# Modeling Beats and Downbeats with a Time-Frequency Transformer

###### Abstract

Transformer is a successful deep neural network (DNN) architecture that has
shown its versatility not only in natural language processing but also in
music information retrieval (MIR). In this paper, we present a novel
Transformer-based approach to tackle beat and downbeat tracking. This approach
employs SpecTNT (Spectral-Temporal Transformer in Transformer), a variant of
Transformer that models both spectral and temporal dimensions of a time-
frequency input of music audio. A SpecTNT model uses a stack of blocks, where
each consists of two levels of Transformer encoders. The lower-level (or
spectral) encoder handles the spectral features and enables the model to pay
attention to harmonic components of each frame. Since downbeats indicate bar
boundaries and are often accompanied by harmonic changes, this step may help
downbeat modeling. The upper-level (or temporal) encoder aggregates useful
local spectral information to pay attention to beat/downbeat positions. We
also propose an architecture that combines SpecTNT with a state-of-the-art
model, Temporal Convolutional Networks (TCN), to further improve the
performance. Extensive experiments demonstrate that our approach can
significantly outperform TCN in downbeat tracking while maintaining comparable
result in beat tracking.

Index Terms—  Beat, Downbeat, Transformer, SpecTNT

##  1 Introduction

Beat is regarded as one of the fundamental rhythmic units perceived by humans,
and a downbeat is conceived to occur at the first beat of a bar (measure),
which usually indicates the start of a chord. Beat and downbeat tracking is
concerned with developing algorithms to detect the beats and downbeats as
pulse signals in music audio. This is a well-defined problem in music
information retrieval (MIR) and has been a regular task in MIREX challenge
111https://www.music-ir.org/mirex/wiki/MIREX_HOME for over a decade.

In recent years, researchers proposed to employ deep neural network
(DNN)-based methods to model beats and downbeats, which are regularly
repeating events in the audio sequence. Böck et al. [1] successfully used a
recurrent neural network (RNN) with input mel spectrogram features to jointly
model beats and downbeats. Such a system requires annotated data to train and
then predict accordingly. This has set up a data-driven paradigm for the
subsequent systems, where improvement can be expected by either involving more
training data or developing a more advanced machine learning algorithm.

Compared to beats, predicting downbeats still remains a challenge, because it
requires a more complicated awareness of harmonic, timbral, and structural
context, which may be related to chords, instrumentation, and percussive
arrangements [2]. To better model these aspects, one need to explore more
local spectral information, and at the same time allow the important local
information to be exchangeable among different temporal positions far apart in
the audio sequence, since a measure may require a longer audio context to be
discovered.

Recently, Transformer architecture [3, 4] has brought lots of attention due to
its remarkable performance in natural language processing. For song-level
classification tasks in MIR, researchers treat it as a temporal encoder to
aggregate the temporal features to represent a music audio sequence [5, 6]. As
a replacement of RNN, it also works well for chord recognition [7, 8].
However, these prior works did not attempt to make full exploitation of
Transformer to model the spectral information that interacts with the temporal
encoder as a whole. On the other hand, Transformer is data-hungry in nature.
For many MIR tasks, annotations are very expensive, and the size of annotated
data is typically small. Therefore, training an effective Transformer-based
model without modification and data augmentation is likely infeasible.

In this paper, we propose to use SpecTNT (Spectral-Temporal Transformer in
Transformer) [9], a variant of Transformer that models the spectrogram along
both time and frequency axes. To the best of our knowledge, this is the first
successful attempt to apply Transformer-based model to beat and downbeat
tracking. The basic principle of SpecTNT stems from the interaction between
two levels of Transformer encoders, namely _spectral Transformer_ and
_temporal Transformer_. The former is responsible for extracting the spectral
features for each frame. Whereas the latter is capable of exchanging the local
information over the time axis. Thanks to the hierarchical design [9], SpecTNT
permits a smaller number of parameters compared to the original Transformer
[4], so it can maintain good generalization ability for MIR tasks with a
smaller size of training data. To further improve the performance, we also
propose a novel structure that combines SpecTNT with a state-of-the-art (SOTA)
model, temporal convolutional networks (TCN). We evaluate our proposed systems
on various public datasets. Our results demonstrate that SpecTNT can achieve
SOTA performance in downbeat tracking on most of the datasets while
maintaining comparable performance in beat tracking. The combined structure of
SpecTNT with TCN can further boost the performance.

##  2 Related Work

Early approaches in beat tracking mostly focused on incorporating heuristic
information into the systems. For example, Dixon et al. [10] proposed to
estimate the tempo and beats by detecting salient rhythmic events at various
metrical levels. Klapuri et al. [11] proposed a probabilistic model to process
the music accents and estimate beat patterns. On the other hand, musical
events such as chord changes and drum patterns were explored to facilitate the
analysis of beat structures [12]. A two-stage approach was also proposed by
[13] to estimate tempo and beat sequentially, where the process attempted to
replicate human ability to tap along music.

More recently, data-driven approaches have been widely used [1], since they
enable a system to automatically learn the relevant information to detect
beats from annotated audio. Specifically, DNN architectures such as RNN [14]
and TCN [15] were employed to effectively model the long sequence of beat
events from audio. Multi-task approaches were also introduced to include more
types of annotations in the learning system, such as tempo and downbeat [16,
17, 18, 19], owing to their high relevance to beats, and each task was
improved by training multiple tasks altogether. These endeavors have improved
beat tracking significantly, and the evaluation scores can easily achieve over
90% on many benchmarks.

As noted in the previous section, downbeat tracking still remains a challenge.
Several prior instances tackled this problem by using musical features such as
time-signature [20], onset positions [21], chords [22], and spectral
difference in beats [23]. With machine learning approaches, we believe it can
be improved by introducing a novel model architecture with more training data.

##  3 Proposed Method

###  3.1 System Overview

The system pipeline follows the standard setting [14, 17] as depicted in
Figure 1. In this work, our contribution is to introduce two novel DNN models
in the pipeline, and the rest of the modules remain as conventional. For
feature extraction, we apply trainable harmonic filters to convert the input
audio into a harmonic representation [24]. This step helps capture useful
harmonic information and has proven beneficial in music auto-tagging [24].
Then, we define the DNN model to contain three modules: front-end, main model,
and a linear layer. The DNN model predicts the likelihoods of beat, downbeat,
and non-beat at each time step and forms three corresponding activation
functions. Finally, a dynamic Bayesian network (DBN) [25] decodes the
activation functions to output the timestamps of beats and downbeats. In what
follows, we will detail three options of DNN models (two proposed and one
baseline models) used in this work.

Fig. 1: Pipeline for the beat and downbeat tracking system.

(a) SpecTNT

(b) TCN

(c) SpecTNT-TCN

Fig. 2: Model architecture overview.

###  3.2 Spectral-Temporal Transformer in Transformer (SpecTNT)

An input harmonic representation is first processed by the front-end
convolutional layers (called _ResNet_ [26]). Then, we stack 3 residual units,
each uses 256 feature maps and a kernel size of 3. This design is similar to
that proposed in [6].

The main SpecTNT module (see Fig. 2(a)) is formed by stacking multiple SpecTNT
blocks. A SpecTNT block consists of a _spectral encoder_ and a _temporal
encoder_. We use 64 feature maps with 4 attention heads for the spectral
encoder to extract the spectral features into a set of _frequency class
tokens_ (FCTs) at each time step. Through the attention mechanism, a FCT can
characterize useful harmonic and timbral components, which may better
represent a chord or instrumentation. For the temporal encoder, we use 256
feature maps and 8 attention heads. It enables the important local spectral
information to be exchangeable via FCTs to pay attention to the beat/downbeat
positions. More details can be found in [9]. Finally, we stack 5 SpecTNT
blocks, followed by a linear layer to output three activation functions for
beat, downbeat, and non-beat.

