancement in conductance due to the partial Kondo screening [206, 187]
is evident also from Fig.7.8. Subsequently to the gradual increment in the
valence transition, the conductance remains at maximum value for the whole
$N=8$ charge regime from $T_{K}^{e}$ till $T=0$. In this case, the couplings
have opposite nature namely $\delta_{e}\sim\pi/2$ and $\delta_{o}\sim 0$.
Hence, we have $\mathcal{G}^{C}(T=0)\sim\sin^{2}(\delta_{e}-\delta_{o})\sim
1$.

###  7.5 Conclusion

Nanoscale junctions involving real molecules require a description
encompassing strong interactions, molecular geometry and its orbital structure
on the same footing. The molecular geometry embodies certain structural
symmetries that determine the transport characteristics of the contributing
orbitals in contact with the lead electronic wave functions. This interplay
between the molecular symmetry and Kondo type strong interactions, between
both molecular electrons and conduction electrons of the leads may direct to a
plethora of rich characteristics in the device.
In this chapter, we have shown that the simple single-benzene molecule coupled
to source and drain electrode can host Kondo enhanced conductance due to
additional pseudospin degrees. According to the system configuration, we have
found the following cases:

  1. 1.

$N=7$ charge sector: $S=1/2$ Kondo effect leading to $(i)$ a conductance node
that is tunable via the application of a back gate voltage and is determined
by the underlying higher $SU(4)$ symmetry of the full lead-coupled transistor,
$(ii)$ first order quantum phase transition representing a molecular current
switch from blockade to maximally enhanced conductance;

  2. 2.

$N=8$ charge sector: $(i)$ two-stage $S=1$ perfectly screened Kondo effect,
$(ii)$ partially screened $S=1$ Kondo model leading to an effective single-
channel underscreened Kondo model.

In particular, we demonstrate the interplay between molecular symmetry and
Kondo effect driven strong interactions via NRG calculation on the full
benzene ring to determine the conductance characteristics, and comprehend
these using effective models.
These theoretical investigation demonstrate the high-tuned conductive states
of the lead-benzene-lead junction via external voltages. The coupled system
embraces the transistor functionality where small $V_{gate}$ variation leads
to different tunnelling regime in the system. Hence, the single-molecule
benzene coupled to leads is a promising candidate for future engineering of
nanoelectronic devices.
As remarkable results, the effective model both in first and in second order
perturbation expansion in $\hat{H}_{hyb}$ correctly predicts the $\omega=T=0$
conductance behaviour in the mixed-valence and in the Coulomb blockade charge
sectors, respectively. Furthermore, from second order results we calculate the
exchange couplings. The advantage of the effective model is to reduce the
electronic complexity from the bare model and to create suitable configuration
under PC that is absent in the microscopic model. In this regards, even
elaborate systems can be studied through minimal element of a generalized
Fermion class of effective two-lead under PC models.

##  Chapter 8 Two-Channel Charge-Kondo physics in graphene quantum dots

Nanoelectronic quantum dot devices exploiting the charge-Kondo paradigm have
been established as versatile and accurate analogue quantum simulators of
fundamental quantum impurity models. In particular, hybrid metal-semiconductor
dots connected to two metallic leads realize the two-channel Kondo (2CK)
model, in which Kondo screening of the dot charge pseudospin is frustrated.
Here, we consider theoretically a two-channel charge-Kondo device made instead
from graphene components, realizing a pseudogapped version of the 2CK model.
We solve the model using Wilson’s Numerical Renormalization Group method, and
uncover a rich phase diagram as a function of dot-lead coupling strength,
channel asymmetry, and potential scattering. The complex physics of this
system is explored through its thermodynamic properties, scattering
$\mathrm{T}$-matrix, and experimentally measurable conductance. We find that
the strong coupling pseudogap Kondo phase persists in the channel-asymmetric
two-channel context, while in the channel-symmetric case frustration results
in a novel quantum phase transition. Remarkably, despite the vanishing density
of states in the graphene leads at low energies, we find a _finite_ linear
conductance at zero temperature at the frustrated critical point, which is of
non-Fermi liquid type. Our results suggest that the graphene charge-Kondo
platform offers a unique possibility to access multichannel pseudogap Kondo
physics.
The results of this chapter are published in the paper [2] \- here in this
version we highlight more physical explanations to adhere this material with
concepts and techniques we presented in previous chapters. As an outline, we
set the work with respect to the existing literature and define our
objectives. Then, we present the model and methods employed prior to the main
section, the results, where we address our findings. We conclude suggesting
future extension of this work.

###  8.1 Introduction

The Kondo effect [37] was originally discussed in the context of local
magnetic impurities such as Fe, embedded in non-magnetic metallic hosts like
Au. By progressively decreasing the temperature $T$, experimental measurements
revealed an unexpected resistivity minimum, attributed to enhanced electronic
scattering from the impurity local moments [38]. The low-energy physics of
such systems is explained by the deceptively simple Kondo model, which
features a single spin-$\tfrac{1}{2}$ local moment exchange coupled to a
featureless bath of metallic, non-interacting conduction electrons. The ‘Kondo
effect’ refers to the universal physics of this model, appearing at $T\sim
T_{K}$ with $T_{K}$ an emergent low-energy scale called the Kondo temperature,
in which the impurity spin is dynamically screened by a surrounding many-body
entanglement cloud of conduction electrons [10].
Since then, variants of the basic Kondo effect that arise when magnetic
impurities are embedded in unconventional host materials have been studied.
Examples include ferromagnets [207] and superconductors [208], as well as
topological materials such as topological insulators [209] or Dirac/Weyl
semimetals [210]. However, local moments in graphene have attracted the most
attention [129, 211, 212]. In neutral graphene, the Dirac point is at the
Fermi level [213] and so a spin-$\tfrac{1}{2}$ impurity couples to a bath of
conduction electrons with a density of states (DoS) featuring a low-energy
pseudogap $\rho(\omega)\sim|\omega|^{r}$ with $r=1$. This has dramatic
consequences for the resulting Kondo physics [211] due to the depletion of
low-energy degrees of freedom in graphene that can participate in screening
the impurity spin.
Deeper insights into strongly correlated electron physics and Kondo have been
gained from tunable circuit realizations of fundamental models in
nanoelectronics devices, made possible by remarkable recent advances in nano-
fabrication and characterization techniques [214, 215]. This provides a route
to probing and manipulating quantum matter at the nanoscale in a way that
would be impossible in bulk systems. In particular, semiconductor quantum dots
(QDs) have been shown to behave like artificial atoms [35], with the extreme
quantum confinement producing a discrete orbital structure and strong
electron-electron interactions. Coupling such quantum dots to metallic
electrodes at quantum point contacts (QPCs) gives rise to the Kondo effect at
low temperatures [120, 12, 121], with a single local moment trapped on the dot
facilitating spin-flip scattering of lead conduction electrons. In such
devices, entanglement spreads across the QD and the leads in a macroscopic
Kondo cloud [216, 217], producing the famous Kondo resonance in electrical
conductance [112]. Quantum transport properties of QDs can be tuned in situ by
applying gate voltages to control the QPC transmissions and dot potential. A
bias voltage can drive the system out of equilibrium.
Quantum dot devices also allow more complex quantum impurity models to be
realized experimentally by controlling the microscopic interactions. As such,
they constitute a versatile platform to study a rich range of physics [215],
including quantum phase transitions (QPTs) [218, 174], emergent symmetries
[219, 220], and non-Fermi liquid (NFL) physics [83, 221, 82, 222]. The two
channel Kondo (2CK) model [75] is a famous example which embodies the
frustration of Kondo screening of a single impurity by two distinct channels
of metallic conduction electrons, and displays all these features [80]. The
standard 2CK model Hamiltonian reads,

$\hat{H}^{2CK}=\hat{H}_{leads}+J_{1}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{1}+J_{2}\hat{\mathbf{S}}\cdot\hat{\mathbf{s}}_{2}\leavevmode\nobreak\
,$ (8.1)

where
$\hat{H}_{leads}=\sum_{\alpha\sigma\mathbf{k}}\epsilon_{\mathbf{k}}^{\phantom{\dagger}}c_{\alpha\sigma\mathbf{k}}^{\dagger}c_{\alpha\sigma\mathbf{k}}^{\phantom{\dagger}}$
describes two leads $\alpha=1,2$, each with spin $\sigma=\uparrow,\downarrow$
electrons with momentum $\mathbf{k}$. The Eq.8.1 is indeed the two-channel
version of the Kondo model we introduce in Eq.2.51. In the original
formulation of 2CK model, the dispersion $\epsilon_{\mathbf{k}}$ is taken to
be linear at low energies such that the electronic DoS of the leads at the
impurity position is flat. The metallic flat band approximation is typically
employed for the free conduction electrons,
$\rho(\omega)\sim\sum_{\mathbf{k}}\delta(\omega-\epsilon_{\mathbf{k}})\equiv\overline{\rho}\theta(|\omega|-D)$,
describing a flat density of states $\overline{\rho}$ inside a band of half-
width $D$. In Eq. 8.1, $\hat{\mathbf{S}}$ is a spin-$\tfrac{1}{2}$ operator
for the impurity and $\hat{\mathbf{s}}_{\alpha}$ is a spin-$\tfrac{1}{2}$
operator for the spin density in lead $\alpha$ at the impurity position, such
that $\hat{H}^{2CK}$ possesses $SU(2)$ spin symmetry. The metallic 2CK model
supports a QPT, with an non-Fermi liquid critical point at $J_{1}=J_{2}$ [80].
Signatures of the critical point in this model have been observed
experimentally in semiconductor quantum dot devices [83, 221].
More recently, a new nanoelectronics paradigm has emerged, based on charge-
Kondo quantum dots [84, 223, 85, 224, 225]. In the standard set-up, a large QD
is coupled at QPCs to source and drain leads. These devices realize
anisotropic multichannel Kondo models through Matveev’s mapping [226, 227] of
the macroscopic charge states of the large QD to an effective pseudospin that
is flipped by electronic tunnelling at the QPCs. This approach has led to
unprecedented control over the frustrated 2CK state, and has uncovered the
full renormalization group (RG) flow diagram through transport measurements
[84].
Motivated by these developments, in this chapter we consider combining the
two-channel charge-Kondo set-up of Ref. [84] with the pseudogap Kondo physics
of graphene in Ref. [211], to realize a novel two-channel pseudogap charge-
Kondo effect. We envisage a charge-Kondo device made from graphene components,
see Fig. 8.1, such that the dot charge pseudospin is coupled to two channels
of conduction electrons, each with the characteristic linear pseudogap DoS of
graphene. This work is a theoretical exploration of such a system and its
phase diagram. We characterize the phases and phase transitions through
thermodynamic quantities, and focus on experimentally-relevant physical
observables such as the conductance. However, we do not claim to address the
practical complexities that will inevitably arise in the experimental
realization of a graphene charge-Kondo device.
We note that the generic properties of fully spin- and channel-symmetric two-
channel pseudogap Kondo models were discussed in Ref. [228], although the
$r=1$ linear pseudogap case relevant to graphene was not analysed in detail
and a device realization was not proposed. Furthermore, our charge-Kondo
implementation leads to crucial differences in the model and transport
measurement set-up which have not previously been considered. These
differences and our new results are highlighted in the following.
Before proceeding with the modelling and the results, we give a short overview
to the electronic structure we study in this work.

#### Graphene density of states

In the general scenario, the energy spectrum in the Kondo model - see
discussion in Sec.2.2 \- can be analysed by the power-law density of states
defined for a conduction bandwidth $D$ as:

$\rho(\omega)=\rho\biggr{\lvert}\frac{\omega}{D}\biggr{\rvert}^{r}\quad\text{at}\leavevmode\nobreak\
|\omega|<D\leavevmode\nobreak\ .$ (8.2)

According to the sign of the exponent $r$, the system undergoes to different
behaviour. At $-1<r<0$, the DoS diverges at the Fermi level $\epsilon_{F}$
leading to a critical or Van-Hove singularity and the system is characterised
by power-law Kondo model; at $r\geq 0$, there are two special cases i.e. $r=0$
metallic and $r=1$ pseudogap spectrum. The former case describes the standard
set-up composed by metallic leads with continuum energy spectrum and finite
$\rho(\omega)$ at the Fermi level. It is customary to approximate the
conduction band structure into wide-flat band limit where
$\rho(\omega)\equiv\overline{\rho}$ is constant within the conduction
bandwidth $D$ and the Kondo temperature is defined as
$T_{K}=\sqrt{DJ}e^{(-1/\overline{\rho}J)}$. The latter case, the DoS vanishes
at the Fermi level. In this special case, the system has very few electronic
states available for scattering events in the vicinity of $\epsilon_{F}$ \-
that is the relevant energy scale for Kondo effect - and the formation of the
Kondo singlet entails a modification of $\rho(\omega)$.
An example of material with zero $\rho(0)$ where we set the Fermi level at
$\epsilon_{F}=0$, is graphene [213]. It is a two dimensional lattice of carbon
atoms with two atoms $A,B$ per unit cell defining two sublattices on the
hexagonal Brillouin zone. Hopping between nearest
$\braket{..\leavevmode\nobreak\ ;\leavevmode\nobreak\ ..}$ and next-nearest
$\braket{\braket{..\leavevmode\nobreak\ ;\leavevmode\nobreak\ ..}}$ neighbours
is described in the tight-biding Hamiltonian, namely:

$\hat{H}=-t\sum_{\braket{i;j}\sigma}(a^{\dagger}_{i\sigma}b_{j\sigma}+h.c.)-t^{\prime}\sum_{\braket{\braket{i;j}}\sigma}(a^{\dagger}_{i\sigma}a_{j\sigma}+b^{\dagger}_{i\sigma}b_{j\sigma}+h.c)\leavevmode\nobreak\
,$ (8.3)

with $a_{i\sigma}$ ($b_{j\sigma}$) Fermionic operator acting on electron in
$i$ ($j$) site of sublattice $A$ ($B$). Once the Hamiltonian is diagonalised,
we obtain the valence $\epsilon_{-}$ and the conduction $\epsilon_{+}$
dispersion bands, that is:

$\displaystyle\epsilon_{\pm}(\mathbf{k})=\pm
t\sqrt{3+f_{\mathbf{k}}}-t^{\prime}f_{\mathbf{k}}\leavevmode\nobreak\ ,$ (8.4)
$\displaystyle\text{with}\quad
f_{\mathbf{k}}=2\cos{(\sqrt{3}k_{y}a_{1})}+4\cos{\left(\frac{\sqrt{3}}{2}k_{y}a_{1}\right)}\cos{\left(\frac{3}{2}k_{x}a_{1}\right)}\leavevmode\nobreak\
,$

where $a_{1}=1.21$Å is the bond length [229]. At the Dirac points identified
by the wavevectors $\mathbf{K},\mathbf{K}^{\prime}$, the two bands touch the
Brillouin zone and the energy dispersion in Eq. 8.4 becomes linear in the
form:

$\epsilon_{\pm}(\mathbf{q})\approx\pm
v_{F}|\mathbf{q}|+\mathcal{O}(q/K)^{2}\leavevmode\nobreak\ ,$ (8.5)

with $\mathbf{q}=\mathbf{k}-\mathbf{K}(or\leavevmode\nobreak\
\mathbf{K}^{\prime})$ and Fermi velocity $v_{F}=3ta_{1}/2$.
In the special case of charge-neutral graphene, all the valence states are
filled and all the conduction ones are empty. This symmetric band structure
sets the Fermi level at the Dirac points and the DoS vanishes linearly at the
Fermi level as:

$\rho(\omega)=\frac{2}{\sqrt{3}\pi t^{2}}|\omega|^{2}\leavevmode\nobreak\ .$
(8.6)

Because of the similar behaviour in the electronic structure at Fermi level
for charge-neutral graphene layer and for pseudogap Kondo model, we can study
a pseudogap Kondo graphene system whose DoS in the low-energy spectrum given
by Eq. 8.6 is now replaced by Eq. 8.2 for $r=1$ and for higher energy levels
the DoS is derived from the energy bands dispersion in Eq. 8.4. In practice we
choose $t^{\prime}=0$, for which case the pristine graphene host possesses
particle-hole symmetry that is $\rho(\omega)=\rho(-\omega)$.

