# Face Sketch Synthesis Style Similarity:
A New Structure Co-occurrence Texture Measure

Deng-Ping Fan1, ShengChuan Zhang2, Yu-Huan Wu1,
Ming-Ming Cheng1†, Bo Ren1, Rongrong Ji2 and Paul L Rosin3

(2018)

###### Abstract.

Existing face sketch synthesis (FSS) similarity measures are sensitive to
slight _image degradation_ (_e.g.,_ noise, blur). However, human perception of
the similarity of two sketches will consider both structure and texture as
essential factors and is not sensitive to slight (“pixel-level”) mismatches.
Consequently, the use of existing similarity measures can lead to better
algorithms receiving a lower score than worse algorithms. This unreliable
evaluation has significantly hindered the development of the FSS field. To
solve this problem, we propose a novel and robust style similarity measure
called Scoot-measure (Structure CO-Occurrence Texture Measure), which
simultaneously evaluates “block-level” spatial structure and co-occurrence
texture statistics. In addition, we further propose 4 new meta-measures and
create 2 new datasets to perform a comprehensive evaluation of several widely-
used FSS measures on two large databases. Experimental results demonstrate
that our measure not only provides a reliable evaluation but also achieves
significantly improved performance. Specifically, the study indicated a higher
degree (78.8%) of correlation between our measure and human judgment than the
best prior measure (58.6%). Our code will be made available.

Face Sketch Synthesis (FSS), Structure CO-Occurrence Texture, Scoot-measure,
Style Similarity

††copyright: rightsretained††doi: 10.475/123_4††isbn:
123-4567-24-567/08/06††conference: ACM Multimedia conference; October 2018;
Seoul, Korea††journalyear: 2018††price: 15.00††submissionid: 163-A12-B3

##  1\. Introduction

\begin{overpic}[width=433.62pt]{FirstExample.pdf} \put(82.0,100.0){$3^{rd}$}
\put(62.0,100.0){$2^{nd}$} \put(42.0,100.0){$1^{st}$} \end{overpic} Figure 1.
Inaccuracy of current evaluation measures. We compare the ranking of face
sketches synthesized by 3 face sketch synthesis (FSS) algorithms: GAN (Isola
et al., 2017), FCN (Zhang et al., 2015b), and MRF (Wang and Tang, 2009).
According to the human ranking (last row), the GAN ranks first, followed by
the FCN and MRF sketches. The GAN synthesizes the face of both structure and
texture most similarly, with respect to the sketch drawn by the artist. The
FCN captures the structure but lost lots of textures. The MRF almost
completely destroyed the structure of the face. However, the most common
measures (SSIM (Wang et al., 2004), FSIM (Zhang et al., 2011b), VIF (Sheikh
and Bovik, 2006), and GMSD (Xue et al., 2014)) fail to rank the sketches
correctly. Only our measure (third row) correctly ranked the results.

Please take a look at Fig. 1 in which you can see several synthesized face
sketches (GAN (Isola et al., 2017), FCN (Zhang et al., 2015b), MRF (Wang and
Tang, 2009)). Which one do you think is closer to the ground-truth (GT) sketch
drawn by the artist? While this comparison task seems trivial for humans, to
date the most common measures (_e.g.,_ FSIM (Zhang et al., 2011b), SSIM (Wang
et al., 2004), VIF (Sheikh and Bovik, 2006), and GMSD (Xue et al., 2014))
cannot reliably rank the algorithms according to their similarity to humans.
This is the problem that we address in this paper, and to solve it we propose
a novel measure that does much better than existing measures in terms of
ranking algorithm syntheses compared to the GT sketch.

Face sketch synthesis (FSS) has been gaining popularity in the design
community (_e.g.,_ face artistic styles synthesis) (Berger et al., 2013; Wang
et al., 2014) and is being used for digital entertainment (Gao et al., 2008;
Yu et al., 2012; Zhang et al., 2017b), and multimedia surveillance law
enforcement (_e.g.,_ sketch based face recognition) (Tang and Wang, 2004;
Best-Rowden et al., 2014). In such applications, the comparison of a
synthesized sketch against a GT sketch is crucial in evaluating the quality of
a face sketch synthesis algorithm.

\begin{overpic}[width=346.89731pt]{Framewrok.pdf} \put(-10.0,26.0){GT Sketch:}
\put(-10.0,8.0){Synthesis:} \end{overpic} Figure 2. The Scoot-measure
framework for FSS style similarity evaluation between GT sketch and synthesis.
We first quantize the two input sketches to a set of grades. Then, we divide
them into blocks to consider their spatial structure. Thirdly, we extract
their “block-level” co-occurrence texture statistics. Finally, the style
similarity can be computed.