###  3.3 Baseline TCN

We adopt the TCN architecture [17, 15] with slight modifications as our
baseline model (see Fig. 2(b)), since it is regarded as the current SOTA. We
re-implemented the model following [17]. Readers can refer to the codes
recently released by the authors in [27].

The TCN front-end is composed of three convolutional layers, each having a
filter size of 20 and kernels with various sizes as described in [17]. Each
convolutional layer also applies a max-pooling to downsample the frequency
dimension. After the front-end, the main model contains 11 dilated
convolutional layers. Each convolutional layer has two dilated convolution
blocks, and each block has a twice the dilation rate of the another. This
design has resulted in better performance according to [17]. Finally, a linear
layer is used to output the three activation functions.

###  3.4 Combining SpecTNT and TCN

SpecTNT and TCN are fundamentally different in many aspects, so we believe
they can be complementary to each other if they are combined. From our pilot
study, we also found that modeling beats and downbeats jointly can result in a
sub-optimal model for either beat or downbeat tracking. Since the amount of
downbeat annotations is less than that of beat annotations, downbeat optimum
will require more epochs to attain. However, longer training may lead to worse
performance for beat, because the model over-fits the beat annotations. We
will discuss this later in Section 4.4. To this end, we propose a fused
architecture, called _SpecTNT-TCN_ (see Fig. 2(c)). We hope it can gain more
flexibility during training so as to facilitate a joint optimum for beat and
downbeat tracking simultaneously.

Specifically, SpecTNT-TCN first uses 2 SpecTNT blocks at the front, and then
arranges a two-stream structure that allocates a TCN branch and 3 succeeding
SpecTNT blocks in parallel. Because SpecTNT requires shorter input, as
discussed in Section 4.1, we divide the input into smaller chunks at the
SpecTNT front-end. The two branches have their own independent linear layers,
and the final output is the average of the two branch outputs. The TCN branch
follows the same TCN module described in Section 3.3. To sum up, during
training, parameters of each branch are updated according to its own loss and
the same SpecTNT parameters at the front are updated according to the added
losses. Final outputs are the average of the outputs from each branch. From
our observations, such design is better than directly merging the outputs of
two full models (i.e., averaging the two outputs of Fig. 2(a) and Fig. 2(b)).

##  4 Experiments

###  4.1 Implementation Details

For data augmentation, we adopt a similar method to that proposed in [17] for
training. Specifically, the hop size to construct the STFT representation is
randomly changed during training using a normal distribution with the
estimated tempo from the training data as the mean and $5\%$ standard
deviation. This can result in more samples of various tempi. Then, multiple
triangular filters are applied to STFT to obtain the harmonic representation.
Following [17], we apply the same target widening strategy for the frame-wise
beat and downbeat labels, where the neighbouring frames of an annotated frame
are set to be positive, but with a lower weight of 0.5. We use a weighted
binary cross-entropy to compare between each target and prediction.

Based on our pilot study, we determine the input length to be 6 seconds for
SpecTNT and 24 seconds for TCN. Although several counterpart systems (e.g.,
[18] and [17]) use 6-second, we found 24-second works better for TCN. In
SpecTNT-TCN, the input is 24-second, where it is later divided into four
6-second chunks for the SpecTNT front module (see Fig. 2(c)). For STFT, we use
sampling rate of 16000 Hz and a window length of 1024.

We apply random sampling to load a random chunk into a batch, instead of
sequentially loading a (6- or 24-second) chunk from the beginning of each
training song. That is, we first enumerate all the valid training chunks into
a list by using a sliding window (6- or 24-second) with 1-second hop size on
every song, and the last chunk of a song shall not exceed the length of the
song. Then, the data loader draws a sample uniformly from the list during
training. Therefore, a batch can include chunks from different locations of
different songs. This technique leads to faster and better convergence.

We use PyTorch 1.8 and the Adam optimizer [28] with 0.001 learning rate and
80% weight decay. The numbers of parameters for TCN, SpecTNT, and SpecTNT-TCN
are 86,679, 4,637,392, and 4,692,896, respectively. We train and test the
models using 4 Tesla-V100-SXM2-32GB GPUs with batch sizes of 128, 128, and 32
for SpecTNT, TCN, and SpecTNT-TCN, respectively. Each epoch runs 500 mini-
batches of training before a validation. Using the aforementioned
specifications, TCN, SpecTNT, and SpecTNT-TCN take about 5.8 minutes, 10
minutes, and 6.5 minutes, respectively, to train 500 batches, and 6 seconds,
12.6 seconds, and 15.1 seconds, respectively, to inference 100 30-second audio
clips. The best model is selected based on the validation set for testing.

###  4.2 Experiment Setting

We use nine public datasets to evaluate the proposed methods. Since the
beat/downbeat tracking datasets are commonly small in size, we tried to
collect as many as possible and combined them to conduct training and
evaluation in an 8-fold cross-validation manner following [17]. These datasets
include _Beatles_ [29], _Ballroom_ [30], _SMC_ [30], _Hainsworth_ [31],
_Simac_ [32], _HJDB_ [33], _RWC-Popular_ [34], and _Harmonix Set_ [35]. An
additional dataset, _GTZAN_ [36], is used for testing only.

We note that the _Harmonix Set_ is comparably larger than other datasets,
containing 912 pop/rock songs with more than 56 hours of beat- and downbeat-
annotated audio content. Our copy is slightly different from the original one
in [35]. The original audio files corresponding to the annotations are not
available, but reference YouTube links are provided. Our investigation
indicated there are many errors when aligning the annotations with the YouTube
audio, so we searched for the audio and manually adjusted the annotations to
ensure the labels and timestamps are sensible and aligned to the audio.

For evaluation, we report the standard metrics: _F1_ , _CMLt_ , and _AMLt_ ,
with a tolerance window of 70 ms as defined in [29]. The resulting scores are
computed using the mir_eval package [37].

(a) Spectral attention

(b) Temporal attention

Fig. 3: Visualization of the attention maps for a _GTZAN_ example,
“pop.00098.wav” from 6 to 12 seconds. Upper subplot displays one of the time-
synchronized harmonic representations.

###  4.3 Attention Map Visualization

To better understand the attention mechanism in SpecTNT when modeling beats
and downbeats, we visualize one of the attention heads of the last SpecTNT
block during the inference. For the spectral attention (see Fig. 3(a)), the
frequency attention scores that generate the FCT of each frame are obtained.
It can be seen that the attention map successfully characterizes the harmonic
components. The spectral attention map emphasizes on the bass notes between
200 and 400 Hz which matches the bass line contour in the harmonic
representation. For the temporal attention (see Fig. 3(b)), we display the
time-by-time self-attention map from one of the 8 attention heads that we
consider mostly relevant to the downbeat modeling. The red dashed rectangles
highlight the downbeat locations. We can see that almost every frame attends
to the downbeat locations, i.e., each column vector (a frame) has higher
scores at the downbeat times. This will help generate the output embeddings
that emphasize the downbeats, and predict higher probabilities for them
accordingly.

###  4.4 Results and Discussion

The beat and downbeat tracking results are presented in Table 1 and Table 2,
respectively. We include the results from prior literature for reference,
although they are not necessarily comparable to ours, as the training data and
experimental settings are different. The performance scores of _Harmonix Set_
for Böck et al. [18] were evaluated on the predictions made by the madmom
package [14], which uses the method proposed in [18]. _Harmonix Set_ was not
included in the training set of madmom.

