# NO PITCH LEFT BEHIND: ADDRESSING GENDER UNBALANCE
IN AUTOMATIC SPEECH RECOGNITION THROUGH PITCH MANIPULATION

###### Abstract

Automatic speech recognition (ASR) systems are known to be sensitive to the
sociolinguistic variability of speech data, in which gender plays a crucial
role. This can result in disparities in recognition accuracy between male and
female speakers, primarily due to the under-representation of the latter group
in the training data. While in the context of hybrid ASR models several
solutions have been proposed, the gender bias issue has not been explicitly
addressed in end-to-end neural architectures. To fill this gap, we propose a
data augmentation technique that manipulates the fundamental frequency ($f0$)
and formants. This technique reduces the data unbalance among genders by
simulating voices of the under-represented female speakers and increases the
variability within each gender group. Experiments on spontaneous English
speech show that our technique yields a relative WER improvement up to 9.87%
for utterances by female speakers, with larger gains for the least-represented
$f0$ ranges.

Index Terms—  Gender Bias, Data Augmentation, ASR.

##  1 INTRODUCTION

As society increasingly relies on automated systems for processing textual and
audio data, ensuring unbiased and inclusive outputs has become a critical
concern. One area of language technology that has garnered significant
attention in this regard is automatic speech recognition (ASR), where inherent
biases and errors persist despite remarkable advancements in the task.
Sociolinguistic variability, determined by demographic factors such as age,
gender, ethnicity, and dialect [1], has historically posed challenges for ASR
models, which are biased toward more frequent patterns in their training data
[2, 3, 4]. Among demographic factors, gender111 In this paper, we use “gender”
as a simplified approximation for sex-related differences within a binary
framework (female and male) to address the anatomical factors contributing to
phonetic variability. We acknowledge that phonetic variability among gender
groups can also arise from social factors that do not neatly fit into this
binary framework. However, in this study, we do not investigate these social
aspects.  is one of the most prominent, as female voices are often
disadvantaged in favor of better recognition performance on male voices. This
phenomenon has been observed in various instances such as YouTube’s automatic
captions [5],222However, gender bias was not observed in a second study [2].
hybrid HMM-DNN architectures applied to French broadcast data [6], and DNN
systems used for English read speech [7]. Although there have been sporadic
reports of higher recognition accuracy for female voices [8], these outliers
were attributed to the consistent adherence of female speakers to standard
pronunciations, as opposed to the higher proportion of disfluencies among male
speakers within specific datasets [8].

The key factor contributing to this discrepancy between female and male voices
is the under-representation of the former in the training data [6]. However,
efforts to address the gender-related performance gap have not explicitly
focused on tackling the data imbalance; rather, specific training strategies
have been explored. In the case of HMM and hybrid ASR systems, a common
approach is the creation of gender-dependent models [9, 10, 11], which are
designed to handle a specific gender without requiring to generalize across
them, thereby reducing the input space and potentially improving performance.
Additionally, speaker normalization [12, 13] and speaker adaptive [14, 15]
techniques have been employed to address speaker variability without
explicitly considering gender. Despite these efforts, the problem of gender
bias in ASR systems persists [6, 7] and the specific issue of data imbalance
has not been addressed with the advent of DNNs.

Along this direction, we propose a data augmentation technique specifically
tailored to enhance the robustness of ASR models to the less-represented
female voices. Inspired by other works on pitch333Throughout the paper, we
employ the term “pitch” to denote the perception of vocal height, with no
implicit reference to any specific acoustic parameter (e.g., the fundamental
frequency). perturbation strategies [16, 17, 18], our approach is based on the
manipulation of the fundamental frequency ($f0$) and the formants of a speech
segment, which are critical acoustic parameters distinguishing male and female
speech [19, 20]. The objective of these perturbations is to shift the pitch of
a segment spoken by a speaker of one gender towards characteristics typical
either of the same gender or of the opposite gender, thereby increasing the
variability within gender groups, in particular females, and decreasing the
representation gap between the two groups. Experiments conducted on English
spontaneous speech demonstrate the effectiveness of our method in improving
the performance of both male and female gender groups. Notably, the method
outperforms a baseline system trained on the original non-manipulated data,
yielding relative improvements for both genders (up to 9.87% for the under-
represented female group). Lastly, a focused analysis of the outputs unveils
that the highest gains are achieved for $f0$ values that are less represented
in the training data, emphasizing the potential of our method to build more
inclusive and fair systems.

##  2 RELATED WORKS

Data augmentation refers to increasing the amount and variability of the
training data by generating additional examples. By doing so, data
augmentation techniques aim at increasing the generalization capability of the
trained models so that they are more robust to a wider range of inference-time
inputs and conditions. This is particularly important for neural systems,
which require large training datasets and risk overfitting [21].

In ASR, various data augmentation techniques have been explored, in particular
after the advent of end-to-end neural systems. They mostly involve altering
the original audio or its feature representation to generate instances with
distinct acoustic characteristics. In particular, we can isolate four
approaches: noise augmentation, time-scale modification, time and frequency
masking, and pitch perturbation. Noise augmentation consists in adding various
types of noise to the clean speech signals in order to enhance the robustness
of acoustic models to background noise present in real-world environments [22,
23]. As such, they are unrelated to the robustness of a system with respect to
different speaker voices, which is the focus of our work. Time-scale
modification involves altering the speech tempo – possibly of only a part of
the original utterance, e.g. to simulate prolongation of vowels [24] – while
preserving the spectral envelope, and therefore the timbre, of the original
signal [25, 26]. For this reason, they do not address the variability of the
speaker pitch. Lastly, the most popular method is based on time and frequency
masking, which consists in randomly masking time segments and frequency bins
in the spectrogram, so as to enhance the model ability to handle noise and
irrelevant temporal and spectral variations in the audio signal [27].
Specifically, thanks to its effectiveness and efficiency, SpecAugment [27] has
become a standard component in training recipes of the most popular toolkits
used to develop state-of-the-art ASR systems, like
NeMo,444https://github.com/NVIDIA/NeMo.
ESPnet,555https://github.com/espnet/espnet.
Fairseq,666https://github.com/facebookresearch/fairseq. and
SpeechBrain.777https://github.com/speechbrain/speechbrain. However,
SpecAugment is not specifically designed to consistently obtain the vocal
variability required to train gender-fair models.

More in line with our objective of increasing gender variability, pitch
perturbation modifies the spectral envelope of the speech signal, introducing
variations in the vocal characteristics of the speaker to improve the
generalization capability of the model across speakers and natural pitch
variations. This is typically done through the so-called “vocal tract length
perturbation” (VTLP) scheme, which applies a warping function to the frequency
axis of the spectrum to alter the pitch while keeping the audio duration
unchanged [16, 28, 29, 26]. Various warping equations can be used for pitch
perturbation. The piecewise linear rule and the bilinear rule are commonly
employed methods [30], but non-linear functions based on the tonotopic
distance between formants and $f0$ have also been proposed [17]. Often, the
warping is also performed by modifying the original linear predictive coding
(LPC) spectrum [18, 31, 32, 33]. In some cases, a random perturbation may be
applied to each pole of the LPC synthesis filter [34] or to each segment of
the spectrum [35]. Regardless of the type of warping function and the features
to which they are applied, pitch manipulation techniques have demonstrated
effectiveness in addressing specific challenges in ASR, such as low-resource
languages [28, 29], disordered speech [26], and child speech [18, 31, 32, 35,
34, 33, 17]. However, it has not been specifically employed to address the
issue of gender bias, as we do here, and it typically involves a warping of
all the frequencies, while here we only focus on the acoustic parameters that
mostly differentiate males and females ($f0$ and formants).

##  3 PITCH MANIPULATION

It is widely recognized that male and female voices exhibit distinct acoustic
characteristics attributed to differences in vocal tract length and anatomy.
Two crucial parameters for distinguishing between male and female voices are
$f0$ and formants [19, 20]. Therefore, to fulfill our objective of enhancing
the representation of gender groups in the training data, and specifically the
under-represented female voices, we devise a data augmentation approach that
revolves around perturbing these two acoustic parameters in the audio segments
of the training data. Specifically, we propose two policies that aim at
increasing the diversity of training data of each gender group and rebalancing
the female/male voice ratio. In the following, we first describe our two
policies (§3.1), then we delve deep into the technical description of how the
audio manipulations are performed (§3.2).

###  3.1 Policies

We define two policies, namely Opposite and Random, to determine whether an
audio sample should be altered and how. The policies are applied independently
over the training epochs to each sample, so as to maximize the variability of
the training data and avoid additional storage requirements (as the
manipulated audio segments are computed on-the-fly and never saved). Both
policies rely on knowing the speaker’s gender for each utterance. In our
experiments (see §4.1), we use training data that already includes information
about the speakers’ gender. However, when this information is not available,
it can be inferred based on a simple spectral analysis (e.g., by considering
the averaged $f0$ value and comparing it to the typical $f0$ ranges for male
and female speakers).

