11institutetext: Bayer AG, Machine Learning Research, Berlin, Germany
11email: {marco.bertolini, djork-arne.clevert<EMAIL_ADDRESS>
# Explaining, Evaluating and Enhancing Neural Networks’ Learned
Representations

Marco Bertolini  Djork-Arné Clevert  Floriane Montanari

###### Abstract

Most efforts in interpretability in deep learning have focused on (1)
extracting explanations of a specific downstream task in relation to the input
features and (2) imposing constraints on the model, often at the expense of
predictive performance. New advances in (unsupervised) representation learning
and transfer learning, however, raise the need for an explanatory framework
for networks that are trained without a specific downstream task. We address
these challenges by showing how explainability can be an aid, rather than an
obstacle, towards better and more efficient representations. Specifically, we
propose a natural aggregation method generalizing attribution maps between any
two (convolutional) layers of a neural network. Additionally, we employ such
attributions to define two novel scores for evaluating the informativeness and
the disentanglement of latent embeddings. Extensive experiments show that the
proposed scores do correlate with the desired properties. We also confirm and
extend previously known results concerning the independence of some common
saliency strategies from the model parameters. Finally, we show that adopting
our proposed scores as constraints during the training of a representation
learning task improves the downstream performance of the model.

###### Keywords:

Explainable AI Representation Learning Convolutional Neural Networks

##  1 Introduction

The steadily growing field of representation learning [18, 1] has been
investigating possible answers to the question “how can we learn good
representations?”. This has led to the rise of self-supervised learning and
transfer learning [48]. In practice, the quality of a representation is
measured by the performance on the downstream task of interest. This can lead
to a proliferation of models that potentially overfit the few benchmark
datasets typically used as downstream tasks.

In this work, we would like to step away from such traditional evaluations and
instead come back to the seminal work by [7], questioning “what makes one
representation better than another?”. Our proposal arises from the field of
explainable AI (XAI) [17, 35, 12, 5]. In XAI, the primary goal is usually to
provide an interpretation of a model’s predictions either in terms of the
input features or in terms of some user-defined high-level concepts [21]. Here
as well, the field has suffered from the same endpoint-bias propelling model
development, as XAI methods are in most cases designed to extract explanations
for a specific downstream task. This is clearly a limitation: such methods
cannot be directly applied to “explain” embeddings learned in a self-
supervised manner.

In this paper, we develop an XAI framework that does not rely on any
downstream network or task, and we then show how we can adopt it to evaluate
the quality of an embedding. This requires a generalization of the concept of
explanation: what we try to achieve here is to visualize and compare
embeddings learned by any neural network.

Our explainability framework is local (we provide a method to infer
explanations for a single data input), independent of any downstream task, and
flexible (the framework is model-agnostic even though most of our examples and
experiments are given for CNNs applied to image analysis). The scores we
derive are global and can interpret a whole embedding vector.

###  1.1 Related Work

To visualize global concepts learned by trained CNNs, one can employ
techniques such as filter visualization [45, 27, 39] or activation
maximization techniques. Those aim at extracting visual patterns that maximize
an internal neuron or predefined direction in the weight space [28, 29, 30].
In this context, [41] visualizes hidden neurons to gain insights on the
property of transfer learning.

In the subfield of local explanations, saliency methods are commonly used to
generate and visualize attributions for specific inputs. In our work, we
compare three common strategies: Vanilla Gradients [4, 39],
Activations$\times$Gradients (which generalizes Gradient$\times$Input [38] to
any hidden layer), and Grad-CAM [37]. Although our techniques can be similarly
applied to any saliency method, the three above suffice to show our proof of
concept, and avoid the computational overhead that comes with, for example,
Integrated Gradients [40]. [9] proposed a variation of the Grad-CAM method for
Embedding Networks. Their approach however fundamentally differs from ours, as
the authors explicitly utilize a downstream loss function for computing
attributions. The authors of [8], instead, extend saliency methods to
visualize relevant features in latent space for generative models.

An alternative strategy to increase model interpretability is to restrict the
architecture to intrinsically interpretable ones [11] or to impose other
constraints during the training process [46, 47, 20, 26]. One frequently used
constraint is to enforce the disentanglement of the learned representation.
However, this increased interpretability through constraints often comes at
the cost of predictive performance [46, 26]. We will show that this need not
be the case: our XAI-derived scores do help the model achieve higher
predictive power.

Several works [3, 32, 6, 42, 14] demonstrated that neural networks learn high-
level concepts in the hidden layers. In particular, [21] showed that (1) such
concepts can be represented as directions in the latent space of hidden
layers, and (2) deeper layers tend to learn more complex concepts as opposed
to shallow layers. The disadvantage of their approach is that such meaningful
directions are learned with the aid of additional labelled data. Our proposal
generalizes this basic idea to learned embeddings by aggregating the concepts,
learned in hidden layers, that contribute to activate the target
representation. Our approach is partly motivated by the work [15]. The authors
show that, often, single filters fail to capture high-level concepts, and that
filter aggregation is instead required.

###  1.2 Contributions

We can summarize our main contributions as follows:

  * •

We propose a simple and flexible aggregation scheme for explaining and
visualizing embeddings of (convolutional) neural networks (Section 2). To the
best of our knowledge, the present work is the first systematic attempt to do
so without any consideration of a downstream task or loss.

  * •

We propose in Section 3 metrics to quantify the quality of these embeddings.

  * •

We show that these metrics correlate with desired embedding properties
(specifically, informativeness and disentanglement) (Section 4.1, 4.2, 4.3)
and detect effects like dataset drift (Section 4.4).

  * •

We show that training with the proposed metrics as constraints improves the
quality of the embeddings (Section 4.5).

##  2 Visualizing Explanations of Embeddings through Aggregation

Figure 1: Schematic description of the embedding explanation framework,
including how the noise and variance scores are calculated.

In this section, we present a simple and intuitive procedure for visualizing
latent embeddings of a neural network.

Let $\Psi$ denote a representation learning task, that is,
$\Psi:\mathcal{M}\rightarrow\mathbb{R}^{N}$, where
$\mathcal{M}\simeq\mathbb{R}^{d}$ is the manifold describing the input data.
For $\mathbf{x}\in\mathcal{M}$, $\Psi(\mathbf{x})$ denotes the target
embedding, which is supposed to be a meaningful, lower-dimensional
representation of the data. Relevant examples are, for instance, the
bottleneck layer of an autoencoder network, or the output of the last
convolutional layer in a image analysis NN. To fully characterize the concepts
captured in the learned representation, and to mitigate the influence of the
input features on the attribution map (as we shall see in Section 4.1), it is
insightful to consider attributions that probe latent dimensions. Thus, we
describe our task as
$\Psi:\mathcal{M}\xrightarrow{\phi_{p}}\mathcal{M}_{p}\xrightarrow{\xi_{p}}\mathbb{R}^{N}$,
where $\mathcal{M}_{p}$ is the input space of the $p^{\text{th}}$ layer in the
network (in this notation, $\mathcal{M}_{0}\equiv\mathcal{M}$). According to
our assumption of a convolution structure,
$\mathcal{M}_{p}\simeq\mathbb{R}^{C_{p}\times d_{p}}$, where $C_{p}$ is the
number of channels and $d_{p}$ is the spatial dimension of the latent
representation.

###  2.1 Attributions for Embeddings

To generate explanations for embeddings without any reference to a downstream
task, we regard each target embedding component $\Psi(\mathbf{x})_{i}$, for
$i=1,\dots,N$, as a stand-alone output node. Consequently, we interpret its
corresponding saliency attributions
$\partial\Psi(\mathbf{x})_{i}/\partial\mathbf{x}$ as feature importance for
the activation of the corresponding latent component. This approach can be
straightforwardly extended to any intermediate layer, where we regard the
attribution values
$\partial\xi_{p}(\mathbf{x})_{i}/\partial\mathbf{\phi}_{p}(\mathbf{x})\in\mathcal{M}_{p}$
as importance weights for the activation of the $i^{\text{th}}$ representation
component with respect to the $p^{\text{th}}$ layer neuron activations
$\partial\mathbf{\phi}_{p}(\mathbf{x})$.

In general, $\dim\mathcal{M}_{p}\neq\dim\mathcal{M}$, as the above procedure
will generate, for each $i$, a map for each of the $C_{p}$ channels in the
feature layer $p$. To generate a unique map for the attribution, we aggregate
the maps
$\partial\xi_{p}(\mathbf{x})_{i}/\partial\mathbf{\phi}_{p}(\mathbf{x})$ along
the channel dimension of layer $p$ by means of a map
$\mathcal{C}:\mathcal{M}_{p}\mapsto\mathbb{R}^{d_{p}}$, such that
$\mathcal{C}:\partial\xi_{p}(\mathbf{x})_{i=1,\dots,N}/\partial\mathbf{\phi}_{p}(\mathbf{x})\mapsto(\widehat{A}_{\Psi}^{p})_{i=1,\dots,N}\in\mathbb{R}^{d_{p}\times
N}$.

In most representation learning applications $N\sim(10^{2},10^{3})$, and it is
unfeasible to examine saliency maps for all latent dimensions $i$. A simple
solution to generate one single map per input sample is to aggregate the
component-wise attribution maps via
$\mathcal{E}:(\widehat{A}_{\Psi}^{p})_{i}\in\mathcal{M}\mapsto\widehat{A}_{\Psi}^{p}\in\mathcal{M}_{p}$.

It is desirable to generate the final attribution in the original input space,
in order to formulate hypothesis about explanations on the input features.
However, it can happen that $d_{0}>d_{p}$, for instance due to the presence of
pooling layers. When this is the case, we apply an upscaling map
$\mathcal{U}:\mathbb{R}^{d_{p}}\rightarrow\mathbb{R}^{d_{0}}$. Given the
property of a convolutional layer that the value of each superpixel depends on
its receptive field, the simplest choice for $\mathcal{U}$ is to consider the
map in $\mathbb{R}^{d_{p}=H_{p}\times W_{p}}$ as a low-resolution map in
$\mathbb{R}^{d=H\times W}$. This is the choice we implemented in all our
examples and experiments.111 Another upsampling scheme, for instance, involves
performing a transposed convolution of the lower-dimensional matrix with a
fixed Gaussian kernel to obtain a full-resolution map [25].

Putting all together, given a learning task $\Psi$, we define the attribution
method $A_{\Psi}^{p}$ as a map

$\displaystyle\small\mathcal{M}\xrightarrow[\text{forw.}]{\Psi}\mathbb{R}^{N}\xrightarrow[\text{back}]{\partial\xi_{p}}\mathcal{M}_{p}^{N}\xrightarrow[\text{aggr.}]{\mathcal{C}}\mathbb{R}^{d_{p}\times
N}\xrightarrow[\text{aggr.}]{\mathcal{E}}\mathbb{R}^{d_{p}}\xrightarrow[\text{upscale}]{\mathcal{U}}\mathcal{M}~{}.$

We depicted this chain of operations in Figure 1. In the preceding discussion,
to keep notations simple, we explicitly considered vanilla gradients
$\partial\xi_{p}$ as the reference saliency method. This however can be
replaced with any available saliency method. Figure 2 shows, for two input
samples, examples of embedding attributions for different saliency and
aggregation strategies. We notice that for deeper layers the attributions
focus on some high-level concepts (e.g., eyes and nose for Figure 2b).

