# Calibrate your listeners!
Robust communication-based training for pragmatic speakers

Rose E. Wang1, Julia White 2, Jesse Mu1, Noah D. Goodman1,3
Departments of 1Computer Science, 2Electrical Engineering and 3Psychology
Stanford University
{rewang, jiwhite, muj<EMAIL_ADDRESS>

###### Abstract

To be good conversational partners, natural language processing (NLP) systems
should be trained to produce contextually useful utterances. Prior work has
investigated training NLP systems with communication-based objectives, where a
neural listener stands in as a communication partner. However, these systems
commonly suffer from semantic drift where the learned language diverges
radically from natural language. We propose a method that uses a population of
neural listeners to regularize speaker training. We first show that language
drift originates from the poor uncertainty calibration of a neural listener,
which makes high-certainty predictions on novel sentences. We explore
ensemble- and dropout-based populations of listeners and find that the former
results in better uncertainty quantification. We evaluate both population-
based objectives on reference games, and show that the ensemble method with
better calibration enables the speaker to generate pragmatic utterances while
scaling to a large vocabulary and generalizing to new games and
listeners.111The accompanying code can be found here:
https://github.com/rosewang2008/calibrate_your_listeners.

##  1 Introduction

To be good conversational partners, language models (LMs) should learn to
produce fluent utterances that serve the needs of their listeners Grice
(1975). However, they are often trained to capture the statistical, rather
than communicative, properties of language via supervised learning Bengio et
al. (2003); Radford et al. (2019). Consider the reference game Lewis (1969) in
Figure 1, where the goal of the speaker is to disambiguate a target image from
distractor images. A naive LM may generate the literal description “the red
shape”; this is semantically accurate, but fails to disambiguate the referent
among its context. A more pragmatically useful description is “the red
circle”.

(a) Communication-based objectives, where a speaker is trained with a listener
reward model.

(b) Traditional LM objectives, where a speaker is trained from a corpus of
image descriptions.

Figure 1: The goal of the speaker (blue agent) is to generate context-aware
utterances to refer to the image 1 and disambiguate from images 2 & 3\. Our
work uses (a) communication-based training objectives rather than (b) non
communication-based objectives which rely on a corpus of ground-truth
captions.

Prior work in training communicative NLP systems has used explicit models of
pragmatics Goodman and Frank (2016) to fine-tune traditional LMs Monroe et al.
(2017); Andreas and Klein (2016); Vedantam et al. (2017), or has trained LMs
with external reward signals that indicate the contextual utility of an
utterance. Human preferences are an ideal source of supervision for the latter
approach Stiennon et al. (2020); Ziegler et al. (2020), but are expensive to
collect. One promising avenue is communication-based training (or _self
play_), where a speaker learns to communicate with a _learned model_ of a
listener Lazaridou et al. (2017); White et al. (2020). However, this approach
commonly suffers from the problem of _semantic drift_. Our use of “semantic
drift” follows the communication-based training literature Lazaridou et al.
(2020); Lee et al. (2019): the speaker produces utterances that satisfy the
listener but diverge from the semantics and conventions of natural language.

It can be difficult to diagnose the nature of semantic drift in complex, open
domain models. In this paper, we isolate one form of semantic drift that
occurs when scaling communication-based training to larger domains: by
increasing the size of the vocabulary available to the speaker and listener,
the speaker fails to generalize to new listeners. We identify miscalibrated
listener uncertainty as the source of this problem: the listener is highly
confident in the interpretation of utterances outside its training domain, so
the speaker overfits and produces nonsensical utterances that fail to
generalize. We propose to correct the calibration problem by using
_populations_ of listeners, drawing from both computational and cognitive
science work that suggests deficiencies in just training with a single
listener Graesser et al. (2020); Raviv et al. (2019); Wagner et al. (2003) and
improvements in emergent communication protocols by regularization effects of
ensembled models Li and Bowling (2019); Tieleman et al. (2019). We find that
ensemble-based populations of listeners are better calibrated, helping
speakers avoid semantic drift and generalize to new games and listeners.

##  2 Approach

We study the problem of learning a pragmatic speaker for reference games with
the ShapeWorld Kuhnle and Copestake (2017) dataset. A reference game
$(\mathbf{I},t)$ consists of $n$ images $\mathbf{I}=(i_{1},\dots,i_{n})$ and a
target image $i_{t}$, with the index $t$ known only to the speaker. The
speaker $S$ must produce an utterance $u$ which allows the listener $L$ to
identify the target $t$ given the images.

Formally, given $u$, the listener $L$ is a distribution over possible targets
in a reference game:

$L(t\mid u)\propto\exp(f_{\theta}(i_{t})^{\top}g_{\theta}(u))$ (1)

where $f_{\theta}$ and $g_{\phi}$ are the listener’s image and language
encoders, respectively.

The speaker is then trained to produce an utterance for the listener given a
game and desired target. Specifically, $S$ is parameterized by a gated
recurrent neural network (GRU) Cho et al. (2014):

$S(u\mid\mathbf{I})=p_{\text{GRU}}(u|f_{\eta}(i_{t}),f_{\eta}(i_{1})\dots,f_{\eta}(i_{n-1})).$
(2)

where $f_{\eta}$ is the speaker’s image encoder used to initialize the GRU
hidden state. The speaker is trained to maximize the listener’s probability of
selecting the correct target given its utterance. Formally, the speaker’s loss
over a game $G$ is

$\mathcal{L}(S;L,\mathbf{I},t)=-\log L(t\mid\hat{u}),\,\hat{u}\sim
S(u\mid\mathbf{I}).$ (3)

This informative communication objective has been used several different ways
in the past. Rational Speech Act models of pragmatic language use Goodman and
Frank (2016) adopt it as a definition of speaker behavior, rather than an
objective for training. Studies of emergent multi-agent communication
Lazaridou et al. (2017) use this objective to jointly train the speaker and
listener, or alternatively train a pragmatic speaker against a fixed listener
White et al. (2020); Lazaridou et al. (2020). We adopt the setting of White et
al. (2020): we first train listener models on separate splits of ShapeWorld
reference games, then train pragmatic speakers with the fixed listener models
as the communication objective. The listeners and speakers do not share data
splits among each other. Training listener(s) are used as the internal
listener model for the speaker in Equation 3, and other validation listeners
are held-out for speaker evaluation. All the speakers and listeners are
trained on 15000 randomly generated reference games with a single target and 2
distractor images. More details on the model architecture, training, data, and
the supplementary code can be found in Appendices A and B.