Opposite. This policy aims at rebalancing the amount of female/male training
samples by shifting both the $f0$ and formants to values that are typical of
the opposite gender. Specifically, we control the female/male data ratio by
means of two probabilities, $p_{f\rightarrow{}m}$ and $p_{m\rightarrow{}f}$,
which respectively determine the probability of manipulating a segment uttered
by a female voice (toward a male voice) and the probability of manipulating a
segment uttered by a male voice (toward a female voice). As such, by choosing
$p_{f\rightarrow{}m}<p_{m\rightarrow{}f}$, we can increase the proportion of
female training samples.

Random. Under this policy, the vocal traits of the speaker are manipulated to
obtain either a voice typical of the opposite gender or a voice typical of the
same gender group. This decision is taken with a 50% probability for each
option. In the first case, both $f0$ and formant shifting are performed, like
in the Opposite policy. In the second case, where manipulation takes place
within the same gender group, we only shift the $f0$ within each gender-
specific frequency range without altering the formants, as preserving the
formant characteristics contributes to maintaining the distinct vocal traits
of each gender. The selection of the target gender for augmentation is done
randomly and independently from the original speaker gender. This approach
aims to reduce the skew in the female/male training data ratio, promoting
gender balance and simultaneously increasing the overall variability of the
data within each gender group. However, since only the portion of the training
data that undergoes manipulation becomes balanced, while the unmodified part
remains biased, this policy does not achieve perfect balancing of the training
data between gender groups. The extent of data that undergoes manipulation is
determined by the single hyperparameter $p_{r}$, i.e. the probability of
applying augmentation to a sample: the higher $p_{r}$ is, the more balanced
the training data is.

###  3.2 $f0$ and Formants Shifting

Once the chosen policy has defined how the input signal should be altered, our
data augmentation technique applies the $f0$ and formant shifting to the raw
waveform, before extracting the features that are passed to the ASR model (see
§4.2). The two operations are described below and have been implemented
through Praat [36].

$f0$ Shifting. First of all, we estimate the $f0$ values of the original
audio, defining the $F\mathit{0}$ contour and its median
$\tilde{f\mathit{0}}$. We then define a value representing the $f0$ median
($\tilde{f\mathit{0}}^{\prime}$) of the new manipulated audio.
$\tilde{f\mathit{0}}^{\prime}$ is sampled from a normal distribution whose
mean and standard deviation depend on the target gender defined by the chosen
policy. For female voices, we use 250 Hz as the mean and 17 as the standard
deviation so that the sampled value is between 199 Hz and 301 Hz with 99.7%
probability. For male voices, the mean is 140 Hz and the standard deviation is
20 to obtain a 99.7% probability in the range [80, 200] Hz. Once we have these
values, we compute the scaling factor $\alpha$ as the ratio
$\tilde{f\mathit{0}^{\prime}}/\tilde{f\mathit{0}}$, by which we scale
$F\mathit{0}$ to obtain the new target contour $F\mathit{0}^{\prime}$. Lastly,
we alter the original waveform by integrating $F\mathit{0}^{\prime}$ with the
Time-Domain Pitch-Synchronous Overlap and Add (TD-PSOLA) synthesis [37], as
this approach manipulates the fundamental frequency, without modifying the
entire spectrum.

Formant Shifting. This effect is obtained by manipulating the sampling
frequency. Specifically, the sampling frequency is scaled by a factor $\beta$,
which determines a scaling of all the frequencies and the duration. $\beta$ is
set to 1.2 if the target gender is female, 0.8 if it is male. Lastly, $f0$ and
duration are scaled back to the original value using TD-PSOLA (while formants
remain manipulated), and the sound is restored to the original sampling
frequency via sinc interpolation [38].

##  4 EXPERIMENTAL SETTINGS

###  4.1 Data and Evaluation Metrics

For training, we utilize the MuST-C corpus [39], which is based on TED talks
data and includes English audio, transcripts, and text translations into other
languages. Specifically, we focus on the largest section, English-Spanish (en-
es), and use its audio recordings (504h) along with their corresponding
transcripts to train our ASR models. The selection of MuST-C is driven by
several factors. First, it provides reliable manual annotation of speaker
gender, which eases the application of our data augmentation technique and the
evaluation of the performance on each gender group data. Second, it exhibits a
notable disparity between male and female segments, with approximately 70%
being male and 30% female, which is representative of the currently-available
data, making it an ideal testbed for evaluating our approach.888Librispeech
[40], for instance, is designed to be balanced in terms of female/male voices,
hence representing an unrealistic and artificial setting. Furthermore, MuST-C
comprises spontaneous speech segments delivered by speakers in front of live
audiences, and in public speech there is typically a wide use of $f0$
variations and a broader range of $f0$ values, due to its richer repertoire of
argumentative and emotionally colored elements [41]. This characteristic helps
to ensure that, also in a non-augmented condition, the models are trained on
various pitch variations.

We use the MuST-C dev set as validation set, and we evaluate on the official
tst-COMMON and tst-HE sets. We used both test sets to maximize the pool of
evaluation data available. Notably, tst-COMMON is larger (27 talks) but does
not exhibit a balanced distribution in terms of speakers’ gender (30% of
segments uttered by female and 70% by male speakers). In contrast, tst-HE (12
talks) offers a more balanced representation of male and female speakers (45%
of segments are uttered by female speakers, and 55% by male speakers),
therefore being particularly suitable to study the effectiveness of our
approach. The evaluation involves computing the word error rate
(WER),999Computed with JiWER 2.3.0: https://github.com/jitsi/jiwer. both on
the overall test sets and separately for each gender group. We also report the
relative WER decrease (WERR) with respect to the baseline.

We encode text into BPE [42] using SentencePiece [43] with 8,000 vocabulary
size, as per [44]. As audio features, we use log-compressed mel-filterbanks
(80 channels), computed over windows of 25 ms with a stride of 10 ms using
pykaldi.101010https://github.com/pykaldi/pykaldi.

###  4.2 Models

Our architecture is made of two 1D convolutional layers with stride 2, which
reduce the length of the input sequences by a factor of 4 as per [45], 12
Conformer [46] encoder layers, and 6 Transformer [47] decoder layers. The
embeddings have 512 features and the inner feed-forward layers 2048. We set
dropout to 0.1 and minimize label-smoothed cross entropy [48] with the Adam
optimizer [49] and Noam learning rate scheduler [47], where the maximum
learning rate is $2e^{-3}$. The objective function also includes the sum with
a Connectionist Temporal Classification (CTC) loss [50]. As default data
augmentation strategy (even when we adopt our Opposite and Random policies),
we apply SpecAugment [27] to every input utterance, masking 1 segment over the
frequency axis with maximum length 27, and 1 segment over the time axis with
maximum length 100. We train our models until the validation loss does not
decrease for 10 epochs, and average the five checkpoints around the best on
the validation set. The whole training takes $\sim$24 hours on 4 V100 GPU
(16GB of VRAM). Our code is based on fairseq-ST, and we use the padding-safe
Conformer implementation by [51].111111Source code available at
https://github.com/hlt-mt/FBK-fairseq/.

###  4.3 Hyperparameters

The two policies of our data augmentation technique, Opposite and Random,
require the definition of respectively two ($p_{f\rightarrow{}m}$ and
$p_{m\rightarrow{}f}$) and one ($p_{r}$) hyperparameters. To set their value
for our experiments, we choose those with the lowest overall WER on the
validation set.

$p_{f\rightarrow{}m}$ | $p_{m\rightarrow{}f}$ | WER ($\downarrow$)
---|---|---
30% | 70% | 10.92
70% | 30% | 11.08
50% | 50% | 11.30
70% | 70% | 11.36
100% | 100% | 11.91
Table 1:  WER on the dev set for Opposite with different
($p_{f\rightarrow{}m}$, $p_{m\rightarrow{}f}$) pairs.

$p_{r}$ | WER ($\downarrow$)
---|---
30% | 11.21
40% | 11.16
50% | 10.89
60% | 11.12
70% | 11.05
80% | 11.02
100% | 11.21
Table 2:  WER on the dev set for Random with different $p_{r}$ values.