Following [6, 3, 21], latent dimensions contain meaningful directions that can
be learned through linear classifiers and that correspond to humanly-
understandable concepts. The attribution maps
$(\widehat{A}^{p}_{\Psi}{})_{i,c}:\mathcal{M}\rightarrow\mathbb{R}^{d_{p}}$,
where $c=1,\dots,C_{p}$, is to be understood as quantifying the concept(s)
learned by the model in the $c^{\text{th}}$ filter of the $p^{\text{th}}$
layer. Thus, in our approach, we regard the channel aggregation map
$\mathcal{C}$ as a “concept” aggregation map. On the other hand, aggregation
along the latent embedding dimension is dictated by the need to “summarize”
explanations across the whole representation. Given a downstream task
$\Theta:\mathbb{R}^{N}\rightarrow\mathbb{R}$, however, each embedding
dimension $\Psi_{i}$ acts simply as an input feature. With regard to
explainability, we interpret the explanations for $\Psi_{i=1,\dots,N}$ as a
basis of explanations for $\Theta$. Namely, explanations $A_{\Theta}$ for the
task $\Theta$ would act as weights for the more elementary explanations
$A_{\Psi_{i}}$.

Figure 2: Example of embeddings for (a) different attribution schemes
($\mathcal{C}=\mathcal{A}_{\text{abs}}$,
$\mathcal{E}=\mathcal{A}_{\text{var}}$) and (b) aggregation strategies
(Activations $\times$ Gradients). Below each map we report its corresponding
sparsity score, given as the average value over the normalized 2D attribution
map. We report in the appendix further examples and combinations of
attribution and aggregation schemes.

###  2.2 Aggregation Schemes

Here we review possible aggregation strategies for both the channel and
embedding dimensions. To formalize our discussion, we represent the set of
attribution maps as a multiset $\mathcal{X}$, that is, a set allowing for
repeating elements. Both the channel ($\mathcal{C}$) and embedding
($\mathcal{E}$) maps can be thought of as an aggregation map
$\mathcal{A}(\mathcal{X})$ over a multiset $\mathcal{X}$. We also propose a
new aggregation strategy, given by $\mathcal{A}=\text{Var}[\mathcal{X}]$, that
yields sparser attribution values.

#### Mean aggregation.

The simplest aggregation method, is the mean aggregator,
$\mathcal{A}_{\text{mean}}(\mathcal{X})=\frac{1}{|\mathcal{X}|}\sum_{x\in\mathcal{X}}x$.
The mean aggregator captures the distribution of elements in a multiset [44],
but not the multiset itself. For example, for $\mathcal{X}_{1}=\\{-1,1,0,0\\}$
and $\mathcal{X}_{2}=\\{0,0,0,0\\}$ we obtain
$\mathcal{A}_{\text{mean}}(\mathcal{X}_{1})=\mathcal{A}_{\text{mean}}(\mathcal{X}_{2})=0$.
Such aggregations “with signs” potentially suffer from the risk of signal
cancellation.

#### Abs aggregation.

Another simple aggregator is given by the mean absolute aggregator,
$\mathcal{A}_{\text{abs}}(\mathcal{X})=\frac{1}{|\mathcal{X}|}\sum_{x\in\mathcal{X}}|x|$.
The mean absolute aggregator captures the distribution of elements in a
multiset without signs. Consequently, the resulting attributions tend to be
quite dense. In fact, $\mathcal{A}_{\text{abs}}=0$ if and only if all elements
in $\mathcal{X}$ are zero. This feature could be detrimental in our setting,
as sparsity is often a desired property in explainability, and especially
challenging to achieve in post-hoc explanations.

#### Variance aggregation.

To remedy this lack of sparsity, we introduce a novel aggregation scheme, the
variance aggregator, $\mathcal{A}_{\text{var}}=\text{Var}[\mathcal{X}]$. The
variance aggregator is somewhat complementary to the mean aggregator. In fact,
for $\mathcal{X}_{1}=\\{-1,1,0,0\\}$, we have
$\mathcal{A}_{\text{var}}(\mathcal{X}_{1})=\frac{2}{3}$ while
$\mathcal{A}_{\text{mean}}(\mathcal{X}_{1})=0$. On the other hand, for
$\mathcal{X}_{2}=\\{1,1,1,0\\}$,
$\mathcal{A}_{\textit{var}}(\mathcal{X}_{2})=\frac{1}{4}$ while
$\mathcal{A}_{\text{mean}}(\mathcal{X}_{2})=\frac{3}{4}$. Thus, the variance
aggregator assigns higher values to multisets with a wider range of values and
penalizes biased multisets where all values concentrate around its mean,
regardless of its magnitude. We derive in the Supplementary Information (SI) a
simple argument that shows, under a few basic assumptions, that the variance
aggregator yields a sparser map than the mean absolute aggregator. We provide
evidence that our claim extends to real data by displaying different
aggregation methods in Figure 2b. For each attribution map, we computed a
measure of sparsity as the average value over the (normalized to the range
$[0,1]$) 2D attribution map. For the mean aggregator, the sparsity is computed
on the absolute value of the map. As claimed, for all layers, the variance
produces a sparser map than the mean absolute aggregator.

#### Comments and practical advices.

Finally, we provide practical guidance as to the combinations of aggregation
schemes for the maps $\mathcal{C},\mathcal{E}$. We find the combination
$\mathcal{C}=\mathcal{A}_{\text{abs}}$, $\mathcal{E}=\mathcal{A}_{\text{var}}$
(and viceversa) especially insightful to identify the prevailing features
dominating the representation, while
$\mathcal{E}\equiv\mathcal{C}=\mathcal{A}_{\text{abs}}$ is helpful to detect
features that are not captured by the embedding (see Figure 2). We find
instead $\mathcal{E}\equiv\mathcal{C}=\mathcal{A}_{\text{mean}}$ potentially
useful when the embedding explanations are combined in a generalized chain
rule with a downstream task $\Theta$ as mentioned above, especially when it is
desirable to distinguish feature contributing positively or negatively. When
$\mathcal{C}\equiv\mathcal{E}=\mathcal{A}_{\text{var}}$, we observed that the
map is often too sparse.

##  3 Explainability-derived Scores

In this section, by employing the embedding explanations described in the
previous section, we derive useful metrics to quantify desired properties of a
learned latent representation: (1) informativeness, the ability of a latent
representation to capture meaningful features of the input sample, and (2)
disentanglement, the property quantifying the degree to which latent features
are independent of one another. We will show that our proposed scores
correlate with the desired properties as well as with the more standard
approach of predictive performance on a downstream task.

###  3.1 Informativeness: the Noise Score

The motivating idea of the noise score is that attributions for an informative
representation should, on average, substantially differ from attributions
generated from a less meaningful representation. Explicitly, let
$x\in\mathbb{R}^{d}$ be a (meaningful) input image, and let
$x_{\text{noise}}\in\mathbb{R}^{d}$ be such that $x_{\text{noise}}\sim
U(0,1)^{d}$. We define the noise score for representation task $\Psi$ and
layer $p$ as the Mean Absolute Error (MAE) between (normalized) attributions
of a meaningful data input and a noise-generated input

$\displaystyle
N_{A^{p}_{\Psi}}(x,x_{\text{noise}})=\overline{|A^{p}_{\Psi}(x)-A^{p}_{\Psi}(x_{\text{noise}})|}~{}.$
(1)

We schematically depicted the above definition in Figure 1. The attribution
maps in (1) are normalized to the range $[0,1]$ to ensure that the score does
not merely reflect the magnitude of the respective attribution values. With
such a normalization, we have $\sup N_{A^{p}}(x,y)=1$, which occurs if
$A^{p}_{\Psi}(x)-A^{p}_{\Psi}(y)=\pm 1$, while $\inf N_{A^{p}}(x,y)=0$, which
occurs when $A^{p}_{\Psi}(x)\equiv A^{p}_{\Psi}(y)$.

We will show in the next section that (1) distinguishes different
representations for which we know their relative informativeness. However, it
is often convenient to have a global reference value $N^{0}$, to establish an
intrinsic metric for informativeness. Since we do not know the underlying
distribution of $A^{p}_{\Psi}$, we can sample a benchmark value as follows:

$\displaystyle
N^{0}_{A^{p}_{\Psi}}=\overline{|A^{p}_{\Psi}(y_{\text{noise}})-A^{p}_{\Psi}(x_{\text{noise}})|}~{},$
(2)

where $y_{\text{noise}}\sim U(0,1)^{d}$.

#### The score curve.

Given a representation task $\Psi$, we can compute (1) and (2) for each
intermediate layer $p$ of the network. The resulting noise score curve
provides a more complete picture of how effectively the different layers
organize information in the target representation.

###  3.2 Conciseness: the Variance Score

A second beneficial property of a representation is disentanglement [13, 33,
10]. Disentanglement, in the sense of decomposability [24], implies that the
latent features are independent of one another. As a consequence, we also
intuitively expect the attributions for each of the latent features not to
overlap among each other significantly. Of course, a pair of latent features
$\Psi_{1,2}$ can be independent but have very similar attribution maps for a
given input, as they might encode independent concepts that happen to overlap
in the given sample. Still, we expect the majority of the input space to be
represented by some high-level concepts, and a decomposable representation
should cover a large portion of it. Since our choice of measuring
disentanglement is correlated but differs from the more standard definition,
we coin a new term: the conciseness of a representation is a measure of the
variance of attribution maps across the latent features.

Explicitly, let $x\in\mathbb{R}^{d}$ be a (meaningful) sample input. We define
the variance score for the representation task $\Psi$ at layer $p$ as the mean
variance of the channel-aggregated (but not embedding-aggregated) attribution
maps $(\widehat{A}^{p}_{\Psi})_{i}(x)$, i.e.,

$\displaystyle
V_{A^{p}_{\Psi}}(x)=\overline{\text{Var}\widehat{A}^{p}_{\Psi}(x)}~{}.$ (3)

Figure 1 illustrates how (3) relates to the aggregation strategies of Section
2. The above definition enables a direct comparison of embeddings: a
representation $\Psi$ is more concise than representation $\Psi^{\prime}$ if
$V_{A_{\Psi}^{p}}(x)>V_{A_{\Psi^{\prime}}^{p}}(x)$.

In order to determine the intrinsic conciseness of a representation, we recall
Popoviciu’s inequality [31] for a bounded random variable $Z$, which reads
$\text{Var}Z\leq\frac{(\sup Z-\inf Z)^{2}}{4}$, yielding in our case
$V_{A^{p}_{\Psi}}(x)\leq 1/4$, assuming that each dimension map
$(\widehat{A}^{p}_{\Psi})_{i}$ is normalized in the range $[0,1]$. The only
random variable $Z$ whose variance assumes the maximum value is such that
$Z=0,1$ each with probability $\frac{1}{2}$. In terms of our map, this would
represent that a “superpixel” is either considered on/off in each
representation component (with equal distribution). Note that this is
independent from the number of drawings, or in our case the embedding size.
Hence, (3) is suitable for comparing representations of any size, as we will
do in Section 4.3.

##  4 Experiments and Results

In this section we describe our experiments and illustrate the insights we
gain from the metrics we defined in the previous section. We apply our
techniques to three widely used architectures: Inception V3 [42], ResNet18
[19] and AlexNet [22].222Resnet18 has a BSD 3-clause license, Inception V3 has
an MIT license, AlexNet has an Apache V2 license. For each of these, the
target representation will be the output of the last convolutional (or
eventually pooling) layer.

We conduct several tests to assess the sensitivity of our scores to
informativeness, disentanglement and predictive performance. Specifically, we
compare the values of our scores for a given task and model to the one
computed from networks with random weights. In addition of being natural
benchmarks for the desired properties listed above, randomly initialized
networks do correspond to non-trivial representations [43]. Thus, our scores
can be interpreted in this case as probing the model architecture as a prior
to the learning task [36, 3].