Table 1: Average speaker accuracy $\pm$ standard error (middle 4 columns) and speaker-generated tokens overlapping with ShapeWorld tokens at test time (left-most column, “token overlap”) in % averaged over 10 seeds. Accuracy is reported with either the training or validation listener (train $L$, val $L$), and on either training or validation set (train $\mathcal{D}$, val $\mathcal{D}$). Gray cells indicate the speaker unable to generalize to new listeners and emitting novel tokens.  | token space | train $L$-train $\mathcal{D}$ | train $L$-val $\mathcal{D}$ | val $L$-train $\mathcal{D}$ | val $L$-val $\mathcal{D}$ | token overlap
---|---|---|---|---|---|---
| small | 99 $\pm$ 1.40 | 90 $\pm$ 1.40 | 80 $\pm$ 1.40 | 80 $\pm$ 1.40 | 100 $\pm$ 0.0
large | 98 $\pm$ 0.00 | 94 $\pm$ 0.01 | 37 $\pm$ 0.06 | 36 $\pm$ 0.06 | 1 $\pm$ 0.6

##  3 The problem of semantic drift

White et al. (2020) found successful pragmatic language production from a
model trained to communicate informatively in the ShapeWorld domain. We first
show that this success hinges on the restricted domain of utterances used. We
consider two settings where the speaker learns to select utterances for two
listeners: both are trained on the ground-truth ShapeWorld data, but one only
has access to the ShapeWorld vocabulary (15 tokens), and one has access to the
entire GPT-2 vocabulary (51k tokens; Radford et al. 2019). Given either
listener, we train the speaker using Equation 2. We chose GPT-2 vocabulary, a
byte-pair encoding (BPE) Sennrich et al. (2016), to access a large
tokenization space as we hope to extend communication-based training to using
GPT-2 as the generation model. Since BPE is the typical choice for most large
pretrained language models, it is representative of large vocabulary spaces
and introduces the challenges we might encounter in communication-based
training for large speakers.

In Table 1, we report speaker communication accuracy with either the listener
used for speaker training (train $L$) or a separately trained held-out
listener (val $L$), and either reference games seen during training (train
$\mathcal{D}$) or held-out games (val $\mathcal{D}$). To measure the extent
the speaker uses domain-relevant vocabulary, we additionally report the
percentage of speaker-generated tokens that overlap with ShapeWorld captions,
such as “blue circle” or “red square”.

The results reveal two noticeable discrepancies between the small and large
vocabulary settings. First, speakers trained over the large vocabulary space
can successfully communicate with the training listener on new reference games
– that is, they generalize to new contexts. However, they are unable to
generalize to held-out listeners and achieve only random chance accuracy
($33\%$). This pattern of overfitting to the training listener despite
generalizing to new contexts with the training listener indicates that the
speaker suffers a form of semantic drift Lazaridou et al. (2020); Lee et al.
(2019).

This semantic drift is shown most clearly in the second discrepancy: speakers
trained with large vocabularies rarely use tokens related to shapes and colors
(i.e. the domain-relevant tokens), whereas those trained with small
vocabularies do so (by design). Upon qualitative examination of the speaker’s
utterances (Table 2), we observe some examples of this drift: the speaker’s
language deviates from natural language and the ShapeWorld domain that the
neural listeners were trained on.

##  4 Diagnosing semantic drift

We hypothesize that the observed semantic drift arises from poor uncertainty
calibration in the neural listener, which the speaker then overfits to. If
this is correct, then populations of listeners—which have been shown to be
better calibrated Beluch et al. (2018); Li and Bowling (2019); Tieleman et al.
(2019)—may yield more useful listeners for communication-based training. To
investigate this hypothesis we first explore the calibration of both single
listeners and populations of listeners.

We compare the uncertainty measurements of three different internal listener
implementations. One baseline is single-$\mathbf{L_{0}}$, which approximates
the internal listener $L$ with a single neural listener, i.e. exactly the
setup in Section 3 where existing work uses a single neural listener for
communication-based training. The second is an ensemble-based population of
$n$ neural listeners; this is equivalent to substituting $L(t\mid
u)=\frac{1}{n}\sum_{j=1}^{n}L^{j}(t\mid u)$) in Equation 3 where each listener
$L^{j}$ is initialized randomly. They are trained on different data splits and
different random seeds, and each achieves at least 92% accuracy on training
and validation. While all listeners generalize to new in-distribution
utterances, they generalize differently out-of-distribution. It is these
disagreements among the ensemble that improve calibration compared to a single
neural network like single-$L_{0}$. The third is the dropout-based population,
where we use a single listener to approximate a population of $n$ listeners
via MC-dropout Gal and Ghahramani (2016); this is equivalent to substituting
$L(t|u)=\frac{1}{n}\sum_{j=1}^{n}L^{\texttt{dropout}}(t\mid u,d_{j})$ in
Equation 3 where $d_{j}$ is a randomly sampled dropout mask. In our
experiments, we use a dropout rate of $p=0.1$ but found no differences with
using larger rates while maintaining high listener accuracy; please refer to
Appendix B for more details on the dropout implementation.

Figure 2: Listener uncertainty (entropy) by domain overlap. Scores come from
either single $L_{0}$, dropout-based population, or an ensemble-based
population. Dashed line is an idealized listener, for reference.

(a)  Ensemble-based population training

(b)  Dropout-based population training

Figure 3: Accuracy and standard error for speaker trained with a (a) listener ensemble of size $n=\\{1,10,20,30\\}$ or (b) dropout listener with $n=\\{1,10,20,30\\}$ number of passes. $n$ represents the population size. Results averaged over 10 seeds.  Target |  Distractors
---|---
|
Method
gt | red shape
single $L_{0}$ | suppressed suppressed Mix
| suppressed Pagan Mix Mix
| Pagan Mix
dropout | suppressed suppressed
| suppressedops suppressed Mix
| Pagan imprison Mix
ensemble | rect rect rect rect
Target |  Distractors
|
Method
gt | gray shape
single $L_{0}$ | ESEESE FDAWHAT
| Macedichickarios
| Macedichick
dropout | ariosichickarios FDA FDAarios
| brainsichick Maced
ensemble | gray gray gray
Table 2: Examples of generated utterances. “gt” refers to the ground truth
ShapeWorld utterance associated to the target image (left-most image). We show
the utterance generated by the speaker trained with either the single $L_{0}$,
dropout-based population, or ensemble-based population method. Based on token
overlap and the reference game context, the ensemble-based population method
results in more pragmatic and useful utterances.

Intuitively, the ideal listener should be uncertain about utterances that are
different from their training domain. To measure this, we evaluate listener
uncertainty scores on utterances with varying degrees of overlap with their
training domain. Specifically, for high-overlap utterances, we sample messages
directly from the ground truth ShapeWorld language. For medium-overlap
utterances, we sample language from a well-calibrated neural speaker (Table 2,
ensemble). For low-overlap utterances, we sample from a poorly-calibrated
speaker (Table 2, single $L_{0}$). Refer to Appendix C for a complete
discussion on utterance sampling. For these utterance types, the ideal
listener should have maximum entropy over targets for low-overlap utterances,
and zero entropy for high-overlap utterances. In Figure 2, we plot the
listener’s uncertainty over utterances with varying domain overlap. We see, as
hypothesized, that the single listener is poorly calibrated, yielding very low
entropy for low-overlap utterances. Somewhat surprisingly, dropout-based
populations do not help either. The ensemble-based populations, however, do
show better calibration and are closer to the idealized listener: they are
uncertain about out-of-domain utterances.

##  5 Countering semantic drift

Following our hypothesis, a well-calibrated listener should lead the speaker
to generalize to new listeners and new contexts while reasoning pragmatically
over a large vocabulary. We show that this is indeed the case in the following
two experiments.

