# Machine learning application to Fermi-LAT data: sharpening all-sky map and
emphasizing variable sources

Shogo Sato  Faculty of Science and Engineering, Waseda University, 3-4-1
Ohkubo, Shinjuku, Tokyo, 169-8555, Japan  Jun Kataoka  Faculty of Science and
Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo, 169-8555, Japan
Soichiro Ito  Faculty of Science and Engineering, Waseda University, 3-4-1
Ohkubo, Shinjuku, Tokyo, 169-8555, Japan  Jun’ichi Kotoku  Teikyo University,
2-11-1 Kaga, Itabashi, Tokyo, 173-8605, Japan  Masato Taki  Rikkyo University,
3-34-1, Nishi-ikebukuro, Toshima, Tokyo, 171-8501, Japan  Asuka Oyama  Teikyo
University, 2-11-1 Kaga, Itabashi, Tokyo, 173-8605, Japan  Takaya Toyoda
Faculty of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku,
Tokyo, 169-8555, Japan  Yuki Nakamura  Faculty of Science and Engineering,
Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo, 169-8555, Japan  Marino
Yamamoto  Faculty of Science and Engineering, Waseda University, 3-4-1 Ohkubo,
Shinjuku, Tokyo, 169-8555, Japan

(Received December, 7, 2020; Accepted March 31, 2021)

###### Abstract

A novel application of machine-learning (ML) based image processing algorithms
is proposed to analyze an all-sky map (ASM) obtained using the Fermi Gamma-ray
Space Telescope. An attempt was made to simulate a one-year ASM from a short-
exposure ASM generated from one-week observation by applying three ML based
image processing algorithms: dictionary learning, U-net, and Noise2Noise.
Although the inference based on ML is less clear compared to standard
likelihood analysis, the quality of the ASM was generally improved. In
particular, the complicated diffuse emission associated with the galactic
plane was successfully reproduced only from one-week observation data to mimic
a ground truth (GT) generated from a one-year observation. Such ML algorithms
can be implemented relatively easily to provide sharper images without various
assumptions of emission models. In contrast, large deviations between
simulated ML maps and GT map were found, which are attributed to the
significant temporal variability of blazar-type active galactic nuclei (AGNs)
over a year. Thus, the proposed ML methods are viable not only to improve the
image quality of an ASM, but also to detect variable sources, such as AGNs,
algorithmically, i.e., without human bias. Moreover, we argue that this
approach is widely applicable to ASMs obtained by various other missions;
thus, it has the potential to examine giant structures and transient events,
both of which are rarely found in pointing observations.

methods: statistical — techniques: image processing

††journal: ApJ

##  1 Introduction

All-sky monitoring in astronomy has begun a new era in which continuous
monitoring of persistent sources as well as automated searches to discover new
transient is possible. Moreover, the all-sky monitor is a unique approach that
reveals the giant structures that are rarely seen in pointing observations. In
the 1990s, ROSAT provided the first precise all-X-ray sky map below 2 keV
(Snowden et al., 1997). The observation started in July 1990 and lasted for
approximately six months. The RXTE-ASM (Bradt et al., 1993) and MAXI onboard
the International Space Station (Matsuoka et al., 2009) carry one-dimensional
slat/shadow cameras extending the maximum energy window to 10 keV, which also
makes it possible to monitor the brightest X-ray sources on a daily or hourly
basis. More recently, eROSITA on the Spectrum Roentgen Gamma (Merloni et al.,
2012) observatory, launched in June 2019, has provided a new, sharp view of
hot and energetic processes taking place across the universe. For gamma rays,
the Fermi Gamma-ray Space Telescope was launched in 2008 (Atwood et al.,
2009). It can observe 30-MeV to 100-GeV gamma-ray sky with unprecedented
sensitivity. Owing to a large field of view of 2.4 str, the large-area
telescopes (LATs) onboard Fermi can monitor the sky every 3-hrs.

Along with the continuous monitoring of persistent sources, various missions
were planned for detecting transient sources, such as gamma-ray bursts (GRBs)
and/or flaring activity of active galactic nuclei (AGNs). For example, BATSE
onboard CGRO (Fishman et al., 1992) carried eight scintillation detectors to
cover the entire sky, localizing more than 2700 GRBs in the 10-year
observation. HETE2 (Kippen et al., 2003) and Swift (Gehrels et al., 2004) are
also missions dedicated to finding and providing quick localization of GRBs by
means of prompt follow-up X-ray and optical/UV observations. In general, these
all-sky observations provide a large amount of data; thus, it is not easy to
pick up interesting events or phenomena without overlooking some. Moreover,
when the exposures for specific fields are very short, detection of persistent
sources and/or transient events may suffer from low photon statistics,
especially when the background dominates the signal. Development of a new
protocol dedicated to the analysis of all-sky monitoring is required to
discriminate real and unreal events effectively and quickly without any human
bias.

In this context, various analytical methods have been developed for source
identification and modeling of foreground diffuse gamma-ray emission (DGE). In
DGE modeling, the cosmic-ray (CR) intensities and spectra are modeled by
considering their propagation in the galaxy (Ackermann et al., 2008, 2012) .
In contrast, Bayesian-inference-based analysis reconstructs flux contributions
to the diffuse and point-like sources (Guglielmetti et al., 2009). In
addition, the D3PO-inference algorithm reaches de-noising, de-convoluting, and
decomposing (Selig & Enßlin, 2015). For variable source detection, the Fermi
data are processed using automated science processing (ASP) in the LAT
Instrument Science Operations Center (ISOC) (Cameron, 2012). ASP searches for
flares by calculating the flux using maximum-likelihood fitting. Apparently,
this is the most accurate approach when sources on the catalog are filtered to
some extent; however, developing and installing such an algorithm is
challenging for individual users. In this context, aperture photometry light
curves (Lenain, 2017) and Fermi all-sky variability analysis (FAVA) (Ackermann
et al., 2013; Abdollahi et al., 2017) are simple and fast analytical methods
for variable source search. These methods are convenient; however, the quality
of the images is not improved.

As for the imaging analysis, significant progress has been made in machine
learning (ML), especially deep learning, to find feature values in images of
interest, for application in face recognition systems. In medical
applications, similar approaches have already been implemented to determine
lesion regions — for example, based on computed tomography images (Kalinovsky
et al., 2017). An obvious advantage of using ML is that it reads images in a
short time with high accuracy without any human bias in the most ideal cases.
However, similar ML techniques have been implemented only very recently in
astronomical research. For example, a new computer program, Morpheus, has been
developed to analyze astronomical image data pixel-by-pixel to identify and
classify all galaxies and stars (Hausen & Robertson, 2020). In addition,
spectral modeling of a type II supernova based on the ML approach is being
developed (Vogl, C. et al., 2020). Application of ML techniques to all-sky
survey data obtained by X-ray and gamma-ray satellite missions is also
crucial, not only to best allocate resources for the data analysis, but also
to examine new astronomical phenomena that have yet to be explored.

In this study, we proposed a new ML approach (1) to improve the quality of
low-statistic images and (2) find variable sources and possibly even transient
sources automatically from all-sky gamma-ray maps obtained with Fermi-LAT. In
particular, short observation data (one-week integration) were prepared as
input, and long- observation data (one-year integration) were predicted using
three independent ML algorithms — dictionary learning (Kreutz-Delgado et al.,
2003), U-net (Ronneberger et al., 2015), and Noise2Noise (Lehtinen et al.,
2018) — for detailed/quantitative comparison with actual observational data.
The remainder of this article is organized as follows. In Section 2, the
details of the Fermi-LAT analysis and data processing before the input ML
algorithms are presented. In Section 3, the architectures of the three ML
algorithms are described. The results of ML predictions are presented in
Section 4. In Section 5, all-sky maps (ASMs) generated by ML algorithms and
ground-truth (GT) map are compared in detail in selective regions to examine
the possible origin of the mismatch. In Section 6, the conclusions are
presented.

##  2 OBSERVATIONS AND ANALYSIS

###  2.1 Fermi-LAT observations

In this study, ML based algorithms were applied to a gamma-ray ASM observed by
Fermi-LAT. The first objective was to fabricate a high-quality ASM with
sufficient photon statistics such that the ASM uncertainties are small
compared to the counting uncertainties of shorter time-scale maps when only a
low-quality ASM is known and/or provided as an input. As discussed in section
3 , dictionary learning and U-net require the image pairs of short-observation
map and long-observation map for image prediction. Moreover, because the
Noise2Noise demands image pairs with different noise patterns, two types of
short-observation maps, corresponding to the first and second week in each
year, were prepared. Therefore, a long-term observation map and two types of
short-term observation maps in the energy range from 100 MeV to 300 GeV were
created at the beginning of each year for years 2009 to 2019’ In this study,
all energy values were incorporated into a single map. We produced the ASMs
using gamma rays in the P8R3_SOURCE event selection and the associated
P8R3_SOURCE_V2 instrument response functions (IRFs). In addition, to correct
the ASM for exposure, we calculated an exposure map with one energy bin using
gtexpcube2, which is the standard analysis tool provided with Fermi-LAT
ScienceTools. We also divided the ASM by the generated exposure map. In
addition, the zenith angle cut was set to 90∘ to remove gamma rays from the
Earth’s limb in the analysis. In this study, ASMs were produced in the
galactic coordinates in the Hammer-Aitoff projection. We generated count maps
with a resolution of 0.1 deg2 per pixel.

The ML algorithms assume that all sources are persistent over one year,
keeping their fluxes exactly comparable to what has been observed in the first
week. Thus, a large deviation of the ML maps and the GT map may suggest that
variable sources are included in regions of interest (ROIs). In such cases, a
light curve was generated by using binned analysis to investigate the flux
variability for specific gamma-ray sources. For each source, a 15∘ $\times$
15∘ ROI centered at its 4FGL catalog (Abdollahi et al., 2020) was chosen, and
a single power-law model was applied to the data. When creating the light
curve, the prefactor and the index of the central source were free, while the
parameters of the other sources within the ROI were fixed. As the background
model, the isotropic and diffuse backgrounds were modeled using
ISO_P8R3_SOURCE_V 2_v1.txt and gll_iem_v07.fits, respectively. Moreover, only
the flux bins with test static (TS) $>$ 9 (corresponding to $\sim$ 3$\sigma$)
were used for each light curve.

###  2.2 Preprocessing for ML

The following four processing steps were applied to each ASM obtained in
Section 2.1 before training the ML algorithms:

  1. 1.

Applying Gaussian blurring

  2. 2.

Applying the natural logarithm function

  3. 3.

Standardization of the pixel values in each map

  4. 4.

Data augmentation

First, Gaussian blurring was performed for each ASM to reduce the number of
pixels whose value equals zero. The blurring process is necessary to prevent
ML algorithms from learning only map smoothing. Subsequently, the natural
logarithm function was applied to these maps owing to the high variance of
pixel values; then, the pixel values were normalized from 0 to 1. Finally,
random sampling was performed for data augmentation. In this process, the same
regions were extracted from the ASM pairs with short-observation and long-
observation maps, and the extracted images were randomly flipped. The size of
the extracted images was 64 $\times$ 64 pixels, which corresponds to
$6^{\circ}\times 6^{\circ}$ in the galactic center. The random sampling was
repeated 2000 times for each ASM pair. Detailed information about the
components of the datasets is presented in Table 1.

