# What Makes Quantization for Large Language Models Hard?
An Empirical Study from the Lens of Perturbation

Zhuocheng Gong1, Jiahao Liu2, Jingang Wang2, Xunliang Cai2, Dongyan
Zhao1,4222Corresponding authors: Dongyan Zhao and Rui Yan., Rui
Yan3222Corresponding authors: Dongyan Zhao and Rui Yan.

###### Abstract

Quantization has emerged as a promising technique for improving the memory and
computational efficiency of large language models (LLMs). Though the trade-off
between performance and efficiency is well-known, there is still much to be
learned about the relationship between quantization and LLM performance. To
shed light on this relationship, we propose a new perspective on quantization,
viewing it as perturbations added to the weights and activations of LLMs. We
call this approach “the lens of perturbation”. Using this lens, we conduct
experiments with various artificial perturbations to explore their impact on
LLM performance. Our findings reveal several connections between the
properties of perturbations and LLM performance, providing insights into the
failure cases of uniform quantization and suggesting potential solutions to
improve the robustness of LLM quantization. To demonstrate the significance of
our findings, we implement a simple non-uniform quantization approach based on
our insights. Our experiments show that this approach achieves minimal
performance degradation on both 4-bit weight quantization and 8-bit
quantization for weights and activations. These results validate the
correctness of our approach and highlight its potential to improve the
efficiency of LLMs without sacrificing performance.

## Introduction

Recently, large language models (LLMs) have gained a lot of attention in the
deep learning community due to their remarkable performance (Touvron et al.
2023; OpenAI 2022). However, their size and computational demands pose
challenges for deployment on resource-constrained devices or in real-time
applications. To address this issue and improve the accessibility and
efficiency of these models, researchers have explored quantization techniques
that enable high-precision data to be represented in lower-precision formats.
These techniques can significantly reduce the memory footprint and
computational requirements of large models, making them more practical for
deployment and use.

Figure 1: Performance of quantized LLMs on Lambada (Paperno et al. 2016) on
different model families (BLOOM (Scao et al. 2022), OPT (Zhang et al. 2022),
and LLAMA (Touvron et al. 2023)), the parameter number scaling from 350M to
66B. We implement two uniform quantization settings: one is to quantize both
weights and activations to 8 bits (W8A8), and the other is to perform 4-bit
channel-wise for weights only (W4A16).

Quantizing large language models presents unique challenges that require
careful consideration. First, LLMs can be incredibly large, with billions of
parameters and multiple layers, resulting in significant quantization error
accumulation. Second, due to the large model scale, training can be extremely
costly, making post-training quantization (PTQ) the most appealing approach
for LLMs, as it requires much less or even zero calibration data to perform
(Bondarenko, Nagel, and Blankevoort 2021; Wu, Yao, and He 2023; Xiao et al.
2023). In summary, effectively quantifying LLMs requires a tailored approach
that balances model complexity, computational resources, and performance.
Despite recent studies investigating the quantization of LLMs, a comprehensive
investigation into the performance of quantization has not been satisfactorily
undertaken. Specifically, exploring the impact of different quantization
strategies on different model families during the quantization process would
be beneficial to the development of LLM quantization.

In this paper, our goal is to investigate the impact of quantization on LLMs
in a systematic manner, with a focus on understanding the challenges of
quantizing large language models. Specifically, we examine the effects of
zero-shot post-training uniform quantization on LLMs, as it is a practical and
representative quantization setting that requires no additional training or
data. To begin, we apply standard uniform quantization to various LLMs and
analyze the resulting performance degradation (as shown in Figure 1). Our
findings indicate that the impact of quantization varies significantly across
different model families and sizes. The BLOOM and LLAMA model families exhibit
greater robustness to quantization, while OPT models are more sensitive to
quantization. Furthermore, we observe that the scale of the model is a crucial
factor that affects the quality of quantization. Finally, we find that
activation quantization has a significant effect on performance.

To account for the various factors that impact quantization, it is important
to establish an analytical framework. This is where ”the lens of perturbation”
comes to play. We view quantization as the addition of small perturbations to
the weights and/or activations of the model. By perturbing these components
and measuring the resulting performance degradation, we can analyze the impact
of quantization on LLMs. This approach enables us to gain insights into how
quantization influences the model’s performance in different ways. To be more
specific, we subject LLMs to different kinds of artificial perturbations and
investigate which perturbation causes the most performance degradation. By
experimenting with artificial perturbations, we can gain insights into the
robustness of the LLM quantization method and identify potential areas for
improvement.

By using the lens of perturbation, we are able to delve deeper into the
mechanism of quantization and identify several factors that impact the
performance of LLM quantization. Our first finding is that the ability of data
to withstand perturbations is closely tied to its magnitude. Specifically,
larger values can endure more significant perturbations with minimal
performance degradation, while smaller values can only tolerate minor
perturbations. Our second observation pertains to the outlier phenomenon,
which involves abnormally large values in weights and activations (Bondarenko,
Nagel, and Blankevoort 2021). These insights obtained through the lens of
perturbation explain why uniform quantization may not always be optimal.
Uniform quantization treats all weights and activations equally, without
considering their sensitivity to perturbations, and is obsessed with outlier
values. Instead, we propose a potential improvement by taking into account the
sensitivity to perturbations. Through experiments, we demonstrate that this
modification significantly reduces the quantization error, where uniform
quantization falls short. These findings provide a more nuanced understanding
of the impact of quantization on LLMs and offer a promising direction for
improving the quantization process.

In summary, our contributions are as follows:

  * •

We propose a new perspective of investigating uniform quantization for LLMs,
namely the lens of perturbation. This lens provides new insights into the
challenges of quantization for LLMs.

  * •

We conduct a comprehensive comparison of the quantization performance of LLMs
across various model scales and three model families (LLAMA, BLOOM, and OPT),
using three quantization settings (W4A16, W8A8, and W4A8)

  * •

Drawing on the insights through the lens of perturbation, we conduct
preliminary experiments to leverage the properties of quantization-friendly
perturbations and explore the potential of non-uniform quantization for LLMs,
which significantly reduces quantization error. These experiments demonstrate
that non-uniform quantization can significantly reduce quantization error.

Figure 2: A toy example of 4-bit uniform quantization with different scale
factors $\alpha$. The top line shows a choice of $\alpha$ that is too large,
leading to too much information lost in the quantization process. The middle
represents the most common choice, where $\alpha$ is set to the maximum value
of the tensor. The bottom line uses a smaller $\alpha$, which results in less
perturbation at the cost of clipping out-of-range values.

## Related Works

### Large Language Model Quantization

Quantization, which represents the weights and activations of neural networks
with low-bit precision, has been widely studied in computer vision and natural
language processing (NLP) communities (Gholami et al. 2021). Recently, some
researchers explore quantizing large language models to reduce the deployment
cost (Shen et al. 2020; Wu et al. 2022; Kim et al. 2021; Bondarenko, Nagel,
and Blankevoort 2021; Wu, Yao, and He 2023; Liu et al. 2023; Dettmers et al.
2023; Xiao et al. 2023). One major factor distinguishing LLM quantization is
that LLMs with their enormous parameters are too expensive to train. Thus,
most current approaches choose few-shot or zero-shot post-training
quantization, which directly quantizes the already-trained model without much
calibration data. These methods mostly build on uniform quantization, the most
widespread, practical, and easy-to-use quantization strategy. Up to now, it
has been validated that quantizing the weight of LLMs to 8 or even 4 bits is
not too hard (Dettmers and Zettlemoyer 2023), however, activation quantization
is still a major obstacle. Multiple studies have revealed the outlier
phenomenon in weights and activations of transformer-based models, a tiny
fixed set of embedding dimensions consistently exhibits large values across
multiple layers (Bondarenko, Nagel, and Blankevoort 2021; Puccetti et al.
2022). Statistical analysis of LLMs shows that activations suffer from
outliers several orders of magnitude larger than weights (Xiao et al. 2023).
As a result, under the setting of uniform quantization, activations encounter
much larger quantization loss. Some methods have been proposed to alleviate
this problem, for example, by separating the quantization of outlier
dimensions and regular dimensions (Gong et al. 2023), or by partially
migrating outlier effect from activations to weights (Xiao et al. 2023) .
Recently, some researchers have analyzed why outliers appear and proposed a
method to remove outliers and produce quantization-friendly models
(Bondarenko, Nagel, and Blankevoort 2023). While most existing works target
proposing new quantization methods for LLMs, some focus on empirical analysis
of LLM quantization, among which Yao et al. (2023) thoroughly compares the
effect of different quantization schemes, model families, and quantization
coverage, e.g.

## Preliminaries

### Uniform Quantization

Quantization is a technique that involves mapping high-precision values into
low-precision discrete values. This process reduces the memory and
computational requirements of the model, making it more efficient to deploy on
devices with limited resources. Given a float-point tensor X (either a weight
tensor or activation tensor), the quantization and de-quantization process can
be viewed as:

$\textbf{X}^{\mathbb{Z}}=Q(\textbf{X}),\;\;\;\widetilde{\textbf{X}}=Q^{-1}(\textbf{X}^{\mathbb{Z}})$
(1)

where $\textbf{X}^{\mathbb{Z}}$ is the quantized tensor, and
$\widetilde{\textbf{X}}$ is the recovered tensor. In this paper, we target the
setting of uniform quantization (Hubara et al. 2017), which is the most common
quantization because of its efficient implementation and satisfactory
performance. The quantization function $Q(\cdot)$ of uniform quantization is
defined as a rounding-to-nearest operation over the scaled input:

$Q_{\text{uni}}(\textbf{X})=\text{clip}\left(\left\lfloor\frac{\textbf{X}}{\alpha}\cdot
2^{b}\right\rceil+z;0,2^{b}-1\right)$ (2)

where $b\in\mathbb{N}$ is the bit-width, $\alpha\in\mathbb{R}$ is the scale
factor, and $z\in\mathbb{N}$ is zero-point. $\lfloor\cdot\rceil$ denotes the
round-to-nearest-integer operator. The corresponding de-quantization function
is as follows:

$Q_{\text{uni}}^{-1}({\textbf{X}})=(\textbf{X}^{\mathbb{Z}}-z)\cdot\alpha/2^{b}$
(3)

In addition to the quantization strategy, the quality of quantization is also
influenced by the bit-width and quantization granularity. The former is
straightforward to comprehend: the number of quantization bins increases
exponentially with the bit-width. For example, 8-bit quantization has 256
unique bins for each X. The quantization granularity determines the size of X.
Tensor-wise quantization involves quantizing the entire tensor. Channel-wise
or group-wise quantization, on the other hand, divides the tensor along a
specific dimension and then quantizes each channel or group independently.
Fine-grained granularity typically leads to better performance.