###  8.2 Model, methods and observables

We consider a two-channel charge-Kondo (CK) device in which both dot and leads
are made from graphene, as illustrated in Fig. 8.1. The large dot is tunnel-
coupled to leads $\alpha=1,2$ at QPCs with transmission $\tau_{\alpha}$ which
can be controlled in situ by gate voltages. A plunger gate voltage $V_{g}$
controls the dot potential and hence the dot filling. A decoherer is
interjected between the leads via an Ohmic contact on the dot (black bar in
Fig. 8.1) which gives rise to a long dwell time and an effective continuum dot
level spectrum - this was achieved in the experiments of Ref. [84] using a
metallic component. This results in two effectively independent electronic
reservoirs around each of the two QPCs; these form the two independent
channels in the 2CK model. However, tunnelling events onto and off the dot are
correlated by the large dot charging energy, $E_{C}$. The whole device is
operated in a strong magnetic field so that the electrons are effectively
spinless - that is, the Zeeman splitting is the largest energy scale in the
problem.

Figure 8.1: Schematic of the two-channel graphene charge-Kondo quantum dot
system. A net current flows from source to drain graphene leads through the
large graphene dot in response to a bias voltage. A gate voltage $V_{g}$
controls the dot filling. The black bar denotes the decoherer.

Before proceeding with the implementation of charge-Kondo model composed of
graphene, we would like to give a qualitative explanation of the mathematical
operations we apply below. Both in the standard set-up with metallic elements
and in the version we propose here with graphene material, the charge-Kondo
paradigm comprises a large central nanostructure coupled to two leads, source
and drain, through quantum point contacts. The nanostructure in the original
work is semiconductor quantum dot - but the following features are also
applicable to graphene dot. The dot presents electronic levels with negligible
level spacing $\delta E$ such that it is considered a bath itself. Finite
charge transfer is measured both in the lead-dot junction and in the coupled
lead-dot-lead system such that leads are probes for dynamical quantities. Upon
appropriate selection of gate voltage and temperature scale, only two
macroscopic degenerate charge ground-states on the dot are energetically
favourable for single-electron tunnelling through QPCs. These identify the
charge pseudospin states and any higher energy states are discarded from the
spin flipping processes. Furthermore, because of finite charging energy
$E_{c}$ on the dot, spin-flip events are confined to charge pseudospin-$1/2$
states and the charge-Kondo model is linked via exact mapping to the standard
spin Kondo model. This significant aspect allows to determine the coupling
parameters exchange coupling $J$, potential scattering $W$ from the
macroscopic tunnelling matrices in the charge basis model and to map those
into the spin basis model - rather than deriving the couplings from
perturbative method as the Schrieffer-Wolff transformation - we explain it in
Sec.2.2. The CK model is in fact a gate-controlled device where experimentally
controlled voltages give direct access to the essential low-energy physics of
the system.
We continue now with the model Hamiltonian for the device illustrated in Fig.
8.1. It is given by,

$\displaystyle\hat{H}=\sum_{\alpha=1,2}\sum_{\mathbf{k}}\sum_{\sigma=L,D}\epsilon_{\mathbf{k}}c^{\dagger}_{\alpha\sigma\mathbf{k}}c_{\alpha\sigma\mathbf{k}}+\sum_{\alpha=1,2}\sum_{\mathbf{k}\mathbf{k}^{\prime}}\Big{(}$
$\displaystyle J_{\alpha}(\hat{Q}^{+}c^{\dagger}_{\alpha D\mathbf{k}}c_{\alpha
L\mathbf{k}^{\prime}}+c^{\dagger}_{\alpha L\mathbf{k}^{\prime}}c_{\alpha
D\mathbf{k}}\hat{Q}^{-})+$ (8.7) $\displaystyle+W_{\alpha}(c^{\dagger}_{\alpha
L\mathbf{k}}c_{\alpha L\mathbf{k}^{\prime}}+c^{\dagger}_{\alpha
D\mathbf{k}}c_{\alpha
D\mathbf{k}^{\prime}})\Big{)}+E_{C}(\hat{N}_{D}-N_{g})^{2}\leavevmode\nobreak\
,$

we know discuss one by one the terms appearing in Eq.8.7. The first term is
$\hat{H}_{leads}$ and it describes the distinct conduction electron reservoirs
around each QPC labelled by $\alpha=1,2$, and with $\sigma=L,D$ corresponding
to lead or dot electrons - rather than physical spin $\uparrow$,$\downarrow$.
For graphene components, we use the energy dispersion $\epsilon_{\mathbf{k}}$
around the Fermi level as in Eq. 8.5 or at higher energy as in Eq. 8.4. The
second term comprises the exchange coupling $J_{\alpha}$. The terms
proportional to it describe electronic tunnelling at the QPCs between leads
and dot. The tunnelling matrix elements $J_{\alpha}$ are related to the bare
(unrenormalized) QPC transmissions via [230]
$\tau_{\alpha}(\omega)=4\pi^{2}\rho_{\alpha L}(\omega)\rho_{\alpha
D}(\omega)J_{\alpha}^{2}/[1+\pi^{2}\rho_{\alpha L}(\omega)\rho_{\alpha
D}(\omega)J_{\alpha}^{2}]^{2}$, which are in general energy-dependent for
structured leads. States of the isolated dot with a macroscopic number of
electrons $N_{D}$ are denoted $|N_{D}\rangle$, with corresponding (total
charge) dot number operator $\hat{N}_{D}=\sum_{\alpha\mathbf{k}}c_{\alpha
D\mathbf{k}}^{\dagger}c_{\alpha
D\mathbf{k}}^{\phantom{\dagger}}\equiv\sum_{N_{D}}N_{D}\ket{N_{D}}\bra{N_{D}}$.
Recalling that the dot is macroscopic, it can host a Brillouin zone and the
corresponding Block wavefunctions with creation $c_{\alpha
D\mathbf{k}}^{\dagger}$ (annihilation $c_{\alpha D\mathbf{k}}$) operator.
Tunnelling at the QPCs changes the dot charge, which we describe [226] using
the charge raising and lowering pseudocharge operators
$\hat{Q}^{\pm}=\sum_{N_{D}}\ket{N_{D}\pm 1}\bra{N_{D}}$. The ladder operator
definition ensures only unitary charge variation occurs in the system. The
third term proportional to $W_{\alpha}$ identifies the charge potential
scattering at QPCs. The last term is the $\hat{H}_{charge}$: the dot has a
finite charging energy that depends on the filling via the term proportional
to $E_{C}$. The filling can be adjusted by tuning $N_{g}$ in Eq. 8.7, which is
controlled in experiment by the gate voltage
$V_{gate}=V_{gate}^{0}-2E_{C}N_{g}/e$. We define $\delta
V_{gate}=-2E_{C}(N_{g}-N_{D}^{0}-\tfrac{1}{2})/e$ such that the macroscopic
dot charge states $\ket{N_{D}^{0}}$ and $\ket{N_{D}^{0}+1}$ are degenerate at
$\delta V_{gate}=0$.
The physics described by the model Hamiltonian in Eq. 8.7 is robust to
variation in $V_{gate}$ provided that any $\delta V_{gate}$ is the smallest
energy scale in the system, namely $\delta V_{gate}\ll E_{C}$. In the special
case where $V_{gate}\rightarrow V_{gate}^{0}$, $N_{g}$ takes only half-integer
values: the system is set at the charge degeneracy crossover between
$\ket{N_{D}^{0}},\ket{N_{D}^{0}+1}$ dot ground-states. At the energy window
$\delta E\ll k_{B}T\ll E_{C}$, only these two degenerate charge states are
energetically accessible for tunnelling through QPCs and any higher charge
states at $N<N_{D}^{0},\leavevmode\nobreak\ N>N_{D}^{0}+1$ are neglected.
Within these conditions, the pseudocharge ladder dot operators are converted
into pseudospin-$1/2$ operators, namely:

$\displaystyle\hat{Q}^{+}\rightarrow\hat{S}^{+}_{D}=\ket{N_{D}^{0}+1}\bra{N_{D}^{0}}\quad,\quad\hat{Q}^{-}\rightarrow\hat{S}^{-}_{D}=\ket{N_{D}^{0}}\bra{N_{D}^{0}+1}\leavevmode\nobreak\
,$ (8.8)
$\displaystyle\hat{S}^{z}_{D}=\tfrac{1}{2}(\ket{N_{D}^{0}+1}\bra{N_{D}^{0}+1}-\ket{N_{D}^{0}}\bra{N_{D}^{0}})\leavevmode\nobreak\
,$

where we also introduce the pseudospin operator $z$-component. Thus, the
$\hat{S}^{+}_{D}$ flips the dot charge pseudospin from down spin to up spin
while $\hat{S}^{-}_{D}$ flips it back and the second term in Eq. 8.7 regulates
the spin-$1/2$ flipping between dot and leads via QPCs - as it does the Kondo
exchange term, see Eq.2.51. Finally, we perform a trivial relabelling
$\sigma=\\{L,D\\}\to\\{\uparrow,\downarrow\\}$ the physical spin-$1/2$ states
such that the electronic operators become $c_{\alpha L\mathbf{k}}\to
c_{\alpha\uparrow\mathbf{k}}$ and $c_{\alpha D\mathbf{k}}\to
c_{\alpha\downarrow\mathbf{k}}$. With this, we map the model in Eq. 8.7 into
the effective pseudogap two-channel charge-Kondo model (2CCK) studied in this
paper:

$\boxed{\begin{aligned}
\hat{H}^{2CCK}=\sum_{\alpha=1,2}\sum_{\mathbf{k}}\sum_{\sigma=\uparrow,\downarrow}\epsilon_{\mathbf{k}}c^{\dagger}_{\alpha\sigma\mathbf{k}}c_{\alpha\sigma\mathbf{k}}+\sum_{\alpha=1,2}\sum_{\mathbf{k}\mathbf{k}^{\prime}}\Big{(}&J_{\alpha}(\underbrace{\hat{S}_{D}^{+}c^{\dagger}_{\alpha\downarrow\mathbf{k}}c_{\alpha\uparrow\mathbf{k}^{\prime}}}_{\hat{S}_{D}^{+}\hat{s}_{\alpha}^{-}}+\underbrace{c^{\dagger}_{\alpha\uparrow\mathbf{k}^{\prime}}c_{\alpha\downarrow\mathbf{k}}\hat{S}_{D}^{-})}_{\hat{s}_{\alpha}^{+}\hat{S}_{D}^{-}}+\\\
&+W_{\alpha}\sum_{\sigma=\uparrow,\downarrow}c^{\dagger}_{\alpha\sigma\mathbf{k}}c_{\alpha\sigma\mathbf{k}^{\prime}}\Big{)}+e\delta
V_{gate}\hat{S}^{z}_{D}\end{aligned}}\leavevmode\nobreak\ ,$ (8.9)

where this mapping is exact and it breaks down only at $k_{B}T\approx E_{C}$
when multiple pairs of charge states are available for spin-flip process. A
schematic of the spin flux in the 2CCK device described by $\hat{H}^{2CCK}$
Hamiltonian is given in Fig.8.2. The model Hamiltonian in Eq.8.9 is a variant
of the famous 2CK model, Eq. 8.1 \- but with a few important differences.
Firstly, the DoS of the conduction electrons described by $\hat{H}_{leads}$
first term is not metallic, but has a low-energy $r=1$ pseudogap. Hence, it is
the same as Eq.8.7 since $\hat{H}$ there and $\hat{H}^{2CCK}$ describe the
very same model. Secondly, tunnelling at the QPCs give an effective
anisotropic exchange coupling between the dot charge pseudospin and the
conduction electrons. The $SU(2)$ symmetry of Eq. 8.1 is broken in Eq. 8.9
since the $z$-component of the coupling is missing. However, we find that this
effective spin anisotropy is RG irrelevant in the two-channel pseudogap Kondo
problem (just as for the single-channel [38, 231] and two-channel [80]
metallic case, as well as the single-channel pseudogap case [232]). Only the
spin-flip terms are important for Kondo, and these are captured by the
effective model. It should also be emphasized that the effective exchange
couplings $J_{\alpha}$ originate from the QPC tunnellings; there is no
underlying Anderson model, so the $J_{\alpha}$ need not be perturbatively
small. In fact, since they are related to the QPC transmissions, they can
become large simply by opening the QPCs [84]. This is important because Kondo
physics is only realized in the pseudogap model at relatively large bare
coupling strengths. We remark also that the $s_{\alpha}^{\pm}$, $S_{D}^{\pm}$
spin-$1/2$ operators are acting on conduction and dot electrons, respectively.
Thirdly, we have an additional potential scattering term $W_{\alpha}$ that is
spin state invariant and identifies the particle-hole symmetry regime in the
coupled system. This is traditionally omitted in Eq. 8.1 because potential
scattering is RG irrelevant in the metallic Kondo problem [38]. However, we
must keep it because potential scattering is known to be important in the
single-channel pseudogap Kondo model [232]. Indeed, we find that it is
strongly RG relevant in our two-channel pseudogap variant, Eq. 8.9. Finally,
the gate voltage $\delta V_{g}$ appears as an effective impurity magnetic
field. In order to reach the charge degeneracy crossover at the edge of the
Coulomb blockade valley as we know we need $\delta V_{gate}\equiv 0$ \- thus,
this last term does not appear in the energy regime of our interest.