For the Opposite policy, we test the following values for
$(p_{f\rightarrow{}m},p_{m\rightarrow{}f})$: (30%, 70%), (70%, 30%), (50%,
50%), (70%, 70%), (100%, 100%). As we can see in Table 2, the best WER is
obtained with (30%, 70%). As 70% of MuST-C is made of male voices, with this
combination the amount of female/male voices in the training data is perfectly
balanced. For the Random policy, we test all values between 30% and 80% with a
10% interval for $p_{r}$. Results in Table 2 show that we obtain the lowest
WER with $p_{r}$ at 50%. This means that the training data still contains more
male samples (parity is reached only at 100%), but the unbalance is halved
with respect to the original training data, with a distribution of around
40%-60% in favor of male speakers. Hence, we provide results and analyses
based on Opposite policy with $(p_{f\rightarrow{}m},p_{m\rightarrow{}f})=$
(30%, 70%), and Random policy with $p_{r}=$ 50%.

| tst-COMMON | tst-HE
---|---|---
| WER ($\downarrow$) | WERR% ($\uparrow$) | WER ($\downarrow$) | WERR% ($\uparrow$)
| Overall | F | M | Overall | F | M | Overall | F | M | Overall | F | M
Baseline | 10.10 | 8.51 | 10.79 | 0.00 | 0.00 | 0.00 | 7.53 | 8.72 | 6.59 | 0.00 | 0.00 | 0.00
\+ VTLP | 10.30 | 8.61 | 11.03 | -1.98 | -1.18 | -2.22 | 8.24 | 9.04 | 7.61 | -9.43 | -3.67 | -15.48
Random | 9.65* |  7.67* |  10.50* | 4.46 | 9.87 | 2.69 | 7.28 |  8.10* | 6.62 | 3.32 | 7.11 | -0.04
Opposite | 9.72* | 7.70* | 10.59 | 3.76 | 9.52 | 1.85 | 7.63 | 8.40 | 7.02* | -1.32 | 3.67 | -6.52

Table 3:  WER and WERR on the MuST-C tst-COMMON and tst-HE sets of the en-es
section with (Random and Opposite) and without (Baseline and Baseline + VTLP)
our data augmentation method. Results are reported also on the female (F) and
male (M) portions. * indicates that the improvement or the degradation of our
augmentation techniques over Baseline

is statistically significant (bootstrap resampling [52] with 95% CI and 1,000
samples).

| tst-COMMON | tst-HE
---|---|---
| WER ($\downarrow$) | WERR% ($\uparrow$) | WER ($\downarrow$) | WERR% ($\uparrow$)
| Overall | F | M | Overall | F | M | Overall | F | M | Overall | F | M
Random | 9.65 | 7.67 | 10.50 | 4.46 | 9.87 | 2.69 | 7.28 | 8.10 | 6.62 | 3.32 | 7.11 | -0.04
\- Formant Shifting | 9.80 | 7.98 | 10.58 | 2.97 | 6.23 | 1.95 | 7.37 | 8.33 | 6.62 | 2.13 | 4.47 | -0.04
\- Gender Switching | 10.14 | 8.46 | 10.86 | -0.40 | 0.59 | -0.65 | 7.57 | 8.33 | 6.97 | -0.53 | 4.47 | -5.77
Table 4:  Ablation study for the Random policy with WER and WERR (compared to
Baseline) on the en-es MuST-C tst-COMMON and tst-HE sets. Results are also
reported on the female (F) and male (M) portions.

##  5 RESULTS

###  5.1 Main Results

In Table 3, we report the WER and WERR results computed on the MuST-C tst-
COMMON and tst-HE sets, both for the Random and the Opposite policies. We
compare them to two baselines: one (Baseline) without any pitch perturbation
strategy; the other (Baseline + VTLP) with VTLP.121212 Following [16], we warp
each utterance with a factor sampled from a normal distribution (mean of 1 and
standard deviation of 0.1), but constrained within the range [0.9, 1.1], and a
boundary frequency of 4800 Hz. We implemented the latter to assess whether our
gender-specific approach outperforms generic pitch perturbation methods.

Firstly, it is worth noting that our Baseline outperforms by a margin of 0.42
the best published result (to the best of our knowledge) on tst-COMMON [51],
which achieved a WER score of 10.52. This demonstrates the validity of our
experimental settings. Secondly, VTLP does not provide any benefit and, in
fact, Baseline + VTLP increases the WER for both test sets and across genders.
Regarding the two gender subgroups, all systems have higher WER for female
speakers than for male speakers in tst-HE, as expected, whereas the trend is
the opposite in tst-COMMON. We speculate that this difference may be due to
factors like speakers’ accents. In tst-HE, most (10) of the 12 speakers are
indeed American, one is British (in the female subset) and one is Pakistani
(in the male subset). In contrast, tst-COMMON features a broader range of
accents, with only one-third of the speakers being American in both gender
groups. The male subgroup includes speakers with diverse accents, from
Northern Europe to Kenya, while three out of the five non-American female
speakers are native English speakers (Canadian and Australian). This accent
variability can significantly affect performance.

Looking at the comparison between Baseline and our solution, we notice that
both our methods (Random and Opposite) always improve the recognition accuracy
in the female subsets. In the male subsets, instead, Baseline is outperformed
on tst-COMMON, while on tst-HE it is on par with Random. The different
behavior between the two sets on male speakers may be motivated by the wider
range of variations in the tst-COMMON set, where non-native speakers are
highly represented. Interestingly, even in tst-COMMON, where the WER on female
samples is lower than that of male samples, the gains are always higher on the
female portion.

Among the policies, Random proves to be consistently the best on all test sets
and genders. Its superior scores demonstrate that, although Opposite produces
a training set with a more balanced 50%-50% distribution compared to Random’s
40%-60% distribution, the higher variability introduced by Random, which also
alters samples within the same gender group, proves to be more effective for
both female and male speech. The WER differences between Random and Baseline
are marginal in the male portions, showing limited yet statistically
significant gains on tst-COMMON (2.69% WERR), and comparable results on tst-HE
(-0.04% WERR). Conversely, the improvements in the female segments are
statistically significant, with error reductions ranging from 7.11% to 9.87%.
These gains translate into an overall WERR of 4.46%-3.32%.

All in all, we can conclude that our data augmentation technique with the
Random policy improves the recognition accuracy of female voices without
negatively affecting the recognition of male voices (where it also provides
marginal gains). This not only produces benefits to the overall performance
but leads to fairer and more robust systems, even when data for one gender is
scarce.

###  5.2 Ablation Study

We conduct an ablation study on our best policy (Random) to better understand
how each design choice impacts the results. The scores are reported in Table
4. In particular, we first remove the formant shifting operation (\- Formant
Shifting), to assess whether the $f0$ shifting alone is enough. Second, in
addition to shifting only the $f0$, we always sample the new $f0$ median
frequency from the range corresponding to the same gender (\- Gender
Switching) instead of randomly picking it from either the female or male
frequency range.

Looking at the overall WER, we notice that both formant shifting and gender
switching contribute to increasing recognition accuracy. Specifically, formant
shifting is critical mostly in the female subset, where it accounts for
$\sim$37% of the relative improvement with respect to the baseline (WERR
decreases from 9.87 to 6.23 on tst-COMMON and from 7.11 to 4.47 on tst-HE).
The male subset, instead, seems not to benefit from formant shifting: on tst-
HE the WER is identical, and on tst-COMMON the gap is minimal.

When we also force choosing an $f0$ frequency typical of the same gender (\-
Gender Switching), the overall WER gains vanish, demonstrating that small $f0$
variations are not enough to improve the performance of ASR systems. For
female voices, slight improvements are still noticeable compared to Baseline
(0.59% for tst-COMMON and 4.47% for tst-HE). However, the large drops with
respect to \- Formant Shifting attest that rebalancing the training data
distribution by shifting across genders is essential to yield considerable
improvements. For male voices, instead, scores are consistently lower also
compared to Baseline (negative WERR). Thus, $f0$ shifting within the same
gender group proves to be counter-productive for the most represented category
and at least $f0$ shifting across genders is required to retain the
performance.

Overall, the ablation study emphasizes the need for perturbing both $f0$ and
formants, as well as the importance of shifting voices from one gender to the
other. Our approach would not yield improvements without these design choices.

$80$$100$$120$$140$$160$$180$$200$$220$$240$$260$$280$$300$$320$$-10$$-5$$0$$5$$10$$15$$20$Average
$f0$ (Hz)WERR%FM Fig. 1:  WERR of Random compared to Baseline on tst-COMMON
and tst-HE (combined) for female (F) and male (M) speakers across various
average $f0$ ranges.
$80$$100$$120$$140$$160$$180$$200$$220$$240$$260$$280$$300$$320$$0$$2000$$4000$$6000$$8000$Average
$f0$ (Hz)Num. wordsFM Fig. 2:  Distribution of the number of words across the
average $f0$ ranges for tst-COMMON and tst-HE (combined), divided between
female (F) and male (M) speakers.

##  6 IMPACT ACROSS FREQUENCY RANGES

To better understand the source of recognition gains of our Random policy, we
also analyse performance variations as the average $f0$ varies across
segments. Figure 1 illustrates the WERR values of Random compared to Baseline
for the segments of tst-COMMON and tst-HE (combined), grouped by average $f0$
ranges. As a reference, in Figure 2 we also display the word counts per
segments grouped by average $f0$ in the two combined test sets.

Positive WERRs are observed across all ranges, with only three isolated
negative values, specifically around 90 Hz and 170 Hz for male voices, and 290
Hz for female voices, which are likely outliers caused by the limited amount
of test data in those ranges (see Figure 2). However, significant improvements
(around or higher than 10%) were observed in specific ranges, such as 100-110
Hz, 180 Hz and 220 Hz for male voices, and 200-230 Hz and 270-280 Hz for
female voices. Interestingly, these notable improvements do not align with the
median of the $f0$ distributions for men and women. Instead, they correspond
to the boundary of the male and female $f0$ ranges, describing an antiphase-
like pattern with respect to the distribution curves. In particular, the male
area spanning 200-220 Hz and the female area spanning 270-280 Hz represent
less typical frequencies. Additionally, the 200-220 Hz range is a span where
the two gender $f0$ distributions overlap, showing considerable and consistent
gains on both genders.