### The Lens of Perturbation

While the quantized low-precision values are expected to be close to their
high-precision counterparts, they usually may not be exactly equal due to the
loss of precision. The extent of deviation from the high-precision values is
influenced by the bit-width and the quantization scheme used. One way to view
quantization is as perturbations added to the original values, which can be
expressed as:

$\textbf{X}=\widetilde{\textbf{X}}+\Delta,$ (4)

where $\Delta$ is the difference matrix between the quantized and original
tensors, representing the noise introduced by the quantization process. We
call this perspective as the lens of perturbation.

Figure 3: An illustration of the relationship between the scale factor and the
intensity of perturbation. To calculate the perturbation intensity, we use the
L2 norm of $\Delta$, which represents the distance between the high-precision
tensor and the quantized one.

### The Choice of the Scale Factor

In uniform quantization, the scale factor $\alpha$ plays a critical role as it
determines the scaling of original values. Usually, the scale factor is
computed by optimizing the performance of a few calibration samples. However,
in the zero-shot setting, it is often set to the absolute maximum value,
represented $\alpha_{\text{max}}=\max(|\textbf{X}|)$.

Figure 2 illustrates how the choice of scale factor impacts uniform
quantization. If the scale factor is greater than $\max(|\textbf{X}|)$, values
are overscaled, resulting in an insufficient utilization of the range of
integers. Conversely, if the scale factor is smaller than
$\max(|\textbf{X}|)$, quantized values exceed the range so out-of-range values
would be clipped. In the example depicted in Figure 2, the maximum value of
the input tensor Smaller scale factors result in smaller quantization
intervals and thus slighter perturbations for in-range values. Figure 3
provides a rough relationship between the intensity of perturbation and the
scale factor. Throughout this paper, we use $\alpha_{\text{max}}$ as the scale
factor for uniform quantization, unless otherwise specified.

##### Uniform Quantization under the Lens of Perturbation

Before conducting experiments, we first examine uniform quantization through
the lens of perturbation. For simplicity, we initially disregard the clipping
operation for out-of-range values and later discuss how it affects
quantization. In uniform quantization, high-precision values are mapped to a
series of equal interval bins. Through the lens of perturbation, we make the
following observations: (1) the intensity of perturbation is positively
correlated with the scale factor (as depicted in Figure 3, the larger the
scale factor, the greater the intensity of perturbation); (2) due to the
uniform intervals, the magnitude of perturbation is not dependent on the
magnitude of the values being quantized.

## Looking into LLM Quantization through the Lens of Perturbation

In this section, we investigate the relationships between various types of
perturbations and their impact on performance degradation. Analyzing the
statistical properties of perturbation can provide a more profound
comprehension of the effects of quantization. Our goal is to determine which
type of perturbation is more quantization-friendly (leading to less
performance degradation).

Figure 4: Lambada (Paperno et al. 2016) performance comparison on different
types of perturbations. Here No perturbation represents the vanilla LLM
(without quantization and without perturbation). Uniform quantization
represents for W8A8 uniform quantization. For a fair comparison, all
artificial perturbations are set to have the same variance with the native
perturbation caused by Uniform quantization. The perturbation methods are run
four times with different random seeds.

### What Kind of Perturbation Is More Quantization-Friendly?

To explore the correlation between perturbations and their impact on
performance, we propose to use manually constructed perturbations instead of
relying on the natural perturbations caused by quantization, which can be
intricate and inconvenient to manipulate. Our approach involves controlling
the properties of artificial perturbations and examining how they influence
the model’s behavior.

Our approach constructs perturbations from two perspectives: the distribution
of the perturbation (i.e., the type of distribution from which the
perturbation is sampled) and the magnitude of the perturbation (i.e., whether
the perturbation’s size is proportional to the input’s size). By manipulating
these properties, we can enhance our comprehension of how perturbations impact
the model’s performance. We consider three types of distributions for our
perturbations:

  * •

Gaussian Perturbation $\Delta_{\text{G}}\sim\mathcal{N}(\mu,\sigma^{2})$:
where the perturbations are sampled from a Gaussian distribution.

  * •

Uniform Perturbation $\Delta_{\text{U}}\sim\mathcal{U}(-c,c)$: where the
perturbations follow the uniform distribution between $-c$ and $c$.

  * •

Rademacher Perturbation $\Delta_{\text{R}}\sim\mathcal{R}$: where the
perturbations have constant sizes but random signs.

Apart from the distribution, we also take into account whether the magnitude
of perturbations should be linked to the magnitude of the input (i.e., whether
larger values should have greater (or lesser) perturbations). In this sense,
the three methods mentioned above are classified as “magnitude-independent
perturbation,” signifying that the perturbation’s magnitude is not contingent
on the input’s magnitude To investigate the impact of perturbation magnitude,
we introduce two “magnitude-aware perturbation” settings, in which the
perturbation’s size is dynamically determined by the value’s size.

  * •

Positively-correlated Perturbation $\Delta_{\text{M+}}$: the positive
correlation setting scales the perturbation magnitude positively with the
input, meaning that higher weights have larger perturbations, formally,
$\delta\propto|\textbf{x}|$.

  * •

Negatively-correlated Perturbation $\Delta_{\text{M-}}$: conversely, the
negative correlation setting scales the perturbation magnitude inversely with
the input, meaning that higher weights lead to smaller perturbations,
formally, we choose $\delta\propto(|\textbf{x}|+\epsilon)^{-1}$.

We conducted experiments using different types of perturbations on the
BLOOM-560M, BLOOM-3B, LLAMA-7B, and LLAMA-13B models. Due to the high
computational cost, we did not experiment with more LLMs. Nevertheless, we are
confident that our current experiments provide sufficient observations to draw
reliable conclusions. To simulate the W8A8 quantization setting, we add
perturbations to both weights and activations in this experiment. To ensure a
fair comparison between artificial perturbations and quantization
perturbations, we control the perturbation intensity (measured by the L2 norm
of $\Delta$) to be the same for each matrix being quantized.

#### Observations

The results of perturbations are shown in Figure 4. As expected, perturbations
lead to non-negligible decreases in performance, regardless of the properties
of the perturbations or the backbone models. The extent of performance
degradation varies across different model families. In the case of the BLOOM
model family, artificial perturbations exhibit similar performance to standard
uniform quantization, whereas LLAMA models experience a significant loss in
performance. This contrast highlights the distinctive characteristics of
different LLM model families, which warrants further investigation. Regarding
our primary objective of establishing a link between the properties of
perturbations and performance degradation, we have made an interesting
discovery. We have found that the sampling distribution has little impact on
performance, as all three distributions we tested exhibit comparable
performance. On the other hand, the magnitude of perturbations plays a crucial
role. Specifically, the Positively-correlated perturbation
$\Delta_{\text{M+}}$ (where larger values have larger perturbations) exhibits
a dominant advantage over all perturbations on all models, while the
negatively-correlated perturbation $\Delta_{\text{M+}}$ (where smaller values
have larger perturbations) almost completely destroys performance. This
phenomenon has led us to a significant finding: larger values in LLM
activations and weights are more robust to perturbations, whereas smaller
values are more sensitive to perturbations.

### Clipping as a Special Perturbation

Figure 5: Effects of clipping on Lambada dataset (left) and Wikitext-2 (Merity
et al. 2016) and C4 (Raffel et al. 2019) (averaged ppl.) (right). We compare
three options of $k$, which determines the clipping threshold:
$k\in\\{3,5,10\\}$. It is worth noting that even with the most stringent
clipping setting, where $k=3$, on average, only less than 0.1% of values are
affected by clipping.

The experiment conducted on artificial perturbations mentioned above has not
considered the effect of the clipping operation, which can have a considerable
impact on realistic quantization scenarios, however. According to the formula
for uniform quantization (Eq. 2), if $\alpha$ is smaller than
$\max(|\textbf{X}|)$, out-of-range values will be clipped, and the deviation
caused by the clipping operation is much greater than that caused by regular
perturbations. Through the lens of perturbation, clipping can be viewed as an
exceedingly large perturbation applied to out-of-range values. Our current
finding shows that larger values can tolerate much greater perturbations.
Therefore, the question arises as to whether a perturbation as large as
clipping is acceptable in terms of model performance.