First, we show that with increasing population size $n$, the ensemble-based
population objective closes the accuracy gap between training and validation
listeners, while still generalizing to new reference games (Figure 3). This is
not the case for the dropout listener with increasing the number of passes
$n$. These results are averaged on 10 seeds, for $n$ of $\\{1,10,20,30\\}$.
With ensemble-based population $n=30$, we found that the speaker token overlap
increased to $48.4\pm 12.3\%$; this is a much larger token overlap than the
original speaker trained on single-$L0$ ($1.0\pm 0.6$ from Table 1). We show
utterance examples from speakers trained with the population objectives in
Table 2.

| speaker type |  $d(z_{\text{GT}}$, SUM$(z_{\text{S}}))$ |  $d(z_{\text{GT}}$, FIRST$(z_{\text{S}}))$
---|---|---|---
| Limited | 9.2 $\pm$ 3.74 | 6.43 $\pm$ 2.54
Calibrated | 10.37 $\pm$ 2.41 | 8.52 $\pm$ 0.50
Miscalibrated | 13.03 $\pm$ 3.62 | 8.72 $\pm$ 0.44
Table 3:  Euclidean distance $d$ in GloVe embedding space between the ground
truth utterances $z_{\text{GT}}$ and speaker utterances $z_{\text{S}}$. Lower
distances indicate that the speaker’s utterance is close in topicality to the
ground truth. The speakers are either trained only on the Shapeworld
vocabulary (Limited), trained with an ensemble-based listener population
(Calibrated), or trained with single-$L0$ (Miscalibrated). The speaker
utterance embeddings are either calculated as a sum of the uttered words
(SUM$(z_{\text{S}})$) or as the embedding of the first uttered word
(FIRST$(z_{\text{S}})$).

To explore the semantic content of utterance distributions, we investigate the
topic overlap between different speakers and the ground truth utterances. We
find that speakers trained with well-calibrated listeners produce messages
which are semantically similar to the ground truth utterances, according to
distances in GloVe Pennington et al. (2014) embedding space (Table 3). We
embed the ground truth utterances by taking the sum of embeddings over the
ground truth utterance. We denote this sum as $z_{\text{GT}}$. We compare
these to the utterance embeddings of three speakers: one speaker which is
constrained to Shapeworld vocabulary (used in White et al. (2020) and in
Section 3), one that is trained with well-calibrated listeners (ensemble-based
population of size 30), and another that is trained with the poorly calibrated
single-$L0$ listener. The speaker’s embeddings, $z_{{\text{S}}}$, are
calculated either as a sum of uttered word embeddings or as the embedding of
the first utterance word; we include the results from the latter because
miscalibrated speakers often repeat words and taking only the sum might
distort the calculated embedding distance. The distance reported in Table 3 is
the Euclidean distance between $z_{\text{GT}}$ and $z_{{\text{S}}}$. The
results show that the calibrated speaker utters words that are similar to
ground truth utterances, though not quite as similar as the limited speaker.
The miscalibrated speaker utters words that are less similar to the ground
truth utterances as indicated by the higher distance scores.

##  6 Conclusion

The true objective for language use is communication. Our work highlights the
importance of well-calibrated listeners in communication-based training. We
show that it’s important to understand the properties of speakers optimized to
communicate with different listeners. We have found that naive communication-
based training over unconstrained vocabularies is subject to a pernicious form
of semantic drift that arises from poorly calibrated listener models. The
overconfidence of listeners is exploited by the speaker during training and,
as a result, speakers acquire niche linguistic properties, like conventions
that fail to generalize to other listeners Hawkins et al. (2017); Graesser et
al. (2020). By contrast, an ensemble of listeners shows better calibration.
Speakers trained to communicate with these listeners avoid semantic drift,
generalizing to new games and new listeners. Future research on communication-
based training for language models will thus benefit from listener ensembles
or other methods for training listeners with properly calibrated uncertainty.

##  7 Acknowledgements

REW and JM are supported by the National Science Foundation Graduate Research
Fellowship. This work was supported in part by the Stanford HAI Hoffman–Yee
project ‘Towards grounded, adaptive communication agents.’ We thank CoCoLab
and the anonymous reviewers for their helpful feedback on the paper, and Ke
Alexander Wang for helpful input on Figure 1.

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##  Appendix A Software and data for reproducibility

The accompanying code can be found here:
https://github.com/rosewang2008/calibrate_your_listeners. It includes code for
training the models, instructions, the data used for this work, and pretrained
$L$ models.

##  Appendix B Implementation and training

### Model details

The listener and speaker have a separate vision encoder, $f_{\theta}$ and
$f_{\phi}$, which are convolutional neural networks with 4 blocks, each block
containing a 64-filter 3x3 convolutional layer, batch normalization, ReLU
activation, and 2x2 max pooling. Images are represented as 64×64×3 input,
which produce 1024-d representations. For speakers, the image embeddings are
projected down to the GRU hidden state size for hidden state initialization in
$p_{\text{GRU}}$. For listeners, we take the dot product of the image
embeddings and utterance embeddings to produce the target probability. The
listener’s language encoder $g(\cdot)$ is also a gated RNN. The listener and
speaker’s GRU has two layers. It takes in an embedding dimension of 512 and
has a hidden size of 100. The token space varies in our experiments: the small
vocabulary setting has 15 tokens as in White et al. (2020) and the large
setting has about 51k tokens from the GPT2 tokenizer Wolf et al. (2020);
Radford et al. (2019). The speaker’s maximum sequence length is 10 tokens.

For the dropout-based listener, we apply a dropout layer with dropout rate
$p=0.1$ in the listener’s image and language encoders,
$f_{\theta},g_{\theta}$. Dropout is applied after every convolutional block
except the last block in the image encoder. Dropout is applied on the outputs
of each GRU layer except for the last layer. We experimented with higher
dropout rates ($p=0.2,0.3$) while maintaining high listener validation
accuracy during listener evaluation, and found no differences in the reported
experimental findings. Specifically, we found that using listeners with higher
dropout rates did not change the uncertainty measurements shown in Figure 2.
Speakers trained with listeners of higher dropout rates still obtained
training and validation accuracies that were still within the margin of error
reported in Figure 3.

The listeners in the ensemble-based population do not share any encoders and
are trained on separate models.

### Training

All the models were trained on 100 epochs, with the Adam optimizer, a batch
size of 32, using the Gumbel Softmax trick Jang et al. (2017). The learning
rate is 0.001 for speakers and 0.01 for listeners. All the listeners after
training reach at least 92% validation accuracy, and are used for speaker
training or validation.

##  Appendix C Domain overlap in utterances

Intuitively, we want utterances with low domain overlap to be utterances that
are unrelated ShapeWorld domain (e.g. those sampled from the speaker trained
with the single-$L_{0}$ method as shown in Table 2), and utterances with high
domain overlap to be captions correct for a given target image and perfectly
related to the ShapeWorld domain and correct (e.g. those directly from the
ShapeWorld dataset). Utterances with medium domain overlap should be an
interpolation between low and high domain overlap. In our setting that is
concerned with semantic language drift, we define overlap to be with respect
to the percentage of utterance tokens that are ShapeWorld-related tokens.

