# CAT: Causal Audio Transformer for Audio Classification

###### Abstract

The attention-based Transformers have been increasingly applied to audio
classification because of their global receptive field and ability to handle
long-term dependency. However, the existing frameworks which are mainly
extended from the Vision Transformers are not perfectly compatible with audio
signals. In this paper, we introduce a Causal Audio Transformer (CAT)
consisting of a Multi-Resolution Multi-Feature (MRMF) feature extraction with
an acoustic attention block for more optimized audio modeling. In addition, we
propose a causal module that alleviates over-fitting, helps with knowledge
transfer and improves interpretability. CAT obtains higher or comparable
state-of-the-art classification performance on ESC50, AudioSet and
UrbanSound8K datasets, and can be easily generalized to other Transformer-
based models.

###### Index Terms:

audio classification, transformer, causal inference, attention

##  I Introduction

Audio classification is the task of analyzing audio recordings and assigning
the corresponding labels to the audio sample. Having witnessed the huge
success of Transformers [1] in the field of Natural Language Processing (NLP)
and Computer Vision (CV), they have been adapted to the audio modality and
have obtained the state-of-the-art performances. Part of the success is
attributed to the global receptive fields in Transformers to capture long-
range context in the audio signals. The existing audio transformer models
inherit the structure of the famous Vision Transformer (ViT) [2], mainly
because one of its most widely-used features, the Mel-Spectrogram, is of the
same format as an image. Instead of letting the x- and y-axis carry the
spatial information in image modeling, the x-axis of the Mel-Spectrogram
denotes temporal information while the y-axis carries the discrete frequency
information for audio inputs[3, 4, 5, 6, 7, 8].

While achieving superior performance, there are still open challenges in these
trendy audio Transformers: (1) The commonly used acoustic representations
leverage different time-frequency transformations and contain acoustic
semantics of various scales and different granularities [6, 5, 4], which can
hardly be effectively captured by ViTs [2] using conventional self-attentions,
patch sampling and embeddings; (2) The successful designs in visual tasks,
such as resnet [9], mixup [10], are shown to be more prone to over-fitting and
less generalizable in the acoustic domain; (3) Feature selection and
representation learning are critical in computer vision, yet have been often
overlooked in acoustic modeling.

In the face of the challenges, we propose a Causal Audio Transformer (CAT),
which includes Multi-Resolution Multi-Filter (MRMF) feature extraction, an
acoustic attention and a causal module. Spectrograms, as standard inputs in
audio models, are sampled using Fourier Transform, leading to a natural trade-
off between the temporal resolution and the frequency resolution. CAT balances
the trade-off by extracting comprehensive temporal-frequency feature patches
in multiple resolutions and filters which is later combined with 3D positional
embeddings. Then, the acoustic attention is proposed to effectively extract
semantics from such representations, taking the feature patches as inputs. The
patches from different filters are evenly distributed among the attention
heads, while we calculate the pair-wise attentions among patches from
different resolutions but within the same time frames, allowing information
exchange in various granularities. We further introduce a causal module to
establish the necessary and sufficient relationship between the learned
representation and predicted labels based on counterfactual reasoning[11, 12].
We extend [12] to the context of audio classification where a lower bound of
Probability of Necessity and Sufficiency (PNS) in terms of interventional
distribution is provided. Since such a lower bound can only be estimated from
the true distribution, we propose a causal module that learns a mapping from
the interventional distribution to the observational dataset (i.e., the one we
have), to alleviate over-fitting, improve interpretability and enable better
knowledge transfer.

CAT achieves SOTA performance on ESC50 [13], AudioSet [14] and UrbenSound8K
[15]. To sum up, our contributions are:

  * •

Causal Audio Transformer (CAT), with Multi-Resolution Multi-Filter (MRMF)
features extraction and an acoustic attention for acoustic modeling.

  * •

Causal loss with reconstruction block, that explicitly measures the feature
quality using Probability of Necessity and Sufficiency (PNS), alleviates over-
fitting, and improves knowledge transfer across different datasets.

Fig. 1: Left: Overview of proposed CAT; Right: the detailed structure of
proposed acoustic attention.

##  II Related Work

Audio classification: from CNN to Transformer. Audio classification is the
task of predicting labels of soundtracks. The great success of large-scale
training originates from the rise of CNN models in audio classification[8, 16,
17, 18, 19, 20]. More recently, with attention mechanism triumphs in the field
of Natural Language Processing (NLP) and Computer Vision (CV), self-attention-
based transformers are first applied to acoustic classification by AST [5],
where the spectrogram is used as the input of a ViT [2] backbone. Multiple
subsequent studies focus on improving the model efficiency: PaSST [4] proposes
a patch-out mechanism, and HTS-AT[6] adopts a hierarchical backbone [21]. To
further boost the performance, MBT[3] introduces an visual-acoustic fusion and
PLSA[7] introduces a model-agnostic framework. However, the network structures
in most recent studies are heavily borrowed from vanilla transformers
originally proposed for NLP and CV tasks, and are more prone to over-fitting
and less generalizable to acoustic data. Therefore, in this paper, we
introduce the acoustic attention which incorporates MRMF features specialized
for acoustic modeling. The proposed acoustic attention can be easily extended
to various Transformer blocks.

Causal inference in feature representation. The concept of causality is first
introduced in graphical probabilistic models[11]. Although causal inference is
a relatively new concept in audio classification, it has shown advances in
interpretable machine learning and representation learning. Modeling causality
among generating factors significantly encourages the learning of
representative features[22]. [23] adopts counterfactual information that helps
knowledge transfer across different domains. [12] proves a lower bound of the
learned representation being a necessary and sufficient condition for label
prediction. However, such a lower bound is in the interventional setting, and
thus cannot be estimated directly without knowing the true distribution. We
base our design on [12] and extend it to a causal module that maps the lower
bound to the observational dataset in the context of audio classification.

##  III Causal Audio Transformer

In this section, we introduce our Causal Audio Transformer (CAT). CAT first
extracts MRMF feature patches with 3D positional embedding. The feature
patches are then sent as input to the acoustic attention (Section III-A). Then
a causal module is proposed to alleviate over-fitting and improve
interpretability and knowledge transfer (Section III-B).

###  III-A Transformer For Acoustic Modeling

Image modeling is largely different from acoustic modeling because a 2D image
only carries spatial information along its axis, whereas acoustic features
convey both temporal and frequency information. Therefore, it is not trivial
to apply the conventional vision transformer for efficient acoustic modeling.
We bridge the gap by extracting MRMF features and acoustic attention.

Multi-resolution multi-filter feature extraction. Due to the nature of 1D
Fourier Transformation (FT), there is a trade-off between frequency and
temporal resolution in a temporal-frequency representation (e.g. spectrogram).
We propose to extract spectrograms of different temporal resolutions as:

$x_{\text{MR}}=\\{\text{FFT}_{\omega}(s)\\},\omega\in\\{\omega_{1},\omega_{2},...,\omega_{K}\\}$
(1)

where s denotes the input audio wave and $\text{FFT}_{\omega}$ denotes the 1D
FFT with a window size of $\omega$, $\\{*\\}$ represents a set. For each raw
input $x$, $x_{\text{MR}}$ contains k spectrograms, each of size
$\mathrm{R}^{T_{i}\times F},i=1,2,...,K$. $T_{i}$ represents the number of
time frames, and $F$ is the number of frequency bins. Such multi-resolution
spectrograms share similar intuitions with [24, 25], but ours is paired with
acoustic attention and 3D positional embedding, as introduced in the next
section, so that the information can be more effectively taken use of. We
further allow spectrograms under multiple filters to serve as parallel inputs
to different attention heads. For example, we could pass a copy of
$x_{\text{MR}}$ to the Mel-filter bank, obtaining the MRMF features
$x_{\text{MRMF}}\in\mathbb{R}^{\\{T_{i}\\}\times K\times F\times 2}$ as
follows:

$x_{\text{MRMF}}=\\{\text{mel}(x_{\text{MR}}),x_{\text{MR}}\\}$ (2)

Spectrograms of different resolutions have different sizes. There are multiple
ways to align them. One possibility is to concatenate them along the temporal
dimension, forming a super spectrogram. Another method is to project the
spectrograms to the same temporal dimension. We adopt the later setting based
on better empirical results.

Patching and 3D positional embedding. The input $x_{\text{MRMF}}$ are
patchified and aggregated with windows ${Wi}\in\mathrm{R}^{1\times K\times
F\times 1}$, and we use linear projections $\\{\xi\\}$ to match the dimension
to $M$. Different from the square-shaped input patching in computer vision
[2], the proposed patch $x_{p}$ contains frequency information of various
resolutions. Such design aggregates and preserves the important multi-scale
frequency information, which is critical for audio classification. Inspired by
3D positional embedding used in video transformer [26], we propose an acoustic
3D positional embedding to make the network aware of features extracted with
different window sizes and filters:

$pe=(g([pe_{1},pe_{3}])\otimes\mathbf{1}_{F})^{T}+pe_{2}\otimes\mathbf{1}_{T}$
(3)

where $g$ is a linear projection, $pe_{1}$ is a $T$ dimensional sinusoid time
embedding and $pe_{2}$ a $F$ dimensional frequency sinusoid embedding,
following [1]. $pe_{3}$ is the one-hot encoding of resolutions.
$\mathbf{1}_{F}$ is $F$-dimensional all-one vector.

Acoustic attention. We consequently design the acoustic attention to
incorporate MRMF feature patches in audio Transformers. Specifically, multi-
filtered inputs are fed into different heads and processed with separate
scaled dot-production (Figure 1) so that attentions with and without Mel-
filtering are calculated independently. Note that the proposed acoustic
attention works with various attention kernels (e.g. multi-head attention [1],
SWIN-attention [21], etc.). We use the SWIN kernel for our experiments.

###  III-B Causal module for better representation

The idea of encouraging the learned hidden representation to be a necessary
and sufficient cause of label prediction is first proposed in [12], by
maximizing a lower bound under intervention. [12] demonstrated it helps build
models with less over-fitting and more robustness. However, such a bound
requires an estimation of the unobserved confounder $C$, making it hard to
succeed in empirical setups. In this paper, we introduce a causal module with
a causal loss that applies addition constraints on bijective mappings to
resolve the issue. In this section, we first introduce the preliminaries from
a causal perspective, then introduce the proposed module.

Preliminaries Let $X$ denote the training data and $Y$ be the counterfactual
label function. we define a function $f$ that maps $X$ to its latent
representation $Z$, where $Z=f(X)$. The probability of sufficiency and
necessity of $\mathbb{I}\\{Z=z\\}$ for $\mathbb{I}\\{Y=y\\}$ is defined as
[12]:

$\text{PNS}_{Z=z,Y=y}=P(Y(Z\neq z)\neq y,Y(Z=z)=y).$ (4)

where $Y(Z=z)$ reasons about what the label would be if $Z=z$. To combat over-
fitting and ensure the feature is representative, our goal is to maximize the
PNS during training. [12] established the lower-bounded of PNS as:

$\text{PNS}\geq P(Y=y|do(Z=z))-P(Y\neq y|do(Z\neq z))$ (5)

where $do$ notation[11] stands for intervention of $Z$.

However, without knowing the true distribution, the corresponding label change
under the certain intervention of latent representation $Z$ cannot be directly
estimated. Instead of solving the problem through deconfounding, we derive an
estimation of the lower bound as:

$\footnotesize\text{PNS}\geq=\int P(Y|X)[P(X|f(X)=z)-P(X|f(X)\neq z)]dX$ (6)

We could observe from a probabilistic view that maximizing $P(Y|X)$ is the
classification objective, and $P(X|f(X)=z)$ is the probability of inferring
$X$ from a specific latent representation $Z=z$. Given that we want to
maximize such a lower bound of PNS, a feasible strategy is to maximize both
$P(Y|X)$ and $P(X|f(X)=z)-P(X|f(X)\neq z)$. The former is consistent with the
classification objective, while the second term indicates that we need an
accurate and bijective matching from $Z$ to $X$.