Quant. | Bits | Bloom | OPT | Llama
---|---|---|---|---
Methods | (W/A) | 560M | 1.1B | 3B | 7.1B | 350M | 1.3B | 6.7B | 13B | 30B | 66B | 7B | 13B | 30B | 65B
Wiketext-2 (ppl$\downarrow$) |  |  |  |  |  |  |  |  |  |  |  |  |  |
full | 16/16 | 19.36 | 24.33 | 11.83 | 10.32 | 15.89 | 11.35 | 8.84 | 8.32 | 7.93 | 7.14 | 19.7 | 13.3 | 10.3 | 10.18
Uniform | 4/16 | 24.19 | 29.97 | 13.41 | 11.45 | 15.95 | 11.34 | 8.86 | 8.30 | 7.93 | 8.65 | 25.05 | 14.98 | 11.78 | 11.42
Non uniform | 4/16 | 22.02 | 27.67 | 12.87 | 10.98 | 15.89 | 11.35 | 8.84 | 8.30 | 7.93 | 7.14 | 20.92 | 15.55 | 11.03 | 11.29
Uniform | 8/8 | 26.14 | 36.88 | 14.98 | 15.64 | 18.28 | 12.2 | 18.83 | 3300 | 1328 | 2254 | 100.81 | 52.41 | 57.91 | 26.88
Non uniform | 8/8 | 19.78 | 24.42 | 11.90 | 10.36 | 15.95 | 11.51 | 8.86 | 8.34 | 7.93 | 7.38 | 22.28 | 16.14 | 12.62 | 15.13
Uniform | 4/8 | 33.31 | 44.56 | 17.28 | 17.52 | 18.33 | 12.16 | 18.58 | 3352 | 1262 | 2379 | 453.05 | 75.81 | 711 | 1095
Non uniform | 4/8 | 22.67 | 27.72 | 12.91 | 11.03 | 15.92 | 11.49 | 8.86 | 8.32 | 7.93 | 7.38 | 21.02 | 15.58 | 11.07 | 12.01
C4 (ppl$\downarrow$) |  |  |  |  |  |  |  |  |  |  |  |  |  |
full | 16/16 | 27.83 | 22.98 | 18.08 | 15.83 | 22.94 | 16.20 | 12.91 | 12.38 | 11.78 | 9.57 | 7.67 | 7.14 | 6.50 | 6.12
Uniform | 4/16 | 34.22 | 27.41 | 20.56 | 17.52 | 22.98 | 16.20 | 12.95 | 12.38 | 11.78 | 10.25 | 9.12 | 7.94 | 7.09 | 6.83
Non uniform | 4/16 | 31.84 | 25.39 | 19.55 | 16.75 | 22.98 | 16.20 | 12.91 | 12.38 | 11.78 | 10.21 | 8.23 | 7.56 | 6.79 | 6.34
Uniform | 8/8 | 34.84 | 35.81 | 21.59 | 20.64 | 26.88 | 17.94 | 98.88 | 6116 | 2790 | 4931 | 20.05 | 18.00 | 19.36 | 18.22
Non uniform | 8/8 | 28.33 | 23.08 | 18.14 | 15.89 | 22.98 | 16.42 | 12.95 | 12.40 | 11.78 | 10.21 | 7.68 | 7.16 | 6.52 | 6.13
Uniform | 4/8 | 43.44 | 45.19 | 25.09 | 22.8 | 26.92 | 17.86 | 100.81 | 6116 | 2812 | 5191 | 39.09 | 28.94 | 30.39 | 28.05
Non uniform | 4/8 | 32.53 | 25.48 | 19.62 | 16.81 | 22.98 | 16.42 | 12.95 | 12.38 | 11.78 | 10.34 | 8.26 | 7.58 | 6.81 | 6.38
Lambada (accuracy$\uparrow$) |  |  |  |  |  |  |  |  |  |  |  |  |  |
full | 16/16 | 64.59 | 69.40. | 73.98 | 77.39 | 67.57 | 75.35 | 81.25 | 80.71 | 81.99 | 83.01 | 88.58 | 89.42 | 90.31 | 90.82
Uniform | 4/16 | 58.08 | 65.00 | 72.85 | 76.69 | 63.50 | 59.01 | 75.19 | 77.28 | 69.89 | 41.01 | 85.77 | 87.06 | 88.09 | 88.77
Non uniform | 4/16 | 61.88 | 68.27 | 71.68 | 76.96 | 64.37 | 70.79 | 79.22 | 79.44 | 80.55 | 81.63 | 87.08 | 88.64 | 89.98 | 90.29
Uniform | 8/8 | 59.77 | 61.90 | 68.62 | 66.87 | 64.61 | 72.91 | 41.75 | 5.38 | 3.37 | 0.84 | 86.03 | 84.10 | 65.00 | 80.45
Non uniform | 8/8 | 64.70 | 68.99 | 73.77 | 77.39 | 67.65 | 74.76 | 80.94 | 80.47 | 81.84 | 82.49 | 88.52 | 89.46 | 90.37 | 90.32

Table 1: Comparison of uniform and non-uniform method on three datasets. In
addition to evaluating W4A16 and W8A8 quantization, we also evaluate the much
more challenging w4a8 quantization setting where weights are quantized to 4
bits and activations to 8 bits. For this experiment, we have not processed the
entire test set for Wikitext-2 and C4. Instead, we use the first 2000 samples
to calculate perplexity.

To discuss the impact of extreme perturbations caused by clipping, we design
and experiment with this special perturbation:

  * •

Clipping Perturbation $\Delta_{\text{C}}$: where large values exceeding the
pre-set threshold are clipped to the threshold. We set the clipping threshold
as $\mu_{\textbf{X}}+k\cdot\sigma_{\textbf{X}}$.

Figure 3 illustrates the effect of clipping perturbations on performance.
Overall, clipping has a considerably more adverse impact than regular
perturbations. This finding aligns with the outlier phenomenon, which posits
that extremely large values are crucial for LLM performance. Clipping these
values results in significant information loss, thus leading to severe
performance degradation. Another interesting observation is the comparison
between LLAMA and BLOOM, where models from the LLAMA family seem to be more
vulnerable to clipping.

Figure 6: Illustration of uniform quantization and its non-uniform adaptation.
The key difference is that non-uniform quantization utilizes two mutually
inverse transformations to shape the input in a more quantization-friendly
manner, one before the scaling operation and the other after the scaling-back
operation.

Quant. Methods | Bits | Wikitext-2 | C4
---|---|---|---
Full Precision | 16/16 | 19.70 | 7.67
Non-uniform (Ours) | 4/16 | 20.92 | 8.23
RTN (alias of Uniform Quant.) | 4/16 | 25.05 | 9.12
GPTQ (Frantar et al. 2022) | 4/16 | 21.09 | 8.48
AWQ (Lin et al. 2023) | 4/16 | 20.77 | 8.03

Table 2: Comparison of different methods on LLaMA-7b.

## Improving LLM Quantization

Having identified the preferred types of perturbations, we can now explore how
these properties can be leveraged to improve existing quantization methods.
Specifically, we have discovered that smaller values in a tensor should be
subject to smaller perturbations, while larger values can handle greater
perturbations, and that the magnitude of the perturbations should be
constrained and clipping should be avoided. These properties can be useful in
designing a quantization method that takes them into account.

To this end, we have implemented a quantization method to verify the
effectiveness of the properties we discovered. Our goal is to ensure that
smaller values in a tensor are subject to smaller perturbations, while larger
values bear more perturbations. We achieve this by applying a nonlinear
transformation before quantization, which amplifies smaller values around zero
and shrinks larger values. The approach is straightforward: perform a
nonlinear transformation before quantization, and inverse transform it back
after quantization, as illustrated in Figure 6. For simplicity, we select
$f(x)=x^{1/3}$ as the nonlinear transformation as it effectively amplifies
values near zero and shrinks larger values.

Method | Humanities | STEM | Social | Other | Avg.
---|---|---|---|---|---
|  |  | Sciences |  |
Full Prec. | 34.80 | 35.12 | 30.29 | 38.07 | 37.95
Uniform | 31.63 | 34.13 | 27.55 | 33.38 | 33.92
Non-uniform | 34.43 | 33.29 | 31.48 | 38.40 | 35.83

Table 3: LLaMA-7B 4-shot ICL results on MMLU.

We conducted experiments on three model families: BLOOM, OPT, and LLAMA. Table
1 summarizes results on Lambada, Wikitext-2, and C4. In the W4A16 setting,
where activations are not quantized, uniform quantization performs well on
most of the models and tasks, suggesting that weight quantization is not
overly challenging. However, when it comes to quantizing activations, uniform
quantization exhibits weaknesses, while the non-uniform method maintains near-
lossless performance. We observe that for OPT models, as the model scales up,
standard min-max uniform quantization struggles to perform accurately with
explosive perplexity scores, whereas the non-uniform method maintains the
model’s performance close to that of the unquantized model. Another
interesting finding is that for non-uniform quantization, as the model size
scales up, the performance gap between before and after quantization
decreases, indicating that larger models are easier to quantize. This
phenomenon contrasts with observations made in uniform quantization, where
model performance deteriorates as the model size grows.

We conducted more experiments on LLaMA-7b. Table 2 shows the comparison of
non-uniform quantization and some recent post-training quantization methods
and Table 3 is the performance on MMLU (Hendrycks et al. 2021), an advanced
benchmark for evaluating the abilities of LLMs.

##### Limitations and Practical Value

Quantization offers two sources of efficiency improvements: reduced memory
costs through the use of low-precision formats, which accelerates data
transfer in Memory/GPUs (memory efficiency), and faster matrix multiplication
of low-precision formats (computation efficiency). However, the latter
advantage is only realized when both the weight and activation are quantized
to low precision. In the commonly used W4A16 setting, matrix calculation is
still performed with high precision. Despite non-uniform quantization being a
dominant approach in signal processing, it has not received sufficient
attention in the realm of neural network compression. One major obstacle to
its application in model compression is efficiency concerns. Although our non-
uniform quantization method demonstrates reliable performance, it is important
to note that the introduction of nonlinear transformation entails additional
computations. Consequently, non-uniform quantization only offers memory
efficiency without computation efficiency. However, as current LLMs are not
computation-constrained but rather memory-constrained, the model wastes a
significant amount of time waiting for variables to be loaded on specific
hardware rather than performing calculations. This imbalance of memory
capacity and computation is particularly outstanding when generating long
text, as the loading/offloading of KV-Cache for each generated token
significantly slows down inference. This scenario is where most of the current
quantization approaches glow.

##### Further Insights

Through our experimentation with the non-uniform quantization method, which
amplifies smaller values and compresses larger ones, we have confirmed a
general principle regarding model quantization. Specifically, larger values
can tolerate more error, while smaller values are more sensitive to larger
errors. Based on this observation, we have identified the limitations of
uniform quantization, which employs uniformly distributed quantization
intervals to scale all values of varying sizes. We propose that is preferable
to establish dense bins for small values and sparser bins for large values. We
believe that this finding is highly thought-provoking and warrants further
exploration in future research on model quantization. There are two potential
avenues to pursue: one is to implement hardware-friendly non-uniform
quantization for efficient deployment, while the other entails shaping the
weight and activation distributions through training to make them easier to
quantize. Some prior research has already been conducted in this direction
(Bondarenko, Nagel, and Blankevoort 2023).

## Conclusion

Our work introduces a new perspective on quantization, which we refer to as
“the lens of perturbation”. Using this approach, we conduct a comprehensive
investigation of uniform quantization on LLMs, evaluating the performance of
various models under different quantization settings. We also probe the models
with manually constructed perturbations, which provide valuable insights into
how quantization impacts the model performance. These insights help us
understand the difficulties associated with quantization for LLMs. Based on
our findings, we propose a non-uniform quantization approach that
significantly reduces the performance degradation caused by quantization. We
hope that our study, conducted through the lens of perturbation, will
contribute to a better understanding of the challenges associated with
quantization for LLMs and inspire the development of more efficient and
effective quantization methods.

## Acknowledgments

This work is supported by National Key R&D Program of China (No.
2022YFC3301900) and National Natural Science Foundation of China (NSFC Grant
No. 62122089). We sincerely thank all reviewers for their valuable comments
and suggestions, which are crucial for improving our work.

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††thanks: These authors have contributed equally to this work.††thanks: These
authors have contributed equally to this work.