We use the ShapeWorld image captions as the high domain overlap because these
captions consist of only tokens related to the ShapeWorld setting, i.e.
$100\%$ overlap. We use utterances sampled from the speaker trained with
ensemble-based population for the medium overlap utterances; these utterances
generally use ShapeWorld-related tokens but in repeated fashion (see Table 2
for an example) and have $51.6\%$ overlap. We use utterances sampled from the
speaker trained with the single-$L_{0}$ method with $0\%$ token overlap for
the low domain overlap, as we found that this condition applied to a wide
majority of utterances sampled from this speaker. We note that using token
overlap as a proxy for domain overlap is an imperfect measurement as it
doesn’t factor in whether an utterance is actually appropriate for a desired
target image. For example, an utterance like “red green shape” uses
ShapeWorld-related tokens, but may not be an ideal utterance for a red-shape
target image. Nonetheless, since our work is concerned with semantic language
drift that uses novel and out-of-domain tokens, we found this to be the best
measurement of domain overlap compared to alternatives like n-gram overlap in
tokens.


#  Two ${}^{9}\mathrm{Li}$ clusters connected with two valence neutrons in
${}^{20}\mathrm{C}$

Naoyuki Itagaki  Tokuro Fukui  Yukawa Institute for Theoretical Physics, Kyoto
University, Kitashirakawa Oiwake-Cho, Kyoto 606-8502, Japan  Junki Tanaka
RIKEN Nishina Center for Aaccelerator-Based Science, 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan  Yuma Kikuchi  Tokuyama College of Technology,
Gakuendai, Shunan, Yamaguchi 745-8585, Japan

###### Abstract

Many preceding works have shown in ${}^{11}\mathrm{Li}$ the presence of the
halo structure comprised of the weakly bound two neutrons around
${}^{9}\mathrm{Li}$, and it is intriguing to see how this halo structure
changes when another ${}^{9}\mathrm{Li}$ approaches. In this study, we
introduce a four-body model for ${}^{20}\mathrm{C}$ with two
${}^{9}\mathrm{Li}$ clusters and two valence neutrons. The recent development
of the antisymmetrized quasi cluster model (AQCM) makes it possible to
generate $jj$-coupling shell-model wave functions from $\alpha$ cluster
models. Here, $jj$-coupling shell model wave function of ${}^{9}\mathrm{Li}$
is regarded as a cluster, which corresponds to the subclosure configuration of
$p_{3/2}$ for the neutrons, and we discuss how the two neutrons connect two
${}^{9}\mathrm{Li}$ clusters. Until now, most of the clusters in the
conventional models have been limited to the closures of the three-dimensional
harmonic oscillators, such as ${}^{4}\mathrm{He}$, 16O, and 40Ca; however,
owing to AQCM, it is feasible to utilize the $jj$-coupling shell model wave
functions as plural subsystems quite easily. The appearance of a rotational
band structure with a cluster structure around the four-body threshold energy
is discussed.

##  I Introduction

The ${}^{4}\mathrm{He}$ nuclei have been known to have large binding energy
(28.3 MeV) distinctly in the light mass region. In contrast, it is known that
the relative interaction between ${}^{4}\mathrm{He}$ nuclei is weak.
Therefore, they can serve as subsystems in some of the light nuclei, called
$\alpha$ cluster structure Brink (1966); Freer  _et al._ (2018). The $\alpha$
cluster structure has been studied for decades, and one of the most famous
examples is the second $0^{+}$ state of ${}^{12}\mathrm{C}$ with a developed
three-$\alpha$ cluster structure Hoyle (1954); Freer and Fynbo (2014), which
is called the Hoyle state. Many cluster models have proven to be capable of
describing various properties of the Hoyle state Fujiwara  _et al._ (1980);
Tohsaki  _et al._ (2001).

In most of the conventional cluster models, however, only the states
corresponding to the closure of the three-dimensional harmonic oscillator,
such as ${}^{4}\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$,
have been treated as subsystems called clusters. In these cases,
unfortunately, the contribution of the non-central interactions (spin-orbit
and tensor interactions), which are quite important in the nuclear systems,
vanishes after the antisymmetrization of the wave functions. This is the
consequence of the fact that the closure configurations of the major shells
are spin-zero systems, and the non-central interactions do not contribute to
such spinless systems. In actual nuclear systems, on the contrary, the
contribution of the spin-orbit interaction is essential, which enables to
explain the observed magic numbers; the subclosure configurations of the
$jj$-coupling shell model ($f_{7/2}$, $g_{9/2}$, and $h_{11/2}$) correspond to
the observed magic numbers of $28$, $50$, and $126$ Mayer and Jensen (1955).
If we enlarge the model space of the cluster models and open the path to
another symmetry, indeed this spin-orbit interaction works as a driving force
to break the $\alpha$ clusters, for instance in ${}^{12}\mathrm{C}$ Itagaki
_et al._ (2004).

Therefore, the important task is to include the spin-orbit contribution by
extending the traditional cluster models; we proposed the antisymmetrized
quasi cluster model (AQCM) Itagaki (2016); Itagaki  _et al._ (2006a); Masui
and Itagaki (2007); Yoshida  _et al._ (2009); Itagaki  _et al._ (2011); Suhara
_et al._ (2013); Itagaki  _et al._ (2016); Matsuno  _et al._ (2017); Matsuno
and Itagaki (2017); Itagaki and Tohsaki (2018); Itagaki  _et al._ (2018,
2020a, 2020b). This method allows the smooth transformation of the $\alpha$
cluster model wave functions to $jj$-coupling shell model ones. We call the
clusters that feel the spin-orbit contribution after this transformation quasi
clusters. The conventional $\alpha$ cluster models cover the model space of
closure of major shells (corresponding to the magic numbers of 2, 8, and 20),
and now the subclosure configurations of the $jj$-coupling shell model,
$p_{3/2}$, $d_{5/2}$, $f_{7/2}$, $g_{9/2}$ $etc.$, are covered by our AQCM
Itagaki  _et al._ (2016).

The achievement of AQCM allows us to use $jj$-coupling shell model wave
functions as subsystems of the nuclei, which is the beginning of a new cluster
model. At first, we have shown the possibility of ${}^{14}\mathrm{C}$,
${}^{6}\mathrm{He}$, and ${}^{9}\mathrm{Li}$ as clusters Itagaki  _et al._
(2020a, b), where subclosure configuration of $p_{3/2}$ plays an important
role. The cluster structures of ${}^{16}\mathrm{Be}$
(${}^{8}\mathrm{He}$+${}^{8}\mathrm{He}$), ${}^{17}\mathrm{B}$
(${}^{8}\mathrm{He}$+${}^{9}\mathrm{Li}$), ${}^{18}\mathrm{C}$
(${}^{9}\mathrm{Li}$+${}^{9}\mathrm{Li}$), ${}^{24}\mathrm{C}$ (three
${}^{8}\mathrm{He}$), and ${}^{42}\mathrm{Ar}$ (three ${}^{14}\mathrm{C}$),
have been investigated around the corresponding threshold energies.