Figure 3: Average noise score curves (over 50 input samples) for ResNet18
trained on ImageNet [34] for different attribution strategies and
$\mathcal{C}=\mathcal{E}=\mathcal{A}_{\text{abs}}$.

###  4.1 Detecting Model’s Parameter (in)dependence

We begin by comparing the noise score (1) for a fixed network architecture and
fixed task $\Psi$. Score curves for different attribution strategies are
presented in Figure 3. For Activations $\times$ Gradients, the noise score
fail to distinguish between the trained and the untrained network in early
layers. This agrees with the findings of [2], where the authors show that upon
visual inspection the defining structure of the input is still recognizable
for a network with randomly initialized weights. Our analysis shows that this
does not occur for explanations in deeper layers, as the two curves separate
further as we go deeper in the network. Thus, while Activations $\times$
Gradients is potentially misleading in shallow layers (due to the bias
introduced by the input values), it is a meaningful attribution method in
deeper layers (we defer explicit examples of such maps to the SI).

For Vanilla Gradients, we notice instead that the two curves (trained and
untrained) are well separated even in the early layers. This confirms the
analysis of [2], where the authors found a significant discrepancy between the
corresponding saliency maps. The difference is however less conspicuous in
comparison to the benchmark curve computed from (2). This shows that Vanilla
Gradients is more sensitive to weight parameters than input parameters (we
recall that the $N^{0}$ benchmark curve (2) is computed for a trained
network). This is not the case for Activations $\times$ Gradients, where the
noise curves for the untrained network and the benchmark exhibit a very
similar behaviour.

Finally, we observe an overall poor performance of Grad-CAM, except in the
very deep layers, where the noise score successfully distinguishes the trained
network from both benchmarks [2]. This is consistent with the usual practice
of considering the output of the last convolution layer for a Grad-CAM
visualization.

###  4.2 Detecting Downstream Performance

Figure 4: Average noise scores ($25$ samples) for ResNet18 networks with
layer-wise randomly initiated weights. Each data point is obtained by
averaging the noise score for each layer of the noise curve. $x$-axis values
correspond to the re-initialized layer’s number.

Table 1: Classification performance for single layer randomization.
All=completely re-initialized network, None=full trained network.

Layer | Top 1 Acc | Top 5 Acc
---|---|---
All | 0.0 | 0.7
5 | 0.0 | 1.1
6 | 0.2 | 0.9
7 | 0.7 | 1.8
8 | 0.3 | 2.8
9 | 1.8 | 4.2
10 | 0.2 | 0.3
11 | 0.2 | 0.3
12 | 63.7 | 85.1
14 | 4.3 | 10.1
17 | 30.0 | 54.3
None | 74.2 | 91.8

The noise score (1) aims at quantifying the amount of information captured by
the target embedding. It is therefore expected to correlate with a given
downstream task performance. We test this expectation by comparing
classification performance and noise scores of embeddings obtained through an
independent layer randomization. Explicitly, we fix the weights of all the
layers to their trained value except for one layer, whose weights we
randomize. We report in Table 1 the classification performance for the
ResNet18 architecture, and in Figure 4 we depict the corresponding embeddings’
noise scores, defined as the average value of the corresponding noise curve.

Table 1 shows that layer randomizations have different effects on the overall
performance of the representation, as some networks (12,17) are still
predictive. In Figure 4 we observe that embeddings which are still predictive
exhibits a higher average noise score. The Spearman rank-order correlation
coefficients between average noise score and downstream performance are $0.89$
(Top5) and $0.91$ (Top1), showing a very strong correlation between the noise
score and the downstream performance of the representations.

###  4.3 Detecting Representations’ Conciseness

Increasing the dimension of a representation increments its expressive power.
However, large embeddings incur in the risk of noisy and redundant
representations, which in turn have detrimental effects on the learning
efficacy.

Figure 5: Variance curves for three popular networks trained on Imagenet.
($\mathcal{C}=\mathcal{A}_{\text{mean}}$, attr.= Activations $\times$
Gradients.) The blue dashed line depicts the theoretical maximum value.

Here we provide evidence that the variance score (3) is suited to detect such
pathology. Namely, we compute the variance curve for a given task (image
classification on the ImageNet dataset [34]) and attribution scheme for
different architectures. The corresponding variance curves are reported in
Figure 5. The attributions $\widehat{A}_{\Psi}^{p}$ have been rescaled to the
values $0,1$ (after normalization) according to the threshold $0.5$. The
variance curves of the untrained models indicate the conciseness prior of the
pure model architecture, which is affected, for example, by the choice of
final non-linearity. In fact, we can assume that randomly initialized networks
will tend to maximize representation conciseness, compatible with the
architecture prior. In our experiments, we observe that the AlexNet
architecture has a strong effect on the variance of the embedding (in average
around $75\%$ of the upper bound), while for Inception V3 and ResNet18 both
curves approach the theoretically maximum value of $1/4$ (see Section 3.2).

Next, we observe that for the trained Inception V3 and ResNet18, whose target
representation dimensions are $2048$ and $512$ respectively, the variance
score is in average very close to its upper bound. On the other hand, the
trained AlexNet (embedding dimension 9216) shows a strong decline from the
randomly initialized version. This shows that the lack of variance in the
embedding explanations is not merely due to the architecture inductive bias
(as in the case for the random weights network), but also on the
representation learning itself. It is expected that such a high embedding size
induces redundancy and therefore more entangled embeddings. Accordingly, our
experiment provides experimental evidence that the conciseness of a
representation correlates with the desired property of disentanglement.

###  4.4 Detecting Dataset Drift

We can employ the noise score to detect the amount of drift from the training
dataset. To test this, we train a simple autoencoder model on the MNIST
dataset [23].333MNIST dataset is available under the terms of the Creative
Commons Attribution-Share Alike 3.0 license. The encoder network consists of
two convolutional layers followed by a linear layer. We refer to the SI for
further details concerning hyperparameters, training details and choice of
architecture. We generate a drift by considering shifted data samples
(normalized in range $[0,1]$)
$\mathbf{x}_{\lambda}=\mathbf{x}+\lambda\mathbf{y}$, where $\mathbf{x}$ is a
sample from the original dataset, and the drift is generated by a normal
distribution $\mathbf{y}\sim\mathcal{N}(\mu=0,\sigma=1)$ and controlled by the
parameter $\lambda$. We present in Figure 6 the relative difference of the
noise scores for the second convolutional layer for various values of
$\lambda$. Explicitly, this is defined as
$(N(\mathbf{x})-N(\mathbf{x}_{\lambda}))/N(\mathbf{x})$. Here we denote
$N(\mathbf{x})=N_{A_{\Psi}^{p}}=(\mathbf{x},\mathbf{x}_{noise})$ for
simplifying notation. Quantitatively, we obtain a Spearman coefficient of
$\rho=0.988$ between the relative noise scores and the corresponding drift
parameter $\lambda$. Our results confirm that the noise score is negatively
correlated with the shift amount. Hence, the noise score can be used as a
metric for dataset drift detection or, more generally, to quantify the
representation quality degradation for datasets unseen during training time.

###  4.5 Enhancing Representations with Score Constraints

We further test the usefulness of our scores by incorporating them as
constraints during model training. We train an autoencoder model $\Omega$
(with encoder $\Psi$) on the MNIST dataset [23] with a linear layer accessing
the bottleneck embedding as a 10-class classifier. The training loss is
therefore
$\mathcal{L}_{\text{train}}(\mathbf{x},y)=\mathcal{L}_{\text{rec}}(\mathbf{x})+\mathcal{L}_{\text{class}}(\mathbf{x},y)-\lambda_{1}N_{A_{\Psi}}(\mathbf{x},\mathbf{y}_{\text{noise}})-\lambda_{2}V_{A_{\Psi}}(\mathbf{x})$,
where
$\mathcal{L}_{\text{rec}}(\mathbf{x})=|\mathbf{x}-\Omega(\mathbf{x})|^{2}$ is
the autoencoder reconstruction error and $\mathcal{L}_{\text{class}}$ is the
classification loss (Cross Entropy Loss). The two coefficients $\lambda_{1,2}$
control the ratios between the model loss and the score constraints. The
encoder network $\Psi$ consists of two convolutional layers followed by a
fully connected layer, whose output is our target bottleneck representation.
The scores are computed with respect to the second convolutional layer. In
order not to affect excessively the training speed, we applied the scores
every 20 mini-batches. We did not perform any hyperparameter optimization: we
trained the model for a fixed number of epochs (dependent on the bottleneck
dimension), and we evaluate our results on a held-out test set. We refer to
the SI for further details concerning hyperparameters, architecture and
training details. We compare the classification test performance of the model
for various values of the coefficients $\lambda_{1,2}$, and we report our
findings in Table 2. We note that, overall, the score constraints led to
representations that achieved a higher test accuracy on the classification
task. We further notice that representation learning benefited from the
variance score constraint for higher embedding size: here, redundancy and
entanglement affect the learning, and forcing more concise representations
reduces these undesirable effects. On the contrary, the noise score constraint
has bigger impact on the models with particularly small bottlenecks, for which
having a very expressive representation is key. Finally, training including
both scores outperforms the original model in all the cases.

Figure 6: Correlation between noise score and dataset drift relized by
introduction of noise shift.

Table 2: Classification accuracies for score-constrained embeddings on the
MNIST dataset. The performances are averaged over 4 runs. NS = noise score, VS
= variance score.

Model | $\lambda_{1}$ | $\lambda_{2}$ |  dim$=2$ |  dim$=3$ |  dim$=5$ |  dim$=10$
---|---|---|---|---|---|---
orig. | 0 | 0 | $88.3\pm 0.7$ | $93.74\pm 0.3$ | $97.2\pm 0.2$ | $97.7\pm 0.2$
NS | 0.01 | 0 | $89.1\pm 0.5$ | $94.3\pm 0.4$ | $97.4\pm 0.1$ | $97.8\pm 0.2$
NS | 0.1 | 0 | $87.8\pm 0.0$ | $94.3\pm 0.2$ | $97.1\pm 0.3$ | $97.8\pm 0.3$
NS | 1.0 | 0 | $89.3\pm 0.5$ | $95.1\pm 0.5$ | $97.3\pm 0.1$ | $97.9\pm 0.1$
VS | 0 | 0.01 | $88.6\pm 0.9$ | $94.8\pm 0.8$ | $97.6\pm 0.1$ | $98.0\pm 0.2$
VS | 0 | 0.1 | $88.5\pm 1.0$ | $94.5\pm 0.7$ | $97.4\pm 0.2$ | $98.0\pm 0.2$
VS | 0 | 1.0 | $88.7\pm 0.9$ | $94.7\pm 0.6$ | $97.5\pm 0.2$ | $97.8\pm 0.2$
NS+VS | 0.1 | 0.1 | $88.5\pm 2.1$ | $94.6\pm 0.8$ | $97.4\pm 0.2$ | $97.8\pm 0.2$

##  5 Conclusions

In this work, we have proposed aggregation schemes to generalize gradient
attribution methods to any two intermediate layers of a neural network. In
particular, this framework can be used to probe the explanations of embeddings
with respect to intermediate feature maps without the need for a downstream
task or a direct mapping back to the input space. We derived two useful
metrics: the noise score can detect the informativeness of a representation,
and this in turn correlates with downstream task performance; the variance
score can be used to probe the disentanglement of an embedding. Both scores
can be used as constraints for representation learning, and were shown in a
proof-of-concept experiment to boost the performance of the learned
representation on the downstream task of interest.