# Symmetry-enforced ideal lantern-like phonons in ternary nitride Li6WN4

Xiaotian Wang  School of Physical Science and Technology, Southwest
University, Chongqing 400715, China.  Feng Zhou  School of Physical Science
and Technology, Southwest University, Chongqing 400715, China.  Tie Yang
School of Physical Science and Technology, Southwest University, Chongqing
400715, China.  Minquan Kuang  School of Physical Science and Technology,
Southwest University, Chongqing 400715, China.  Zhi-Ming Yu
<EMAIL_ADDRESS>Key Lab of Advanced Optoelectronic Quantum
Architecture and Measurement (MOE), Beijing Key Lab of Nanophotonics $\&$
Ultrafine Optoelectronic Systems, and School of Physics, Beijing Institute of
Technology, Beijing 100081, China.  Gang Zhang<EMAIL_ADDRESS>Institute of High Performance Computing, Agency for Science, Technology and
Research (A*STAR), 138632, Singapore.

###### Abstract

Condensed matter systems contain both fermionic and bosonic quasiparticles.
Owing to the constraint imposed by the Fermi level, ideal material candidate
for the emergent particles with higher-dimensional degeneracy manifold (i.e.,
nodal lines and nodal surfaces) has not been found in electronic systems. This
paper demonstrates that according to the first-principle calculations and
symmetry analysis, realistic ternary nitride Li6WN4 features ideal (nearly
flat) nodal-surface and nodal-line structures in its phonon spectra. These
nodal degeneracies are shaped like lanterns, and their existence is guaranteed
by nonsymmorphic symmetry. The corresponding topological phonon surface state
covers exactly half the surface Brillouin zone (BZ) and can thereby be
distinguished from those of conventional nodal-line and nodal-surface
semimetals. The results of our study demonstrate the existence of ideal
lantern-like phonons in realistic materials, which enriches the classification
of topological quantum phases and provides a good basis for investigating the
interaction between nodal-line and nodal-surface phonons in a single material.

Condensed matter systems add1 ; add2  contain both fermionic and bosonic
quasiparticles. The existence of the former in electronic structures add3 ;
add4 ; add5 ; add6 ; add7 ; add8 ; add8a  has been widely predicted and
verified; by contrast, the existence of topological bosonic excitations add9 ;
add10 ; add11 ; add12 ; add13  has been rarely reported. According to the
dimension of the degeneracy manifold, topological semimetals can be classified
into nodal-point add14 ; add15 ; add16 ; add17 ; add18 ; add19 ; add20 ,
nodal-line add21 ; add22 ; add23 ; add24 ; add25 , and nodal-surface
semimetals add26 ; add27 ; add28 . Both nodal-line and nodal-surface
semimetals exhibit many intriguing phenomena add28 ; add30 ; add31 ; add32 .
For example, it has been predicted that the nodal lines have drumhead-like
surface states that cover a finite region in the surface Brillouin zone (BZ)
or torus surface states spanning over the entire BZ. Although the nodal
surface cannot have an intrinsic Chern number, it can have an induced one
add33 , which can be any integer value, depending on the chiral particles in
the BZ. To determine the properties of nodal lines or nodal surfaces, the line
or surface should preferably have a relatively flat energy dispersion. In
addition, in electronic systems, the line or surface should be close to the
Fermi level. Therefore, ideal nodal semimetals (in particular, ideal nodal-
surface semimetals) are still missing in electronic systems.

Phonons, which are the basic emergent kind of bosons in crystalline lattices,
can also display nontrivial degeneracy in their spectra. Because the Pauli
exclusion principle does not apply to phonons, it provides a feasible basis
for investigating bosonic excitations in a wide frequency range without the
rigorous constraint of the fermion level. Researchers are currently searching
for ideal topological phases in phonon systems add9 ; add13 ; add34 ; add35 ;
add36 ; add37 ; add38 ; add39 ; add40 ; add41 ; add42 ; add43 ; add44 ; add45
. In particular, the existence of Weyl, Dirac, triple point, nodal line
phonons in the phonon spectra of several materials has been predicted, and the
existence in some of them has been experimentally confirmed add38 ; add45 .
However, the existence of nodal-surface phonons has not been reported.

_A natural question is whether ideal nodal-surface phonons can exist in
realistic materials._ In this paper, we answer this question affirmatively.
_For the first time_ , based on first-principle calculations and symmetry
analysis, we predict the existence of nodal-surface phonons in ternary
nitridotungstate Li6WN4, which has been experimentally synthesized add46  via
the solid state reaction of lithium subnitride (Li3N) with W in a nitrogen
atmosphere. The Li6WN4 material has two nodal surfaces in the $k_{x}$= $\pi$
and $k_{y}$= $\pi$ planes, respectively, the existence of which is guaranteed
by nonsymmorphic screw-rotational symmetry and time-reversal symmetry.
Remarkably, the two nodal surfaces are ideal in that (i) the two bands forming
the nodal surfaces are well separated from other bands and in that (ii) the
nodal surface is approximately flat in energy, with energy variations of less
than 0.25 THz (approximately 1.00 meV). These characteristics have not been
observed for other proposed nodal surface materials. Moreover, the two bands
that form the nodal surfaces become degenerate in the Z-A path of the $k_{z}$=
$\pi$ planes, thereby leading to two symmetry-enforced nodal lines.
Eventually, the two nodal surfaces and two nodal lines in Li6WN4 result in
exotic nodal-lantern phonon excitations, as illustrated in Fig. 2(c) and Fig.
S1. Hence, this work presents novel topological phases of phonon systems,
predicts ideal material candidate, and paves the way for investigating nodal-
lantern phonons in experiments.

The density-functional theory add47  was used to calculate the ground state of
this material, and the GGA-PBE formalism add48  was used for the exchange-
correlation functional. In addition, the projector augmented-wave method was
used for the interactions between ions and valence electrons, and the energy
cutoff was set to 600 eV. A $\Gamma$-centered $k$-mesh of 5$\times$5$\times$6
size was used to sample the BZ. Moreover, lattice dynamic calculations were
performed to obtain the phonon dispersion of Li6WN4 at equilibrium lattice
constants in the PHONOPY package add49  with density-functional perturbation
theory. The topological characteristics of the [001] phonon surface states
were calculated by constructing a Wannier tight-binding Hamiltonian for
phonons add50 .

Figure 1: Crystal structure of Li6WN4 material with P42/nmc type structure
under different viewsides. (b) Bulk BZ and high-symmetry points.

As shown in Fig. 1a, Li6WN4 is a tetragonal material with space group (SG) No.
137 (P42/nmc). The atomic positions are as follows: W is located at the 2a
(0.25, 0.75, 0.25), N at the 8g (0.25, 0.995, 0.05), Li (1) at the 4d (0.25,
0.25, 0.338), and Li (2) at the 8f (0.537, 0.463, 0.25) Wyckoff positions. The
crystal structure of Li6WN4 is completely relaxed according to the first-
principle calculations; the computed lattice constants a = b = 6.71 Å and c =
4.94 Å agree well with the experimental results (a = b = 6.675 Å and c = 4.928
Å).

The symmetry operators of Li6WN4 are generated via fourfold screw rotation
$S_{4z}$=$\\{$${C_{4z}|00\frac{1}{2}}$$\\}$, twofold screw rotation $S_{2x}$
=$\\{$${C_{2x}|\frac{1}{2}\frac{1}{2}0}$$\\}$, spatial inversion
$\\{$${I|\frac{1}{2}\frac{1}{2}0}$$\\}$, and time-reversal symmetry
${\cal{T}}$. Hence, Li6WN4 has three mirror symmetries: $M_{x}$, $M_{y}$, and
$M_{z}$. Because we are interested in the phonon spectrum of Li6WN4, the spin-
orbit coupling effect is neglected, indicating that the 2$\pi$ rotation equals
1 and ${\cal{T}}^{2}=1$.

In this work, the topological signature of the phonon dispersion of Li6WN4 was
studied. The calculated phonon band along the high-symmetry paths (see Fig.
1b) is shown in Fig. 2a. The absence of imaginary frequency modes in the
phonon dispersion indicates that Li6WN4 is dynamically stable. We focused on
the two phonon bands appearing in the 22.5-24.5 THz range; they are well
separated from other energy bands. Evidently, the two phonon bands become
twofold degenerate along the X-M-A-R-X and Z-A paths. For clarity, we divided
the doubly degenerate phonon bands in the 22.5-24.5 THz range into the two
regions R2 and R3, as shown in Fig. 2b. The degeneracies in the R2 and R3
regions are discussed separately.

First, the double degeneracies along the X-M-A-R-X path were studied (R2 in
Fig. 2b). All these high-symmetry paths lie in the BZ boundary, e.g., in the
$k_{x}$= $\pi$ plane (see Fig. 2c). In Fig. 3a and b, we divided the A-X paths
into 5 parts and selected some other symmetry points along the A-X path: b1,
b2, b3, and b4. The phonon dispersion along the a1-b1-a1${}^{{}^{\prime}}$,
a2-b2-a2${}^{{}^{\prime}}$, a3-b3-a3${}^{{}^{\prime}}$, and
a4-b4-a4${}^{{}^{\prime}}$ paths are shown in Fig. 3c and Fig. S2,
respectively. Evidently, two non-degenerated phonon bands linearly cross at
b1, b2, b3, and b4.

In fact, the two phonon bands become degenerate in the entire $k_{x}$= $\pi$
plane (see the schematic diagram in Fig. 3a), which leads to the creation of a
nodal surface. The nodal surface is guaranteed by nonsymmorphic $S_{2x}$
symmetry and ${\cal{T}}$ symmetry because any generic point in the $k_{x}$=
$\pi$ plane is invariant under ${\cal{T}}S_{2x}$ symmetry and
${({\cal{T}}S_{2x})}^{2}=-1$ in the $k_{x}$=$\pi$ plane. Owing to the fourfold
screw rotation $S_{4z}$, there must exist another nodal surface in the
$k_{y}$=$\pi$ plane. It should be noted that while the z-direction represents
a screw rotation axis, there is no nodal surface in the $k_{z}$=$\pi$ plane,
which can be directly inferred from the phonon dispersion along the R-Z path.
This is because $S_{4z}^{2}$ is equivalent to a symmorphic operator and
${({\cal{T}}S_{4z}^{2})}^{2}=1$ in the $k_{z}$=$\pi$ plane. However, although
the existence of two nodal surfaces in the $k_{x}$=$\pi$ and $k_{y}$=$\pi$
planes is guaranteed owing to the symmetry characteristics, the energy
dispersion of the surface is not constrained by these symmetries.