Figure 8.2: Schematic of the spin flux in two-channel charge Kondo device with
model Hamiltonian derived in Eq.8.9 where $\alpha=1,2$ indicates the
conduction electron reservoir are distinct per quantum point contact (QPC) and
$\sigma=\uparrow,\downarrow$ the real spin flavour.

Another important difference in terms of the experimental realization is the
nature of the transport measurement - meaning the spin current relevant to
Eq.8.9 compared to the charge current in Eq.8.7, as we discuss more now. As
illustrated in Fig. 8.1, a series current of spinless electrons is measured
between the physical source and drain leads through the dot, in response to a
bias voltage. But in the mapped spin model, this is an unconventional
measurement: we effectively apply a bias between leads $\alpha=1,2$ but only
to the $\sigma=\uparrow$ conduction electrons. Even though there is no charge
current possible between leads in the original 2CK model Eq. 8.1, the charge-
Kondo set-up Eq. 8.9 allows an effective spin current to be measured - see
schematic in Fig.8.2.
The $ac$-linear response electrical conductance through the device is defined,

$\mathcal{G}^{C}(\omega,T)=\frac{\langle\hat{I}_{2\uparrow}\rangle}{V_{bias}}\biggr{\rvert}_{V_{bias}\rightarrow
0}\leavevmode\nobreak\ ,$ (8.10)

due to an oscillating bias described by $\hat{H}_{bias}=-eV_{bias}\cos(\omega
t)\hat{N}_{1\uparrow}$ with $ac$ frequency $\omega$ \- see more discussion in
Sec.3.2. Here
$\hat{I}_{\alpha\uparrow}=-e\tfrac{d}{dt}\hat{N}_{\alpha\uparrow}$ is the
current operator for lead $\alpha$ and $\sigma=\uparrow$ while
$\hat{N}_{\alpha\uparrow}=\sum_{\mathbf{k}}c_{\alpha\uparrow\mathbf{k}}^{\dagger}c_{\alpha\uparrow\mathbf{k}}^{\phantom{\dagger}}$.
We obtain the ac linear conductance from the Kubo formula [102],

$\displaystyle\mathcal{G}^{C}(\omega,T)=\frac{-\mathit{Im}\langle\langle\hat{I}_{1\uparrow};\hat{I}_{2\uparrow}\rangle\rangle_{\omega,T}}{\omega}\equiv-2\pi\mathcal{G}_{0}\omega{\mathit{Im}}\langle\langle\hat{N}_{1\uparrow};\hat{N}_{2\uparrow}\rangle\rangle_{\omega,T}\leavevmode\nobreak\
,$ (8.11)

where $\langle\langle\cdot\leavevmode\nobreak\ ;\leavevmode\nobreak\
\cdot\rangle\rangle$ denotes a retarded real-frequency correlation function
evaluated at equilibrium, and $\mathcal{G}_{0}=e^{2}/h$ is the conductance
quantum per channel or spin and $\hbar=1$. The second equality in Eq. 8.11
follows from equations of motion and is found to greatly improve the accuracy
of numerical calculations [1], as we derive in Eq.4.81 and see the complete
discussion of the improved version of the Kubo formula in Sec.4.3. Note that
the system is not in proportionate coupling, and so correlated electron
transport coefficients cannot be expressed in terms of a Landauer-type formula
[105] involving the dot spectral function.
In addition to the conductance, we explore the phase diagram and RG fixed
points (FPs) of the model using physical thermodynamic observables. We define
the dot contribution to a thermodynamic quantity $\Omega$ at temperature $T$
as $\Omega_{D}(T)=\Omega(T)-\Omega^{0}(T)$, where $\Omega(T)$ is calculated
for the full lead-dot-lead system, while $\Omega^{0}(T)$ is calculated only
for the free conduction electrons without the dot pseudospin. For the entropy
$S_{D}(T)$ we use $S^{(0)}=-\partial F^{(0)}/\partial T$, with
$F^{(0)}=-k_{B}T\ln Z^{(0)}$ the free energy and $Z$ partition function to
have:

$S_{D}(T)=S_{tot}(T)-S^{0}_{tot}(T)\leavevmode\nobreak\ .$ (8.12)

Recently this entropy has been extracted experimentally in similar quantum dot
devices by exploiting a Maxwell relation connecting the entropy change for a
process to measurable changes in the dot charge [233, 224]. For the magnetic
susceptibility $k_{B}T\chi_{D}(T)$, at zero field $\delta V_{g}=0$, we
evaluate:

$k_{B}T\chi_{D}^{(0)}=\braket{(\hat{S}^{z}_{tot})^{2}}^{(0)}-(\braket{\hat{S}^{z}_{tot}}^{(0)})^{2}\leavevmode\nobreak\
,$ (8.13)

with $\hat{S}^{z}_{tot}$ the $z$-projection of the total spin of the system.
The role of particle-hole asymmetry will be assessed through the conduction
electron excess charge:

$N_{\alpha}=\langle\hat{N}_{\alpha}\rangle-\langle\hat{N}_{\alpha}\rangle^{0}\leavevmode\nobreak\
,$ (8.14)

where we have defined $\hat{N}_{\alpha}=\sum_{\sigma}\hat{N}_{\alpha\sigma}$.
Dynamics of the system are characterized by the channel-resolved
$\mathrm{T}$-matrix, which describes how conduction electrons are scattered
from the dot pseudospin. The $\mathrm{T}$-matrix equation, see definition in
Eq.2.36, reads,

$\mathbb{G}_{\alpha\beta}(\omega,T)-\delta_{\alpha\beta}\mathbb{G}^{0}_{\alpha\beta}(\omega)=\mathbb{G}^{0}_{\alpha\alpha}(\omega)\mathbb{T}_{\alpha\beta}(\omega,T)\mathbb{G}^{0}_{\beta\beta}(\omega)\leavevmode\nobreak\
,$ (8.15)

where $\mathbb{G}_{\alpha\beta}(\omega,T)$ and
$\mathbb{G}_{\alpha\beta}^{0}(\omega)$ are, respectively, the full and free
retarded electronic Green’s functions at the dot position. Due to decoherence
between the QPCs resulting in separately conserved charge in each channel in
Eq. 8.9, we have $\mathbb{T}_{\alpha\beta}=\mathbb{T}_{\beta\alpha}=0$ and the
$\mathrm{T}$-matrix equation becomes channel-diagonal. Furthermore,
$-\tfrac{1}{\pi}{\rm Im}\mathbb{G}^{0}_{\alpha\alpha}(\omega)=\rho(\omega)$ is
the free graphene DoS. We introduce now the $\mathrm{T}$-spectral function for
channel $\alpha$:

$t_{\alpha}(\omega,T)=-\frac{\mathit{Im}\mathbb{T}_{\alpha\alpha}(\omega,T)}{\pi}\leavevmode\nobreak\
,$ (8.16)

and we observe that $t_{\alpha}(\omega)$ is sensitive to the scattering rate
in each channel. The definition in Eq.8.16 returns nonzero results for
vanishing density of states, compare it with the expression in Eq.3.68, as we
have a pseudogap spectrum in this model.
Before we continue with the results, we discuss the NRG applied to the two-
channel charge-Kondo model in Eq.8.9 \- compare with Sec.2.3.

#### Numerical Renormalization Group

The two-channel pseudogap charge-Kondo model, Eq. 8.9, is solved using
Wilson’s Numerical Renormalization Group (NRG) technique [10, 70, 74], which
provides numerically-exact access to the physical quantities discussed in the
previous section.
Here we go briefly through the NRG for this model, see also the schematic in
Fig.2.4. The first step is the logarithmic discretization of the conduction
electron DoS, and subsequent mapping of $\hat{H}_{leads}$ to Wilson chains
[10, 70],

$\hat{H}_{leads}\to\hat{H}^{WC}_{leads}=\sum_{\alpha,\sigma}\leavevmode\nobreak\
\sum_{n=0}^{\infty}t_{n}\left(f^{\dagger}_{\alpha\sigma n}f_{\alpha\sigma
n+1}^{\phantom{\dagger}}+f^{\dagger}_{\alpha\sigma n+1}f_{\alpha\sigma
n}^{\phantom{\dagger}}\right)\leavevmode\nobreak\ .$ (8.17)

The dot then couples to the end of the Wilson chains, at site $n=0$. The
logarithmic discretization is parametrized by $\Lambda$, with the continuum
description being recovered as $\Lambda\to 1$ (in this work we use a standard
choice of $\Lambda=2.5$). The key feature of the Wilson chain is the behaviour
of the hopping parameters $t_{n}$. For the metallic flat band,
$t_{n}\sim\Lambda^{-n/2}$ at large $n$ [10]. This exponential energy-scale
separation down the chain justifies a numerical scheme based on iterative
diagonalization and truncation: starting from the dot, successive sites of the
Wilson chain are coupled into the system and this intermediate Hamiltonian is
diagonalized. Only the lowest $M_{K}$ eigenstates at iteration $n$ are used to
construct the Hamiltonian at iteration $n+1$. High-energy states discarded at
a given iteration do not affect the retained low-energy states at later
iterations because of the ever-decreasing couplings $t_{n}$. This constitutes
an RG procedure since the physics of the system at successively lower energy
scales is revealed as more Wilson orbitals are added. The computational
complexity is constant as new Wilson orbitals are added (rather than
exponentially growing) because the same number $M_{K}$ of states is kept at
each step. Importantly, it was shown in Ref. [234] that although the detailed
structure of the Wilson chain coefficients are modified in the pseudogap DoS
case, the energy scale separation down the chain is maintained, and hence the
NRG can still be used in this case. We use the exact graphene DoS in this work
rather than a pure pseudogap, and keep $M_{K}=6000$ states at each iteration.
Dynamical quantities are calculated using the full-density-matrix NRG approach
[74, 235], established on the complete Anders-Schiller basis [72].

###  8.3 Results

In this section, initially we present in Sec.8.3.1 a summary of our results by
means of the phase-diagram of the system couplings and the fixed point
analysis on the $3D$ version of RG-flow diagram, then we detail our findings
with respect to the thermodynamics in Sec. 8.3.2 and dynamical quantities in
Sec. 8.3.3 according to the system degree of freedom.
In the following we confine our attention to the charge-degeneracy point
$\delta V_{g}=0$ and we also assume $W>0$ such that the Kondo effect can
happen. We also introduce the channel-asymmetry parameter
$\Delta=J_{2}/J_{1}\equiv W_{2}/W_{1}$ and discuss the physics in the space of
$J\equiv J_{1}$, $W\equiv W_{1}$ and $\Delta$. Note that $\Delta=0$
corresponds to the situation in which channel $\alpha=2$ is decoupled on the
level of the bare model, while for $0<\Delta<1$ both channels are coupled to
the dot but channel $1$ couples more strongly. $\Delta=1$ describes the
frustrated two-channel situation. We confine our attention to regime
$0\leq\Delta\leq 1$ but it should be noted that $\Delta>1$ simply corresponds
to stronger coupling for channel $2$, and results follow from the duality
$1\leftrightarrow 2$ and $\Delta\leftrightarrow 1/\Delta$. A last observation:
the reason behind the same numerical value $\Delta$ for the couplings ratio is
because we want to deform in equal proportion each dimension of the phase-
diagram. In principle, the parameters are all independent but in practice
these are not and any variation has to be controlled at equal footing in any
axis.
Before we start presenting our findings, we give below a list of abbreviations
we use in the next parts: FP i.e. fixed-point, LM i.e. local moment, (F)ALM
i.e. (frustrated) asymmetric local moment, (F)ASC i.e. (frustrated) asymmetric
strong coupling, (F)SSC i.e. (frustrated) symmetric strong coupling and (F)ACR
i.e. (frustrated) asymmetric critical.

####  8.3.1 Overview: phase-diagram and fixed-point analysis

In Fig.8.3 we show the quantitative phase-diagram in the $(J,W)$ plane for
different $\Delta$, with the exact phase boundaries obtained with NRG. The
phase in each region depends on: $(i)$ particle-hole symmetry where symmetric
(asymmetric) FP indicates zero (finite) $W$, $(ii)$ channel asymmetry
parameter $\Delta$ varying from asymmetric channel configuration at
$0\leq\Delta<1$ to symmetric at $\Delta=1$ with corresponding frustrated FPs.
The region outside the curves is the local moment phase characterised by the
dot spin uncoupled from both leads whereas the region inside each curve
corresponds to asymmetric strong coupling phase where the Kondo effect
develops as the dot spin is quenched by the surrounding conduction electrons.
On the curve, the value pairs $(J_{CR},W_{CR})$ identify the critical phase
boundary.

Figure 8.3: Full NRG phase diagram for $0\leq\Delta\leq 1$. The enclosed
region in each case is the Kondo-screened ASC phase (frustrated FASC for
$\Delta=1$); the exterior region is the unscreened LM phase. Inset shows
asymptotic behaviour of phase boundaries, see text.

From the non-perturbative NRG results displayed in Fig.8.3, we deduce the $3D$
version of the RG-flow diagram in the space of $(J,W,\Delta)$, see schematic
in Fig.8.4, and it gives a good overview of the physics of Eq. 8.9. We discuss
first the phase-diagram and then go through the fixed-point analysis.

Figure 8.4: $3D$ RG-flow diagram for the pseudogap 2CCK model Eq. 8.9, in the
space of effective exchange coupling $J$, potential scattering $W$ and channel
asymmetry $\Delta$. Stable (unstable) FPs are denoted as blue (red) squares.
Couplings flow from $\Delta=1$ to $\Delta=0,+\infty$ are indicated with
magenta lines. $\Delta=0$ ($+\infty$) is the pure single-channel for
$\alpha=1$ ($\alpha=2$) strongest channel and $\Delta=1$ is frustrated two-
channel model. For an explanation of the FPs, see text.