In this study, we further add neutrons and discuss in ${}^{20}\mathrm{C}$ the
${}^{9}\mathrm{Li}$+${}^{9}\mathrm{Li}$ cluster configuration as an example.
It has been known in ${}^{11}\mathrm{Li}$ that the two neutrons have neutron
halo structure around ${}^{9}\mathrm{Li}$ Tanihata  _et al._ (1985). It is
intriguing to see how such structure is affected when another
${}^{9}\mathrm{Li}$ approaches to the halo neutrons. In the Be isotopes, it
has been extensively discussed that neutrons perform molecular-orbital motion
around $\alpha$ clusters; very developed $\alpha$-$\alpha$ cluster structure
appears when two neutrons occupy the $\sigma$ orbit Itagaki and Okabe (2000);
Ito  _et al._ (2008). This idea can be extended to the linear-chain structure
of three $\alpha$ clusters in the C isotopes Itagaki  _et al._ (2001); Zhao
_et al._ (2015). If the neutrons in the halo state perform molecular orbital
motion around two ${}^{9}\mathrm{Li}$ clusters with large distances, it paves
the way to a novel binding mechanism in the excited states of the neutron-rich
nuclei.

This paper is organized as follows. The framework is described in Sec. II. The
results are shown in Sec. III. The conclusions are presented in Sec. IV.

##  II framework

###  II.1 Basic feature of AQCM

AQCM allows the smooth transformation of the cluster model wave functions to
the $jj$-coupling shell model ones. In AQCM, each single particle is described
by a Gaussian form as in many other cluster models including the Brink model
Brink (1966),

$\phi^{\tau,\sigma}\left(\bm{r}\right)=\left(\frac{2\nu}{\pi}\right)^{\frac{3}{4}}\exp\left[-\nu\left(\bm{r}-\bm{\zeta}\right)^{2}\right]\chi^{\tau,\sigma},$
(1)

where the Gaussian center parameter $\bm{\zeta}$ is related to the expectation
value of the position of the nucleon, and $\chi^{\tau,\sigma}$ is the spin-
isospin part of the wave function. For the size parameter $\nu$, here we use
$\nu=0.20\,\mathrm{fm}^{-2}$, which gives the optimal $0^{+}$ energy of
${}^{12}\mathrm{C}$ within a single AQCM basis state. The Slater determinant
is constructed from these single-particle wave functions by antisymmetrizing
them.

Next, we focus on the Gaussian center parameters
$\left\\{\bm{\zeta}_{i}\right\\}$. As in other cluster models, here four
single-particle wave functions with different spin and isospin sharing a
common $\bm{\zeta}$ value correspond to an $\alpha$ cluster. This cluster wave
function is transformed into $jj$-coupling shell model based on the AQCM. When
the original value of the Gaussian center parameter $\bm{\zeta}$ is $\bm{R}$,
which is real and related to the spatial position of this nucleon, it is
transformed by adding the imaginary part as

$\bm{\zeta}=\bm{R}+i\Lambda\bm{e}^{\text{spin}}\times\bm{R},$ (2)

where $\bm{e}^{\text{spin}}$ is a unit vector for the intrinsic-spin
orientation of this nucleon. The control parameter $\Lambda$ is associated
with the breaking of the cluster, and with a finite value of $\Lambda$, the
two nucleons with opposite spin orientations have the $\bm{\zeta}$ values,
which are complex conjugate with each other. This situation corresponds to the
time-reversal motion of two nucleons. After this transformation, the $\alpha$
clusters are called quasi clusters. We can generally create the single-
particle orbits of the $jj$-coupling shell model by taking the limits of
$\bm{R}\to 0$ and $\Lambda\to 1$.

###  II.2 Wave function for ${}^{20}\mathrm{C}$

The total wave function for ${}^{20}\mathrm{C}$ is the superposition of
different Slater determinants, $\\{\Phi_{i}\\}$,

$\Psi_{J^{\pi}}=\sum_{i,K}c^{K}_{i}P_{J^{\pi}}^{K}\Phi_{i}.$ (3)

All Slater determinants are projected to the eigen states of parity and
angular momentum by using the projection operator $P_{J^{\pi}}^{K}$,

$P_{J^{\pi}}^{K}=P^{\pi}\frac{2J+1}{8\pi^{2}}\int
d\Omega\,{D_{MK}^{J}}^{*}R\left(\Omega\right).$ (4)

Here ${D_{MK}^{J}}$ is the Wigner $D$-function and $R\left(\Omega\right)$ is
the rotation operator for the spatial and spin parts of the wave function.
This integration over the Euler angle $\Omega$ is numerically performed. The
operator $P^{\pi}$ is for the parity projection
($P^{\pi}=\left(1+P^{r}\right)/\sqrt{2}$ for the positive-parity states, where
$P^{r}$ is the parity-inversion operator), which is also performed
numerically. The coefficients $\left\\{c^{K}_{i}\right\\}$ are obtained
together with the energy eigenvalue $E$ when we diagonalize the norm and
Hamiltonian ($H$) matrices, namely by solving the Hill-Wheeler equation.

$\displaystyle\sum_{i,j,K,K^{\prime}}$
$\displaystyle(<\Phi_{i}|(P_{J^{\pi}}^{K^{\prime}})^{\dagger}HP_{J^{\pi}}^{K}|\Phi_{j}>$
(5)
$\displaystyle-E<\Phi_{i}|(P_{J^{\pi}}^{K^{\prime}})^{\dagger}P_{J^{\pi}}^{K}|\Phi_{j}>)c^{K}_{j}=0.$

This angular momentum projection enables to generate different $K$ number
states as independent basis states from each Slater determinant.

Each Slater determinant consists of the antisymmetrized product of single-
particle wave functions.

$\displaystyle\Phi_{i}={\cal A}\\{$
$\displaystyle\phi^{\tau_{1},\sigma_{1}}\left(\bm{r}_{1},\bm{\zeta}_{1}\right)\phi^{\tau_{2},\sigma_{2}}\left(\bm{r}_{2},\bm{\zeta}_{2}\right)\cdots$
(6)
$\displaystyle\cdots\phi^{\tau_{19},\sigma_{19}}\left(\bm{r}_{19},\bm{\zeta}_{19}\right)\phi^{\tau_{20},\sigma_{20}}\left(\bm{r}_{20},\bm{\zeta}_{20}\right)\\}_{i}.$

Here, the single-particle wave functions from
$\phi^{\tau_{1},\sigma_{1}}\left(\bm{r}_{1},\bm{\zeta}_{1}\right)$ to
$\phi^{\tau_{9},\sigma_{9}}\left(\bm{r}_{9},\bm{\zeta}_{9}\right)$ belong to
one ${}^{9}\mathrm{Li}$ cluster, whereas those from
$\phi^{\tau_{10},\sigma_{10}}\left(\bm{r}_{10},\bm{\zeta}_{10}\right)$ to
$\phi^{\tau_{18},\sigma_{18}}\left(\bm{r}_{18},\bm{\zeta}_{18}\right)$ are for
another ${}^{9}\mathrm{Li}$. For each ${}^{9}\mathrm{Li}$ cluster, we
introduce the subclosure configuration of $(s_{1/2})^{2}(p_{3/2})^{4}$ for the
neutrons. For the proton part, the last protons in two ${}^{9}\mathrm{Li}$
clusters are introduced as time-reversal configurations, $\left|3/2\
3/2\right\rangle$ and $\left|3/2\ -3/2\right\rangle$. Then these two
${}^{9}\mathrm{Li}$ clusters are separated with the relative distance of
$d_{i}$, which is randomly generated with equal probability between 0.5 fm and
5.0 fm.