Reconstruction and causal loss We argue that such mapping can be accomplished
by conducting a reconstruction module and a “causal” loss, as shown in Figure
1. We purpose a simple reconstruction block that runs in parallel with a
classification layer so that $X$ and $Z$ approximate bijective relationship
measured by the reconstruction loss $l_{rs}$:

$l_{rs}=\lVert\Phi(Z)-X\rVert_{2}$ (7)

where $\Phi$ is parameterized by a reconstruction block as in Figure 1.

With the underlying requirement being satisfied, we further minimize the
causal loss ( [12] Equation 25 )as :

$lc=-\sum_{j=1}^{d}log\text{PNS}_{n}(f_{j}(X),Y|f_{-j}(X))$ (8)

where

$\text{PNS}(f_{j}(X),Y|f_{-j}(X))=\prod_{i=1}^{n}\text{PNS}_{f_{j}(X_{i}),Y_{i}|f_{-j}(X_{i})}$
(9)

The objective of CAT is the sum of a cross-entropy loss $l_{\theta}$, a
reconstruction loss, and a causal loss:

$L=\operatorname*{\arg\\!\min}(l_{\theta}+l_{c}+l_{rs})$ (10)

##  IV Experiments

###  IV-A Dataset

We evaluate CAT on three datasets. AudioSet[14]: contains over two million
10-second audio clip with 527 labels. We report mean average precision (mAP)
on evaluation set following [5]. ESC50[13]:  contains 50 categories, 2000
audio recordings. We report the averaged accuracy of 5-fold cross-validate and
mAP of 5 rounds of experiments following [5]. UrbanSound8K[15]: contains 8732
labeled sound excerpts of 10 classes. We report the average accuracy of
10-fold cross-validation following [27].

###  IV-B Implementation details

Following previous recipe [8, 5, 4], we use 32kHz as sample rate for all
experiments. We generate both the spectrogram and mel-spectrograms in a window
of 25ms and hop length of 10ms. In training, we adopt the inverse proportional
weighting[28] to balance the AudioSet training data, the mixup of 0.5 [29] is
used in all of our experiments.

###  IV-C Main Results

Table I: Evaluation on AudioSet and ESC50 dataset. We report mAP for AudioSet
and top1 average accuracy for ESC50.* denotes the ensemble version of the
model.

Models | Modality | AudioSet %wo | ESC50 %
---|---|---|---
PANN [8] | Audio | 0.434 | 94.7
AST [5] | Audio | 0.347 | 95.6
HTS-AT [6] | Audio | 0.471 | -
CAT | Audio | 0.478 | 96.4
PSLA* [7] | Audio | 0.474 | -
AST*[5] | Audio | 0.475 | 95.7
HTS-AT*[6] | Audio | 0.487 | 97.0
PaSST*[4] | Audio | 0.496 | 96.8
CLAP [30] | Audio+Text | - | 96.7
AnT[31] | Audio+Video | - | 95.7
CAT* | Audio | 0.489 | 96.9
CAT + PANN* | Audio | 0.491 | 97.2

Audioset We show CAT performance on AudioSet in Table I. Our model
significantly outperforms previous work based on ConvNets [8], as well as SOTA
transformer-based models by a noticeable margin (+$0.7\%$ comparing to [6]).
Note that both our CAT and HTS-AT share a similar SWIN backbone, the superior
performance demonstrates that our proposed audio-transformer with MRMF and
acoustic attention, plus the causal module are effective. It is worth
mentioning that following the same ensemble protocol, the CAT achieves
performance comparable to SOTA ensemble models. By simply ensemble the CAT
with PANN [8], we achieve performance comparable to the previous 9-model
ensemble [4] results. This also shows the CAT provides complementary
information to ConvNet based model.

ESC50 The CAT achieves SOTA performance on ESC50 dataset as well (Table I). We
observe a similar pattern that the proposed CAT outperforms previous SOTA [5,
6] by 0.8% and 0.2% with and without ensemble; demonstrating the effectiveness
of the proposed design. Note that ensemble CAT with other acoustic model[8]
achieves performance comparable to previous work trained on additional
modality, e.g. CAT is 0.7% better than AnT [30] which utilizes visual-acoustic
information. It can be anticipated that incorporating multimodality features
into CAT may further improve performance.

###  IV-D Ablations

Table II: Ablation on ESC50 and AudioSet. CL denotes causal loss and US8K
denotes urbanSound8K.

| AudioSet | ESC50
---|---|---
w/o CL | 0.474 | 95.9
w. CL | 0.478 | 96.4

(a)

Pre-train | AudioSet | ESC50
---|---|---
N/A | 0.451 | 89.4
ImageNet | 0.478 | 96.4

(b)

| AST[5] | CAT
---|---|---
- | 88.6 | 88.7
\+ MRMF | 88.9 | 89.1
\+ CL | 89.1 | 89.4

(c)

pre-train | val | w/o CL | w. CL
---|---|---|---
US8K | ESC50 | 84.1 | 86.7
ESC50 | US8K | 95.9 | 96.1

(d)

We perform ablations on ESC50 and AudioSet.

Effectiveness of causal loss.  Table 1(a) show that the causal loss helps with
performance on datasets of different scale.

Impact of pre-training. The imagenet pre-training helps with the performance
on both datasets (Table 1(b)), however, the impact is more significant on
ESC50. This is due to that the transformer generally requires large-scale data
to train.

Generalization We show that the proposed MRMF and causal loss generalize well
to different backbone (e.g. backbone w [21] and w/o [5] hierarchy), and
consistently improve the performance in Table 1(c).

Acoustic embedding quality We show that the proposed causal loss is able to
eliminate noise and thus the feature transfer better across datasets (Table
1(d)). We perform pre-training on small-scale datasets to ensure the
performance gain is from proposed causal loss instead of data overlapping.

##  V Conclusion and Future Work

In this paper, we propose Causal Audio Transformer, an acoustic transformer
designed for audio classification. CAT has MRMF features, an acoustic
attention and a causal module. We empirically show that the proposed CAT
achieves SOTA performance on multiple datasets. Our ablation also demonstrates
the MRMF and causal loss generalize well to different backbones.

## References

  * [1] Ashish Vaswani and et al.,  “Attention is all you need,”  Advances in neural information processing systems, vol. 30, 2017\.
  * [2] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al.,  “An image is worth 16x16 words: Transformers for image recognition at scale,”  arXiv preprint arXiv:2010.11929, 2020.
  * [3] Arsha Nagrani, Shan Yang, Anurag Arnab, Aren Jansen, Cordelia Schmid, and Chen Sun,  “Attention bottlenecks for multimodal fusion,”  Advances in Neural Information Processing Systems, vol. 34, pp. 14200–14213, 2021.
  * [4] Khaled Koutini, Jan Schlüter, Hamid Eghbal-zadeh, and Gerhard Widmer,  “Efficient training of audio transformers with patchout,”  arXiv preprint arXiv:2110.05069, 2021.
  * [5] Yuan Gong, Yu-An Chung, and James Glass,  “Ast: Audio spectrogram transformer,”  arXiv preprint arXiv:2104.01778, 2021.
  * [6] Ke Chen and et al.,  “Hts-at: A hierarchical token-semantic audio transformer for sound classification and detection,”  in ICASSP 2022-2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2022, pp. 646–650.
  * [7] Yuan Gong, Yu-An Chung, and James Glass,  “Psla: Improving audio tagging with pretraining, sampling, labeling, and aggregation,”  IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 29, pp. 3292–3306, 2021.
  * [8] Qiuqiang Kong, Yin Cao, Turab Iqbal, Yuxuan Wang, Wenwu Wang, and Mark D Plumbley,  “Panns: Large-scale pretrained audio neural networks for audio pattern recognition,”  IEEE/ACM Transactions on Audio, Speech, and Language Processing, vol. 28, pp. 2880–2894, 2020.
  * [9] Alex Krizhevsky, Geoffrey Hinton, et al.,  “Learning multiple layers of features from tiny images,”  2009\.
  * [10] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz,  “mixup: Beyond empirical risk minimization,”  arXiv preprint arXiv:1710.09412, 2017.
  * [11] Judea Pearl,  Causality,  Cambridge university press, 2009.
  * [12] Yixin Wang and Michael I Jordan,  “Desiderata for representation learning: A causal perspective,”  arXiv preprint arXiv:2109.03795, 2021.
  * [13] Karol J Piczak,  “Esc: Dataset for environmental sound classification,”  in Proceedings of the 23rd ACM international conference on Multimedia, 2015, pp. 1015–1018.
  * [14] Jort F Gemmeke, Daniel PW Ellis, Dylan Freedman, Aren Jansen, Wade Lawrence, R Channing Moore, Manoj Plakal, and Marvin Ritter,  “Audio set: An ontology and human-labeled dataset for audio events,”  in 2017 IEEE international conference on acoustics, speech and signal processing (ICASSP). IEEE, 2017, pp. 776–780.
  * [15] Justin Salamon, Christopher Jacoby, and Juan Pablo Bello,  “A dataset and taxonomy for urban sound research,”  in Proceedings of the 22nd ACM international conference on Multimedia, 2014, pp. 1041–1044.
  * [16] Paulo Lopez-Meyer, Juan A del Hoyo Ontiveros, Hong Lu, and Georg Stemmer,  “Efficient end-to-end audio embeddings generation for audio classification on target applications,”  in ICASSP 2021-2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2021, pp. 601–605.
  * [17] Jaehun Kim,  “Urban sound tagging using multi-channel audio feature with convolutional neural networks,”  Proceedings of the Detection and Classification of Acoustic Scenes and Events, 2020.
  * [18] Loris Nanni, Gianluca Maguolo, Sheryl Brahnam, and Michelangelo Paci,  “An ensemble of convolutional neural networks for audio classification,”  Applied Sciences, vol. 11, no. 13, pp. 5796, 2021.
  * [19] Hardik B Sailor, Dharmesh M Agrawal, and Hemant A Patil,  “Unsupervised filterbank learning using convolutional restricted boltzmann machine for environmental sound classification.,”  in InterSpeech, 2017, vol. 8, p. 9.
  * [20] Jonathan J Huang and Juan Jose Alvarado Leanos,  “Aclnet: efficient end-to-end audio classification cnn,”  arXiv preprint arXiv:1811.06669, 2018.
  * [21] Ze Liu and et al.,  “Swin transformer: Hierarchical vision transformer using shifted windows,”  in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp. 10012–10022.
  * [22] Mengyue Yang, Furui Liu, Zhitang Chen, Xinwei Shen, Jianye Hao, and Jun Wang,  “Causalvae: Disentangled representation learning via neural structural causal models,”  in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2021, pp. 9593–9602.
  * [23] Sharmila Reddy Nangi, Niyati Chhaya, Sopan Khosla, Nikhil Kaushik, and Harshit Nyati,  “Counterfactuals to control latent disentangled text representations for style transfer,”  in Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers), 2021, pp. 40–48.
  * [24] Boqing Zhu, Changjian Wang, Feng Liu, Jin Lei, Zhen Huang, Yuxing Peng, and Fei Li,  “Learning environmental sounds with multi-scale convolutional neural network,”  in 2018 International Joint Conference on Neural Networks (IJCNN). IEEE, 2018, pp. 1–8.
  * [25] Emad M Grais, Hagen Wierstorf, Dominic Ward, and Mark D Plumbley,  “Multi-resolution fully convolutional neural networks for monaural audio source separation,”  in International Conference on Latent Variable Analysis and Signal Separation. Springer, 2018, pp. 340–350.
  * [26] Yanyi Zhang and et al.,  “Vidtr: Video transformer without convolutions,”  in Proceedings of the IEEE/CVF International Conference on Computer Vision, 2021, pp. 13577–13587.
  * [27] Avi Gazneli, Gadi Zimerman, Tal Ridnik, Gilad Sharir, and Asaf Noy,  “End-to-end audio strikes back: Boosting augmentations towards an efficient audio classification network,”  arXiv preprint arXiv:2204.11479, 2022.
  * [28] Pavel Izmailov, Dmitrii Podoprikhin, Timur Garipov, Dmitry Vetrov, and Andrew Gordon Wilson,  “Averaging weights leads to wider optima and better generalization,”  arXiv preprint arXiv:1803.05407, 2018.
  * [29] Yuji Tokozume, Yoshitaka Ushiku, and Tatsuya Harada,  “Learning from between-class examples for deep sound recognition,”  arXiv preprint arXiv:1711.10282, 2017.
  * [30] Benjamin Elizalde, Soham Deshmukh, Mahmoud Al Ismail, and Huaming Wang,  “Clap: Learning audio concepts from natural language supervision,”  arXiv preprint arXiv:2206.04769, 2022.
  * [31] Yanpeng Zhao, Jack Hessel, Youngjae Yu, Ximing Lu, Rowan Zellers, and Yejin Choi,  “Connecting the dots between audio and text without parallel data through visual knowledge transfer,”  arXiv preprint arXiv:2112.08995, 2021.