Figure 1:  U-Net architecture with two downsampling layers: black arrows
denote shortcut path

Figure 2:  ML algorithms

architecture for Noise2Noise: the architecture of SRResNet (left) and residual
block (right) are shown

Figure 3:  Results of applying ML algorithms to ASM obtained with Fermi-LAT.
The input map, blurred map, and GT MAP are shown (top); ML maps generated by
each ML algorithm (dictionary learning, U-net, and Noise2Noise) are shown
(middle), and relative-ratio maps, which were calculated by dividing the GT
MAP by each ML map are also shown (bottom).

Hereinafter, “input map” and “GT map” refer to the ASMs, which cover the time
interval MJD 58486.034–-58493.026 (2009 January 3–9) and MJD
58486.034–-58857.048 (2009 January 2–-2020 January 9), respectively. The input
map was blurred with a 11 pixel $\times$ 11 pixel Gaussian kernel. In input
map, standard deviation, which defines the smoothing effect, equals to 3.0.
Additionally, we will use ”blurred map” to refer to blurred versions of the
ASM, which were blurred with a 15 pixel $\times$ 15 pixel Gaussian blur kernel
with a standard deviation equal to 100. “ML maps” refers to an ASM generated
from the input map using three ML based algorithms (dictionary learning, U-net
and Noise2Noise). We also note that the ML algorithm applied in this study
only suggests the source variability in the corresponding image pixels.
Therefore, detailed follow-up analysis such as drawing light curves of
candidate sources was necessary to check the variability, as described in
detail in Section 5.2.

Table 1: Components of datasets prepared for training and testing of ML processing | Training | Validation | Test
---|---|---|---
Sample number | 18000 | 2000 | 1
Observation year | 2009-2017 | 2018 | 2019
Extracted size(pixel) | $64\times 64$ | $64\times 64$ | $3600\times 1800$
Extracted size(degree) | $6^{\circ}\times 6^{\circ}$ | $6^{\circ}\times 6^{\circ}$ | $360^{\circ}\times 180^{\circ}$

##  3 PREDICTION ALGORITHMS

In this study, three ML algorithms were used for prediction: dictionary
learning, U-net, and Noise2Noise. The purpose and theory of these algorithms
are detailed in the following sections.

###  3.1 Dictionary learning

Dictionary learning is generally used for dimensional reduction in the field
of signal-processing (Tosic & Frossard, 2011). Unlike the deep-learning
algorithms, dictionary learning is based on linear algebra; hence, one can
sequentially trace the obtained results using this technique. Dictionary
learning involves three steps: preparing a pair of databases, creating
dictionaries, and generating images. First, five patches with 16 $\times$ 16
pixels were extracted randomly from each training sample of a short-
observation map. The patches were arranged in columns and aligned to create a
database of short-observation data ($A_{\mathrm{short}}$). In the same manner
as $A_{\mathrm{short}}$, a database of long-observation data
($A_{\mathrm{long}}$) was also created. Then, the arrays, called the
“dictionary,” were calculated. In this study, the dictionary from short-
observation data ($D_{\mathrm{short}}$) and long-observation data
($D_{\mathrm{long}}$) were created from $A_{\mathrm{short}}$ and
$A_{\mathrm{long}}$ by sparse representation. After the dictionaries were
calculated, the test samples from input map were represented by
$D_{\mathrm{short}}$ and sparse matrix $C$. Finally, $D_{\mathrm{short}}$ was
replaced with $D_{\mathrm{long}}$ to generate images corresponding to the GT
map. The details of the calculation in dictionary learning are provided
elsewhere (Sato et al., 2020). In this study, scikit-learn (version 0.21.2),
which is a library of ML algorithms used in Python, was used for dictionary
learning and sparse coding.

###  3.2 U-net architecture

Originally, a U-net was developed for masking the medical images (Han & Ye,
2018; Dong et al., 2017; Zhou et al., 2018). An example of the algorithm
architecture is illustrated in Figure 1. U-net consists of two paths: the
contracting path and expanding path. Two convolution layers and a max pooling
layer are repeated four times in the contracting path. In this study, 3
$\times$ 3 convolution was used with leaky rectified linear unit (ReLU)
activation and batch normalization. The number of channels was doubled for
each downsampling. The expanding path consists of two convolution layers and a
deconvolution layer with 3 $\times$ 3 filter sizes. The ReLU function was used
as the activation function after the convolution layers except the last one.
After the last convolutional layer, a hyperbolic tangent function was used. In
this study, the ”GeForce 1080Ti” graphics processing unit (GPU) processer was
employed, and the U-net was implemented with TensorFlow (version 2.1.0) and
Keras (version 2.3.1), which are libraries of deep-learning algorithms used in
Python.

###  3.3 Noise2Noise architecture

Noise2Noise, which is a semisupervised learning algorithms, is used to
reconstruct clean signals from corrupted ones (Wu et al., 2019; Hasan et al.,
2021; Díaz Baso, C. J. et al., 2019). This algorithms can perform signal
denoising without the need for clean signal datasets. Therefore, Noise2Noise
is applicable to cases where clean signals are difficult to obtain. The
superresolution residual network (SRResNet) (Ledig et al., 2017) used in this
study consists of a convolution layer with parametric rectified linear unit
(PReLU) activation, six layers of residual block, two layers of convolution
layer with batch normalization, and concatenated layer, as depicted in Figure
2 (left). Figure 2 (right) depicts the details of the residual block. The
residual block consists of a convolution layer, batch normalization, and PReLU
activation. In this study, the filter number and size in the convolution layer
were set as 64 and 3 $\times$ 3, respectively. The software and GPU processer
were used in the same way as for U-net in Section 3.2.

Figure 4:  Extracted part of input map, blurred map and GT map are shown
(top), and the same parts of ML maps are displayed (bottom) Figure 5:
Evaluation results in RMSE (left) and SSIM (right) based on the GT map (one-
year map). Maps with observation periods of one week, four weeks, eight weeks,
12 weeks, and 16 weeks were evaluated and plotted in blue. The result of
applying ML maps are also shown. Figure 6:  Example of 1D projection histogram
from input map, U-net map, and GT map. Extracted region (top) and 1D histogram
(bottom) are shown. Representative sources in 4FGL catalog whose average flux
are larger than $4\times 10^{-9}$ are indicated.

##  4 Results

###  4.1 Improvement of image quality

To confirm the improvement of maps in detail, close-ups of the various regions
in Figure 3 were extracted, as shown in Figure 4. Image noise resulting from
low photon statistics is reduced by ML algorithms; thus, the parts of the
point source are enhanced. Moreover, feature of the diffuse gas emission
around the galactic plane is also well reproduced.

To evaluate the image improvement in ML maps, we applied a root mean squared
error (RMSE), which is the standard evaluation index in computer vision. The
RMSE represents the squared error of each pixel value in the two images and
was calculated using the following equation:

$\mathrm{RMSE}=\sqrt{\frac{1}{n}\Sigma_{\mathrm{i}=1}^{\mathrm{n}}(V_{\mathrm{i}}^{\mathrm{image1}}-V_{\mathrm{i}}^{\mathrm{image2}})^{2}},$
(1)

where $\mathrm{n}$ and $V_{\mathrm{i}}$ denote the number of image pixels and
the $i$-th pixel value in each input image, respectively. An RMSE that is
closer to zero represents a smaller error between the input images. Similarly,
we also computed the structural similarity (SSIM) between the produced maps
and the GT map. The SSIM, which represents visual similarity, was calculated
using equation 2.

$\mathrm{SSIM}=\frac{(2\mu_{x}\mu_{y}+\mathrm{C}_{1})(2\sigma_{xy}+\mathrm{C}_{2})}{(\mu_{x}^{2}+\mu_{y}^{2}+\mathrm{C}_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+\mathrm{C}_{2}),},$
(2)

where $\mu$ and $\sigma$ denote the average value and standard deviation of
the image, respectively. $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ are small
constants for avoiding instability in case that $\mu_{x}^{2}+\mu_{y}^{2}$ or
$\sigma_{x}^{2}+\sigma_{y}^{2}$ is close to zero. Details of the constants are
described in Wang et al.(Wang et al., 2004). Figure 5 depicted the results of
RMSE and SSIM evaluation. In Figure 5, the horizontal axis represents the
observation time, and the vertical axis represents the RMSE or SSIM values. We
discussed the evaluation results in Section 5.3.

Figure 7:  Example of projection histogram and division: 1D projection (top)
and the division values obtained by dividing GT map by the values of ML maps
are shown (bottom), respectively. Orange circles show the large difference
between the 1D projection calculated from the ML maps and the projection from
GT map Figure 8:  Corresponding part of large division value (PKS2052-47):
input map, blurred map, and GT map are shown (top), the same parts of the ML
maps are displayed (bottom), and the part with a large difference is
surrounded by an orange circle

###  4.2 Comparing ML maps to GT map

Figure 9:  Negative case of projection histogram and division: the extracted part (top) and the 1D projection (middle) are shown, the division values obtained by dividing GT map by the values of ML maps are shown (bottom), and orange circles show the large difference between the 1D projection calculated from ML maps and the projection from GT map Figure 10:  Corresponding part of large division value (PKS1824-582): input map, blurred map, and GT map are shown (top), the same parts of the ML maps are displayed (bottom), and the part with a large difference is surrounded by an orange circle Table 2: List of source IDs and positions corresponding to the circled part in Figures 7 and 9 region | assoc4FGL | source name | source type | redshift | RA | Dec
---|---|---|---|---|---|---
|  |  |  |  | J2000 | J2000
region1 | 3C454.3 | J2253.9$+$1609 | AGN/FSRQ | 0.859 | 343.4906 | 16.1482
region2 | PKS2052-47 | J2056.2$-$4714 | AGN/FSRQ | 1.489 | 314.0715 |  $-$47.2369
region3 | PKS0454-234 | J0522.9$-$3628 | AGN/FSRQ | 1.003 | 74.2608 |  $-$23.4145
region4 | PKS1824-582 | J1829.2$-$5813 | AGN/FSRQ | 1.531 | 277.3108 |  $-$58.2323

For source detection, we compared the excess of pixel values in the ML maps
corresponding to the source position, as already listed/known in the 4FGL
catalog. Figure 6 (bottom) shows an example of a projection histogram, the
region of which is shown in top panel. In Figure 6, 15 representative sources
with the highest average flux are indicated. The signal-to-noise ratio (S/N)
in the projection was improved by ML algorithms. However, unlike input and GT
maps based on photon statistics, pixel values in the ML maps lose any physical
meaning; thus, we could not evaluate test statistics (TS), which is often
applied in the analysis of Fermi LAT to evaluate the significance of source
detection. We will develop such advanced algorithms for ML maps in the future;
however, it is beyond the scope of this paper.

