# Using Adapters to Overcome Catastrophic Forgetting in End-to-End Automatic
Speech Recognition

###### Abstract

††Research supported by Research Foundation Flanders (FWO) under grant
S004923N of the SBO programme.

Learning a set of tasks in sequence remains a challenge for artificial neural
networks, which, in such scenarios, tend to suffer from Catastrophic
Forgetting (CF). The same applies to End-to-End (E2E) Automatic Speech
Recognition (ASR) models, even for monolingual tasks. In this paper, we aim to
overcome CF for E2E ASR by inserting adapters, small architectures of few
parameters which allow a general model to be fine-tuned to a specific task,
into our model. We make these adapters task-specific, while regularizing the
parameters of the model shared by all tasks, thus stimulating the model to
fully exploit the adapters while keeping the shared parameters to work well
for all tasks. Our method outperforms all baselines on two monolingual
experiments while being more storage efficient and without requiring the
storage of data from previous tasks.

Index Terms—  end-to-end automatic speech recognition, continual learning,
transformer, conformer, adapters

##  1 Introduction

Artificial Neural Networks (ANN) have considerably transformed Automatic
Speech Recognition (ASR). With End-to-End (E2E) models, they have become ASR’s
state of the art. However, while ANNs are very successful at learning single
tasks (e.g. learning one accent of English) or multiple tasks jointly (e.g.
learning multiple accents of English simultaneously), they experience great
difficulty when learning a set of tasks in sequence (e.g. learning English
accents one by one). If nothing is done to prevent it, ANNs will in that case
suffer from Catastrophic Forgetting (CF) [1]. Measures must be taken to allow
ANNs to learn tasks in sequence (without CF), i.e. to enable them to learn
continually, called Continual Learning (CL).

Within the image classification community, CL has received much attention.
Consequently, many algorithms to enable CL, called CL methods, have been
proposed. They can be categorized into three groups [2]: (a) regularization-
based methods, such as [3, 4], add a regularization term to the loss to
prevent forgetting; (b) rehearsal-based methods store a set of representative
samples of previous tasks to rehearse during the training of new tasks - e.g.
[5, 6]; (c) architectural-based methods, e.g. [7, 8], update the architecture
of the model when new tasks arrive, thus preventing forgetting. Contrary to
image classification, CL for ASR has received little attention. [9, 10] focus
on CL for Hidden Markov Model based ASR models. [11] applies [12] to a E2E ASR
model. [13, 14] implement, respectively, four and nine CL methods for E2E ASR.
Finally, [15] focuses on semi-supervised E2E ASR, aiming to overcome CF in the
wav2vec 2.0 [16] model. To do this, they insert adapters [17], which they make
task-specific, and adapt to the new task, while freezing the shared
parameters. Consequently, CL remains a major challenge for E2E ASR, which
means that its models cannot be extended to new languages, dialects, or simply
new data that becomes available, without suffering from catastrophic
forgetting; unless all previous data is re-introduced during training, a
strategy which soon becomes too expensive, both in terms of storage and
training time. CL should enable E2E ASR models to exploit all the data at its
disposal, whenever it becomes available, without suffering from CF.

In this paper, we address this challenge for E2E ASR. To this end, we use,
similar to [15], adapters, which we insert into the model and make task-
specific, meaning that each task uses its own adapters. In addition, we
protect the shared parameters and thus prevent CF. We test our method on two
monolingual experiments and find that it outperforms all baselines, while
being more storage efficient and without requiring the storage of some past
data.

##  2 Continual Learning for E2E ASR

###  2.1 E2E ASR Model

The model is a hybrid encoder-decoder E2E model, consisting of a Conformer
[18] or Transformer [19] encoder and Transformer decoder. As the model is
hybrid, a CTC loss is computed on the encoder’s output, in addition to a
decoder cross-entropy (CE) loss: their weights are $c=0.3$ and $1-c=0.7$,
respectively. The output of the model is a probability distribution over $v$
word pieces. We do not use an external Language Model (LM). We denote ${\cal
L}(X,y;\theta)$ the loss of the model with parameters
$\theta\in\mathbb{R}^{N}$ ($N$ is the number of parameters),
$X\in\mathbb{R}^{F\times f}$ the input utterance consisting of $F$ frames of
dimension $f$, and $y$ the ground truth transcription of $w$ word pieces.

###  2.2 Objective of Continual Learning

Given $D_{1},D_{2},...,D_{T}$, the labeled data (containing pairs $(X,y)$) of
$T$ tasks, the objective is to learn these tasks in sequence such that:

$\theta^{T}=\operatorname*{arg\,min}_{\theta}\sum_{i=1}^{T}\sum_{(X,y)\in
D_{i}}{\cal L}(X,y;\theta)$ (1)

where $\theta^{T}$ is adapted from $\theta^{T-1}$, the parameters of the model
after training $T-1$ tasks. When learning task $T$, the model no longer has
access to data from previous tasks. The objective is thus to learn the new
task $T$ from $D_{T}$, while retaining the performance on the previous tasks
$1$ to $T-1$ without having access to $D_{1},...,D_{T-1}$. Thus, the model
should, on the one hand, have the ability to learn the new task well, called
forward transfer [20]; on the other hand, it should be able to retain its
performance on previous tasks, called knowledge retention [20]. Forward
transfer and knowledge retention are two desiderata of the continually trained
model. In addition, we add, like [20], fixed model capacity, which demands
that the storage requirements of the model do not increase with the number of
tasks.

###  2.3 Inserting Adapters

Similar to [15], we consider adapters [17], small architectures inserted to
fine-tune a model trained on a general task to specific tasks. The adapters
from [15] were originally used in Neural Language Processing (NLP) in a LM
[17]. Since the adapters consist of few parameters, the adaptation of the pre-
trained LM to the specific NLP tasks is very easy and fast. In ASR, adapters
have been used to train multilingual models [21, 22], with each language
having its own adapters. In [23], the adapters of different languages are put
in parallel and the model must learn which adapters to use. In [24], adapters
are used to adapt an ASR model to specific accents unseen during training. In
semi-supervised ASR, [25] uses adapters to fine-tune the pre-trained wav2vec
2.0 model to downstream tasks.

Adapters are attractive for CL because they allow the model to exploit task-
specific parameters for each task, while only slightly increasing the number
of parameters, thus approximately satisfying fixed model capacity. For that
reason, we, too, consider adapters, namely the same adapters as in [22], where
they were used for multilingual ASR with a hybrid CTC-Transformer. These
adapters consist of: (i) a layer-norm layer; (ii) a down projection to
dimension $d$; (iii) a ReLU activation; (iv) an up projection to the original
dimension; (v) a skip connection between the adapter’s input and output. As in
[22], we insert adapters between the attention and feedforward sublayers of
the Transformer layers. For the Conformer encoder, we insert the adapters
between the convolution module and last feedforward layer. Since preliminary
experiments showed that CF mainly occurs in the encoder for monolingual tasks,
which, intuitively, makes sense since the tasks are part of the same language,
we insert the adapters only in the encoder layers.

For the dimension $d$ of the adapters, we consider $d=32$. With $d=32$, the
number of parameters per adapter is $N_{a}=17k$. By introducing adapters, the
model consists, after learning $T$ tasks, of parameters
$\theta^{T}=\\{\theta^{T,S},\alpha^{1},\alpha^{2},...,\alpha^{T}\\}$, with
$\theta^{T,S}\in\mathbb{R}^{N}$ the shared parameters and
$\alpha^{j}\in\mathbb{R}^{N_{a}}$ the adapter parameters for task $j\leq T$.
For a new task $t$, the adapter parameters $\alpha^{t}$ are initialized from
$\alpha^{t-1}$.

###  2.4 Protecting the Shared Parameters

As the number of task-specific parameters in the adapters is small, most of
the model is still shared across tasks and susceptible to CF. [15] deals with
this by freezing the shared parameters. We, too, consider this method, called
A/Freeze. In addition, we consider A/CFT (Cautious Fine-Tuning) which consists
of two stages: 1) train the adapters of the new task while freezing the shared
parameters (as in A/Freeze); 2) adapt the entire model with a ten times
smaller learning rate, i.e. if $lr_{\text{ft}}$ is the learning rate to adapt
a trained model to a new task, then use $0.1lr_{\text{ft}}$. In the first
stage, the new task’s adapters are trained to work well with the ’old’ shared
model, thus updating the shared parameters cautiously might improve on the new
task while preventing forgetting of the old tasks.

###  2.5 Exploiting the Adapters

Since the adapters are task-specific, the model should, when learning a new
task, exploit the adapters as much as possible. While guaranteed for
subsequent tasks (as the shared parameters are either frozen or regularized,
while the adapter parameters are not), this is not the case for the initial
task which the model learns. Ideally, when training the model for the first
time, the model should keep the shared parameters ’general’, while the
adapters should be used to store the ’task-specific information’. To stimulate
this, we apply weight decay to the shared parameters (but not to the adapter
parameters) during training of the first task. For the weight
$\lambda_{wd}=10^{\omega}$ of the weight decay loss, we take the largest
$\omega<0$ for which the model learns the new task at most $1\%$ worse than
the model without weight decay.