Comparing the results among SpecTNT, TCN, and SpecTNT-TCN in Table 1, we
observe that SpecTNT is generally better than TCN, and that SpecTNT-TCN
outperforms SpecTNT in most cases. This demonstrates that SpecTNT is a
promising DNN architecture for this task. In particular, SpecTNT and SpecTNT-
TCN perform significantly better on _SMC_ , one of the most challenging
datasets in beat tracking. Our qualitative inspections indicate that SpecTNT
is superior in handling expressive timing. For instance, it performs better in
cases such as “SMC_00111” (with a pause) and “SMC_00165” (with varying tempo).
However, it may suffer from non-trivial percussive patterns that yield phase
errors (e.g., “SMC_00213”). Other than that, the results also show that
SpecTNT seems less successful on _Ballroom_ and _Hainsworth_. This could be
attributed to the sub-optimal issue mentioned in Section 3.4. Further
investigations reveal that SpecTNT tends to over-fit beat annotations of these
two datasets early during training, and then the performance drops slightly in
the later epochs. This is expected, because _Hainsworth_ covers more diverse
genres (e.g., classical, folk, and jazz), and _Ballroom_ is compiled with old-
school dance music. These two datasets provide relatively less examples for
their specific genres, as compared to a much larger dataset, _Harnmonix set_ ,
which offers fairly more examples of pop-style in the training set. Since our
model selection criterion considers the joint performance of beat and
downbeat, a model that performs much better in downbeat than in beat is picked
eventually (and it requires more epochs). Nevertheless, SpecTNT-TCN is more
robust to this issue and shows improvements over SpecTNT and TCN.

| F1 | CMLt | AMLt | F1 | CMLt | AMLt
---|---|---|---|---|---|---
| RWC-POP | Harmonix Set
Böck et al. [18] | .943 | - | - |  .933† |  .841† |  .938†
TCN (baseline) | .947 | .922 | .952 | .946 | .895 | .942
SpecTNT | .953 | .925 | .957 | .947 | .896 | .943
SpecTNT-TCN | .950 | .925 | .958 | .953 | .939 | .959
| SMC | Beatles
Böck et al. [18] | .516 | .406 | .575 | .918 | - | -
Böck et al. [17] | .544 | .443 | .635 | - | - | -
TCN (baseline) | .560 | .474 | .621 | .933 | .870 | .933
SpecTNT |  .602⋆ |  .515⋆ | .661 | .940 | .898 | .929
SpecTNT-TCN |  .605⋆ |  .514⋆ | .663 | .943 | .896 | .938
| Ballroom | Hainsworth
Davies et al. [15] | .933 | .881 | .929 | .874 | .795 | .930
Böck et al. [17] | .962 | .947 | .961 | .902 | .848 | .930
TCN (baseline) | .940 | .870 | .957 | .860 | .849 | .915
SpecTNT | .927 | .856 | .939 | .866 | .865 | .914
SpecTNT-TCN |  .962⋆ |  .939⋆ | .967 | .877 | .862 | .915

Table 1: Beat tracking results using an 8-fold cross validation, where ⋆
denotes statistical significance compared to TCN (baseline), and † denotes the
predictions made by madmom [14].

| F1 | CMLt | AMLt | F1 | CMLt | AMLt
---|---|---|---|---|---|---
| RWC-POP | Harmonix Set
Böck et al. [18] | .861 | - | - |  .804† |  .747† |  .873†
TCN (baseline) | .930 | .928 | .938 | .873 | .839 | .908
SpecTNT | .941 | .929 | .957 |  .897⋆ | .862 |  .924⋆
SpecTNT-TCN | .945 | .939 | .959 |  .908⋆ |  .872⋆ |  .928⋆
| Ballroom | Beatles
Fuentes et al. [38] | .830 | - | - | .860 | - | -
Böck et al. [17] | .916 | .913 | .960 | .837 | .742 | .862
TCN (baseline) | .841 | .788 | .937 | .843 | .767 | .851
SpecTNT |  .884⋆ |  .835⋆ | .937 | .867 |  .810⋆ | .860
SpecTNT-TCN |  .937⋆ |  .927⋆ | .968 | .870 |  .812⋆ | .865
| Hainsworth |
Böck et al. [17] | .722 | .696 | .872 |  |  |
TCN (baseline) | .682 | .683 | .852 |  |  |
SpecTNT |  .729⋆ |  .734⋆ | .879 |  |  |
SpecTNT-TCN |  .748⋆ |  .738⋆ | .870 |  |  |

Table 2: Downbeat tracking results using an 8-fold cross validation, where ⋆
denotes statistical significance compared to TCN (baseline), and † denotes the
predictions made by madmom [14].

| Beat | Downbeat
---|---|---
| F1 | CMLt | AMLt | F1 | CMLt | AMLt
Böck et al. [17] | .885 | .813 | .931 | .672 | .640 | .832
Transformer | .853 | .741 | .887 | .667 | .617 | .843
TCN (baseline) | .879 | .802 | .911 | .702 | .660 | .859
SpecTNT | .883 | .809 | .906 |  .745⋆ |  .696⋆ |  .875⋆
SpecTNT-TCN | .887 | .812 | .920 |  .756⋆ |  .715⋆ |  .881⋆

Table 3: Beat and downbeat tracking result on (test-only) _GTZAN_ , where ⋆
denotes statistical significance compared to TCN (baseline).

From the downbeat tracking results in Table 2, similar observations can also
be made. The performance difference between TCN and SpecTNT is more obvious.
Such results are in line with our research motivation that SpecTNT can help
capture harmonic and timbral information and improve downbeat tracking. The
improvement varies across different datasets. Our proposed models perform
better than the baseline by a wider margin for non-pop-centric datasets:
_Ballroom_ and _Hainsworth_. In particular, the performance difference between
SpecTNT (or SpecTNT-TCN) and TCN are statistically significant in terms of
_F1_ and _CMLt_. As mentioned before, SpecTNT may sacrifice its beat tracking
performance for downbeat, as the performance gain of downbeat is larger.
Qualitatively, we also found SpecTNT can avoid quite a few phase errors as
compared to TCN.

Lastly, we present the results of _GTZAN_ in Table 3. In addition to the
models presented above, we also include the results from a regular Transformer
[6], which contains only the temporal encoder [4]. As it is non-hierarchical
and without the spectral encoder, we treat the comparison as the ablation
study. It is clear that the regular Transformer does not work well in this
case, most likely because the training data is insufficient. Whereas SpecTNT’s
design can handle this well by leveraging spectral encoders in a stacked
architecture. Moreover, we once again see similar results to previous
experiments when comparing among SpecTNT, SpecTNT-TCN, and TCN.

##  5 Conclusion

In this paper, we present a novel Transformer-based model, SpecTNT, to model
beats and downbeats from audio signals. The design of spectral and temporal
encoders enables a promising DNN approach, showing SOTA results in downbeats
tracking.

For future work, we will try to mitigate the sub-optimal issue (see Section
4.4) with better solutions, as we expect it is currently the limitation for
SpecTNT to improve beat tracking. We also plan to study the possibility to
include additionally relevant MIR tasks (e.g., music structure [39], chords,
and melody) into a unified multi-task learning framework and model them
jointly with SpecTNT.