# Quantum vacuum, rotation, and nonlinear fields

Antonino Flachi  Department of Physics, & Research and Education Center for
Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521,
Japan<EMAIL_ADDRESS>Matthew Edmonds  ARC Centre of Excellence in
Future Low-Energy Electronics Technologies, School of Mathematics and Physics,
University of Queensland, St Lucia, QLD 4072, Australia<EMAIL_ADDRESS>Department of Physics, & Research and Education Center for Natural Sciences,
Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan

###### Abstract

In this paper, we extend previous results on the quantum vacuum or Casimir
energy, for a noninteracting rotating system and for an interacting
nonrotating system, to the case where both rotation and interactions are
present. Concretely, we first reconsider the noninteracting rotating case of a
scalar field theory and propose an alternative and simpler method to compute
the Casimir energy based on a replica trick and the Coleman-Weinberg effective
potential. We then consider the simultaneous effect of rotation and
interactions, including an explicit breaking of rotational symmetry. To study
this problem, we develop a numerical implementation of zeta function
regularization. Our work recovers previous results as limiting cases and shows
that the simultaneous inclusion of rotation and interactions produces
nontrivial changes in the quantum vacuum energy. Besides expected changes
(where, as the size of the ring increases for fixed interaction strength, the
angular momentum grows with the angular velocity), we notice that the way
rotation combines with the coupling constant amplifies the intensity of the
interaction strength. Interestingly, we also observe a departure from the
typical massless behavior where the Casimir energy is proportional to the
inverse size of the ring.

##  I Introduction

Consider a periodic array of identical particles at low temperature confined
to a one-dimensional ring by means of a particular trapping potential. Then,
making such a system rotate at a constant velocity, i.e. uniformly, is a
trivial exercise. What we mean by “trivial” here is that an observer at rest
in the laboratory frame cannot detect that the device is rotating: the system
is invariant under a rotation of reference frames. Put in a more elegant way,
the gauge invariance associated with rotation is preserved. Obviously, if this
gauge invariance is in some way broken, then the above is no longer true and
rotation acquires a physical, in principle measurable, element or, in other
words, the observer at rest in the laboratory frame can detect rotation. It is
nontrivial, in fact interesting, to build up models where this symmetry can be
broken; however, in $1+1$ dimensions the Coleman-Hohenberg-Mermin-Wagner
theorem prevents this from happening dynamically Mermin:1966 ; Hohenberg:1968
; Coleman:1973 . The simplest and most natural way to break this gauge
invariance is to do this classically by enforcing nontrivial boundary
conditions at one point along the ring, thus breaking the periodicity of the
array: this is equivalent to an explicit symmetry breaking. A simple way to
achieve this is by placing an impurity somewhere along the ring, thus breaking
the uniformity of the array. A pictorial explanation of this is given in
Fig.1.

The “device” we have described above, whether rotating or static, is
susceptible to a Casimir force, arising from the deformations of its (quantum)
vacuum due to the compactness of the topology of the setup, i.e., the boundary
conditions Milton:2001 ; Bordag:2009 . If the ring is static (i.e., non
rotating) and periodic (i.e., periodic boundary conditions are imposed at any
one point along the ring), then quantum fluctuations are massless (i.e., long-
range) and induce a Casimir force that scales with the inverse size of the
ring. Then, the arguments given in the previous paragraph imply that making
such a periodic ring rotate should not change the Casimir force. However, if
we change the boundary conditions into nonperiodic ones, something interesting
happens: rotation becomes “physical”, quantum fluctuations have to obey
nontrivial boundary conditions, and as a result the Casimir force should
respond to variations in angular velocity.

A computation of the Casimir energy for a noninteracting scalar field theory
can be found in Refs. Chernodub:2012em ; Schaden:2012 , while a computation of
the Casimir energy for a scalar field theory with quartic interactions can be
found in Ref. Bordag:2021 . A calculation of the one-loop effective action for
a non-relativstic nonlinear sigma model with rotation can be found in Ref.
Corradini:2021yha . Here, we consider the simultaneous effect of both rotation
and interactions.

The paper is organized as follows. After introducing the notation by
illustrating the calculation of the Casimir energy for a real scalar field, we
proceed by reviewing the free-field rotating case and illustrate an
unconventional and computationally convenient way to calculate the Casimir
energy. We then move on to the interacting, rotating case and discuss how both
factors alter rather nontrivially the Casimir energy.

Figure 1: Both panels represent a ring rotating with an angular velocity
$\Omega$. On the left, periodic boundary conditions are imposed at any one
point, and the two observers (represented by the two reference frames $O$ and
$O^{\prime}$, the latter rotating with the ring) cannot detect that the ring
is rotating. The ring on the right has an impurity (represented by a black
dot) that breaks the rotational symmetry. In this second case, for the
observer $O^{\prime}$ at rest with the ring (i.e., rotating with the ring) the
position of the impurity would not change (i.e. the boundary conditions are
time independent), while for the laboratory observer $O$, rotation is evident
and the boundary conditions imposed at the impurity are time dependent.

##  II Free case

To introduce our notation, let us consider the case of a noninteracting
single-component, real scalar field, whose Lagrangian density, in absence of
rotation, is

$\displaystyle\mathcal{L}={1\over
2}\left[\left({\partial\phi\over\partial\hat{t}}\right)^{2}-{1\over
R^{2}}\left({\partial\phi\over\partial\hat{\varphi}}\right)^{2}\right].$ (1)

$R$ is the radius of the ring and $\hat{\varphi}\in\left[0,2\pi\right)$. In
the first part of this section we simply repeat the calculation of Refs.
Chernodub:2012em ; Schaden:2012 ; in the second part of this section, we carry
out the calculation of the vacuum energy in a simpler way. The field $\phi$ is
assumed to satisfy certain boundary conditions at the end points,
$\hat{\varphi}=0,~{}2\pi$. Taking for concreteness Dirichlet boundary
conditions, we have

$\displaystyle\phi(\hat{t},\hat{\varphi})\Big{|}_{\hat{\varphi}=0}=\phi(\hat{t},\hat{\varphi})\Big{|}_{\varphi=2\pi}=0.$
(2)

In this case the system is static and a potential barrier at
$\hat{\varphi}=0=2\pi$ enforces the above Dirichlet boundary conditions.
Changing the potential barrier will change the boundary conditions. The
Casimir energy is defined as

$\displaystyle E=\int_{0}^{2\pi R}dx\rho(x),$ (3)

where $\rho(x)=\langle T_{00}(x)\rangle$ is the energy density component of
the energy-momentum tensor in the vacuum state. The expression above can be
written as the regularized sum over the zero-point energy levels $E_{n}$ of
the fluctuations,

$\displaystyle E=\sum_{n}{}^{{}^{\prime}}E_{n},$ (4)

which defines the Casimir energy. The prime in the sum is a reminder that the
summation is divergent and requires regularization. Throughout our paper we
adopt zeta function regularization Toms:2012 . A straightforward calculation
of the Casimir energy gives Bordag:2009

$\displaystyle E=-\pi/(24\times L),$ (5)

showing that the energy scales as the inverse size of the ring $L=2\pi R$.
This is the one-dimensional scalar equivalent of the usual electromagnetic
Casimir energy; see e.g. Ref. Milton:2001 . The numerical coefficient and the
sign in Eq. (5) depend on the boundary conditions, on the features of the
vacuum fluctuations (e.g., spin) and of the background (e.g., an external
potential or a nontrivial background geometry in more than one dimension). In
the present case the Casimir force is attractive.

##  III Spinning the ring

In this section we describe what happens to the Casimir energy when the ring
is rotating. In the first part of this section we review some known results,
Refs. Chernodub:2012em ; Schaden:2012 , and compute the Casimir energy using
the standard way of summing over the eigenvalues and regularizing the sum. In
the second part of the section we show how it is possible to obtain the same
result for the Casimir energy in a different way using a replica trick applied
to functional determinants.

Following the standard approach, we first pass from the laboratory frame
(where the boundary conditions are time dependent) to a corotating frame
(where the boundary conditions are simply Eq. (2)) by performing the following
coordinate transformation,

$\displaystyle t=\hat{t},~{}~{}~{}\varphi=\hat{\varphi}+\Omega t.$ (6)
$\displaystyle{\partial\over\partial t}$ $\displaystyle\to$
$\displaystyle{\partial\over\partial\hat{t}}-\Omega{\partial\over\partial\hat{\varphi}},$
$\displaystyle{\partial\over\partial\varphi}$ $\displaystyle\to$
$\displaystyle{\partial\over\partial\hat{\varphi}},$

that leads to the following Lagrangian density:

$\displaystyle{L}={1\over 2}\left[\left({\partial\phi\over\partial
t}+\Omega{\partial\phi\over\partial\varphi}\right)^{2}-{1\over
R^{2}}\left({\partial\phi\over\partial\varphi}\right)^{2}\right].$ (7)

The equation of motion following from the above Lagrangian density are

$\displaystyle 0=\left(\left({\partial\over\partial
t}-\Omega{\partial\over\partial\varphi}\right)^{2}-{1\over
R^{2}}{\partial^{2}\over\partial\varphi^{2}}\right)\phi.$ (8)

Imposing Dirichlet boundary conditions means

$\displaystyle\phi\left(t,\varphi\right)\Big{|}_{\varphi=0,2\pi}=0.$ (9)

Notice that the above boundary conditions are time independent for a
corotating observer.