###  2.6 Decoding without Task Label

One disadvantage of the adapter-based methods is that, even at inference time,
they require the task label of the utterance, to know which adapters to use.
When task labels are not available, one way to infer them is to send the
utterance $T$ times (if the model has seen $T$ tasks) through the model, using
adapter parameters $\alpha^{i}$ for each task $i\leq T$. We can then consider
the output (out of the $T$ outputs) with the highest likelihood. Using the
hybrid CTC/decoder output, we refer to this as conf-infer. As this might
become slow as $T$ increases, we consider another alternative, avg-apt, in
which we send utterances of which the task is not known through average
adapters, thus with parameters $\tilde{\alpha}=1/T\sum_{i=1}^{T}\alpha^{i}$.

##  3 Experimental set-up

Table 1: Results after training the six CV tasks in sequence. WER for each task (evaluated on final model) and, AVG, BWT, FWT and COV are reported. ’NTL’ refers to ’No task label’. If ✓, the model does not require the task label at inference time; ✗ indicates that it does. Methods with ’A/’ in their name use adapters. Best models are in bold.  |  | WER per task | Summary
---|---|---|---
Model | NTL | T1–US | T2–ENG | T3–AUS | T4–IND | T5–SCO | T6–IRE | AVG$\downarrow$ | BWT$\uparrow$ | FWT$\uparrow$ | COV$\uparrow$
Sep. Model | ✗ | 17.3 | 10.8 | 10.6 | 16.7 | 12.1 | 11.4 | 13.14 | 0.0 | 0.0 |  100.0$\%$
Sep. Enc. | ✗ | 17.3 | 11.3 | 11.3 | 17.2 | 12.4 | 10.6 | 13.34 | 0.0 | -0.2 | $90.3\%$
Sep. Enc.-FF | ✗ | 17.3 | 11.6 | 11.2 | 17.5 | 13.2 | 10.8 | 13.61 | 0.0 | -0.5 |  77.7$\%$
Fine-Tuning | ✓ | 19.4 | 12.7 | 14.0 | 20.6 | 13.4 | 11.4 | 15.25 | -2.5 | 0.0 |  0.0$\%$
LWF | ✓ | 19.0 | 12.2 | 13.7 | 20.7 | 12.6 | 11.1 | 14.88 | -2.3 | +0.2 |  17.5$\%$
ER | ✓ | 18.1 | 12.0 | 13.0 | 18.6 | 12.7 | 10.8 | 14.20 | -1.3 | +0.1 |  49.8$\%$
KD | ✓ | 18.5 | 12.3 | 14.0 | 19.8 | 12.9 | 10.7 | 14.67 | -1.9 | +0.1 |  27.3$\%$
A/Freeze | ✗ | 16.6 | 12.1 | 11.3 | 19.5 | 14.2 | 11.2 | 14.15 | 0.0 | -1.3 |  52.0$\%$
A/CFT | ✗ | 17.2 | 11.3 | 11.2 | 17.0 | 13.1 | 11.0 | 13.45 | -0.5 | 0.0 | 85.0$\%$
xx\+ avg-apt | ✓ | 16.3 | 11.8 | 12.6 | 17.8 | 13.4 | 10.6 | 13.75 | -0.2 | -0.6 |  71.1$\%$
xx\+ conf-infer | ✓ | 16.5 | 11.4 | 11.3 | 17.4 | 12.6 | 10.7 | 13.31 | -0.0 | -0.1 | 91.7$\%$

For all our experiments, we use the ESPnet library [26].

Data. For the first experiment, we consider the Common Voice (CV) 7.0 English
dataset [27], which is split into six dialects learned in the following order:
United States (US), England (ENG), Australia (AUS), India (IND), Scotland
(SCO), Ireland (IRE). US and ENG are the largest tasks, with 350k and 116k
utterances, respectively; the rest of the tasks are smaller and their order is
randomized. For the second experiment, we use the Corpus Gesproken Nederlands
(CGN) [28] dataset, consisting of Dutch speech from Belgium (BE) and the
Netherlands (NL). To obtain four tasks, we first split the data into NL and
BE; then we split both NL and BE into ’clean’ and spontaneous speech,
referring to the former tasks as NLc and BEc and to the latter as NLs and BEs,
for NL and BE respectively. The tasks, between 120k and 245k utterances, are
learned in this order, which starts with the largest ’clean’ task and
maximizes the interference between tasks (by interleaving NL and BE).

Model. For the CV experiments, the model consists of 12 Conformer encoder
layers of attention dimension 256 and feedforward dimension 2048, and 6
Transformer decoder layers of the same dimensions. With an output of 5000 word
pieces, generated on US with Sentence Piece [29], the number of parameters is
46.8M. The adapters for one task contain 204k parameters, i.e. $0.43\%$ of the
model. For the CGN experiments, the model is the hybrid CTC-Transformer from
[30], with 12 Transformer encoder and 6 Transformer decoder layers of the same
dimensions as in the CV model. The output are 300 word pieces, generated on
NLc. The number of parameters is 27.4M; the adapters are thus $0.75\%$ of the
model. For the CV and CGN model, it thus takes, respectively, 230 and 132
tasks for the number of parameters of the model to double, which illustrates
that adapters introduce task-specific parameters while still approximately
satisfying fixed model capacity. For both models, as in [14], the learning
rate to adapt the model to a new task ($lr_{\text{ft}}$) is ten times smaller
than the learning rate for the initial task (US or NLc). For both experiments,
$\omega=-5$ for the weight decay (see Sec. 2.5).

Baselines. We consider following baselines: (i) Fine-Tuning: naively learns
the new task, suffering from CF; (ii) Separate Model (Sep. Model): each task
$t$ trains a separate model $\theta^{t}$, initialized from $\theta^{t-1}$
(Sep. Model thus stores $t$ models); (iii) Separate Encoder (Sep. Enc.): same
as Sep. Model, but the decoder is shared and frozen (after the first task);
(iv) Separate Encoder-Feedforward (Sep. Enc-FF): same as Sep. Enc., but all
attention (and convolution) modules are also shared and frozen; (v) Learning
Without Forgetting (LWF) [31]: regularization-based method which uses the new
task’s data to distill knowledge from the old to new model - LWF was the best
memory-free CL method from [13, 14]; (vi) Knowledge Distillation (KD): a
rehearsal-based method and the best CL method from [14] \- works the same as
LWF but with a memory to distill knowledge from old to new model; (vii)
Experience Replay (ER): a rehearsal-based method and second-best CL method
from [14] \- trains jointly on the current task’s mini-batch and a mini-batch
sampled from a memory. None of these methods uses adapters. We call (ii)-(iv)
’bounds’ rather than ’baselines’, since they violate fixed model capacity (by
introducing a large number of task-specific parameters) and are used to see
what the adapter-based methods are capable of. For (iii)-(iv), we consider
making (the feedforward layers of) the encoder task-specific, since
preliminary experiments showed that these layers are most susceptible to
forgetting. The rehearsal-based baselines have access to a memory of 2000 and
1000 uniformly sampled utterances for the CV and CGN experiments,
respectively. The size of the memory is fixed, even as tasks are added. For
the hyper-parameters of the baselines, we use the values of [14].

Metrics. To assess the models, we consider four metrics, based on WER
(expressed in percentages). Average WER (AVG), the most important metric, is
computed by taking the average of WERs on all seen tasks after learning the
last one. Backward Tranfer (BWT) [5] measures the extent to which knowledge
retention is achieved. Negative BWT indicates forgetting and thus the average
increase of WER of previous tasks compared to when they were first learned.
Forward Transfer (FWT) [5] measures the extent to which forward transfer is
achieved. Positive FWT indicates average decrease of WER of new tasks,
compared to Fine-Tuning. Coverage (COV) [10] measures to which extent the
models close the gap between what we consider as worst case (Fine-Tuning) and
as best case (Sep. Model) in terms of AVG. 100$\%$ COV would mean that the AVG
of the model equals Sep. Model’s, for 0$\%$ COV it equals Fine-Tuning’s.