This work is the first thorough study on how XAI techniques can be fruitfully
implemented to evaluate and enhance representation learning independently of a
downstream task. It would be interesting to apply our framework beyond the
convolutional realm, for instance, to graph representation learning or neural
machine translation. Our proposed variance aggregator can be seen as a first
step towards explainability from the point of view of projection pursuit [16]:
it would be of interest to explore, both theoretically and experimentally,
more general aggregation strategies based on higher momenta (e.g.,
lopsidedness). Moreover, our proposed score constraints should be tested in
more complex learning problems. One limitation of our current approach is the
increase of computational time due to the computation of the scores and the
extra back-propagations during training. It would be important to derive
“lighter” scores, which estimate (1) and (3) but with a lower memory and
computation footprint. We plan to return to these issues in future work.

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##  Appendix 0.A Variance Aggregator is sparser than Mean Absolute Aggregator

We present a simple argument supporting our claim that the variance aggregator
$\mathcal{A}_{\text{var}}$ is sparser than the mean absolute aggregator
$\mathcal{A}_{\text{abs}}$. Let $\mathcal{X}$ be a multiset whose elements
assume values in the range $[0,1]$. In our case, this can be achieved by
normalizing the multiset. Given the positiveness of $\mathcal{X}$, the mean
absolute aggregator simply reduces to the mean of $\mathcal{X}$, i.e.,
$\mathcal{A}_{\text{abs}}=\mathbb{E}[\mathcal{X}]$. Now we have that

$\displaystyle\mathbb{E}[\mathcal{X}]^{2}=\mathbb{E}[\mathcal{X}^{2}]-\text{Var}[\mathcal{X}]\leq\mathbb{E}[\mathcal{X}]-\text{Var}[\mathcal{X}]\quad\Longrightarrow\quad\text{Var}[\mathcal{X}]\leq\mathbb{E}[\mathcal{X}](1-\mathbb{E}[\mathcal{X}])~{},$
(4)

where we used the fact that
$\mathbb{E}[\mathcal{X}^{2}]\leq\mathbb{E}[\mathcal{X}]$ since $x^{2}\leq x$
for $x\in[0,1]$. Next, let us assume that $\text{Var}[Z]>\mathbb{E}[Z]$, then

$\displaystyle\mathbb{E}[\mathcal{X}]<\mathbb{E}[\mathcal{X}](1-\mathbb{E}[\mathcal{X}])\quad\Longrightarrow\quad\mathbb{E}[\mathcal{X}]<0~{},$
(5)

which violates the assumption of positiveness of $\mathcal{X}$. Thus, we have
shown that
$\mathcal{A}_{\text{var}}=\text{Var}[\mathcal{X}]\leq\mathbb{E}[\mathcal{X}]=\mathcal{A}_{\text{abs}}$.
In particular, the equality is achieved only for the multiset $\mathcal{X}$
for which all elements equal zero, where
$\mathcal{A}_{\textit{var}}=\mathcal{A}_{\text{abs}}=0$. This simple argument
shows that under some basic assumptions, given the same underlying
distribution $\mathbb{E}[\text{Var}Z]<\mathbb{E}[\mathbb{E}[Z]]$, and
therefore the variance aggregation induces a sparser attribution map.

##  Appendix 0.B Architecture, Hyperparameter and Training Strategy

###  0.B.1 Detecting Dataset Drift

In Section 4.4 of the main text we report the results of training an
autoencoder model on the MNIST dataset, showing that the noise score captures
the effect of dataset drift. The encoder model consists of $3$ convolutional
layers of size $[8/5/1/2,4/5/1/2,2/5/1/2]$.444 The notation U/K/S/P completely
defines a convolutional layer: U=number of units, K=kernel size, S=stride,
P=padding. After each convolutional layer we apply batch normalization and max
pooling (kernel=2). This produces a 32-dimensional bottleneck representation.
The decoder network consists of 3 (transposed) convolutional layers of size
$[4/5/1/2,8/5/1/2,1/5/1/2]$. A sampling interpolation (scale factor=2)
precedes each convolutional layer, and we apply batch normalization after the
first two convolutional layers. We used ReLU as activation function, except
for the final output of the decoder, where instead we apply a sigmoid function
to make sure the final output is in the range $[0,1]$. We use Mean Squared
Error (MSE) as our reconstruction loss. We use the Adam optimizer with a fixed
learning rate of $10^{-3}$, and we train for 5 epochs.

###  0.B.2 Training with Scores as Constraints

In section 4.5 of the main text we report our experiments regarding training
an autoencoder model on the MNIST dataset with our scores as constraints. The
encoder model consists of $2$ convolutional layers of size $[8/5/1/2,4/5/1/2]$
followed by a fully connected layer of size $s$. We apply batch normalization
after each convolutional layer. We conducted several experiments for
$s=2,3,5,10$. We used softplus as the encoder activation function. This is
because when the score constraints enter the training loss, we need to compute
second derivatives with respect to the weights, which vanish if the activation
functions are piece-wise linear. The classification branch of the network is a
10-dimensional fully connected layer accessing the bottleneck. The decoder
network consists of two fully connected layers of size $[32,28\times 28]$. We
used ReLU as a decoder activation function, and MSE as our reconstruction
loss. We use the Adam optimizer with a fixed learning rate of $10^{-3}$, and
we train for a fixed number of epochs $n_{\text{ep}}(s)$ depending on the
bottleneck size: $n_{\text{ep}}(2)=20$, $n_{\text{ep}}(3)=15$,
$n_{\text{ep}}(5)=10$, and $n_{\text{ep}}(10)=5$.

###  0.B.3 Computation Time

We report in Table 3 the training time for the autoencoder models, whose
results we presented in Section 4.5 of the main text. When our scores are
added to the training as constraints, we observe an increasing computational
time cost when the dimension of the target bottleneck embedding $\Psi$
increases. This is mainly due to the computation of the scores themselves, as
this involves a full backpropagation for each of the latent dimensions of
$\Psi$. We observe that the variance score has a lesser computational
footprint than the noise score. One reason is that the former does not require
the computation of attributions for the noise-generated input. We expect that
this alone introduces a factor of 2 on the computing time, and thus it cannot
explain the full difference. We hypothesize that some inefficiency occurs when
we compute attributions for the noise input after having computed attributions
for the meaningful input. In fact, in order to keep track of gradients, we set
the option create_graph=True in the function torch.autograd.grad in PyTorch,
and this might lead to an overall heavy computational derivative graph.

Table 3: Execution time $[\frac{\text{seconds}}{\text{epoch}}]$ for the models
reported in Section 4.5 of the main text. All the models were trained on a
single NVIDIA Tesla V100-16GB GPU.
Model | dim$=2$ | dim$=3$ | dim$=5$ | dim$=10$
---|---|---|---|---
orig. | 12 | 12 | 12 | 12
VS | 18 | 18 | 19 | 22
NS | 134 | 225 | 483 | 1569
NS+VS | 135 | 222 | 500 | 1567

##  Appendix 0.C Further Embedding Visualization Examples

In the main text we restricted, for space reasons, to just a few examples
regarding the possible combinations of attribution and aggregation strategies.
In Figure 7 below we provide a more complete set of embedding visualization
for the two input pictures of Figure 2 of the main text. Specifically, for
each example, we list, for each of the attribution schemes (Vanilla Gradients,
Activations$\times$Gradients, Grad-CAM), all possible combinations (for our
channel and embedding aggregation maps $\mathcal{C}$ and $\mathcal{E}$,
respectively) of basic aggregation strategies we discussed in the main text
($\mathcal{A}_{\text{mean}}$, $\mathcal{A}_{\text{abs}}$ ,
$\mathcal{A}_{\text{var}}$). Note that when
$\mathcal{C}=\mathcal{A}_{\text{var}},\mathcal{A}_{\text{abs}}$, then
$\mathcal{E}=\mathcal{A}_{\text{mean}}$ acts equivalently to
$\mathcal{E}=\mathcal{A}_{\text{abs}}$.

Figure 7: Embedding visualization for two examples for different attribution
schemes and aggregation strategies.


# Quantum vacuum effects in non-relativistic quantum field theory

Matthew Edmonds<EMAIL_ADDRESS>ARC Centre of Excellence in Future Low-
Energy Electronics Technologies, School of Mathematics and Physics, University
of Queensland, St Lucia, QLD 4072, Australia  Research and Education Center
for Natural Sciences, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa
223-8521, Japan  Antonino Flachi<EMAIL_ADDRESS>Department of
Physics, Keio University, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan
Research and Education Center for Natural Sciences, Keio University, 4-1-1
Hiyoshi, Yokohama, Kanagawa 223-8521, Japan  Marco Pasini
<EMAIL_ADDRESS>Department of Mathematics, Computer Science, and
Physics, University of Udine, Via delle Scienze 206 I-33100 Udine, Italy
Research and Education Center for Natural Sciences, Keio University, 4-1-1
Hiyoshi, Yokohama, Kanagawa 223-8521, Japan

###### Abstract

Nonlinearities in the dispersion relations associated with different
interactions designs, boundary conditions and the existence of a physical cut-
off scale can alter the quantum vacuum energy of a nonrelativistic system
nontrivially. As a material realization of this, we consider a 1D-periodic
rotating, interacting non-relativistic setup. The quantum vacuum energy of
such a system is expected to comprise two contributions: a fluctuation-induced
quantum contribution and a repulsive centrifugal-like term. We analyze the
problem in detail within a complex Schödinger quantum field theory with a
quartic interaction potential and perform the calculations non-perturbatively
in the interaction strength by exploiting the nonlinear structure of the
associated nonlinear Schrödinger equation. Calculations are done in both zeta-
regularization, as well as by introducing a cut-off scale. We find a generic,
regularization-independent behavior, where the competition between the
interaction and rotation can be balanced at some critical ring-size, where the
quantum vacuum energy has a maxima and the force changes sign. The inclusion
of a cut-off smoothes out the vacuum energy at small distance but leaves
unaltered the long distance behavior. We discuss how this behavior can be
tested with ultracold-atoms.

##  I Introduction

In quantum field theory, the canonical quantization scheme does not fix the
order of non-commuting operators in the Hamiltonian, leaving a residual
divergent “zero-point energy” contribution to the energy density (in natural
units):

$\displaystyle\mathscr{E}={1\over 2}\sum_{n}\omega_{n},$ (1)

with $\omega_{n}$ representing the frequencies of the quantum fluctuations.
Wick’s normal ordering is then used to enforce a specific order of operators’
products, resulting in the subtraction of this infinite shift from the vacuum
expectation value (vev) of the Hamiltonian that will then vanish. This has the
consequence that the quantum vacuum, so defined, does not carry energy, linear
or angular momentum. Such a procedure is usually justified by saying that a
constant shift in the energy cannot be measured, although this view is not
entirely tenable as any finite energy is, in principle, measurable due to its
gravitational effect. In relativistic quantum field theory, a better
justification follows from the fact that the expectation value of the
Hamiltonian in the noninteracting vacuum (i.e., in absence of external fields
or interactions) must vanish for the Hamiltonian, a generator of the Poincaré
group, to satisfy the correct commutation rules. Then, the usual notion of a
noninteracting vacuum as a state devoid of energy follows, justifying the use
of normal ordering [1, 2].