To find the correspondence between the ML maps and GT map, the 1D projections
along the same galactic latitude $l$ were compared for various regions. An
example of a projection histogram obtained by GT map, U-net map, and
Noise2Noise map is depicted in Figure 7 (top). The extracted part is the same
as in Figure 6. The projection was created by integrating the pixel values for
15∘ in the direction of galactic latitude $l$. For a quantitative comparison,
the values in the ML maps were divided by those in the GT map. Figure 7
(bottom) indicates the ratio of the ML maps to the GT map. Three regions with
large differences are marked by orange circles. Figure 8 shows close-ups
around region 2 to illustrate such a difference. While no evident sources
exist in the input map and ML maps, a bright point source, which is identified
as PKS2052-47, is found only in the GT map. In the opposite case shown in
Figure 7, there exist regions in which the ratio is smaller than unity, as
represented by region 4 in Figure 9. Figure 10 shows close-ups around region
4. The details of regions with apparent ”mismatch” are listed in Table 2.

##  5 DISCUSSION

###  5.1 Comparing various ML algorithms

In this study, three types of ML algorithms were applied: dictionary learning,
U-net, and Noise2Noise. Dictionary learning is based on linear algebra and can
be used to trace the calculation process sequentially. However, dictionary
learning has a low representation capability compared with other deep-learning
algorithms and reflects a relatively low-quality image compared with other
algorithms presented in Figure 4. Thus, dictionary learning is less effective
than other deep-learning algorithms, for the purpose of this study. In the
U-net architecture, the input and output layers are connected by two paths: an
extended path and a shortcut connection. Since these two paths are considered
to capture detailed features and context, U-net can successfully predict long-
observation map from the input map with high precision (see Figures 4).
However, in Noise2Noise, the algorithm captures features of statistical noise
by comparing two types of short-observation maps. Therefore, the algorithm is
used to generate the ASM by neglecting the statistical noise, which can
sometimes overestimate the significance of each point source. Thus, U-net may
be superior to Noise2Noise for predicting a long-observation map. In this
study, U-net was considered to be the most reliable/precise for predicting the
long-observation map. Furthermore because these three ML algorithms are based
on completely different theories, ML maps are more reliable if the outputs
from each ML algorithm are similar to each other.

Figure 11:  Fermi-LAT light curve and variation of photon index for each
source shown in Table 2 over the time interval from January 2019 to December
2019 in one-week time bins. In each graph, only flux bins with test statistic
TS $>$ 9 is plotted. Because PKS2052-47 had too few photons and the TS did not
exceed 9, the data for the first two months from the beginning of 2019 were
used as the first bin. Similarly, the light curve for PKS1824-582 has few bins
with TS $>$ 9 throughout the year, so its light curve data were binned with
one-month intervals, while only the first bin was binned with one-week data
because the number of photons was sufficient for the first week. Top panel:
changes in the $100$MeV $<E<300$ GeV flux. The orange line shows the initial
one-week flux. Middle panel: changes in the power-law photon index. Bottom
panel: changes in the TS.

###  5.2 Origin of mismatch between GT map and ML maps

In this section, a possible reason for the mismatch between the GT map and ML
maps, as shown in Figures 7 and 9, is discussed. First, the regions of the
largest mismatch, indicated as orange circles in Figures 7 and 9, are
discussed. The ML maps are generated based on the hypothesis that the sources
are persistent over one year, which is, however, not the case for variable
and/or transient sources. Thus, the observed mismatch between the ML maps and
GT map is likely associated with the variable and transient sources, such as
the flaring activity of AGNs. To confirm this assumption, the sources listed
in Table 2 are analyzed in more detail with the Fermi analysis tool, as
described in section 2.1. The light curves around each source are shown in
Figure 11. For example, a variable source in region 1 corresponds to a flat
spectrum radio quasar (FSRQ) 3C454.3 with a redshift of $z$ = 0.859 (Table 2).
As shown in Figure 11, the flux of 3C454.3 in the first week was much smaller
than the average flux, shown as an orange line, integrated over a year.
Similarly, regions 2 and 3 were identified as two FSRQs; PKS2052-47 and
PKS0454-234, respectively. Contrary to the sources in which flux substantially
increased after the first week, the average flux over the year is smaller than
the first-week’s flux in the case of region 4, which corresponds to FSRQ
PKS1824-582. Obviously, such temporal variability in fluxes explains the
observed mismatch between the ML maps and GT map, as shown in Figures 7 and 9.
As can be seen in this figure’, point sources with increasing photon flux
appear observed to be brighter in Figure 3 (bottom). On the contrary, point
sources with decreasing flux appear faint.

However, there is a systematic difference between the GT map and ML maps
around the galactic longitude 270∘ $<$ $l$ $<$ 330∘ in Figure 7 (bottom),
which cannot be explained either by the statistical fluctuation or combination
of variable sources. We also confirm the difference in Figure 3 (bottom). Such
a systematic gradient is thought to be caused by the nonuniformity of the
exposures inherent in the one-week observation, which were used as input for
ML algorithms. Although exposure map correction is performed based on the
position and angle of the satellite, such nonuniformity affects the
reconstructed counting map, especially in the case of short observation times.
In fact, we confirmed that such a gradient is reduced if two-week ASM maps are
used as input, but the optimization of the time duration for the input maps is
beyond the scope of this study. Additionally, ASMs in this paper were produced
with no energy binning, hence the exposure map might not correct the actual
exposure. This is because the accuracy of integrating the exposure over the
energy band depends on the smoothness of the source spectrum across the energy
band, and these are quite large energy bands.

###  5.3 Quality of ML maps

As shown in Figure 5, the ML-based algorithms were effective in improving the
ASM quality. In particular, U-net was the best among the three ML algorithms.
The RMSE and SSIM assessment of the U-net map were equivalent to approximately
10 and 12 weeks, respectively. Considering that the exposure-map and variable
source were incorrect, the image quality was improved by more than 10 weeks.
Despite the significant improvement, faint source detection was considered
difficult for our algorithms due to image blur. Some peaks around the source
location were blurred, as shown in Figure 6. In general, for ML processing,
the loss function must be defined. “Training ML algorithms” refer to parameter
optimization, which could be used for minimizing the loss function. The RMSE,
which is one of the standard loss functions, was used in this study. However,
in the case of using MSE or RMSE, it is known that high-frequency components
are difficult to reproduce. Due to the difficulty of point-source detection,
some schemes are necessary to improve the detection efficiency. For example,
adopting the sum of point source detectability and RMSE as the loss function
may improve the performance. Other approaches are mentioned in Section 5.5.

###  5.4 Comparison with existing method

Compared with other techniques described in section1, our ML-based method has
the following three advantages:

  1. 1.

The ML algorithm can be easily implemented by end users. Moreover, any
assumption or modeling of foreground diffuse emission is unnecessary. Owing to
its simplicity, the ML all-sky map can be obtained within less than 10 min.

  2. 2.

Signal-to-noise on the image is improved both in terms of RMSE and SSIM,
although the optimum algorithms to detect faint point sources remain a future
work.

  3. 3.

By comparing the ML and GT maps, variable sources can be detected
automatically in ASM in either cases (increasing and/or decreasing) of fluxes.

Conversely, the ML method generally loses the physical properties of the
source images, such as the energy of each photon and the statistical
significance. Thus, one should consider such an ML-based image as a starting
point for finding possible interesting sources. These sources would be chosen
for subsequent detailed analysis based on the standard analysis tool released
by Fermi-LAT collaboration. Detailed modeling of DGE, as well as evaluation of
the test statistic (TS), is essential even after the application of ML methods
presented in this paper.

In this context, the D3PO inference algorithm, which is one of the latest
modeling techniques, demonstrated excellent noise reduction. However, analysis
with complex models is required, which is labor-intensive and computationally
expensive. Our ML-based method, on the contrary, is model-independent, and can
be completed by simple implementation, as mentioned above. While the method is
cost-effective, ML maps lose physical properties, such as energy spectrum.

Finally, we discussed the detection variable source in comparison to existing
methods. For source variability in the 4FGL catalog, the variability index,
such as $\mathrm{TS}_{\mathrm{var}}$ (Nolan et al., 2012), was calculated from
light curves using one-year bin data. For bright and variable sources, light
curves are automatically updated daily for cataloged or well-known sources. In
contrast, our ML methods have the potential to automatically detect
serendipitous variable sources. However, as mentioned earlier, our ML-based
technique loses any physical properties of the sources; hence, a detailed
analysis is necessary. In this context, FAVA, which is a simple and effective
method for detecting variable sources, was employed. However, as FAVA detects
the deviation between the map expected from long-term observations and the one
from short-term observations, the S/N does not change. Our algorithms improved
the S/N around the point source; hence, the variable source detection has the
potential to achieve higher accuracy than FAVA.

###  5.5 Future prospects

In this study, an initial one-week map obtained with Fermi-LAT in 2019 was
prepared as a test sample. However, the prediction method is applicable not
only to gamma-ray sky maps but also to other maps obtained in different energy
bands, such as radio, optical, and X-ray ASMs. Moreover, one can set any time
interval and duration depending on the research purpose, astrophysical
phenomenon, and source. For example, one can use a time scale that is as short
as minutes, to detect various transient sources, such as GRBs, without human
bias at any point by applying ML algorithms. Longer time scales of weeks to
months are more favorable for searching tidal disruption events (Burrows et
al., 2011; Gezari et al., 2012) and/or supernova.

In this research, a one-week map was used as input map to demonstrate the
concept and performance of various ML algorithms as applied to Fermi-LAT data.
In planned future work, an attempt will be made to find transient sources by
setting ASMs generated at different time scales, to widely cover scales of
minutes to days as input. Moreover, another interesting possibility is that
one can predict longer observation maps if the observations are continued,
even after the mission has been completed. For example, if a one-year ASM is
used as the input, it may be possible to predict a 100-year ASM from existing
data. Clearly, such predictions may or may not be correct, as shown in Section
5; however, it can provide strong motivation for future missions and important
scientific aspects to be explored in the future. Similar ML approaches have
been proposed to confirm the giant but weak emission structure of diffuse gas,
such as the Fermi bubble (Su et al., 2010) and NPS/Loop I (Kataoka et al.,
2018), which was proposed using a Gaussian mixture algorithms (Müller et al.,
2018).

For future research directions, three different approaches are promising. The
first is to train the ML algorithm separately for the point source and DGE
regions. In this study, although we used ASM data for training, the DGE and
point source regions were separated to train the ML algorithms. The second
approach is to apply other ML algorithms, especially deep-learning algorithms.
In this study, U-net and Noise2Noise were considered to be effective for
improving the ASM obtained by Fermi-LAT. However, it is necessary to search
other ML algorithms, such as the generative adversarial network (GAN) type or
Fourier convolutional neural network (FCNN)-type for more practical
applications. By using the FCNN-type algorithms, it may be possible to obtain
the features represented in the frequency domain. Another approach is to use a
large amount of data for training. In this study, only actual observational
data were prepared. The amount of training data can be easily increased using
simulation data.