To date, the most widely-used (Wang et al., 2018; Wang et al., 2017c; Peng et
al., 2016; Zhang et al., 2017a, 2015a, 2016a; Wang et al., 2017b; Zhang et
al., 2016b; Wang et al., 2014) evaluation measures are that of Structural
SIMilarity (SSIM) (Wang et al., 2004), Visual Information Fidelity (VIF)
(Sheikh and Bovik, 2006), Feature SIMilarity (FSIM) (Zhang et al., 2011b), and
Gradient Magnitude Similarity Deviation (GMSD) (Xue et al., 2014). These
measures were originally designed for pixel-level Image Quality Assessment
(IQA) which aims to detect types of image degradation such as Gaussian blur,
jpeg, and jpeg 2000 compression. Therefore these measures should be sensitive
to slight (_e.g.,_ pixel-level) change in image. Psychophysics (Zucker et al.,
1989) and prior works (_e.g.,_ style for lines of drawing) (Grabli et al.,
2004; Zucker, 2005; Freeman et al., 2003) indicate that style is an essential
factor in sketches. Note that human perception of the style similarity of two
sketches will consider both of structure and texture (Wang et al., 2016) as
the essential factor rather than being sensitive to “pixel-level” mismatches.
As a consequence, current FSS measures often provide unreliable evaluation due
to the different nature of their task. Consequently, a better algorithm may
receive a lower score than a worse algorithm (see Fig. 1). This unreliable
evaluation has significantly hindered the development of the FSS field.
However, designing a reliable style similarity measure is difficult. Ideally,
such a measure should:

  * •

be insensitive to slight mismatches since real-world sket-ches drawn by
artists do not exactly match each pixel to the original photos (Sec. 4.1, Sec.
4.2);

  * •

hold good holistic-content capture capability, and should for example score a
complete state-of-the-art (SOTA) sketch higher than the results of only
preserving light strokes (Sec. 4.3);

  * •

obtain a high correlation to human perception so that the sketch can be
directly used in various subjective applications, especially for law
enforcement and entertainment (Sec. 4.4).

As far as we know, no previous measures can meet all these goals
simultaneously and provide reliable evaluation results.

To this end, we propose a new and robust style similarity measure called
Scoot-measure (Structure CO-Occurrence Texture Measure), which simultaneously
evaluates “block-level” spatial structure and co-occurrence texture
statistics. We experimentally demonstrate that our measure offers a reliable
evaluation and achieves significantly improved performance. Specially, the
experiment indicated a higher degree (78.8%) of correlation between our
measure and the human judgment than the best prior measure (58.6%). Our main
contributions are:

1) We propose a simple and robust style similarity measure for face sketch
synthesis that provides a unified evaluation capturing both structure and
texture. We experimentally show that our measure is significantly better than
most of the current widely used measures and some alternatives including GLRLM
(Galloway, 1974), Gabor (Gabor, 1946), Canny (Canny, 1987), and Sobel (Sobel,
1990).

2) To assess the quality of style similarity measures, we further propose 4
new meta-measures based on 3 hypotheses and build two new human ranked face
sketches datasets. These two datasets contain 676, and 1888 synthesized face
sketches, respectively. We use the two datasets to examine the ranking
consistency between current measures and human judgment.

##  2\. Related Works

As discussed in the introduction, a reliable similarity evaluation measure in
FSS does not exist (Wang et al., 2016). Previous researchers just used the
standard measures in IQA to evaluate the similarity of FSS.

Wang et al.  (Wang et al., 2004) proposed the SSIM index. Since the structural
information reflects the object structure features in the scenes, SSIM
computes the structure similarity, the luminance and contrast comparison using
a sliding window on the local patch.

Sheikh and Bovik (Sheikh and Bovik, 2006) proposed the VIF measure which
evaluates the image quality by quantifying two kinds of information. One is
obtained via the human visual system (HVS) channel, with the input ground
truth and the output reference image information. The other is obtained via
the distortion channel, called distortion information. And the result is the
ratio of these two kinds of information.