Even without calling gravity into question [3], a variety of quantum vacuum
phenomena, most notably the Casimir effect [4], clearly demonstrates some
level of inadequacy of the above definition of an empty physical vacuum tout
court. In the original version of the Casimir effect, for example, this was
evident owing to the imposition of boundary conditions on the quantum
fluctuations of the electromagnetic field in the presence of perfectly
conducting, parallel plates, resulting in an attractive force between the
plates. More general (and realistic) situations are not different, as boundary
conditions result from quantum fields existing in interaction with other
fields, and modify the spectrum of the quantum fluctuations, thus changing the
zero-point energy.

These arguments converge into Casimir’s definition of the energy of the
quantum vacuum ${E}_{vac}$ as the difference between the zero-point energies
in the presence, $E\left[{\mathcal{\partial}B}\right]$, and in absence,
$E\left[{\emptyset}\right]$, of boundaries:

$\displaystyle{E}_{vac}=E\left[{\mathcal{\partial}B}\right]-E\left[{\emptyset}\right].$
(2)

Such a definition is compatible with the vanishing of the vev of the
Hamiltonian in the noninteracting vacuum (i.e., no boundary) and gives a
calculable recipe (within any regularization scheme) of the quantum vacuum
energy in response to changes in external conditions [2, 5, 6]. This view on
the complexity of the vacuum has been vindicated during the past quarter of a
century by many successful experiments starting with [7, 8] (see also Ref. [9]
for a recent additional list of examples of applications to nanophotonics,
nanomechanics, and chemistry).

A less explored question concerns the quantum vacuum energy in non-
relativistic systems (see, for some discussions, Refs.[10, 11, 12, 13, 15,
14]). The answer might seem simple, since in a non-relativistic context there
is no issue associated with antiparticles or the ordering of the operators,
suggesting that the zero-point energy can be safely ignored. However, this is
not the case in general. Even from the vantage point of the original Casimir
effect, the story remains subtle because the quantum vacuum energy emerges
from deformations of the electromagnetic quantum fluctuations, and no simple
non-relativistic limit can be taken: the photon is massless and propagates at
the speed of light.

However, in a non-relativistic set-up one can imagine emergent degrees of
freedom, constrained by boundaries, and how these could give rise to non-
trivial quantum vacuum phenomena and a number of works have explored such
question, particularly in the context of quantum liquids and Bose-Einstein
condensation where (1) contributes to the zero temperature thermodynamic
potential (on top of the classical ground state contribution); see, for
example, Refs. [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].

There are at least two reasons why in a non-relativistic setting the situation
is far from obvious. The first is that any time we are in the presence of
interactions and non-trivial boundary conditions, the frequencies $\omega_{n}$
in (1) develop a non-trivial dependence on the ground state of the system.
This can be seen using the background field method (see Ref. [11]), although
computing the frequencies within this framework becomes a hard task. Earlier
calculations relying on a perturbative expansion around small coupling exist
[27, 28, 29] and more recently Ref. [30] has developed a way to compute the
quantum vacuum energy for a relativstic $1+1$ dimensional scalar field theory
without relying on expansions in powers of the interaction strength (see also
Refs. [32, 33, 34, 31]). The second reason has to do with the regularization.
In the relativistic case, the quantum vacuum energy emerges from the summation
of the entire spectrum as in (1); this summation is divergent and must be
regularized. A subtlety with this is due to the existence of a physical cut-
off that may alter the spectral sum in (1). Within a lattice approach this
should be possible (see Ref. [12]), however it is not at all obvious how to do
this within an effective field theory approach. It is certainly an interesting
question to ask whether any remnant of the quantum vacuum energy remains in
the non-relativistic limit.

When the model under consideration is nonlinear, the difference in the
dispersion relation due to the presence of interactions, the presence of
external forcing (e.g., rotation), a physical cut-off scale and boundary
conditions are all factors that together conjure to induce intricate behaviors
in the quantum vacuum energy. Here we look at the above questions within the
paradigmatic nonlinear Schrödinger equation. Our approach to compute the
quantum vacuum energy exploits the integrability structure of the nonlinear
Schrödinger equation associated with our problem. The calculations are done
both using zeta-regularization including the contribution from the whole
spectrum, as well as a more physical regularization scheme where the spectral
sums are modulated by a frequency dependent window-function that suppresses
the contribution of the high-energy modes, leaving a dependence on a physical
cross-over scale. As we shall see the two methods lead to compatible results,
with the only expected consequence of the cut-off being that of regularizing
the vacuum energy at short distance. In conclusion, we will describe how our
predictions can, in principle, be measured experimentally with cold-atom
rings.

##  II Non-relativistic Schrödinger model

We shall consider a system of non-relativistic interacting bosons, described
by a complex Schrödinger quantum field
$\Phi={(\phi_{1}+i\phi_{2})}/{\sqrt{2}}$, with
$\phi_{1},\phi_{2}\in\mathbb{R}$, confined to a 1D ring of radius $R$,
rotating with constant angular velocity $\Omega$. We assume that the
periodicity of the ring is externally broken by the presence of a barrier that
we describe by imposing Dirichlet boundary conditions at one point on the
ring. The Lagrangian density is

$\begin{split}\mathcal{L}&=\frac{i}{2}\left(\Phi^{\dagger}\dot{\Phi}-\Phi\dot{\Phi}^{\dagger}\right)+\frac{i}{2}\Omega\left(\Phi^{\dagger}\Phi^{\prime}-\Phi\Phi^{\dagger\prime}\right)-\\\
&\frac{1}{2mR^{2}}\Phi^{\dagger\prime}\Phi^{\prime}-\frac{\lambda}{4}\left(\Phi^{\dagger}\Phi\right)^{2},\end{split}$
(3)

where $0\leq\varphi\leq 2\pi$, $x=R\varphi$, $\dot{}=d/dt$ and
${}^{\prime}=d/d\varphi$. We adopt units of $\hbar=1$. Expression (3)
represents the Lagrangian density of an observer co-rotating with the ring. In
this reference frame, boundary conditions for the co-rotating observer are
time-independent [35, 36, 37]. The following nonlinear Schrödinger equation
can be derived from (3):

$\begin{split}i\dot{\Phi}=-i\Omega\Phi^{\prime}-\frac{1}{2mR^{2}}\Phi^{\prime\prime}+\frac{\lambda}{2}\left|\Phi\right|^{2}\Phi.\end{split}$
(4)

The normal mode decomposition can be carried out by looking for stationary
solutions of the form

$\begin{split}\Phi\left(t,\varphi\right)=e^{-i\omega_{p}t}f_{p}\left(\varphi\right).\end{split}$
(5)

This allows us to write the original equation (4) as

$\begin{split}0=\frac{1}{2mR^{2}}f_{p}^{\prime\prime}+i\Omega
f_{p}^{\prime}-\left(\frac{\lambda}{2}\left|f_{p}\right|^{2}-\omega_{p}\right)f_{p}.\end{split}$
(6)

To solve (6) we decompose $f_{p}$ as

$\begin{split}f_{p}\left(\varphi\right)=\rho\left(\varphi\right)e^{i\alpha\left(\varphi\right)},\,\,\,\,\,\,\,\,\,\,\,\,\text{with}\,\,\rho\left(\varphi\right),\alpha\left(\varphi\right)\in\mathbb{R},\end{split}$
(7)

that leads to

$\displaystyle\hskip-28.45274pt0$ $\displaystyle=$
$\displaystyle\frac{\rho^{\prime\prime}}{2mR^{2}}+\left(\omega_{p}-\frac{\alpha^{\prime\,2}}{2mR^{2}}-\Omega\alpha^{\prime}\right)\rho-\frac{\lambda}{2}\rho^{3},~{}~{}~{}$
(8) $\displaystyle\hskip-28.45274pt0$ $\displaystyle=$
$\displaystyle\frac{\alpha^{\prime\prime}\rho}{2mR^{2}}+\frac{1}{mR^{2}}\alpha^{\prime}\rho^{\prime}+\Omega\rho^{\prime}.$
(9)

The above system of equations can be solved analytically, first obtaining
$\alpha^{\prime}$ in terms of $\rho$ from Eq. (9),

$\displaystyle\alpha^{\prime}=\beta=\frac{C}{\rho^{2}}-mR^{2}\Omega,$ (10)

($C$ is an integration constant) and then substituting $\alpha^{\prime}$ in
Eq. (8); this gives rise to a cubic nonlinear equation in $\rho$ that can be
solved in terms of Jacobi elliptic functions. Imposition of the boundary
conditions selects the solution as a Jacobi $\mathtt{sn}$ function and leads
to the following quantization conditions for the eigenfrequencies. The
procedure is straightforward but lengthy. For completeness we give all the
details in the Appendix and refer the reader to Refs. [38, 39, 40, 41, 42] for
further details on elliptic equations. The solution can be written as

$\begin{split}\Phi\left(t,\varphi\right)=A_{n}\,e^{-i\omega_{n}t}e^{-i\left(mR^{2}\Omega\varphi-\pi/4\right)}\mathtt{sn}\left(q_{n}\varphi,k_{n}\right),\end{split}$
(11)

with the normalization factor $A_{n}$ expressed in terms of elliptic integrals
of first and second kind, $K(z)$ and $E(z)$ respectively,

$\displaystyle A_{n}^{2}=\frac{k_{n}^{2}}{2\pi
R\left(1-{E\left(k_{n}\right)}/{K\left(k_{n}\right)}\right)}.$ (12)

The momentum $q_{n}$ and the elliptic modulus $k_{n}$ are quantized according
to the following relations

$\displaystyle q_{n}$ $\displaystyle=$
$\displaystyle\frac{n}{\pi}K\left(k_{n}\right),~{}~{}~{}n\in\mathbb{N},$ (13)
$\displaystyle\lambda mR\frac{\pi}{4n^{2}}$ $\displaystyle=$ $\displaystyle
K\left(k_{n}\right)\left(K\left(k_{n}\right)-E\left(k_{n}\right)\right),$ (14)

where (13) comes from the periodicity of the solution and (14) is derived from
the first integral of the equation of motion. Finally, the eigenfrequencies
are given by

$\displaystyle\omega_{n}=\left(1+k_{n}^{2}\right)q_{n}^{2}/(2mR^{2})-{m}R^{2}\Omega^{2}/2.$
(15)

Details on how to derive Eqs. (11), (12), (13), (14) and (15) are given in the
Appendix.

##  III Quantum vacuum energy and spectral asymptotics

Figure 1: (color online) Comparison between (15) and the eigenvalues computed
numerically. Panel (a) shows the absolute difference between the asymptotic
and exact nonlinear eigenvalues, while (b) shows individual datasets for fixed
$R$. Coloured data in (b) were obtained numerically, while the grey dashed
lines were calculated from Eq.(15). Here $\Omega Rml/\hbar=0.5$ and
$\lambda=ml/\hbar^{2}=0.5$ throughout.