##  4 Results

Table 2: Results after learning the four CGN tasks in sequence. WER for each task (evaluated on final model) and, AVG, BWT, FWT and COV are reported. ’NTL’ refers to ’No task label’. If ✓, the model does not require the task label at inference time; ✗ indicates that it does. Methods with ’A/’ in their name use adapters. Best models are in bold. |  | WER per task | Summary
---|---|---|---
Model | NTL | T1–NLc | T2–VLc | T3–NLs | T4–VLs | AVG$\downarrow$ | BWT$\uparrow$ | FWT$\uparrow$ | COV$\uparrow$
Sep. Model | ✗ | 23.6 | 23.5 | 45.4 | 40.5 | 33.25 | 0.0 | 0.0 |  100.0$\%$
Sep. Enc. | ✗ | 23.6 | 24.9 | 47.3 | 42.9 | 34.69 | 0.0 | -1.9 |  84.2$\%$
Sep. Enc.-FF | ✗ | 23.6 | 24.6 | 48.4 | 45.3 | 35.48 | 0.0 | -3.0 |  75.5$\%$
Fine-Tuning | ✓ | 39.3 | 27.3 | 62.2 | 40.5 | 42.32 | -12.1 | 0.0 |  0.0$\%$
LWF | ✓ | 36.8 | 26.9 | 59.7 | 40.4 | 40.96 | -10.2 | -0.0 |  15.0$\%$
ER | ✓ | 32.3 | 26.4 | 56.6 | 41.8 | 39.29 | -7.3 | -0.8 |  33.5$\%$
KD | ✓ | 29.0 | 25.0 | 53.1 | 41.5 | 37.12 | -4.6 | -0.5 |  57.4$\%$
A/Freeze | ✗ | 22.9 | 27.2 | 51.0 | 50.0 | 37.78 | 0.0 | -6.3 |  50.1$\%$
A/CFT | ✗ | 25.9 | 24.0 | 48.8 | 39.9 | 34.64 | -2.4 | +0.3 | 84.7$\%$
xx\+ avg-apt | ✓ | 27.1 | 24.6 | 53.1 | 40.8 | 36.38 | -3.7 | -0.7 |  65.6$\%$
xx\+ conf-infer | ✓ | 26.7 | 24.9 | 51.2 | 40.1 | 35.72 | -3.4 | -0.1 | 72.8$\%$
Table 3: Ablation of the effect of weight decay (Sec. 2.5) on A/Freeze, and of the smaller learning rate and the two stages in A/CFT (Sec. 2.4). The results are the first adaptation of CGN. Model | AVG$\downarrow$ | BWT$\uparrow$ | FWT$\uparrow$
---|---|---|---
A/Freeze | 25.07 | 0.0 | -3.8
$\rightarrow$ no weight decay | 25.75 | 0.0 | -4.9
A/CFT | 23.20 | +0.0 | -0.1
$\rightarrow$ with regular learning rate | 23.87 | -2.7 | +1.3
$\rightarrow$ in one stage | 24.44 | -3.2 | +0.7

We discuss the results of the CV experiments, shown in Table 1. First, it can
be seen that adapters indeed improve the model’s ability to learn continually
and adapt to new tasks. A/Freeze, while only training 200k parameters per task
($0.43\%$ of the model), is able to close the gap between Fine-Tuning and Sep.
Model by more than $50\%$. While the gap with Sep. Enc.-FF is still clear, it
is relatively small, given that Sep. Enc.-FF trains 25.2M parameters or
$53.9\%$ of the model. Second, while freezing the shared parameters, as in
A/Freeze, results in zero forgetting (by definition), the performance can be
improved by regularizing but adapting the shared parameters. Indeed, A/CFT
outperforms A/Freeze, achieving a much better FWT, even if at the cost of a
slightly worse BWT. It closes the gap between worst and best case by 85.0$\%$.
Consequently, A/CFT outperforms all baselines by a large margin, including
Sep. Enc.-FF, while having only 200k task-specific parameters, compared to the
latter’s 25.2M. In addition, A/CFT even comes very close to Sep. Enc.’s
performance, which has 33.5M task-specific paramaters or 71.5$\%$ of the
model. Third, with their performance, A/Freeze and A/CFT outperform all
baselines, the latter even by a large margin. Nevertheless, the adapter-based
methods require task labels, while the baselines do not, making the comparison
for scenarios in which the task labels are unknown at inference time unfair.
In Sec. 2.6, we proposed two ways to overcome the adapter-based methods’ need
for task labels. Applied to A/CFT, we find that the deterioration in
performance (compared to the case where the task label is known), is small.
For avg-apt, there is indeed a small deterioration, though A/CFT with avg-apt
still closes the gap by more than $70\%$, outperforming all baselines. For
conf-infer, there is even an improvement, with A/CFT now outperforming Sep.
Enc. and closing the gap between worst and best case by $91.7\%$, almost
matching the performance of Sep. Model. Nevertheless, a disadvantage of conf-
infer is that the time to infer the task increases with the number of tasks.

We proceed with the CGN experiments, of which the results are shown in Table
2. First, we note that A/Freeze and A/CFT are again very effective to enable
CL, both reaching a very similar COV as in the CV experiments. In addition,
the gap with the baselines is similar, except that KD is now slightly better
than A/Freeze. A/CFT once again outperforms all baselines by a large margin.
Moreover, it even outperforms Sep. Enc.-FF, which has 12.6M task-specific
parameters or 46.0$\%$ of the model, and matches the performance of Sep. Enc,
which even has 17.7M task-specific parameters or 64.5$\%$ of the model.
Second, for the case when the task labels are not known at inference time, the
results remain similar to those from the CV experiments. A/CFT with avg-apt
experiences a slightly larger deterioration as for CV, though still reaching a
COV of 65.6$\%$ and outperforming all baselines. For conf-infer, the
deterioration is smaller, reaching a COV of $72.8\%$, which is again slightly
worse than for CV, where it even outperformed A/CFT with known task labels.

Satisfying the Desiderata. In Sec. 2.2, we stated three desiderata of a CL
method. Given their very small number of task-specific parameters, we
concluded, in Sec. 3, that the adapter-based methods approximately achieve
fixed model capacity. From the results of the experiments, it is also clear
that A/CFT, of all methods, is best able to combine knowledge retention and
forward transfer, and thus to learn new tasks well with minimal forgetting of
old tasks.

CL without a memory. [13, 14] found that a memory of representative utterances
from previous tasks was necessary to achieve high performance in CL for E2E
ASR. Even when this memory was small, the gap between the rehearsal-based
methods (using this memory) and the regularization-based methods (not using
this memory) was large. However, we find that this need for a memory can be
overcome by introducing adapters. The combination of task-specific adapters
and regularizing the shared model results in much better performance, even
when the task labels are not known at inference time. Note that the adapter-
based methods are also more storage efficient. Storing the memory of 2000
utterances for CV and 1000 utterances for CGN requires, expessed in number of
models, 1.66 and 1.54 models, respectively. In other words, the adapter-based
methods can learn 382 and 204 tasks for CV and CGN, respectively, before they
match the storage requirements of the rehearsal-based methods. To conclude,
compared to the rehearsal-based methods, the previous state-of-the-art, the
adapter-based methods 1) perform much better; 2) are much more storage
efficient; 3) do not require the storage of a memory, which may not be
allowed.

Ablation. Table 3 shows an ablation study on the first adaptation (from T1–NLc
to T2–VLc) of CGN. First, it illustrates the effectiveness of applying weight
decay during the initial task (Sec. 2.5). The model with weight decay is
better able to exploit the adapters and learn new tasks, achieving a higher
FWT. Second, it illustrates the importance of A/CFT’s: 1) lower learning rate
during the second stage; 2) training in two stages, i.e. by first training the
adapters while freezing the shared model, before adapting the entire model.
Without these two aspects, A/CFT suffers from CF; while with these two
aspects, A/CFT is able to learn the new task without forgetting.

##  5 Conclusion

In this paper, we considered adapters to overcome CF in (monolingual) E2E ASR.
This was done by making the adapters task-specific, allowing the model to
exploit the adapter parameters to learn the new tasks. Using adapters
introduced very few parameters per task, yet it turned out to be very
effective. While freezing the remaining shared parameters already worked
decently, A/CFT, which proceeds in two stages while using a smaller learning
rate in the second stage, further improves the performance greatly,
outperforming all baselines on our two experiments, using two different
datasets and models. On these experiments, A/CFT even outperforms or matches
the performance of baselines which have between 60 and 125 times more task-
specific parameters. To overcome the need for task labels at inference time,
we proposed two methods – averaging adapters and considering the confidence –
which both resulted in, at worst, a very small deterioration in performance.
Overall, thus, our method A/CFT greatly improves previous state-of-the-art,
while satisfying to a much higher extent the desiderata of a CL method. In
addition, A/CFT is much more storage efficient, compared to previous state-of-
the-art, without requiring the storage of utterances from previous tasks in a
memory, which may not always be allowed or desired.

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# The freezing phase transition in hard core lattice gases on triangular
lattice with exclusion up to seventh next-nearest neighbor

Asweel Ahmed A Jaleel<EMAIL_ADDRESS>The Institute of Mathematical
Sciences, C.I.T. Campus, Taramani, Chennai 600113, India  Homi Bhabha National
Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
Department of Physics, Sadakathullah Appa College, Tirunelveli, Tamil Nadu
627011, India  Dipanjan Mandal<EMAIL_ADDRESS>Department of
Physics, University of Warwick, Coventry CV4 7AL, United Kingdom  Jetin E.
Thomas<EMAIL_ADDRESS>The Institute of Mathematical Sciences, C.I.T.
Campus, Taramani, Chennai 600113, India  Homi Bhabha National Institute,
Training School Complex, Anushakti Nagar, Mumbai 400094, India  R. Rajesh
<EMAIL_ADDRESS>The Institute of Mathematical Sciences, C.I.T. Campus,
Taramani, Chennai 600113, India  Homi Bhabha National Institute, Training
School Complex, Anushakti Nagar, Mumbai 400094, India

###### Abstract

Hard core lattice gas models are minimal models to study entropy driven phase
transitions. In the $k$-NN lattice gas, a particle excludes all sites upto the
$k$-th next-nearest neighbors from being occupied by another particle. As $k$
increases from one, it extrapolates from nearest neighbor exclusion to the
hard sphere gas. In this paper, we study the model on the triangular lattice
for $k\leq 7$ using a flat histogram algorithm that includes cluster moves.
Earlier studies had focused on $k\leq 3$. We show that for $4\leq k\leq 7$,
the system undergoes a single phase transition from a low-density fluid phase
to a high-density sublattice-ordered phase. Using partition function zeros and
non-convexity properties of the entropy, we show that the transitions are
discontinuous. The critical chemical potential, coexistence densities, and
critical pressure are determined accurately.