Physiological and psychological studies found that the perceived features of
human vision show great consistency with the phase consistency of Fourier
series at different frequencies. Therefore, Zhang et al.  (Zhang et al.,
2011b) choose the phase congruency as the primary feature. Since phase
congruency is not affected by the contrast, which is a key point of image
quality, the image gradient magnitude is chosen as the second feature. Then
they proposed a low-level feature similarity measure called FSIM.

Recently, Xue et al.  (Xue et al., 2014) devised a simple measure named
gradient magnitude similarity deviation (GMSD), where the pixel-wise gradient
magnitude similarity is utilized to obtain image local quality, and the
standard deviation of the overall GMS map is calculated as the final image
quality index. Their measure achieves the SOTA performance compared with the
other measures in IQA.

However, all the above-mentioned measures were originally designed for IQA
which focuses on pixel-level image degradation, and so they are sensitive to
slight resizing (see Fig. 7) of the image. Although some of them consider
structure similarity, the structure is based on pixel-wise rather than pair-
wise measurements. By observing sketches we found that the pair-wise co-
occurrence texture is more suitable to capture the style similarity (also see
our discussion in Sec. 5) than the pixel-wise texture.

##  3\. Scoot measure

In this section, we introduce our novel measure to evaluate the style
similarity for FSS. The important property of our measure is that it captures
the pair-wise co-occurrence texture statistics in the “block-level” spatial
structure. As a result, our Scoot-measure works better than the current
widely-used measures. The framework is shown in Fig. 2.

\begin{overpic}[width=433.62pt]{PressureLevel.pdf} \put(3.0,0.0){(a) Outline}
\put(30.0,0.0){(b) Light} \put(55.0,0.0){(c) Middle} \put(82.0,0.0){(d) Dark}
\end{overpic} Figure 3. Creating various tones of the stroke by applying
different pressure on the tip. (a) is the outline of the sketch. Strokes are
added to this outline by perturbing the path of the tip to generate various
sketches (b, c, d). From left to right these stroke tones are light, middle
(in hair regions), dark.

###  3.1. Co-occurrence Texture

\begin{overpic}[width=368.57964pt]{PencilGrades.pdf} \put(2.0,14.0){9H}
\put(12.0,14.0){8H} \put(21.0,14.0){7H} \put(31.0,14.0){6H}
\put(41.0,14.0){5H} \put(52.0,14.0){4H} \put(63.0,14.0){3H}
\put(73.0,14.0){2H} \put(84.0,14.0){H} \par\put(95.0,14.0){F}
\put(2.0,1.0){HB} \put(13.0,1.0){B} \put(22.0,1.0){2B} \put(32.0,1.0){3B}
\put(42.0,1.0){4B} \put(53.0,1.0){5B} \put(63.0,1.0){6B} \put(73.0,1.0){7B}
\put(83.0,1.0){8B} \put(93.0,1.0){9B} \end{overpic} Figure 4. Pencil grades
and their strokes.

####  3.1.1. Motivation.

To uncover the secrets of face sketch and explore quantifiable factors in
style, we observed the basic principles of the sketch and noted that the
“graphite pencil grades” and the “pencil’s strokes” are the two fundamental
elements in the sketch.

_Graphite Pencil Grades._ In the European system, “H” & “B” stand for “hard” &
“soft” pencil, respectively. Fig. 4 illustrates the grade of graphite pencil
from 9H-9B. Sketch images are expressed through a limited medium (graphite
pencil) which provides no color. Illustrator Sylwia Bomba (2015) said that “if
you put your hand closer to the end of the pencil, you have _darker_ markings.
Gripping further up the pencil will result in _lighter_ markings.” In
addition, after a long period of practice, artists will form their own _fixed
pressure_ style. In other words, the marking of the stroke can be varied by
varying the pressure on the tip (Fig. 3 (c - d)). It is important to note that
different pressures on the tip will result in various types of marking which
is one of the quantifiable factors called gray tone.

_Pencil Strokes._ Because all of the sketches are generated by moving a tip on
the paper. Hence, different paths of the tip along the paper will create
various stroke shapes. One example is shown in Fig. 5; different spatial
distributions of the stroke have produced various textures (_e.g.,__sparse_ or
_dense_). Thus, the stroke tone is another quantifiable factor.

####  3.1.2. Analysis

In this section, we analyse the two quantifiable factors: gray tone and stroke
tone.