The two valence neutrons,
$\phi^{\tau_{19},\sigma_{19}}\left(\bm{r}_{19},\bm{\zeta}_{19}\right)$ and
$\phi^{\tau_{20},\sigma_{20}}\left(\bm{r}_{20},\bm{\zeta}_{20}\right)$, are
introduced to have opposite spin directions (spin-up and spin-down). Their
Gaussian center parameters ($\bm{\zeta}_{19}$ and $\bm{\zeta}_{20}$) are
generated by using random numbers $\left\\{r_{k}\right\\}$, which have the
probability distribution $P\left(\left|r_{k}\right|\right)$ proportional to
$\exp\left[-\left|r_{k}\right|/\sigma\right]$,

$P\left(\left|r_{k}\right|\right)\propto\exp\left[-\left|r_{k}\right|/\eta\right].$
(7)

The value of $\eta$ is chosen to be $1.5\,\mathrm{fm}$ which corresponds to
the standard deviation of 3.67 fm. After generating $\left\\{r_{k}\right\\}$,
we multiply the sign factor to each $r_{k}$, which allows $r_{k}$ to be
positive and negative with equal probability. The resultant random numbers are
applied to all three ($x$, $y$, $z$) components of the Gaussian center
parameters of the two valence neutrons.

###  II.3 Hamiltonian

The Hamiltonian consists of the kinetic energy and potential energy terms. For
the potential part, the interaction consists of the central
($\hat{V}_{\text{central}}$), spin-orbit ($\hat{V}_{\text{spin-orbit}}$), and
Coulomb terms. For the central part, the Tohsaki interaction Tohsaki (1994) is
adopted. This interaction has finite ranges for the three-body terms in
addition to two-body terms, which is designed to reproduce both saturation
properties and scattering phase shifts of two $\alpha$ clusters. For the spin-
orbit part, we use the spin-orbit term of the G3RS interaction Tamagaki
(1968), which is a realistic interaction originally developed to reproduce the
nucleon-nucleon scattering phase shifts.

The Tohsaki interaction consists of two-body ($V^{\text{(2)}}$) and three-body
($V^{\text{(3)}}$) terms:

$\hat{V}_{\text{central}}=\frac{1}{2}\sum_{i\neq
j}V^{\text{(2)}}_{ij}+\frac{1}{6}\sum_{i\neq j,j\neq k,i\neq
k}V^{\text{(3)}}_{ijk},$ (8)

where $V^{\text{(2)}}_{ij}$ and $V^{\text{(3)}}_{ijk}$ have three ranges,

$\displaystyle V^{\text{(2)}}_{ij}=$
$\displaystyle\,\sum_{\alpha=1}^{3}V^{\text{(2)}}_{\alpha}\exp\left[-\frac{\left(\vec{r}_{i}-\vec{r}_{j}\right)^{2}}{\mu_{\alpha}^{2}}\right]\left(W^{\text{(2)}}_{\alpha}-M^{\text{(2)}}_{\alpha}P^{\sigma}P^{\tau}\right)_{ij},$
(9) $\displaystyle V^{\text{(3)}}_{ijk}=$
$\displaystyle\,\sum_{\alpha=1}^{3}V^{\text{(3)}}_{\alpha}\exp\left[-\frac{\left(\vec{r}_{i}-\vec{r}_{j}\right)^{2}}{\mu_{\alpha}^{2}}-\frac{\left(\vec{r}_{i}-\vec{r}_{k}\right)^{2}}{\mu_{\alpha}^{2}}\right]$
$\displaystyle\times\left(W_{\alpha}^{\text{(3)}}-M_{\alpha}^{\text{(3)}}P^{\sigma}P^{\tau}\right)_{ij}\left(W_{\alpha}^{\text{(3)}}-M_{\alpha}^{\text{(3)}}P^{\sigma}P^{\tau}\right)_{ik}.$
(10)

Here, $P^{\sigma}P^{\tau}$ represents the exchange of the spin-isospin part of
the wave functions of interacting two nucleons. The physical coordinate for
the $i$th nucleon is $\vec{r}_{i}$. The details of the parameters are shown in
Ref. Tohsaki (1994), but we use F1’ parameter set for the Majorana parameter
($M_{\alpha}^{\text{(3)}}$) of the three-body part introduced in Ref. Itagaki
(2016).

The G3RS interaction Tamagaki (1968) is a realistic interaction, and the spin-
orbit term has the following form;

$\hat{V}_{\text{spin-orbit}}=\frac{1}{2}\sum_{i\neq j}V^{ls}_{ij},$ (11)

where

$V^{ls}_{ij}=\left(V_{ls}^{1}e^{-d_{1}\left(\vec{r}_{i}-\vec{r}_{j}\right)^{2}}-V_{ls}^{2}e^{-d_{2}\left(\vec{r}_{i}-\vec{r}_{j}\right)^{2}}\right)P\left({}^{3}O\right){\vec{L}}\cdot{\vec{S}}.$
(12)

Here, $\vec{L}$ is the angular momentum for the relative motion between the
$i$th and $j$th nucleons, and $\vec{S}$ is the sum of the spin operator for
these two interacting nucleons. The operator $P\left({}^{3}O\right)$ stands
for the projection onto the triplet-odd state. The strength of the spin-orbit
interactions is set to $V_{ls}^{1}=V_{ls}^{2}=1800\,\mathrm{MeV}$, which
allows consistent description of ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$
Itagaki (2016).

##  III Results

###  III.1 Energy convergence of ${}^{20}\mathrm{C}$

Figure 1:  Energy curves for the $0^{+}$ states of ${}^{20}\mathrm{C}$
calculated with the ${}^{9}\mathrm{Li}$+${}^{9}\mathrm{Li}$+$n$+$n$ model.
Horizontal axis shows the number of Slater determinants superposed. Energies
are measured from the four-body threshold.

We start the discussion with the $0^{+}$ energy curves of ${}^{20}\mathrm{C}$
calculated with the present ${}^{9}\mathrm{Li}$+${}^{9}\mathrm{Li}$+$n$+$n$
model. The horizontal axis of Fig. 1 shows the number of Slater determinants
superposed, and for each Slater determinant, the distances between two
${}^{9}\mathrm{Li}$ ($d$) and the positions for the Gaussian center parameters
for the two valence neutrons ($\bm{\zeta}_{19}$ and $\bm{\zeta}_{20}$) are
randomly generated. The energies are measured from the four-body threshold.
This calculation is based on the bound state approximation and it could be
possible that some of the obtained states above the neutron threshold are
continuum states, but the states with flat energies after rapid convergence
are candidates for the resonance states. The lowest $0^{+}$ state converges to
$-31.4$ MeV, and experimentally, the ground state of ${}^{20}\mathrm{C}$ is
located at $-28.5$ MeV from the threshold. The obtained $0^{+}$ energies are
summarized in Table LABEL:properties.