##  I Introduction

Hard core lattice gases (HCLGs), a collection of particles on lattices that
interact only through excluded volume interactions, are the simplest models to
show phase transitions. In particular, they are minimal models to study fluid
to solid freezing transitions induced by increasing density [1, 2]. Since the
interaction energies are either zero or infinity, temperature plays no role in
such phase transitions. The phases and the nature of the transitions depend
only on the shape of the excluded volume. To understand this dependence as
well as the order in which the phases occur with increasing density, the phase
diagram of many different shapes have been studied in detail in literature.
Examples include hard hexagons [3, 4], triangles [5], squares [6, 7, 8],
rectangles [9, 10], tetraminos [11], rods [12, 13], etc.

Among the different shapes, a subclass of shapes of particular interest is the
$k$-NN HCLG model in which the first $k$ next-nearest neighbors of a particle
are excluded from being occupied by another particle. The case $k=1$
corresponds to nearest neighbor exclusion while the limit $k\to\infty$
corresponds to the well-studied hard sphere problem in the continuum. Also,
the model can be thought of as limiting cases of spin models with long ranged
interaction [14, 15, 16, 17]. Introduced in the 1950s [18, 19, 20, 6, 21, 22,
23], the $k$-NN model, in addition to its relevance to critical phenomena and
the hard sphere gas, has found applications in diverse areas of research as
well as direct experimental realizations. Examples of applications include
adsorption on surfaces [24, 25, 26, 27, 28, 29, 30], glass transitions [31,
32, 33], phase transitions in closely related $BM_{n}$ model [34], attractive
gases [35, 36] and in combinatorial problems [37]. Some direct experimental
realizations include porphyrins adsorbed on the Au(111) surface [38, 39, 40],
adsorption of chlorine on a silver (100) surface [24] and adsorption of
selenium on a nickel (100) surface [41].

In two dimensions, the phase diagram for the $k$-NN model for different $k$
has been studied on the square, honeycomb and triangular lattices using
different techniques such as transfer matrix calculations, high and low-
density expansions, Monte Carlo simulations, etc. We summarize the known
results below.

On the square lattice, the model has been studied for $k\leq 11$, and a
summary of known results may be found in Refs. [42, 43, 44]. For
$k=1,3,6,7,8,9$, the model undergoes a single phase transition from a low-
density fluid phase to a high-density sublattice-ordered phase. The transition
is continuous for the $1$-NN model and first order otherwise. The $2$-NN and
$5$-NN models reduce to $2\times 2$ and $3\times 3$ hard square problems
respectively and the high-density phase has columnar order. The transition
belongs to the Ashkin-Teller universality class for the $2$-NN model and is
discontinuous for the $5$-NN model. The $4$-NN, $10$-NN and $11$-NN models
show multiple phase transitions. The presence of multiple transitions are due
to a sliding instability present in the high-density phase and also appear for
$k>11$ [43, 45].

On the honeycomb lattice, the model has been studied for $k\leq 5$ [46, 47,
48]. The $1$-NN model shows a single continuous transition, belonging to the
Ising universality class, from a fluid to sublattice-ordered phase. The $2$-NN
model shows a first order transition while the $3$-NN model, surprisingly,
does not undergo any transition [46, 47]. The $4$-NN model shows a single
continuous phase transition that belongs to the three state Potts model. The
$5$-NN model has been shown to undergo two discontinuous transitions. However,
these transitions show non-standard scaling with system size [46].

The $k$-NN model on the triangular lattice, the main focus in this paper, has
been studied for $k\leq 5$. The triangular lattice is of particular interest
as it is a better approximation to the continuum than the square and honeycomb
lattices. Thus, one would expect that phases seen in the continuum like the
hexatic phase would be most easily seen on the triangular lattice. The $1$-NN
model on the triangular lattice is the hard hexagon model. It is the only
exactly solvable HCLG model that undergoes a transition from a disordered to
sublattice-ordered phase which belongs to the three state Potts model [3, 4,
49]. The $2$-NN model on triangular lattice is known to show a single second
order transition which belongs to four state Potts model universality class
[50, 51, 52]. The $3$-NN model has been studied in detail recently and it
undergoes a single discontinuous transition to a sublattice-ordered phase [53,
52, 54]. Using tensor renormalization group (TRG) methods, the $4$-NN and
$5$-NN models were also studied. However, the phenomenology for the $4$-NN and
$5$-NN models is not well-established. The TRG results give a wide range of
values for the critical chemical potential, and the nature of the transitions
are also not clear [52]. Thus, compared to the square and honeycomb lattices,
the $k$-NN model on the triangular lattice is less studied, and it would be of
interest to determine the phase diagram for the model for $k\geq 4$.

An additional motivation for studying the $k$-NN model on the triangular
lattice comes from the algorithmic point of view. In general, Monte Carlo
studies of HCLG models of particles with large excluded volume suffers from
equilibration issues especially at densities close to full packing. When only
local moves are allowed, the system gets stuck in long-lived metastable
states. Algorithms that include cluster moves can overcome these issues
partially. One such algorithm is the grand canonical transfer matrix based
strip cluster update algorithm (SCUA) that updates the configurations on
strips that span the lattice in one attempt [55, 13]. This algorithm has been
very useful in obtaining the phase diagram of the $k$-NN model on square [43]
and honeycomb [46] lattices as well as that of many other differently shaped
particles [13, 56, 57, 58, 59]. However, the SCUA algorithm has difficulty in
equilibrating systems which undergo a strong first order transition close to
full packing [53]. More recently, there have been two other algorithms that
have proved useful. One is the TRG method which was used to obtain the phase
diagram for the $k$-NN model on the triangular lattice for $k\leq 5$ [52] and
interacting hard equilateral triangles [47]. The other is the strip cluster
Wang Landau (SCWL) algorithm that combines the SCUA algorithm with flat
histogram methods, and thus determines the density of states for all densities
[60, 53]. A comparison of the efficacies of the TRG method and SCWL algorithm
would be useful.

In this paper, we study the $k$-NN model on the triangular lattice using flat
histogram Monte Carlo simulations (SCWL algorithm) for $1\leq k\leq 7$. We
benchmark our simulations against the known exact solution of the $1$-NN
model. For the $2$-NN model, we obtain improved estimates for the critical
density. For the $4$-NN to $7$-NN models, we show, using partition function
zeros, that the system undergoes a single phase transition from a low-density
fluid phase to a high-density sublattice-ordered phase. Based on non-convexity
of the numerically obtained entropy, we show that the transitions are
discontinuous. The estimates for the critical parameters are summarized in
Table 3. In addition, we show that, compared to the TRG method, the SCWL
algorithm is able to obtain more accurate results for larger $k$.

The remainder of the paper is organized as follows. In Sec. II, we define the
$k$-NN model and outline the SCWL algorithm adapted to the $k$-NN model on the
triangular lattice. In Sec. III, we first benchmark the simulations with known
results from the $1$-NN and $2$-NN models. We also describe the results for
the $2$-NN model as well for $4$-NN to $7$-NN models. Section IV contains a
summary and discussion of the results.

##  II  Model and Monte Carlo algorithm

###  II.1 $k$-NN hard core lattice gas

Consider a $L\times L$ triangular lattice with periodic boundary conditions. A
lattice site can be occupied by utmost one particle. In the $k$-NN exclusion
model, a particle excludes all the sites up to the $k$-th nearest neighbors
from being occupied by another particle. Figure 1 shows the excluded neighbors
of a particle for $k=1,2,\ldots,7$. The number of excluded sites
$N_{\mathrm{excl}}$ and the maximum number density $n_{\mathrm{max}}$ for a
given $k$ are given in Table 1.

Figure 1:  (Color online) The neighbors of the central site (black color) are labeled according to the distance from it. The number in the circles denote the range of exclusion $k$. In the $k$-NN exclusion model, a particle excludes all the sites with label less than or equal to $k$ from being occupied by another particle. Table 1: For each $k$, the number of neighbors that are excluded from being occupied by a particle $N_{\mathrm{excl}}$, the maximum number density $n_{\mathrm{max}}$, the orientation of the rows and the range of exclusion $\delta$, as defined for the evaporation and deposition algorithm (see text in Sec. II.2), are tabulated. $k$ | $N_{\mathrm{excl}}$ | $n_{\mathrm{max}}$ | orientation of rows | $\delta$
---|---|---|---|---
1 | 6 | $1/3$ | 0, $\pi/3$ , $2\pi/3$ | 1
2 | 12 | $1/4$ | 0, $\pi/3$ , $2\pi/3$ | 1
3 | 18 | $1/7$ | 0, $\pi/3$ , $2\pi/3$ | 2
4 | 30 | $1/9$ | 0, $\pi/3$ , $2\pi/3$ | 2
5 | 36 | $1/12$ | 0, $\pi/3$ , $2\pi/3$ | 3
6 | 42 | $1/13$ |  $\pi/6$, $\pi/2$ , $5\pi/6$ | 2
7 | 54 | $1/16$ | 0, $\pi/3$ , $2\pi/3$ | 3

###  II.2 Monte Carlo Algorithm

We study the $k$-NN model using the strip cluster Wang Landau (SCWL) algorithm
introduced recently [53, 60]. For the sake of completeness, as well as for
explaining the generalization of the algorithm to general $k$, we briefly
describe the algorithm below.