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# Two-channel charge-Kondo physics in graphene quantum dots

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

###### Abstract

Nanoelectronic quantum dot devices exploiting the charge-Kondo paradigm have
been established as versatile and accurate analog 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. In
this article, a two-channel charge-Kondo device made instead from graphene
components is considered, realizing a pseudogapped version of the 2CK model.
The model is solved using Wilson’s Numerical Renormalization Group method,
uncovering 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 T-matrix,
and experimentally measurable conductance. The strong coupling pseudogap Kondo
phase is found to persist 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, a _finite_ linear conductance is found 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.

##  I Introduction

The Kondo effect [1] was originally discussed in the context of magnetic
impurities such as Fe, in non-magnetic metallic hosts like Au. By
progressively decreasing the temperature, experimental measurements revealed
an unexpected resistivity minimum, attributed to enhanced electronic
scattering from the impurity local moments [2]. 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 [3].

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 [4] and superconductors [5], as well as
topological materials such as topological insulators [6] or Dirac/Weyl
semimetals [7]. However, local moments in graphene have attracted the most
attention [8, 9]. In neutral graphene, the Dirac point is at the Fermi level
[10] 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 [9] 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
nanofabrication and characterization techniques [11, 12]. 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 [13], with the
extreme quantum confinement producing a discrete level 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 [14, 15, 16], 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’ [17, 18], producing the famous Kondo resonance in electrical
conductance [19]. 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 [12],
including quantum phase transitions (QPTs) [20, 21], emergent symmetries [22,
23], and non-Fermi liquid (NFL) physics [24, 25, 26, 27]. The two channel
Kondo (2CK) model [28] 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 [29]. The standard 2CK
model Hamiltonian reads,

$\hat{H}_{\rm 2CK}=\hat{H}_{\rm
leads}+J_{1}\hat{\boldsymbol{S}}\cdot\hat{\boldsymbol{s}}_{1}+J_{2}\hat{\boldsymbol{S}}\cdot\hat{\boldsymbol{s}}_{2}\;,$
(1)

where $\hat{H}_{\rm
leads}=\sum_{\alpha\sigma\boldsymbol{k}}\epsilon_{\boldsymbol{k}}^{\phantom{\dagger}}c_{\alpha\sigma\boldsymbol{k}}^{\dagger}c_{\alpha\sigma\boldsymbol{k}}^{\phantom{\dagger}}$
describes two leads $\alpha=1,2$, each with spin $\sigma=\uparrow,\downarrow$
electrons with momentum $\boldsymbol{k}$. In the original formulation of 2CK
model, the dispersion $\epsilon_{\boldsymbol{k}}$ 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,
$\rho(\omega)\sim\sum_{\boldsymbol{k}}\delta(\omega-\epsilon_{\boldsymbol{k}})\equiv\rho_{0}\Theta(|\omega|-D)$,
describing a flat density of states $\rho_{0}$ inside a band of half-width
$D$. In Eq. 1, $\hat{\boldsymbol{S}}$ is a spin-$\tfrac{1}{2}$ operator for
the impurity and $\hat{\boldsymbol{s}}_{\alpha}$ is a spin-$\tfrac{1}{2}$
operator for the spin density in lead $\alpha$ at the impurity position, such
that $\hat{H}_{\rm 2CK}$ possesses SU(2) spin symmetry. The metallic 2CK model
supports a QPT, with an NFL critical point at $J_{1}=J_{2}$ [29]. Signatures
of the critical point in this model have been observed experimentally in
semiconductor quantum dot devices [24, 25].

More recently, a new nanoelectronics paradigm has emerged, based on charge-
Kondo quantum dots [30, 31, 32, 33, 34]. In the standard setup, a large QD is
coupled at QPCs to source and drain leads. These devices realize anisotropic
multichannel Kondo models through Matveev’s mapping [35, 36] of the
macroscopic charge states of the large QD to an effective pseudospin that is
flipped by electronic tunneling 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
[30].

Motivated by these developments, in this paper we consider combining the two-
channel charge-Kondo setup of Ref. [30] with the pseudogap Kondo physics of
graphene in Ref. [9], to realize a novel two-channel pseudogap charge-Kondo
effect. We envisage a charge-Kondo device made from graphene components (see
Fig. 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. [37], although the $r=1$
linear pseudogap case relevant to graphene was not analyzed in detail and a
device realization was not proposed. Furthermore, our charge-Kondo
implementation leads to crucial differences in the model and transport
measurement setup which have not previously been considered. These differences
and our new results are highlighted in the following.

##  II Model, Methods, and Observables

We consider a two-channel charge-Kondo device in which both dot and leads are
made from graphene, as illustrated in Fig. 1. We note that graphene quantum
dots have been the topic of active experimental study recently [38, 39, 40].
We envisage a 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. 1) which gives rise to a long dwell time and an
effective continuum dot level spectrum (this was achieved in the experiments
of Ref. [30] 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, tunneling 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 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.

The model Hamiltonian for the device illustrated in Fig. 1 is given by,

$\begin{split}\hat{H}=\hat{H}_{\rm
leads}+\sum_{\alpha=1,2}~{}\sum_{\mathbf{k}\mathbf{k}^{\prime}}\Big{[}&J_{\alpha}(\hat{Q}^{+}c^{\dagger}_{\alpha
D\mathbf{k}}c_{\alpha L\mathbf{k}^{\prime}}+c^{\dagger}_{\alpha
L\mathbf{k}^{\prime}}c_{\alpha D\mathbf{k}}\hat{Q}^{-})\\\
+&W_{\alpha}(c^{\dagger}_{\alpha L\mathbf{k}}c_{\alpha
L\mathbf{k}^{\prime}}+c^{\dagger}_{\alpha D\mathbf{k}}c_{\alpha
D\mathbf{k}^{\prime}})\Big{]}\\\ +&E_{c}(\hat{N}_{D}-N_{g})^{2}\;,\end{split}$
(2)

where $\hat{H}_{\rm
leads}=\sum_{\alpha\sigma\boldsymbol{k}}\epsilon_{\boldsymbol{k}}^{\phantom{\dagger}}c_{\alpha\sigma\boldsymbol{k}}^{\dagger}c_{\alpha\sigma\boldsymbol{k}}^{\phantom{\dagger}}$
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 dispersion $\epsilon_{\boldsymbol{k}}^{\pm}=\pm
t\sqrt{1+4\cos^{2}\tfrac{a}{2}k_{x}+4\cos\tfrac{a}{2}k_{x}\cos\tfrac{\sqrt{3}a}{2}k_{y}}$
for the two bands, assumed here to be independent of $\alpha$ and $\sigma$,
with nearest neighbour tunneling matrix element $t\approx 2.8$ eV and lattice
constant $a\approx 2.46$ Å [10]. The resulting DoS
$\rho_{\alpha\sigma}(\omega)\equiv\rho(\omega)$ has a bandwidth $D=3t$ and
possesses a linear pseudogap $\rho(\omega)\sim|\omega|$ for $|\omega|\ll t$.