###  III.2 Principal quantum number of the harmonic oscillator

Table 1:  Properties of $0^{+}$ states of ${}^{20}\mathrm{C}$ obtained by superposing Slater determinants. The energies are measured from the ${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}+n+n$ threshold ($E$ (MeV)). Expectation values of the principal quantum number for the proton part ($N$-proton) and neutron part ($N$-neutron) are listed together with those of the single-particle parity of the protons ($spp$-proton) and neutrons ($spp$-neutron). |  $E$ (MeV) |  $N$-proton |  $N$-neutron |  $spp$-proton |  $spp$-neutron
---|---|---|---|---|---
1 | $-31.37$ | $4.19$ | $18.61$ | $-1.90$ | $1.85$
2 | $-21.56$ | $4.43$ | $18.92$ | $-1.78$ | $1.87$
3 | $-20.34$ | $4.90$ | $19.82$ | $-1.56$ | $1.64$
4 | $-13.02$ | $4.83$ | $19.86$ | $-1.58$ | $1.06$
5 | $-10.75$ | $5.60$ | $21.41$ | $-1.27$ | $-0.79$
6 | $-8.03$ | $4.71$ | $19.62$ | $-1.59$ | $1.49$
7 | $-5.03$ | $5.68$ | $21.44$ | $-1.29$ | $1.04$
8 | $-2.96$ | $6.12$ | $21.79$ | $-0.35$ | $0.08$
9 | $-2.02$ | $5.77$ | $21.68$ | $-1.13$ | $1.15$
10 | $1.38$ | $5.96$ | $22.13$ | $-1.09$ | $0.89$

We discuss the property of each state obtained after superposing the basis
states. Particular focus is placed on the point which states have the
character of ${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}$ cluster configuration. One
of the physical quantities, which characterize each state, is the expectation
value of the principal quantum number $\hat{N}$ of the harmonic oscillator,

$\hat{N}=\sum_{i}\bm{a}^{\dagger}_{i}\cdot\bm{a}_{i}.$ (13)

The summation can be taken independently for the proton part and neutron part.
In Table LABEL:properties, the column “$N$-proton” stands for the expectation
value of $\hat{N}$ for the protons and “$N$-neutron” for the neutrons. The
lowest Pauli-allowed values are 4 for the protons (two are in the $s$-shell
and four are in the $p$-shell) and 18 for the neutrons (two are in the
$s$-shell, six are in the $p$-shell, and six are in the $sd$-shell). The
lowest $0^{+}$ state has the values of 4.19 (protons) and 18.61 (neutrons)
fairly close to the lowest Pauli-allowed values, which indicates that the
state has pure shell-model character. However, the excited states have much
larger values.

It should be stressed that the principal quantum number $N$ for the protons
($N$-proton) is governed by the distance between two ${}^{9}\mathrm{Li}$
clusters, since protons are only in the ${}^{9}\mathrm{Li}$ clusters and the
internal wave functions of each ${}^{9}\mathrm{Li}$ is frozen except for the
antisymmetrization effect. Thus, the $N$-proton values listed in Table
LABEL:properties is considered to contain information for the distance between
two ${}^{9}\mathrm{Li}$ clusters, although the expectation values are obtained
after superposing Slater determinants with different
${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$ distances. To extract the contribution
of the ${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}$ core part, in Fig. 2, we show the
$N$ values of ${}^{18}\mathrm{C}$ without the two valence neutrons as a
function of the distance between two ${}^{9}\mathrm{Li}$ clusters (Fig. 2(a):
protons and Fig. 2(b): neutrons). At the zero-limit for the distance $d$
between two ${}^{9}\mathrm{Li}$, the values are 4 (protons) and 14 (neutrons),
The values increase to 4.9 (protons) and 15.7 (neutrons) at $d=3$ fm, and they
grow to 5.6 and 16.9 at $d=3.5$ fm (they further increase to 6.6 and 18 at
$d=4$ fm).

According to Table LABEL:properties, the fifth state at $-10.75$ MeV, seventh
state at $-5.03$ MeV, eighth state at $-2.96$ MeV, ninth state at $-2.02$ MeV,
and tenth state at $1.38$ MeV have the $N$-proton values of $5.5-6.0$. Thus,
it is quite likely that these states have the cluster structure with the
${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$ distances of $3.5-4.0$ fm. Among them,
the eighth state at $-2.96$ MeV has the largest $N$-proton value of 6.12, and
we consider this state as a candidate for the cluster state representing these
states. The $N$-proton value of 6.12 corresponds to the
${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$ distances of $3.75$ fm in
${}^{18}\mathrm{C}$ without the valence neutrons (Fig. 2(a)). In our preceding
work for ${}^{18}\mathrm{C}$ Itagaki  _et al._ (2020b),
${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}$ cluster state has been shown to appear
slightly below the corresponding threshold energy. Therefore, it is rather
reasonable to find here the appearance of the cluster state in
${}^{20}\mathrm{C}$ after adding two valence neutrons to ${}^{18}\mathrm{C}$
below the four-body threshold. This state shows rapid energy convergence in
Fig 1, which is considered to be a possible resonance state.

If we assume that the $0^{+}$ state of ${}^{20}\mathrm{C}$ at $-2.96$ MeV,
which is the candidates for the cluster state, has the
${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$ distances of about 3.75 fm, the
neutrons in the ${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}$ core part must have the
$N$-neutron value around 17.75 (Fig. 2(b)). Therefore, we can deduce the
contribution of the two valence neutrons in the state at $-2.96$ MeV by
subtracting 17.75 from the $N$-neutron value of 21.79 listed in Table
LABEL:properties. The two valence neutrons are estimated to have the $N$ value
of about 4.0 in the cluster state, and each valence neutron shares the value
of $\sim$2.0. Thus, it turns out that valence neutrons remain in two-node
orbits as in the ground state.

Figure 2:  Principal quantum number $N$ of the harmonic oscillator for the
$0^{+}$ state of ${}^{18}\mathrm{C}$ as a function of the distance between two
${}^{9}\mathrm{Li}$ clusters. (a): protons and (b): neutrons.

###  III.3 Single-particle parity

To discuss the properties of the two valence neutrons in ${}^{20}\mathrm{C}$
in more detail, next we calculate the single-particle parity. In this
framework, the single-particle orbits introduced are non-orthogonal, and
therefore we cannot directly discuss the parity of each orbit. Another
difficulty of directly discussing the parity of each single-particle comes
from the superposition of Slater determinants performed here. Nevertheless, we
can get insights for the single-particle parity from the expectation value of
the single-particle operator, which is the sum of the parity inversion
operator for each nucleon Itagaki  _et al._ (2006b),

$\hat{O}^{spp}=\sum_{i}P^{r}_{i}.$ (14)

Here $P^{r}_{i}$ is the parity-inversion operator for the $i$-th nucleon. The
eigenvalues of $P^{r}_{i}$ is 1 and $-1$, for the positive-parity orbit and
negative-parity orbit, respectively. The summation can be taken over the
protons or neutrons independently.

We calculate the expectation value of this single-particle parity for the
$0^{+}$ states of ${}^{20}\mathrm{C}$ listed in Table LABEL:properties, which
are obtained by superposing Slates determinants. In Table LABEL:properties,
the column “$spp$-proton” is for the protons and “$spp$-neutron” is for the
neutrons. The ground state has the values of $-1.90$ (protons) and 1.85
(neutrons) indicating that the state has the character of the lowest shell-
model state. In the lowest shell-model state, two protons and two neutrons are
in the $s$-shell with positive-parity, four protons and six neutrons are in
the $p$-shell with negative-parity, and six neutrons are in the $sd$-shell
with positive-parity, and hence, the single-particle parity becomes $2-4=-2$
for the protons and $2-6+6=2$ for the neutrons. However, the candidates for
the cluster state (eighth states at $-2.96$ MeV in Table LABEL:properties
identified as cluster state in the previous subsection) has quite different
values; $-1.09$ ($spp$-proton) and 0.08 ($spp$-neutron).