##  6 CONCLUSION

A novel application of ML algorithms to analyze ASMs obtained with the Fermi-
LAT was demonstrated in this paper. An ASM corresponding to a long-
observation map was successfully generated from short- observation map by
using three types of ML algorithms. ML based methods make it possible to
improve image quality, not only for point sources, but also for diffuse gas
structures. Furthermore, these methods are useful for detecting variable
sources automatically without human bias by analyzing large deviations between
the simulated ML map and GT map. In addition, the method can be extended to
other satellite data, even if the satellite has completed its observations.
Thus, in future work, the proposed method will be applied to other satellite
data, such as eROSITA on SRG and/or BATSE onboard CGRO.

We thank an anonymous referee for his/her careful advice to improve the
manuscript. This research was supported by Japan Society for the Promotion of
Science (JSPS) KAKENHI [Grant Number 20H00669].

## References

  * Abdollahi et al. (2017) Abdollahi, S., Ackermann, M., Ajello, M., et al. 2017, ApJ, 846, 34, doi: 10.3847/1538-4357/aa8092
  * Abdollahi et al. (2020) Abdollahi, S., Acero, F., Ackermann, M., et al. 2020, The Astrophysical Journal Supplement Series, 247, 33, doi: 10.3847/1538-4365/ab6bcb
  * Ackermann et al. (2008) Ackermann, M., Jóhannesson, G., Digel, S., et al. 2008, AIP Conference Proceedings, 1085, 763, doi: 10.1063/1.3076791
  * Ackermann et al. (2012) Ackermann, M., Ajello, M., Atwood, W. B., et al. 2012, The Astrophysical Journal, 750, 3, doi: 10.1088/0004-637x/750/1/3
  * Ackermann et al. (2013) Ackermann, M., Ajello, M., Albert, A., et al. 2013, ApJ, 771, 57, doi: 10.1088/0004-637X/771/1/57
  * Atwood et al. (2009) Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, The Astrophysical Journal, 697, 1071, doi: 10.1088/0004-637X/697/2/1071
  * Bradt et al. (1993) Bradt, H. V., Rothschild, R. E., & Swank, J. H. 1993, Astronomy and Astrophysics Supplement Series (ISSN 0365-0138), 97, 355
  * Burrows et al. (2011) Burrows, D. N., Kennea, J. A., Ghisellini, G., et al. 2011, Nature, 476, 421–424, doi: 10.1038/nature10374
  * Cameron (2012) Cameron, R. A. 2012, in Observatory Operations: Strategies, Processes, and Systems IV, ed. A. B. Peck, R. L. Seaman, & F. Comeron, Vol. 8448, International Society for Optics and Photonics (SPIE), 459 – 468, doi: 10.1117/12.926550
  * Díaz Baso, C. J. et al. (2019) Díaz Baso, C. J., de la Cruz Rodríguez, J., & Danilovic, S. 2019, A&A, 629, A99, doi: 10.1051/0004-6361/201936069
  * Dong et al. (2017) Dong, H., Yang, G., Liu, F., Mo, Y., & Guo, Y. 2017, in Medical Image Understanding and Analysis, ed. M. Valdés Hernández & V. González-Castro (Cham: Springer International Publishing), 506–517
  * Fishman et al. (1992) Fishman, G., Meegan, C., Wilson, R., et al. 1992, Astronomy and Astrophysics Supplement Series, 97, 17
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, The Astrophysical Journal, 611, 1005, doi: 10.1086/422091
  * Gezari et al. (2012) Gezari, S., Chornock, R., Rest, A., et al. 2012, Nature, 485, 217–220, doi: 10.1038/nature10990
  * Guglielmetti et al. (2009) Guglielmetti, F., Fischer, R., & Dose, V. 2009, MNRAS, 396, doi: 10.1111/j.1365-2966.2009.14739.x
  * Han & Ye (2018) Han, Y., & Ye, J. C. 2018, IEEE Transactions on Medical Imaging, 37, 1418, doi: 10.1109/TMI.2018.2823768
  * Hasan et al. (2021) Hasan, A. M., Mohebbian, M. R., Wahid, K. A., & Babyn, P. 2021, IEEE Transactions on Radiation and Plasma Medical Sciences, 5, 235, doi: 10.1109/TRPMS.2020.3002178
  * Hausen & Robertson (2020) Hausen, R., & Robertson, B. E. 2020, The Astrophysical Journal Supplement Series, 248, 20, doi: 10.3847/1538-4365/ab8868
  * Kalinovsky et al. (2017) Kalinovsky, A., Liauchuk, V., & Tarasau, A. 2017, ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, XLII-2/W4, 13, doi: 10.5194/isprs-archives-XLII-2-W4-13-2017
  * Kataoka et al. (2018) Kataoka, J., Sofue, Y., Inoue, Y., et al. 2018, Galaxies, 6, 27, doi: 10.3390/galaxies6010027
  * Kippen et al. (2003) Kippen, R. M., Woods, P. M., Heise, J., et al. 2003, AIP Conference Proceedings, 662, 244, doi: 10.1063/1.1579349
  * Kreutz-Delgado et al. (2003) Kreutz-Delgado, K., Murray, J. F., Rao, B. D., et al. 2003, Neural Computation, 15, 349, doi: 10.1162/089976603762552951
  * Ledig et al. (2017) Ledig, C., Theis, L., Huszar, F., et al. 2017, Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network.  https://arxiv.org/abs/1609.04802
  * Lehtinen et al. (2018) Lehtinen, J., Munkberg, J., Hasselgren, J., et al. 2018, ArXiv, abs/1803.04189
  * Lenain (2017) Lenain, J.-P. 2017, FLaapLUC: Fermi-LAT automatic aperture photometry light curve.  http://ascl.net/1709.011
  * Matsuoka et al. (2009) Matsuoka, M., Kawasaki, K., Ueno, S., et al. 2009, Publications of the Astronomical Society of Japan, 61, 999, doi: 10.1093/pasj/61.5.999
  * Merloni et al. (2012) Merloni, A., Predehl, P., Becker, W., et al. 2012, arXiv: High Energy Astrophysical Phenomena
  * Müller et al. (2018) Müller, A., Hackstein, M., Greiner, M., et al. 2018, Astronomy and Astrophysics, 620, A64, doi: 10.1051/0004-6361/201833604
  * Nolan et al. (2012) Nolan, P. L., Abdo, A. A., Ackermann, M., et al. 2012, ApJS, 199, 31, doi: 10.1088/0067-0049/199/2/31
  * Ronneberger et al. (2015) Ronneberger, O., Fischer, P., & Brox, T. 2015, in Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, ed. N. Navab, J. Hornegger, W. M. Wells, & A. F. Frangi (Cham: Springer International Publishing), 234–241
  * Sato et al. (2020) Sato, S., Kataoka, J., Kotoku, J., et al. 2020, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 969, 164034, doi: https://doi.org/10.1016/j.nima.2020.164034
  * Selig & Enßlin (2015) Selig, M., & Enßlin, T. A. 2015, Astronomy and Astrophysics, 574, A74, doi: 10.1051/0004-6361/201323006
  * Snowden et al. (1997) Snowden, S. L., Egger, R., Freyberg, M. J., et al. 1997, The Astrophysical Journal, 485, 125, doi: 10.1086/304399
  * Su et al. (2010) Su, M., Slatyer, T. R., & Finkbeiner, D. P. 2010, The Astrophysical Journal, 724, 1044–1082, doi: 10.1088/0004-637x/724/2/1044
  * Tosic & Frossard (2011) Tosic, I., & Frossard, P. 2011, IEEE Signal Processing Magazine, 28, 27, doi: 10.1109/MSP.2010.939537
  * Vogl, C. et al. (2020) Vogl, C., Kerzendorf, W. E., Sim, S. A., et al. 2020, Astronomy and Astrophysics Supplement Series, 633, A88, doi: 10.1051/0004-6361/201936137
  * Wang et al. (2004) Wang, Z., Bovik, A. C., Sheikh, H. R., & Simoncelli, E. P. 2004, IEEE Transactions on Image Processing, 13, 600, doi: 10.1109/TIP.2003.819861
  * Wu et al. (2019) Wu, D., Gong, K., Kim, K., Li, X., & Li, Q. 2019, in Medical Image Computing and Computer Assisted Intervention – MICCAI 2019, ed. D. Shen, T. Liu, T. M. Peters, L. H. Staib, C. Essert, S. Zhou, P.-T. Yap, & A. Khan (Cham: Springer International Publishing), 741–749
  * Zhou et al. (2018) Zhou, Z., Rahman Siddiquee, M. M., Tajbakhsh, N., & Liang, J. 2018, in Deep Learning in Medical Image Analysis and Multimodal Learning for Clinical Decision Support, ed. D. Stoyanov, Z. Taylor, G. Carneiro, T. Syeda-Mahmood, A. Martel, L. Maier-Hein, J. M. R. Tavares, A. Bradley, J. P. Papa, V. Belagiannis, J. C. Nascimento, Z. Lu, S. Conjeti, M. Moradi, H. Greenspan, & A. Madabhushi (Cham: Springer International Publishing), 3–11


††thanks: Present address:Laboratoire des Solides Irradiés, CEA/DRF/lRAMIS,
Ecole Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau F-91128,
France

# Electronic thermal resistivity and quasi-particle collision cross-section in
semi-metals

Adrien Gourgout  Laboratoire de Physique et d’Étude des Matériaux (ESPCI Paris
- CNRS - Sorbonne Université)
PSL Research University, 75005 Paris, France  Arthur Marguerite  Laboratoire
de Physique et d’Étude des Matériaux (ESPCI Paris - CNRS - Sorbonne
Université)
PSL Research University, 75005 Paris, France  Benoît Fauqué  JEIP, USR 3573
CNRS, Collège de France, PSL Research University, 11, Place Marcelin
Berthelot, 75231 Paris Cedex 05, France  Kamran Behnia  Laboratoire de
Physique et d’Étude des Matériaux (ESPCI Paris - CNRS - Sorbonne Université)
PSL Research University, 75005 Paris, France

###### Abstract

Electron-electron collisions lead to a T-square component in the electrical
resistivity of Fermi liquids. The case of liquid 3He illustrates that the
thermal resitivity of a Fermi liquid has a T-square term, expressed in
m$\cdot$W-1. Its natural units are $\hbar/k_{F}E_{F}^{2}$. Here, we present a
high-resolution study of the thermal conductivity in bismuth, employing
magnetic field to extract the tiny electronic component of the total thermal
conductivity and resolving signals as small as $\approx 60\mu$K. We find that
the electronic thermal resistivity follows a T-square temperature dependence
with a prefactor twice larger than the electric T-square prefactor. Adding
this information to what has been known for other semi-metals, we find that
the prefactor of the T-square thermal resistivity scales with the square of
the inverse of the Fermi temperature, implying that the dimensionless fermion-
fermion collision cross-section is roughly proportional to the Fermi
wavelength, indicating that it is not simply set by the strength of the
Coulomb interaction.