The terms proportional to $J_{\alpha}$ describe electronic tunneling at the
QPCs between leads and dot. The tunneling matrix elements $J_{\alpha}$ are
related to the bare (unrenormalized) QPC transmissions via [41]
$\tau_{\alpha}(\omega)=4\pi^{2}\rho_{\alpha L}(\omega)\rho_{\alpha
D}(\omega)J_{\alpha}^{2}/[1+\pi^{2}\rho_{\alpha L}(\omega)\rho_{\alpha
D}(\omega)J_{\alpha}^{2}]^{2}$, 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 dot number
operator $\hat{N}_{D}=\sum_{\alpha\boldsymbol{k}}c_{\alpha
D\boldsymbol{k}}^{\dagger}c_{\alpha
D\boldsymbol{k}}^{\phantom{\dagger}}\equiv\sum_{N_{D}}N_{D}|N_{D}\rangle\langle
N_{D}|$. Tunneling at the QPCs changes the dot charge, which we describe [35]
using the charge raising and lowering operators
$\hat{Q}^{\pm}=\sum_{N_{D}}\ket{N_{D}\pm 1}\bra{N_{D}}$. 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. 2, which is
controlled in experiment by the gate voltage $V_{g}=V_{g}^{0}-2E_{c}N_{g}/e$.
We define $\delta V_{g}=-2E_{c}(N_{g}-N_{D}^{0}-\tfrac{1}{2})/e$ such that the
macroscopic dot charge states $|N_{D}^{0}\rangle$ and $|N_{D}^{0}+1\rangle$
are degenerate at $\delta V_{g}=0$. Potential scattering at the QPCs is
described by the term proportional to $W_{\alpha}$.

Provided $k_{\rm B}T,e\delta V_{g}\ll E_{c}$, only the lowest two dot charge
states $|N_{D}^{0}\rangle$ and $|N_{D}^{0}+1\rangle$ are accessible and
relevant for transport. In this case the dot charge operators become effective
pseudospin-$\tfrac{1}{2}$ operators,
$\hat{Q}^{+}\to\hat{S}_{D}^{+}=|N_{D}^{0}+1\rangle\langle N_{D}^{0}|$ and
$\hat{Q}^{-}\to\hat{S}_{D}^{-}=|N_{D}^{0}\rangle\langle N_{D}^{0}+1|$. Thus
$\hat{S}_{D}^{+}$ flips the dot charge pseudospin from $\Downarrow$ to
$\Uparrow$ while $\hat{S}_{D}^{-}$ flips it back. We also introduce the
pseudospin operator
$\hat{S}_{D}^{z}=\tfrac{1}{2}(\ket{N_{D}^{0}+1}\bra{N_{D}^{0}+1}-\ket{N_{D}^{0}}\bra{N_{D}^{0}})$.
Finally, we perform a trivial relabelling
$\sigma=\\{L,D\\}\to\\{\uparrow,\downarrow\\}$ 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
arrive at the effective pseudogap two-channel charge-Kondo (2CCK) model
studied in this paper:

$\begin{split}\hat{H}_{2CCK}=&\sum_{\alpha=1,2}~{}\sum_{\mathbf{k}\mathbf{k}^{\prime}}\Big{[}J_{\alpha}(\underbrace{\hat{S}_{D}^{+}c^{\dagger}_{\alpha\downarrow\mathbf{k}}c_{\alpha\uparrow\mathbf{k}^{\prime}}}_{\hat{S}_{D}^{+}\hat{s}_{\alpha}^{-}}+\underbrace{c^{\dagger}_{\alpha\uparrow\mathbf{k}^{\prime}}c_{\alpha\downarrow\mathbf{k}}\hat{S}_{D}^{-})}_{\hat{s}_{\alpha}^{+}\hat{S}_{D}^{-}}\\\
&+W_{\alpha}\sum_{\sigma}c^{\dagger}_{\alpha\sigma\mathbf{k}}c_{\alpha\sigma\mathbf{k}^{\prime}}\Big{]}+e\delta
V_{g}\hat{S}^{z}_{D}+\hat{H}_{\rm leads}\;.\end{split}$ (3)

This model is a variant of the famous 2CK model, Eq. 1 – but with a few
important differences. Firstly, the DoS of the conduction electrons described
by $\hat{H}_{\rm leads}$ is not metallic, but has a low-energy $r=1$
pseudogap. Secondly, tunneling at the QPCs give an effective anisotropic
exchange coupling between the dot charge pseudospin and the conduction
electrons. The SU(2) symmetry of Eq. 1 is broken in Eq. 3 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 [2, 42] and two-channel [29] metallic case, as
well as the single-channel pseudogap case [43]). 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 tunnelings; 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 [30]. This is important because Kondo physics is only realized in the
pseudogap model at relatively large bare coupling strengths. Thirdly, the gate
voltage $\delta V_{g}$ appears as an effective impurity magnetic field.
Finally, we have an additional potential scattering term $W_{\alpha}$. This is
traditionally omitted in Eq. 1 because potential scattering is RG irrelevant
in the metallic Kondo problem [2]. However, we must keep it because potential
scattering is known to be important in the single-channel pseudogap Kondo
model [43]. Indeed, we find that it is strongly RG relevant in our two-channel
pseudogap variant, Eq. 3.

Another important difference in terms of the experimental realization is the
nature of the transport measurement. As illustrated in Fig. 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. 1, the charge-Kondo setup Eq. 3 allows an effective
spin current to be measured.

The ac linear response electrical conductance through the device is defined,

$G_{C}(\omega,T)=\frac{\langle\hat{I}_{2\uparrow}\rangle}{V_{bias}}\biggr{\rvert}_{V_{bias}\rightarrow
0}$ (4)

due to an oscillating bias described by $\hat{H}_{\rm
bias}=-eV_{\textit{bias}}\cos(\omega t)\hat{N}_{1\uparrow}$ with ac frequency
$\omega$. Here
$\hat{I}_{\alpha\uparrow}=-e\tfrac{d}{dt}\hat{N}_{\alpha\uparrow}$ is the
current operator for lead $\alpha$ (and $\sigma=\uparrow$) while
$\hat{N}_{\alpha\uparrow}=\sum_{\boldsymbol{k}}c_{\alpha\uparrow\boldsymbol{k}}^{\dagger}c_{\alpha\uparrow\boldsymbol{k}}^{\phantom{\dagger}}$.
We obtain the ac linear conductance from the Kubo formula [44],

$\displaystyle G_{C}(\omega,T)=\frac{-{\rm
Im}\langle\langle\hat{I}_{1\uparrow};\hat{I}_{2\uparrow}\rangle\rangle_{\omega,T}}{\omega}\equiv
2\pi G_{0}\omega{\rm
Im}\langle\langle\hat{N}_{1\uparrow};\hat{N}_{2\uparrow}\rangle\rangle_{\omega,T}~{},$
(5)

where $\langle\langle\cdot~{};~{}\cdot\rangle\rangle$ denotes a retarded real-
frequency correlation function evaluated at equilibrium, and $G_{0}=e^{2}/h$
is the conductance quantum ($\hbar=1$). The second equality in Eq. 5 follows
from equations of motion and is found to greatly improve the accuracy of
numerical calculations [45]. 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 [46] 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 thermodynamical 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 $F^{(0)}=-k_{\rm B}T\ln Z^{(0)}$ the free energy.
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 [47, 33]. For the magnetic
susceptibility $k_{\rm B}T\chi_{D}(T)$ we evaluate $k_{\rm
B}T\chi^{(0)}=\braket{(\hat{S}^{z}_{tot})^{2}}^{(0)}-(\braket{\hat{S}^{z}_{tot}}^{(0)})^{2}$
at zero field ($\delta V_{g}=0$), 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’
$N_{\alpha}=\langle\hat{N}_{\alpha}\rangle-\langle\hat{N}_{\alpha}\rangle^{0}$
with $\hat{N}_{\alpha}=\sum_{\sigma}\hat{N}_{\alpha\sigma}$. Dynamics of the
system are characterized by the channel-resolved T-matrix, which describes how
conduction electrons are scattered from the dot pseudospin. The T-matrix
equation reads,

$\mathbb{G}_{\alpha\beta}(\omega,T)-\mathbb{G}^{0}_{\alpha\beta}(\omega)=\mathbb{G}^{0}_{\alpha\alpha}(\omega)\mathbb{T}_{\alpha\beta}(\omega,T)\mathbb{G}^{0}_{\beta\beta}(\omega)~{},$
(6)

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. 3), we have
$\mathbb{G}_{\alpha\beta},\mathbb{G}^{0}_{\alpha\beta},\mathbb{T}_{\alpha\beta}\propto\delta_{\alpha\beta}$
and the 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. In the following we consider the spectrum of the
T-matrix for channel $\alpha$, defined as
$t_{\alpha}(\omega,T)=-\tfrac{1}{\pi}{\rm
Im}\mathbb{T}_{\alpha\alpha}(\omega,T)$.