Figure 2:  (a) Phonon dispersion of Li6WN4 along $\Gamma$-X-M-A-
R-X-$\Gamma$-Z-A-R-Z paths; (b) enlarged frequencies of Li6WN4 with nodal
surface phonons (see R2) and Weyl nodal line (WNL)-phonons (See R3);(c)
Schematic diagram of nodal phonon states of R2 and R3 in 3D BZ.

To the best of our knowledge, the existence of an ideal topological nodal-
surface semimetal with a flat nodal surface state has not been reported add26
. Fortunately, the nodal-surface phonons in Li6WN4 are very flat in energy
with energy variations of less than 0.25 THz (approximately 1.00 meV). More
importantly, the phonon band structure is “clean” in the sense that the bands
forming the surface are well separated from the other bands. Another feature
of the nodal-surface phonon is that it exhibits linear dispersion along the
direction normal to the surface; in addition, the linear energy range is
considerable large for most points on the surface, as indicated in Fig. 2b. To
present these characteristics more clearly, the calculated energy dispersion
for points on the surface along the transverse direction is presented in Fig.
3c and Fig. S2c, showing the linear energy range for the path away from the
surface is considerable large.

Figure 3:  (a) Schematic diagram of nodal surface states in $k_{x}$=$\pi$ and
$k_{y}$=$\pi$ planes; (b) some symmetry points of [011] plane; a1, a2, a3, and
a4 and b1, b2, b3, and b4 are located at equal distances between
R${}^{{}^{\prime}}$(A) and $\Gamma$(X); (c) phonon dispersion along
a4-b4-a4${}^{{}^{\prime}}$ paths; (d) shape of one pair of Z-centered WNLs in
$k_{z}$=$\pi$ plane.

In the next step, the degeneracies along the Z-A path are discussed (see R3 in
Fig. 2b). This is an essential nodal line protected by $M_{z}$ symmetry and
twofold rotation along the (110) direction. The low-energy effective
Hamiltonian expanding around a generic point on the Z-A path can be written as
follows add51 :

$\mathcal{H}=c_{1}+c_{2}\sigma_{3}(k_{x}-k_{y})+c_{3}\sigma_{1}k_{z},$ (1)

where $\sigma_{i}$ (i = 1, 2, 3) is the Pauli matrix, and the model parameter
$c_{i}$ depends on the material and $k_{x(y)}$; the momentum is measured from
the generic point on the Z-A path. To examine the nontrivial topological
behavior of this nodal line, we calculate the Berry phase for a closed loop
surrounding the line, expressed as:

$P_{B}=\oint_{c}\textbf{\emph{A}}(\textbf{\emph{k}})\cdot d\textbf{\emph{k}}$
(2)

with
$\textbf{\emph{A}}(\textbf{\emph{k}})=-i\left\langle\varphi(\textbf{\emph{k}})\left|\nabla_{\textbf{\emph{k}}}\right|\varphi(\textbf{\emph{k}})\right\rangle$
the Berry connection and $\varphi(\textbf{\emph{k}})$ the periodic part of the
Bloch function. According to our results, we obtain $P_{B}=\pi$. Then one
knows that the nodal line along the Z-A path is topologically nontrivial, and
should result in interesting surface states. As previously mentioned, owing to
the $S_{4z}$ symmetry, there are two nodal lines along the diagonal and
clinodiagonal directions, respectively (see Fig. 2c). The shape of the nodal
lines in the $k_{z}$=$\pi$ plane is shown in Fig. 3d, where one pair of nodal
lines can be clearly observed (see the white lines). According to Fig. 2b, the
nodal-line phonon is relatively flat in energy, and its energy variation is
less than 0.3 THz (approximately 1.24 meV).

Remarkably, the ideal nodal-surface phonon and ideal nodal-line phonon
constitute a hitherto unobserved type of bosonic excitation, namely, a nodal-
lantern phonon, as illustrated in Fig. 2(c) and Fig. S1. Furthermore, as the
existence of the two nodal surfaces and two nodal lines is guaranteed by
symmetry, any 3D material with SG 137 must have nodal-lantern phonons as long
as two of its phonon bands are separated from the others.

Figure 4: (a) projection of WNL onto [001] surface and some surface symmetry
points
$\bar{A},\bar{A}_{1},\bar{A}_{2},\bar{A}_{3},\bar{B},\bar{B}_{1},\bar{B}_{2},\bar{B}_{3}$
of [001] surface; (b) schematic diagram of surface state of [001] surface.
Values 0 and $\pi$ are Zak phases for lines normal to [001] surface.(c) [001]
phonon surface states of $\mathrm{Li}_{6}\mathrm{WN}_{4}$ along
$\bar{B}-\bar{B}_{1}$ and $\bar{B}_{2}-\bar{B}_{3}$ surface paths;

In the next step, the phonon surface states corresponding to the nodal-lantern
phonon are studied. Because the two boundaries of the BZ (i.e., the
$k_{x}=\pi$ and $k_{y}=\pi$ planes) are covered by the nodal surface, the
[001] surface is the only surface that can experience a clear surface state.
The corresponding surface BZ and some surface symmetry points
$\left(\bar{A},\bar{A}_{1},\bar{A}_{2},\bar{A}_{3},\bar{B},\bar{B}_{1},\bar{B}_{2},\bar{B}_{3}\right)$
of the [001] surface are shown in Fig. 4a. The calculated projected spectrum
for the [001] surface along four surface paths $\bar{B}-\bar{B}_{1}$,
$\bar{B}_{1}-\bar{B}_{2}$, $\bar{B}_{2}-\bar{B}_{3}$, and
$\bar{B}_{3}-\bar{B}$ are shown in Fig. 4c. Interestingly, evident [001]
phonon surface states (highlighted by white arrows) can be found along the
$\bar{B}-\bar{B}_{1}$ and $\bar{B}_{2}-\bar{B}_{3}$ surface paths. However,
the phonon surface states have disappeared along the $\bar{B}_{1}-\bar{B}_{2}$
and $\bar{B}_{3}-\bar{B}$ paths. We use the Zak phase to explain this peculiar
surface state. Here, the relevant Zak phase is the Berry phase along a
straight line parallel to the $\Gamma$-Z direction and across the bulk BZ:

$Z\left(k_{x},k_{y}\right)=\sum_{n\in
occ}\oint_{c}A_{z}(\textbf{\emph{k}})dk_{z},$ (3)

where $A_{z}$ is the z-component of the Berry connection. The summation is
performed for all the occupied bands. Owing to $M_{z}$ symmetry, the Zak phase
is quantized to 0 or $\pi$, which correspond to two topologically distinct
phases. A $\pi$ Zak phase generally indicates the existence of nontrivial
topological surface state add52 . Because the two essential nodal lines along
the Z-A path have nontrivial $\pi$ Berry phases, their projections onto the
[001] surface divides the surface BZ into four regions. Two regions have
$Z$=0, and two regions have $Z$=$\pi$, as illustrated in Fig. 4b. Because each
region occupies a quarter of the surface BZ, the topological surface state
covers exactly half the surface BZ, which is consistent with the calculated
results in Fig. 4c.

Before closing the article, we would like to present some important remarks
about the topological signatures of the Li6WN4 phonon dispersion: (i) Although
the existence of nodal surface states in the fermionic electronic structures
of realistic materials has been predicted add26 ; add27 ; add28 ; add33 , the
proposed nodal surface states of phonons have not been studied by other
researchers. This paper presents the existence of ideal phononic nodal surface
states in realistic materials _for the first time_. (ii) The predicted nodal
surface is ideal in that it is nearly flat in energy with energy variations of
less than 0.25 THz; in addition, the two relevant phonon bands are the only
“clean” bands in the 22.5-24.5 THz range. (iii) The shape of the nodal
structure in Li6WN4 (e.g., the nodal surface in the $k_{x}$=$\pi$ and
$k_{y}$=$\pi$ planes and the nodal lines along the Z-A path) resembles that of
a traditional lantern (see Fig. S1). We would like to point out that the
nodal-lantern phonon is protected by symmetry and has never been reported
before. These results can be used to investigate the entanglement between
nodal-line and nodal-surface phonons.

In summary, based on first-principle calculations and symmetry analysis, a new
topological phase was discovered: the nodal-lantern phonons in ternary nitride
Li6WN4 are composed of two nodal-surface phonons in the $k_{x}$=$\pi$ and
$k_{y}$=$\pi$ planes and one pair of nodal-line phonons in the $k_{z}$=$\pi$
plane. The existence of nodal-lantern phonons in Li6WN4 is protected by
nonsymmorphic symmetries and time-reversal symmetry. Moreover, the phonon
surface states in the [001] surface of this material were investigated;
according to the results, the topological surface states of the nodal-lantern
phonons cover exactly half the surface BZ. The presented results extend the
concept of nodal surfaces to phonon systems, propose the existence of a
undiscovered type of bosonic excitations, and predict the ideal material
candidate.

_Acknowledgments_ X.T. Wang is grateful for the support from the National
Natural Science Foundation of China (No. 51801163) and the Natural Science
Foundation of Chongqing (No. cstc2018jcyjA0765).

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# Casimir Self-Interaction Energy Density of Quantum Electrodynamic Fields

Alexandre Tkatchenko  Dmitry V. Fedorov  Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg City, Luxembourg

###### Abstract

Quantum electrodynamic fields possess fluctuations corresponding to transient
particle/antiparticle dipoles, which can be characterized by a non-vanishing
polarizability density. Here, we extend a recently proposed quantum scaling
law to describe the volumetric and radial polarizability density of a quantum
field corresponding to electrons and positrons and derive the Casimir self-
interaction energy (SIE) density of the field, $\bar{E}_{\rm{SIE}}$, in terms
of the fine-structure constant. The proposed model obeys the cosmological
equation of state $w=-1$ and the magnitude of the calculated
$\bar{E}_{\rm{SIE}}$ lies in between the two recent measurements of the
cosmological constant $\Lambda$ obtained by the Planck Mission and the Hubble
Space Telescope.