First, we describe the “canonical” method. Assuming that any solution to Eq.
(8) can be written as

$\displaystyle\psi(t,\varphi)=\sum_{n}a_{n}e^{-i\omega_{n}t}f_{n}(\varphi)+a^{\dagger}_{n}e^{i\omega_{n}t}f^{*}_{n}(\varphi).$
(10)

with $\omega_{n}\geq 0$, and substituting the above expression into the
original equation for $\psi$, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\sum_{n}\omega_{n}^{2}e^{-i\omega_{n}t}f_{n}(\varphi)-\left(\Omega^{2}-R^{-2}\right)\sum_{n}e^{-i\omega_{n}t}f^{\prime\prime}_{n}(\varphi)$
(11)
$\displaystyle-2i\Omega\sum_{n}\omega_{n}e^{-i\omega_{n}t}f^{\prime}_{n}(\varphi)$

The above assumption is verified if the modes $f_{n}(\varphi)$ are a complete
and orthogonal set of solutions satisfying the equation

$\displaystyle
0=\left(1-\beta^{2}\right)f^{\prime\prime}_{n}(\varphi)-2i\beta\tilde{\omega}_{n}f^{\prime}_{n}(\varphi)+\tilde{\omega}_{n}^{2}f_{n}(\varphi).$
(12)

We have defined, for brevity of notation,

$\displaystyle\beta^{2}$ $\displaystyle=$ $\displaystyle\Omega^{2}R^{2},$
$\displaystyle\tilde{\omega}_{n}$ $\displaystyle=$ $\displaystyle
R\omega_{n}.$ (13)

Equation (12) can be solved exactly and after imposing the boundary conditions
(9) on the general solutions, one finds through simple algebra that the
following quantization condition holds

$\displaystyle\sin^{2}\left({2\pi\tilde{\omega}_{n}\over
1-\beta^{2}}\right)=0,$ (14)

which gives the following spectrum of the quantum fluctuations

$\displaystyle\omega_{n}={n\over 2R}\left(1-\beta^{2}\right),$ (15)

with $n\in\mathbb{N}$. The corresponding eigenfunctions can be written as
follows

$\displaystyle\phi_{n}(t,\varphi)={1\over\sqrt{\pi
R}}e^{-i\omega_{n}t}e^{\imath\varphi{\beta\omega_{n}R\over
1-\beta^{2}}}\sin\left({n\varphi\over 2}\right),$ (16)

with the prefactor coming from the requirement that the modes are normalized.
The above solutions for the modes $\phi_{n}(t,\varphi)$ correspond to the
modes in the corotating frame. To go back to the stationary-laboratory frame,
we can perform the inverse coordinate transformation,

$\displaystyle t$ $\displaystyle\to$ $\displaystyle
t,~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{and}~{}~{}~{}~{}~{}~{}~{}~{}~{}\varphi\to\varphi+\Omega
t$

to get

$\displaystyle\phi_{n}(t,\varphi)={1\over\sqrt{\pi
R}}e^{-i\omega_{n}t}e^{\imath\left[\varphi+\Omega
t\right]_{2\pi}{\beta\omega_{n}R\over 1-\beta^{2}}}\sin\left({n\over
2}\left[\varphi+\Omega t\right]_{2\pi}\right),$

with $\left[u\right]_{2\pi}\equiv u\left[\mbox{mod($2\pi$)}\right]$, thus
explicitly indicating that the solutions have a $2\pi$ periodicity. These
solutions and the method we have described are those of Refs. Chernodub:2012em
; Schaden:2012 .

A direct way to compute the Casimir energy is to perform the regularized sum
over the eigenvalues. It is straightforward to write

$\displaystyle E_{r}$ $\displaystyle=$ $\displaystyle{1\over
2}\lim_{s\to-1}\sum_{n=1}^{\infty}\left({n\over
2R}\left(1-\beta^{2}\right)\right)^{-s}$ (17) $\displaystyle=$
$\displaystyle-{1\over 2\times 12\times 2R}\left(1-\beta^{2}\right),$ (18)

where the factor $1/12$ comes from the summation over $n$ yielding
$\zeta(-1)=-1/12$. Here, $\zeta(s)$ defines the Riemann zeta function.

As clearly discussed in Refs. Chernodub:2012em ; Schaden:2012 , the Casimir
energy $E_{s}$ in the stationary-laboratory frame is related to the Casimir
energy $E_{r}$ in the rotating frame by the following relation,

$\displaystyle E_{s}-E_{r}={\bf\Omega L},$ (19)

from which, by taking the inverse Legendre transform of $E_{r}$, one can
obtain the angular momentum dependence of the ground state energy:

$\displaystyle L$ $\displaystyle=$ $\displaystyle-{\partial
E_{r}\over\partial\Omega}=-{1\over 24}\Omega R.$ (20)

It follows that

$\displaystyle E_{s}=-{1\over 48R}\left(1+\beta^{2}\right).$ (21)

As noted in Ref. Schaden:2012 , the above quantity is the quantum vacuum
energy and it should be augmented by a classical contribution proportional to
the moment of inertia, $I$, yielding the total energy to be

$\displaystyle E_{s}=-{1\over 48R}\left(1+\beta^{2}\right)+{1\over
2}I\Omega^{2}.$ (22)

While Ref. Chernodub:2012em  had initially and correctly argued that the
vacuum energy lowers the moment of inertia of such a system, Ref. Schaden:2012
later argued that this never happens, at least within the range of validity of
a semiclassical treatment.

##  IV Functional determinants and the replica trick

Before including interactions, we will illustrate an alternative way to
compute the Casimir energy for the free case with almost no calculation by
using a replica trick to obtain directly the energy $E_{s}$ in the laboratory
frame without passing through the solution of the mode equation. This is
interesting for two reasons. First, it is simple. Second, it offers at least
in some cases a simple way to compute the quantum vacuum energy in stationary
(although simple) backgrounds, which is usually complicated due to the mixing
of space and time components in the metric tensor that yields nonlinear
eigenvalue problems Fursaev ; Fursaev:2011 .

The method we outline below is valid in setups more general than what we
discuss here, assuming that there are no parity breaking interactions. The
replica trick here stems from the fact that the Casimir energy of the rotating
system must be an even function of the rotational velocity, i.e. it does not
change if we invert the direction of rotation. This simple physical
consideration allows us to greatly simplify the calculation.

Rather than following the canonical approach of summing over the zero-point
energies, we pass through the Coleman-Weinberg effective potential from which
one can extract the Casimir energy. The Coleman-Weinberg effective potential
can be obtained starting from the following functional determinant

$\displaystyle\delta\Gamma=\log\det\left(\left({\partial\over\partial
t}-\Omega{\partial\over\partial\varphi}\right)^{2}-{1\over
R^{2}}{\partial^{2}\over\partial\varphi^{2}}\right).$ (23)

The problem with the expression above is that the presence of first order
derivatives in time makes the eigenvalue problem nonlinear Fursaev . Here we
will bypass the nonlinearities using the following replica trick. First of
all, we can express the determinant as

$\displaystyle\delta\Gamma=\log A=\log\det\left(L_{+}\times L_{-}\right)$ (24)

having defined

$\displaystyle L_{\pm}\left(\Omega\right)={\partial\over\partial
t}-\alpha_{\pm}{\partial\over\partial\varphi}$ (25)

with

$\displaystyle\alpha_{\pm}=\Omega\pm{1\over R}$ (26)

This gives

$\displaystyle
A=\det\left(L_{+}\left(\Omega\right)\right)\times\det\left(L_{-}\left(\Omega\right)\right).$
(27)

The replica trick relies on the assumption that inverting the sense of
rotation $\Omega\to-\Omega$, does not change the effective action. This is a
physical assumption, which is valid in our case. If parity breaking terms are
present, one can modify the trick by adding up a residue. In our case, we can
express Eq. (27) as

$\displaystyle A$ $\displaystyle=$
$\displaystyle\left(\det\left(L_{+}\left(\Omega\right)L_{-}\left(-\Omega\right)\right)\times\left(L_{-}\left(\Omega\right)L_{+}\left(-\Omega\right)\right)\right)^{1/2}$
(28) $\displaystyle=$ $\displaystyle\det\left(\left({\partial^{2}\over\partial
t^{2}}-{1\over\rho_{+}^{2}}{\partial^{2}\over\partial\varphi^{2}}\right)\left({\partial^{2}\over\partial
t^{2}}-{1\over\rho_{-}^{2}}{\partial^{2}\over\partial\varphi^{2}}\right)\right)^{1/2}$

where

$\displaystyle\rho_{\pm}^{-1}=\left|\Omega\pm{1\over R}\right|.$ (29)

From the above relation we can factorize the effective action as follows:

$\displaystyle\delta\Gamma$ $\displaystyle=$ $\displaystyle{1\over
2}\log\det\left({\partial^{2}\over\partial
t^{2}}-{1\over\rho_{+}^{2}}{\partial^{2}\over\partial\varphi^{2}}\right)$ (30)
$\displaystyle+{1\over 2}\log\det\left({\partial^{2}\over\partial
t^{2}}-{1\over\rho_{-}^{2}}{\partial^{2}\over\partial\varphi^{2}}\right).$

Notice that we have commuted the operators $L_{+}$ and $L_{-}$. This practice
requires some justification since a noncommutative or Wodzicki residue may
appear under some circumstances in the trace functionals of formal
pseudodifferential operators. In the present case it makes no difference, but
in more general situations care should be paid about this point
Elizalde:1997nd ; Dowker:1998tb ; McKenzie-Smith:1998ocy .

The replica trick has allowed us to express the effective action for the
rotating system in terms of the effective action for an analogous system
without rotation but with an effective radius,

$\displaystyle\delta\Gamma_{\pm}$ $\displaystyle=$ $\displaystyle{1\over
2}\log\det\left({\partial^{2}\over\partial
t^{2}}-{1\over\rho_{\pm}^{2}}{\partial^{2}\over\partial\varphi^{2}}\right).$
(31)

The above is nothing but the integral over the volume of the Coleman-Weinberg
potential for a free theory; combining the $\delta\Gamma_{\pm}$ terms as in
Eq. (30) should return the Casimir energy after renormalization. We outline
the computation of the $\delta\Gamma_{\pm}$ terms for completeness. Setting

$\displaystyle x_{\pm}=\rho_{\pm}\varphi$ (32)

we have

$\displaystyle\delta\Gamma_{\pm}={1\over
2}\log\det\left({\partial^{2}\over\partial t^{2}}-{\partial^{2}\over\partial
x_{\pm}^{2}}\right),$ (33)

with boundary conditions imposed at $x_{\pm}=0$ and $x_{\pm}=2\pi\rho_{\pm}$.
For Dirichlet boundary conditions, the eigenvalues of the above differential
operators are

$\displaystyle E^{(\pm)}_{n}=\pi n/x_{\pm}.$ (34)

Using zeta regularization, we get for the effective potential $V_{\pm}$ (i.e.,
the effective action divided by the volume)

$\displaystyle V_{\pm}={1\over 2x_{\pm}}\sum_{n}E^{(\pm)}_{n}.$ (35)

One can then show that

$\displaystyle V_{\pm}=-{1\over 96\pi\rho_{\pm}^{2}}.$ (36)

The contribution to the Casimir energy from the $+$ and $-$ terms summed up
gives

$\displaystyle E_{s}=2\pi R\left(V_{+}+V_{-}\right)=-{1\over 2\times
24R}{\left(1+R^{2}\Omega^{2}\right)},$ (37)

where the factor $1/2$ comes from the overall $1/2$ in the factorization of
the determinant. The above formula coincides with Eq. (21), and leads to an
attractive force. Adding additional flat directions, orthogonal to the
rotating ring, turning it into a rotating cylinder, does not change the
attractive nature of the force for the physically relevant regime of $\Omega
R\ll 1$. If we change the boundary conditions to periodic, then the sums in
the above zeta function $\zeta_{\pm}$ extend over $n\in\mathbb{Z}$, resulting
in a total Casimir energy independent of the rotational velocity $\Omega$, as
we argued earlier.

##  V Adding rotation and interactions

Now, we get to the physically novel part of this work, that is working out the
vacuum energy in the presence of rotation and interactions. The nonrotating
case has been worked out in Ref. Bordag:2021  and here those results will be
reobtained as a limiting case of our more general expressions. In the
following, we shall consider a complex scalar theory with quartic interactions
and start from the Lagrangian density

$\displaystyle{L}={1\over 2}\left[\left({\partial\phi\over\partial
t}+\Omega{\partial\phi\over\partial\varphi}\right)^{2}-{1\over
R^{2}}\left({\partial\phi\over\partial\varphi}\right)^{2}+{\lambda\over
4}\phi^{4}\right],~{}~{}~{}$ (38)

from which the equations of motion can be easily obtained,

$\displaystyle 0=\left(\left({\partial\over\partial
t}-\Omega{\partial\over\partial\varphi}\right)^{2}-{1\over
R^{2}}{\partial^{2}\over\partial\varphi^{2}}+{\lambda}\phi^{2}\right)\phi.$
(39)

The computation of the Casimir energy in the present case can be carried out
in a similar manner as we did in the free case (i.e., going over the mode
decomposition, finding the spectrum of the fluctuations, and performing the
renormalized sum over the zero-point energies); however, differently from
before, in general the Casimir energy cannot be obtained in analytical form.
The reason will become clear in a moment.