###  II.1 Numerical Renormalization Group

The two-channel pseudogap charge-Kondo model, Eq. 3, is solved using Wilson’s
Numerical Renormalization Group (NRG) technique [3, 48, 49], which provides
numerically-exact access to the physical quantities discussed in the previous
section.

The first step is the logarithmic discretization of the conduction electron
DoS, and subsequent mapping of $\hat{H}_{\rm leads}$ to Wilson chains [3, 48],

$\hat{H}_{\rm leads}\to\hat{H}^{\rm disc}_{\rm
leads}=\sum_{\alpha,\sigma}~{}\sum_{n=0}^{\infty}t_{n}\left(f^{\dagger}_{\alpha\sigma
n}f_{\alpha\sigma n+1}^{\phantom{\dagger}}+f^{\dagger}_{\alpha\sigma
n+1}f_{\alpha\sigma n}^{\phantom{\dagger}}\right)~{}.$ (7)

The dot then couples to the end of the Wilson chains, at site $n=0$. The
logarithmic discretization is parameterized 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 $n$ [3]. 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. [50] 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
[49, 51], established on the complete Anders-Schiller basis [52].

##  III Results and Discussion

Having introduced the model and methods, we now discuss our NRG results in
detail, starting with an overview of the phase diagram, RG flow diagram, and
fixed point analysis. In the following we confine our attention to the charge-
degeneracy point $\delta V_{g}=0$. 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 $\Delta\leftrightarrow 1/\Delta$. We also
assume $W>0$.

###  III.1 Overview and phase diagram

The schematic RG flow diagram in the space of $(J,W,\Delta)$ shown in the left
panel of Fig. 2 was deduced from non-perturbative NRG results, and gives a
good overview of the physics of Eq. 3. In the right panel we show the
quantitative phase diagram in the $(J,W)$ plane for different $\Delta$, with
the exact phase boundaries obtained with NRG.

Figure 2: Left: RG flow diagram for the pseudogap 2CCK model Eq. 3, in the
space of effective exchange coupling $J$, potential scattering $W$ and channel
asymmetry $\Delta$. Stable (unstable) FPs denoted as blue (red) squares.
$\Delta=0$ ($1$) is the pure single-channel (frustrated two-channel) model.
For an explanation of the FPs, see text. Right: Full NRG phase diagram for
different $\Delta$. 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.

We first briefly recapitulate the results for the single-channel (1CK) case
obtained here for $\Delta=0$ (see Fig. 2: front plane in RG diagram on the
left, and turquoise line in the phase diagram on the right). The basic physics
are well-known from previous studies of the $r=1$ pseudogap Anderson and Kondo
models [50, 53, 54, 43, 55, 56, 9], although note that our graphene charge-
Kondo setup gives a spin-anisotropic model, and we use the full graphene DoS
rather than a pure pseudogap. At $W=0$ there is no Kondo effect: the symmetric
strong coupling (SSC) FP is unstable and a finite potential scattering is
required to screen the dot pseudospin. In this case the system flows to weak
coupling and eventually to the free local moment (LM) FP with asymptotically
decoupled conduction electrons. At finite $W$ an LM phase with an emergent
particle-hole symmetry can be realized, in which the dot and leads decouple at
low temperatures (that is $W,J\to 0$ under RG). However, at sufficiently
strong bare $W$ the model supports a QPT to an asymmetric strong coupling
(ASC) Kondo state, in which the dot pseudospin is screened and a single hole
forms in the bath (that is $W,J\to\infty$ under RG). However the coupling $J$
must overcome a critical threshold value (for Eq. 3 $J_{CR}^{\rm min}\simeq
1.81D$). For $J<J_{CR}^{\rm min}$ no Kondo state is possible at any $W$. For
$J\geq J_{CR}^{\rm min}$, the ASC FP is stable for
$W^{-}_{CR}(J)<W<W^{+}_{CR}(J)$. In the lower branch we find
$W^{-}_{CR}(J)\sim 1/J$ at large $J\gg J_{CR}^{\rm min}$, such that
infinitesimal particle-hole symmetry breaking $W\to 0$ is required at large
bare coupling $J\to\infty$. Thus, although SSC is unstable, the system can
flow arbitrarily close to it, before ultimately crossing over to either ASC or
LM. The transition between ASC and LM is first-order [43, 55], and controlled
by a particle-hole asymmetric critical FP denoted ACR. The full NRG phase
boundary for our model at $\Delta=0$ is shown as the turquoise line in the
right panel of Fig. 2, and shows an interesting re-entrant behaviour back into
the LM phase at large $W$ (we are not aware of a detailed discussion of this
in the literature, even though the same behaviour arises in the regular spin-
isotropic pseudogap Kondo model). This is physically intuitive since $J$ and
$W$ work antagonistically: at very large $W$ the bath 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 $J_{\rm eff}\sim t_{0}^{2}J/W^{2}$,
which is consistent with $W^{+}_{CR}(J)\sim J$ for the upper branch of the
phase boundary. This is indeed confirmed by NRG calculations.

Table 1: Classification of FPs according to their physical observables, with * denoting unstable FPs. 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⋆ | $\ln 4$ | $1/8$ | $-1/4$ |  $1/\omega\ln^{2}(\lambda_{CR}/\omega)$ | $const$

The main focus of this paper is the situation when the coupling to the second
channel is switched on, $\Delta>0$, where we find several differences from the
pure 1CK case. We discuss $0<\Delta<1$ first. Importantly, we find that the
same LM and ASC phases are accessible, with $\Delta$ flowing to zero under RG
flow on reducing the temperature or energy scale. Therefore, even though both
channels are initially coupled to the dot (at high temperatures we have a free
channel degree of freedom $\alpha=1,2$), any channel asymmetry leads
asymptotically to the decoupling of the less strongly coupled channel
$\alpha=2$ (this can be regarded as ‘channel freezing’ at low temperatures).
In the ASC phase, the dot flows to strong coupling with the more strongly
coupled channel $\alpha=1$; while in the LM phase, channel $\alpha=1$ also
eventually decouples, leaving a free dot pseudospin and free conduction
electrons. This is indicated by the flow arrows towards the front plane in the
RG diagram, Fig. 2 (left).