In the present model, the protons are only in the
${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}$ core part as mentioned before. Thus, the
calculated single-particle parity of the protons ($spp$-proton) is governed by
the distance between two ${}^{9}\mathrm{Li}$ clusters. The correspondence
between the distance between two ${}^{9}\mathrm{Li}$ clusters and $spp$-proton
can be clarified in ${}^{18}\mathrm{C}$ by removing the two valence neutrons.
Figure 3 depicts the single-particle parity for the $0^{+}$ state of
${}^{18}\mathrm{C}$ as a function of the distance between two
${}^{9}\mathrm{Li}$ clusters, where the dotted line is for the protons
($spp$-proton) and the dashed-line is for the neutrons ($spp$-neutron). For
the protons (dotted line), the value is around $-2$ at small relative
distances because the last proton in each ${}^{9}\mathrm{Li}$ occupies
$p$-shell-like one-node orbit. However the value starts slightly deviating
from $-2$ with increasing the distance, suggesting partial excitation of the
protons to two-node orbits with positive-parity, while the total parity of the
two-${}^{9}\mathrm{Li}$ system is always projected to positive. The value for
the protons is $-1.60$ at the $d=3.5$ fm and it becomes $-0.97$ at $d=4$ fm.

We can compare this result with the $spp$-proton values of ${}^{20}\mathrm{C}$
listed in Table LABEL:properties, where the two valence neutrons are added and
the states with different ${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$ distances
are superposed. In the previous subsection, we have identified that the eighth
state in Table LABEL:properties as the candidates for the cluster state with
the ${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$ distances of $\sim$$3.75$ fm. The
obtained $spp$-proton values of ${}^{20}\mathrm{C}$ is $-0.35$ for the eighth
state, quite different from $-2$. The obtained value is consistent with that
for the states having finite ${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$
distances, suggesting the prominent cluster structure.

Next, we discuss the single-particle parity of the neutron part. It is
intriguing to point out that in ${}^{18}\mathrm{C}$ without the two valence
neutrons, the single-particle parity of the neutrons is always zero
independent of the ${}^{9}\mathrm{Li}$-${}^{9}\mathrm{Li}$ distance (Fig. 3
dashed line). This is coming from the fact that the neutrons in both
${}^{9}\mathrm{Li}$ have identical (subclosure) configurations except for the
central positions of the clusters. The antisymmetrization effect allows the
linear combination of the single-particle orbits, and the combinations of the
two orbits with good parity around the center of left or right cluster always
create a pair of positive-parity and negative-parity orbits. Thus, the single-
particle parity for the neutrons of ${}^{18}\mathrm{C}$ is zero, and in
${}^{20}\mathrm{C}$, the $spp$-neutron values purely show the contribution of
the two valence neutrons. In Table LABEL:properties, the ground state and low
excited states have the value close to 2 indicating that the two valence
neutrons are in the $sd$-shell-like positive-parity orbits. However, the
candidate for the cluster state (eighth state in Table LABEL:properties) have
the value of 0.08 completely different from 2. The result indicates that the
valence neutrons have more complex characters than “pure two-node orbits”,
which was not evident when we discussed the principal quantum number $N$ in
the previous subsection. One of the possible explanations is that the orbits
of the valence neutrons have mixed components of the $p$-shell, $sd$-shell,
and $pf$-shell orbits.

Figure 3:  Single-particle parity for the $0^{+}$ state of ${}^{18}\mathrm{C}$
as a function of the distance between two ${}^{9}\mathrm{Li}$ clusters. Dotted
line is for the protons and dashed-line is for the neutrons.

###  III.4 Rotational band structure of ${}^{20}\mathrm{C}$

The same calculation can be performed also for the $2^{+}$ and $4^{+}$ states
of ${}^{20}\mathrm{C}$. The eighth states in Table LABEL:properties at $-2.96$
MeV, which is a candidate for the cluster structure, serves as a band head
state of a rotational band structure as shown in Fig. 4, where the solid,
dotted, and dashed lines are for $0^{+}$, $2^{+}$, and $4^{+}$ states,
respectively. The $2^{+}$ state at $-2.29$ MeV and $4^{+}$ states at $0.33$
MeV are connected with the $0^{+}$ state at $-2.96$ MeV and classified into a
rotational band judging from the similarity of the calculated properties
(principal quantum number and single-particle parity) and electromagnetic
transition probabilities among the states. We have not considered the energy
of each state in this classification, but eventually, typical level spacings
of the rotational band structure comes out around the
${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}+n+n$ threshold energy. The rotational
band shows small level spacings between $0^{+}$ and $2^{+}$ reflecting the
large deformation.

Figure 4:  Rotational band structure of ${}^{20}\mathrm{C}$ measured from the
${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}+n+n$ threshold energy. Solid, dotted, and
dashed lines are for $0^{+}$, $2^{+}$, and $4^{+}$ states.

##  IV Conclusions

In this study, we discussed the structure of ${}^{20}\mathrm{C}$ by
introducing a four-body cluster model with
${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}+n+n$ configurations. The recent
development of the antisymmetrized quasi cluster model (AQCM) enables us to
utilize $jj$-coupling shell-model wave functions as plural subsystems quite
easily. Until now, many works have shown in ${}^{11}\mathrm{Li}$ the presence
of the halo structure comprised of the weakly bound two neutrons around
${}^{9}\mathrm{Li}$, and our study was motivated by a question how this halo
structure changes when another ${}^{9}\mathrm{Li}$ approaches.

In our preceding work for ${}^{18}\mathrm{C}$ Itagaki  _et al._ (2020b), it
has been discussed that ${}^{9}\mathrm{Li}+{}^{9}\mathrm{Li}$ cluster state
appears below the threshold, and therefore, it is natural in
${}^{20}\mathrm{C}$ to see the appearance of the cluster states below the
threshold after adding the two valence neutrons. The candidate for the cluster
state was identified by calculating the principal quantum number and single-
particle parity. The candidate for the cluster state serves as a band head
state and forms a rotational band structure, where the members are collected
by the similarity of the properties and electromagnetic transition
probabilities among the states. The states have small level spacings
reflecting the large deformation.

As future work, we investigate the possibility of molecular-orbital states. In
the present analysis, we discussed that atomic-orbit states, where neutron(s)
sticks to one of the two centers, but in the molecular-orbit state, each
valence neutron rotates around two centers equally with good parity. Such
molecular-obit states are expected to appear around the threshold energy.
Also, we will connect the calculation to the nuclear reaction and discuss how
we can populate these states in the actual experiment.

###### Acknowledgements.

The authors would like to thank the discussion with Dr. M. Sasano (RIKEN). The
numerical calculations have been performed using the computer facility of
Yukawa Institute for Theoretical Physics, Kyoto University.

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