In SCWL, rejection free cluster moves are combined with Wang Landau flat
histogram algorithm [61, 62] for estimating the density of states $g(N,L$),
where $N$ is the number of particles. We first outline the Wang Landau
protocol.

The weight of a configuration with $N$ particles is taken to be inversely
proportional to $g(N,L)$. A histogram $H(N)$ records the number of times
configurations with $N$ particles are visited during the simulations.
Initially $H(N)=0$ and $S(N)=0$, where $S(N)$ is the entropy

$S(N)=\ln g(N,L).$ (1)

The system evolves via an evaporation-deposition algorithm (described below).
After every microstep, if there are $N$ particles, then, $H(N)=H(N)+1$ and
$S(N)=S(N)+f$, where $f$ is a modification factor. The system is evolved till
the histogram becomes reasonably flat, i.e., $H_{min}(N)>c\times H_{max}(N)$,
where $H_{min}$ and $H_{max}$ are the minimum and maximum value of the
histogram respectively, and $c$ is a constant. In our simulations, we choose
$c=0.80$. This is one iteration of Wang Landau algorithm. The modification
factor $f$ is halved for the next iteration and histogram $H(N)$ is reset to
zero. Initially $f$ is chosen to be equal to 1. The iterations are continued
till $f$ is less than a predefined limit, $f_{min}$. In our simulations
$f_{min}=2^{-22}$.

We now outline the cluster move for evaporating and depositing particles [60,
53]. A row is selected at random. The orientations of the rows depend on $k$
and are as tabulated in Table 1. Some of the lattice sites in the row are
excluded from occupation due to particles in nearby rows. These excluded sites
break up the row into open segments. We note that the orientations of the
rows, as given in Table 1, are chosen such that the filling of these segments
are independent.

Choose a segment at random. Let this segment have $\ell$ sites. Remove all the
particles in the segment and let there be $N_{0}$ particles remaining in the
system. Along this row, there has to be a minimum of $\delta$ empty sites
between two particles. The value of $\delta$ for different $k$ are given in
Table 1. It is possible to occupy $0,1,\ldots,n^{*}$ particles, where
$n^{*}=[(\ell+\delta)/(\delta+1)]$. The refilling is done in two steps: first
we determine the number of particles $n$ that should be deposited and second
we choose a random configuration from all possible ways of placing $n$
particles in $\ell$ sites.

The probability of choosing $n$ particles to deposit is

${\rm
Prob}_{o}(\ell,n)=\frac{C_{o}(\ell,n)/g(N_{0}+n,L)}{\sum_{i=0}^{n^{*}}C_{o}(\ell,i)/g(N_{0}+i,L)},$
(2)

where $C_{o}(\ell,n)$ is the number of ways of placing $n$ particles on a
segment of length $\ell$ sites with open boundary conditions. If there are no
sites which are excluded in an entire row, then we have to use periodic
boundary condition for deposition. Then the probability ${\rm
Prob}_{p}(\ell,n)$ of choosing $n$ particles to deposit is

${\rm
Prob}_{p}(\ell,n)=\frac{C_{p}(\ell,n)/g(N_{0}+n,L)}{\sum_{i=0}^{n^{*}}C_{p}(\ell,i)/g(N_{0}+i,L)},$
(3)

where $C_{p}(\ell,n)$ is the number of ways of placing $n$ particles on a ring
of $\ell$ sites and $n^{*}=[\ell/(\delta+1)]$.

Determining $C_{o}(\ell,n)$ and $C_{p}(\ell,n)$ is a straightforward
enumeration problem and are given by

$\displaystyle C_{o}(\ell,n)$ $\displaystyle=$
$\displaystyle\frac{(\ell+\delta-n\delta)!}{(\ell+\delta-n\delta)!n!},~{}n=0,1,\ldots,\left[\frac{\ell+\delta}{\delta+1}\right],$
(4) $\displaystyle C_{p}(\ell,n)$ $\displaystyle=$
$\displaystyle\frac{\ell(\ell-n\delta-1)!}{(\ell-n\delta-n)!n!},~{}n=0,1,\ldots,\left[\frac{\ell}{\delta+1}\right].$
(5)

After determining $n$, we fill the segment iteratively from one end to the
other. For open segments, the probability $P_{o}(\ell,n)$ that the first site
is empty is given by

$P_{o}(\ell,n)=\frac{C_{o}(\ell-1,n)}{C_{o}(\ell,n)}=\frac{\ell+\delta-n\delta-n}{\ell+k-n\delta}.$
(6)

If the first site is empty, the $\ell$ is reduced by one, keeping $n$ the
same, and the step is repeated. If the first site is occupied, then
$\ell\to\ell-\delta$, $n\to n-1$, and the step is repeated. If the segment has
periodic boundary conditions, the probability $P_{p}(\ell,n)$ that the first
$\delta$ sites are empty is given by

$P_{p}(\ell,n)=\frac{C_{o}(\ell-\delta,n)}{C_{p}(\ell,n)}=\frac{\ell-n\delta}{\ell}.$
(7)

If the first $\delta$ sites are empty, then it reduces to an open segment of
length $\ell-\delta$ with $n$ particles. Else, one of the first $\delta$
sites, chosen at random, is occupied, and it reduces to an open segment of
length $\ell-2\delta-1$ with $n-1$ particles.

$C_{0}(\ell,n)$, $C_{p}(\ell,n)$, $P_{o}(\ell,n)$ and $P_{p}(\ell,n)$ do not
change during the simulation and are therefore evaluated once in the beginning
and stored as a look up table.

The values of entropy, order parameter, susceptibility etc., reported in the
paper are those calculated at the last Wang Landau iteration.

##  III Results

We now describe the results from the flat histogram simulations. For $k=1$,
the hard hexagon model, an exact solution is available, and we benchmark our
simulations against the known results. For $k=2$, we compare our results with
those from earlier simulations. For $k=3$, our recent results can be found in
Ref. [53]. For $k=4,5,6,7$, we present new results.

We start by defining different thermodynamic quantities. Knowing the density
of states $g(N,L)$, the grand canonical partition function
$\mathcal{L}(\mu,L)$, the pressure $P$, and the average values of any
observable $\mathcal{O}$ are given by

$\displaystyle\mathcal{L}(\mu,L)$ $\displaystyle=$
$\displaystyle\sum_{N=0}^{N_{max}}g(N,L)e^{\mu N},$ (8) $\displaystyle P(\mu)$
$\displaystyle=$ $\displaystyle\frac{1}{L^{2}}\ln\mathcal{L}(\mu,L),$ (9)
$\displaystyle\langle\mathcal{O}\rangle$ $\displaystyle=$
$\displaystyle\frac{\sum_{N}\mathcal{O}(N)g(N,L)e^{\mu
N}}{\mathcal{L}(\mu,L)}.$ (10)

The pressure in the canonical ensemble, $\widetilde{P}(\rho)$, is calculated
as [63, 64]

$\widetilde{P}(\rho)=\int_{0}^{\rho}[1-\phi(\rho)]\frac{\partial}{\partial\rho}\left[\frac{\rho}{1-\phi(\rho)}\right]d\rho.$
(11)

where $\phi(\rho)$ is the mean fraction of sites where a new particle cannot
be placed because of exclusion due to existing particles.

For all the models, we will show that the model undergoes a single transition
from a disordered fluid phase to a high-density solid phase. In the solid
phase, the particles occupy one of $N_{\mathrm{sub}}$ sublattices. It is clear
that $N_{\mathrm{sub}}=1/n_{\mathrm{max}}$, where the maximum number density
$n_{\mathrm{max}}$ for each $k$ is as tabulated in Table 1. We define the
sublattice order parameter for a given $k$ to be

$q_{k}=\left|\sum_{j=1}^{N_{\mathrm{sub}}}\rho_{j}\exp\left[\frac{2\pi
i(j-1)}{N_{\mathrm{sub}}}\right]\right|,$ (12)

where $\rho_{j}$ is the density of particles in sublattice $j$.
Compressibility $\kappa_{k}$ and susceptibility $\chi_{k}$ are

$\displaystyle\kappa_{k}$ $\displaystyle=$ $\displaystyle
L^{2}\left[\langle\rho_{k}^{2}\rangle-\langle\rho_{k}\rangle^{2}\right]$ (13)
$\displaystyle\chi_{k}$ $\displaystyle=$ $\displaystyle L^{2}(\langle
q_{k}^{2}\rangle-\langle q_{k}\rangle^{2}).$ (14)

For the numerical analysis, it is useful to define an associated quantity
$t_{k}$ defined as

$t_{k}=\frac{\partial\ln\langle q_{k}\rangle}{\partial\mu}.$ (15)

From finite size scaling [65, 66, 67] , the above thermodynamic quantities at
the transition point scale with $L$ as

$\begin{split}\kappa_{k}(\mu_{c})&\sim L^{\alpha/\nu},\\\ \langle
q_{k}(\mu_{c})\rangle&\sim L^{-\beta/\nu},\\\ \chi_{k}(\mu_{c})&\sim
L^{\gamma/\nu},\\\ t_{k}&\sim L^{1/\nu},\end{split}$ (16)

where $\alpha$, $\beta$, $\gamma$, and $\nu$ are critical exponents. For a
first order transition in two dimensions, $\nu=1/2$ and
$\alpha/\nu=\beta/\nu=\gamma/\nu=2$.