Overall, although improvements are observed across all ranges, some notable
peaks are located at the boundaries of the typical $f0$ values for male and
female voices. As these frequency spans are less common, they are less
represented in the data. The random perturbations introduced by our data
augmentation strategy increase the coverage of these less-likely ranges and of
a more diverse set of pitch variations. This results in enhanced recognition
accuracy, especially in regions where, regardless of gender, data might be
limited.

##  7 CONCLUSION

We presented a data augmentation technique designed to enhance the robustness
of ASR models to female voices, which are often under-represented in speech
training data. Our method involves manipulating critical acoustic parameters,
namely the fundamental frequency and formants, which distinguish male and
female speech. By introducing perturbations to these parameters, we generate
voice variants for both genders, using samples from both the same and opposite
genders. This approach increases the variability within each gender group and
helps balancing the gender distribution in the training data. Experimental
results on English public speech demonstrate that our technique significantly
improves the recognition accuracy of female voices, with relative WER
improvements up to 9.87%. The improvement does not come at the detriment of
performance on male voices, which indeed features limited benefits.
Additionally, when analyzing the WERR values across different $f0$ ranges, we
observed larger gains in regions considered less typical or potentially
ambiguous for both genders. As our approach effectively addresses gender bias
toward fairer ASR systems, our future plan is to extend it to closely related
tasks, like speech translation, where the problem is also attested and
pervasive [53].

##  8 ACKNOWLEDGEMENTS

This work is part of the project “Bias Mitigation and Gender Neutralization
Techniques for Automatic Translation”, which is financially supported by an
Amazon Research Award AWS AI grant. We also acknowledge the support of the
PNRR project FAIR - Future AI Research (PE00000013), under the NRRP MUR
program funded by the NextGenerationEU.

## References

  * [1] Peter Trudgill,  Sociolinguistic Variation and Change,  Edinburgh University Press, Edinburgh, 2001.
  * [2] Rachael Tatman and Conner Kasten,  “Effects of Talker Dialect, Gender & Race on Accuracy of Bing Speech and YouTube Automatic Captions,”  in Interspeech, 2017, pp. 934–938.
  * [3] Laureano Moro-Velázquez, Jaejin Cho, et al.,  “Study of the Performance of Automatic Speech Recognition Systems in Speakers with Parkinson’s Disease,”  in Interspeech, 2019, pp. 3875–3879.
  * [4] Siyuan Feng, Olya Kudina, et al.,  “Quantifying Bias in Automatic Speech Recognition,” 2021.
  * [5] Rachael Tatman,  “Gender and Dialect Bias in YouTube’s Automatic Captions,”  in 1st ACL Workshop on Ethics in Natural Language Processing, 2017\.
  * [6] Mahault Garnerin, Solange Rossato, and Laurent Besacier,  “Gender Representation in French Broadcast Corpora and Its Impact on ASR Performance,”  in 1st International Workshop on AI for Smart TV Content Production, Access and Delivery, 2019, p. 3–9.
  * [7] Mahault Garnerin, Solange Rossato, and Laurent Besacier,  “Investigating the Impact of Gender Representation in ASR Training Data: a Case Study on Librispeech,”  in 3rd Workshop on Gender Bias in Natural Language Processing, 2021, pp. 86–92.
  * [8] Martine Adda-Decker and Lori Lamel,  “Do Speech Recognizers Prefer Female Speakers?,”  in Interspeech, 2005, pp. 2205–2208.
  * [9] Yochai. Konig and Nelson Morgan,  “GDNN: A Gender-Dependent Neural Network for Continuous Speech Recognition,”  in IJCNN, 1992, pp. 332–337.
  * [10] Waleed Abdulla and Nikola K. Kasabov,  “Improving Speech Recognition Performance through Gender Separation,”  Changes, vol. 9, no. 10, 2001.
  * [11] Amer Sallam and Sreedhar Bhukya,  “Effect of Gender on Improving Speech Recognition System,”  International Journal of Computer Applications, vol. 179, no. 14, pp. 22–30, 2018.
  * [12] Diego Giuliani, Matteo Gerosa, and Fabio Brugnara,  “Improved Automatic Speech Recognition through Speaker Normalization,”  Computer Speech & Language, vol. 20, no. 1, pp. 107–123, 2006\.
  * [13] Hari Krishna, Sivanand Achanta, and Anil K. Vuppala,  “Incorporating Speaker Normalizing Capabilities to an End-to-End Speech Recognition System,”  in 6th Workshop on Spoken Language Technologies for Under-Resourced Languages, 2018, pp. 172–176.
  * [14] Peter Bell, Joachim Fainberg, et al.,  “Adaptation Algorithms for Neural Network-Based Speech Recognition: An Overview,”  IEEE Open Journal of Signal Processing, vol. 2, pp. 33–66, 2021\.
  * [15] Jinyu Li,  “Recent Advances in End-to-End Automatic Speech Recognition,”  APSIPA Transactions on Signal and Information Processing, vol. 11, no. 1, 2022.
  * [16] Navdeep Jaitly and Geoffrey E. Hinton,  “Vocal Tract Length Perturbation (VTLP) Improves Speech Recognition,”  in ICML, 2013.
  * [17] Gary Yeung, Ruchao Fan, and Abeer Alwan,  “Fundamental Frequency Feature Warping for Frequency Normalization and Data Augmentation in Child Automatic Speech Recognition,”  Speech Communication, vol. 135, pp. 1–10, 2021.
  * [18] Hemant K. Kathania, Sudarsana Reddy Kadiri, et al.,  “Study of Formant Modification for Children ASR,”  in ICASSP, 2020, pp. 7429–7433.
  * [19] Ralph O. Coleman,  “A Comparison of the Contributions of Two Voice Quality Characteristics to the Perception of Maleness and Femaleness in the Voice,”  Journal of Speech & Hearing Research, vol. 19(1), pp. 168–180, 1976.
  * [20] James M. Hillenbrand and Michael J. Clark,  “The Role of f0 and Formant Frequencies in Distinguishing the Voices of Men and Women,”  Attention Perception & Psychophysics, vol. 71(5), pp. 1150–1166, 2009.
  * [21] Xue Ying,  “An Overview of Overfitting and its Solutions,”  Journal of Physics: Conference Series, vol. 1168, no. 2, 2019.
  * [22] Awni Hannun, Carl Case, et al.,  “Deep Speech: Scaling up End-to-End Speech Recognition,” 2014.
  * [23] Mark J. F. Gales, Anton Ragni, et al.,  “Support Vector Machines for Noise Robust ASR,”  in ASRU, 2009, pp. 205–210.
  * [24] Tohru Nagano, Takashi Fukuda, et al.,  “Data Augmentation Based on Vowel Stretch for Improving Children’s Speech Recognition,”  in ASRU, 2019, pp. 502–508.
  * [25] Tom Ko, Vijayaditya Peddinti, et al.,  “Audio Augmentation for Speech Recognition,”  in Interspeech, 2015, pp. 3586–3589.
  * [26] Mengzhe Geng, Xurong Xie, et al.,  “Investigation of Data Augmentation Techniques for Disordered Speech Recognition,”  in Interspeech, 2020, pp. 696–700.
  * [27] Daniel S. Park, William Chan, et al.,  “SpecAugment: A Simple Data Augmentation Method for Automatic Speech Recognition,”  in Interspeech, 2019, pp. 2613–2617.
  * [28] Anton Ragni, Kate M. Knill, et al.,  “Data Augmentation for Low Resource Languages,”  in Interspeech, 2014, pp. 810–814.
  * [29] Naoyuki Kanda, Ryu Takeda, and Yasunari Obuchi,  “Elastic Spectral Distortion for Low Resource Speech Recognition with Deep Neural Networks,”  in ASRU, 2013, pp. 309–314.
  * [30] Chanwoo Kim, Minkyu Shin, et al.,  “Improved Vocal Tract Length Perturbation for a State-of-the-Art End-to-End Speech Recognition System,”  in Interspeech, 2019, pp. 739–743.
  * [31] Hemant K. Kathania, Avinash Kumar, and Mikko Kurimo,  “Vowel Non-Vowel Based Spectral Warping and Time Scale Modification for Improvement in Children’s ASR,”  in ICASSP, 2021, pp. 6983–6987.
  * [32] Hemant K. Kathania, Sudarsana R. Kadiri, et al.,  “A Formant Modification Method for Improved ASR of Children’s Speech,”  Speech Communication, vol. 136, pp. 98–106, 2022.
  * [33] Mohit Dua, Virender Kadyan, et al.,  “Spectral Warping and Data Augmentation for Low Resource Language ASR System under Mismatched Conditions,”  Applied Acoustics, vol. 190, 2022.
  * [34] Alexander Johnson, Ruchao Fan, et al.,  “LPC Augment: an LPC-based ASR Data Augmentation Algorithm for Low and Zero-Resource Children’s Dialects,”  in ICASSP, 2022, pp. 8577–8581.
  * [35] Vishwanath Pratap Singh, Hardik Sailor, et al.,  “Spectral Modification Based Data Augmentation For Improving End-to-End ASR For Children’s Speech,”  in Interspeech, 2022, pp. 3213–3217.
  * [36] Paul Boersma,  “Praat, a System for Doing Phonetics by Computer,”  Glot International, vol. 5, no. 9/10, pp. 341–345, 2001.
  * [37] Eric Moulines and Francis Charpentier,  “Pitch-Synchronous Waveform Processing Techniques for Text-to-Speech Synthesis Using Diphones,”  Speech Communication, vol. 9, no. 5–6, pp. 453–467, 1990.
  * [38] Claude E. Shannon,  “A Mathematical Theory of Communication,”  Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, 1948\.
  * [39] Roldano Cattoni, Mattia A. Di Gangi, et al.,  “MuST-C: A Multilingual Corpus for End-to-End Speech Translation,”  Computer Speech & Language, vol. 66, pp. 101155, 2021.
  * [40] Vassil Panayotov, Guoguo Chen, et al.,  “Librispeech: An ASR Corpus Based on Public Domain Audio Books,”  in ICASSP, 2015, pp. 5206–5210.
  * [41] Eva Strangert,  “Prosody in Public Speech: Analyses of a News Announcement and a Political Interview,”  in Interspeech, 2005, pp. 3401–3404.
  * [42] Rico Sennrich, Barry Haddow, and Alexandra Birch,  “Neural Machine Translation of Rare Words with Subword Units,”  in ACL, 2016, pp. 1715–1725.
  * [43] Taku Kudo and John Richardson,  “SentencePiece: A Simple and Language Independent Subword Tokenizer and Detokenizer for Neural Text Processing,”  in EMNLP, 2018, pp. 66–71.
  * [44] Mattia A. Di Gangi, Marco Gaido, et al.,  “On Target Segmentation for Direct Speech Translation,”  in AMTA, 2020, pp. 137–150.
  * [45] Changhan Wang, Yun Tang, et al.,  “Fairseq S2T: Fast Speech-to-Text Modeling with Fairseq,”  in AACL-IJCNLP, 2020, pp. 33–39.
  * [46] Anmol Gulati, James Qin, et al.,  “Conformer: Convolution-augmented Transformer for Speech Recognition,”  in Interspeech, 2020, pp. 5036–5040.
  * [47] Ashish Vaswani, Noam Shazeer, et al.,  “Attention Is All You Need,”  in NeurIPS, 2017, vol. 30.
  * [48] Christian Szegedy, Vincent Vanhoucke, et al.,  “Rethinking the Inception Architecture for Computer Vision,”  in CVPR, 2016, pp. 2818–2826.
  * [49] Diederik P. Kingma and Jimmy Ba,  “Adam: A Method for Stochastic Optimization,”  in ICLR, 2015.
  * [50] Alex Graves, Santiago Fernández, et al.,  “Connectionist Temporal Classification: Labelling Unsegmented Sequence Data with Recurrent Neural Networks,”  in ICML, 2006, p. 369–376.
  * [51] Sara Papi, Marco Gaido, et al.,  “When Good and Reproducible Results are a Giant with Feet of Clay: The Importance of Software Quality in NLP,” 2023.
  * [52] Philipp Koehn,  “Statistical Significance Tests for Machine Translation Evaluation,”  in EMNLP, 2004, pp. 388–395.
  * [53] Luisa Bentivogli, Beatrice Savoldi, et al.,  “Gender in Danger? Evaluating Speech Translation Technology on the MuST-SHE Corpus,”  in ACL, 2020, pp. 6923–6933.