We start discussing the most significant aspect in Fig.8.3: in the phase-
diagram we find a finite ASC region at any $\Delta$ and this area does not
disappear even at the asymptotes, as it follows from our analysis. The upper
branch asymptote equation is $W^{U}_{CR}(J,\Delta)=a_{U}(\Delta)J_{CR}$
fitting and for all $\Delta$ we find $a_{U}(\Delta)\equiv a_{U}\approx 1.0$
such that $W^{U}_{CR}\leavevmode\nobreak\ \propto\leavevmode\nobreak\ J_{CR}$
has linear behaviour. In the lower branch asymptote at $\Delta=0$, considering
the effective exchange coupling expression we would expect the curve to decay
as $W^{L}_{CR}\leavevmode\nobreak\ \propto\leavevmode\nobreak\ J_{CR}^{-1}$.
However, our numerical analysis breaks down at $10^{2}\lesssim J$. This
numerical limitation prevents from resolving completely the asymptotic
behaviour at $\Delta=0$ and we obtain from the numerical
$W^{L}_{CR}\leavevmode\nobreak\ \propto\leavevmode\nobreak\ J_{CR}^{-1.35}$.
On the contrary, at $0<\Delta\leq 1$ we find no numerical restrictions and the
lower branch asymptote equation is $W^{L}_{CR}(J,\Delta)=a_{L}(\Delta)J_{CR}$.
We find $a_{L}$ increases with increasing $\Delta$, as shown in the inset in
Fig. 8.3. However, $a_{L}(\Delta)/a_{U}<1$ for all $\Delta$ and the ratio
reaches its maximum $\approx 0.2$ as $\Delta\to 1$. This asymptotically proves
the upper and lower phase boundaries never cross, and the ASC phase persists
out to infinite $J$ and $W$.
We analyse now the system at $\Delta=0$, that is the turquoise line in the
phase-diagram in Fig.8.3 and the front plane in the $3D$ RG-flow in Fig.8.4.
The basic physics of this single-channel Kondo model are well-known from
previous studies of the $r=1$ pseudogap Anderson and Kondo models [234, 236,
237, 232, 238, 239, 211, 212], although note that our graphene charge-Kondo
set-up gives a spin-anisotropic model, and we use the full graphene DoS rather
than a pure pseudogap.
At particle-hole symmetry regime, the system shows no spin screening except
for arbitrary small but finite $W$ in the limit $J\rightarrow+\infty$
corresponding to symmetric strong coupling FP (SSC). In this sense, the
$\Delta=0$ curve does not have a $W^{min}$. Once particle-hole symmetry is
broken, we find on the curve a $J^{min}$: for Eq. 8.9, $J_{CR}^{min}\simeq
1.81D$. For $J<J_{CR}^{min}$ no Kondo state is possible at any $W$ value. The
$J^{min}$ value denotes both the minimum exchange coupling required to
activate the Kondo singlet formation and a critical value at the transition
between LM and ASC regions. Inside the ASC phase, we distinguish two phases:
$(i)$ $W<W^{L}_{CR}(J^{min}_{CR})$ implies the exchange coupling is not large
enough to compensate the rather small $W$, $(ii)$ $W>W^{U}_{CR}(J^{min}_{CR})$
implies $W$ is very large and suppresses the effective spin-exchange. Under RG
scaling the effective exchange coupling between $f_{1\sigma 0}$ and
$f_{1\sigma 1}$ lead orbitals becomes very small at large $W$ in this region.
Hence, in both cases the system enters in LM phase and for $J\geq
J_{CR}^{min}$. The ASC phase is stable for $J\geq J_{CR}^{min}$ and within
$W_{CR}^{L}(J)<W<W_{CR}^{U}(J)$. This behaviour found from the analysis of the
full NRG phase boundary deserves more comments. That is recognised as re-
entrant behaviour back into the LM phase at large $W$: only suitable
intermediate couplings bring the system inside the ASC phase. We are not aware
of a discussion of this in the literature, but the physical arguments we
describe for re-entrant LM behaviour in the single and two-channel spin-
anisotropic pseudogap Kondo model are also valid for the regular spin-
isotropic single-channel pseudogap Kondo case. NRG data for the full phase
boundary for the single channel spin-isotropic Kondo model confirm
qualitatively the same curve we found in Fig.8.3. This re-entrant LM behaviour
is physically intuitive since $J$ and $W$ work antagonistically: at very large
$W$ the lead orbital $f_{1\sigma 0}$ becomes depopulated, and hence the
exchange coupling to that site $J$ gets switched off. Perturbative arguments
suggest that the residual coupling to the $f_{1\sigma 1}$ bath orbital is then
$J_{eff}\sim t_{0}^{2}J/W^{2}$, which is consistent with
$W^{U}_{CR}\leavevmode\nobreak\ \propto\leavevmode\nobreak\ J_{CR}$. This is
indeed confirmed by NRG calculations.
The main focus of this work is the situation when the coupling to the second
channel is switched on, hence we have $\Delta>0$, and we find several
differences form the pure single-channel model.
The system at $0<\Delta<1$ presents finite threshold values of the couplings
to realize ASC physics - see curves from light blue to light violet in Fig.8.3
and the intermediate volume connecting $\Delta=1$ towards $\Delta=0$ plane
through magenta lines in the $3D$ RG-flow in Fig.8.4. Those are
$J^{min}_{CR}(\Delta)>0$ \- which increases slightly from $\simeq 1.81D$ at
$\Delta=0$ up to $\simeq 2.47D$ as $\Delta\to 1$ \- and
$W^{min}_{CR}(\Delta)>0$ \- which reaches its maximum value $\simeq D$ as
$\Delta\to 1$. For $W^{min}_{CR}(J_{min}(\Delta))>W^{min}$ and
$J^{min}_{CR}(\Delta)>J^{min}$ we observe again the re-entrant LM phase with
stable ASC phase within
$W^{L}_{CR}(J^{min}_{CR}(\Delta))<W<W^{U}_{CR}(J^{min}_{CR}(\Delta))$.
Finally, see magenta curve in Fig.8.3 and the middle plane $\Delta=1$ in
Fig.8.4, an analogue phase-digram topology to $0<\Delta<1$ persists also in
the system at $\Delta=1$ channel frustrated regime.
The different FPs we identify from the full NRG phase-diagram can be
schematically represented in the RG-flow diagram, see Fig.8.4. The FPs are
classified according to particle-hole symmetry (asymmetry) and channel
asymmetry parameter $\Delta$. We can further distinguish the FPs based on the
type of Fermi liquid system. As it will be clear from the $\mathrm{T}$-matrix
spectral function analysis in Sec. 8.3.3, both LM and ASC phases are regular
Fermi liquids but the ACR PF is of non-Fermi liquid type. In the RG-flow the
system undergoes simultaneous perturbations in exchange coupling $J$,
potential scattering $W$ and asymmetry parameter $\Delta$. The different FPs
at $(J,W,\Delta)$ \- which are understood as the values taken under RG at the
FPs themselves - and their trajectories are presented in the $3D$ version of
RG-flow, see Fig.8.4. The set of FPs located on the $\Delta=0$ plane mirrors
the ones at $\Delta=+\infty$ for the mentioned duality $1\leftrightarrow 2$.
The $\Delta=1$ plane is also characterised by additional channel frustration.
We explain now the various FPs and their interconnection under RG.
In the $\Delta=0$ plane, that correspond to the standard single-channel Kondo
model, we find the stable LM FP at $(0,0,0)$ and the stable ASC FP at
$(\infty,\infty,0)$. The unstable ACR FP at $(J_{CR},W_{CR},0)$ represents the
crossover between LM and ASC FPs where the flow moves to. This is a
$1^{st}$-order level crossing transition meaning that the degrees of freedom
at the CR phase are the addition of those at LM and ASC phases [232, 238]. At
the unstable SSC FP located in $(\infty,0,0)$, the system can flow arbitrarily
close to it before ultimately crossing over to either LM or ASC FPs. The
latter ASC FP is accessible from SSC due to the lack of finite $W^{min}$. At
$(0,\infty,0)$ we find the unstable asymmetric LM FP (ALM): the existence of
finite $J^{min}$ prevents from direct flowing from ALM towards ASC and the ALM
FP is not directly reachable under RG. However, the eigenspectrum levels in
ALM phase coincide with the levels in the LM FP, hence from ALM the couplings
flow towards LM FP.
In the $\Delta=1$ plane, symmetry dictates a channel degeneracy down to $T=0$
and therefore no channel freezing \- meaning decoupling of the less strongly
coupled channel that is $\alpha=2$ in $0\leq\Delta<1$ case. In this
configuration, we have only one stable FP that is the LM FP at $(0,0,1)$, in
which both channels flow symmetrically to weak coupling and to particle-hole
symmetry. The frustrated ASC (FASC) at $(\infty,\infty,1)$ acquires
instability because of the channel degeneracy: it is a doubled version of ASC
FP in $\Delta=0$ with the Kondo effect and conduction electron hole form in
either channel $\alpha=1$ or $2$. The unstable frustrated ACR (FACR) at
$(J_{CR},W_{CR},1)$ is tricritical because it remains at the crossover between
LM, FASC and ASC phases. It indicates again a $1^{st}$-order quantum phase
transition. Due to finite $W^{min}$, from the unstable frustrated SSC FP
(FSSC) at $(\infty,0,1)$ there is no crossover to FASC phase but from FSSC the
coupling moves only towards LM FP. Due to finite $J^{min}$, from the unstable
frustrated ALM FP (FALM) at $(0,\infty,1)$ there is again no crossover to FASC
phase under RG and couplings flow only towards LM FP.
We note that the frustrated FPs are delicate because they sit precisely on the
separatrix between RG-flow to states with dominant channel $1$ for $\Delta<1$,
and flow to states with dominant channel $2$ for $\Delta>1$. Any finite
perturbation in $|\Delta-1|$ \- as we discuss more in Sec.8.3.2 \- relieves
the channel frustration and leads ultimately to channel freezing on the lowest
energy scales.
Perturbing FPs located on the $\Delta=1$ plane such that $\Delta<1$ leads to
RG-flow continuously connected towards the FPs on $\Delta=0$ plane, see
magenta lines pointing from the middle plane towards the front plane in
Fig.8.4. For all $0<\Delta<1$ \- identified by the intermediate planes $J=0$,
$W=0$, $J=\infty$ and $W=\infty$ \- the stable FPs are all in the $\Delta=0$
plane. Analogously, starting from $\Delta=1$ plane, we find flow arrows
towards the back plane $\Delta=+\infty$ due to perturbation in $\Delta>1$.
Because of relevant perturbation in $\Delta$, under RG, the couplings flow
starting from FPs corresponding to asymmetric channel configurations move away
from $\Delta=1$ plane until ending on single-channel FPs in either $\Delta=0$
or $\Delta=\infty$ planes. In this sense, the single-channel model is
understood as limiting case of the asymmetric two-channel model with the
weakest channel fully decoupled under RG-flow. Any asymmetric two-channel
model represents intermediate instances of the single-channel limit because
any channel asymmetry leads asymptotically to the decoupling of the less
strongly coupled channel. This can be regarded as channel freezing at low
temperatures opposed to free channel degree of freedom $\alpha=1,2$ found at
at high temperatures. As consequence of this insight, with respect to
$\Delta=1$ plane, symmetric RG-flow trajectories move either towards
$\Delta=0$ if $\Delta<1$ and so $\alpha=2$ channel is decoupled or towards
$\Delta=\infty$ if $\Delta>1$ and so $\alpha=1$ channel is decoupled, see Fig.
8.4. Hence, each frustrated FPs find its asymmetric counterpart either on
$\Delta=0$ or $\Delta=\infty$ plane according to the channel strength. This
smooth connection between the two planes fails for the LM FP: $\Delta$ is
defined as ratio of couplings hence on $J=W=0$ axis it is undetermined. We
find more appropriate to identify the $J=W=0$ axis as LM FP line in the limit
$J,W\rightarrow 0$: this axis represents the LM phase at any $\Delta$.
To conclude, giving the importance of the $\Delta$ parameter in attributing
the FPs properties, we classify the system degree of freedom as follows: at
$0\leq\Delta<1$ frozen channel degrees - since all the couplings will
inevitably flow under RG towards the $\Delta=0$ FPs and at $\Delta=1$
frustrated channel degrees of freedom. We use this classification to present
the thermodynamics in the next section.
In Table 8.1, we summarize the FPs discussed above in relation to Figs.
8.3,8.4 and classify them according to their physical properties. These
properties are extracted from the limiting behaviour of the full thermodynamic
and dynamic observables presented in Sec.8.3.2 and Sec.8.3.3, respectively. In
particular, the excess charge for the strongest channel $\alpha=1$ can be
directly computed using NRG via the formula in Eq.8.14. These are discussed
below.

asymmetry | fixed-point | $S_{D}(T=0)$ | $T\chi_{D}(T=0)$ | $N_{1}$ | $t_{1}(\omega,T\rightarrow 0)$ | $\mathcal{G}^{C}(\omega,T\rightarrow 0)$
---|---|---|---|---|---|---
$\forall\Delta$ | LM line | $\ln 2$ | $1/4$ | $0$ | $|\omega|$ | $\omega^{2}$
$0\leq\Delta<1$ | ASC | $0$ | $0$ | $-1$ | $|\omega|$ | $\omega^{2}$
$0\leq\Delta<1$ | ACR⋆ | $\ln 3$ | $1/6$ | $-1/3$ | $1/\omega\ln^{2}(\lambda_{CR}/\omega)$ | $0$
$\Delta=1$ | FASC⋆ | $\ln 2$ | $0$ | $-1/2$ | $|\omega|$ | $\omega^{2}$
$\Delta=1$ | FACR⋆ | $\ln 4$ | $1/8$ | $-1/4$ | $1/\omega\ln^{2}(\lambda_{CR}/\omega)$ | $const$

.

Table 8.1: Classification of FPs according to their physical observables, with
superscript $\star$ denoting unstable FPs.

####  8.3.2 Thermodynamics

The temperature dependence of the dot contribution to entropy $S_{D}(T)$ and
magnetic susceptibility $T\chi_{D}(T)$ are obtained from NRG [70] and
presented in Fig. 8.5 for different channel asymmetries $\Delta$. We focus on
the behaviour near the critical points by fixing $W$ and tuning $J$ across the
transition. From this, information on the fixed points is deduced.