Quantum fields possess transient particle/antiparticle ($p^{-}/p^{+}$) pair
fluctuations or “virtual excitations” Milonni (2013); Zee (2010). For
electrodynamic fields, such virtual excitations correspond to formation of
minute and short-lived dipoles, indicating that the field should possess a
finite _intrinsic_ polarizability density. In this work, we propose a model
for the polarizability density of quantum electrodynamic fields based on an
extension of a recently derived quantum-mechanical scaling law between
polarizability and equilibrium (van-der-Waals) radius Fedorov  _et al._
(2018); Tkatchenko  _et al._ (2021) to $p^{-}/p^{+}$ fluctuations. We then
calculate the finite Casimir self-interaction energy density of such a quantum
field and find that its equation of state matches the one of cosmological dark
energy, $w=-1$, whereas the magnitude of the obtained energy density agrees
with the most recent measurements of the cosmological constant $\Lambda$.

Linear polarization of a quantum field is described by a polarizability
density-density tensor (PDDT) $\bm{\alpha}_{\rm F}({\bf r},{\bf
r^{\prime}},t,t^{\prime})=\delta\bm{\mathcal{P}}({\bf
r},t)/\delta\bm{\mathcal{E}}({\bf r^{\prime}},t^{\prime})$ – a functional
derivative of polarization field $\bm{\mathcal{P}}({\bf r},t)$ with respect to
the electric field $\bm{\mathcal{E}}({\bf r^{\prime}},t^{\prime})$. Since time
is homogeneous, $\bm{\alpha}_{\rm F}({\bf r},{\bf
r^{\prime}},t,t^{\prime})=\bm{\alpha}_{\rm F}({\bf r},{\bf
r^{\prime}},t-t^{\prime})$, which allows switching to the frequency-dependent
quantity $\bm{\alpha}_{\rm F}({\bf r},{\bf r^{\prime}},\omega)$. Upon
integrating $\bm{\alpha}_{\rm F}({\bf r},{\bf r^{\prime}},t,t^{\prime})$ or
$\bm{\alpha}_{\rm F}({\bf r},{\bf r^{\prime}},\omega)$ over the spatial
variable ${\bf r^{\prime}}$, tracing over tensor components, and taking the
infinite-time limit $|t-t^{\prime}|\to\infty$ or the zero-frequency limit
$\omega\to 0$ yields a homogeneous static polarizability density $\alpha_{\rm
F}({\bf r})$. In addition, the following argument suggests that an integral
over ${\bf r^{\prime}}$ leads to a finite value of $\alpha_{\rm F}({\bf r})$.
The production of particle-antiparticle pairs emerging from the vacuum upon
applying an electric field (Schwinger effect) is an established physical fact.
For sufficiently long timescales, the polarizability of a created
$p^{-}/p^{+}$ pair is essentially infinite since the particle and its
antiparticle yield an increasingly growing dipole. In contrast, the
polarizability of bound $p^{-}/p^{+}$ pairs is finite. Reducing the applied
field to zero does not eliminate zero-point fluctuations, hence a quantum
field should possess intrinsic polarizability density even in the absence of
any external field. The observable (di)electric permittivity of vacuum/free
space, $\varepsilon_{0}$ (with units of polarizability density), has a finite
value, yielding further support for our proposal. Obviously, the effects of
vacuum polarization are experimentally measured by applying external fields
Riek  _et al._ (2015); Moskalenko and Ralph (2019); Benea-Chelmus  _et al._
(2019), hence the possibility of observing intrinsic (zero-field) vacuum
polarization is a contentious question at this moment. We are here ultimately
concerned with measurable quantities, such as energy density and the
cosmological constant. Therefore, we use a finite vacuum polarizability
density as a working hypothesis and explore its consequences on experimental
observables. In what follows, the polarizability unit will be normalized by
$4\pi\varepsilon_{0}\,$, to measure the polarizability $\alpha_{\rm F}$ in
terms of the polarizability volume Atkins and Friedman (2005). Consequently,
the two-point polarizability $\alpha_{\rm F}({\bf r},{\bf r^{\prime}})$ has
units of inverse volume and the polarizability density $\alpha_{\rm F}({\bf
r})$ is unitless. The fine-structure constant (FSC) is denoted by $\alpha_{\rm
fsc}\,$.

All polarizable entities experience mutual dispersion forces, which are either
of van der Waals (vdW) or Casimir type London (1930); Casimir (1948); Stone
(1996); Parsegian (2005); Buhmann (2013a, b). Both forces are manifestations
of the same phenomenon related to polarization propagated _via_ the
corresponding gauge field. The broad relevance of dispersion forces has been
recognized for diverse phenomena, ranging from particle physics Vovchenko  _et
al._ (2017), to atoms Derevianko  _et al._ (1999), molecules Hermann  _et al._
(2017), and condensed matter Klimchitskaya  _et al._ (2009); French  _et al._
(2010), and even for cohesion in macroscopic objects of cosmological relevance
Rozitis  _et al._ (2014); Scheeres and Sánchez (2018). The difference between
vdW and Casimir forces is that the latter name is employed when polarization
propagation requires finite time determined by the underlying gauge field
(usually, speed of light).

The exact vdW/Casimir self-interaction energy (SIE) density caused by
particle/antiparticle fluctuations of a quantum field can be computed using
the adiabatic connection fluctuation-dissipation theorem (see Refs. Gunnarsson
and Lundqvist (1976); Langreth and Perdew (1977) for seminal works formulated
in terms of the density response function and Refs. Hermann  _et al._ (2017);
Woods  _et al._ (2016); Venkataram  _et al._ (2017) for a modern formulation
in terms of the polarizability):

$\displaystyle\bar{E}_{\rm SIE}$ $\displaystyle=$
$\displaystyle\frac{\hbar}{2\pi
V}\int_{0}^{\infty}du\int_{0}^{1}d\lambda\int_{V}\int_{V}d\mathbf{r}d\mathbf{r}^{\prime}$
(1) $\displaystyle\times$
$\displaystyle\mathrm{Tr}\\{[{\bm{\alpha}}_{\lambda}(\mathbf{r},\mathbf{r}^{\prime},iu)-{\bm{\alpha}}_{0}(\mathbf{r},\mathbf{r}^{\prime},iu)]\mathbf{T}(\mathbf{r},\mathbf{r}^{\prime},iu)\\}$

where ${\bm{\alpha}}_{0}(\mathbf{r},\mathbf{r}^{\prime},iu)$ is the PDDT
corresponding to _bare_ $p^{-}/p^{+}$ fluctuations, with $u$ as the
oscillation frequency. Then,
${\bm{\alpha}}_{\lambda}(\mathbf{r},\mathbf{r}^{\prime},iu)$ is the PDDT
corresponding to $p^{-}/p^{+}$ fluctuations interacting with strength
$\lambda$ and $\mathbf{T}(\mathbf{r},\mathbf{r}^{\prime},iu)$ is the dipolar
propagator for the bosonic field mediating the interaction (_e.g._ the
retarded dipole potential when considering the electromagnetic field). The
integration volume $V$ is infinite, but in practice it is sufficient to
integrate over $V$ of the same order as the volume of quantum $p^{-}/p^{+}$
fluctuations, _i.e._ of the order of Thomson’s scattering length cubed. We
remark that elementary particles are usually considered to be structureless,
_i.e._ they have zero size. However, elementary particles acquire an effective
orbital size when they are probed by other particles or fields. For example,
the electron orbital has an effective radius of $a_{0}$ (Bohr’s radius) when
interacting with a proton. Similarly, the effective electron orbital radius
becomes $\alpha_{\rm fsc}a_{0}$ in an inelastic interaction with a photon
(Compton scattering). Whereas, in an elastic electron–photon interaction
(Thomson scattering), the electron orbital radius becomes $\alpha_{\rm
fsc}^{2}a_{0}$. Our work proposes a model for the electron/positron field
interacting with the photon field in the low-frequency elastic regime relevant
to vdW/Casimir phenomena. In this case, the Thomson’s scattering radius
($\alpha_{\rm fsc}^{2}a_{0}$) is the most natural choice for the effective
size of the electron (and positron) orbital. Furthermore, Thomson’s scattering
length defines the onset of measurable effects of quantum fluctuations in
quantum field theory Milonni (2013).

The internal polarization dynamics which generate
${\bm{\alpha}}_{\lambda}(\mathbf{r},\mathbf{r}^{\prime},iu)$ for an
interacting quantum field are not known in detail yet. Therefore, we resort to
coarse-grained models for
${\bm{\alpha}}_{\lambda}(\mathbf{r},\mathbf{r}^{\prime},iu)$ taking into
account the main symmetries of a quantum field in its vacuum state, such as
its homogeneity and isotropy. In the retarded Casimir regime, the static
($u=0$) full-potential ($\lambda$=1) approximation
${\bm{\alpha}}(\mathbf{r},\mathbf{r}^{\prime})_{u=0}$ is sufficient. Using
either homogeneity or isotropy of the field, in what follows we propose models
for volumetric and radial polarizability densities of $p^{-}/p^{+}$
fluctuations and use these models for evaluating approximations to Eq. (1).

To simplify the notation, forthcoming derivations will be carried out for the
electron/positron ($e^{-}/e^{+}$) field. Generalization to other quantum
fields will be briefly discussed below. Due to the homogeneity of zero-point
fluctuations, the electron/positron field can be modeled as a condensed
overall-neutral homogeneous medium of electrons and positrons with
$e^{-}/e^{+}$ pairs forming transient dipoles characterized by an isotropic
polarizability that follows a quantum-mechanical scaling law Fedorov  _et al._
(2018); Tkatchenko  _et al._ (2021)

$\alpha_{e^{-}/e^{+}}=\frac{2}{3}\left(\frac{\alpha_{\rm
fsc}^{\nicefrac{{1}}{{3}}}\times
R_{e^{-}/e^{+}}}{a_{0}}\right)^{4}R_{e^{-}/e^{+}}^{3}\quad,$ (2)

where $a_{0}$ is Bohr’s radius and $R_{e^{-}/e^{+}}$ is the equilibrium (vdW)
radius of the $e^{-}/e^{+}$ pair. The quantum-mechanical scaling of the
polarizability $\propto R_{\rm vdW}^{7}$ and its dependence on the fine-
structure constant originates from the dressing of particle/hole excitations
by virtual photons of the electromagnetic field, as discussed in Ref.
Tkatchenko  _et al._ (2021). The four-dimensional renormalization factor
$(\alpha_{\rm fsc}^{\nicefrac{{1}}{{3}}}\times R_{e^{-}/e^{+}}/{a_{0}})^{4}$
corresponds to the polarizability density of the excited electromagnetic field
Szabó  _et al._ (2022). The factor $2/3$ accounts for two contributions: i)
both electrons and positrons are polarizable (factor of $2$), ii) the
$e^{-}/e^{+}$ pair polarizes in a homogeneous field where all polarization
directions are degenerate, akin to the jellium model (factor of 1/3) Nesbet
(1997). As discussed above, we set the equilibrium radius $R_{e^{-}/e^{+}}$ to
the Thomson scattering length, $R_{\rm Th}=\alpha_{\rm fsc}^{2}a_{0}$, at
which the electrostatic self-interaction energy of a particle
(${e^{2}}/{4\pi\varepsilon_{0}R_{\rm Th}}$) equals its rest mass–energy
($m_{e}c^{2}$). Finally, the effective volume corresponding to each
$e^{-}/e^{+}$ pair becomes
$V_{e^{-}/e^{+}}=2\times\frac{4\pi}{3}R_{e^{-}/e^{+}}^{3}\,$, since both
electrons and positrons contribute to the polarizability volume.