Gray tone. To reduce the effect of slight noise and over-sensitivity to
imperceptible gray tone gradient changes in sketches, intensity quantization
can be introduced during the evaluation of gray tone similarity. Inspired by
previous works (Clausi, 2002), we can quantize the input sketch $I$ to $N_{l}$
different grades to reduce the number of intensities to be considered:
$I^{\prime}=\Omega(I)$.

A typical example of such quantization is shown in Fig. 6. Human beings will
consistently rank (b) higher than (c) before and after quantizing the input
sketches when evaluating the style similarity. Although, the quantization
technology may introduces artifacts. However, our experiments (Sec. 5 of the
quantized Scoot measure) show that this process can reduce sensitivity to
minor intensity variations, thus achieving the best overall performance, and
also reducing the computational complexity.

\begin{overpic}[width=260.17464pt]{texture.pdf} \end{overpic} Figure 5. Using
stroke tones to indicate texture. The stroke textures used, from top to
bottom, are: “cross-hatching”, “stippling”. The stroke attributes from left to
right, are: sparse to dense. Images from (Winkenbach and Salesin, 1994).
\begin{overpic}[width=281.85034pt]{Hair.pdf} \put(3.0,0.0){(a) Sketch}
\put(38.0,0.0){(b) GAN} \put(76.0,0.0){(c) LR} \put(-24.0,65.0){original}
\put(-24.0,25.0){quantized} \end{overpic} Figure 6. Different stroke
attributes result in different styles. The first row contains the original
sketches. The second row contains the quantized sketches. The images of the
first column are the GT sketches, and the images of the second and third
columns are the synthesized sketches (GAN (Isola et al., 2017), LR (Wang et
al., 2017c)). Texture of the hair region in (b) is more similar than that in
(c) compared to the GT sketch in (a). Therefore, humans give (b) a higher
score than (c).

Stroke tone. Although we can efficiently capture the gray tone by quantizing
the input sketch to $N_{l}$ grades we notice that the stroke tone (_e.g.,_
dense & sparse) is also important to formulate the sketch style. Stroke tone
and gray tone are not independent concepts; rather, they bear an inextricable
relationship to one another. Both gray tone and stroke tone are innate
properties of the sketches. The gray tone is based on the varying strokes of
gray-scale in a sketch image, while the stroke tone is concerned with the
spatial (statistical) distribution of gray tones.

This is demonstrated in Fig. 6, which presents the ground-truth sketch (a) and
two other synthesis results (b) & (c). These two images are the results of
SOTA face synthesis algorithms. Intuitively, result (b) is better than (c),
since (b) preserves the texture of the hair and details in the face while (c)
presents a smooth result and has lost much of the sketch style. In other
words, (b) and (c) have different stroke patterns. The co-occurrence stroke
pattern of (b) has a higher similarity than the sketch in (c).

####  3.1.3. Implementation.

After confirming the two quantifiable factors, we start to describe our
implementation details. To simultaneously extract statistics about the “stroke
tone” and their relationship to the surrounding “gray tone”, we need to
characterize their _spatial interrelationships_. Theoretically, the sketches
are quite close to textures. As a consequence, our goal is to capture the _co-
occurrence texture_ in the sketch. Such co-occurrence properties help to
capture the patterns of the face sketch (especially in the hair region).
Similar to the general technology (Haralick et al., 1973), we built the co-
occurrence gray tones in a matrix at distance $d=(\Delta x,\Delta y)$ to
encode the dependency of the gray tones. Specifically, this matrix $M$ is
defined as:

(1) $M_{(i,j)|d}=\sum_{y=1}^{H}\sum_{x=1}^{W}\begin{cases}1,\mbox{ if
}I^{\prime}_{x,y}=i\ \mbox{and}\ I^{\prime}_{(x,y)+d}=j\\\ 0,\mbox{
otherwise}\end{cases},$

where $i$ and $j$ denote the gray value; $x$ and $y$ are the spatial positions
in the given quantized sketch $I^{\prime}$; $I^{\prime}_{x,y}$ denotes the
gray value of $I^{\prime}$ at position $(x,y)$; $W$ and $H$ are the width and
height of the sketch $I^{\prime}$, respectively.

To extract the style features we tested the three most widely-used (Haralick,
1979) statistics: Homogeneity, Contrast, and Energy.