However, at finite $\Delta$ the topology of the phase diagram changes – see
Fig. 2 (right). We still have a finite threshold value of the coupling to
realize ASC physics, $J_{CR}^{\rm min}(\Delta)>0$ (which increases slightly
from $\simeq 1.81D$ at $\Delta=0$ up to $\simeq 2.47D$ as $\Delta\to 1$). But
the critical phase boundary now also develops a finite threshold value of the
potential scattering $W_{CR}^{\rm min}(\Delta)>0$ (which reaches its maximum
value $\simeq D$ as $\Delta\to 1$). Even at strong bare coupling $J\to\infty$
a finite $W$ is required to access the ASC phase. In fact, $W_{CR}^{\rm min}$
occurs at an intermediate value of $J$; at large $J$ we find $W^{-}_{CR}\sim
J$ in the lower branch. For even larger $W$, we again have re-entrant LM
behaviour, with an upper branch of the phase boundary. For $\Delta>0$ we
therefore have large-$J$ behaviour $W^{-}_{CR}(J,\Delta)=a_{-}(\Delta)J$ for
the lower branch and $W^{+}_{CR}(J,\Delta)=a_{+}(\Delta)J$ for the upper
branch. Interestingly $a_{+}\approx 1$ independent of $\Delta$, while $a_{-}$
increases with increasing $\Delta$, as shown in the inset to the right panel
of Fig. 2. However, $a_{-}(\Delta)/a_{+}<1$ for all $\Delta$ (the ratio
reaches its maximum $\approx 0.2$ as $\Delta\to 1$) meaning that the upper and
lower phase boundaries never cross, and the ASC phase persists out to infinite
$J$ and $W$. The finite $W_{CR}^{\rm min}$ also implies that there is no
crossover from SSC to ASC for $\Delta>0$.

Finally, we examine the channel-frustrated case $\Delta=1$ (see middle plane
of Fig. 2, left; and purple line of Fig. 2, right). Here symmetry dictates a
channel degeneracy down to $T=0$ and therefore no channel freezing. We find
that the model supports an LM phase in which both channels flow symmetrically
to weak coupling and to particle hole symmetry. However, the ASC FP is
unstable since the Kondo effect and conduction electron hole in ASC occur in
only one of the two channels. Instead we have a frustrated asymmetric strong
coupling (FASC) phase, with a free channel degree of freedom (a doubled
version of the ASC FP, with the Kondo effect and conduction electron hole
forming in either channel $\alpha=1$ or $2$). The critical point separating LM
and FASC in the $\Delta=1$ plane is denoted FACR. 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 $|1-\Delta|$
relieves the channel frustration and leads ultimately to channel freezing on
the lowest energy scales. This QPT is also first-order; FACR is therefore
tricritical since it sits at the boundary between LM, FASC and ASC.

In Table 1 we summarize the FPs discussed above in relation to Fig. 2,
classifying them according to their physical properties. These properties are
extracted from the limiting behaviour of the full thermodynamic and dynamic
observables presented in the following.

Figure 3: 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.

###  III.2 Thermodynamics and fixed points

The temperature dependence of the dot contribution to entropy $S_{D}(T)$ and
magnetic susceptibility $T\chi_{D}(T)$ are obtained from NRG [48] and
presented in Fig. 3 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.

####  III.2.1 Frozen channel degree of freedom: $0\leq\Delta<1$

We first consider the regime $0\leq\Delta<1$ (left and middle columns of Fig.
3). 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. The excess conduction electron charge (not shown) 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 [9]. The FP Hamiltonian in the LM phase is therefore
given by,

$\hat{H}_{LM}=\hat{H}_{2CCK}~{}~{}{\rm
with}~{}~{}J_{1}=J_{2}=W_{1}=W_{2}=V_{g}=0$ (8)

In the ASC phase at $T=0$ we see quenched dot entropy $S_{D}=0$ and
$T\chi_{D}=0$ in all cases, 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
$\alpha=1$ ($W_{1}\to\infty$ 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: as $W_{1}$ grows under RG, the $f_{1\sigma 0}$ Wilson orbital becomes
depopulated, thereby generating an effective coupling between the dot
pseudospin and the Wilson $f_{1\sigma 1}$ orbital, $J_{\textit{eff}}\sim
t_{0}^{2}J_{1}/W_{1}^{2}$. However, the DoS of the $f_{1\sigma 1}$ orbital is
modified by the hole forming at the $n=0$ site. With
$\rho(\omega)\equiv-\tfrac{1}{\pi}{\rm Im}\langle\langle f_{1\sigma
0};f_{1\sigma 0}^{\dagger}\rangle\rangle\sim|\omega|$ at low energies, we find
$\rho_{\textit{eff}}(\omega)\equiv-\tfrac{1}{\pi}{\rm Im}\langle\langle
f_{1\sigma 1};f_{1\sigma 1}^{\dagger}\rangle\rangle\sim 1/|\omega|$ at low
energies. Therefore even though $J_{\textit{eff}}$ is perturbatively small,
the effective DoS is strongly enhanced. The effective dimensionless RG flow
parameter $j_{1}=\rho_{\textit{eff}}J_{\textit{eff}}$ grows under RG and leads
to Kondo screening of the dot. The Kondo scale for this process is strongly
enhanced because of the diverging effective DoS [57], and we find in practice
that throughout the ASC phase $T_{K}\sim D$ (since $J_{CR}^{\rm min}>D$).
However, no hole forms in the weakly coupled channel, and so $j_{2}=\rho
J_{2}$ 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. This is confirmed by
analysis of the NRG level spectrum.

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,

$\begin{split}\hat{H}^{\alpha}_{ASC}=&\hat{H}_{\rm
leads}+J_{\alpha}\left(\hat{S}_{D}^{+}f_{\alpha\downarrow
1}^{\dagger}f_{\alpha\uparrow
1}^{\phantom{\dagger}}+\hat{S}_{D}^{-}f_{\alpha\uparrow
1}^{\dagger}f_{\alpha\downarrow 1}^{\phantom{\dagger}}\right)\\\
&+W_{\alpha}\sum_{\sigma}f_{\alpha\sigma 0}^{\dagger}f_{\alpha\sigma
0}^{\phantom{\dagger}}\qquad{\rm with}\qquad
J_{\alpha},W_{\alpha}\to\infty\;.\end{split}$ (9)

We now consider the situation in the close vicinity of the QPT, by fixing $W$
and tuning $J$. At the critical point itself (orange line, $J=J_{CR}$) we find
$S_{D}=\ln(3)$ and $T\chi_{D}=\tfrac{1}{6}$ at $T=0$. 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 (2 for LM, 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
crossover scale $T^{*}\sim|J-J_{CR}|$, describing the flow from ACR to either
LM or ASC due to a small detuning perturbation. This scale is evident in Fig.
3 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 $T^{*}$. We also checked this behaviour along the entire critical
phase boundary lines $(J_{CR},W_{CR})$ in Fig. 2 for different $\Delta$. We
find,

$\displaystyle T^{*}=b|J-J_{CR}|+c|W-W_{CR}|\;,$ (10)

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
$T/T^{*}$, independent of the combination of bare perturbations that act. The
FP Hamiltonian describing the critical point is,

$\hat{H}^{\alpha}_{ACR}=\left(\frac{1+\hat{\tau}^{z}}{2}\right)\hat{H}_{LM}+\left(\frac{1-\hat{\tau}^{z}}{2}\right)\hat{H}^{\alpha}_{ASC}~{}.$
(11)

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. 11, $\tau^{z}=+1$ gives the doubly-degenerate LM ground state
($\hat{H}_{LM}$ given in Eq. 8) while $\tau^{z}=-1$ gives the ASC ground state
($\hat{H}_{ASC}$ given in Eq. 9). At the ACR FP, the three many-body ground
states are degenerate and uncoupled ($\hat{\tau}$ has no dynamics). Since ACR
is unstable, we also consider the leading RG relevant perturbations to the FP
Hamiltonian, $\delta\hat{H}^{\alpha}_{ACR}\sim T^{*}\hat{\tau}^{z}$, which has
the effect of biasing towards either the LM or ASC ground states on the scale
of $T^{*}$.

The qualitative behaviour of the thermodynamics shown in Fig. 3 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. 2,
left).