With a homogeneous model for
${\bm{\alpha}}(\mathbf{r},\mathbf{r}^{\prime})_{u=0}\equiv\alpha_{e^{-}/e^{+}}\,$,
the Casimir energy density for two interacting $e^{-}/e^{+}$ pairs, as a
second-order approximation to Eq. (1), can be obtained according to Ref.
Casimir and Polder (1948) as

$\bar{E}_{\rm SIE}^{(2)}=-\frac{23\hbar
c}{4\pi}\times\frac{\alpha_{e^{-}/e^{+}}^{2}}{(2R_{e^{-}/e^{+}})^{7}\times
V_{e^{-}/e^{+}}}\quad.$ (3)

Expressing the polarizability in terms of $R_{e^{-}/e^{+}}$ yields

$\bar{E}_{\rm SIE}^{(2)}=-\frac{23\hbar
c}{3072\pi^{2}}\times\left(\frac{\alpha_{\rm
fsc}^{\nicefrac{{1}}{{3}}}}{a_{0}}\right)^{8}\times\left(R_{e^{-}/e^{+}}\right)^{4}\quad,$
(4)

whereas upon substituting the equilibrium vdW radius we obtain

$\bar{E}_{\rm SIE}^{(2)}=-\frac{23\,\alpha_{\rm
fsc}^{\nicefrac{{29}}{{3}}}}{3072\pi^{2}}\times\frac{\hbar c\alpha_{\rm
fsc}}{a_{0}^{4}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}=-\frac{23\,\alpha_{\rm
fsc}^{\nicefrac{{29}}{{3}}}E_{h}}{3072\pi^{2}a_{0}^{3}}}\quad,$ (5)

where $E_{h}=\hbar c\alpha_{\rm fsc}a_{0}^{-1}$ is the Hartree energy. The
$\bar{E}_{\rm SIE}^{(2)}$ contribution amounts to a pairwise interaction
between $e^{-}/e^{+}$ pair fluctuations. Calculating $\bar{E}_{\rm SIE}$ for
the field requires summing the pairwise potential over all possible many-body
geometries of $e^{-}/e^{+}$ fluctuations, which can be approximated as a
lattice sum. Hence, the Casimir SIE density of the electron/positron field is
given by

$\bar{E}_{\rm SIE}=\bar{E}_{\rm SIE}^{(2)}\times N_{\rm eff}\ ,$ (6)

where $N_{\rm eff}$ corresponds to a weighted lattice sum of $R^{-7}$
potential over all possible field arrangements of $e^{-}/e^{+}$ pairs.
Performing a more general treatment of the field below, here we determine
limiting behaviors of $N_{\rm eff}$ as corresponding to body-centered-cubic
(bcc), $N_{\rm eff}^{\rm bcc}=11.05$, and face-centered-cubic (fcc), $N_{\rm
eff}^{\rm fcc}=13.36$, lattices calculated from

$N_{\rm
eff}=\sum\limits_{\mathbf{R}_{j}\neq\mathbf{0}}\frac{(2R_{e^{-}/e^{+}})^{7}}{|\mathbf{R}_{j}|^{7}}\
,$ (7)

where the sum runs over lattice vectors of the corresponding bcc/fcc
structure. The obtained estimates for $N_{\rm eff}$ yield the following
interval of (absolute) values for the Casimir SIE density: $|\bar{E}_{\rm
SIE}|=\\{1.85\times 10^{-23},2.24\times 10^{-23}\\}$ Ha/Bohr3. Remarkably,
this energy density range (barring the sign to be discussed below) seems to
agree well with the recent estimates of the vacuum energy density as given by
the cosmological constant $\Lambda=\\{1.84\times 10^{-23},2.20\times
10^{-23}\\}$ Ha/Bohr3 Ast , when taking two different values of the Hubble
constant measured by either the Planck Mission (67.66 (km/s)/Mpc) Pla  or the
Hubble Space Telescope (74.03 (km/s)/Mpc) Riess  _et al._ (2019).

The uncertainty related to the estimation of $N_{\rm eff}$ can be avoided by
making a derivation of the Casimir SIE density based on collective
fluctuations of concentric spherical field shells, instead of
particle/antiparticle pairs. This model is based on both homogeneity and
isotropy of quantum fields. First, we define the radial polarizability density
of a thin spherical shell with radius $r$ (centered at $r^{\prime}=0$) for the
electron/positron field

$\bar{\alpha}_{e^{-}/e^{+}}(r)=\frac{4\pi
r^{2}}{a_{0}^{3}}\left(\frac{\alpha_{\rm fsc}^{\nicefrac{{1}}{{3}}}\times
r}{a_{0}}\right)^{4}\,r^{3}\quad.$ (8)

The Casimir SIE density for a particle/antiparticle field can now be
calculated by summing the SIE over concentric spherical shells from $r=0$ to
$r=r_{f}$, where $r_{f}$ is the dressed Thomson scattering length. As shown in
Ref. Tkatchenko  _et al._ (2021), the corresponding (vdW) equilibrium length
$R_{e^{-}/e^{+}}$ is renormalized to $r_{f}=\alpha_{\rm
fsc}^{-\nicefrac{{1}}{{3}}}R_{e^{-}/e^{+}}\,$, which gives us
$r_{f}=\alpha_{\rm fsc}^{\nicefrac{{5}}{{3}}}a_{0}\,$. Then, the SIE density
for the $e^{-}/e^{+}$ field, as an approximation for concentric shells to Eq.
(1), becomes Casimir (1948); Zaheer  _et al._ (2010)

$\bar{E}_{\rm{SIE}}=-\frac{3\hbar c}{8\pi
a_{0}^{3}}\int_{0}^{r_{f}}\frac{\bar{\alpha}_{e^{-}/e^{+}}(r)}{r^{4}}\,\mathrm{d}r\quad.$
(9)

Evaluating the integral yields

$\bar{E}_{\rm{SIE}}=-\frac{\hbar c}{4a_{0}^{6}}\left(\frac{\alpha_{\rm
fsc}^{\nicefrac{{1}}{{3}}}}{a_{0}}\right)^{4}r_{f}^{6}\quad.$ (10)

Finally, substituting $r_{f}\,$, we obtain

$\bar{E}_{\rm{SIE}}=-\frac{\hbar c\,\alpha_{\rm
fsc}^{\nicefrac{{34}}{{3}}}}{4a_{0}^{4}}=-\frac{1}{4}\alpha_{\rm
fsc}^{\frac{31}{3}}E_{h}a_{0}^{-3}\quad.$ (11)

In atomic units, $|\bar{E}_{\rm{SIE}}|=2.07\times 10^{-23}$ Ha/Bohr3. This
number lies in between the two recent estimates of the cosmological constant
$\Lambda=\\{1.84\times 10^{-23},2.20\times 10^{-23}\\}$ Ha/Bohr3 Ast . In
cosmology, Planck’s units are often used, setting $\hbar=1$, $c=1$, and $8\pi
G=1$. Expressing Eq. (11) in Planck’s units, we obtain
$|\bar{E}_{\rm{SIE}}|=3.31\times 10^{-122}\,\,l_{\rm P}^{-2}$, where $l_{\rm
P}$ is Planck’s length.

The comparison between Eqs. (6) and (11), allows us to derive an effective
value for $N_{\rm eff}$, as a weighted lattice interaction sum over all
possible field arrangements of particle/antiparticle pairs. Based on Eqs. (5)
and (11), we obtain $N_{\rm
eff}=\bar{E}_{\rm{SIE}}/\bar{E}^{(2)}_{\rm{SIE}}=\frac{768\pi^{2}}{23}\alpha_{\rm
fsc}^{\nicefrac{{2}}{{3}}}\approx 12.40$. This number lies in between the
corresponding values calculated for body-centered-cubic and face-centered-
cubic lattices above.

Although $\bar{E}_{\rm{SIE}}$ given by Eqs. (5) or (11) has a negative sign,
as vdW or Casimir energy should have in a typical geometry, it satisfies the
(cosmological) equation of state of an expanding scalar field
$w=P/\bar{E}_{\rm{SIE}}=-1$, where $P=-dE/dV$ is the pressure. The field has a
tendency to expand because of the linear dependence of the energy on the
volume, $E_{\rm{SIE}}\propto\bar{E}_{\rm{SIE}}V$, according to Eq. (11). We
also remark that matter would experience an effective field with a positive
$\bar{E}_{\rm{SIE}}$, because the polarization of matter, as an excitation of
the field, is measured with respect to the polarization of the vacuum field
which would be described _via_
${\bm{\alpha}}_{0}(\mathbf{r},\mathbf{r}^{\prime},iu)$ in Eq. (1) when
computing the vdW/Casimir interactions between two material objects.

The obtained results rely on the applicability of the fully retarded Casimir-
Polder energy given by Eq. (3) for calculating the SIE of a quantum field. Let
us evaluate its validity for distances comparable to Thomson’s scattering
length. The Casimir-Polder formula is accurate for distances $R$ satisfying
the retarded regime, $R\gg c/\omega_{\rm F}$, where $\omega_{\rm F}$ is a
characteristic frequency. The value of $\omega_{\rm F}$ can be estimated from
the polarizability of a quantum harmonic oscillator Hermann  _et al._ (2017),
as $\alpha_{\rm F}\,\omega_{\rm F}^{2}=e^{2}N/m_{e}$, where $N$ is the
effective particle number contributing to field fluctuations. For the
electron/positron field, we obtain that Eq. (3) is accurate for
$R\gg\alpha_{\rm fsc}^{\nicefrac{{20}}{{3}}}\,a_{0}$. This condition is
strictly satisfied within the context of our work.