##  I Introduction

The collision rate between two electrons, both confined to a thermal window at
the Fermi level, increases with the square of temperature, leading to an
electrical resistivity with quadratic temperature dependence. While this has
been known for many decades [1, 2], the microscopic mechanism by which these
collisions degrade conduction has not been unambiguously identified. The
observation of T-square resistivity in dilute metals with a Fermi surface too
small to allow Umklapp events [3, 4, 5] has initiated a regain of interest in
this issue [6, 7, 8, 9].

A related topic is the amplitude of $A$ in this expression for the electric
resitivity, $\rho$ of a Fermi liquid:

$\rho=\rho_{0}+AT^{2}$ (1)

Here, $\rho_{0}$, the residual resitivity is due to disorder and $AT^{2}$ is
due to e-e scattering. The Kadowaki-Woods (KW) scaling [10, 11, 12, 13, 14],
which links $A$ (in $\mu\Omega$$\cdot$cm$\cdot$K-2) to the Sommerfeld
coefficient (the T-linear electronic specific heat dubbed $\gamma$) works in
dense metals, where there is roughly one mobile electron per atom, but not in
dilute metals (where a mobile electron is shared by thousands of atoms) [4,
8].

A T-square temperature dependence in the resistivity of semi-metallic bismuth
was first observed in 1969 by Hartman [15], who argued that it arises due to
the Coulomb interaction between carriers belonging to different valleys. This
interpretation was subsequently contested by other authors [16, 17] who
proposed an electron-phonon scattering mechanism tailored to produce a
quadratic temperature dependence. A possible source of skepticism resided in
the amplitude of the T-square resitivity prefactor in bismuth ($A=1.2\times
10^{-8}\Omega$.cm.K-2), which was many orders of magnitude larger than was
observed in other metallic elements [10, 18, 8]. In Al, for example, it is
more than 4 orders of magnitude smaller ($A=5.3\times 10^{-13}\Omega$.cm.K-2)
[19]. Nevertheless, this large discrepancy is understandable nowadays [8].
Since the Fermi energy is much lower in bismuth than in Al, the e-e collision
phase space is much larger in the former than in the latter.

Figure 1: a) Thermal conductivity of bismuth as a function of temperature in
Bismuth. Red squares are taken from ref [20], purple triangles are from ref
[21] and black circles represent this work. The solid line is an exponential
fit to the data in the $4K<T<20K$ range. The inset shows the same data between
4K and plotted as ln($\kappa$) vs 1/$T$, demonstrating the exponential
behavior in this temperature range with the solid red line representing a
$\propto exp(-\Delta/k_{B}T)$ fit to the data. b) Thermal conductivity as a
function of magnetic field, applied along the trigonal axis, at $T$=4.5K. The
plateau induced by the magnetic field represents the phononic contribution.
The difference between the zero-field conductivity and this plateau is the
electronic contribution. c) The total thermal conductivity with the plateau
subtracted as a function of magnetic field, at different temperatures. With
increasing temperature, the electronic thermal conductivity decreases in
amplitude and its field dependence broadens.

In this context, scrutinizing electronic thermal conductivity, $\kappa_{e}$,
is insightful. The Wiedemann-Franz (WF) law establishes a correlation between
the amplitude of thermal and electrical conductivities by the Sommerfeld value
($L_{0}=\frac{\pi^{2}}{3}\frac{k_{B}^{2}}{e^{2}}=2.44\times
10^{-8}V^{2}K^{-2}$). Let us define the electronic thermal resistivity as
$WT=\frac{T}{L_{0}\kappa_{e}}$, which also displays a quadratic temperature
dependence:

$WT=(WT)_{0}+BT^{2}$ (2)

The first term is the counterpart of $\rho_{0}$. The strict validity of the WF
law at zero temperature implies that $(WT)_{0}=\rho_{0}$. On the other hand, a
departure from this law is expected at finite temperature leading to $B>A$
(Note that thanks to L0, one can express $A$ and $B$ in the same units).
Thermal transport studies have quantified $B$ in Ni [22], W [23, 24], Al [19],
UPt3 [25], CeRhIn5 [26], WP2 [27]), Sb [7], and most recently in dilute
metallic SrTi1-xNbxO3[9]. These studies confirmed the validity of the WF law
in the zero temperature ($(WT)_{0}=\rho_{0}$) and detected a departure from it
at finite temperature giving rise an inequality between $B$ and $A$. The
thermal prefactor has been found to be larger than the electric prefactor. In
other words, $B>A$, with the B/A ratio varying between 2 and 7 across
different solids. The existence of a bound to this ratio has been a subject of
theoretical debate [28].

Decades ago, MacDonald and Geldart [29] argued that because of the importance
of Umklapp events, thermal resistivity caused by e-e scattering is simpler to
understand than electric resistivity. Exaperimenatlly, in contrast, measuring
thermal resitivity is more difficult than electric resistivity.

A discussion of thermal transport in Fermi liquids brings us to consider the
case of normal liquid 3He [30, 31, 32, 33, 34, 35]. Its thermal conductivity
is proportional to the inverse of temperature at very low temperatures [36,
37]. Now, $\frac{T}{\kappa}\equiv WT$ and therefore the slope of $(\kappa
T)^{-1}$ as a function of $T^{2}$, is equivalent to $B$. It has been recently
noticed that the evolution of $(\kappa T)^{-1}$ with pressure in 3He follows
the scaling seen for the T-square resistivity with $E_{F}$ in metals and
implies that understanding the existence and the amplitude of T-square thermal
resistivity in a Fermi liquid, set by its Landau parameters, does not require
Umklapp events or multiple Fermi pockets [8].

Elemental bismuth has played a major role in the history of metal physics [38,
39]. Its thermal conductivity has been the subject of several studies [20, 21,
40, 41]. However, they had little to say about the temperature dependence of
the electron thermal resitivity, $WT$. This is not surprising, given the
dominance of the phononic component of the thermal conductivity, $\kappa_{ph}$
in this semi-metal, in which one mobile electron is shared by 105 atoms. Since
$\kappa_{ph}$ is several orders of magnitude larger than $\kappa_{e}$, one
needs a remarkably high resolution to extract the accurate temperature
dependence of $\kappa_{e}$ and $WT$.

In this paper, we present a study of thermal conductivity in bismuth. By
employing magnetic field as a tool for separating $\kappa_{ph}$ and
$\kappa_{e}$, we extract the latter and find that the electronic thermal
resistivity in bismuth follows a T-square temperature dependence and $B$ is
three time larger than $A$. In the second part of the paper, we compare the
amplitude of $B$ and $A$ in several semi-metals and find that they are both
proportional to the inverse of the square of the Fermi temperature. Then, we
will compare the data to what is known about thermal transport in other Fermi
liquids (liquid 3He and metals). The natural units of $(\kappa T)^{-1}$
expressed in m$\cdot$W-1, has $\frac{\hbar}{E_{F}}{k_{F}}$ for natural units.
It is $B$ without multiplication by L0),We show that the fermion-fermion
dimensionless cross-section, dubbed $\zeta$ [8], is less than unity in these
semi-metals. This contrasts with 3He and strongly-correlated metals in which
$\zeta\gg 1$. We also show that $\zeta$ steadily increases with carrier
density, a feature hiding behind the empirical success of the extended KW
scaling across various families of Fermi liquids [5].

Figure 2: Thermal (black squares) and electrical (red squares) resistivity as
a function of the temperature squared. Thermal resistivity is expressed in
units of $\mu\Omega$.cm using $L_{0}$ as a conversion factor. Solid lines are
linear fits to the data. Note the identical intercepts and the different
slopes.

##  II Results

The bismuth single crystal used in this study was obtained commercially from
Mateck. The crystal dimensions were $12\times 1.73\times 1.73mm^{3}$ with its
longer dimension along the bisectrix crystalline axis. Its residual
resistivity of 0.7 $\mu\Omega$.cm corresponds to a Room-temperature-Residual-
resistivity ratio (RRR) of $\approx$230\. This is lower than the RRR of larger
crystals obtained from the same source and used in recent transport studies of
bismuth [42, 43]. This size dependence of residual resistivity in bismuth
indicates that at least a subset of carriers in mm-sized crystals are
ballistic. Such a feature was also observed in other elemental semi-metals,
such as antimony [7].

Figure 1a) shows the temperature dependence of the total thermal conductivity,
$\kappa$tot in bismuth, at zero magnetic field, from room temperature down to
low temperature ($T$=0.1K) and up to 300K. As seen in the figure, our data is
close to what was previously reported by previous authors [20, 21]. We used a
standard one-heater-two-thermometers method for measuring thermal conductivity
in a dilution refrigerator and in a 4He cryostat (PPMS). We could not extract
reliable data in the intermediate temperature range between 1.2 K and 3 K.
This is the temperature range at which phonon thermal transport in bismuth
displays hydrodynamic features, such as Poiseuille flow [20] and second sound
[44]. The breakdown of the Fourier heat flow picture in this region and the
very large thermal conductivity of bismuth near this peak makes a reliable
investigation difficult. The absence of data in this temperature range does
not affect the results obtained and discussed in the present study.

Above $\approx 20$ K, the temperature dependence is roughly
$\kappa_{tot}\propto T^{-1}$. This is the so-called kinetic regime [45] where
the lattice thermal conductivity is dominated by phonon-phonon Umklapp
collisions [46]. The scattering time is longer than, but close to
$\hbar/k_{B}T$, a generic feature of phonon heat transport near the Debye
temperature [47, 48]. Below 20 K, the increase in thermal conductivity with
cooling becomes much faster than 1/$T$. As shown in the inset, in this
temperature range $ln\kappa_{total}$ is a linear function of $T^{-1}$,
indicating that thermal conductivity follows $\kappa_{total}\propto
exp(\frac{-\Delta}{T})$, with $\Delta=14.3$ K. This is the so-called Ziman
regime [45], where Umklapp collisions become rare. After peaking around 4 K,
thermal conductivity starts to decrease and at low temperature, it recovers a
T3 temperature dependence corresponding to the Casimir [45, 49] limit.

In order to separate the electronic and phononic contributions to the thermal
conductivity, we focused on the variation of $\kappa_{total}$ a function of
magnetic field following a procedure employed previously [50, 51, 7, 52].
Thanks to the extremely large magnetoresistance of Bi [53], the electronic
thermal conductivity is rapidly suppressed by a magnetic field which does not
affect the phononic thermal conductivity. This can be seen in Figure 1b, which
shows the magnetic field dependence of the thermal conductivity at 4.5 K.
Thermal conductivity is largest at zero magnetic field. The application of a
magnetic field of either polarity generates a plateau above a small field of
0.025 T. This plateau represents the phononic contribution to the total
thermal conductivity, $\kappa_{ph}$. The difference between the amplitude of
this plateau and the zero-field peak represents the electronic contribution
$\kappa$el at this temperature. Comparing the amplitude of the two components
($\kappa$e = 5 W/K.m and $\kappa$ph = 1176 W/K.m at 4.5 K), gives an idea of
the resolution required to separate the two components. We had to resolve a
very small variation of an already small temperature difference. The
temperature difference between the two thermometers measuring the temperature
gradient was about 10 mK, which dropped by 60 $\mu$K, when then plateau was
reached at 0.025 T.