# Lattice Fermionic Casimir effect in a Slab Bag and Universality

Yash V. Mandlecha 111Email address<EMAIL_ADDRESS><EMAIL_ADDRESS>Rajiv V. Gavai 222Email address<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Physics, Indian Institute of Science Education and Research
Bhopal, Bhopal 462066, Madhya Pradesh, India

###### Abstract

We apply the physically more appealing MIT Bag boundary conditions to study
the Casimir effect on the lattice. Employing the formalism of Ref. [1] to
calculate the Casimir energy for free lattice fermions, we show that the
results for the naive, Wilson and overlap fermions match the continuum
expressions precisely in the zero lattice spacing limit, as expected from
universality. In contrast to Ref. [1] where the result for the naive fermions
rapidly oscillates with the lattice size for both, the periodic (P) and anti-
periodic (AP) boundary conditions, no oscillations are observed with the
lattice size. Furthermore, the apparent violation of the universality for
naive fermion in Ref. [1] is shown to be cured by applying suitable series
extrapolation techniques, thus demonstrating that the Casimir energy for the
naive fermions with periodic/antiperiodic boundary conditions agrees with the
results for other free lattice fermions, and can be used to obtain the results
for the Dirac fermion in the zero limit of the lattice spacing.

###### keywords:

Casimir effect , lattice fermions , MIT Bag model

††journal: Elsevier

##  1  Introduction

The Casimir effect [2] arises as a consequence of restricting the allowed
zero-point fluctuations in the vacuum to discrete values due to the
experimental set-up. This vacuum distortion manifests as an attractive force
due to a finite lowering of the vacuum energy. These forces have been measured
between two parallel plates [3], and the phenomenon was confirmed to a high
degree of precision in several experiments [4, 5] recently. One might conclude
quantum phenomena like these to be esoteric, with a limited practical
consequence. But, as characteristic distances get smaller, their effects have
become increasingly significant in nanotechnology, for example, the silicon
integrated circuits based on micro-and nano-electromechanical systems [6]. The
Casimir effect also established the fact that energy density in certain
regions of space is negative relative to the ordinary vacuum energy. This has
exciting consequences, as such effects contribute to the stability of hadrons
and model colour-confinement [7, 8, 9], might make it possible to stabilize a
traversable wormhole [10], and provide significant insights into the
cosmological constant problem [11]. The Casimir effect has also been
generalized to the scalar and fermion fields.

The strong coupling between quark and gluon fields gives rise to various non-
perturbative phenomena in Quantum Chromodynamics. Given the role Casimir
effect plays in the MIT Bag model, it would be interesting to investigate its
role in the confinement of quarks in hadrons and spontaneous symmetry breaking
using the formalism defined in Ref. [1]. Clearly using a spacetime lattice for
such investigations appears to be a natural choice. An important motivation
for studying the Casimir effect for various lattice fermions is that their
form also appears in condensed matter systems, like Dirac semi-metals,
topological insulators and ultracold atom systems, which are currently active
areas of research. Chern Insulators are also shown to exhibit a repulsive
Casimir effect [12, 13]. The negative mass Wilson fermions and overlap
fermions correspond to the bulk and surface fermions of the topological
insulator, respectively [14]. Studying the Casimir effect for these lattice
fermions [15] is equivalent to studying it for corresponding topological
insulators, which can then be experimentally observed for very small lattice
sizes.

The Casimir effect for free lattice fermions was first studied in Ref. [1, 14]
using periodic and antiperiodic boundary conditions. Periodic and anti-
periodic boundary conditions are are often used as a standard theoretical
setup in lattice simulations or condensed matter physics. While the Casimir
energy for the Wilson and overlap fermions exactly matched the continuum
expressions expected for the fermionic Casimir effect in the zero limit of the
lattice spacing, it was observed that the result for the naive fermion rapidly
oscillated and approached two different expressions in the continuum limit for
the odd and even lattice sizes respectively. This lead Ref. [1] to claim that
the naive fermion cannot be used to compute Casimir energy for the Dirac
fermion, which is an apparent violation of the universality,

Motivated by the physically appealing fact that the MIT Bag boundary
conditions [7] prevent the fermionic current from crossing the plates, thereby
ensuring the confinement of fermions within the bag, we adopt these boundary
conditions in this work to study the fermionic Casimir energy. If the fermion
field $\psi$ is subjected to the MIT Bag boundary conditions, and confined
between the two parallel plates placed at $x^{1}=0$ and $x^{1}=d$, $\psi$
satisfies the following equation

$(1+i\gamma^{1})\psi|_{x^{1}=0,d}=0~{}.$ (1)

By substituting the standard positive and negative frequency solutions of the
Dirac equation in the equation for the MIT Bag boundary conditions one obtains
the following transcendental relations for the corresponding momentum $k_{1}$
[16, 17]:

$md~{}\sin(k_{1}d)/k_{1}d+\cos(k_{1}d)=0.$ (2)

For the massless fermion case, this reduces to

$k_{1}d=\left(n+\frac{1}{2}\right)\pi\;\;\;n\in\mathbb{Z}$ (3)

These boundary conditions will be used to calculate Casimir energy for the
lattice fermions in section (2).

The results we obtained from the MIT Bag boundary conditions were encouraging
and precisely matched the continuum results for all lattice fermions,
including the naive fermions, in the zero limit of the lattice spacing. As
expected, doubling was observed for the results for the naive fermion. In
order to investigate whether the choice of boundary conditions is crucial for
our results above, we focused our attention on the periodic and antiperiodic
boundary conditions for naive fermions again. Instead of examining the even
and the odd lattice size series separately we treat them together as one
series and applied known series extrapolation techniques. Contrary to the
claim of Ref. [1] of an apparent violation of the universality for the naive
fermion, we succeeded in demonstrating numerically that the Casimir effect for
Dirac fermions can also be computed from the naive fermion in the zero limit
of the lattice spacing since looked upon this way, the series converges to the
same result as given by other lattice fermions in the continuum limit. We
mention the known continuum results for both massless and massive fermions for
the MIT Bag boundary conditions for reference and comparison.