##### Frozen channel degree of freedom: $0\leq\Delta<1$

We first consider the regime $0\leq\Delta<1$ in the left and middle columns of
Fig. 8.5. Solid lines show the behaviour in the ASC phase for $J>J_{CR}$,
dashed lines for the LM phase with $J<J_{CR}$, and orange line at the critical
point $J=J_{CR}$.
In the LM phase, $S_{D}=\ln(2)$ and $T\chi_{D}=\tfrac{1}{4}$ at $T=0$ in all
cases, characteristic of the asymptotically-free spin-$\tfrac{1}{2}$ dot
pseudospin - compare also with first row in Table 8.1. The excess conduction
electron charge - not shown in the plot - is zero in both channels, suggesting
an emergent particle-hole symmetry. This is confirmed by analysis of the NRG
many-particle level spectrum (finite size spectrum) at the LM FP, which is
identical to that of the free leads. The dot remains unscreened in LM because
of the depleted conduction electron DoS at low energies in graphene [211]. The
FP Hamiltonian in the LM phase is therefore given by,

$\displaystyle\boxed{\hat{H}_{LM}=\hat{H}^{2CCK}}\quad{\rm with}\quad
J_{1}=J_{2}=W_{1}=W_{2}=\delta V_{g}=0\leavevmode\nobreak\ .$ (8.18)

In the ASC phase at $T=0$, the spin degree of freedom is completely quenched
by the conduction electrons. As consequence, the system has quenched dot
entropy $S_{D}=\ln 1=0$ and $T\chi_{D}=0$ because the dot spin is in a singlet
state. These thermodynamical values apply in all ASC cases, see Table 8.1,
characteristic of Kondo singlet formation. However, the conduction electron
excess charge is $N_{1}=-1$ and $N_{2}=0$ for $W>0$, implying hole formation
in the more strongly coupled channel $\alpha=1$ with $W_{1}\to\infty$ under
RG, while the less strongly coupled channel $\alpha=2$ recovers an effective
low-energy particle-hole symmetry, $W_{2}\to 0$ under RG. This suggests the
screening mechanism in the generic two-channel case which we explain now with
the aid of the Wilson chain representation given in Eq. 8.17 and the
dimensionless coupling $j_{\alpha}=J_{eff}\rho_{\alpha n}$ defined on
$\hat{H}_{leads}^{WC}$.
The two-channel model in the NRG scheme corresponds to the graphene-dot
coupled to two Wilson chains. In general, the potential scattering applies
only to the $0^{th}$-site - corresponding to $f_{\alpha\sigma 0}$ Wilson
orbital - in which it creates a hole at $T_{hole}$ temperature. This is
because its effect corresponds to an unoccupied orbital in the physical
system. Once we introduce channel asymmetry, only the channel with the largest
$W$ forms the hole on the $0^{th}$-site. If we take $W_{1}>W_{2}$ as it occurs
in the $\Delta<1$ case, the hole develops only on the $\alpha=1$ channel at
$f_{1\sigma 0}$ Wilson orbital with pseudogap DoS
$\rho_{10}(\omega)=\rho|\omega/D|$ at $n=0$ site as given in Eq. 8.2 with
$r=1$. Because of $f_{1\sigma 0}$ Wilson orbital becomes depopulated, thereby
it generates an effective coupling between the dot pseudospin and the Wilson
$f_{1\sigma 1}$ orbital, $J_{eff}\sim t_{0}^{2}J_{1}/W_{1}^{2}$. Then, the
$0^{th}$-site does not contribute to the Kondo singlet because it has no
electronic levels available to screen the dot spin and the $0^{th}$-site is
like removed. The remaining shortened chain modifies the pseudogap DoS at
$1^{st}$-site into a power-law $\rho_{11}(\omega)=\rho|\omega/D|^{-1}$. The
$1^{st}$-site now has enough electronic states to quench the dot spin and the
Kondo singlet is formed at temperature $T_{K}$. Despite of $J_{eff}$ coupling
on $\alpha=1$ channel being smaller than $J_{2}$ under RG-flow, the
$\rho_{11}(\omega)$ is strongly enhanced such that
$j_{1}=J_{eff}\rho_{11}(\omega)$ is larger than $j_{2}=J_{2}\rho_{20}(\omega)$
at the Fermi level due to hole-singlet mechanism. As consequence, the weakest
$\alpha=2$ channel decouples from the graphene-dot under RG such that the
resulting system is effectively single-channel model. This is explained
because no hole forms in the weakly coupled channel, and so
$j_{2}=\rho_{20}J_{2}$ remains small due to the depleted bare DoS in channel
$2$, and flows under RG to weak coupling. This argument shows that the Kondo
singlet must form in the same channel in which the hole forms and this is
confirmed by analysis of the NRG level spectrum. Furthermore, hole and Kondo
singlet formation on the same chain side is a simultaneous process hence it
manifests at the same energy scale $T_{hole}\equiv T_{K}$. Throughout the ASC
phase $T_{K}\sim D$ since $J_{CR}^{min}>0$. Thus, this Kondo scale for the
screening process is strongly enhanced because of the diverging effective DoS
[240].
By means of this two-channel screening or hole-singlet mechanism, the
$0<\Delta<1$ model represents the system with intermediate hole-singlet stage
and partial decoupling of the weakest channel: once these processes are
completed the system reaches the limiting case of the $\Delta=0$ single-
channel model.
In general we therefore have two distinct ASC phases and two distinct ASC
fixed points, depending on whether $\Delta<1$ or $\Delta>1$. For $\Delta<1$,
the hole-singlet complex forms in channel $\alpha=1$ and channel $\alpha=2$
decouples (FP denoted ASC1), while for $\Delta>1$ (ASC2) it is the other way
around. The ASCα FP Hamiltonian obtained when channel $\alpha$ is more
strongly coupled reads,

$\displaystyle\boxed{\hat{H}^{\alpha}_{ASC}=\hat{H}_{leads}+J_{\alpha}\left(\hat{S}_{D}^{+}f_{\alpha\downarrow
1}^{\dagger}f_{\alpha\uparrow
1}^{\phantom{\dagger}}+\hat{S}_{D}^{-}f_{\alpha\uparrow
1}^{\dagger}f_{\alpha\downarrow
1}^{\phantom{\dagger}}\right)+W_{\alpha}\sum_{\sigma}f_{\alpha\sigma
0}^{\dagger}f_{\alpha\sigma 0}^{\phantom{\dagger}}}\leavevmode\nobreak\ {\rm
with}\leavevmode\nobreak\ J_{\alpha},W_{\alpha}\to\infty\leavevmode\nobreak\
.$ (8.19)

We now consider the situation in the close vicinity of the QPT, by fixing $W$
and tuning $J$. At the critical point itself - indicated in the orange line
for $J=J_{CR}$ \- we find $S_{D}=\ln(3)$ and $T\chi_{D}=\tfrac{1}{6}$ at
$T=0$, compare with the third row in Table 8.1. This suggests a level-crossing
transition in which the critical FP ACR comprises uncoupled sectors
corresponding to LM and ASC. This gives an overall dot ground state degeneracy
of $2+1=3$ states, that is $2$ for LM and $1$ for ASC, consistent with the
$\ln(3)$ entropy, and a magnetic susceptibility
$(\tfrac{1}{4}+\tfrac{1}{4}+0)/3=\tfrac{1}{6}$ corresponding to the average of
$(S^{z})^{2}$ for these three degenerate states. This is further supported by
the conduction electron excess charge $N_{1}=-\tfrac{1}{3}$, since a single
hole appears in channel $1$ for only one of the three degenerate ground states
and $N_{2}=0$ for the decoupled free channel $2$.
The first-order transition is also consistent with the linear emergent
crossover scale $T^{\star}\sim|J-J_{CR}|$, describing the flow from ACR to
either LM or ASC due to a small detuning perturbation. This scale is evident
in Fig. 8.5 by the sequence of lines for different $(J-J_{CR})$. Indeed, one
can cross the QPT by fixing $J$ and tuning $W$ through $W_{CR}$, which also
gives a linear scale $T^{\star}$. We also checked this behaviour along the
entire critical phase boundary lines $(J_{CR},W_{CR})$ in Figs. 8.4,8.3 for
different $\Delta$. We find,

$\displaystyle T^{\star}=b|J-J_{CR}|+c|W-W_{CR}|\;,$ (8.20)

where $b\equiv b(J_{CR},W_{CR},\Delta)$ and $c\equiv c(J_{CR},W_{CR},\Delta)$.
This implies a universal scaling in terms of a single reduced parameter
$T/T^{\star}$, independent of the combination of bare perturbations that act.
The FP Hamiltonian describing the critical point is,

$\boxed{\hat{H}^{\alpha}_{ACR}=\left(\frac{1+\hat{\tau}^{z}}{2}\right)\hat{H}_{LM}+\left(\frac{1-\hat{\tau}^{z}}{2}\right)\hat{H}^{\alpha}_{ASC}}\leavevmode\nobreak\
.$ (8.21)

where the $\alpha$ label denotes the more strongly coupled lead with which the
dot forms the Kondo effect in ASC, and $\hat{\tau}^{z}$ is a Pauli-$z$
operator. In Eq. 8.21, $\tau^{z}=+1$ gives the doubly-degenerate LM ground
state ($\hat{H}_{LM}$ given in Eq. 8.18) while $\tau^{z}=-1$ gives the ASC
ground state ($\hat{H}_{ASC}$ given in Eq. 8.19). At the ACR FP, the three
many-body ground states are degenerate and uncoupled - $\hat{\tau}^{z}$ has no
dynamics. Since ACR is unstable, we also consider the leading RG relevant
perturbations to the FP Hamiltonian,
$\delta\hat{H}^{\alpha}_{ACR}\sim+T^{\star}\hat{\tau}^{z}$, which has the
effect of biasing towards either the LM or ASC ground states on the scale of
$T^{\star}$. In such a way, the ACR ground-state degeneracy is lifted.
We briefly comment about the SSC and ALM FPs Hamiltonian: both present
instability due to relevant correction in the RG-flow of the same type of the
ACR phase, namely
$\delta\hat{H}_{SSC},\delta\hat{H}_{ALM}\sim+T^{\star}\hat{\tau}^{z}$.
We conclude the analysis for system presenting frozen channel degree with two
comments. As first: the qualitative behaviour of the thermodynamics shown in
Fig. 8.5 for $\Delta=0$ and $\Delta=0.8$ is similar, but it should be noted
that both channels are involved for $\Delta\neq 0$ at finite temperatures.
However, the less strongly coupled channel decouples asymptotically because
finite $0<\Delta<1$ flows to $\Delta=0$ under RG on reducing the temperature
or energy scale - see Fig. 8.4. As second, we report that from the
eigenspectrum analysis for both FPs Hamiltonian $\hat{H}^{\alpha}_{ASC}$ and
free Wilson chain, we observe a matching in the energy levels. This confirms
an even number of sites, i.e. $0^{th}$ and $1^{st}$, have to be decoupled from
the chain after the hole-singlet formation. This fact is verified also from
the constant phase-shift computed in NRG.

Figure 8.5: Dot contribution to thermodynamic quantities for the graphene 2CCK
model obtained by NRG. Top row: entropy $S_{D}(T)$; _bottom row:_ magnetic
susceptibility $T\chi_{D}(T)$. Left, middle, and right columns correspond to
$\Delta=0$ (pure 1CCK), $\Delta=0.8$ (asymmetric 2CCK), and $\Delta=1$
(symmetric 2CCK) respectively. Shown for fixed $W=2D$, varying $J$ across the
QPT according to the colour scale, with solid lines for $J>J_{CR}$ in the ASC
(FASC) phase, and dashed lines for $J<J_{CR}$ in the LM phase. Orange lines
show the behaviour at the ACR (FACR) critical point.

##### Frustrated channel degree of freedom: $\Delta=1$

We turn now to the frustrated case $\Delta=1$, with pristine channel symmetry,
see Fig. 8.5, right column. Although the $T=0$ entropy is $S_{D}=\ln(2)$
everywhere except on the phase boundary in the top right panel of Fig. 8.5,
the origin of the ground state degeneracy is different in the two phases
separated by it.
In the LM phase realized for $J<J_{CR}$, no reduction of spin degree of
freedom occurs in the symmetric two-channel model and the fixed-point
Hamiltonian is again given by Eq. 8.18. Thus, we again have a free dot
pseudospin decoupled from two symmetric baths of free conduction electrons;
the $\ln(2)$ entropy here derives from the free dot pseudospin-$\tfrac{1}{2}$
degree of freedom. This is confirmed by the magnetic susceptibility in this
phase, which reaches $T\chi_{D}=\tfrac{1}{4}$, see dashed lines in the bottom
right panel of Fig. 8.5.
The other phase realized for $J>J_{CR}$ is described by the FASC FP: due to
the channel symmetry, the ASC state can form in either channel $\alpha=1,2$.
The $\ln(2)$ entropy in this case derives from the free channel degree of
freedom [228], which embodies the choice of forming the hole-singlet complex
of ASC with either of the two channels. Hence, this is a residual entropy
arising from the frustrated choice of which channel to form the Kondo effect
in and it opposes to the results in the channel asymmetric case. The finite
entropy value is similarly reflected in the $T=0$ value of $T\chi_{D}=0$ in
the FASC phase, see solid lines in the bottom right panel of Fig. 8.5 and
fourth row in Table 8.1, since the dot pseudospin is Kondo screened in both of
the degenerate ground states. Furthermore, we find that the average conduction
electron excess charge in FASC is $N_{\alpha}=\tfrac{1}{2}$ for both channels
- that is, a single hole forms, with equal probability to be in either channel
$1$ or $2$.
A Kondo strong coupling state involving both channels simultaneously is not
stable. To see this, we consider two holes forming symmetrically in the
$f_{1\sigma 0}$ and $f_{2\sigma 0}$ Wilson $n=0$ orbitals with
$W_{1}=W_{2}\leavevmode\nobreak\ ,\leavevmode\nobreak\ J_{1}=J_{2}$. We have
then a channel symmetric version of the hole-singlet mechanism we describe in
ASC phase for $0\leq\Delta<1$. However, the dot entropy is not quenched in
this case, since the ground state of the complex is a spin-doublet. At the end
of the Kondo singlet formation, this effective doublet state couples to the
Wilson $n=2$ orbitals $f_{1\sigma 2}$ and $f_{2\sigma 2}$. But since the DoS
of these sites is again $\rho_{\alpha 2}(\omega)=\rho|\omega/D|$ the effective
local moment cannot be screened and the system flows to the LM FP. The dot
pseudospin can only be screened by an asymmetric ASC state. Channel symmetry
is restored by having two such degenerate states, one in each channel. To sum
up: the resulting Kondo singlet derives from the superposition of the singlet
formation only on $\alpha=1$ channel and only $\alpha=2$ channel due to
channel degeneracy. The final system is in a frustrated channel state. In each
of the two overlapping state, there are residual degrees of freedom from the
unscreened channel i.e. (pseudospin) free channel degrees from the leads which
are added to the spin-$1/2$ degrees from the screened channel.
The FASC FP Hamiltonian comprises a combination of $\hat{H}_{ASC}^{1}$ and
$\hat{H}_{ASC}^{2}$ from Eq. 8.19, controlled by an emergent channel degree of
freedom $\hat{\alpha}$,

$\boxed{\hat{H}_{FASC}=\left(\frac{1+\hat{\alpha}^{z}}{2}\right)\hat{H}^{1}_{ASC}+\left(\frac{1-\hat{\alpha}^{z}}{2}\right)\hat{H}^{2}_{ASC}}\leavevmode\nobreak\
.$ (8.22)