Figure 1c) shows the evolution of the field dependence of
$\kappa_{e}=\kappa_{total}-\kappa_{ph}$ with temperature. The zero field peak
gets broader as the temperature increases, which is consistent with what is
expected given that warming, by reducing mobility, weakens orbital
magnetoresistance. Thus, this procedure allows us to extract $\kappa$e as a
function of temperature.

Figure 3: a) Temperature dependence of the phononic thermal conductivity in Bi
(black open squares) and Sb (red open squares). b) The electronic component of
the thermal conductivity in Bi (black open squares), Sb [7](red open squares)
and WP2 [27] (blue open squares). c) The average mobility, $\mu$, extracted
from $\kappa_{e}/T=L_{0}(n+p)e\mu$ of the three semi-metals as a function of
temperature. d) Normalized Lorenz number as a function of temperature in the
three semi-metals. The WF law is verified at low and high temperature and
there is a downward deviation at the intermediate temperature range.

Having quantified $\kappa_{e}$, we can now plot $WT=L_{0}\frac{T}{\kappa_{e}}$
and compare it with electrical resistivity, $\rho$. The result is shown in
Fig. 2, which presents the two resistivities, electrical and thermal, as a
function of $T^{2}$. One can see that both are roughly linear in $T^{2}$
dependence with an identical intercept but different slopes. To be more
specific, $A$ = 13.6 $n\Omega\cdot$cm$\cdot$K-2 and $B$ = 34.4
$n\Omega\cdot$cm$\cdot$K-2. This is our main new experimental result. Thus,
bismuth joins a club of materials in which a T-square thermal resistivity has
been detected with a prefactor larger than the prefactor of electrical
resistivity [8]. Fingers of both hands are sufficient for counting the members
of this club.

##  III Discussion

It is instructive to compare bismuth with other semi-metals with a much larger
carrier density. Figure 3 compares features of thermal transport in Bi with Sb
[7, 52] and WP2[27]. In Figure 3a), one sees a comparison of $\kappa$ph in Bi
and Sb (In WP2, thermal conductivity is dominated by $\kappa$e and $\kappa$ph
has not been extracted). In Bi and Sb, above 10 K, the amplitude and the
behavior of $\kappa$ph is similar. The most remarkable difference between the
two cases is the position of the peak, which occurs at a lower temperature in
bismuth, which has a lower Debye temperature.

System | n=p (cm-3) | A (n$\Omega$.cm/K2) | $(\kappa T)^{-1}(m/W)$ | B (n$\Omega$.cm/K2) | T${}_{F}^{fermiology}$ (K) | T${}_{F}^{n,\gamma}$ (K) |  $\gamma$ (mJ/mol/K2) | References
---|---|---|---|---|---|---|---|---
Bi | 3.0$\times 10^{17}$ | 12 | 0.014 | 34.4 | 109 - 272 | 339 | 0.0085 | [15, 39, 54]
Sb | 5.5$\times 10^{19}$ | 0.3 | $2.9\times 10^{-4}$ | 0.6 | 844 - 931 | 1292 | 0.112 | [7, 55, 39]
WTe2 | 6.8$\times 10^{19}$ | 4.5 | $4.5\times 10^{-3}$ | 11 | 250 - 500 | 484 | 6 | [56, 57, 58]
WP2 | 2.5$\times 10^{21}$ | 0.017 | $3\times 10^{-5}$ | 0.074 | 2130 - 6920 | 4109 | 2 | [59, 27]
Mo | 1.4$\times 10^{22}$ | 0.00148 | ? | ? | 2070 - 16620 | 13821 | 1.9 | [60, 61]
W | 2.5$\times 10^{22}$ | 8.7$\times 10^{-4}$ | $2.5\times 10^{-6}$ | 6$\times 10^{-3}$ | 2950 - 21940 | 30288 | 0.84 | [23, 60, 62]
Table 1: A comparison of semi-metals with an equal density of electrons and
holes ($n=p$) spanning over four orders of magnitude of $n$. The table lists
$n$, the electrical resistivity T2 prefactor ($A$), the thermal resistivity
prefactor ($(\kappa T)^{-1}$ and $B$, the latter is the former multiplied by
L0), the Fermi temperature (obtained by quantum oscillations or by the
combination of $\gamma$ and $n$) and the Sommerfeld coefficient. Figure 4: a)
The prefector of T-square electrical resitivity, $A$, as a function of the
Fermi temperature in different semimetals. b) Same plot for the prefactor of
T-square thermal resistivity, $B$ (in units of $n\Omega$.cm). In both plots,
the dashed line represents a T${}_{F}^{-2}$ slope. For each case, the Fermi
temperature has a minimum and a maximum value. They correspond to what was
extracted in fermiology studies from the cross sections of the Fermi pockets
and the effective masses found in the analysis of the period and the amplitude
of quantum oscillations.

Figure 3b) compares $\kappa_{e}$ in Bi, Sb and WP2. It shows that the
amplitude differs by three orders of magnitude, which is not surprising, given
the large difference in their carrier concentration. As one can see in table
1, the density of mobile electrons and holes is almost four orders of
magnitude larger in WP2 than in bismuth. In the case of Sb, $\kappa_{e}$
presents a sudden kink at $T\approx 20$ K. Below this temperature, scattering
by phonons ceases to be a source of momentum loss by electrons [52]. A similar
feature has been observed in WTe2 [57], whose carrier density is comparable to
antimony.

Another relevant quantity extracted from the data is the electronic mobility.
In the case of electrical conductivity, $\sigma$, mobility, $\mu$, is defined
by the equation $\sigma=ne\mu$, where $n$ is the carrier density. Analogously,
one can write for thermal transport: $\frac{\kappa_{el}}{T}=L_{0}ne\mu_{th}$.
Using the carrier density of each material (see table 1), Figure 3c) compares
the thermal mobility of carriers in the three semi-metals. One can see that
mobility is largest in Bi, followed by Sb and finally WP2, partially
compensates with a higher electronic mobility. The large mobility of carriers
in Bi, already known from electric resistivity measurements [15, 53] is a
consequence of the fact that long wavelength electrons are less scattered by
spatially confined disorder.

Figure 3c) shows the temperature dependence of the normalized Lorenz ratio
($L$/$L_{0}$) in the three semi-metals. The WF law is satisfied at both ends
and there is a downward deviation at intermediate temperatures. The ratio
becomes as low as 0.2 in Sb and in WP2 and 0.4 in Bi. The curves look similar
in the three cases, despite their absolute (thermal and electrical)
conductivities being orders of magnitude apart.

The quantification of the electronic thermal transport in bismuth permits us
to draw a global picture of the evolution of the T-square prefactors in four
elemental semi-metals (Bi, Sb, Mo, W) together with two newly discovered Weyl
semi-metals (WTe2 and WP2). Table 1 lists their T-square prefactors together
with their other physical properties. Since these metals have several Fermi
pockets with nontrivial morphologies, there is no unique Fermi temperature.
The table lists the results of two procedures. In the first, minimal and
maximal values of the Fermi temperature have been calculated by taking the
effective mass and the frequency of quantum oscillations obtained for each
Fermi pocket in fermiology studies. In the second, the average Fermi
temperature has been extracted using the one-band expression which links,
$\gamma$ to TF through the carrier density
($\gamma=2V_{m}\frac{\pi}{2}k_{B}\frac{n}{T_{F}}$), where $V_{m}$ is the molar
volume and the factor 2 is due to the presence of a hole density as large as
the electron one.

One can immediately notice that the ordinary KW scaling ($A\propto\gamma^{2}$)
does not operate here [5, 8]. To see this, it suffices to compare bismuth with
tungsten. While $A$ is four orders of magnitude larger in Bi, $\gamma$ is a
hundred times larger in W. This is not surprising. Given the wide variety of
their carrier density, one cannot employ the common version of the KW scaling
[8]. Indeed, the electronic specific heat depends both on carrier density and
the degeneracy temperature. For the same reason the $A/\gamma$ scaling fails
[63] in the case of dilute metallic strontium titanate.

On the other hand, an ‘extended’ version of the KW scaling, a proportionality
between $A$ and T${}_{F}^{-2}$, has been shown to be relevant to a wide
variety of Fermi liquids [4, 5]. Figure 4 shows that this is indeed the case
here. Panel a) is a log-log plot of $A$ in these semi-metals as a function of
their Fermi temperature (see also Figure 3 in ref. [8] and Fig.12 in ref. [64]
). Panel b shows the same for $B$. For both prefactors, the available data
scatters around a line representing T${}_{F}^{-2}$. The large extension of the
Fermi temperatures indicates that while this correlation is robust, it should
not be taken too seriously.

Since their conception decades ago, the slope seen in such plots has escaped a
quantitative explanation. The scaling plots by Rice [10] and by Kadowoki and
Woods [11] between $A$ and $\gamma^{2}$ yielded a quantity expressed in
$\Omega$.m.J-2.K4.mol2 and not easy to decipher. Plotting $A$ a s function of
Fermi temperature [4, 5] clarified the situation to some extent. Invoking the
quantum of conductance, could write $A$ as:

$A=\frac{\hbar}{e^{2}}\frac{k_{B}}{T_{F}^{2}}\ell_{quad}$ (3)

Taking this equation as a guide, the slope of $A$ vs. $T_{F}^{2}$ yields a
length scale, $\ell_{quad}$. In the case of elemental semi-metals, what was
noticed first by Rice [10] corresponds to $\ell_{quad}$=1.6nm. In the case of
strongly correlated metals, what was popularized by Kadowaki and Woods [11]
corresponds to $\ell_{quad}$=40 nm. Wang et al. showed that available data for
all Fermi liquids, suggest a link between $A$ and EF leading to a
$\ell_{quad}$ between these two extreme values [5]. Our data for both $A$ and
for $B$ is compatible with this observation. Both plots lead to an
$\ell_{quad}$ of the order of a few nanometers. It remains to figure out what
hides behind the amplitude of $\ell_{quad}$.

Figure 5: a) The prefactor of T-square thermal resistivity in units of mW-1 as
a function of Fermi temperature in semimetals. Also included are three
correlated metals(UPt3 [25, 65], CeRhIn5 [26, 66], and Sr2RuO4 [67, 68, 69]),
a dilute metal (Nb-doped strontium titanate [9, 70]), as well as the normal
liquid 3He under pressure [37]. b) The dimensionless fermion-fermion cross
section $\zeta$ in elemental semi-metals as a function of their wave-
vector.$\zeta$ remains below unity and displays a tendency to enhance with
increasing kF. As shown in the case of 3He and two other metals $\zeta$ can
exceed unity. The largest known $\zeta\approx 60$ occurs for 3He at the onset
of solidification.