After proceeding with the decomposition in normal modes, as in Eq. (10), we
obtain the following nonlinear Schödinger equation

$\displaystyle 0=\left(\left(-{\imath\omega_{p}}-\beta{\partial\over\partial
x}\right)^{2}-{\partial^{2}\over\partial
x^{2}}+{\lambda}f_{p}^{2}\right)f_{p}.$ (40)

Above, we have defined $x=R\varphi$, and the index $p$ is a generic, boundary-
conditions-dependent multi-index that can in principle take continuous and/or
discrete values. Setting $\lambda=0$, which is in the noninteracting limit,
returns Eq. (12).

Notice that Eq. (40) differs from the analogous equation of Ref. Bordag:2021
by the addition of the term proportional to $\beta$; such a contribution
combines with the frequency $\omega_{p}$ making the associated eigenvalue
problem nonlinear, as we anticipated in the previous section.

For notational convenience we first write the mode equation as follows:

$\displaystyle 0=f_{p}^{\prime\prime}-\imath
a_{p}f_{p}^{\prime}+b_{p}f_{p}-\gamma\left|f_{p}\right|^{2}f_{p},$ (41)

where the prime refers to differentiation with respect to $x$ and

$\displaystyle a_{p}$ $\displaystyle=$ $\displaystyle{2\omega\beta\over
1-\beta^{2}}>0,$ (42) $\displaystyle b_{p}$ $\displaystyle=$
$\displaystyle{\omega^{2}\over 1-\beta^{2}}>0,$ (43) $\displaystyle\gamma$
$\displaystyle=$ $\displaystyle{\lambda\over 1-\beta^{2}}.$ (44)

Equation (41) is that of a damped, cubed anharmonic oscillator and can be
explicitly solved only for special values or combinations of the parameters.
For general values of the parameters, the equation is not exactly integrable.
In the present case, thanks to the presence of the imaginary term accompanying
the first derivative, analytic solutions can be obtained. Proceeding by
redefining

$\displaystyle f_{p}(x)$ $\displaystyle=$ $\displaystyle
g_{p}(x)e^{{\imath\over 2}a_{p}x}$ (45)

allows us to write Eq. (41) as

$\displaystyle g_{p}^{\prime\prime}(x)+{\epsilon}^{2}_{p}g_{p}(x)-\gamma
g_{p}^{3}(x)=0,$ (46)

where we have defined

$\displaystyle{\epsilon}^{2}_{p}=\left({a_{p}^{2}\over 4}+b_{p}\right).$ (47)

Equation (46) can be solved exactly in terms of Jacobi elliptic functions. We
have taken advantage of the imaginary term in Eq. (46) to reduce Eq. (41) to a
standard cubic nonlinear Schrödinger equation. This would have not been
possible if the coefficient of the first derivative were any real number.

Since the model we are considering is classically unstable for $\lambda<0$, we
shall focus here on the $\lambda>0$ case. The calculation of the quantization
relations of the energy eigenvalues follows a similar approach to the
nonrotating case; since it is slightly cumbersome and requires some
familiarity with elliptic functions, the actual calculation will be relegated
to the Appendix. Here, we summarize the main results. The solution to Eq. (46)
can be written as follows:

$\displaystyle g_{p}(x)=A\;\mathtt{sn}(qx+\delta,k^{2}),$ (48)

with the coefficients $A$, $\delta$, $q$ and $k^{2}$ to be determined by
imposing the boundary and the normalization conditions, and by use of the
first integral of Eq. (46). As a result, $k^{2}$ (the elliptic modulus) and
the eigenvalues $\epsilon_{n}^{2}$ are quantized according to the following
equations (see the Appendix for details of the computation),

$\displaystyle{\lambda L\over 8(1-\beta^{2})n^{2}}$ $\displaystyle=$
$\displaystyle\Kappa(k^{2}_{n})\left({\Kappa(k^{2}_{n})}-\Epsilon\left(k^{2}_{n}\right)\right),$
(49) $\displaystyle\epsilon_{n}^{2}$ $\displaystyle=$
$\displaystyle{4n^{2}\over L^{2}}\Kappa^{2}(k_{n}^{2})(1+k_{n}^{2}).$ (50)

with $0<n\in\mathbb{N}$, $\Kappa(k^{2})$ and $\Epsilon(k^{2})$ defining the
complete elliptic integral of the first and second kinds, respectively. The
spectrum of the rotating problem is encoded in the two equations (50) and
(49). Thus, for any admissible value of the physical parameters, $L$,
$\lambda$, $\beta$, and any positive integer $n$, Eq.(49) determines the value
of the elliptic modulus $k_{n}^{2}$, which together with Eq. (50) determines
the eigenvalue $\epsilon_{n}$. Notice, that the angular velocity $\Omega$
($\beta=\Omega R={\Omega L\over 2\pi}$) appears with even powers preserving
the symmetry $\Omega\leftrightarrow-\Omega$.

The difference from the nonrotating case is rather nonobvious; rotation enters
the frequencies $\omega_{n}$ via the coefficient $(1-\beta^{2})$ in Eq. (49)
(see Eqs. (42), (43) and (47) for the relation between $\omega_{n}$ and
$\epsilon_{n}$). This is already nontrivial as the coupling constant is
multiplied by the factor $1/(1-\beta^{2})$, implying that the coupling
constant can be effectively amplified if the rotation is large, i.e. when
$\beta$ tends to unity. The second way that rotation enters the frequencies is
via relation (50), which relates nonlinearly to the rotational velocity
$\Omega$ and to the elliptic modulus $k^{2}_{n}$. That is, once the parameters
($\lambda$, $\beta$, $L$) and the quantum number $n$ are fixed, the elliptic
modulus $k^{2}_{n}$ is determined according to Eq. (49). Then, upon
substitution into Eq. (50), $k_{n}^{2}$ enters the frequencies.

Before going into the computation of the vacuum energy, there are a number of
relevant limits to check. The first is the noninteracting limit,
$\lambda\rightarrow 0$. In this case, the first equation (49) gives

$\displaystyle k_{n}^{2}=0,\mbox{$\forall n\in\mathbb{N}$.}$ (51)

Substituting in Eq. (50) we obtain

$\displaystyle\omega_{n}\equiv\omega^{(0)}_{n}={\pi n\over L}(1-\beta^{2}),$
(52)

which coincides with the free rotating result (15). Naturally, taking the
nonrotating limit, $\beta\to 0$, we also recover the nonrotating case.

Corrections to the above leading result are interesting in this case because
of the way that the interaction strength and rotation intertwine. For small
interaction strengths and zero rotation, the elliptic modulus can be obtained
by expanding the right-hand side of Eq. (49) for small argument. Ignoring
terms of order $k_{n}^{4}$ or higher, we find

$\displaystyle{k^{2}_{n}}\approx{\lambda L\over\pi^{2}n^{2}}.$ (53)

Keeping only this first correction, we get for the frequencies

$\displaystyle\omega^{2}_{n}$ $\displaystyle\approx$
$\displaystyle{{\pi^{2}}n^{2}\over L^{2}}\left(1+{3\over 2\pi^{2}}{\lambda
L\over n^{2}}\right),$ (54)

which recovers the result of Ref. Bordag:2021 .

More interesting is the limit of small interaction strength with rotation on.
In this case, relation (49) implies that the relevant expansion parameter is
$\lambda/(1-\beta^{2})$, which plays the role of an effective coupling. When
rotation is small, $\beta\ll 1$, then at leading order we return to the
previous small-$\lambda$ small-$\beta$ case. When the ring is spinning fast,
i.e. $1-\beta^{2}\ll 1$, even for small $\lambda$ the quantity
$\lambda/(1-\beta^{2})$ can in principle be large. Thus we have two physically
relevant limits. The first is for weak coupling and slow rotation, which
yields the following expressions for the frequencies,

$\displaystyle\omega^{2}_{n}{\approx}{{\pi^{2}}n^{2}\over
L^{2}}\left(1{+}{3\over 2\pi^{2}}{\lambda L\over
n^{2}}\right)\left(1{-}\beta^{2}\left(3{-}{2n^{2}\pi^{2}\over
2n^{2}\pi^{2}{+}3\lambda L}\right)\right).$

The other is for weak coupling and fast rotation. In this case, the limit is
trickier to take since in the calculation of the Casimir energy we need to sum
over $n\in\mathbb{N}$. Since the sum extends to infinity, the quantity
${\lambda L/8/(1-\beta^{2})/n^{2}}$ is not necessarily large for any $n$ if
${\lambda L/(1-\beta^{2})}$ is large but finite. The limit of large but finite
effective coupling, ${\lambda/(1-\beta^{2})}$, can be conveniently treated
numerically, and we will do so in the next subsection.

Finally, there is the case of ${\lambda/(1-\beta^{2})}\to\infty$, i.e.
infinitely strong coupling (which as explained can also be realized in the
limit of small coupling and fast rotation); this corresponds in Eq. (49) to
the limit of $k_{n}^{2}\sim 1$. Thus, from Eq. (49) we get, for any
$k_{n}^{2}$,

$\displaystyle\Kappa(k^{2}_{n})={1\over
2}\left(\Epsilon\left(k^{2}_{n}\right)\pm\sqrt{\Epsilon^{2}\left(k^{2}_{n}\right)+{\lambda
L^{2}\over 2n^{2}(1-\beta^{2})}}\right).$

Upon substituting in Eq. (50) we get

$\displaystyle\omega^{2}_{n}$ $\displaystyle=$ $\displaystyle{n^{2}\over
L^{2}}\left(\Epsilon\left(k^{2}_{n}\right)\pm\sqrt{\Epsilon^{2}\left(k^{2}_{n}\right)+{\lambda
L^{2}\over 2n^{2}(1-\beta^{2})}}\right)^{2}\times$ (55)
$\displaystyle\times(1+k_{n}^{2})(1-\beta^{2})^{2}.$

The leading term results from the $k_{n}^{2}\to 1$ limit, yielding

$\displaystyle\omega^{2}_{n}{\approx}{2n^{2}(1{-}\beta^{2})^{2}\over
L^{2}}\left(1{+}\sqrt{1{+}{L^{2}\lambda\over
2n^{2}(1{-}\beta^{2})}}\right)^{2}{+}\dots.$ (56)

This regime can also be computed numerically.

###  V.1 A numerical implementation of zeta regularization

In this section we shall compute the quantum vacuum energy $E_{r}$ given by
the following (formal) expression

$\displaystyle E_{r}={1\over 2}\lim_{s\to
0}\sum_{n}{}^{{}^{\prime}}\omega_{n}^{1-2s}.$ (57)

The prime is a reminder that the sum is divergent and requires regularization.

Before going into the specifics of the computation, it may be useful to
explain the method we use to compute the vacuum energy in more general terms.
The issue at hand is having to compute a summation of the form of Eq. (57)
without any explicit knowledge of the eigenvalues. Knowing the eigenvalues
would be enough to compute the above expression, were the summation
convergent. However, as is common in quantum field theory, divergences are
present and the above brute force approach cannot be used. The procedure
described below is essentially a regularization procedure based on a numerical
implementation of zeta function regularization.

Figure 2: Numerical eigenvalue analysis. (a) Comparison of the polynomial fit
(Eq. (62)) and the numerically computed eigenvalues (Eqs. (49) and (50))
eigenvalues up to cubic order. The individual fitting coefficients are defined
in the legend as
$\tilde{\omega}_{j}=(\tilde{\omega}_{0},\tilde{\omega}_{1},\tilde{\omega}_{2},\tilde{\omega}_{3})$.
(b) Correction $\delta E_{r}$ (Eq. (59)) computed for each of the cases
presented in (a), i.e. for fixed $\lambda=1$, $L=2$ with $\Omega
R=0.2,0.4,0.6,0.8$. Inset: example of the eigenvalue difference
$\omega_{n}-\tilde{\omega}_{n}$ for $\Omega R=0.2$.