###  1.1 Massless fermions in $(3+1)$-dimensional spacetime

In continuum theory, one computes the zero-point energy using dimensional
regularization by summing over the odd integer modes obtained in (3). We
obtain the characteristic Casimir energy for massless fermions in (1+1)- and
(3+1)-dimensional spacetime as [18, 8] ($\hbar=c=\kappa_{B}$=1):

$E^{3+1,\text{cont},f}_{\text{Cas}}(d)=-\frac{7\pi^{2}}{2880d^{3}}\;\;;\;\;E^{1+1,\text{cont},f}_{\text{Cas}}(d)=-\frac{\pi}{24d}$
(4)

###  1.2 Massive fermions in $(3+1)$-dimensional spacetime

One obtains approximate solutions for the Casimir energy of massive fermions
(mf) in continuum theory for two limiting cases, namely $md\ll 1$ and $md\gg
1$. The Casimir force per unit area in the limit $md\gg 1$, which we shall
verify for the lattice fermions is [19, 20]:

$E^{3+1,\text{cont},mf}_{\text{Cas}}(d)=-\frac{3}{32}\sqrt{\frac{m}{\pi^{3}d^{5}}}e^{-2md}$
(5)

where the Casimir energy decays exponentially with $md$.

##  2 Casimir effect for lattice fermion fields in slab-bag

In this letter, out of the $D$ latticized spatial dimensions, we consider only
one ($x^{1}$) compactified by a boundary condition at $x^{1}=0,\;d$. The
corresponding spatial momentum component $p_{1}$ is discretized, while the
other momenta components remain continuous. Initially, modelling time is kept
continuous and temporal components of momentum are unaffected by
latticization. The non-zero lattice spacing is ‘$a$’, with an ultraviolet cut-
off scale ‘$1/a$’ in momentum space. The energy - momentum dispersion relation
for lattice fermions is obtained from the Dirac operator defined in a
relativistic fermion action. The energy is defined as:

$aE(ap)=a\sqrt{\mathcal{D}^{\dagger}\mathcal{D}}$ (6)

The Dirac operator, $\mathcal{D}$ includes the spatial momenta and mass but
not the temporal momenta. The boundary condition obtained in (3) constrains
the allowed modes and ensures that there is no particle current through the
walls, thereby confining the fermion fields inside the slab-bag. The MIT Bag
(B) boundary conditions are realized on the lattice as follows:

$ap_{1}\rightarrow ap^{\text{B}}_{1}(n)=\left(n+\frac{1}{2}\right)\frac{\pi
a}{d}=\left(n+\frac{1}{2}\right)\frac{\pi}{N}$ (7)

Here, $N=d/a$ is the lattice size in the compactified spatial direction. The
discretized momentum component under periodic (P) and antiperiodic (AP)
boundary conditions, on the other hand, is:

$ap_{1}\rightarrow
ap^{\text{P}}_{1}(n)=\frac{2n\pi}{N}\;\;;\;\;ap_{1}\rightarrow
ap^{\text{AP}}_{1}(n)=\frac{(2n+1)\pi}{N}$ (8)

Restriction of the momentum to the Brillouin zone bounds the integer $n$ for
periodic, antiperiodic boundary conditions is, $0\leq n^{\text{P,AP}}<N$, and
MIT Bag boundary conditions is $0\leq n^{\text{B}}<2N$. The Casimir energy on
lattice is also calculated by subtracting the zero-point energy at finite
lattice size $aE_{0}(N\rightarrow\infty)$ from the one at infinite lattice
size $aE_{0}(N)$ [1]:

$\displaystyle aE^{\text{3+1}}_{\text{Cas}}$
$\displaystyle=aE_{0}(N)-aE_{0}(N\rightarrow\infty)$
$\displaystyle=c_{\text{deg}}\int_{\text{BZ}}\frac{d^{2}ap_{\perp}}{(2\pi)^{2}}\left[-\sum_{n}aE(ap_{\perp},ap_{1}(n))+N\int_{\text{BZ}}\frac{dap_{1}}{2\pi}aE(ap)\right]$
(9)

The Casimir energy for the naive fermion is $2^{D}$ times the result for the
Wilson fermion due to the doubling multiplicity for the naive fermion. Here,
$c_{\text{deg}}$ is the degeneracy factor which accounts for the spin of
fermion and the naive fermion doubling. All comparisons are made taking care
of this factor, so that the equivalence between analytic results of naive and
Wilson can be established in (15) and (18) below.

###  2.1 Naive Fermions

The dispersion relation for naive fermions in ($D+1$)-dimensions obtained from
(6) is [21]:

$a^{2}E_{nf}(ap)=\sum_{k=1}^{D}\sin^{2}(ap_{k})+(am_{f})^{2}$ (10)

As in Ref. [1], the ($1+1$)-dimensions is treated analytically. By
substituting (10) in the definition (2), the expression for Casimir energy of
naive fermion with MIT Bag boundary conditions in ($D+1$)-dimensional
spacetime is:

$\displaystyle aE_{\text{Cas}}^{\text{D+1,B,}nf}\equiv
aE_{\text{0}}^{\text{D+1,B,}nf}(N)-aE_{\text{0}}^{\text{D+1,B,}nf}(N\rightarrow\infty)$
(11)
$\displaystyle=c_{\text{deg}}\int_{\text{BZ}}\frac{d^{D-1}ap_{\perp}}{(2\pi)^{2}}\Bigg{[}-\sum_{n}\sqrt{\sin^{2}\frac{(n+1/2)\pi}{N}+\sum_{k=2}^{D}\sin^{2}(ap_{k})+(am_{f})^{2}}\;\;+$
(12)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;N\int_{\text{BZ}}\frac{dap_{1}}{2\pi}\sqrt{\sum_{k=1}^{D}\sin^{2}(ap_{k})+(am_{f})^{2}}\Bigg{]}$

In the case for $D=1$, the series expansions are substituted, and the exact
expression is calculated using the Abel-Plana formulae for finite range [22,
23], to be:

$\displaystyle aE_{\text{Cas}}^{\text{1+1,B,}nf}$
$\displaystyle=\frac{N}{\pi}-\frac{1}{2}\csc(\frac{\pi}{2N})$ (13)
$\displaystyle=-4\left[-\frac{d}{4a\pi}+\frac{1}{8}\left\\{\frac{2d}{a\pi}+\frac{1}{6}\frac{a\pi}{2d}+\frac{7}{360}\left(\frac{a\pi}{2d}\right)^{3}+...\right\\}\right]$
$\displaystyle\Rightarrow E_{\text{Cas}}^{\text{1+1,B,}nf}$
$\displaystyle=-\frac{\pi}{24d}-\frac{7\pi^{3}a^{2}}{5760d^{3}}+\mathcal{O}(a^{4})$
(14)

Therefore, the Casimir energy obtained per unit area in the continuum limit
$a\rightarrow 0$, is:

$\lim_{a\to 0}E_{\text{Cas}}^{\text{1+1,B},nf}=-\frac{\pi}{24d}$ (15)

As expected, taking the naive fermion doubling correction in one spatial
dimension into account, the above result in continuum limit is equivalent to
the expression obtained for Dirac fermion (4). These expressions for Casimir
energy of naive fermions, without the doubling factor are plotted in Fig. 1.
Studies for the naive fermion for $D>1$ are done numerically. Results for
(2+1)- and (3+1)- dimensions have been verified to be exactly equal to the
continuum expressions. Only results for (3+1)-dimensions have been shown in
Fig. 1.

###  2.2 Wilson Fermion

We shall now discuss the Casimir energy for Wilson lattice fermion in
$(1+1)$-dimensional spacetime. The Wilson term proportional to $r$ in the
Dirac operator breaks the chiral symmetry and acts as a momentum dependent
mass term introduced to eliminate fermion doubling. The expression for Casimir
energy of Wilson fermion in ($D+1$) is similar to the one for naive fermion,
where the Wilson Dispersion relation, obtained using (6), is substituted in
(2). Using the Abel-Plana formulae for finite range like in the naive fermion
case, setting $r=1$, the following expressions are obtained:

$\displaystyle aE_{\text{Cas}}^{\text{1+1,B,W}}$
$\displaystyle=\frac{4N}{\pi}-\csc(\frac{\pi}{4N})$ (16)
$\displaystyle=-4\left[-\frac{d}{a\pi}+\frac{1}{4}\left\\{\frac{4d}{a\pi}+\frac{1}{6}\frac{a\pi}{4d}+\frac{7}{360}\left(\frac{a\pi}{4d}\right)^{3}+...\right\\}\right]$
$\displaystyle\Rightarrow E_{\text{Cas}}^{\text{1+1,B,W}}$
$\displaystyle=-\frac{\pi}{24d}-\frac{7\pi^{3}a^{2}}{23040d^{3}}+\mathcal{O}(a^{4})$
(17)

(a) (b) (c) (d)

Figure 1: Casimir energy for massless and positively massive Wilson and naive
fermion with MIT Bag boundary conditions. [(a),(b)] and [(c),(d)] represent
the MIT Bag Casimir energy for naive and Wilson fermion in ($1+1$)- and
($3+1$)-dimension respectively. Subfigure (d) also confirms the exponential
decay of Casimir energy for massive fermions with $N$ on a lattice.