The case of normal liquid 3He sheds light on this issue. Its thermal
conductivity is proportional to the inverse of temperatures as a result of
fermion-fermion scattering [36, 37] (but only at very very low temperatures
[35]). The amplitude of this residual $\kappa T$ term has been carefully
measured as a function of pressure up to the threshold of solidification [37].
This leads to the quantification of the fermion-fermion scattering time,
$\tau_{\kappa}$, which is linked to the Fermi energy by [8, 71]:

$\tau_{\kappa}=\frac{1}{\zeta}\frac{\hbar E_{F}}{(k_{B}T)^{2}}$ (4)

Here, $\zeta$ is a dimensionless parameter, which quantifies the strength of
collision. Often, it is assumed to be close to unity (See for example,
equation 13 of ref. [72], where it is assumed that $\zeta=\frac{\pi}{4}$).
Now, using the expression linking the thermal conductivity and the specific
heat per volume ($\kappa=\frac{1}{3}$Cv${}_{F}^{2}\tau_{\kappa}$), one finds:

$(\kappa T)^{-1}=\frac{9}{2}\frac{\hbar}{k_{F}E_{F}^{2}}\zeta$ (5)

Thus, the natural units of $(\kappa T)^{-1}$ is
$\frac{\hbar}{k_{F}E_{F}^{2}}$, a combination of the Fermi energy and the
Fermi wave-vector. Its amplitude is not merely set by the inverse of the
square of the Fermi energy $E_{F}^{-2}$, but also by the $\zeta/k_{F}$. This
ratio is sets the empirical length scale $\ell_{quad}$.

With these considerations in mind, we have plotted in Figure 5a the variation
of $(\kappa T)^{-1}$ (in units of W/m) as a function of their average Fermi
energy (in units of K), estimated from the carrier density and Sommerfeld
coefficient of each solid. For comparison, the figure also contains the data
for 3He, and a number of correlated metals as well as dilute metallic
strontium titanate. Clearly, the correlation seen for the weakly correlated
semimetals does not hold for other cases, indicating the non-universality of
$\zeta$ and implying that the collision strength is not identical across
various metallic families.

Figure 5b shows the extracted $\zeta$ as a function of the average Fermi
wavelength. The first feature to notice is that $\zeta$ in all semimetals is
of the order of unity or less, as one may expect. What is more mysterious is
the apparent correlation between $\zeta$ and $k_{F}$. We can now see that the
success of the ’extended’ Kadowaki-Woods scaling is due to a rough linearity
between $\zeta$ and $k_{F}$. At present, there is no solid understanding of
this feature. A larger Fermi radius may enhance the collision cross section in
momentum space, but it will reduce it in the real space by making the Fermi
wavelength smaller.

As seen in the figure, in strongly correlated metals and in 3He, $\zeta$ is
much larger than unity. The largest known $\zeta$ is attained by 3He abutting
solidification [8]. 3He atoms are known to interact through a refined version
of Lennard-Jones potential [73]. This is a short-range and, given the distance
between atoms, and attractive interaction. In contrast, the screened Coulomb
interaction between electrons is long-range and repulsive. It presumably
dominates in strongly correlated metals. However, electrons can also interact
attractively by exchanging phonons. Thanks to superconductivity, this is a
widely known, although not quantitatively domesticated, feature of electrons
in metals.

In this context, it is instructive to examine the case of strontium titanate
fits in this picture. The magnitude of the electric permittivity in this
quantum para-electric solid is four orders of magnitude larger than vacuum
permittivity [74]. Therefore, the screening of the Coulomb interaction is very
large. Nevertheless, as seen in Figure 5b, its $\zeta$ is larger than what is
found for semi-metals. This is one strong indication that what drives
electron-electron collision rate cannot be Coulomb interaction alone. Exchange
of phonons between electrons plays play a significant role in generating T2
thermal resistivity in Sb [52]. Theoretical considerations on the origin of
electron hydrodynamics in WTe2 [75] identified phonon exchange as a driver.
Interestingly, early theoretical attempts in understanding the amplitude of
T-square electric resitivity in Al [76] and in noble metals [77] indicated a
crucial role played by phonon exchange.

Quantifying the amplitude of e-e scattering and T2 resistivity in real metals
in which the electronic band structure, the pseudo-potential map and the
phonon spectrum are all known is a challenge to computational condensed matter
physics. We note a recent dynamical mean field theory study devoted to SrVO3
[78].

In summary, we measured thermal conductivity of elemental bismuth and
quantified its electronic and the phononic components. The Wiedemann-Franz is
obeyed at zero temperature, but a downward deviation appears at finite
temperature. By combining our result with what has been reported for other
semi-metals, we drew a picture of the evolution of the amplitude of the
prefactors of the quadratic resistivities. The dimensionless cross section of
fermion-fermion scattering seems to grow linearly with the Fermi wave-vector
and appears not set by Coulomb interaction.

##  IV Acknowledgements

We thank Mikhail Feigel’man, Yo Machida and Zengwei Zhu for stimulating
discussions. This work was supported by a grant from the Île de France Region.
KB acknowledges a stay in KITP, where this paper was partly written, supported
by the National Science Foundation under Grant Nos. NSF PHY-1748958 and NSF
PHY-2309135.