Our approach consists in finding an approximate form for the eigenvalues, say,
$\tilde{\omega}_{n}$, that can be used to compute the summation (57) without
relying on any approximation; this is possible as long as the approximant
$\tilde{\omega}_{n}$ captures the correct asymptotic behavior that causes the
divergence. In the present case, since we can find the eigenvalues numerically
by solving Eqs. (49) and (50) to any desired order and accuracy, we can
proceed to find a suitable $\tilde{\omega}_{n}$ simply by fitting the
eigenvalues. The form of the fitting function is unimportant (as we shall
explain below), as long as the correct asymptotic behavior is captured by the
fitting function. Assuming that this the case, we can write the above
expression for $E_{r}$ as

$\displaystyle E_{r}{=}{1\over 2}\lim_{s\to
0}\sum_{n}\left(\omega_{n}^{1-2s}{-}\tilde{\omega}_{n}^{1-2s}\right){+}{1\over
2}\lim_{s\to 0}\sum_{n}\tilde{\omega}_{n}^{1-2s}.$ (58)

In the first term above the limit ${s\to 0}$ can be safely taken (by
construction); we call this term

$\displaystyle\delta E_{r}={1\over
2}\sum_{n}\left(\omega_{n}-\tilde{\omega}_{n}\right).$ (59)

We note that the above quantity can be computed numerically to any desired
accuracy.

Thus, the Casimir energy (58) can be expressed as

$\displaystyle E_{r}={1\over 2}\lim_{s\to
0}\sum_{n}\tilde{\omega}_{n}^{1-2s}+\delta E_{r}.$ (60)

There is no approximation at this stage: Eqs. (57) and (60) are equivalent.

Now, in order to get the exact Casimir energy $E_{r}$ we are left with the
computation of

$\displaystyle\tilde{E}_{r}={1\over 2}\lim_{s\to
0}\sum_{n}\tilde{\omega}_{n}^{1-2s},$ (61)

which, as detailed below, can be carried out analytically. Once we have
$\tilde{E}_{r}$ and $\delta E_{r}$, these can be combined into an exact result
for $E_{r}$.

There are still two issues to clarify. The first is about the renormalization
of the Casimir energy: $\tilde{E}_{r}$ is still divergent and needs to be
regularized and renormalized. In the present case the regularization, i.e.
extracting the diverging behavior of the sum in $\tilde{E}_{r}$, is
straightforward using zeta regularization. The renormalization also can be
carried out at ease by subtracting from $\tilde{E}_{r}$ the asymptotic (i.e.,
calculated for $L\to\infty$) value of $\tilde{E}_{r}$ (practically, this is
equivalent to normalizing the vacuum energy to zero in the absence of
boundaries, a straightforward procedure in Casimir energy calculations).

The other point to clarify regards the choice of the fitting function. First
of all, one can easily understand that the choice of the fitting function is
not unique, as two different fitting functions may result in differing $\delta
E_{r}$. However, any such difference is irrelevant as it is compensated out in
Eq. (60) due to the fact that we add and subtract the same quantity. Using two
different fitting functions may result in an expedited numerical evaluation
and in an efficient regularization of $\tilde{E}_{r}$.

Where uniqueness is required and two fitting functions cannot differ is in
their asymptotic behavior. This is necessary to properly “treat” (i.e.,
renormalize) the vacuum energy. In order to find the asymptotic behavior one
may proceed heuristically, essentially by trial and error, to find the best
fitting function (this is straightforward to implement numerically in our
case, or in situations where the eigenvalues can be computed numerically to
any desired order and accuracy). In the present case, as discussed in the next
section, we find the leading term in the frequencies $\omega_{n}^{2}$ behaves
as $n^{2}$. A more general approach would be to invoke Weyl’s theorem on the
asymptotic behavior of the eigenvalues of a differential operator. In general
dimensionality, given certain assumptions on the differential operator
involved, the Weyl theorem provides an intuitive rationale for obtaining the
correct asymptotic behavior as the eigenvalues behave as $n^{2/d}$ with $d$
being the dimensionality. To be precise, in its original formulation Weyl’s
law refers to the Laplace-Beltrami operator in $d\geq 2$, however, extensions
of the theorem and to more general manifolds and operators, including in
$d=1$, have been discussed in the literature Kirsten ; Arendt ; Bonder .

To recap, given two fitting function $\omega^{(fit)}_{n}$ and
$\bar{\omega}^{(fit)}_{n}$ with the same asymptotic behavior, as per Eq. (60),
the vacuum energy will not change: any change in the fitting function will
result in different $\tilde{E}_{r}$ and $\delta E_{r}$, with the difference
simply compensated by construction. Of course, as already stated, the
arbitrariness in the choice of the fitting function can be used to achieve a
speedier computation of $\delta E_{r}$ as well as a more manageable expression
for $\tilde{E}_{r}$.

The above method will be concretely applied to our problem in the next
section.

####  V.1.1 The Casimir energy

In order to compute the vacuum energy we shall proceed as follows. First, we
fix the values of the interaction strength $\lambda$, of the rotation
parameter $\beta={L\Omega\over 2\pi}$ and $L$ (the procedure is iterated over
these parameters). Then, we solve Eq. (49) for a sequence of $n\in\mathbb{N}$
up to some large value $n_{\rm max}$. These values of $k_{n}$ are then used to
compute the frequencies $\omega_{n}$ up to $n_{\rm max}$ according to formula
(50).

With this set of frequencies
$\Theta\left(n_{*}\right)=\left\\{\omega_{1},\omega_{2},\dots,\omega_{n_{*}}\right\\}$,
we fit the spectrum with a polynomial,

$\displaystyle\tilde{\omega}^{2}_{n}=\varpi_{0}+\varpi_{1}n+\varpi_{2}n^{2}+\varpi_{3}n^{3}+\dots+\varpi_{p_{*}}n^{p_{*}},$
(62)

with $2\leq p_{*}\in\mathbb{N}$ (the fit is repeated for various values of
$p_{*}$ with the value corresponding to the best fit selected. This is, in
fact, a redundant step as it follows that the best-fit value is $p_{*}=2$).
The next step consists in increasing the value $n_{*}$ and repeating the
process until the coefficients $\varpi_{k}$ converge. We find that the best
fit returns $\varpi_{1}\approx 0$ and $\varpi_{j}\approx 0$ with $j\neq 2$,
indicating that the spectrum is approximately of the form

$\displaystyle\tilde{\omega}_{n}^{2}\approx\varpi_{0}+\varpi_{2}n^{2},$ (63)

where the “$\approx$” symbol should be understood in the sense of numerical
approximation.

As we explained in the preceding section, the form (63) of the spectrum is
anticipated from the expected Weyl-like asymptotic behavior of the
eigenvalues, from which it follows $p_{*}=2$. As explained in the previous
section, we should note that one may chose to proceed using a different
fitting function with the same asymptotic behavior; this will not change the
final result, but the following step should be modified accordingly. The
choice of Eq. (63) is the easiest to handle and the most natural, since the
resulting summation will be of the form of a generalized Epstein-Hurwitz zeta
function (i.e., a type of zeta function associated with quadratic forms) whose
regularization can be carried out using a rearrangement by Chowla and Selberg,
as explained below.

Figure 3: Rotation-interaction heat map. (a) Casimir energy $E_{r}$ (Eq. (68))
in the $(\Omega R,\lambda)$ parameter space. (b) $E_{r}(\lambda)$ for $\Omega
R=0,0.2,0.4,0.6,0.8,0.9$. The size of the ring is $L=20$.

Once we have the eigenvalues $\omega_{n}$, these have to be summed according
to Eq. (57) in order to obtain the Casimir energy of the rotating system.

Figure 4: Casimir energies. (a) $E_{r}$ (Eq. (68)) for varying $L$ for various
fixed interaction strengths $\lambda=1,10,25,50,75,100$. The rotation
parameter is $\Omega R=0.75$. (b) $E_{r}$ for fixed $\lambda=10$ for $\Omega
R=0,0.5,0.6,0.7,0.8,0.9$. Inset: enlarged portion between $L=10^{-3}:10^{-2}$.
(c), (d) $E_{r}$ as a function of the rotation parameter $\Omega R$ for a
fixed ring length $L=20$ in (c) and a fixed interaction strength $\lambda=0.5$
in (d). Here we have fixed $\mu=1$ throughout.

The computation of the Casimir energy is performed using the approximate
formula of Eq. (63) leading to

$\displaystyle\tilde{E}_{r}=\lim_{s\to 0}{\mu^{2s}\over
2}\varpi_{2}^{1/2-s}\sum_{n}\left({\varpi_{0}\over\varpi_{2}}+n^{2}\right)^{1/2-s}.$
(64)

We have added the multiplicative term $\mu^{2s}$ in order to keep the
dimensionality of the above expression to be that of energy Toms:2012 .

The expression (64) can be easily dealt with analytically when the ratio
$\varpi_{0}/\varpi_{2}$ is small, for instance in the noninteracting limit. In
this case, a valid representation can be built by simply using the binomial
expansion of the summand in Eq. (64), yielding

$\displaystyle\hskip-14.22636pt\tilde{E}_{r}{=}-{\sqrt{\varpi_{2}}\over
24}\left\\{1{+}6{\varpi_{0}\over\varpi_{2}}\left[1{-}\gamma_{e}{-}{1\over
2s}{+}{1\over 2}\log\left(\varpi_{2}\over\mu^{2}\right)\right]\right\\}$ (65)

Notice that the noninteracting case corresponds to $\varpi_{0}\to 0$, leading
to

$\displaystyle\tilde{E}_{r}$ $\displaystyle=$
$\displaystyle-{\sqrt{\varpi_{2}}\over 24},$ (66)

implying that

$\displaystyle\lim_{\lambda\to 0}{\varpi_{2}}={\pi^{2}\over
L^{2}}{\left(1-\beta^{2}\right)^{2}}.$ (67)

The above limit can (and has been) verified numerically.

When the ratio $\varpi_{0}/\varpi_{2}$ is not small, we can use the following
(Chowla-Selberg) representation for the sum above (see Ref. Flachi:2008 ):

$\displaystyle\tilde{E}_{r}$ $\displaystyle=$
$\displaystyle-{{\varpi_{0}}^{1/2}\over 4}\left\\{1-{1\over
2}\sqrt{\varpi_{0}\over\varpi_{2}}\left[1-{\gamma_{e}}+{1\over
s}-\log\left(\varpi_{0}\over\mu^{2}\right)\right.\right.$ (68)
$\displaystyle\left.-{\psi(-1/2)}\right]\Big{\\}}-{1\over
2\pi}\sqrt{{\varpi_{0}}}\sigma\left(\sqrt{\varpi_{0}\over\varpi_{2}}\right),$

where

$\displaystyle\sigma(t)=\sum_{p=1}^{\infty}{1\over p}K_{1}\left(2\pi
pt\right).$ (69)

For clarity we have left the diverging pieces $\propto 1/s$ in both expression
for the vacuum energy. These terms are removed by subtracting from the
nonrenormalized vacuum energy its counterpart for $L\to\infty$. We again
stress here that using a different ansatz for the frequencies, for instance
keeping a linear term, will change the expression (64) and, in turn, both
representations for the summations have to be modified accordingly. However,
as we have explained, this does not change the Casimir energy, since the
difference is compensated away by the same change in $\delta E_{r}$.

In order to obtain the exact quantum vacuum energy, we need to add to
$\tilde{E}_{r}$ the term $\delta E_{r}$ that can be calculated numerically,

$\displaystyle\delta E_{r}=\lim_{s\to 0}{\mu^{2s}\over
2}\sum_{n}\left(\omega_{n}^{1-2s}-\tilde{\omega}_{n}^{1-2s}\right).$ (70)

The above is nothing but the sum over the difference between the exact
eigenvalues and the approximated ones. The advantage of using this formulation
is that the above expression is regular in the ultraviolet and the limit $s\to
0$ can be taken without any further manipulation, as explained in the previous
section. With the two pieces in hand the exact Casimir energy can be
calculated as stipulated by Eq. (60).