Here $\hat{\alpha}^{z}$ is a Pauli-$z$ operator that selects ASC1 when
$\alpha^{z}=+1$ and ASC2 when $\alpha^{z}=-1$ \- since no mixed channel terms
can be defined through $\hat{\alpha}^{z}$. Restricting to the symmetric
$\Delta=1$ plane, FASC is stable. However, there is an instability with
respect to breaking channel degeneracy - not shown in the plot, since then
either ASC1 or ASC2 will be selected on the lowest energy scales. A finite
perturbation $|\Delta-1|$ generates a flow from FASC to ASC1 or ASC2; from NRG
we find that this QPT is also first-order. The low-energy scale determining
the emergent crossover, namely

$T_{\Delta}\sim|\Delta-\Delta_{CR}|\equiv|\Delta-1|\leavevmode\nobreak\ ,$
(8.23)

This can be captured in the effective model by including the leading RG
relevant perturbation to the FASC FP,
$\delta\hat{H}_{FASC}\sim+T_{\Delta}\hat{\alpha}^{z}$.
Finally, we consider the quantum critical point in the $\Delta=1$ plane
between LM and FASC. Here we find a level crossing first-order transition,
with entropy $S_{D}=\ln(4)$ and magnetic susceptibility
$T\chi_{D}=\tfrac{1}{8}$ at the FACR FP [228] and see last row in Table 8.1,
which derives from the composition of uncoupled LM and FASC sectors. We have
two spin-$\tfrac{1}{2}$ states from the LM degenerate with two spin-singlet
states with a (pseudospin) free channel degree of freedom in FASC. The excess
conduction electron charge is therefore $N_{\alpha}=-\tfrac{1}{4}$ per
channel. We describe the FACR FP with the Hamiltonian,

$\boxed{\hat{H}_{FACR}=\left(\frac{1+\hat{\tau}^{z}}{2}\right)\hat{H}_{LM}+\left(\frac{1-\hat{\tau}^{z}}{2}\right)\hat{H}_{FASC}}\leavevmode\nobreak\
,$ (8.24)

where $\hat{H}_{LM}$ is given in Eq. 8.18 and $\hat{H}_{FASC}$ in Eq. 8.22,
and we have introduced the operator $\hat{\tau}^{z}$ to distinguish the
sectors, similar to Eq. 8.21. As with ACR, the FP is destabilized by RG
relevant detuning perturbations that favour either LM of FASC ground-state,
which collectively generate the scale $T^{\star}$ given in Eq. 8.20. This
leads to an FP correction
$\delta\hat{H}_{FACR}^{\star}\sim+T^{\star}\hat{\tau}^{z}$. This is shown by
the sequence of lines in the right column of Fig. 8.5. However, FACR is also
destabilized by relieving the channel frustration through the perturbation
$|\Delta-1|$ which generates the scale $T_{\Delta}$, since FACR contains an
FASC sector with this instability. Therefore FACR has a second RG relevant
correction $\delta\hat{H}_{FACR}^{\Delta}\sim+T_{\Delta}\hat{\alpha}^{z}$.
FACR is in this sense tricritical since it sits between LM, FASC and ASC.

####  8.3.3 Dynamics and transport

The FPs analysis, discussed in Sec.8.3.1, includes the FPs coordinates on the
$3D$ RG-flow in Figure 8.4 by means of the parameters $(J,W,\Delta)$ and how
those points are connected from $\Delta=1$ to $\Delta=0$ plane and to
$\Delta=\infty$ as well under channel symmetry breaking perturbations. In the
fixed-point Hamiltonian expressions, see Sec.8.3.2, we find also the FPs
stability property with related relevant perturbation parameters and crossover
energy scales. We now aim to complete the FPs characterization by analysing
dynamical properties. We can also differentiate between Fermi liquid and non-
Fermi liquid states by examining the $\mathrm{T}$-matrix spectral function
given in Eq. 8.16 via NRG in the $\omega\to 0$ limit.
In this final section of the results, we now discuss the low-temperature
behaviour of the scattering $\mathrm{T}$-matrix and linear response
$ac$-electrical conductance in the graphene 2CCK device - see Fig. 8.6. We
first consider the $T=0$ spectrum of the $\mathrm{T}$-matrix as a function of
energy in the top row of Fig. 8.6, for the channel asymmetric case
$\Delta=0.8$ on the left and frustrated case $\Delta=1$ on the right. In all
cases we identify an emergent low-energy scale $\lambda$ (which is $\approx
10^{-4}D$ for the parameters chosen) which characterizes the RG-flow through a
crossover behaviour in the pseudogap dynamics [232, 228].

Figure 8.6: NRG results for dynamics and transport in the graphene 2CCK model
at $T=0$. The legends refers to both top and bottom panels. Top row: Channel-
resolved spectrum of the $\mathrm{T}$-matrix $t_{\alpha}(\omega,0)$. Bottom
row: Linear response ac electrical conductance $G_{C}(\omega,0)$. Left:
channel asymmetry $\Delta=0.8$; Right frustrated case $\Delta=1$. Model
parameters: $J/D=10$ with $W/D=1$ (LM); $12$ (ASC); $12$ (FASC); $\simeq
8.605$ (ACR); $\simeq 8.572$ (FACR).

Deep in the LM phase, see blue solid and dotted lines in Fig. 8.6, the bare
potential scattering $W$ modifies the bare conduction electron pseudogap DoS
of graphene $\rho_{\alpha 0}(\omega)\sim|\omega|$, to give an effective DoS
$\rho_{\alpha 1}(\omega)\sim 1/|\omega|$ \- up to logarithmic corrections -
that corresponds to $\pi/2$ phase shift. This produces leading behaviour in
the $\mathrm{T}$-matrix $t_{\alpha}(\omega,0)\sim 1/|\omega|$, as seen in Fig.
8.6 for $|\omega|\gg\lambda$. However, under RG $W\to 0$ in the LM phase; this
flow is controlled by the scale $\lambda$. Therefore, on the scale of
$\lambda$ the effective DoS returns to $\rho_{\alpha 1}(\omega)\sim|\omega|$
\- indicating now zero phase shift - and hence
$t_{\alpha}(\omega,0)\sim|\omega|$ for $|\omega|\ll\lambda$. Since the
emergent particle-hole symmetry in the LM phase occurs in both channels for
any $\Delta$, compare with Table 8.1, we see the same behaviour for
$t_{1}(\omega,0)$ and $t_{2}(\omega,0)$ for both $\Delta=0.8$ and $1$.
In the ASC phase for $\Delta=0.8$, the weakly coupled channel $\alpha=2$ shows
the same behaviour as LM since it decouples from the dot and gains particle-
hole symmetry. For the strongly coupled channel $\alpha=1$, we have the hole-
singlet mechanism in which both the effective $W,J\to\infty$. Counter-
intuitively we again see similar dynamical behaviour as for LM. This is
because, for $|\omega|\gg\lambda$ we have a developing conduction electron
hole which gives $t_{1}(\omega,0)\sim 1/|\omega|$, while for
$|\omega|\ll\lambda$ the Kondo singlet forming with the $n=1$ Wilson orbital
effectively removes a second site from the bath. The remaining conduction
electrons experience a $\pi$ phase shift from the modified boundary, and the
effective DoS is back to $\rho_{\alpha 2}(\omega)\sim|\omega|$. Therefore in
ASC we also have $t_{1}(\omega,0)\sim|\omega|$ for $|\omega|\ll\lambda$. That
we have identical behaviour for $\Delta=1$ in both channels confirms that FASC
is indeed a superposition of ASC1 and ASC2 as argued above.
On the lowest energy scales we have $t_{\alpha}(\omega,0)\sim|\omega|$ in both
channels, at any $\Delta$, and in either phase. Given the bare DoS
$\rho_{\alpha 0}(\omega)\sim|\omega|$, this confirms that both phases are
regular Fermi liquids (FL) with well-defined, long-lived quasiparticles [38,
232, 238].
More interesting is the behaviour at the critical point (F)ACR, since here we
have both spin and charge fluctuations associated with the degenerate LM and
(F)ASC ground states. A new dynamical scale is generated,
$\lambda_{CR}\sim\lambda^{2}/D$ ($\approx 10^{-8}D$ for the chosen parameters)
which characterizes the low-energy RG-flow [232, 228]. We find from NRG that
in the channel-asymmetric graphene 2CCK model e.g. at $\Delta=0.8$ as shown in
the plot, the $\mathrm{T}$-matrix of the more strongly-coupled channel
$\alpha=1$ diverges at low energies. Specifically, $t_{1}(\omega,0)\sim
1/[|\omega/\lambda_{CR}|\times\ln^{2}(|\omega/\lambda_{CR}|)]$ as $|\omega|\to
0$ see solid green lines in the top panels of Fig. 8.6 [241, 242], indicating
that ACR is an non-Fermi liquid (NFL) FP. We remark that the nature of this
non-Fermi liquid FP is not ascribable to the standard two-channel Kondo
critical fixed-point found in system with Majorana states and fractional
residual impurity entropy at $T=0$. The dynamical crossover, and hence the
minimum in $t_{1}(\omega,0)$ occurs on the scale of
$|\omega|\leavevmode\nobreak\ \propto\leavevmode\nobreak\ \lambda_{CR}$.
However, the weakly coupled channel $\alpha=2$ has FL correlations
$t_{2}(\omega,0)\sim|\omega|$ as $|\omega|\to 0$ in dotted green lines,
confirming that it decouples from the critical complex formed from the dot and
channel $1$. In the frustrated case $\Delta=1$, both channels behave
identically - and both exhibit the same NFL critical divergence at low
energies. This again suggests that FACR comprises two copies of ACR, one in
each channel. The enhanced conduction electron scattering at the critical
point has implications for the conductance, as now shown.
In the bottom row of Fig. 8.6 we plot the $T=0$ dynamical $ac$-conductance as
a function of $ac$-driving frequency $\omega$, for the same set of systems.
The $dc$-conductance is obtained in the $\omega\to 0$ limit, which we consider
first. In the charge-Kondo system, series transport proceeds by the following
mechanism: an electron tunnels from the source lead onto the dot (say at QPC
$\alpha=1$), thus flipping the dot charge pseudospin from $\Downarrow$ to
$\Uparrow$. A second electron then tunnels from the dot to the drain lead (at
QPC $\alpha=2$), thus flipping the dot charge pseudospin back to $\Downarrow$
and resetting the device ready for transport of another electron. A bias
voltage between source and drain produces a net current flow. The amplitude
for such a process depends on the conduction electron density of states
$\rho(\omega)$ and the tunnelling rate at the QPCs. For graphene we have
$\rho(\omega)\sim|\omega|$ at low energies, suggesting that the low-
temperature $dc$ conductance should vanish, since there are not enough low-
energy electrons in the graphene leads to tunnel through the nanostructure. On
the other hand, the tunnelling rate gets renormalized by the interactions -
the energy-dependent scattering at the QPCs is characterized by the
$\mathrm{T}$-matrices discussed above. Indeed, strong renormalization of the
bare QPC transmission at low temperatures due to Kondo physics was measured
experimentally in the metallic leads version of the present system in Ref.
[84].
The measured $dc$-conductance of the graphene 2CCK system involves a subtle
interplay between the conduction electron DoS and interaction-renormalized
scattering rates. We expect the $T=0$ $dc$ conductance to vanish in all
channel-asymmetric systems because the less strongly coupled channel always
decouples on the lowest energy scales. Both leads must remain coupled to
ensure a finite series current. This is indeed seen in the $\omega\to 0$ limit
of each of the curves in the bottom left panel of Fig. 8.6 for $\Delta=0.8$.
However, in the frustrated channel-symmetric case $\Delta=1$, both channels
remain coupled down to $T=0$. Although the scattering rates and bare DoS both
vanish as $\sim|\omega|$ in the LM and FASC phases implying a suppression of
$dc$ conductance, at the critical point FACR the electronic scattering
diverges as $|\omega|\to 0$. We find from NRG that these effects conspire to
give a finite linear $dc$ conductance in this case - see green line in the
bottom right panel of Fig. 8.6.
For an $ac$-bias the conductance is measured as a function of the driving
frequency $\omega$. Conductance resonances are expected when the $ac$
frequency matches the QPC tunnelling rate. At high energies the pseudospin
flip rate in the 2CCK model is given by the bare $J$ or effective $J_{eff}$.
We therefore expect to see a peak in the $ac$ conductance when $|\omega|\sim
J,J_{eff}$; this is observed from NRG results in the LM, ASC and FASC phases
in Fig. 8.6. However, at low energies $|\omega|\ll\lambda$, the pseudospin
flip rate is renormalized and we find
$\mathcal{G}^{C}(\omega,0)\sim\omega^{2}$ in these cases, independent of
$\Delta$. At the critical point ACR for $0<\Delta<1$, both charge and spin
fluctuations give an enhanced $ac$ conductance around $|\omega|\sim\lambda$.
However, channel $\alpha=2$ decouples for $|\omega|\ll\lambda$ and so the
conductance also decays at low frequencies. We find from NRG a slow
attenuation $\mathcal{G}^{C}(\omega,0)\sim-1/\ln|\omega|$ in this regime.
However, in the channel-symmetric case $\Delta=1$ at the critical point FACR,
$\mathcal{G}^{C}(\omega,0)\sim\textit{const}.$ for $|\omega|\ll\lambda_{CR}$.
The finite dynamical conductance here persists down to the $dc$-limit. This is
the smoking gun signature of the NFL frustrated critical point in the graphene
2CCK system.