Therefore, In the limit $a\rightarrow 0$, the Casimir energy obtained per unit
area is:

$\lim_{a\to 0}E_{\text{Cas}}^{\text{1+1,B,W}}=-\frac{\pi}{24d}$ (18)

The exact expressions obtained for the massless naive and Wilson fermion using
MIT Bag boundary conditions in $(1+1)$-dimension are (13) and (16)
respectively. The leading terms of the order $1/N$ in the zero lattice spacing
limit of these analytic results and the numerical results for naive lattice
fermion in ($3+1$)- dimensions are plotted in Fig. 1. Note that the
exponential suppression of Casimir energy for massive fermionic fields, which
is derived for the fermion mass limit $md\gg 1$ in (5), is also verified in
numerical calculations on a lattice as shown in Fig. 1(d) for comparison.

In $(D+1)$-dimensional spacetime, the smaller windows in each plot represent
the coefficient of Casimir energy, $N^{D}\cdot a\cdot E_{\text{Cas}}$, which
is constant in continuum and thus is used to observe the precision of
agreement between the lattice and continuum results. The result obtained for
the Wilson fermion matches with the continuum result (4) exactly.

(a) (c)

Figure 2: Casimir energy for overlap fermion with MIT Bag boundary conditions.
(a),(b) and (c) represent the Casimir energy in ($1+1$)- and ($3+1$)-dimension
respectively.

###  2.3 Overlap Fermion

The analytic calculation of the Casimir effect for overlap fermions in
(1+1)-dimensions turned out to be intractable. Thus, the results for the
overlap fermion are numerically calculated for all dimensions, using the MIT
Bag boundary conditions by substituting the corresponding Dirac operator into
the definition (2). The Dirac operator for overlap fermion is defined as:

$a{\mathcal{D}}_{\text{OV}}\equiv
2M_{0}\times\frac{(1+am_{f})+(1-am_{f})V}{2}$ (19)

where $am_{f}$ is the fermion mass and $V$ is defined as:

$V\equiv\gamma_{5}\;\text{sign}(\gamma_{5}a{\mathcal{D}}_{\text{W}})=\frac{{\mathcal{D}}_{\text{W}}}{\sqrt{{\mathcal{D}}_{\text{W}}^{\dagger}{\mathcal{D}}_{\text{W}}}}$
(20)

with ${\mathcal{D}}_{\text{W}}$ as the previously defined Wilson Dirac
operator with $r=1$ and a negative mass parameter -$M_{0}$. We used
$M_{0}=0.5$, $1.0$ and $1.5$. The results obtained for $(1+1)$\- and
$(3+1)$-dimensions are plotted in Fig. 2. It is easy to notice in ($1+1$)-
dimensions that the overlap fermion $M_{0}=1.0$ case and massless ($am_{f}=0$)
Wilson fermion case have equivalent dispersion relations. The expressions
obtained numerically for overlap fermions in higher dimensions, also match
precisely with the naive and Wilson fermions in the zero lattice spacing limit
$a\rightarrow 0$ and, subsequently, also with the continuum result.

##  3 Naive fermion and Series Extrapolation in large $N$-Limit

While it is reassuring that one obtains the same results for the naive, Wilson
and overlap fermions for the physically appealing MIT bag boundary conditions,
as shown in the section above, it still is a bit disturbing that such
validation of universality is not seen in the results of Ref. [1] which
employed periodic and antiperiodic boundary conditions. We therefore now turn
to examine the naive fermion case with those boundary conditions in an attempt
to shed more light on this problem. The exact expressions for Casimir energy
$aE_{\text{Cas}}$ of massless naive fermion on lattice in ($1+1$)-dimensional
spacetime with periodic and antiperiodic boundary conditions, were first
obtained in Ref [1, 14]. Their results are:

$\lim_{a\to
0}E_{\text{Cas}}^{\text{1+1,P,}nf}=\frac{\pi}{6d}\;\;\;\;\;(\text{odd
}N)\;\;;\;\;\lim_{a\to
0}E_{\text{Cas}}^{\text{1+1,P,}nf}=\frac{2\pi}{3d}\;\;\;\;\;(\text{even }N)$
(21)

for periodic boundary. Similarly, for antiperiodic boundary conditions, the
result is:

$\lim_{a\to
0}E_{\text{Cas}}^{\text{1+1,AP,}nf}=\frac{\pi}{6d}\;\;\;\;\;(\text{odd
}N)\;\;;\;\;\lim_{a\to
0}E_{\text{Cas}}^{\text{1+1,AP,}nf}=-\frac{\pi}{3d}\;\;\;\;\;(\text{even }N)$
(22)

The continuum Casimir energy for massless Dirac fermion using periodic and
antiperiodic conditions in (8) is found to be:

$E^{1+1,\text{cont},\text{P}}_{\text{Cas}}=\frac{\pi}{3d}\;;\;E^{1+1,\text{cont},\text{AP}}_{\text{Cas}}=-\frac{\pi}{6d}$
(23)

Thus, the Casimir energy depends on the type of boundary conditions. Comparing
(4) and (23) one observes a peculiar apparent violation of universality as
first shown in Ref [1]. The results depend on the whether the lattice size is
odd or even, and their respective continuum limits do not agree with (23).
Since $d=N\times a$, where $N$ now can be even or odd, $aE\propto 1/N$ with
differing constants for the continuum result as well as odd/even lattice
results, as seen in (21-23). These curves are shown with appropriate labels in
the panels of Fig. 3 for both periodic and antiperiodic boundary conditions.

(a) (b)

Figure 3: Oscillation of Casimir energy for positively massive and massless
naive fermion for odd and even $N$. [(a),(b)] represent the periodic and
antiperiodic boundary conditions in ($1+1$)-dimension respectively.

In the limit of large lattice size $N$, one can compute the difference between
the Casimir energy for odd and even lattice sizes from Ref. [1], as

$aE^{\text{1+1},nf}_{\text{Cas}}(\text{odd
}N)-aE^{\text{1+1},nf}_{\text{Cas}}(\text{even }N)=\mp 4\tan(\frac{\pi}{4N})$
(24)

for periodic and antiperiodic boundary conditions respectively, which is an
oscillating yet rapidly converging function. The oscillations as $N$ increases
from odd to even are depicted by the black dotted function in Fig. 3 in both
the panels. The continuum expressions are represented by the continuous green
line in Fig. 3 and are seen to average out the oscillations well. This
observation provided us a hint to look for suitable methods to understand the
behaviour of Casimir energy for naive fermion as part of a single series. In
(2), the Casimir energy is expressed as a difference between the integral and
summation of the same function in one spatial dimension on the lattice. It is
the sum-part which contributes to the rapidly oscillating behaviour above. The
Euler-Maclaurin formula333The routine attribute of numerical summation in
Wolfram Mathematica, "NSum" uses the Euler-Maclaurin formulae using
$\texttt{"Method}\rightarrow\texttt{EulerMaclaurin"}$ to extrapolate the
series after a specified number of terms in the summation using the
"NSumTerms" attribute.  is a series acceleration technique often applied in
such cases, where the last term of the series tends to 0 as
$n\rightarrow\infty$, and expresses the Euler-Maclaurin sum of a function as
an infinite series in terms of integral of the same function and higher-order
derivative differences.

(a) (b) (c) (d)

Figure 4: The convergence of Casimir energy for massless naive fermion to the
continuum expression after extrapolation is demonstrated by series
acceleration methods. [(a),(b)] and [(c),(d)] represent the periodic and
antiperiodic boundary conditions for naive fermion in ($1+1$)- and
($2+1$)-dimension respectively.

Employing the Euler-Maclaurin method as implemented in Wolfram Mathematica, we
obtained the results as exhibited in Fig. 4 (a),(b). The oscillatory Casimir
energy data points for $N<15$ are obtained from definition (2) for periodic
and antiperiodic boundary conditions. The extrapolated results are represented
by hollow markers in Fig. 4 for $N>15$, and signal their precise agreement
with the continuum results in (23). The Euler-Maclaurin method of numerical
sum extrapolates the data for points greater than the prescribed number to
terms to be considered using the NSumTerms attribute of Wolfram Mathematica.
In Fig. 4, this attribute is set to 15 for the agreement between the
extrapolated points and continuum results to be clearly visible. Changing the
attribute to higher values, say 20, does not alter the results qualitatively.
We find that the Casimir energy for naive fermion converge to a single
function which is exactly equal to the continuum result. Moreover, these
expressions also match the Wilson and overlap fermion results for both the
periodic and antiperiodic boundary conditions, as expected from universality.
In contrast to Ref. [1], we therefore argue that the continuum limit of the
Casimir energy expression (11) must be treated as a combined oscillatory
expression of the odd and even lattice sizes ($N$).