Both Eqs. (3) and (9) employ the dipole approximation to the field response.
Some authors argue that the response of a quantum field has only dipolar
contributions Sernelius (2009) and in fact the PDDT of an arbitrary field (or
matter) system can be exactly described by an infinite number of dipolar
quantum oscillators Milonni (2013); Hermann  _et al._ (2017). However, we rely
on a coarse-grained representation for the PDDT of a quantum field, which
could make multipolar interactions non-negligible Hermann  _et al._ (2017).
Because of particle/antiparticle symmetry, the quadrupole polarizability
vanishes and the first non-vanishing multipolar polarizability beyond dipole
is the octupolar one. We can estimate the octupolar contribution to
$\bar{E}_{\rm{SIE}}$ by a generalization of Eqs. (8) and (9) to higher
multipoles and using the quantum scaling law for multipole polarizabilities
derived in Refs. Fedorov  _et al._ (2018); Vaccarelli  _et al._ (2021). By
doing so, we find that the contribution from a potential octupolar interaction
term to $\bar{E}_{\rm{SIE}}$ would be vanishingly small (multiplied by
$\alpha_{\rm fsc}^{6}$) compared to the dipolar one in Eq. (9). The tiny
magnitude of the dipole polarizability of pair excitations in a quantum field
$\propto\alpha_{\rm fsc}^{\nicefrac{{46}}{{3}}}$ in Eq. (2) provides an
explanation of why drastic approximations to the exact vdW/Casimir energy
density seem to yield very accurate results when computing the SIE density of
quantum fields in their vacuum state.

Going beyond the electron-positron field, we can also calculate the
$\bar{E}_{\rm SIE}$ for muon-antimuon and tau-antitau virtual pair
fluctuations. For heavier leptons possessing the same charge but an arbitrary
mass $\mu$, one needs to replace
$a_{0}=(4\pi\epsilon_{0})\hbar^{2}/m_{e}e^{2}$ by
$a_{\mu}=(4\pi\epsilon_{0})\hbar^{2}/\mu e^{2}$ in Eq. (2), with a
corresponding scaling of the vdW radius:
$R_{\mu^{-}/\mu^{+}}=(a_{\mu}/a_{0})\times R_{e^{-}/e^{+}}=(m_{e}/\mu)\times
R_{e^{-}/e^{+}}$. These conditions ensure the correct polarizability-mass
scaling for hydrogen-like atoms Szabó  _et al._ (2022):
$\alpha_{\mu^{-}/\mu^{+}}=\alpha_{e^{-}/e^{+}}\times(m_{e}/\mu)^{3}$. Since
both the polarizability and the related volume of a lepton pair are inversely
proportional to the particle mass cubed, the electron-positron, muon-antimuon,
and tau-antitau quantum fields possess the same polarizability density
$\propto$ $\epsilon_{0}$, as they should. Based on Eq. (5), we obtain
$\bar{E}_{\rm SIE}^{(2)}(\mu^{-}/\mu^{+})~{}/~{}\bar{E}_{\rm
SIE}^{(2)}(e^{-}/e^{+})=(a_{0}/a_{\mu})^{4}=(\mu/m_{e})^{4}$. However, the
total Casimir SIE density given by Eq. (6) is a product of the pairwise SIE
density and the weighted lattice sum $N_{\rm eff}\,$. This effective number of
pairwise interacting virtual pairs depends on the lifetime of constituent
particles. Each virtual particle possesses a finite lifetime
$\tau_{\mu}=\tau_{e}\times(m_{e}/\mu)$, as follows from Heisenberg’s
uncertainty principle. For two interacting species, four virtual particles are
present, which requires the renormalization $N_{\rm
eff}(\mu^{-}/\mu^{+})~{}/~{}N_{\rm eff}(e^{-}/e^{+})=(m_{e}/\mu)^{4}$. This
scaling compensates the factor $(\mu/m_{e})^{4}$ in the ratio $\bar{E}_{\rm
SIE}^{(2)}(\mu^{-}/\mu^{+})~{}/~{}\bar{E}_{\rm SIE}^{(2)}(e^{-}/e^{+})$
resulting in the same Casimir SIE density for electron-positron, muon-
antimuon, and tau-antitau fields. Our arguments based on the uncertainty
principle are supported by the effective Lagrangian approach in quantum
electrodynamics (QED), where each fermionic propagator is inversely
proportional to the particle mass and the contributions of Feynman diagrams
for (two-)photon exchange between two virtual pairs scale with $\mu^{-4}$
Greiner and Reinhardt (2009). In particular, the first non-linear term in the
effective Heisenberg-Euler Lagrangian Greiner and Reinhardt (2009); Schwinger
(1951); Dittrich (2014) describing interactions between quadratic field
fluctuations scales as $\mu^{-4}$. This fourth-order QED process is closely
connected to our Casimir model Karimpour  _et al._ (2022); Craig and
Thirunamachandran (1994); Salam (2009), for both Eq. (6) and Eq. (11). Thus,
we conclude that electrodynamic fields corresponding to different leptons
possess the same Casimir SIE density.

The extension of our work to fields other than the ones of charged leptons
would require accounting for propagators other than the electromagnetic one in
Eq. (1) and for the increase of the FSC at high energies Fritzsch (2002).
However, the conceptual idea underlying the presented calculations should be
equally applicable to arbitrary fields. In the current model, the energy
density of a given field depends on two partially compensating factors
determined by the characteristic length of quantum fluctuations $R_{\rm Th}$:
(i) the polarizability density that scales $\propto$ $R_{\rm Th}^{7}$, (ii)
the (pairwise) Casimir interaction that scales $\propto$ $R_{\rm Th}^{-7}$.
Ultimately, a scale-invariant value of SIE density equal to $\Lambda$ for any
quantum field would be the most elegant result and such outcome could arise
from a subtle compensation between the polarizability density and the Casimir
interaction potential.

To embed our work in the context of other models for the cosmological constant
Weinberg (1989), we note that there is a solid body of literature attempting
to connect the cosmological and gravitational constants to zero-point
fluctuations of quantum fields. These attempts go back to seminal
phenomenological proposals by Dicke Dicke (1957), Zel’dovich Zel’dovich (1968)
and Sakharov Sakharov (1967). However, calculations based on the direct
summation of zero-point energies of quantum fields overestimate the observed
cosmological constant by 40 to 120 orders of magnitude Weinberg (1989).
Interestingly, if instead of the quantum scaling law for polarizability in Eq.
(2) we use the classical scaling $\alpha_{e-/e+}\propto R_{e-/e+}^{3}$, the
resultant $\bar{E}_{\rm SIE}$ would overestimate $\Lambda$ by 39 orders of
magnitude, similar to the simplistic direct sum over zero-point energies
Hawking (1983). Modified approaches to calculation of zero-point energies have
been attempted in the past, for example by Puthoff Puthoff (1989, 2002), and
more recently by Leonhardt, who proposed to apply Lifshitz theory with
renormalization and obtained a more reasonable estimate for the energy density
of the electromagnetic vacuum field Leonhardt (2019, 2020).

Instead, the key conceptual advance in our work is to propose that one can
define non-interacting particle/antiparticle fluctuations, with the PDDT given
by ${\bm{\alpha}}_{0}(\mathbf{r},\mathbf{r}^{\prime},iu)$. These fluctuations
then interact through the electromagnetic field, yielding
${\bm{\alpha}}_{\lambda=1}(\mathbf{r},\mathbf{r}^{\prime},iu)$, whose coarse-
grained representation is given by the rather minute homogeneous
polarizability in Eq. (2) or the radial shell polarizability density in Eq.
(8). The Casimir interaction propagated by the respective gauge field then
yields a finite SIE density for any fermionic quantum field. It is pertinent
to draw a parallel between our model and the calculation of many-electron
correlation energies in molecules and materials based on Eq. (1) Dobson and
Gould (2012); Harl and Kresse (2009); Ren  _et al._ (2012). For interacting
electrons in matter, the definition of
${\bm{\alpha}}_{0}(\mathbf{r},\mathbf{r}^{\prime},iu)$ has a high degree of
arbitrariness, as it can be constructed from Hartree-Fock, Kohn-Sham, Wannier
orbitals, or alternatively using quantum harmonic oscillators to model the
response of valence electrons Hermann  _et al._ (2017); Woods  _et al._
(2016). Hence, ${\bm{\alpha}}_{0}(\mathbf{r},\mathbf{r}^{\prime},iu)$ or its
coarse-grained representations are unobservable. In contrast, the interacting
macroscopic polarizability tensor $\int_{\bf r}\int_{{\bf
r}^{\prime}}{\bm{\alpha}}_{\lambda=1}(\mathbf{r},\mathbf{r}^{\prime},iu)\,d{{\bf
r}^{\prime}}d{\bf r}$ is observable in practice. We are not aware of
experimental measurements of the intrinsic polarizability density of the
electron/positron field, however our work makes a verifiable prediction that
quantum fields possess a non-vanishing (albeit quite small) polarizability
density.

In summary, we have proposed a model for quantum electrodynamic fields endowed
with a finite polarizability density arising from zero-point fluctuations of
particle/antiparticle pairs. We then calculated the Casimir self-interaction
energy density of such quantum fields (owing to propagation of fluctuations by
the electromagnetic field) and obtained the expected dark-energy equation of
state and excellent numerical agreement with the measured value of the
cosmological constant $\Lambda$. Obviously, our work leaves many open
questions concerning the polarizability density of more general quantum fields
and its contribution to the vacuum self-interaction energy density. The
preliminary connections identified in this work between the Casimir approach
and the effective Lagrangian formalism in QED suggest a possibility to
calculate the self-interaction energy density directly from first principles
of quantum field theory.

The authors acknowledge financial support from the European Research Council
via the ERC Consolidator Grant “BeStMo(GA 725291)” and the Luxembourg National
Research Fund via the FNR CORE Jr project “PINTA(C17/MS/11686718)”. We thank
Matteo Gori (Uni.lu), Matthieu Sarkis (Uni.lu), Jorge Charry (Uni.lu),
Alejandro Rodriguez (Princeton), Lilia Woods (USF), Andrey Moskalenko (KAIST),
and John F. Dobson (Griffith) for valuable discussions. We also thank the
Anonymous Referee B for the insightful feedback that helped us improve our
presentation. A.T. dedicates this paper to a decade of friendship and
collaboration with Klaus-Robert Müller (TU Berlin).

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