Homogeneity reflects how much the texture changes in local regions, it will be
high if the gray tone of each pixel pair is similar. The homogeneity is
defined as:

(2) $h=\sum_{j=1}^{N_{l}}\sum_{i=1}^{N_{l}}{\frac{M_{(i,j)|d}}{1+|i-j|}},$

Contrast represents the difference between a pixel in $I^{\prime}$ and its
neighbor summed over the whole sketch. This reflects that a low-contrast
sketch is not characterized by low gray tones but rather by low spatial
frequencies. The contrast is highly correlated with spatial frequencies. The
contrast equals 0 for a constant tone sketch.

(3) $c=\sum_{j=1}^{N_{l}}\sum_{i=1}^{N_{l}}{|i-j|^{2}M_{(i,j)|d}}$

Energy measures textural uniformity. When only similar gray tones of pixels
occur in an sketch ($I^{\prime}$) patch, a few elements in $M$ will be close
to 1, while others will be close to 0. Energy will reach the maximum if there
is only one gray tone in a sketch ($I^{\prime}$) patch. Thus, high energy
corresponds to the sketch’s gray tone distribution having either a periodic or
constant form.

(4) $e=\sum_{j=1}^{N_{l}}\sum_{i=1}^{N_{l}}{(M_{(i,j)|d})^{2}}$

CE (Contrast and Energy) Feature. Our experiments (see Tab. 1) show that the
combination of two statistics Contrast and Energy achieves the best average
performance. In this work, we concatenate the two statistics as the CE feature
to represent the style of the input sketch $I^{\prime}_{s}$.

###  3.2. Spatial Structure

Note that the matrix $M$ (Eq. 1) we built is based at the “image-level”.
However, the weakness of this approach is that it only captures the global
statistics, and the structure of the sketch is ignored. Therefore, for
instance, it would be difficult to distinguish men and women with the same
hairstyle.

To holistically represent the spatial structure, we follow the spatial
envelope method (Oliva and Torralba, 2001) to capture the statistics from the
“block-level” _spatial structure_ in the sketch. Firstly, we divide the whole
sketch image into a $k\times k$ grid of $k^{2}$ blocks111 We set $k=4$ to
achieve the best performance in our all experiments.  before extracting the CE
feature. Our experiments demonstrate that the process can help to derive
content information. Second, we compute the co-occurrence matrix $M$ for all
blocks and normalize each matrix such that the sum of its components is 1.
Finally, we concatenate the statistics of all the $k^{2}$ blocks into a
$2k^{2}$ dimension vector $\overrightarrow{\Phi}(I^{\prime}_{s}|d).$

Note that each of the statistics is only based on a single direction (_e.g.,_
$90^{o}$, that is d = (0, 1)), since the direction of the spatial distribution
is also very important to capture the style such as “hair direction”, “the
direction of shadowing strokes”. To exploit this observation for efficiently
extracting the _stroke direction_ style, we compute the average feature
$\overrightarrow{\Psi}(I^{\prime}_{s})$ of four orientations222 Due to the
symmetry of the co-occurrence matrix $M(i,j)$, the statistical features in 4
orientations are actually equivalent to 8 neighbors direction at 1 distance.
Empirically, we set 4 orientations $d_{i}\in\\{(0,1),(-1,1),(-1,0),(-1,-1)\\}$
to achieve the best performance.  vectors to capture more directional
information:

(5)
$\overrightarrow{\Psi}(I^{\prime}_{s})=\frac{1}{4}\sum_{i=1}^{4}{\overrightarrow{\Phi}}(I^{\prime}_{s}|d_{i}),$

where $d_{i}$ denotes the $i$th direction and the average CE feature
$\overrightarrow{\Psi}(I^{\prime}_{s})\in\mathbb{R}^{2k^{2}}$.

###  3.3. Scoot measure

Having obtained the style feature vectors of the GT sketch $Y$ and synthesis
sketch $X$, we use the Euclidean distance between their feature vectors to
evaluate their style similarity. Thus, our style Scoot-measure333 In all of
our experiment, we set $N_{l}=6$ grades to achieve the best performance. can
be defined as:

(6)
$E_{s}=\frac{1}{1+\left\|\overrightarrow{\Psi}(X^{\prime}_{s})-\overrightarrow{\Psi}(Y^{\prime}_{s})\right\|_{2}},$

where $\left\|\cdot\right\|_{2}$ denotes $l_{2}$-norms.
$X^{\prime}_{s},Y^{\prime}_{s}$ denote the quantized $X_{s},Y_{s}$,
respectively. $E_{s}=1$ corresponds to identical style. Using this measure to
evaluate the three synthesized face sketches in Fig. 1, we can correctly rank
the sketches consistent with the human ranking.