####  III.2.2 Frustrated channel degree of freedom: $\Delta=1$

We turn now to the frustrated case $\Delta=1$, with pristine channel symmetry
– see Fig. 3, right column. Although the $T=0$ entropy is $S_{D}=\ln(2)$
everywhere except on the phase boundary (top right panel of Fig. 3), the
origin of the ground state degeneracy is different in the two phases separated
by it. In the LM phase (realized for $J<J_{CR}$), 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 $T\chi_{D}=\tfrac{1}{4}$ (dashed lines, bottom right
panel of Fig. 3). 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 [37], which embodies the choice of forming the hole-
singlet complex of ASC with either of the two channels. This is reflected in
the $T=0$ value of $T\chi_{D}=0$ in the FASC phase (solid lines, bottom right
panel of Fig. 3), 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, consider two holes forming symmetrically in the
$f_{1\sigma 0}$ and $f_{2\sigma 0}$ Wilson $n=0$ orbitals
($W_{1}=W_{2}\to\infty$), and effective Kondo couplings
$J_{1,\textit{eff}}=J_{2,\textit{eff}}\to\infty$ between the dot pseudospin
and the residual Wilson $n=1$ orbitals $f_{1\sigma 1}$ and $f_{2\sigma 1}$,
which have an effective DoS $\rho_{\textit{eff}}(\omega)\sim 1/|\omega|$ – a
channel symmetric version of the usual hole-singlet mechanism in ASC as
described in the previous section. However, the dot entropy is not quenched in
this case, since the ground state of the complex is a spin-doublet. 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 $\sim|\omega|$
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.

The FASC FP Hamiltonian comprises a combination of $\hat{H}_{ASC}^{1}$ and
$\hat{H}_{ASC}^{2}$ from Eq. 9, controlled by an emergent channel degree of
freedom $\hat{\alpha}$,

$\hat{H}_{FASC}=\left(\frac{1+\hat{\alpha}^{z}}{2}\right)\hat{H}^{1}_{ASC}+\left(\frac{1-\hat{\alpha}^{z}}{2}\right)\hat{H}^{2}_{ASC}~{}.$
(12)

Here $\hat{\alpha}^{z}$ is a Pauli-$z$ operator that selects ASC1 when
$\alpha^{z}=+1$ and ASC2 when $\alpha^{z}=-1$. Restricting to the symmetric
$\Delta=1$ plane, FASC is stable. However, there is an instability with
respect to breaking channel symmetry (not shown), since then either ASC1 or
ASC2 will be selected on the lowest energy scales. A finite perturbation
$|1-\Delta|$ 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 crossover
is $T_{\Delta}\sim|1-\Delta|$. 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 [37], 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 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,

$\hat{H}_{FACR}=\left(\frac{1+\hat{\tau}^{z}}{2}\right)\hat{H}_{LM}+\left(\frac{1-\hat{\tau}^{z}}{2}\right)\hat{H}_{FASC}~{},$
(13)

where $\hat{H}_{LM}$ is given in Eq. 8 and $\hat{H}_{FASC}$ in Eq. 12, and we
have introduced the operator $\hat{\tau}^{z}$ to distinguish the sectors,
similar to Eq. 11. As with ACR, the FP is destabilized by RG relevant detuning
perturbations that favour either LM of FASC, which collectively generate the
scale $T^{*}$ given in Eq. 10. This leads to an FP correction
$\delta\hat{H}_{FACR}^{*}\sim T^{*}\hat{\tau}^{z}$. This is shown by the
sequence of lines in the right column of Fig. 3. However, FACR is also
destabilized by relieving the channel frustration through the perturbation
$|1-\Delta|$ 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.

Figure 4: NRG results for dynamics and transport in the graphene 2CCK model at
$T=0$. 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).

###  III.3 Dynamics and transport

We now discuss the low-temperature behaviour of the scattering T-matrix and
linear response ac electrical conductance in the graphene 2CCK device – see
Fig. 4. We first consider the $T=0$ spectrum of the T-matrix as a function of
energy in the top row of Fig. 4, for the channel asymmetric case $\Delta=0.8$
(left) and frustrated case $\Delta=1$ (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 [43, 37].

Deep in the LM phase (blue solid and dotted lines), the bare potential
scattering $W$ modifies the bare conduction electron pseudogap DoS of graphene
$\rho(\omega)\sim|\omega|$, to give an effective DoS
$\rho_{\textit{eff}}(\omega)\sim 1/|\omega|$ ($\pi/2$ phase shift) up to
logarithmic corrections. This produces leading behaviour in the T-matrix
$t_{\alpha}(\omega,0)\sim 1/|\omega|$, as seen in Fig. 4 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 $\sim|\omega|$ (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 $\Delta$, 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 $W,J\to\infty$.
Counterintuitively 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 $\sim|\omega|$. 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
$\rho(\omega)\sim|\omega|$, this confirms that both phases are regular Fermi
liquids, with well-defined (long-lived) quasiparticles [2, 43, 55].

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 [43, 37]. We find from NRG that in
the channel-asymmetric graphene 2CCK model (e.g. at $\Delta=0.8$ as shown) the
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 $|\omega|\to
0$ (solid green lines in the top panels of Fig. 4) [58], indicating that ACR
is an NFL FP. The dynamical crossover, and hence the minimum in
$t_{1}(\omega,0)$ occurs on the scale of $|\omega|\propto\lambda_{CR}$.
However, the weakly coupled channel $\alpha=2$ has FL correlations
$t_{2}(\omega,0)\sim|\omega|$ as $|\omega|\to 0$ (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. 4 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 tunneling 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 tunneling rate gets renormalized by the interactions (the
energy-dependent scattering at the QPCs is characterized by the 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. [30].

The measured dc conductance of the graphene 2CCK 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. 4 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. 4.

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 tunneling rate. At high energies the pseudospin flip rate in
the 2CCK model is given by the bare $J$ (or effective $J_{\text{eff}}$). We
therefore expect to see a peak in the ac conductance when $|\omega|\sim
J,J_{\textit{eff}}$; this is observed from NRG results in the LM, ASC and FASC
phases in Fig. 4. However, at low energies $|\omega|\ll\lambda$, the
pseudospin flip rate is renormalized and we find
$G_{C}(\omega,0)\sim\omega^{2}$ 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
$G_{C}(\omega,0)\sim-1/\ln|\omega|$ in this regime. However, in the channel-
symmetric case $\Delta=1$ at the critical point FACR,
$G_{C}(\omega,0)\sim\textit{const}.$ 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.

##  IV Conclusion and Outlook

In this paper we propose 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 analyze the
thermodynamic and dynamical properties of the model using NRG, and use 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 with 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 tunneling 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.

###### Acknowledgements.

We acknowledge funding from the Irish Research Council through the Laureate
Award 2017/2018 grant IRCLA/2017/169 (AKM/JBR) and the Enterprise Partnership
Scheme grant EPSPG/2017/343 (ELM)

## Abbreviations

The following abbreviations are used in this manuscript:

QD | quantum dot
---|---
QPC | quantum point contact
QPT | quantum phase transition
QCP | quantum critical point
2CK | two channel Kondo
NRG | numerical renormalization group
FL | Fermi liquid
NFL | non-Fermi liquid
DoS | density of states
RG | renormalization group
FP | fixed-point
LM | local moment
(F)ALM | (frustrated) asymmetric local moment
(F)ASC | (frustrated) asymmetric strong coupling
(F)SSC | (frustrated) symmetric strong coupling
(F)ACR | (frustrated) asymmetric critical

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