In Fig. 2 we present an overview of our numerical methodology. Figure 2(a)
shows the eigenvalues of the polynomial fit (Eq. (62)) (gray solid) and the
numerically computed ones (Eqs. (49) and (50)) (open colored symbols),
respectively. The comparison includes terms up to cubic order (viz. Eq. (62))
for fixed interaction strength and ring size, for various values of the
rotation parameter $\Omega R$. Although the (odd) terms $\tilde{\omega}_{1}$
and $\tilde{\omega}_{3}$ have been included, they are many orders of magnitude
smaller than $\tilde{\omega}_{0}$ and $\tilde{\omega}_{2}$, typically
$\tilde{\omega}_{1}\sim 10^{-6}$ and $\tilde{\omega}_{3}\sim 10^{-12}$,
supporting our choice of Eq. (63). We also checked the typical values of
$\tilde{\omega}_{j>3}$, which are even smaller than the preceding terms. Then,
Fig. 2 (b) shows the quantity $\delta E_{r}$, Eq. (59). This quantity is
calculated again for fixed $\lambda=1$, $L=2$ for various $\Omega R$, showing
a monotonic decrease as $n_{\rm max}$ is increased from $n_{\rm
max}=50:10^{4}$. Finally, the inset shows an example of the quantity
$\omega_{n}-\tilde{\omega}_{n}$, displaying a prominent maximum located at
$n\sim 5$. Due to the asymptotic nature of the regularization procedure, the
behavior of this quantity is formally accurate for $n\gg 1$, hence we exclude
the first $\sim 10\%$ of computed eigenvalues from our simulations (shaded
gray), leading to the monotonic decrease of $\delta E_{r}$ toward zero for
increasing $n_{\rm max}$. In the calculations that follow we set $n_{\rm
max}=1000$.

##  VI Results and discussion

Figures (3), (4) and (5) illustrate the results. Figure 3(a) summarizes in a
rotation-interaction heat map how the Casimir energy $E_{r}$ depends on both
the rotational parameter $\Omega R$ and the interaction strength $\lambda$. A
local maxima in the Casimir energy is present for $\Omega R\geq 0$, as
depicted in Fig. 3 where Eq. (68) is computed as a function of $\lambda$ for
fixed values of $\Omega R$. We also found that increasing $L$, the size of the
ring causes a proportional increase to the corresponding Casimir energy. Note
that the existence of the local maxima depends on the size $L$ of the ring,
appearing in the noninteracting $\Omega R=0$ limit for $L\gtrsim 5$. In fact,
the changes in the Casimir energy caused by both rotation and interaction are
rather nontrivial, as we illustrate in Figs. 4(a) and 4(b); Fig. 4(a) shows
the Casimir energy $E_{r}$ as a function of $L$ for various fixed and
increasing interaction strengths $\lambda=1,\dots,100$ with rotation parameter
set to $\Omega R=0.75$; Fig. 4(b) shows $E_{r}$ as a function of $L$ for a
fixed $\lambda=10$ and for increasing $\Omega R=0,\dots,0.9$. While the energy
increases for rings of small sizes and decays to zero for large enough rings,
from both cases it is clear that a departure from the typical $1/L$ dependence
of the energy occurs as a consequence of both interaction and rotation (only
rotation would not cause such a departure). Figures 4(c) and 4(d) illustrate
the increase in the Casimir energy with respect to its nonrotating counterpart
for different values of the interaction strength and for rings of different
sizes. Finally, in Fig. 5 we plot the angular momentum $L_{\rm am}$ calculated
according to Eq. (20): Fig. 5(a) shows $L_{\rm am}$ computed from the data in
Fig. 4(c) as a function of $\Omega R$ for fixed ring length $L=20$ in (a) and
from the data in Fig. 4(d) for fixed interaction strength $\lambda=0.5$ in
(b).

Figure 5:  Angular momentum calculated per Eq. (20). (a) $L_{\rm am}$ computed
from the data in Fig. 4(c) as a function of $\Omega R$ for fixed ring length
$L=20$. (b) $L_{\rm am}$ computed from the data in Fig. 4(d) for fixed
interaction strength $\lambda=0.5$.

The Casimir effect, and more generally calculations of quantum vacuum energies
for interacting field theories, have been considered for many years, starting
with References Ford:1979b ; Toms:1980a ; Peterson:1982  (see also Ref.
Toms:2012 ). Refs. Maciolek:2007 ; Moazzemi:2007 ; Schmidt:2008 ; Flachi:2012
; Flachi:2013 ; Flachi:2017 ; Chernodub:2017gwe ; Chernodub:2018pmt ;
Valuyan:2018 ; Flachi:2017xat ; Flachi:2021 ; Song:2021 ; Flachi:2022  give an
incomplete list of more recent papers that have discussed different aspects of
the Casimir effect and related physics in the presence of field interactions.
Of particular interest to this work are Refs. Flachi:2017 ; Bordag:2021  where
the connection between the Casimir energy and elliptic functions has been
pointed out and used, numerically in Ref. Flachi:2017  and analytically in
Ref. Bordag:2021 , to compute the Casimir energy. Particularly worthy of
notice are in fact the results of Ref. Bordag:2021  that have highlighted
clearly how to compute semianalytically (full analytic results can be obtained
in specific regimes, but in general numerics cannot be avoided) and exactly
(without resorting to any approximation) the quantum vacuum energy dealing
directly with the nonlinear problem. In this work, we have extended those
results by adding rotation, a feature relevant to cold-atomic systems at least
in the long wavelength regime where the perturbations around a Bose-Einstein
condensate evolve according to a relativistic Klein-Gordon equation (see, for
example, Refs Kurita:2008fb ; Barcelo:2010bq ). Complementary to this, the
Casimir effect has also been studied for superfluids in the presence of
vorticity impens_2010 ; mendonca_2020 , as well as near surfaces (Casimir-
Polder interaction) pasquini_2004 ; harber_2005 ; moreno_2010 ; bender_2014 ,
at finite temperature obrecht_2007 , in the presence of disorder moreno_2010a
or with an impurity marino_2017 .

The other interesting aspect of the generalization we have considered here is
that the mixing of interaction and rotation can give rise to a departure from
the usual massless behavior of the Casimir energy for both the rotating and
nonrotating noninteracting cases. Also, from Eq. (49) one notices that
rotation combines with the coupling constant in a way that even when the
interaction strength is small, its effect can be amplified by fast rotation.
Extending the present results to the nonrelativistic Gross-Pitaevski case can
be done along the lines discussed here. More interesting would be
generalizations to higher dimensions since the Coleman-Hohenberg-Mermin-Wagner
theorem does not apply and phase transitions may occur dynamically without
having to introduce any explicit breaking of rotational symmetry, at least in
principle. In this case, not only will the Casimir effect experience a phase
transition in correspondence to a critical value of the coupling constant and
rotation at which symmetry breaking eventually occurs, but it may also become
a proxy of critical quantities. This is of course a technically complicated
problem as in more than one noncompact dimension the separability of the field
equation becomes nontrivial. Obviously, the more interesting aspect to
investigate is a closer connection to cold atoms and to the possibility of
using these as a probe of quantum vacuum effects.

## ACKNOWLEDGEMENTS

A.F.’s research was supported by the Japanese Society for the Promotion of
Science Grant-in-Aid for Scientific Research (KAKENHI, Grant No. 21K03540).
M.E.’s research was supported by the Australian Research Council Centre of
Excellence in Future Low-Energy Electronics Technologies (Project No.
CE170100039) and funded by the Australian Government, and by the Japan Society
of Promotion of Science Grant-in-Aid for Scientific Research (KAKENHI Grant
No. JP20K14376). One of us (A.F.) wishes to thank O. Corradini, G. Marmorini,
and V. Vitagliano for earlier discussions and I. Moss for a comment regarding
the use of a relativistic Klein-Gordon equation in Bose-Einstein condensates
in the long wavelength approximation.

*

##  Appendix A Solutions and quantization of the energy eigenvalues

In order to make the paper self-contained, in this appendix we present the
details of the derivation of Eqs. (49) and (50). The calculations below follow
closely those of Refs. Bordag:2021 ; Carr:2000::1 ; Carr:2000::2 , the
difference being the inclusion of rotation. We should stress that although the
calculations are similar to the nonrotating case (since we have reduced the
equation to the form of Eq. (46)), the nonlinearity generated by the inclusion
of rotation in the associated eigenvalue problem induces nontrivial changes in
the frequencies, as will become clear below.

A general solution to Eq. (46) with $\lambda>0$ can be written in terms of the
Jacobi elliptic function $\mathtt{sn}(qx+\delta,k^{2})$:

$\displaystyle g_{p}(x)=A\;\mathtt{sn}(qx+\delta,k^{2}),$ (71)

where $A$ is a normalization factor, $q$ a parameter, the elliptic modulus
$k^{2}$, and the phase $\delta$; all these parameters are to be fixed by the
boundary conditions, normalization and by a condition deriving from the first
integral of the above equation (46). (For $\lambda<0$ a solution can be
written in terms of $\mathtt{cn}(qx+\delta,k^{2})$ (see Refs. Bordag:2021 ;
Carr:2000::2 ), but we will not consider this case here.)

Dirichlet boundary conditions at $x=0$ are satisfied by $\delta=0$, leaving,
at this stage, the other parameters undetermined; notice that it is possible
to chose a different phase for $\delta$, however, this can be eliminated by a
coordinate transformation. Imposing Dirichlet boundary conditions also at the
other end $x=L$ gives

$\displaystyle f_{p}(x)\Big{|}_{x=0,L}=0,$ (72)

where $L=2\pi R$, or explicitly,

$\displaystyle 0=\mathtt{sn}(qL,k^{2}).$ (73)

The above relation implies a quantization for $q$,

$\displaystyle qL=2\Kappa(k^{2})n,$ (74)

with $0<n\in\mathbb{N}$ and $\Kappa(k^{2})$ being the complete elliptic
integral of the first kind ($\Kappa(k^{2})$ is quarter period of the Jacobi
elliptic function $\mathtt{sn}$). Details on elliptic functions can be found
in Ref. nist , while a discussion of the solutions to Eq. (46) can be found in
Ref. Carr:2000::1  for the case of Dirichlet boundary conditions and
$\lambda>0$ and in Ref. Carr:2000::2  for the case of Dirichlet boundary
conditions and $\lambda<0$. Below we go over the details for our case in a
self-contained manner.