In the following, we illustrate how to compute the quantity (1) for the
present case. A non-renormalized expression for the quantum vacuum energy can
be written as follows (see Ref. [10, 43]):

$\displaystyle\mathscr{E}_{r}(s)={\mu^{s}\over 2}\sum_{n}\omega_{n}^{1-s},$
(16)

where $s\in\mathbb{C}$ is a complex-valued regularization parameter and $\mu$
is a renormalization scale with dimension of energy. The index $r$ is a
reminder that (16) refers to the co-rotating frame. The eigenvalues
$\omega_{n}$ are given in terms of the nonlinear, coupled algebraic equations
(13), (14) and (15). The regularization of (16) is done by finding a
representation that converges in some region of the complex-$s$ plane,
followed by analytical continuation to the physical value $s\to 0$. Here, we
use the spectral asymptotics of the eigenvalues and express (16) as

$\displaystyle\mathscr{E}_{r}(s)=\Delta+\tilde{\mathscr{E}_{r}}(s),$ (17)

where

$\displaystyle\tilde{\mathscr{E}_{r}}(s)={\mu^{s}\over
2}\sum_{n}\left(\omega^{(a)}_{n}\right)^{1-s},$ (18)

and

$\displaystyle\Delta={1\over
2}\sum_{n}\left(\omega_{n}-\omega^{(a)}_{n}\right).$ (19)

The quantity $\omega^{(a)}_{n}$ represents the asymptotic expansion of the
eigenvalues $\omega_{n}$ as a function of the quantum number $n$. If the
asymptotic expansion includes all terms up to $O(1/n^{2})$ as we shall do
here, then $\Delta\sim O(1/n^{2})$, i.e. (19), and thus converges for $s\to 0$
(in formula (19) we have already set $s\to 0$). Such a procedure simply
confines the divergences to $\tilde{\mathscr{E}_{r}}(s)$ that will need
explicit regularization. The first step of the process is to obtain the
asymptotic behavior of the eigenvalues. This can be obtained numerically, but
it is not difficult to find its analytical form. Since the left hand side of
Eq.(14) converges to zero for $n\to\infty$, while the right hand side, as a
function of $k_{n}$, goes to zero only in the limit $k_{n}\to 0$, while
decreasing monotonically for increasing $k_{n}>0$, the right hand side of (14)
for small $k_{n}$ gives the relevant limit to capture the large $n$ asymptotic
behavior $k_{n}^{2}\approx{2\lambda mR/(\pi n^{2})}$. This result used in
conjunction with (13) and (15) allows us to readily extract the leading
asymptotic behavior of $\omega_{n}$:

$\displaystyle\omega^{(a)}_{n}=n^{2}/\eta^{2}+\rho^{2}+O\left(1/n^{2}\right),$
(20)

where $\eta^{2}={8mR^{2}}$ and $\rho^{2}={3\lambda/(8\pi
R)}-{mR^{2}\Omega^{2}/2}$. Fig.1 shows a comparison between the eigenvalues
computed numerically and their asymptotic counterpart. The large-$n$ scaling
of the eigenvalues is consistent with Weyl’s law that in the present case
predicts a leading large-$n$ behavior of the $\omega_{n}$ scaling as $n^{2}$
and independent of $\lambda$ [44]. Using (20),

$\displaystyle\tilde{\mathscr{E}_{r}}(s)={\left({\mu\eta^{2}}\right)^{s}\over
2\eta^{2}}\sum_{n}\left(n^{2}+\eta^{2}\rho^{2}\right)^{1-s},$ (21)

and the Chowla-Selberg representation (See Refs. [45, 46]),

$\displaystyle\begin{split}&\sum_{n=1}^{\infty}\left(n^{2}+\gamma^{2}\right)^{-z}=-{\gamma^{-2z}\over
2}+{\sqrt{\pi}\over 2}{\Gamma(z-1/2)\over\Gamma(z)}\gamma^{1-2z}\\\
&+{2\pi^{z}\over\Gamma(z)}\gamma^{-z+1/2}\sum_{p=1}^{\infty}p^{z-1/2}K_{z-1/2}\left(2\pi
p\gamma\right),\end{split}$ (22)

from which the limit $s\to 0$ can be taken to arrive at the following
regularized expression

Figure 2:  (color online) Quantum vacuum energy and force. (a) and (c) show
the quantum vacuum energy, Eq. (24) for the same $\lambda$ (colour groups) and
varying $\Omega$ (panel a) or $R$ (panel c) (see text labels). (b) and (d)
show the corresponding force, Eq. (25) for each data set. The light grey
shading indicates the parameter regions where the force changes sign from
attractive to repulsive. Throughout the paper, the quantity $l$ represents a
generic unit-scale.

$\displaystyle\tilde{\mathscr{E}_{r}}=\lim_{s\to
0}\tilde{\mathscr{E}_{r}}(s)=-{1\over 4}\left({3\lambda\over 8\pi
R}-{mR^{2}\Omega^{2}\over 2}\right).$ (23)

Thus, the total quantum vacuum energy in the co-rotating frame is given by
$E_{r}=\tilde{\mathscr{E}_{r}}+\Delta$. To get the energy in the laboratory
frame $E_{s}$ one can use $E_{s}-E_{r}=\bf\Omega L$, where $L=-{\partial
E_{r}/\partial\Omega}$ is the angular momentum [31, 47, 35, 36]:

$\displaystyle
E_{s}=\Delta-\Omega{\partial\Delta\over\partial\Omega}-{3\lambda\over 32\pi
R}-{mR^{2}\Omega^{2}\over 8}.$ (24)

The resulting force $F_{s}=-{\partial E_{s}/\partial R}$ is

$\displaystyle F_{s}=-{\partial\Delta\over\partial
R}+\Omega{\partial^{2}\Delta\over\partial\Omega\partial R}-{3\lambda\over
32\pi R^{2}}+{mR\Omega^{2}\over 4}.$ (25)

Ignoring for the time being the contributions from $\Delta$, the above
expression comprises a contribution proportional to $-\lambda/R$ that vanishes
for $\lambda\to 0$ and scales as the inverse of the ring size: this is an
attractive “Casimir-like” contribution. The other contribution
$E_{\mathcal{I}}=\tfrac{1}{2}\mathcal{I}_{\mathcal{R}}\Omega^{2}$ is
proportional to the moment of inertia of the ring $\mathcal{I}_{R}=mR^{2}$
with radius $R$. The vanishing behavior for $\lambda\to 0$ and $\Omega\to 0$
is consistent with the fact that the quantum vacuum energy should vanishes in
the absence of interactions and boundary condtions. The angular velocity
appears as the square of $\Omega$, and this is again consistent with the fact
that our model does not include parity breaking terms, thus the energy should
be symmetric wrt $\Omega\leftrightarrow-\Omega$. The force vanishes at the
critical radius

$\displaystyle R_{crit}\approx\sqrt[3]{3\lambda\over 8\pi m\Omega^{2}},$ (26)

with its sign changing from negative-attractive for $R<R_{crit}$ to positive-
repulsive for $R>R_{crit}$. Interestingly, also the way the force scales with
the ring size changes with the angular velocity: it scales linearly in the
regime of fast rotation, while it scales as the inverse square of the ring
size for slow rotation. The symbol “$\approx$” in (26) indicates that the
contribution of $\Delta$ has been ignored. Units of $\hbar$ are restored in
the numerics. Fig. 2 shows the quantum vacuum energy (panels a and c) as a
function of radius $R$ and rotation strength $\Omega$ respectively, while the
two lower panels (b and d) show the corresponding force associated with each
dataset from (a) and (c). The grey shaded region shows the parameter regime
where the force is repulsive.

Fig. 3 shows heatmaps of Eq. (26) in the $(R,\Omega)$ and $(R,\lambda)$
parameter spaces, (a) and (b) respectively. In panel (a) the interaction
strength is $\lambda ml/\hbar^{2}=10$ while the rotation strength is $\Omega
ml^{2}/\hbar=5$ in (b). The solid blue lines in both panels show the border
between the repulsive regime and the causality limit defined by $\Omega
Rml/\hbar=1$. The red dashed line indicates where the force changes sign,
obtained from Eq. (26). The red data point in each panel corresponds to the
point $(R_{\circ},\Omega_{\circ})$ in (a) and $(R_{\circ},\lambda_{\circ})$ in
(b) where Eq. (26) and the causality limit coincide, and

$\big{(}R_{\circ},\Omega_{\circ}\big{)}=\bigg{(}\frac{3}{8\pi}\frac{\lambda
ml^{2}}{\hbar^{2}},\frac{8\pi}{3}\frac{\hbar^{2}}{m^{2}l^{3}}\frac{1}{\lambda}\bigg{)}.$
(27)

The point defined by Eq. (27) in Fig. 3 shows the maximum rotation strength
where repulsive solutions are obtained, then the model of Eq. (4) is expected
to support a causal repulsive force in the region
$\Omega_{c}<\Omega<R^{-1}(ml/\hbar)$ and $R>R_{\circ}$. Likewise for panel (b)
the causal repulsive regime is defined between $0<\lambda<\lambda_{\rm c}$ and
$0<R<R_{\circ}$. An analysis, qualitatively similar to Fig. 3(b), can be done
for the $(R,\Omega)$ parameter space with constant $\lambda$.

Figure 3:  (color online) Quantum fluctuation induced-force heatmaps. (a)
shows Eq. (25) in the limit $\Delta=0$ in the $(\Omega,R)$ parameter space for
fixed $\lambda ml/\hbar^{2}=10$. The solid blue line indicates the border
between the repulsive and noncausal regions, while the dashed red line
indicates the point at which the force changes sign. (b) shows the magnitude
of the force in the $(\lambda,R)$ parameter space for fixed $\Omega
ml^{2}/\hbar=5$.

A subtle point has to do with how the above results will change in the
presence of a cut-off scale associated with a minimal length scale (e.g., the
inter-atomic separation scale). We address this question by modifying the
regularization procedure to include a frequency dependent window function.
This is implemented by defining

$\displaystyle\tilde{\mathscr{E}_{r}}={1\over
2}\sum_{n}\omega^{(a)}_{n}\sigma_{n}(\ell_{c}),$ (28)

and the residual $\Delta$ as

$\displaystyle\Delta={1\over
2}\sum_{n}\left(\omega_{n}-\omega^{(a)}_{n}\right)\sigma_{n}(\ell_{c}).$ (29)

Here, we choose the window function as follows:

$\displaystyle\sigma_{n}(\ell_{c})=\exp\left({-{\ell_{c}n^{2}/(8mR^{2})}}\right),$
(30)

with the argument of the exponential set by the leading large-$n$ asymptotics
of the spectrum. The cut-off scale $\ell_{c}$ determines how high-frequency
modes are suppressed. The limit $\ell_{c}\to 0$ of (28-29) returns the non-
regularized expression for $\tilde{\mathscr{E}_{r}}$ discussed earlier. While
the choice of $\sigma_{n}(\ell_{c})$ is arbitrary, (30) allows to write (28)
as

$\displaystyle\begin{split}\tilde{\mathscr{E}_{r}}&=&-{\rho^{2}\over
4}+{\rho^{2}\over 4}\theta_{3}\left({\ell_{c}\over\pi\eta^{2}}\right)-{1\over
4\pi\eta^{2}}\theta_{3}^{\prime}\left({\ell_{c}\over\pi\eta^{2}}\right).\end{split}$
(31)

where $\theta_{3}$ is the following Jacobi thetanull function [48],

$\displaystyle\begin{split}\theta_{3}(x)=\sum_{n=-\infty}^{\infty}e^{-\pi
xn^{2}}.\end{split}$ (32)

This choice of regularization has the advantage that the first term of (31)
corresponds to the fully resummed result and the effect of the cut-off is
encoded in the latter two terms of (31). Proving the consistency of the two
approaches, with and without the cut-off, requires care since in the limit
$\ell_{c}\to 0$ the theta function diverges and requires regularization. The
theta function can be regularized by requiring that the cut-off dependent
contribution in (31) vanishes in the limit of $\ell_{c}\to 0$, corresponding
to a subtraction of the divergent contribution. To compute the finite
$\ell_{c}\to 0$ limit of (31) we use the modular transformation

$\displaystyle\begin{split}\theta_{3}\left(x\right)=\sqrt{1/x}~{}\theta_{3}\left({1/x}\right)\end{split}$
(33)

along with the small $x$ expansion of the theta function [48], leading to

$\displaystyle\begin{split}\theta_{3}\left(x\right)\approx{1/\sqrt{x}}+O(\exp(-\pi/x)/\sqrt{x}).\end{split}$
(34)

Using this expression in (31) and removing the divergent part, consistently
with the regularization of the theta function, gives the expected fully
resummed result. The corrections due to the cut-off near $R\sim R_{crit}$ to
the fully resummed result can be estimated assuming that $\ell_{c}\ll
R_{crit}$ and can be computed including higher order corrections in the
expansion of the theta function:

$\displaystyle\theta_{3}\left(x\right)-{1/\sqrt{x}}\approx{2\sigma/\sqrt{x}}+{2\sigma^{4}/\sqrt{x}}+O\left({\sigma^{9}/\sqrt{x}}\right),$
(35)

where $\sigma=\exp\left(-\pi/x\right)$. Using (35) in (31) implies that
corrections to the fully resummed result are exponentially small, that is the
behavior of the vacuum energy is robust against the inclusion of a cut-off
smaller than the critical radius for large enough ring size. For small $R$, we
expect the cut-off to regulate the diverging $1/R$ behavior of leading term in
Eq. (31). Expanding the theta functions in (31) for $\ell_{c}/R^{2}$ large,
gives at leading order

$\displaystyle\begin{split}\tilde{\mathscr{E}_{r}}\approx{\rho^{2}-\eta^{-2}\over
2}e^{-{\ell_{c}/\eta^{2}}}\xrightarrow[R\to 0]{}0,\end{split}$ (36)

which can be contrasted with the $\rho^{2}\sim R^{-2}$ behavior of the vacuum
energy as obtained by full resummation.