Let $\mu_{c}$, $\rho_{c}$ and $P_{c}$ denote the critical chemical potential,
critical density and critical pressure in the thermodynamic limit. For a
finite size system, the deviation of these quantities from the thermodynamic
limit varies as

$x_{c}(L)-x_{c}\sim L^{-1/\nu},~{}~{}x=\mu,\rho,P.$ (17)

For the first order transitions, we use the following analysis. We use the
loci of the zeros of the partition function to determine the number of phase
transitions. The nature of the transition is established from the angle at
which the zeros approach the real axis, as well as the non-convexity
properties of the entropy. $\mu_{c}(L)$ is obtained from the partition
function zeros, analysis of the non-convex (NC) behavior of the entropy (see
Sec. III.4 and Ref. [60, 53] for details), and the value of $\mu$ at which
susceptibility is maximum. The coexistence densities [$\rho_{f}(L)$ and
$\rho_{s}(L)$] of first order transition are also estimated from NC analysis.

To estimate the critical behavior accurately, the relevant thermodynamic
quantities were calculated with a step size of $\Delta\mu=10^{-5}.$ Errors in
each data point are standard errors obtained from 16 independent simulations
using different sequences of random numbers. Errors in the final estimate of
critical parameters are fitting errors.

###  III.1 $1$-NN

The $1$-NN model, or the hard hexagon model, is exactly solvable [3, 4]. There
is a single continuous phase transition from a disordered phase to the
sublattice phase with increasing particle density. In the ordered phase,
particles preferentially occupy one of the three sublattices shown in Fig.
2(a). The critical parameters are
$\mu_{c}=\ln{[\frac{1}{2}(11+5\sqrt{5})]}=2.4060\dots$,
$\rho_{c}=\frac{3}{10}(5-\sqrt{5})=0.82917\dots$, $\alpha=1/3$, $\beta=1/9$,
$\gamma=13/9$, and $\nu=5/6$ [3]. The phase transition belongs to the three
state Potts model universality class. In addition, the density and
compressibility, for $\rho>\rho_{c}$, varies with $\mu$ as [49]

$\displaystyle\mu(\rho)$ $\displaystyle=$
$\displaystyle\log[2(\rho-2)(3-\rho)^{3}]$ (18)
$\displaystyle-\log[(27-108\rho+135\rho^{2}-66\rho^{3}+11\rho^{4})$
$\displaystyle+(-9+15\rho-5\rho^{2})^{3/2}(-1+3\rho-\rho^{2})^{1/2}],$
$\displaystyle\kappa(\rho)$ $\displaystyle=$
$\displaystyle\frac{1}{15}[(3-2\rho)(-1+3\rho-\rho^{2})^{1/2}$ (19)
$\displaystyle(-9+15\rho-5\rho^{2})^{-1/2}-(-1+3\rho-\rho^{2})],$

Figure 2:  Division of lattice sites into sublattices for the (a) $1$-NN model
(3 sublattices) and (b) $2$-NN model (4 sublattices).

To benchmark our flat histogram simulations, we compare the numerically
obtained density and compressibility for $\rho>\rho_{c}$ with the exact
results in Eqs. (18) and (19), as shown in Fig. 3(a) and (b) respectively. The
numerical results are in excellent agreement with the exact results except
near the critical point, where the numerical results deviate from the exact
result for the infinite system due to finite size effects.

Figure 3:  (Color online) The numerically obtained (a) density $\rho$ and (b)
compressibility $\kappa$ for the $1$-NN model are compared with the exact
results as given in Eqs. (18) and (19) for $\rho>\rho_{c}$. Data for different
system sizes are shown.

We estimate the exponents using the scaling laws in Eq. (16), as shown in Fig.
4. We obtain $1/\nu\approx 1.21(1)$, $\beta/\nu\approx 0.134(2)$ and
$\gamma/\nu\approx 1.737(6)$. These estimates are in good agreement with the
exact value of the exponents $1/\nu=1.2$, $\beta/\nu=0.133\dots$ and
$\gamma/\nu=1.733\dots$. We determine the critical chemical potential
$\mu_{c}$ and critical density $\rho_{c}$ by extrapolating the $\mu_{c}(L)$
and $\rho_{c}(L)$ to infinite systems size using Eq. (17). $\mu_{c}(L)$ is
identified as the $\mu$ at which $\chi$ is maximum and $\rho_{c}(L)$ is the
corresponding density. The extrapolation is shown in Fig. 5. We obtain
$\mu_{c}\approx 2.4064(6)$ and $\rho_{c}\approx 0.826(4)$, which are in good
agreement with the exact results $2.4060\dots$ and $0.82917\dots$.

Figure 4: (Color online) The power law scaling of (a) $t_{1}^{\mathrm{max}}$,
(b) $\chi_{1}^{\mathrm{max}}$, and (c) $\langle q_{1}(\mu_{c})\rangle$ with
system size $L$ for the $1$-NN model. The solid straight lines are best fits
to the data. Figure 5:  (Color online) The extrapolation of $\mu_{c}(L)$ and
$\rho_{c}(L)$ (inset) to infinite system size using the scaling in Eq. (17).
The data are for the $1$-NN model and we have chosen $\nu=5/6$. In the
extrapolation of $\rho_{c}(L)$ we have used the data for only $L\geq 96$.

In addition to these checks, we also confirm that the entropies of the fully
packed state and the state with one vacancy are reproduced correctly in the
simulations. Based on the above results, we conclude that the SCWL algorithm
can accurately obtain the entropy for all densities.

###  III.2  $2$-NN

The $2$-NN model is known to exhibit a single continuous phase transition from
a disordered phase to a sublattice-ordered phase. In the ordered phase, the
particles preferentially occupy one of the 4 sublattices shown in Fig. 2(b).
Due to the breaking of the four-fold symmetry, the phase transition is
expected to belong to universality class of the four state Potts model. The
exact values of critical exponents for four state Potts model are
$\alpha=2/3$, $\beta=1/12$, $\gamma=7/6$ and $\nu=2/3$ [68].

The $2$-NN model has been studied earlier using different numerical methods.
The known estimates for $\mu_{c}$ are $\approx 1.75989$ from transfer matrix
scaling [50], $1.75682(2)$ using Monte Carlo simulations [51], $1.75587$ using
transfer matrix methods [52] and $1.75398$ using tensor renormalization group
method [52]. The corresponding values for $\rho_{c}$ are $0.74856$ [50], and
$0.72(2)$ [51]. The numerical estimation of the critical exponents have
convergence issues due to logarithm corrections, both additive and
multiplicative, that are present for the four state Potts model [69]. The
earlier estimates for the critical exponents of the $2$-NN model were
$\nu=0.72$ and $\beta/\nu=1.115$ from transfer matrix scaling [50], and
$1/\nu=1.51(1)$ and $\beta/\nu=0.1257(7)$ from Monte Carlo simulations [51].

We estimate the critical exponents for the $2$-NN model using the scaling laws
in Eq. (16), as shown in Fig. 6. We obtain $1/\nu=1.48(2)$, $\beta/\nu\approx
0.124(3)$ and $\gamma/\nu\approx 1.812(8)$. The known exact value of the
exponents are $1/\nu=1.5$, $\beta/\nu=0.125$ and $\gamma/\nu=1.75$. The value
of $\gamma/\nu$ is overestimated, possibly because of ignoring logarithmic
corrections.

Figure 6:  The power law scaling of (a) $t_{2}^{\mathrm{max}}$, (b)
$\chi_{2}^{\mathrm{max}}$, and (c) $\langle q_{2}(\mu_{c})\rangle$ with system
size $L$ for the $2$-NN model. The solid straight lines are best fits to the
data.

We determine the critical chemical potential $\mu_{c}$ and critical density
$\rho_{c}$ by extrapolating $\mu_{c}(L)$ and $\rho_{c}(L)$ to infinite system
size using Eq. (17). The extrapolation is shown in Fig. 7. We obtain
$\mu_{c}=1.7568(4)$ and $\rho_{c}=0.7419(5)$. Our estimate for $\mu_{c}$ is
consistent with earlier estimates (see above). For $\rho_{c}$, the estimate is
closer to the transfer matrix estimate of $0.74856$ [50] rather than the more
recent Monte Carlo estimate of $0.72(2)$ [51]. From the inset of Fig. 7, it
can be seen that even for the smallest system size $\rho_{c}(L)>0.725$.