A similar extrapolation works for the $(2+1)$-dimensions as shown in Fig 4
(c),(d). However, some lattice points are seen to be off the mark as the
numerical computation involves calculating the integral of a sum approximated
by an expression of higher-order derivatives of the function $|\sin(2\pi
n/N)|$, over a continuous variable. This function is not differentiable at the
lattice points and thus leads to this discrepancy. Nonetheless, it is
sufficiently clear that the oscillating series indeed converges to the same
continuum expression as $N$ increases.

In [14], it was shown that the negative mass Wilson fermions behave
differently according to phases of the fermion mass and represent the bulk
fermions, whereas the overlap fermions represent the surface fermions in a
topological insulator. They also exhibit oscillation of Casimir energy on odd
and even lattice sizes in certain phases. In all such phases, it is observed
that these oscillatory results approach different expressions in the continuum
limit. We have verified that applying the same principle of treating both odd
and even $N$ as part of the same series, and applying the Euler-Maclaurin
series extrapolation method for the Casimir energy of the negative mass Wilson
fermions also leads to the correct result in the continuum limit. Therefore,
it is shown that the apparent universality violations in all known cases so
far are cured when the rapid oscillations between odd and even $N$ are treated
as a single oscillatory function.

##  4 Conclusion

In this paper, we first studied the continuum limit of the Casimir effect for
common lattice Dirac fermions in the light of the MIT Bag model. In section
(2), the MIT Bag boundary conditions were realized on a lattice for the first
time and used to calculate the Casimir effect for lattice fermions as per the
formalism developed in Ref. [1]. Following the studies done in Ref. [1, 14],
analytic expressions for the Casimir energy of naive (with doubling
correction) and Wilson fermions are obtained, along with the numerical results
for overlap fermions with MIT Bag boundary conditions. These expressions
matched the continuum results for Dirac fermions exactly as expected from
universality. Results in higher dimensions are also similar to the
($1+1$)-dimensions. Moreover, unlike the Casimir energy oscillations in the
naive fermion case, seen for both periodic and antiperiodic boundary
conditions, MIT Bag boundary conditions do not exhibit any such oscillations,
neither do the Dirac fermions in the continuum, nor the other lattice fermions
like Wilson and overlap fermions. Interestingly, negative mass Wilson fermions
also exhibit such oscillatory behaviour [14] for a certain range of mass
parameter $am_{f}$, suggesting it to be possibly related to specific structure
of the dispersion relations.

In Ref. [1, 14], the analytic expressions of Casimir energy for odd and even
lattice sizes for naive fermion were obtained and interpreted separately. This
led to different results for odd and even lattice sizes, which in turn did not
agree with the continuum results, suggesting an apparent universality
violation. We noticed that the difference between the Casimir energy for odd
and even lattice sizes given in (24), converges rapidly. This led us to
propose that the results obtained from two analytic expressions formed a
single oscillatory series. In the continuum limit of this single oscillatory
series, we obtained the same result as for other lattice fermions agreeing
with the continuum results, thus restoring universality. Thus, Casimir energy
for naive fermions can also be used to obtain results in the continuum limit,
in contrast to the claim in [1, 14]. Employing the Euler-Maclaurin series
extrapolation method, better suited for strongly oscillating series like the
one obtained, we showed the convergence of the oscillating analytic expression
between odd and even lattice sizes to the same expression as continuum. This
makes even the naive fermions with periodic/antiperiodic boundary conditions
suitable for such calculations.

##  5 Acknowledgements

RVG gratefully acknowledges the support of the Department of Atomic Energy,
Government of India through Raja Ramanna Fellowship. YVM gratefully
acknowledges the support of the INSPIRE fellowship, Department of Science and
Technology, Government of India.

## References

  * [1] T. Ishikawa, K. Nakayama, K. Suzuki, Casimir effect for lattice fermions, Phys. Lett. B 809 (2020) 135713.  arXiv:2005.10758, doi:10.1016/j.physletb.2020.135713.
  * [2] H. B. G. Casimir, On the Attraction Between Two Perfectly Conducting Plates, Indag. Math. 10 (1948) 261–263.
  * [3] G. Bressi, G. Carugno, R. Onofrio, G. Ruoso, Measurement of the casimir force between parallel metallic surfaces, Phys. Rev. Lett. 88 (2002) 041804.  doi:10.1103/PhysRevLett.88.041804.
  * [4] S. K. Lamoreaux, Demonstration of the casimir force in the 0.6 to $6\mu m$ range, Phys. Rev. Lett. 78 (1997) 5–8.  doi:10.1103/PhysRevLett.78.5.
  * [5] U. Mohideen, A. Roy, Precision measurement of the casimir force from 0.1 to $0.9{\mu}m$, Phys. Rev. Lett. 81 (1998) 4549–4552.  doi:10.1103/PhysRevLett.81.4549.
  * [6] T. Gong, M. R. Corrado, A. R. Mahbub, C. Shelden, J. N. Munday, Recent progress in engineering the casimir effect – applications to nanophotonics, nanomechanics, and chemistry, Nanophotonics 10 (1) (2021) 523–536.  doi:10.1515/nanoph-2020-0425.
  * [7] K. Johnson, The M.I.T. Bag Model, Acta Phys. Polon. B 6 (1975) 865.
  * [8] K. A. Milton, The Casimir effect: Physical manifestations of zero-point energy, World Scientific, 2001.  doi:10.1142/4505.
  * [9] M. N. Chernodub, V. A. Goy, A. V. Molochkov, Nonperturbative Casimir Effects in Field Theories: aspects of confinement, dynamical mass generation and chiral symmetry breaking, PoS Confinement2018 (2019) 006.  arXiv:1901.04754, doi:10.22323/1.336.0006.
  * [10] M. S. Morris, K. S. Thorne, U. Yurtsever, Wormholes, time machines, and the weak energy condition, Phys. Rev. Lett. 61 (1988) 1446–1449.  doi:10.1103/PhysRevLett.61.1446.
  * [11] G. Mahajan, S. Sarkar, T. Padmanabhan, Casimir effect confronts cosmological constant, Physics Letters B 641 (1) (2006) 6–10.  doi:https://doi.org/10.1016/j.physletb.2006.08.026.
  * [12] Y. Araki, T. Kimura, A. Sekine, K. Nomura, T. Z. Nakano, Phase structure of topological insulators by lattice strong-coupling expansion, PoS LATTICE2013 (2014) 050.  arXiv:1311.3973, doi:10.22323/1.187.0050.
  * [13] P. Rodriguez-Lopez, A. G. Grushin, Repulsive casimir effect with chern insulators, Phys. Rev. Lett. 112 (2014) 056804.  doi:10.1103/PhysRevLett.112.056804.
  * [14] T. Ishikawa, K. Nakayama, K. Suzuki, Lattice-fermionic Casimir effect and topological insulators, Phys. Rev. Res. 3 (2) (2021) 023201.  arXiv:2012.11398, doi:10.1103/PhysRevResearch.3.023201.
  * [15] L. Susskind, Lattice Fermions, Phys. Rev. D 16 (1977) 3031–3039.  doi:10.1103/PhysRevD.16.3031.
  * [16] S. G. Mamaev, N. N. Trunov, Vacuum expectation values of the energy-momentum tensor of quantized fields on manifolds with different topologies and geometries III, Sov. Phys. J. 23 (1980) 551–554.  doi:10.1007/BF00891938.
  * [17] S. Bellucci, A. A. Saharian, Fermionic casimir effect for parallel plates in the presence of compact dimensions with applications to nanotubes, Phys. Rev. D 80 (2009) 105003.  doi:10.1103/PhysRevD.80.105003.
  * [18] R. D. M. Paola, R. B. Rodrigues, N. F. Svaiter, Casimir Energy of Massless Fermions in the Slab-Bag, Modern Physics Letters A 14 (34) (1999) 2353–2361.  doi:10.1142/s0217732399002431.
  * [19] E. Elizalde, F. C. Santos, A. C. Tort, The Casimir energy of a Massive Fermionic field confined in a ($d+1$)-dimensionsal Slab-Bag, International Journal of Modern Physics A 18 (10) (2003) 1761–1772.  doi:10.1142/s0217751x03014186.
  * [20] M. B. Cruz, E. R. B. de Mello, A. Y. Petrov, Fermionic casimir effect in a field theory model with lorentz symmetry violation, Phys. Rev. D 99 (2019) 085012\.  doi:10.1103/PhysRevD.99.085012.
  * [21] C. Gattringer, C. B. Lang, Quantum chromodynamics on the lattice, Vol. 788, Springer, Berlin, 2010.  doi:10.1007/978-3-642-01850-3.
  * [22] A. A. Saharian, Generalized Abel-Plana formula as a renormalization tool in quantum field theory with boundaries, PoS IC2006 (2006) 019.  arXiv:hep-th/0609093, doi:10.22323/1.031.0019.
  * [23] A. A. Saharian, The generalized abel-plana formula with applications to bessel functions and casimir effect, arXiv: High Energy Physics - Theory (2007).