###  8.4 Discussion

In this chapter we proposed a charge-Kondo quantum dot device made from
graphene components that realizes a linear-pseudogap two-channel Kondo model.
This exotic system has a complex phase diagram in the space of dot-lead
coupling strength, potential scattering, and channel asymmetry. We analysed
the thermodynamic and dynamical properties of the model using NRG, and used
this to gain a detailed understanding of the renormalization group flow and
fixed points. We classify the fixed points and construct the corresponding
fixed-point Hamiltonians, together with their leading corrections. In
particular, we uncover a channel-frustrated Kondo phase, with a non-Fermi
liquid quantum critical point at the first-order quantum phase transition.
Despite the depleted electronic density of the neutral graphene leads at low
energies, critical fluctuations give rise to diverging scattering rates at the
critical point, which produce a finite conductance even as $T\to 0$.
The model supports other interesting but as yet unexplored regimes. We have
confined attention to the dot charge degeneracy point $\delta V_{g}=0$; but
finite $\delta V_{g}$ appears in the effective model like a magnetic field on
the dot pseudospin. One could also investigate the effect of doping/gating the
graphene so that the Fermi level is not at the Dirac point. This will give
rise to a quantum phase transition between metallic 2CK and pseudogap 2CK,
with the competition controlled by the ratio of the doping to the Kondo
temperature. Other physical quantities could also be investigated, such as
thermoelectric transport on including a temperature gradient between leads.
The pseudogap 2CCK model we have studied theoretically here is likely a
simplified description of any real graphene charge-Kondo nanoelectronics
device. There may be complexities and subtleties in an experimental
realization which are not included in our model or analysis. For example, we
have assumed that the conduction electrons on both leads and dot have the same
DoS. In particular, gate voltage tuning of the dot to achieve charge-
degeneracy, and the addition of the decoherer, may affect the dot electronic
DoS. However, we do not expect our basic results to be qualitatively modified
by this because the Kondo exchange interaction derives from tunnelling at the
QPC and hence involves the DoS of both lead and dot - see expression for
$\tau(\omega)$. Therefore even if only the lead DoS is pseudogapped at low
energies, an effective pseudogap Kondo model should still result. To make
quantitative connection to experiment, such effects would have to be taken
into account, as well as the possible involvement of more than just 2 dot
charge states - that is, relaxing the condition $T\ll E_{C}$. We believe the
predicted conductance signature of the frustrated quantum critical point
should however still be observable in experiment.

##  Chapter 9 Discussion and conclusion

#### Main results

In this conclusion chapter we aim to summarise our most important
contributions to the field of mesoscopic transport in interacting
nanostructures obtained in Chapters $4-8$.
Chapter $4$, Improved calculations for quantum transport in interacting multi-
orbital nanostructures, contains all the analytical derivations of the new
sets of formulae found in this work.
In Sec.4.1, we introduce the concept of structured leads, both for equivalent
and for inequivalent leads, see definition in Eq.4.1 \- such that we go beyond
the usual wide flat band limit approximation in Eq.3.13 and the standard
choice of metallic leads. We find there is no Landauer-type of formula for
structured, inequivalent leads whereas we find an expression for generic
interacting models under proportionate coupling (PC) with equivalent leads in
Eq.4.16 and for noninteracting models with inequivalent leads in Eq.4.26. The
summary of our results is given in Table 4.1. In Sec.4.2, we follow two
distinct approaches to derive an alternative Meir-Wingreen current formula. In
the first case, we attempt to calculate an expression for
$G^{<}_{dd\sigma}(\omega)$ containing an explicit $V_{bias}$ dependence such
that the correlator can be readily employed in linear response transport. This
calculation fails, since the final equation for $G^{<}_{dd\sigma}(\omega)$ in
Eq.4.30 not only does not present explicitly the voltage parameter but it is
also calculated for higher order impurity correlator $F^{<}_{\sigma}(\omega)$
in Eq.4.34 which cannot be further simplified. In the second case, taking
insight from the Ng ansatz, we derive a corrected Fermi distribution out of
equilibrium condition $f^{corr}(\omega)$ under perturbation expansion up to
order $\mathcal{O}(V_{bias},U^{2})$. The final expression of
$f^{corr}(\omega)$ in Eq.4.67 contains corrections to the standard Ng ansatz
to order in $V_{bias}$ and $U$, compared with Table 4.2, and it reduces to the
equilibrium distribution $f^{Eq}(\omega)$ in the case of symmetrically-coupled
equivalent leads. This result can be used to derive the $G^{<}_{dd}(\omega)$
correlator in Eq.4.71. By means of this expression, each term in the Meir-
Wingreen formula has an explicit $V_{bias}$ dependence and we can evaluate the
conductance under linear response. In Sec.4.3, we calculate an improved Kubo
formula for electric and heat conductance, see Eqs.4.81,4.82 respectively. We
find that the implementation in NRG of the derived analytical expression for
$\mathcal{G}^{C}(\omega,T)$ shows better numerical precision. We
systematically demonstrate this for the AIM, see plot in Fig.4.1. The improved
electrical Kubo formula shows finite numerically-stable values in the dc-
regime and identifies correctly the crossover between LM and Fermi liquid
regime even for surprisingly inexpensive NRG calculations (modest number of
kept states $M_{K}$ for any given discretization $\Lambda$) - unlike the Kubo
standard formula. We perform also a similar investigation for the heat Kubo
formula $\mathcal{G}^{E}(\omega,T)$ in NRG for both standard and improved
versions. Regardless the choice of $M_{K},\Lambda$ parameters,
$\mathcal{G}^{E}(\omega,T)$ in NRG implementation fails completely to capture
the expected physics, see plot in Fig.4.2. Our understanding of the issue is
that in the NRG protocol, the Wilson chain does not represent a true thermal
reservoir. We conclude that it is yet unsolved how to obtain the calculation
of $\mathcal{G}^{E}(\omega,T)$ in NRG for genuinely non-PC impurity models. In
Sec.4.4, within the framework of extended impurity model, see schematic in
Fig.4.3, starting from noninteracting model we derive an alternative Fisher-
Lee formula using $\mathrm{T}$-matrix in Eq.4.103. Then, turning on
interactions in the model but confining the study at zero energy and
temperature scale, we derive an alternative Oguri formula using renormalized
$\mathrm{T}$-matrix in Eq.4.105. In general, formulations obtained from the
calculation of the $\mathrm{T}$-matrix, instead of diagrammatic expansion,
facilitate derivations and present a tangible quantity to be measured in
experiments.
In Chapter $5$, Effective models: emergent proportionate coupling, we study
the RG-flow of $\mathrm{T}$-matrix spectrum via NRG using the effective
parameters derived from the effective low energy model showing an emergent PC
condition.
We analyse two prominent cases. In Sec.5.1, we consider a non-PC two-channel
Kondo model in Coulomb blockade regime for both even and odd charge ground-
states. The most important result occurs in the case of the odd charge ground-
state with half-integer spin $S=1/2$ state in sd-symmetry regime. We succeed
in analytically deriving an expression for the electrical conductance
$\mathcal{G}^{dc}(0,0)$ in terms only of renormalised potential scatterings,
see Eq.5.53. We numerically verify it in comparison to the fully temperature
and energy dependent improved Kubo formula $\mathcal{G}^{ac}(\omega,T)$ using
NRG, see plots in Fig,5.5(a),(b). In Sec.5.2, we consider a non-PC two-channel
Kondo model in mixed-valence regime for singlet-doublet and doublet-triplet
crossovers. The main output from this analysis is the fact that we can map
effective models to minimal one- or two-site system satisfying an emergent PC
property, see Eq.5.114. It is straightforward to generalise this modelling to
higher ground-states spin. Our findings are supported by NRG results of the
$\mathrm{T}$-matrix spectrum.
In Chapter $6$, Application and comparison of quantum transport techniques, we
give a comprehensive numerical test of the analytical formulations derived in
this thesis. The system of choice is the TQD, as it can be tuned to a regime
and configuration of difficult - if not inaccessible - implementation using
literature methods.
For the TQD in PC configuration, we show via NRG the improved Kubo formula,
$\mathcal{G}^{C}(\omega,T)$ compared to the standard formulation, performs
higher accuracy results in the dc-regime despite being computationally cheaper
with respect to regular Kubo, see Fig.6.2. Furthermore, in Sec.6.2 for the TQD
in PC, using Meir-Wingreen formula we discover that there is a revival of the
Kondo effect in a strong magnetic field, see Fig.6.5, which was unknown - to
the best of our knowledge. In Sec.6.3 for the TQD in non-PC, we compare in
Fig.6.6 the various transport methods studied in this thesis. We find that all
formulations agree with energy and temperature scale with the Oguri formula,
that is low temperature dc-regime. However, only the improved Kubo formula for
electrical conductance $\mathcal{G}^{C}(\omega,T)$ manages to display the
behaviour in line of the expected Kondo scale whereas the Landauer-Büttiker
formula and Ng ansatz fail to do so. Interestingly, Landauer-Büttiker formula
is closer to the real scale than the Ng ansatz - although we have not checked
if Landauer formula consistently outperforms the Ng prediction for other
models.
In Chapter $7$, Transport in molecular junction: benzene transistor, we
investigate the practicality of the benzene molecule in a break junction as an
electronic switch with a back gate voltage actuator.
We derive the effective model $\hat{H}_{eff}$ at the mixed-valence transition
between singlet-doublet and doublet-triplet points. The $\hat{H}_{eff}$
presents an effective PC configuration that allows to be readily evaluated
using the Meir-Wingreen formula under PC, see corresponding conductance
formulae in Eqs.7.33,7.27 for singlet-doublet and doublet-triplet crossovers.
In particular, the analytical prediction at zero energy and zero temperature
scale in the singlet-doublet MV regime is in great agreement with the
numerical calculation using improved Kubo formula in NRG, see Fig.7.3. Then,
we compute in NRG the molecular entropy and the electrical conductance using
improved Kubo formula in both configurations, see Fig.7.4. Our major findings
are in the $N=7$ charge sector. In the meta case, we discover the appearance
of the $SU(4)$ Kondo effect and consequent conductance node. This result
persists till $T=0$, see Fig.7.6. In the para case, we observe a level
crossing quantum phase transition corresponding to a sudden conductance switch
from maximally enhanced to identically vanishing. Again, this feature is seen
till $T=0$, see Fig.7.8. This latter property is ideal for the application in
electronic circuits where the molecule plays the role of a transistor.
In Chapter $8$, Two-channel charge-Kondo physics in graphene quantum dots, we
propose a device to study Kondo physics in graphene by means of a two-channel
charge-Kondo paradigm.
We compute the entire phase diagram numerically with NRG. The most prominent
observation from our studies is the re-entrant LM behaviour with persistence
of the strong coupling phase even asymptotically, see Fig.8.3. We give the
fixed-point Hamiltonians for each phase and the leading perturbation order for
all fixed-point, as classified in the $3D$ RG-flow diagram in Fig.8.4.
Surprisingly, in the symmetric two-channel set-up, we find finite electrical
conductance down to $\omega\to 0$ at the frustrated asymmetric critical fixed-
point, see Fig.8.6. This conductance behaviour is a signature of exotic non-
Fermi liquid physics that constitutes a new frustrated asymmetric critical
fixed-point.

#### Thinking forward

In this work we give our contribution to the following question: how to
provide well-defined and numerically controlled quantum transport solutions
for elaborate systems without diminishing the rich physics of the bare model.
Our general strategy is to decompose complicated interacting systems under
nonequilibrium condition into simpler exact analytical expressions which
limiting cases at $\omega\to 0$ and $T\to 0$ can be benchmarked numerically.
In this thesis we employ NRG as the numerical method to validate our
analytical predictions. However, because the class of quantum transport
problems we solved in this work is under linear response theory, we envisage
other numerical methods based on the linear response principle are amenable
for testing the alternative and the improved quantum transport formulae
derived in this thesis. Furthermore, these new formulae are defined through
physical quantities which can be probed in realistic experimental devices.
Another crucial aspect of our strategy is to allow the maximum flexibility of
the components in the coupled system. About the leads, we designed formulae
considering: $(i)$ structured object with energy and spin dependence, $(ii)$
arbitrary coupling configuration. About the impurity, we formulate it in terms
of arbitrary $(i)$ type of local interactions, $(ii)$ orbital complexity,
$(iii)$ number of sites. The flexibility of our formulations opens to the
applicability of our results not only to strongly correlated many-body systems
but also to organic single-molecules with functionality as single-molecule
transistors (e.g. p-Phenylene [243]) and nanoparticles network (e.g. to mimic
programmable logic gates [244] for near term neuromorphic computers).

#### What can be done next

An important future direction of this work is the extension of these new
transport formulations to multi-channel models. With this respect, we suppose
two main issues would arise.
The first is the analytical complexity to extend our alternative and improved
formulae to an arbitrary number of channels. It may result that some
formulations do not have a counterpart in systems with more than two channels.
The second is related to the numerical implementation with NRG of multi-
channel systems as the complexity grows exponentially with the number of
channels. To tackle this problem, we propose to carry future investigation in
multi-channel systems by means of impurity solvers with more favourable
scaling.

##  Appendix A The Meir-Wingreen current formula

In this appendix we derive the electrical current formula in the time-
independent case, as originally introduced by Meir and Wingreen [105]

$\displaystyle\hat{I}=\frac{ie}{2h}\sum_{\sigma}\bigintsss d\omega Tr\bigg{[}$
$\displaystyle\Big{(}f^{\mu_{L}}(\omega-\mu_{L})\Gamma^{L}-f^{\mu_{R}}(\omega-\mu_{R})\Gamma^{L}\Big{)}\Big{(}G_{dd\sigma}^{R}(\omega)-G_{dd\sigma}^{A}(\omega)\Big{)}$
(A.1)
$\displaystyle+\Big{(}\Gamma^{L}-\Gamma^{L}\Big{)}G_{dd\sigma}^{<}(\omega)\bigg{]}\leavevmode\nobreak\
,$

that is used to study transport in strongly interacting mesoscopic
nanostructure. We refer for the nonequilibrium theory through the Keldysh
formalism in the Sec.2.1. Further technical readings can be found in the
references [15, 22, 23, 24]. The physics of this formula is discussed in the
corresponding Sec.3.3.

We start with the general impurity model Hamilonian in Eq.3.1, here we write
down its explicit form as used in the original derivation with a two-channel
model, namely

$\hat{H}=\sum_{\alpha\mathbf{k}\sigma}\epsilon_{\alpha\mathbf{k}}c^{\dagger}_{\alpha\mathbf{k}\sigma}c_{\alpha\mathbf{k}\sigma}+\hat{H}^{U}_{imp}\\{d^{\dagger}_{n\sigma};d_{n\sigma}\\}+\sum_{n\mathbf{k}\sigma}\Big{(}V_{\alpha\mathbf{k}}c^{\dagger}_{\alpha\mathbf{k}\sigma}d_{n\sigma}+V^{\star}_{\alpha\mathbf{k}}d_{n\sigma}^{\dagger}c_{\alpha\mathbf{k}\sigma}\Big{)}\leavevmode\nobreak\
,$ (A.2)

where $c^{\dagger}_{\alpha\mathbf{k}\sigma}c_{\alpha\mathbf{k}\sigma}$ is set
of spinful electron operators acting on $\alpha\in L,R$ leads and
$d_{n\sigma}^{\dagger},d_{n\sigma}$ is set of spinful electron operators
acting in the central region for $n$ number of degree of freedom. This
impurity or central region contains the one-body Coulomb potential to
represent the interaction part. The hybridization between lead and central
region is described by coupling strength $V_{\alpha\mathbf{k}}$.
The Hamiltonian model as defined in Eq.A.2 is used to derive a steady state
current \- that is the key input to calculate the time-independent
conductance. We will develop the analytical derivation of the Meir-Wingreen
formula as calculated in the literature.