##  4\. Experiment

In this section, we compare our Scoot-measure with 8 measures including 4
popular measures (FSIM, SSIM, VIF, GMSD) and 4 alternative baseline measures
(GLRLM (Galloway, 1974), Gabor (Gabor, 1946), Canny (Canny, 1987), Sobel
(Sobel, 1990)) which related with texture or edge. For the four baseline
measures, we only replace our CE feature (describe in Sec. 3.1.3) with these
features.

Meta-measure. To test the quality of our measure, we adopt the meta-measure
methodology. The meta-measures consist of assuming some plausible hypotheses
about the results and assessing how well each measure reflects these
hypotheses (Pont-Tuset and Marques, 2013; Fan et al., 2017). We design 4 novel
meta-measures based on our plausible hypotheses (Sec. 1) and the experiment
results are summarized in Tab. 2.

Dataset. All of the meta-measures were performed on 2 widely used databases:
CUFS (Wang and Tang, 2009) and CUFSF (Zhang et al., 2011a). The CUFS database
includes 606 face photo-sketch pairs consisting of 3 sub-sections: CUHK
Student (188 pairs), XM2VTS (295 pairs) and Purdue AR (123 pairs). The CUFSF
database contains 1194 pairs of persons from the FERET database (Phillips et
al., 2000). There is one face photo and one GT sketch drawn by the artist for
each person in both CUFS and CUFSF databases. Following (Wang et al., 2018),
we use 338 pairs (CUFS) and 944 pairs (CUFSF) test set images to conduct our
experiments.

\begin{overpic}[width=433.62pt]{RankingStability.pdf} \put(5.0,36.0){(a)
Photo} \put(32.0,36.0){(b) GT} \put(51.0,36.0){(c) Differenced}
\put(77.0,36.0){(d) downsized} \par\put(6.0,0.0){(e) GT} \put(28.0,0.0){(f)
Resized} \put(56.0,0.0){(g) GAN} \put(83.0,0.0){(h) LR} \end{overpic} Figure
7. Meta-measure 1. (a) is the original photo. (b) is the GT sketch drawn by
the artist. (c) is the boundary difference between (a) & (b). (d) is the
downsized image from (b). The ranking of an evaluation measure should be
insensitive to slight resizing of the GT. While GT (e) & 5 pixels scaled-down
GT (f) differ slightly, widely-used measures (_e.g.,_ VIF, SSIM, and GMSD)
switched the ranking order of the two synthesized sketches (g) & (h) when
using the different versions of GT referenced (e, f). In contrast, our Scoot
measure consistently ranked (g) higher than (h).
\begin{overpic}[width=411.93767pt]{MM1.pdf} \end{overpic} Figure 8. Meta-
measure 1 results: The lower the result is, the more stable an evaluation
measure is to slight resizing.

Sketch Synthesis Results. Sketch synthesis results were generated for each
test image using 10 SOTA models (FCN (Zhang et al., 2015b), GAN (Isola et al.,
2017), LLE (Liu et al., 2005), LR (Wang et al., 2017c), MRF (Wang and Tang,
2009), MWF (Zhou et al., 2012), RSLCR (Wang et al., 2018), SSD (Song et al.,
2014), DGFL (Zhu et al., 2017), BFSS (Wang et al., 2017a)).

###  4.1. Meta-measure 1: Stability to Slight Resizing

The first meta-measure specifies that the rankings of synthetic results should
not change much with slight changes in the GT sketch. Therefore, we perform a
slight 5 pixels downsizing of the GT by using nearest-neighbor interpolation.

Fig. 7 gives an example. The hair of the GT sketch in (b) drawn by the artist
has a slight size discrepancy compared to the photo (a). We observe that about
5 pixels deviation (Fig. 7 c) in the boundary is common. While the two
sketches (e) & (f) are almost identical, commonly-used measures including
SSIM, VIF, and GMSD switched the ranking of the two synthesized results (g, h)
when using (e) or (f). However, our measure consistently ranked (g) higher
than (h).

Here, we applied the $\theta=1-\rho$ (Best and Roberts, 1975) measure to test
the measure ranking stability before and after the GT downsizing was
performed. Fig. 8 and Tab. 2 show the results: the lower the result is, the
more stable an evaluation measure is to slightly downsizing. We can see a
significant ($\approx$ 88% and 92%) improvement over the existing SSIM, FSIM,
GMSD, and VIF measures in both the CUFS and CUFSF databases. These
improvements are mainly because our evaluation measure considers “block-level”
features rather than “pixel-level”.