##  IV Conclusions

The behavior of the quantum vacuum energy of an interacting non-relativistic
system is far from trivial. Here we have looked at an example of this using a
nonlinear Schrödinger quantum field theory and computed the quantum vacuum
energy and force without resorting to any perturbative expansion in the
coupling constant, simply relying on the exact integrability of the nonlinear
problem. The novel results are summarized in the “phase diagram” of Fig. 3
which shows how the fluctuation-induced force as a function of rotation and
interaction strength separates into a noncausal region plus an attractive-
repulsive region. This behavior arises from the stabilization between an
attractive Casimir-like component and a repulsive centrifugal one.

An interesting potential connection seems evident between our quantum field
theoretical set-up and the area of ultracold atoms. A possibly relevant
example is the set-up of Ref. [49], which consists a 23Na BEC confined in a
ring of size $R\sim 20\mu$m. Considering a quasi-1D approximation the
interaction strength $\lambda$ can be expressed in terms of the scattering
length $a_{s}$ and the transverse length scale $l$ as $\lambda=g/(\pi l^{2})$
(here $g=4\pi\hbar^{2}a_{s}/m$ defines the atomic interaction with
$a_{s}=50a_{0}$ for 23Na) [50]. Taking $l\sim 2\mu$m to ensure that $l\ll R$
and assuming a condensate of $N=2\times 10^{3}$ atoms (species other than
23Na, e.g. 87Rb, can have larger atom numbers and different scattering
lengths), allows us to arrive at a dimensionless interaction strength $\lambda
ml/\hbar^{2}\sim 4a_{s}N/l$, which, using the above parameter values, gives
$4a_{s}N/l\sim 10$, a value close to that used in Fig. (3-a). The force
$F_{s}$ can also be estimated in a similar manner, using Eq. (25) and the
above definitions we obtain a dimensionless force

$\displaystyle\begin{split}F_{s}ml^{3}/\hbar^{2}=-(3a_{s}Nl/8\pi
R^{2})+m^{2}l^{3}R\Omega^{2}/4\hbar^{2}.\end{split}$ (37)

Using a rotation speed of $\Omega{\sim}2\pi{\times}25$Hz from the experiment
of Ref. [51], we obtain $F_{s}ml^{3}/\hbar^{2}\sim 0.1$, modest but
potentially large enough to be observable in a future experiment. Using these
values a ring of size $R\sim 20\mu$m would fall in the causal repulsive region
of Fig. 3(a), and $\Omega ml^{2}/\hbar\sim 0.2$ favoring lower rotation
frequencies. In this work Dirichlet boundary conditions have been used, which
could be simulated using a weak-link as realized in the BEC ring experiments
of Refs. [52, 53].

The physical system described in this work has potential applications in
atomtronics [54], facilitating an additional opportunity to explore the
fundamental physics associated with the quantum vacuum. Extensions to systems
of fermions [55] or with multiply-connected geometries [56] offers additional
avenues to explore the effects described in this work in uncharted scenarios.

## 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). A.F. and M.E. acknowledge support from the University of
Queensland (SMP Accelerator Grant). We would like to thank G. Marmorini for
discussions.

*

## Derivation of the solutions and of the spectrum

### Solutions

In this appendix, we will illustrate how to solve the system (9)-(10) that we
re-write here for convenience:

$\displaystyle\frac{1}{2mR^{2}}\rho^{\prime\prime}+\left(\omega_{p}-\frac{\alpha^{\prime\,2}}{2mR^{2}}-\Omega\alpha^{\prime}\right)\rho-\frac{\lambda}{2}\rho^{3}=0$
(38)
$\displaystyle\frac{1}{2mR^{2}}\alpha^{\prime\prime}\rho+\frac{1}{mR^{2}}\alpha^{\prime}\rho^{\prime}+\Omega\rho^{\prime}=0.$
(39)

The second equation can be solved by separation of variables, replacing
$\alpha^{\prime}\left(\varphi\right)=\beta\left(\varphi\right)$,

$\begin{split}\beta^{\prime}\rho+2\beta\rho^{\prime}+2mR^{2}\Omega\rho^{\prime}=0.\end{split}$
(40)

Integrating yields

$\begin{split}\int\frac{d\beta}{2\beta+2mR^{2}\Omega}=-\int\frac{d\rho}{\rho},\end{split}$
(41)

leading to

$\begin{split}\alpha^{\prime}=\beta=\frac{C}{\rho^{2}}-mR^{2}\Omega,\end{split}$
(42)

where ${C}$ is an integration constant. Substituting (42) allows to express
(38) as

$\begin{split}\rho^{\prime\prime}+F_{1}\frac{1}{\rho^{3}}+F_{2}\rho+F_{3}\rho^{3}=0.\end{split}$
(43)

where we have defined

$\begin{split}\begin{dcases}F_{1}=-C^{2}\\\
F_{2}\equiv\epsilon_{p}=2mR^{2}\left(\omega_{p}+\frac{m}{2}R^{2}\Omega^{2}\right)\\\
F_{3}=-\lambda mR^{2}\end{dcases}\end{split}$ (44)

The first integral can be obtained by multiplying both sides of (43) by
$\rho^{\prime}$ and integrating

$\begin{split}\rho^{\prime\,2}-F_{1}\frac{1}{\rho^{2}}+F_{2}\rho^{2}+\frac{F_{3}}{2}\rho^{4}+H=0,\end{split}$
(45)

where $H$ is an integration constant. Finally, multiplying both sides for
$\rho^{2}$ and changing variables,

$\begin{split}\begin{dcases}s=\rho^{2}\\\
\frac{s^{\prime}}{2}=\rho\rho^{\prime},\end{dcases}\end{split}$ (46)

gives

$\begin{split}\left(s^{\prime}\right)^{2}=4F_{1}-4Hs-4F_{2}s^{2}-2F_{3}s^{3}.\end{split}$
(47)

The above equation corresponds to the differential equation of an undamped
quadratic anharmonic oscillator, whose canonic form can be obtained by
differentiating with respect to $\varphi$ and dividing by $2s^{\prime}$,

$\begin{split}s^{\prime\prime}=-2H-4F_{2}s-3F_{3}s^{2}.\end{split}$ (48)

Following Ref. [38] (see also Refs. [39, 40, 41, 42]), we can write Eq. (47)

$\displaystyle\left(s^{\prime}\right)^{2}$ $\displaystyle=$ $\displaystyle
d\left(s-\alpha_{1}\right)\left(s-\alpha_{2}\right)\left(s-\alpha_{3}\right)$
(49) $\displaystyle=$
$\displaystyle-d\alpha_{1}\alpha_{2}\alpha_{3}+d\left(\alpha_{1}\alpha_{2}+\alpha_{2}\alpha_{3}+\alpha_{3}\alpha_{1}\right)s-$
$\displaystyle-d\left(\alpha_{1}+\alpha_{2}+\alpha_{3}\right)s^{2}+ds^{3}$

with $\alpha_{1},\alpha_{2},\alpha_{3}$ being roots of RHS polynomial, and

$\begin{split}\begin{dcases}\alpha_{1}+\alpha_{2}+\alpha_{3}=-2\frac{F_{2}}{F_{3}}\\\
\alpha_{1}\alpha_{2}+\alpha_{2}\alpha_{3}+\alpha_{3}\alpha_{1}=2\frac{H}{F_{3}}\\\
\alpha_{1}\alpha_{2}\alpha_{3}=2\frac{F_{1}}{F_{3}}.\end{dcases}\end{split}$
(50)

The advantage of expressing Eq. (47) as Eq. (49) is that the latter is one of
the standard nonlinear ordinary differential equation, whose solutions can be
expressed in terms of Jacobi elliptic functions as:

$\begin{split}s\left(\varphi\right)=\alpha_{3}-\left(\alpha_{3}-\alpha_{2}\right)\mathtt{sn}^{2}\left(q\varphi,k\right)\end{split}$
(51)

with

$\begin{split}k=\sqrt{\frac{\alpha_{3}-\alpha_{2}}{\alpha_{3}-\alpha_{1}}},\end{split}$
(52)

being the elliptic modulus and

$\begin{split}q=\sqrt{\frac{F_{3}}{2}\left(\alpha_{3}-\alpha_{1}\right)}.\end{split}$
(53)

Throughout the paper, we adopt for the Jacobian elliptic functions the
notation of NIST Digital Library of Mathematical Functions [57], according to
which

$\displaystyle\mathtt{dn}^{2}\left(x,m\right)+m^{2}\,\mathtt{sn}^{2}\left(x,m\right)=1$
(54a) $\displaystyle
E\left(m\right)=\int_{0}^{\frac{\pi}{2}}\sqrt{1-m^{2}\sin^{2}\left(t\right)}dt$
(54b) $\displaystyle
K\left(m\right)=\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{1-m^{2}\sin^{2}\left(t\right)}}dt;$
(54c)

In the above expressions, $\mathtt{dn}$ and $\mathtt{sn}$ define,
respectively, the Jacobi delta amplitude and elliptic sine, and
$K\left(m\right)$ and $E\left(m\right)$ define, respectively, the complete
elliptic integrals of the first and second kind.

Using (46) and (51), the general solution can be expressed as follows

$\begin{split}\rho\left(\varphi\right)=\sqrt{\alpha_{3}-\left(\alpha_{3}-\alpha_{2}\right)\mathtt{sn}^{2}\left(q\varphi,k\right)}.\end{split}$
(55)

The solution depends on the physical parameters $m,R,\Omega,\lambda,\omega$
and the integration constants $C,H$ through the roots
$\alpha_{1},\alpha_{2},\alpha_{3}$ and through $k$, and $q$.