## References

  * Landau and Pomerantschuk [1936] L. Landau and I. Pomerantschuk, Über eigneschaften der metalle bei sehr niedrigen temperaturen, Phys. Z. Sowjet 10, 649 (1936).
  * Baber [1937] W. Baber, The contribution to the electrical resistance of metals from collisions between electrons, Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences 158, 383 (1937).
  * van der Marel  _et al._ [2011] D. van der Marel, J. L. M. van Mechelen, and I. I. Mazin, Common fermi-liquid origin of ${T}^{2}$ resistivity and superconductivity in $n$-type SrTiO3, Phys. Rev. B 84, 205111 (2011).
  * Lin  _et al._ [2015] X. Lin, B. Fauqué, and K. Behnia, Scalable $T^{2}$ resistivity in a small single-component fermi surface, Science 349, 945 (2015).
  * Wang  _et al._ [2020] J. Wang, J. Wu, T. Wang, Z. Xu, J. Wu, W. Hu, Z. Ren, S. Liu, K. Behnia, and X. Lin, T-square resistivity without umklapp scattering in dilute metallic Bi2O2Se, Nature Communications 11, 3846 (2020).
  * Kumar  _et al._ [2021] A. Kumar, V. I. Yudson, and D. L. Maslov, Quasiparticle and nonquasiparticle transport in doped quantum paraelectrics, Phys. Rev. Lett. 126, 076601 (2021).
  * Jaoui  _et al._ [2021] A. Jaoui, B. Fauqué, and K. Behnia, Thermal resistivity and hydrodynamics of the degenerate electron fluid in antimony, Nature Communications 12, 195 (2021).
  * Behnia [2022a] K. Behnia, On the origin and the amplitude of T-square resistivity in Fermi liquids, Annalen der Physik 534, 2100588 (2022a).
  * Jiang  _et al._ [2023] S. Jiang, B. Fauqué, and K. Behnia, $T$-square dependence of the electronic thermal resistivity of metallic strontium titanate, Phys. Rev. Lett. 131, 016301 (2023).
  * Rice [1968] M. Rice, Electron-electron scattering in transition metals, Physical Review Letters 20, 1439 (1968).
  * Kadowaki and Woods [1986] K. Kadowaki and S. Woods, Universal relationship of the resistivity and specific heat in heavy-fermion compounds, Solid state communications 58, 507 (1986).
  * Tsujii  _et al._ [2003] N. Tsujii, K. Yoshimura, and K. Kosuge, Deviation from the Kadowaki–Woods relation in Yb-based intermediate-valence systems, Journal of Physics: Condensed Matter 15, 1993 (2003).
  * E. Hussey [2005] N. E. Hussey, Non-generality of the Kadowaki–Woods ratio in correlated oxides, Journal of the Physical Society of Japan 74, 1107 (2005).
  * Jacko  _et al._ [2009] A. C. Jacko, J. O. Fjærestad, and B. J. Powell, A unified explanation of the Kadowaki–Woods ratio in strongly correlated metals, Nature Physics 5, 422 (2009).
  * Hartman [1969] R. Hartman, Temperature dependence of the low-field galvanomagnetic coefficients of bismuth, Phys. Rev. 181, 1070 (1969).
  * Uher and Pratt [1977] C. Uher and W. P. Pratt, High-precision, ultralow-temperature resistivity measurements on bismuth, Phys. Rev. Lett. 39, 491 (1977).
  * Kukkonen [1978] C. A. Kukkonen, ${T}^{2}$ electrical resistivity due to electron-phonon scattering on a small cylindrical fermi surface: Application to bismuth, Phys. Rev. B 18, 1849 (1978).
  * Uher [2004] C. Uher, Thermal conductivity of metals, in  _Thermal Conductivity: Theory, Properties, and Applications_ (Springer US, Boston, MA, 2004) pp. 21–91.
  * Garland and Van Harlingen [1978] J. Garland and D. Van Harlingen, Low-temperature electrical and thermal transport properties of pure aluminum, Journal of Physics F: Metal Physics 8, 117 (1978).
  * Kopylov and Mezhov-Deglin [1973] V. Kopylov and L. Mezhov-Deglin, Study of kinetic coefficients of bismuth at helium temperatures, Soviet Journal of Experimental and Theoretical Physics 65, 720 (1973).
  * Boxus  _et al._ [1981] J. Boxus, C. Uher, J. Heremans, and J. P. Issi, Size dependence of the transport properties of trigonal bismuth, Phys. Rev. B 23, 449 (1981).
  * White and Tainsh [1967] G. White and R. Tainsh, Electron scattering in nickel at low temperatures, Physical Review Letters 19, 165 (1967).
  * Wagner  _et al._ [1971] D. Wagner, J. Garland, and R. Bowers, Low-temperature electrical and thermal resistivities of tungsten, Physical Review B 3, 3141 (1971).
  * Trodahl [1973] H. J. Trodahl, The thermopower of pure tungsten below 9 K, Journal of Physics F: Metal Physics 3, 1972 (1973).
  * Lussier  _et al._ [1994] B. Lussier, B. Ellman, and L. Taillefer, Anisotropy of heat conduction in the heavy fermion superconductor UPt3, Physical Review Letters 73, 3294 (1994).
  * Paglione  _et al._ [2005] J. Paglione, M. Tanatar, D. Hawthorn, R. Hill, F. Ronning, M. Sutherland, L. Taillefer, C. Petrovic, and P. Canfield, Heat transport as a probe of electron scattering by spin fluctuations: The case of antiferromagnetic CeRhIn5, Physical Review Letters 94, 216602 (2005).
  * Jaoui  _et al._ [2018] A. Jaoui, B. Fauqué, C. W. Rischau, A. Subedi, C. Fu, J. Gooth, N. Kumar, V. Süß, D. L. Maslov, C. Felser, _et al._ , Departure from the wiedemann–franz law in WP2 driven by mismatch in T-square resistivity prefactors, npj Quantum Materials 3, 64 (2018).
  * Li and Maslov [2018] S. Li and D. L. Maslov, Lorentz ratio of a compensated metal, Phys. Rev. B 98, 245134 (2018).
  * MacDonald and Geldart [1980] A. H. MacDonald and D. J. W. Geldart, Electron-electron scattering and the thermal resistivity of simple metals, Journal of Physics F: Metal Physics 10, 677 (1980).
  * Abrikosov and Khalatnikov [1959] A. A. Abrikosov and I. M. Khalatnikov, The theory of a Fermi liquid (the properties of liquid 3He at low temperatures), Reports on Progress in Physics 22, 329 (1959).
  * Nozières and D. [1966] P. Nozières and P. D.,  _The Theory of Quantum Liquids_ (CRC Press, 1966).
  * Wolfle [1979] P. Wolfle, Low-temperature properties of liquid 3He, Reports on Progress in Physics 42, 269 (1979).
  * Vollhardt and Wolfle [1990] D. Vollhardt and P. Wolfle,  _The superfluid phases of helium 3_ (CRC Press, 1990).
  * Dobbs [2000] E. R. Dobbs,  _Helium Three_ (Oxford University Press, 2000).
  * Behnia and Trachenko [2024] K. Behnia and K. Trachenko, How heat propagates in liquid 3he, Nature Communications 15, 1771 (2024).
  * Wheatley [1975] J. C. Wheatley, Experimental properties of superfluid ${}^{3}\mathrm{He}$, Rev. Mod. Phys. 47, 415 (1975).
  * Greywall [1984] D. S. Greywall, Thermal conductivity of normal liquid 3He, Physical Review B 29, 4933 (1984).
  * Edelman [1976] V. S. Edelman, Electrons in bismuth, Advances in Physics 25, 555 (1976).
  * Issi [1979] J. Issi, Low temperature transport properties of the group v semimetals, Australian Journal of Physics 32, 585 (1979).
  * Pratt and Uher [1978] W. Pratt and C. Uher, Thermal conductivity of bismuth at ultralow temperatures, Physics Letters A 68, 74 (1978).
  * Behnia  _et al._ [2007] K. Behnia, M.-A. Méasson, and Y. Kopelevich, Nernst effect in semimetals: The effective mass and the figure of merit, Physical Review Letters 98, 076603 (2007).
  * Kang  _et al._ [2022] W. Kang, F. Spathelf, B. Fauqué, Y. Fuseya, and K. Behnia, Boundary conductance in macroscopic bismuth crystals, Nature Communications 13, 189 (2022).
  * Spathelf  _et al._ [2022] F. Spathelf, B. Fauqué, and K. Behnia, Magneto-seebeck effect in bismuth, Phys. Rev. B 105, 235116 (2022).
  * Narayanamurti and Dynes [1972] V. Narayanamurti and R. C. Dynes, Observation of second sound in bismuth, Phys. Rev. Lett. 28, 1461 (1972).
  * Beck  _et al._ [1974] H. Beck, P. F. Meier, and A. Thellung, Phonon hydrodynamics in solids, physica status solidi (a) 24, 11 (1974).
  * Berman [1976] R. Berman,  _Thermal Conduction in Solids_ , Oxford studies in physics (Clarendon Press, 1976).
  * Behnia and Kapitulnik [2019] K. Behnia and A. Kapitulnik, A lower bound to the thermal diffusivity of insulators, Journal of Physics: Condensed Matter 31, 405702 (2019).
  * Mousatov and Hartnoll [2020] C. H. Mousatov and S. A. Hartnoll, On the planckian bound for heat diffusion in insulators, Nature Physics 16, 579 (2020).
  * Casimir [1938] H. Casimir, Note on the conduction of heat in crystals, Physica 5, 495 (1938).
  * White and Woods [1958] G. K. White and S. B. Woods, The thermal and electrical resistivity of bismuth and antimony at low temperatures, The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics 3, 342 (1958).
  * Uher and Goldsmid [1974] C. Uher and H. J. Goldsmid, Separation of the electronic and lattice thermal conductivities in bismuth crystals, physica status solidi (b) 65, 765 (1974).
  * Jaoui  _et al._ [2022] A. Jaoui, A. Gourgout, G. Seyfarth, A. Subedi, T. Lorenz, B. Fauqué, and K. Behnia, Formation of an electron-phonon bifluid in bulk antimony, Phys. Rev. X 12, 031023 (2022).
  * Collaudin  _et al._ [2015] A. Collaudin, B. Fauqué, Y. Fuseya, W. Kang, and K. Behnia, Angle dependence of the orbital magnetoresistance in bismuth, Phys. Rev. X 5, 021022 (2015).
  * Zhu  _et al._ [2011] Z. Zhu, B. Fauqué, Y. Fuseya, and K. Behnia, Angle-resolved landau spectrum of electrons and holes in bismuth, Phys. Rev. B 84, 115137 (2011).
  * Fauqué  _et al._ [2018] B. Fauqué, X. Yang, W. Tabis, M. Shen, Z. Zhu, C. Proust, Y. Fuseya, and K. Behnia, Magnetoresistance of semimetals: The case of antimony, Phys. Rev. Mater. 2, 114201 (2018).
  * Zhu  _et al._ [2015] Z. Zhu, X. Lin, J. Liu, B. Fauqué, Q. Tao, C. Yang, Y. Shi, and K. Behnia, Quantum oscillations, thermoelectric coefficients, and the Fermi surface of semimetallic WTe2, Physical Review Letters 114, 176601 (2015).
  * Xie  _et al._ [2023] W. Xie, F. Yang, L. Xu, X. Li, Z. Zhu, and K. Behnia, Purity-dependent Lorenz number, electron hydrodynamics and electron-phonon coupling in WTe2 (2023), arXiv:2312.16178 [cond-mat.mes-hall] .
  * Callanan  _et al._ [1992] J. E. Callanan, G. Hope, R. D. Weir, and E. F. Westrum, Thermodynamic properties of tungsten ditelluride (WTe2) I. The preparation and lowtemperature heat capacity at temperatures from 6 K to 326 K, The Journal of Chemical Thermodynamics 24, 627 (1992).
  * Schönemann  _et al._ [2017] R. Schönemann, N. Aryal, Q. Zhou, Y.-C. Chiu, K.-W. Chen, T. J. Martin, G. T. McCandless, J. Y. Chan, E. Manousakis, and L. Balicas, Fermi surface of the Weyl type-II metallic candidate ${\mathrm{WP}}_{2}$, Phys. Rev. B 96, 121108 (2017).
  * Desai  _et al._ [1984] P. D. Desai, T. K. Chu, H. M. James, and C. Y. Ho, Electrical resistivity of selected elements, Journal of Physical and Chemical Reference Data 13, 1069 (1984).
  * Koelling  _et al._ [1974] D. D. Koelling, F. M. Mueller, A. J. Arko, and J. B. Ketterson, Fermi surface and electronic density of states of molybdenum, Phys. Rev. B 10, 4889 (1974).
  * Girvan  _et al._ [1968] R. Girvan, A. Gold, and R. Phillips, The de Haas-van Alphen effect and the fermi surface of tungsten, Journal of Physics and Chemistry of Solids 29, 1485 (1968).
  * McCalla  _et al._ [2019] E. McCalla, M. N. Gastiasoro, G. Cassuto, R. M. Fernandes, and C. Leighton, Low-temperature specific heat of doped $\mathrm{SrTi}{\mathrm{o}}_{3}$: Doping dependence of the effective mass and kadowaki-woods scaling violation, Phys. Rev. Mater. 3, 022001 (2019).
  * Nakajima  _et al._ [2024] Y. Nakajima, Y. Akahama, and Y. Machida, Thermoelectric response across the semiconductor-semimetal transition in black phosphorus, Phys. Rev. Res. 6, 013125 (2024).
  * Joynt and Taillefer [2002] R. Joynt and L. Taillefer, The superconducting phases of ${\mathrm{UPt}}_{3}$, Rev. Mod. Phys. 74, 235 (2002).
  * Hegger  _et al._ [2000] H. Hegger, C. Petrovic, E. G. Moshopoulou, M. F. Hundley, J. L. Sarrao, Z. Fisk, and J. D. Thompson, Pressure-induced superconductivity in quasi-2d ${\mathrm{CeRhIn}}_{5}$, Phys. Rev. Lett. 84, 4986 (2000).
  * Maeno  _et al._ [1997] Y. Maeno, K. Yoshida, H. Hashimoto, S. Nishizaki, S.-i. Ikeda, M. Nohara, T. Fujita, A. Mackenzie, N. Hussey, J. Bednorz, and F. Lichtenberg, Two-dimensional fermi liquid behavior of the superconductor Sr2RuO4, Journal of the Physical Society of Japan 66, 1405 (1997).
  * Mackenzie and Maeno [2003] A. P. Mackenzie and Y. Maeno, The superconductivity of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ and the physics of spin-triplet pairing, Rev. Mod. Phys. 75, 657 (2003).
  * Hassinger  _et al._ [2017] E. Hassinger, P. Bourgeois-Hope, H. Taniguchi, S. René de Cotret, G. Grissonnanche, M. S. Anwar, Y. Maeno, N. Doiron-Leyraud, and L. Taillefer, Vertical line nodes in the superconducting gap structure of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, Phys. Rev. X 7, 011032 (2017).
  * Collignon  _et al._ [2019] C. Collignon, X. Lin, C. W. Rischau, B. Fauqué, and K. Behnia, Metallicity and superconductivity in doped strontium titanate, Annual Review of Condensed Matter Physics 10, 25 (2019).
  * Behnia [2022b] K. Behnia, On the dynamic distinguishability of nodal quasi-particles in overdoped cuprates, SciPost Phys. 12, 200 (2022b).
  * Poduval  _et al._ [2023] P. P. Poduval, K. Laubscher, and S. Das Sarma, Apparent kondo effect in moiré transition metal dichalcogenide bilayers: Heavy fermions versus disorder, Phys. Rev. B 108, 085405 (2023).
  * Godfrin and Krotscheck [2022] H. Godfrin and E. Krotscheck, The dynamics of quantum fluids (2022), arXiv:2206.06039 [cond-mat.other] .
  * Müller and Burkard [1979] K. A. Müller and H. Burkard, SrTi${\mathrm{O}}_{3}$: An intrinsic quantum paraelectric below 4 K, Phys. Rev. B 19, 3593 (1979).
  * Vool  _et al._ [2021] U. Vool, A. Hamo, G. Varnavides, Y. Wang, T. X. Zhou, N. Kumar, Y. Dovzhenko, Z. Qiu, C. A. C. Garcia, A. T. Pierce, J. Gooth, P. Anikeeva, C. Felser, P. Narang, and A. Yacoby, Imaging phonon-mediated hydrodynamic flow in WTe2, Nature Physics 17, 1216 (2021).
  * MacDonald [1980] A. H. MacDonald, Electron-phonon enhancement of electron-electron scattering in al, Phys. Rev. Lett. 44, 489 (1980).
  * MacDonald  _et al._ [1981] A. H. MacDonald, R. Taylor, and D. J. W. Geldart, Umklapp electron-electron scattering and the low-temperature electrical resistivity of the alkali metals, Phys. Rev. B 23, 2718 (1981).
  * Abramovitch  _et al._ [2024] D. J. Abramovitch, J. Mravlje, J.-J. Zhou, A. Georges, and M. Bernardi, Respective roles of electron-phonon and electron-electron interactions in the transport and quasiparticle properties of srvo3 (2024), arXiv:2404.07772 [cond-mat.mtrl-sci] .