###  4.2. Meta-measure 2: Rotation Sensitivity

\begin{overpic}[width=433.62pt]{rotate.pdf} \par\put(6.0,0.0){(a) GT}
\put(26.0,0.0){(b) Rotated GT} \put(56.0,0.0){(c) GAN} \put(80.0,0.0){(d) MWF}
\end{overpic} Figure 9. Meta-measure 2. The ranking of an evaluation measure
should be insensitive to slight rotation of the GT. While GT (a) and $5^{o}$
rotated GT (b) has the same style, all of the current measures switched the
ranking order of the two synthesized sketches (c) & (d), relying on the
different GT referenced. Oppositely, our Scoot measure consistently ranked (c)
higher than (d).

In real-wold situations, sketches drawn by artists may also have slight
rotations compared to the original photographs. Thus, our second meta-measure
verifies the sensitivity of GT rotation for the evaluation measure. We did a
slight counter-clockwise rotation ($5^{o}$) for each GT sketch. Fig. 9 shows
an example. When the GT (a) is switched to the slightly rotated GT (b), the
ranking results should not change much.

We got the ranking results for each measure by using GT sketches and slightly
rotated GT sketches separately and applied the same measure ($\theta$) as
meta-measure 1 mentioned to evaluate the rotation sensitivity. The sensitivity
results are shown in Fig. 10, Tab. 2. Thanks to our use of “block-level”
statistics, our measure again significantly outperforms the current measures
over the CUFS and CUFSF databases.

\begin{overpic}[width=411.93767pt]{MM2.pdf} \end{overpic} Figure 10. Meta-
measure 2 results: The lower the result is, the more stable an evaluation
measure is to slight rotation.

###  4.3. Meta-measure 3: Content Capture Capability

The third meta-measure describes that a good evaluation measure should not
only evaluate the style similarity but also can capture the content
similarity. Fig. 11 presents an example. We expect that an evaluation measure
should prefer the SOTA synthesized result over the light strokes444 To test
the third meta-measure, we use a simple threshold of gray scale (_e.g.,_ 170)
to separate the sketch (Fig. 11 GT) into darker strokes & lighter strokes. The
lighter strokes image loses the main texture features of the face (_e.g.,_
hair, eye, beard), resulting in an incomplete sketch.  results. To our
surprise, only our measure gives the correct order.

We use the _mean score_ of 10 SOTA synthesis algorithms (FCN, GAN, LLE, LR,
MRF, MWF, RSLCR, SSD, DGFL, BFSS). The mean score is robust to situations in
which a certain model generates a poor result. We recorded the number of times
the mean score of SOTA synthesized algorithms is higher than a light stroke’s
score. The results are shown in Fig. 12 & Tab. 2. In the CUFS database we can
see a great improvement over the most widely-used measures. A slight
improvement is also achieved for the CUFSF database.

\begin{overpic}[width=325.215pt]{ContentCaptureAbility.pdf} \put(10.0,0.0){(a)
GT} \put(37.0,0.0){(b) Synthesis} \put(75.0,0.0){(c) Light} \end{overpic}
Figure 11. Meta-measure 3. We use a simple threshold to separate the GT sketch
(a) into dark strokes and light strokes (c). From the style similarity
perspective, a good evaluation measure should prefer the SOTA synthesized
sketch (b) which contains the main texture and structure over the result only
contains light strokes and has lost the main facial features (c).
\begin{overpic}[width=411.93767pt]{MM3.pdf} \end{overpic} Figure 12. Meta-
measure 3 results: The higher the result, the stronger is the holistic content
capture capability. Our measure achieves the best performance in both the CUFS
and CUFSF databases.

The reason why other measures fail in this meta-measure is shown in Fig. 11.
In terms of pixel-level matching, it is obvious that the regions where dark
strokes are removed are totally different from the corresponding parts in (a).
But at other positions the pixels are identical to the GT. Previous measures
only consider “pixel-level” matching, and will rank the light strokes sketch
higher. However, the synthesized sketch (b) is better than the light one (c)
in terms of both style and content.

###  4.4. Meta-measure 4: Human Judgment

The fourth meta-measure specifies that the ranking result according to an
evaluation measure should agree with the human ranking. As far as we know,
there is no such human ranking sketch database publicly available.