### Steady state current definition

As initial step, we seek for an expression for the steady state current. In
the system current is conserved i.e.
$\hat{I}_{L}(t)=-\hat{I}_{R}(t)\Rightarrow\hat{I}(t)$. Hence, without loss of
generality, we define the current with respect to left lead as

$\hat{I}_{L}(t)\equiv-e\langle\dot{N}_{L}(t)\rangle\leavevmode\nobreak\ ,$
(A.3)

that is calculated from the expectation value of time-derivative of the number
operator $\hat{N}_{L}$. We evaluate the current using the Heisenberg equation
of motion i.e. $\dot{\Psi}(t)=i[H,\Psi(t)]$ such that we obatin

$-i\langle\dot{N_{L}}(t)\rangle=\langle[\hat{H}(t),\hat{N}_{L}(t)]\rangle\leavevmode\nobreak\
,$ (A.4)

where $\hat{H}$ indicates the full Hamiltonian expression given in eq.A.2.
It can be shown the equal time Green’s function $G(t,t)$ is equivalent to
calculate the Green’s function evaluated in static expectation value:
$G(t,t)\equiv-i\langle\Psi\Psi^{\dagger}\rangle$ and this equality is also
consistent with the fact we are interested in steady state transport. At this
stage, we note the steady state condition indicates also time-translational
invariance in the system. Its propagator is a the function of time difference
only $G(t-t^{\prime})$ and we can apply Fourier transformation to bring the
final result into energy domain. Hence, in this derivation, we refer to time-
translational, steady-state Green’s functions.
Taking into account these considerations, we compute the current in its static
expectation value

$\hat{I}=-e(-i)\langle\dot{N}_{L}\rangle=e\langle[\hat{H},\hat{N}_{L}]\rangle\leavevmode\nobreak\
,$ (A.5)

where we note the number operator $\hat{N}$ commutes in general with
$\hat{H}_{leads},\hat{H}_{imp}$. At the end of the calculation, we can restore
the time parameters by transforming the operators into interacting
representation. After applying the equation of motion technique (EoM) see
Eq.2.26, the result is a combination of anti-Hermitian operators. By means of
the Keldysh-contour, see Fig.2.1, we recognize the calculated Green’s function
expression is in lesser type form. In particular, we find the so-called mixed
lesser Green’s function namely

$G_{n,\alpha\mathbf{k}\sigma}^{<}(t-t^{\prime})=i\langle
c^{\dagger}_{\alpha\mathbf{k}\sigma}(t^{\prime})d_{n\sigma}(t)\rangle\leavevmode\nobreak\
,$ (A.6)

since both lead and nanostructure operators are included hence the subscripts
in the Green’s function are group accordingly. We also note, that as it is
written, $G_{n,\alpha\mathbf{k}\sigma}^{<}(t-t^{\prime})$ is a time-ordered
Green’s function.
In conclusion, we obtain the steady state current equation with respect to
left lead

$\hat{I}_{L}(t)=2e\mathit{Re}\Big{[}\sum_{n\alpha\mathbf{k}\sigma}V_{\alpha\mathbf{k}}G_{n,\alpha\mathbf{k}\sigma}^{<}(t,t^{\prime})\Big{]}$
(A.7)

where the time-ordered operator current is given by the real part of the
hybridization term $V$ multiplied for the mixed lesser $G^{<}$ This expression
in Eq.A.7 requires the calculation of the equilibrium form $G^{<}$, once
transient phenomena are completed. However, the problem is set in the out-of-
equilibrium regime. We will study in the next section the connection between
equilibrium and nonequilibrium quantities in order to calculate the time-
translational invariant lesser Green’s function $G^{<}(t-t^{\prime})$.

### Mixed $G^{<}$ in nonequilibrium form

In the previous section, we have derived the expression for the steady state
current in time domain, see Eq.A.7. We seek now an expression for the lesser
Green’s function $G_{n,\alpha\mathbf{k}\sigma}^{<}$ in nonequilibrium, steady
state.
Taking the standard results from nonequilibrium perturbation theory, at $T=0$
it can be shown the formal structure of equilibrium theory is the same as the
nonequilibrium one. Due to this crucial correspondence at zero temperature, we
can adopt and apply the familiar equilibrium calculation techniques to the
corresponding nonequilibrium expressions. Thus, we make use of the equation of
motion method to calculate Green’s functions and of the perturbation expansion
with its diagrammatic interpretation. The conversion to nonequilibrium theory
is simply provided by substituting the real-time axis with the semi-closed
path Keldysh-contour and by changing the time-ordered operators in real-time
variables with the contour-ordered ones in complex-time variables.
Having these concepts in mind, we start evaluating
$G_{n,\alpha\mathbf{k}\sigma}^{<}(t-t^{\prime})$ that is the equilibrium mixed
lesser Green’s functions defined on real-time contour. As we will see later,
the Fourier transformed expression of $G^{<}$ can be inserted in Eq.A.7 and
the final current expression is depending only on energy arguments. Hence, the
lesser Green’s function from EoM calculation results in

$\sum_{n\alpha\mathbf{k}\sigma}\Big{(}-i\frac{d}{d{t^{\prime}}}-\epsilon_{\alpha\mathbf{k}}\Big{)}G_{n,\alpha\mathbf{k}\sigma}^{<}(t-t^{\prime})=\sum_{nm\alpha\mathbf{k}\sigma}V^{\star}_{\alpha\mathbf{k}}G_{nm\sigma}^{<}(t-t^{\prime})\leavevmode\nobreak\
,$ (A.8)

where we made use of noninteracting leads to get a closed set of EoM and the
Green’s function on the right hand side is identified as the equilibrium
lesser Green’s function of the central-region

$\sum_{nm\sigma}G_{nm\sigma}^{<}(t-t^{\prime})\equiv i\langle
d^{\dagger}_{n\sigma}(t^{\prime});d_{m\sigma}(t)\rangle\leavevmode\nobreak\ .$
(A.9)

As we will see in later steps, an important aspect of the Meir Wingreen
approach is that only the local Green’s function $G_{nm}$ will appear in the
final formula.
In order to identify and then to invert the term multiplying upfront
$G_{n,\alpha\mathbf{k}\sigma}^{<}$ in Eq.A.8, we need to bring the expression
from time to frequency domain by applying the Fourier transform and we get

$\sum_{n\alpha\mathbf{k}\sigma}G_{n,\alpha\mathbf{k}\sigma}^{<}(\omega)=\sum_{nm\alpha\mathbf{k}\sigma}G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{R}(\omega)V^{\star}_{\alpha\mathbf{k}}G_{nm\sigma}^{<}(\omega)\leavevmode\nobreak\
,$ (A.10)

where we can uniquely identify
$G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{R}$ as the exact retarded
Green’s function of the uncoupled leads - hence, when the system is at
$t\rightarrow-\infty$.
We can transform back this expression to real time domain using the
convolution theorem [101] and finally we calculate the time-ordered mixed
lesser equilibrium Green’s function

$\sum_{n\alpha\mathbf{k}\sigma}G_{n,\alpha\mathbf{k}\sigma}^{<}(t-t^{\prime})=\sum_{nm\alpha\mathbf{k}\sigma}V^{\star}_{\alpha\mathbf{k}}G_{nm\sigma}^{<}(t-t^{\prime})G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{R}(t-t^{\prime})\leavevmode\nobreak\
,$ (A.11)

where

$G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{R}(t-t^{\prime})\equiv-i\theta(t-t^{\prime})\langle\\{c_{\alpha\mathbf{k}\sigma}(t),c^{\dagger}_{\alpha\mathbf{k}\sigma}(t^{\prime})\\}\rangle=(-id_{t}-\epsilon_{\alpha\mathbf{k}})$
(A.12)

is the exact time-independent retarded Green’s function of isolated leads. The
expression given in Eq.A.11 is the explicit version of Eq.A.8, since now all
the elements are identified and $G_{n,\alpha\mathbf{k}\sigma}^{<}$ takes a
proper analytical form.
We conclude now by showing the correspondence at zero temperature of Eq.A.11
with its nonequilibrium counterpart. As we know already, the mixed lesser
nonequilibrium Green’s function in complex-time arguments
$G_{n,\alpha\mathbf{k}\sigma}^{<}(\tau,\tau^{\prime})$ has the same formal
structure as $G_{n,\alpha\mathbf{k}\sigma}^{<}(t-t^{\prime})$ derived in
Eq.A.11. However, due to the replacement of the real-time variables, the real-
time axis is substituted by the Keldysh-contour $\mathcal{C}_{\tau}$ where the
contour-ordered Green’s functions are integrated on. Hence, the mixed lesser
nonequilibrium Green’s function reads

$\sum_{n\alpha\mathbf{k}\sigma}G_{n,\alpha\mathbf{k}\sigma}^{<}(\tau,\tau^{\prime})=\sum_{nm\alpha\mathbf{k}\sigma}V^{\star}_{\alpha\mathbf{k}}\int_{\mathcal{C}_{\tau}}d\tau_{1}G_{nm\sigma}^{<}(\tau,\tau_{1})G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{R}(\tau_{1},\tau^{\prime})\leavevmode\nobreak\
,$ (A.13)

where performing this integral on general contour is subject of the next
section.

For the sake of conciseness in the equations, we will omit the upfront
summation over parameters and consider it as implicitly performed.

### Analytic continuation from the Keldysh-contour to real-time axis

As next step, we directly evaluate of the nonequilibrium
$G^{<}(\tau,\tau^{\prime})$ Eq.A.13 and we attempt it through two different
approaches.
The first one consists in deforming the original Keldysh-contour
$\mathcal{C}_{\tau}$ in Eq.A.13 into two sub-contours such that the
integration is split over these new branches. By considering the new time-
label order, we are able to recognise the type of Green’s functions. We
manipulate the integral on complex-time variables in Eq.A.13 using the Keldysh
identities in Eq.2.16 to turn it into into the form of mixed lesser
equilibrium Green’s function

$G_{n,\alpha\mathbf{k}\sigma}^{<}(t-t^{\prime})=V^{\star}_{\alpha\mathbf{k}}\int_{-\infty}^{\infty}dt_{1}\Big{(}G_{nm\sigma}^{R}(t-t_{1})G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{<}(t_{1}-t^{\prime})+G_{nm\sigma}^{<}(t-t_{1})G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{A}(t_{1}-t^{\prime})\Big{)}\leavevmode\nobreak\
,$ (A.14)

where the mixed $G^{<}$ is written again as time-ordered Green’s function.
This result is expected: the process of deformation the complex-time contour
and subsequent extrapolation of real-time variables, is a projection of
complex variables onto real-time axis. We note that, on a practical level,
manipulating contour-ordered Green’s functions might bring to cumbersome
calculation - suggesting different methods should be explored.
The second approach is developed in the Langreth theorem [27]: the Keldysh-
contour modification is completely avoided by introducing a specific matrix
representation of the nonequilibrium $G$ that encapsulates the internal
structure of the Keldysh-contour and absorbs the integration over
$\mathcal{C}_{\tau}$. The outcome is a successive matrix multiplication rules
that can be directly evaluated as integrand in real-time integration. In this
procedure, from the complex-time contour we have developed an analytical
continuation to the real-time axis. To the purpose of this derivation, we
employ the multiplicative rules explained in the Langreth theorem for
Fermionic models only. For completeness, we mention that the Langreth theorem
is also applicable to Bosonic and mix of Bosonic-Fermionic systems [245].
We proceed now with a quantitative discussion of the Langreth theorem for
analytical continuation. We start with writing the contour-ordered Green’s
functions as matrix with $dim(2\times 2)$ and its matrix elements are
identified according to which branch the time labels are located on. We take
as example the lesser Green’s function $G^{<}(\tau,\tau^{\prime})$ in matrix
form becomes

$\mathbb{G}^{<}=\begin{pmatrix}G^{R}&G^{<}\\\
0&G^{A}\end{pmatrix}\leavevmode\nobreak\ .$ (A.15)

According to type of Green’s function we aim to calculate, we write its
factors in the corresponding matrix form and we perform their matrix
multiplication. At the end, we select the matrix element corresponding to the
desired type of Green’s function. For instance, if we want to calculate the
correlators product $G^{<}G_{0}^{R}$ in Eq.A.13 into the corresponding matrix
multiplication rule we proceed as follows

$\displaystyle\mathbb{G}_{n,\alpha
k}^{<}=\mathbb{G}^{<}_{nm\sigma}\cdot\mathbb{G}^{<}_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}$
$\displaystyle=\begin{bmatrix}G^{R}&G^{<}\\\
0&G^{A}\end{bmatrix}_{nm\sigma}\cdot\begin{bmatrix}G_{0}^{R}&G_{0}^{<}\\\
0&G_{0}^{A}\end{bmatrix}_{\alpha\mathbf{k}\alpha\mathbf{k}\sigma}$ (A.16)
$\displaystyle=\begin{bmatrix}G_{nm\sigma}^{R}G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{R}&G_{nm\sigma}^{R}G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{<}+G_{nm\sigma}^{<}G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{A}\\\
0&G_{nm\sigma}^{A}G_{0,\alpha\mathbf{k}\alpha\mathbf{k}\sigma}^{A}\end{bmatrix}$
$\displaystyle\equiv\begin{bmatrix}G^{R}&G^{<}\\\
0&G^{A}\end{bmatrix}_{n,\alpha\mathbf{k}\sigma}\leavevmode\nobreak\ ,$

where in the second line, the top right element represents the lesser function
analytical continuation on the real-time axis of the product
$G^{<}(\tau,\tau^{\prime})G^{R}_{0}(\tau,\tau^{\prime})$ on
$\mathcal{C}_{\tau}$ contour as given in eq.A.13. As we expected, it is the
same result of Eq.A.14 obtained by direct integration on the Keldysh contour:
the matrix representation incorporates the contour calculation on the modified
branches. Hence, from the matrix multiplication
$\mathbb{G}^{<}\cdot\mathbb{G}_{0}^{<}$ we take the top right quantity as the
lesser element of the matrix and we plug it in directly as integrand of
corresponding real-time integration:

$\displaystyle
G^{<}(\tau,\tau^{\prime})=\int_{\mathcal{C}_{\tau}}G^{<}G_{0}^{R}\quad\Leftrightarrow\quad
G^{<}(t,t^{\prime})=\int_{\mathcal{C}_{t}}\big{(}G^{R}G_{0}^{<}+G^{<}G_{0}^{A}\big{)}\leavevmode\nobreak\
.$ (A.17)