Figure 7:  (Color online) The extrapolation of $\mu_{c}(L)$ and $\rho_{c}(L)$
(inset) to infinite system size using the scaling in Eq. (17). The data are
for the $2$-NN model and we have chosen $\nu=2/3$.

###  III.3  $3$-NN Model

The $3$-NN model has been studied using matrix methods and series expansion
[23], tensor renormalization group method [52], evaporation-deposition
algorithms [54] and the SCWL algorithm [53]. The model shows a single first
order transition from a disordered phase to a sublattice-ordered phase. Based
on SCWL algorithm, we recently found that $\mu_{c}=4.4641(3)$ and critical
pressure is $P_{c}=0.6397(1)$. Both phases coexist between densities
$\rho_{f}=0.8482(1)$ and $\rho_{s}=0.9839(2)$ [53]. A detailed discussion of
the analysis as well as a summary of past work can be found in Ref. [53].

###  III.4  $4$-NN

Compared to $1$-NN, $2$-NN and $3$-NN models, there is not much known for
$k\geq 4$. The $4$-NN model has been studied earlier using tensor
renormalization group [52]. It was concluded that the $4$-NN model shows a
continuous phase transition with $\mu_{c}$ between $2.65$ and $2.7$.

We now present our results for the $4$-NN model in detail. We will follow a
similar analysis for larger $k$.

At full packing, the particles occupy one of the $N_{\mathrm{sub}}=9$
sublattices shown in Fig. 8(a). We first show that there is only one phase
transition from a disordered fluid phase to the high-density sublattice-
ordered phase.

Figure 8:  Division of lattice sites into sublattices for the (a) $4$-NN model
(9 sublattices) and (b) $5$-NN model (12 sublattices).

To do so, we examine the loci of zeros of the the partition function in the
complex $z$-plane. Figure 9 shows the zeros for $L=216$. The partition
function zeros (PFZ) pinch the real axis at only one point, showing the
presence of only a single phase transition. The locus forms a circle, hence
the angle it makes with the real axis is $\pi/2$, suggesting that the
transition is discontinuous [70, 71].

Figure 9:  Zeros of the grand canonical partition function in the complex
z-plane ($z=\mathrm{e}^{\mu}$) for the $4$-NN model. The locus of the zeros
forms a circle and pinches the positive real axis only at one point. The data
are for $L=216$

The first order nature of the transition can be further established by showing
that the entropy is non-convex in certain regions [60, 53]. Figure 10 shows
the non-convex part of the entropy for $L=36$ and the corresponding convex
envelope, shown by the solid line. Analysis of the non-convexity gives precise
estimate of critical parameters of the first order transition [60, 53]. Let
$N_{f}$ and $N_{s}$ denote the number of particles at the endpoints of the
convex envelope in the fluid side and sublattice side respectively (see Fig.
10). Then $\rho_{f}(L)=N_{f}/L^{2}\times N_{\mathrm{sub}}$ and
$\rho_{s}(L)=N_{s}/L^{2}\times N_{sub}$ are the densities at the boundaries of
the coexistence regime. The critical chemical potential

$\mu_{c}(L)=-\frac{S(N_{s})-S(N_{f})}{N_{s}-N_{f}}$ (20)

is the slope of the convex envelope. The system size dependent critical
parameters $\rho_{f}(L)$, $\rho_{s}(L)$ and $\mu_{c}(L)$, thus obtained, are
tabulated in Table 2 for different system sizes. The convergence to the
infinite system size is rapid.

Figure 10: The non-convex part of the entropy and the corresponding convex envelope construction for the $4$-NN model. The black solid arrows show location of $N_{f}$ and $N_{s}$. Significance of $N_{f}$ and $N_{s}$ is explained in text.  Table 2: Critical parameters from non-convexity in entropy of $4$-NN model. The last row shows the results of extrapolation to infinite $L$ using Eq. (17) with $\nu=1/2$ (see Fig. 11). $L$ | $\rho_{f}$ | $\rho_{s}$ | $\mu_{c}$
---|---|---|---
108 | 0.7319(1) | 0.9094(1) | 2.8593(1)
117 | 0.7331(1) | 0.9090(1) | 2.8606(1)
126 | 0.7340(1) | 0.9089(1) | 2.8632(1)
135 | 0.7347(1) | 0.9085(1) | 2.8632(1)
144 | 0.7356(1) | 0.9086(1) | 2.8643(1)
153 | 0.7358(1) | 0.9081(1) | 2.8644(1)
162 | 0.7364(1) | 0.9080(1) | 2.8651(1)
171 | 0.7366(1) | 0.9077(1) | 2.8655(1)
180 | 0.7374(1) | 0.9077(1) | 2.8661(1)
189 | 0.7376(1) | 0.9076(1) | 2.8663(1)
198 | 0.7379(1) | 0.9076(1) | 2.8666(1)
207 | 0.7382(1) | 0.9075(1) | 2.8669(1)
216 | 0.7384(1) | 0.9072(1) | 2.8671(1)
225 | 0.7386(1) | 0.9073(1) | 2.8675(1)
$\infty$ | 0.7404(1) | 0.9067(1) | 2.8696(1)

Figure 11 shows the extrapolation of $\mu_{c}(L)$ to infinite system size
using Eq. (17) with $\nu=2$. We determine $\mu_{c}(L)$ from the maximum of
$\chi$, NC analysis and the position of the PFZ closest to the origin. We
obtain $\mu_{c,4}=2.8696(2)$ from NC analysis, and $\mu_{c,4}=2.8697(2)$ from
susceptibility and PFZ. The coexistence densities are found to
$\rho_{f,4}=0.7404(2)$ and $\rho_{s,4}=0.9067(2)$ from NC (extrapolation not
shown). The critical pressure is estimated to be $P_{c,4}=0.3262(2)$.

Figure 11:  Variation of the critical chemical potential $\mu_{c}(L)$, with
system size L for $4$-NN model. The data are obtained from non-convexity (NC)
analysis, susceptibility $\chi$ and partition function zeros (PFZ). The error
in each data point was obtained from 16 independent simulations. Solid and
dashed lines are the best linear fit to the data.

Further results for the dependence of density, order parameter, Binder
cumulant, and pressure with chemical potential $\mu$ can be found in the
Supplemental Material [72] under section $4$-NN.

Note that our analysis shows that the transition is first order. This is in
contradiction to that obtained from TRG method where it was concluded that the
transition is continuous. Also, our estimate of $\mu_{c}=2.8696(2)$ is outside
the range of $2.65$ and $2.7$ estimated using TRG in Ref. [52]. The better
accuracy is presumably due to the SCWL algorithm accessing the high-density
states in an efficient manner.

###  III.5  $5$-NN Model

In the $5$-NN model, at full packing, the particles occupy one of the
$N_{\mathrm{sub}}=12$ sublattices shown in Fig. 8(b). In earlier work it was
argued, based on the tensor renormalization group, that the model exhibits a
single first order transition with $\mu_{c}$ roughly in the range
$4.4<\mu_{c}<4.5$ [52].

The analysis we present for the $5$-NN model (and higher $k$) is same as that
of the $4$-NN model, and we only summarize the results. We simulate system
sizes up to $L=180$. We first show that there is only one phase transition
from a disordered fluid phase to the high-density sublattice-ordered phase.
Figure 12 shows the PFZ for $L=180$. The locus of the zeros form a circle and
pinch the real axis at only one point. We conclude that there is only one
phase transition in the system. Since the locus is a circle, the angle of
approach of the leading zeros is $\pi/2$, suggesting that the transition is
discontinuous. We note that for smaller $L$, the PFZ of $5$-NN model also
forms an inner circle. However, as $L$ increases, these two circles merge and
become indistinguishable.

Figure 12:  Zeros of the grand canonical partition function in the complex
z-plane ($z=\mathrm{e}^{\mu}$ ) for $5$-NN model. The locus of the zeros forms
a circle and pinches the positive real axis only at one point.

We find that the entropy is non-convex, giving direct proof of the first order
nature of the transition. From the NC analysis, we obtain $\rho_{f}(L)$,
$\rho_{s}(L)$, and $\mu_{c}(L)$ for different system sizes. Figure 13 shows
the extrapolation to infinite system size of $\mu_{c}(L)$ obtained from non-
convexity analysis, PFZ and the peak position of susceptibility. The critical
chemical potential was estimated to be $\mu_{c,5}=4.720(3)$ from NC analysis
and PFZ data, and $\mu_{c,5}=4.720(1)$ from susceptibility. The endpoints of
the coexistence regime were similarly found to be $\rho_{f,5}=0.916(3)$ and
$\rho_{s,5}=0.988(3)$ from the NC analysis. We obtain the critical pressure to
be $P_{c,5}=0.3942(2)$.

Figure 13:  Variation of the critical chemical potential $\mu_{c}(L)$, with
system size L for $5$-NN model. The data are obtained from non-convexity (NC)
analysis, susceptibility $\chi$ and partition function zeros (PFZ). The error
in each data point was obtained from 16 independent simulations. Solid and
dashed lines are the best linear fit to the data.

Further results for the dependence of density, order parameter, Binder
cumulant, and pressure with chemical potential $\mu$ can be found in the
Supplemental Material [72] under section $5$-NN.