#### Boundary conditions, normalization and spectrum

Having to deal with the solution (55) in all its algebraic complexity (the
explicit dependence of the solutions on the physical parameters is intricate)
can be bypassed by directly imposing the boundary conditions. In this paper,
we shall focus on the case of Dirichlet boundary conditions,

$\begin{split}\Phi\left(t,0\right)=\Phi\left(t,2\pi R\right)=0,\end{split}$
(56)

which implies (using Eqs. (6) and (8) in the main text)

$\begin{split}\rho\left(0\right)=0.\end{split}$ (57)

Exploiting the fact that

$\begin{split}\mathtt{sn}\left(0,k\right)=0,\,\,\,\,\,\,\,\,\,\forall
k\in\mathbb{R}\end{split}$ (58)

we can write

$\begin{split}\rho\left(0\right)=\sqrt{\alpha_{3}-\left(\alpha_{3}-\alpha_{2}\right)\mathtt{sn}^{2}\left(0,k\right)}=0,\end{split}$
(59)

which implies

$\displaystyle\alpha_{3}=0.$ (60)

This simplifies the solution to

$\begin{split}\rho\left(\varphi\right)=\sqrt{\alpha_{2}}\,\mathtt{sn}\left(\sqrt{-\frac{F_{3}}{2}\alpha_{1}}\varphi,\sqrt{\frac{\alpha_{2}}{\alpha_{1}}}\right).\end{split}$
(61)

Notice that the constraint $\alpha_{3}=0$ also fixes the value of the
integration constant $C$: specifying $\alpha_{3}=0$ in equation (49) gives

$\begin{split}\left(s^{\prime}\right)^{2}&=d\left(\alpha_{1}\alpha_{2}\right)s-d\left(\alpha_{1}+\alpha_{2}\right)s^{2}+ds^{3}\end{split}$
(62)

where no constant terms appear. Comparing the above equation with Eq. (47)
implies that the constant term proportional to $F_{1}$ in Eq. (47) has to
vanish:

$\begin{split}F_{1}\equiv-C^{2}=0.\end{split}$ (63)

This simplifies considerably the system of equations (50), which become

$\begin{split}\begin{dcases}\alpha_{1}+\alpha_{2}=-2\frac{F_{2}}{F_{3}}\\\
\alpha_{1}\alpha_{2}=2\frac{H}{F_{3}}.\\\ \end{dcases}\end{split}$ (64)

Imposing the remaining boundary condition at $\varphi=2\pi$, i.e.,

$\begin{split}\rho\left(2\pi\right)=0\end{split}$ (65)

implies that

$\begin{split}\mathtt{sn}\left(\sqrt{-\frac{F_{3}}{2}\alpha_{1}}\varphi,\sqrt{\frac{\alpha_{2}}{\alpha_{1}}}\right)=0,\end{split}$
(66)

from which, using the the property of the Jacobi $\mathtt{sn}$ function

$\begin{split}\mathtt{sn}\left(2nK\left(m\right)+2ilK\left(1-m\right),m\right)=0\;\;\;\;\;\;\;\;\forall
n,l\in\mathbb{Z},\end{split}$ (67)

we arrive at

$\begin{split}-\frac{F_{3}}{2}\alpha_{1}=\frac{n}{\pi}K\left(\sqrt{\frac{\alpha_{2}}{\alpha_{1}}}\right).\end{split}$
(68)

$K\left(x\right)$ represents the elliptic integral of the first kind as
defined in (54c).

Summarizing, we have

$\displaystyle\alpha_{1}+\alpha_{2}=-2\frac{F_{2}}{F_{3}}$ (69a)
$\displaystyle\alpha_{1}\alpha_{2}=2\frac{H}{F_{3}}$ (69b)
$\displaystyle-\frac{F_{3}}{2}\alpha_{1}=\frac{n}{\pi}K\left(\sqrt{\frac{\alpha_{2}}{\alpha_{1}}}\right)$
(69c)

that, together with the normalization condition, give us 4 independent
equations for 4 variables $(\alpha_{1},\alpha_{2},H,\omega)$. In practice, the
above set of equations defines the quantization condition of the eigenvalues
$\omega_{p}$. It is possible to show that the system admits a unique solution
but it is easier to proceed in an alternative and faster way. Using the
condition $F_{1}=0$ directly in equation (43) leads to

$\begin{split}\rho^{\prime\prime}+F_{2}\rho+F_{3}\rho^{3}=0.\end{split}$ (70)

This step allows us to write the solution in a simpler form

$\begin{split}\rho\left(\varphi\right)=A\,\mathtt{sn}\left(q\varphi,k\right).\end{split}$
(71)

Using (71) in Eq. (70) gives

$\displaystyle
2k^{2}q^{2}\mathtt{sn}^{3}\left(q\varphi,k\right)-\left(1+k^{2}\right)q^{2}\mathtt{sn}\left(q\varphi,k\right)=$
$\displaystyle-
F_{2}\,\mathtt{sn}\left(q\varphi,k\right)-F_{3}A^{2}\,\mathtt{sn}\left(q\varphi,k\right).$
(72)

From which we obtain, by matching the coefficients of the like powers of
$\mathtt{sn}\left(q\varphi,k\right)$, the following relations

$\displaystyle F_{3}=-\frac{2k^{2}q^{2}}{A^{2}}$ (73a) $\displaystyle
F_{2}=\epsilon_{p}=\left(1+k^{2}\right)q^{2}.$ (73b)

It is interesting to notice that the quantization condition (69c) becomes

$\begin{split}q_{n}=\frac{n}{\pi}K\left(k\right),~{}~{}~{}n\in\mathbb{N}.\end{split}$
(74)

(The same quantization conditions would have arisen imposing the boundary
conditions directly on the simpler solutions, confirming the validity of the
procedure). Solving the remaining conditions (73a),(73b) for $q,k$ we obtain
the following relations

$\displaystyle q_{n}^{2}=\frac{A_{n}^{2}F_{3}+2F_{2}}{2}$ (75a) $\displaystyle
k_{n}^{2}=-\frac{A_{n}^{2}F_{3}}{A_{n}^{2}F_{3}+2F_{2}}$ (75b) $\displaystyle
q_{n}=\frac{n}{\pi}K\left(k_{n}\right),$ (75c)

where we have defined $q\rightarrow q_{n}$, $k\rightarrow k_{n}$ and
$A\rightarrow A_{n}$ to make the dependence on $n$ explicit.
The computation of the “normalization” coefficients $A_{n}$ is carried out
using the non-relativistic normalization condition,

$\begin{split}\braket{\phi}{\phi}=\int_{V}\,dV\phi^{\star}\left(x\right)\phi\left(x\right)=1,\end{split}$
(76)

which gives

$\begin{split}1&=A_{n}^{2}R\int_{0}^{2\pi}\mathtt{sn}^{2}\left(q_{n}\varphi,k_{n}\right)d\varphi\\\
&=2\pi\frac{A_{n}^{2}R}{k_{n}^{2}}-\frac{A_{n}^{2}R}{k_{n}^{2}}\int_{0}^{2\pi}\mathtt{dn}^{2}\left(q_{n}\varphi,k_{n}\right)d\varphi\end{split}$
(77)

where we have used the Jacobi identity (54a). Using the definition of the
Jacobi epsilon function

$\begin{split}&\mathcal{E}\left(x,k\right)=\int_{0}^{x}\mathtt{dn}^{2}\left(t,k\right)\,dt\end{split}$
(78)

and the following relation (which comes from a combination of quasi-addition
and quasi-periodic formulas [57]),

$\begin{split}&\mathcal{E}\left(nK\left(k\right),k\right)=nE\left(k\right)\end{split}$
(79)

it is possible to simplify the normalization condition as

$\begin{split}1&=\frac{A_{n}^{2}R}{k_{n}^{2}}\left(2\pi-\frac{\pi}{nK\left(k_{n}\right)}\mathcal{E}\left(2nK\left(k_{n}\right),k_{n}\right)\right)\\\
&=\frac{2\pi
A_{n}^{2}R}{k_{n}^{2}}\left(1-\frac{E\left(k_{n}\right)}{K\left(k_{n}\right)}\right)\end{split}$
(80)

or equivalently

$\begin{split}A_{n}^{2}=\frac{k_{n}^{2}}{2\pi
R\left(1-\frac{E\left(k_{n}\right)}{K\left(k_{n}\right)}\right)}.\end{split}$
(81)

Using relations (75a), (75b) and (75c), it takes simple steps to arrive at the
following relation

$\begin{split}-\frac{F_{3}}{R}\frac{\pi}{4n^{2}}=K\left(k_{n}\right)\left(K\left(k_{n}\right)-E\left(k_{n}\right)\right)\end{split}$
(82)

that, along with (73b), closes the quantization condition for $\omega_{n}$,
$k_{n}$ and $q_{n}$.

#### Complete solution

In order to find the complete solution,

$\begin{split}f_{p}\left(\varphi\right)=\rho\left(\varphi\right)e^{i\alpha\left(\varphi\right)},\end{split}$
(83)

we shall need to find the phase $\alpha\left(\varphi\right)$. Using (42) and
(63) it is easy to arrive at the following expression

$\begin{split}\alpha\left(\varphi\right)=-mR^{2}\Omega\varphi+\Xi\end{split}$
(84)

where $\Xi$ is an integration constant. Using (84), (71), along with Eqs. (6),
and (8) from the main text, we arrive at

$\begin{split}\Phi\left(t,\varphi\right)=A_{n}\,e^{-i\omega_{n}t}e^{-i\left(mR^{2}\Omega\varphi-\Xi\right)}\mathtt{sn}\left(q_{n}\varphi,k_{n}\right).\end{split}$
(85)

The quantity $\Xi=\pi/4$ is a phase and the factor $\exp(i\Xi)$ corresponds to
a rotation of the phase $\alpha$, leaving the EOM unaltered.

#### Noninteracting limit

The noninteracting limit $\lambda\to 0$ simplifies Eq. (14) from the main text
to

$\begin{split}K\left(k_{n}\right)\left(K\left(k_{n}\right)-E\left(k_{n}\right)\right)=0.\end{split}$
(86)

Using the fact that $K\left(m\right)>0$ for any $m\in\left[0,1\right)$, the
above condition further simplifies to $K\left(m\right)=E\left(m\right)$, hence
$m=0$, i.e. $k_{n}\to 0$ for $\lambda\to 0$. Setting $k_{n}=0$ in the equation
for the eigenvalues and using the properties
$\mathtt{sn}\left(x,0\right)=\sin\left(x\right)$ and $K\left(0\right)=\pi/2$
yields

$\displaystyle\omega_{n}=\frac{n^{2}}{4}\frac{1}{2mR^{2}}-\frac{m}{2}R^{2}\Omega^{2}.$
(87)

Taking similar steps in the eigenfunctions, leads to

$\displaystyle\Phi\left(t,\varphi\right)=A_{n}\,e^{-i\omega_{n}t}e^{-i\left(mR^{2}\Omega\varphi-\Xi\right)}\sin\left(\frac{n}{2}\varphi\right),$
(88)

which reduce to ordinary plane waves for $\Omega\to 0$. For comparison, see
Refs. [35, 36, 15].

#### Solutions in the laboratory frame

The solutions in the stationary-laboratory frame can be obtained by performing
the inverse coordinate transformation: $t\to t$ and $\varphi\to\varphi+\Omega
t$, leading to

$\displaystyle\Phi\left(t,\varphi\right)$ $\displaystyle=$ $\displaystyle
A_{n}\,e^{-i\omega_{n}t}e^{-i\left(mR^{2}\Omega\left[\varphi+\Omega
t\right]_{2\pi}-\Xi\right)}\mathtt{sn}\left(q_{n}\left[\varphi+\Omega
t\right]_{2\pi},k_{n}\right),$

with $\left[u\right]_{2\pi}\equiv u\left[\mbox{mod($2\pi$)}\right]$, which
encodes the $2\pi$ periodicity of the solutions.

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