To obtain the remaining parameters $k^{2}$ and $A$, we proceed as follows.
After imposing the boundary conditions at $x=0,L$ on the general solution, we
can obtain the first integral associated with Eq. (46), multiply Eq. (46) by
$g_{p}^{\prime}(x)$, and integrate by parts,

$\displaystyle\left(g_{p}^{\prime}(x)\right)^{2}+{\epsilon}^{2}_{p}g_{p}^{2}(x)-{1\over
2}\gamma g_{p}^{4}(x)=C.$ (75)

with $C$ an undetermined constant. At this point we substitute the general
solution (48) in the first integral above and equate like powers of
$\mathtt{sn}$. This leads to the following relations:

$\displaystyle{{\epsilon}^{2}_{p}}$ $\displaystyle=$
$\displaystyle(1+k^{2})q^{2},$ (76) $\displaystyle A^{2}$ $\displaystyle=$
$\displaystyle 2q^{2}k^{2}{1-\beta^{2}\over\lambda},$ (77) $\displaystyle C$
$\displaystyle=$ $\displaystyle A^{2}q^{2}.$ (78)

The last piece needed is the normalization of the solution, which allows us to
fix the parameter $k^{2}$,

$\displaystyle 1$ $\displaystyle=$
$\displaystyle\int_{0}^{L}\left|g_{p}(x)\right|^{2}dx$ $\displaystyle=$
$\displaystyle{2qL\over\gamma
L}\left({qL}-\mathcal{E}\left(\mathtt{am}(qL,k^{2}),k^{2}\right)\right),$

where $\mathcal{E}\left(x,k^{2}\right)$ is the Jacobi epsilon function and
$\mathtt{am}(x,m)$ is the Jacobi amplitude. The Jacobi epsilon function can be
defined as

$\displaystyle\mathcal{E}\left(x,k^{2}\right)=\int_{0}^{x}\mathtt{dn}(z,k^{2})dz;$
(79)

the Jacobi amplitude is related to the Jacobi function by the following
definition:

$\displaystyle\mathtt{am}(x,k^{2})=\int_{0}^{x}\mathtt{dn}(x^{\prime},k^{2})dx^{\prime};$
(80)

The function $\mathtt{dn}(x,k^{2})$ is the Jacobi delta amplitude, related to
the $\mathtt{sn}$ function by

$\displaystyle\mathtt{dn}(x,k^{2})=\sqrt{1-k^{4}\sin^{2}\left(\mathtt{am}(x,k^{2})\right)}.$
(81)

The Jacobi amplitude satisfies the following relation,

$\displaystyle\mathtt{am}(2n\Kappa(k^{2}),k^{2})=n\pi$ (82)

which allows the normalization condition to be rewritten as

$\displaystyle 1$ $\displaystyle=$ $\displaystyle{4\Kappa(k^{2})n\over\gamma
L}\left({2\Kappa(k^{2})n}-\mathcal{E}\left(\mathtt{am}(2\Kappa(k^{2})n,k^{2}),k^{2}\right)\right).$

Using the identity

$\displaystyle\mathcal{E}\left(n\pi,k^{2}\right)=2n\Epsilon\left(k^{2}\right),$
(83)

with $\Epsilon\left(k^{2}\right)$ being the complete elliptic integral of the
second kind, together with the periodicity of the Jacobi $\mathtt{sn}$, Eq.
(74), we arrive at the following relation

$\displaystyle{\lambda L\over 8(1-\beta^{2})n^{2}}$ $\displaystyle=$
$\displaystyle\Kappa(k^{2}_{n})\left({\Kappa(k^{2}_{n})}-\Epsilon\left(k^{2}_{n}\right)\right),$
(84)

Notice that the generic index $p$ is now quantized, $p\to n$ and so is the
elliptic parameter $k^{2}$. Substituting $k^{2}\to k_{n}^{2}$ and $q\to
q_{n}$, as per (74), we can also get the quantization relation for the
eigenvalues $\epsilon_{n}$:

$\displaystyle\epsilon_{n}^{2}={4n^{2}\over L^{2}}\Kappa^{2}(k^{2})(1+k^{2}).$
(85)

## References

  * (1) N. Mermin and H. Wagner, _Absence of ferromagnetism or antiferromagnetism in one-dimensional or two-dimensional isotropic Heisenberg models_ , Phys. Rev. Lett. 17 (1966) 1133.
  * (2) P. Hohenberg, _Existence of Long-Range Order in One and Two Dimensions_ , Phys. Rev. 158 (1967) 383.
  * (3) S. R. Coleman, _There are no Goldstone bosons in two-dimensions_ , Commun. Math. Phys. 31 (1973) 259.
  * (4) K.A. Milton, _The Casimir Effect. Physical Manifestations of Zero-Point Energy_ , World Scientific 113 (2002) 1.
  * (5) M. Bordag, G.L. Klimchitskaya, U. Mohideen, V.M. Mostepanenko, _Advances in the Casimir Effect_ , Oxford University Press, International Series of Monographs on Physics, (2009).
  * (6) M. N. Chernodub, _Rotating Casimir systems: magnetic-field-enhanced perpetual motion, possible realization in doped nanotubes, and laws of thermodynamics,_ Phys. Rev. D87 (2013) 025021 (2012).
  * (7) M. Schaden, _Quantization and Renormalization and the Casimir Energy of a Scalar Field Interacting with a Rotating Ring,_ arXiv:1211.2740 [quant-ph].
  * (8) M. Bordag, _Vacuum Energy for a Scalar Field with Self-Interaction in (1 + 1) Dimensions_ , Universe 7(3) (2021) 55
  * (9) O. Corradini, A. Flachi, G. Marmorini, M. Muratori and V. Vitagliano, _Bosons on a rotating ring with free boundary conditions,_ J. Phys. A 54, no.40, 405401 (2021)
  * (10) David J. Toms, _The Schwinger Action Principle and Effective Action_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (2012)
  * (11) D.V. Fursaev, _Spectral Asymptotics of Eigenvalue Problems with Non-linear Dependence on the Spectral Parameter_ , Class. Quant. Grav. 19 (2002) 3635-3652; Erratum-ibid. 20 (2003) 565
  * (12) D. Fursaev, D. Vassilevich, _Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory_ , Springer (2011)
  * (13) E. Elizalde, L. Vanzo and S. Zerbini, _Zeta function regularization, the multiplicative anomaly and the Wodzicki residue,_ Commun. Math. Phys. 194, 613-630 (1998)
  * (14) J. S. Dowker, _On the relevance of the multiplicative anomaly,_ [arXiv:hep-th/9803200 [hep-th]]
  * (15) J. J. McKenzie-Smith and D. J. Toms, _There is no new physics in the multiplicative anomaly,_ Phys. Rev. D 58, 105001 (1998)
  * (16) K. Kirsten, _Spectral Functions in Mathematics and Physics_ , Chapman and Hall/CRC (2001).
  * (17) W. Arendt, R. Nittka, W. Peter, F. Steiner, _Weyl’s Law: Spectral Properties of the Laplacian in Mathematics and Physics_ , in Mathematical Analysis of Evolution, Information, and Complexity, Wiley (2009).
  * (18) J.F. Bonder, J.P. Pinasco, _Asymptotic behavior of the eigenvalues of the one-dimensional weighted $p$-Laplace operator_, Ark. Matematik 41 (2003) 267.
  * (19) A. Flachi, T. Tanaka, _Vacuum polarization in asymptotically anti-de Sitter black hole geometries,_ Phys. Rev. D78 (2008) 064011
  * (20) L. Ford, _Casimir Effect for a Self-Interacting Scalar Field_ , Proc. R. Soc. Lond. A 368 (1979) 305
  * (21) D. J. Toms, _Casimir effect and topological mass_ , Phys. Rev. D 21 (1980) 928
  * (22) C. Peterson, T.H. Hansson, and K. Johnson, _Loop diagrams in boxes_ , Phys. Rev. D 26 (1982) 415
  * (23) A. Maciolek, A. Gambassi, and S. Dietrich, _Critical Casimir effect in superfluid wetting films_ , Phys. Rev. E76 (2007) 031124
  * (24) R. Moazzemi, M. Namdar, and S.S. Gousheh, _The Dirichlet Casimir effect for $\phi^{4}$ theory in $(3+1)$-dimensions: a new renormalization approach_, JHEP09 (2007) 029
  * (25) F.M. Schmidt and H.W. Diehl, _Crossover from Attractive to Repulsive Casimir Forces and Vice Versa_ , Phys. Rev. Lett. 101 (2008) 100601
  * (26) A. Flachi, _Interacting Fermions, Boundaries, and Finite Size Effects_ , Phys. Rev. D 86 (2012) 104047
  * (27) A. Flachi, _Strongly Interacting Fermions and Phases of the Casimir Effect_ , Phys. Rev. Lett. 110 (2013) 060401
  * (28) A. Flachi, M. Nitta, S. Takada and R. Yoshii, _Sign Flip in the Casimir Force for Interacting Fermion Systems_ , Phys. Rev. Lett. 119 (2017) 031601
  * (29) M. N. Chernodub, V. A. Goy and A. V. Molochkov, _Casimir effect and deconfinement phase transition,_ Phys. Rev. D 96, no.9, 094507 (2017)
  * (30) M. N. Chernodub, V. A. Goy, A. V. Molochkov and H. H. Nguyen, _Casimir Effect in Yang-Mills Theory in D=2+1,_ Phys. Rev. Lett. 121, no.19, 191601 (2018)
  * (31) M.A. Valuyan, _The Dirichlet Casimir energy for the $\phi^{4}$ theory in a rectangle_, Eur. Phys. J. Plus 133 (2018) 401
  * (32) A. Flachi, M. Nitta, S. Takada and R. Yoshii, _Casimir force for the ${\mathbb{C}}P^{N-1}$ model,_ Phys. Lett. B 798, 134999 (2019)
  * (33) A. Flachi, V. Vitagliano, _The Casimir effect for nonlinear sigma models and the Mermin–Wagner–Hohenberg–Coleman theorem_ , J. Phys. A: Math. Theor. 54 (2021) 265401
  * (34) P.T. Song, N. Van Thu, _The Casimir Effect in a Weakly Interacting Bose Gas_ , J. Low Temp. Phys. 202 (2021) 160
  * (35) A. Flachi, _Quantum vacuum phenomena in various backgrounds_ , Int. J. Mod. Phys. A 37 (2022) 2241008
  * (36) Y. Kurita, M. Kobayashi, T. Morinari, M. Tsubota and H. Ishihara, _Spacetime analogue of Bose-Einstein condensates: Bogoliubov-de Gennes formulation,_ Phys. Rev. A 79, 043616 (2009)
  * (37) C. Barcelo, L.J. Garay and G. Jannes, _The two faces of quantum sound,_ Phys. Rev. D 82, 044042 (2010)
  * (38) F. Impens, A. M. Contreras-Reyes, P. A. Maia Neto, D. A. R. Dalvit, R. Guérout, A. Lambrecht and S. Reynaud, _Driving quantized vortices with quantum vacuum fluctuations,_ Europhys. Lett. 92 (2010) 40010
  * (39) J. T. Mendonça, H. Terças, J. D. Rodrigues and A. Gammal, _Casimir Force between Two Vortices in a Turbulent Bose–Einstein Condensate,_ Atoms 8 (2020) 77
  * (40) T. A. Pasquini, Y. Shin, C. Sanner, M. Saba, A. Schirotzek, D. E. Pritchard, and W. Ketterle, _Quantum Reflection from a Solid Surface at Normal Incidence,_ Phys. Rev. Lett. 93 (2004) 223201
  * (41) D. M. Harber, J. M. Obrecht, J. M. McGuirk, and E. A. Cornell, _Measurement of the Casimir-Polder force through center-of-mass oscillations of a Bose-Einstein condensate,_ Phys. Rev. A 72 (2005) 033610
  * (42) G. A. Moreno, D. A. R. Dalvit and E. Calzetta, _Bragg spectroscopy for measuring Casimir–Polder interactions with Bose–Einstein condensates above corrugated surfaces,_ New J. Phys. 12 (2010) 033009
  * (43) H. Bender, C. Stehle, C. Zimmermann, S. Slama, J. Fiedler, S. Scheel, S. Y. Buhmann, and V. N. Marachevsky, _Probing Atom-Surface Interactions by Diffraction of Bose-Einstein Condensates,_ Phys. Rev. X 4 (2014) 011029
  * (44) J. M. Obrecht, R. J. Wild, M. Antezza, L. P. Pitaevskii, S. Stringari, and E. A. Cornell, _Measurement of the Temperature Dependence of the Casimir-Polder Force,_ Phys. Rev. Lett. 98 (2007) 063201
  * (45) G. A. Moreno, R. Messina, D. A. R. Dalvit, A. Lambrecht, P. A. Maia Neto, and S. Reynaud, _Disorder in Quantum Vacuum: Casimir-Induced Localization of Matter Waves,_ Phys. Rev. Lett. 105 (2010) 210401
  * (46) J. Marino, A. Recati, and I. Carusotto, _Casimir Forces and Quantum Friction from Ginzburg Radiation in Atomic Bose-Einstein Condensates,_ Phys. Rev. Lett. 118 (2017) 045401
  * (47) Editors: F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, _NIST Handbook of Mathematical Functions_ , Cambridge university press (2010).
  * (48) L.D. Carr, C.W. Clark, & W.P. Reinhardt, _Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity_ , Phys. Rev. A 62 (2000) 063610
  * (49) L.D. Carr, C.W. Clark, & W.P. Reinhardt, _Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity_ , Phys. Rev. A 62 (2000) 063611