Note that our analysis shows that the transition is first order, consistent
with preliminary results obtained using TRG [52]. Also, our estimate of
$\mu_{c}=4.720(3)$ is outside the range of $4.4<\mu_{c}<4.5$ obtained in Ref.
[52].

###  III.6  $6$-NN Model

In the $6$-NN model, at full packing, the particles occupy one of the
$N_{\mathrm{sub}}=13$ sublattices shown in Fig. 14. The lattice can be divided
into 13 sublattices in two different ways, which we call as type-A and type-B,
as shown in Fig. 14. The nature and location of the transition is not known
for this model as it has not been studied earlier.

Figure 14:  Division of lattice sites into sublattices for the $6$-NN model
(13 sublattices). The division can be done in two equivalent ways as shown in
(a) and (b)

We simulate system sizes upto $L=169$. We first show that there is only one
phase transition from a disordered fluid phase to the high-density sublattice-
ordered phase. In Fig. 15 we show the PFZ for $L=169$. The locus of the zeros
form a circle and pinch the real axis at only one point. We conclude that
there is only one phase transition in the system. Since the locus is a circle,
the angle of approach of the leading zeros is $\pi/2$, suggesting that the
transition is discontinuous.

Figure 15:  Zeros of the grand canonical partition function in the complex
z-plane ($z=\mathrm{e}^{\mu}$ ) for $6$-NN model. The locus of the zeros forms
a circle and pinches the positive real axis only at one point.

We find that the entropy is non-convex, giving a direct proof of the first
order nature of the transition. From the NC analysis, we obtain $\rho_{f}(L)$,
$\rho_{s}(L)$, and $\mu_{c}(L)$ for different system sizes. Figure 16 shows
the extrapolation to infinite system size of $\mu_{c}(L)$ obtained from non-
convexity analysis, PFZ and the peak position of susceptibility. The critical
chemical potential is estimated to be $\mu_{c,6}=4.2574(4)$ from non-convexity
analysis and susceptibility and $\mu_{c,6}=4.2575(4)$ from PFZ data. The
endpoints of the coexistence regime were similarly found to be
$\rho_{f,6}=0.7898(1)$ and $\rho_{s,6}=0.9818(2)$ from the NC analysis. We
obtain the critical pressure to be $P_{c,6}=0.3287(1)$.

Figure 16:  Variation of the critical chemical potential $\mu_{c}(L)$, with
system size L for $6$-NN model. The data are obtained from non-convexity (NC)
analysis, susceptibility $\chi$ and partition function zeros (PFZ). The error
in each data point was obtained from 16 independent simulations. Solid and
dashed lines are the best linear fit to the data.

Further results for the dependence of density, order parameter, Binder
cumulant, and pressure with chemical potential $\mu$ can be found in the
Supplemental Material [72] under section $6$-NN.

###  III.7  $7$-NN Model

In the $7$-NN model, at full packing, the particles occupy one of the
$N_{\mathrm{sub}}=16$ sublattices shown in Fig. 8(b). The nature and location
of transition are not known for this model as it is not studied earlier.

Figure 17:  Division of lattice sites into sublattices for the $7$-NN model
(16 sublattices).

We simulate system sizes up to $L=192$. We first show that there is only one
phase transition from a disordered fluid phase to the high-density sublattice-
ordered phase. In Fig. 18 we show the PFZ for $L=192$. The locus of the zeros
form a circle and pinch the real axis at only one point. We conclude that
there is only one phase transition in the system. Since the locus is a circle,
the angle of approach of the leading zeros is $\pi/2$, suggesting that the
transition is discontinuous.

Figure 18:  Zeros of the grand canonical partition function in the complex
z-plane ($z=\mathrm{e}^{\mu}$ ) for $7$-NN model. The locus of the zeros forms
a circle and pinches the positive real axis only at one point.

We find that the entropy is non-convex, giving a direct proof of the first
order nature of the transition. From the NC analysis, we obtain $\rho_{f}(L)$,
$\rho_{s}(L)$, and $\mu_{c}(L)$ for different system sizes. Figure 19 shows
the extrapolation to infinite system size of $\mu_{c}(L)$ obtained from non-
convexity analysis, PFZ and the peak position of susceptibility. The critical
chemical potential was estimated to be $\mu_{c,7}=3.9326(8)$ from non-
convexity analysis and $\mu_{c,7}=3.9328(8)$ from susceptibility and
$\mu_{c,7}=3.9329(7)$ from PFZ data. The endpoints of the coexistence regime
were similarly found to be $\rho_{f,7}=0.7469(4)$ and $\rho_{s,7}=0.9709(3)$
from the NC analysis. We obtain the critical pressure to be
$P_{c,7}=0.2471(1)$.

Figure 19:  Variation of the critical chemical potential $\mu_{c}(L)$, with
system size L for $7$-NN model. The data are obtained from non-convexity (NC)
analysis, susceptibility $\chi$ and partition function zeros (PFZ). The error
in each data point was obtained from 16 independent simulations. Solid and
dashed lines are the best linear fit to the data.

Further results for the dependence of density, order parameter, Binder
cumulant, and pressure with chemical potential $\mu$ can be found in the
Supplemental Material [72] under section $7$-NN.

##  IV Summary and Discussion

In this paper, we studied the phase transitions in the $k$-NN hard core
lattice gas model on the triangular lattice. In this model, a particle
excludes all the sites up to the $k$-th next-nearest neighbors from being
occupied by another particle. We obtain the phase diagram and quantify the
critical behavior for $k\leq 7$ using the SCWL flat histogram algorithm, which
incorporates rejection free cluster moves to evolve the system.

We use the $1$-NN model or the hard hexagon model, for which an exact solution
is known, to benchmark our simulations. For the $2$-NN model, we obtain better
estimates for the critical density. We show that the $4$-NN to $7$-NN models
undergo only a single first order phase transition from a low-density
disordered phase to a high-density sublattice-ordered phase. For each of these
models, we obtain accurate estimates of the critical chemical potential, the
densities of the disordered and sublattice phases in the coexistence regime,
and the critical pressure. The quantitative results are summarized in Table 3.
The phase diagram in the $\mu$-$k$ plane and the $\rho$-$k$ plane are shown in
Fig. 20. Though $\mu_{c}$ on an average increases with $k$ [see Fig. 20(a)],
we are unable to quantify the increase based only on results for $k\leq 7$.
For the coexistence densities, with increasing $k$, we observe that $\rho_{s}$
tends to $1$ while $\rho_{f}$ saturates around $0.8$ [see Fig. 20(b)].

Table 3: Summary of the coexistence densities of the disordered phase ($\rho_{f}$), the sublattice phase ($\rho_{s}$), the critical chemical potential ($\mu_{c}$) and critical pressure ($P_{c}$) for different $k$. For $k=1$, the results are exact [3, 4], and the results for $k=3$ are taken from Ref. [53].  $k$ | $\rho_{f}$ | $\rho_{s}$ | $\mu_{c}$ | $P_{c}$
---|---|---|---|---
1 | - | 0.82917 | 2.4060 |
2 | - | 0.7419(5) | 1.7568(4) |
3 | 0.8482(1) | 0.9839(2) | 4.4641(3) | 0.6397(1)
4 | 0.7404(2) | 0.9067(2) | 2.8696(2) | 0.3262(2)
5 | 0.916(3) | 0.988(3) | 4.720(3) | 0.3942(2)
6 | 0.7898(1) | 0.9818(2) | 4.2574(4) | 0.3287(1)
7 | 0.7469(4) | 0.9709(3) | 3.9315(6) | 0.2471(1)
Figure 20:  The phase diagram of the $k$-NN model in the (a) $\mu$-$k$ plane
and (b) $\rho$-$k$ plane.

The results obtained in the paper allow us to compare the efficacies of the
TRG method and SCWL algorithm. The TRG method has been used to study the
$k$-NN model on the triangular lattice [52]. The method yields good results
for $k=1,2,3$. However, the results for $4$-NN and $5$-NN differ considerably
from the results that we obtain using SCWL. For the $4$-NN model, TRG method
predicts a continuous transition. However, we show clearly that the transition
is first order. In addition, the extrapolated critical chemical potential
obtained in this paper does not lie in the range predicted by TRG both for
$k=4$ and $k=5$. It would be of interest to see how the TRG method can be
improved to be able to study HCLG of particles with large excluded volume.

For infinite $k$, the $k$-NN model resembles the hard sphere model for which
an intermediate hexatic phase separates a fluid and orientationally ordered
high-density phase [73, 74, 75, 76, 77, 78]. Thus, one would expect that
signatures of these continuum phases are seen in the $k$-NN model for large
enough $k$. For the $k$ that we have studied, there is only a single phase
transition, and it does not appear that the continuum limit can be obtained
for a $k$ that is computationally tractable.

The SCWL algorithm, as seen in this paper, appears well-suited to study phase
transitions in HCLG models. The power of the flat histogram method is even
more when applied to density of states which depend on more than one variable.
A viable example is the binary gas, the simplest of which is the mixture of
$1$-NN and $0$-NN particles which shows a non-trivial phase diagram with a
tri-critical point [79, 80, 81, 82, 83, 84]. Similarly, multi-dimensional
density of states appear in HCLG with additional attractive interactions [35,
36]. These are promising areas for future study.

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