Source images. As mentioned in Sec. 4, our source sketch synthesized results
are from the two widely-used datasets: CUFS and CUFSF and 10 SOTA sketch
synthesis algorithms. Thus, we have 3380 (10 $\times$ 338) and 9440 (10
$\times$ 944) synthesized results for CUFS and CUFSF, respectively.

Human judgment. For each photo of the CUFS and CUFSF databases, we have 10
sketches synthesized by 10 SOTA algorithms. Due to the similarity of the
synthesized results of some algorithms, it is difficult for viewers to judge
the ranking. Following (Wang et al., 2016), we trained the viewers to assess
synthesized sketch quality based on two criterion: texture similarity and
content similarity. To minimize the ambiguity of human ranking, we asked 5
viewers to select 2 synthesized sketches for each photo through the following
3 stage processes:

i) We let the first group of viewers select 4 out of 10 sketches for each
photo. The 4 sketches should consist of two good and two bad ones. Thus we are
left with 1352 (4 $\times$ 338) and 3776 (4 $\times$ 944) sketches for CUFS
and CUFSF, respectively.

ii) For the 4 sketches in each photo, the second group of viewers is further
asked to select 3 sketches for which they can rank them easily. Based on the
voting results of viewers, we pick out the 3 most selected sketches.

iii) The last group of viewers are asked to pick out a pair of sketches that
are most obvious to rank. Note that we have 5 volunteers involved in the whole
process for cross-checking the ranking. Finally, we create two new human
ranked datasets: CUFS-Ranked and CUFSF-Ranked (see the _supplementary_ for all
images in these two datasets) 555They include 776 & 1888 human ranked images,
respectively. The two datasets will be made publicly available.. Fig. 13 shows
the human ranked examples.

\begin{overpic}[width=433.62pt]{HumanRanking.pdf} \put(1.0,45.0){GT}
\put(1.0,25.0){$1^{st}$} \put(1.0,8.0){$2^{nd}$} \end{overpic} Figure 13.
Meta-measure 4. Sample images from our human ranked database. The first row is
the GT sketch, followed by the first and second ranked synthesis result.
Please refer the supplementary material for more images.
\begin{overpic}[width=411.93767pt]{MM4.pdf} \end{overpic} Figure 14. Meta-
measure 4 results: The higher the result, the more consistency an evaluation
measure is to human judgment. Our measure achieves the best performance in
both CUFS and CUFSF databases.

Here, we examined the the consistency between human ranking and evaluation
measure ranking. Results are given in Fig. 14, which shows a significant
(about 26.3%) improvement over the best prior measure in CUFS. This
improvement is due to our consideration of style similarity which human
perception considers as an essential factor when evaluating sketches.

##  5\. Discussion

In this section, we will discuss the combination of statistics in Sec. 5.1 and
other alternative features in Sec. 5.2. These discussions indicate that the
pair-wise “co-occurrence” properties are quite important for the style
similarity evaluation for FSS.

Table 1. Quantitative comparison of different combination of 3 statistical features on our 4 meta-measures. MM: meta-measure. The best balanced performance is highlighted in bold. These differences are all statistically significant at the $\alpha<0.05$ level. | CUFS | CUFSF
---|---|---
Measure | MM1 | MM2 | MM3 | MM4 | MM1 | MM2 | MM3 | MM4
$h$ | 0.007 | 0.005 | 61.5% | 77.5% | 0.003 | 0.003 | 79.1% | 77.8%
$e$ | 0.200 | 0.104 | 98.5% | 73.1% | 0.044 | 0.026 | 99.2% | 77.4%
$c$ | 0.010 | 0.007 | 54.4% | 74.6% | 0.009 | 0.006 | 64.7% | 73.4%
$c+h$ | 0.011 | 0.007 | 60.1% | 74.6% | 0.007 | 0.005 | 78.1% | 73.7%
c $+$ e | 0.037 | 0.025 | 95.9% | 76.3% | 0.012 | 0.008 | 97.5% | 78.8%
$h+e$ | 0.156 | 0.088 | 97.9% | 75.7% | 0.030 | 0.017 | 98.8% | 80.3%
$h+e+c$ | 0.034 | 0.024 | 95.9% | 76.3% | 0.011 | 0.008 | 97.4% | 78.7%

###  5.1. Statistical Feature Selection

According to Sec. 3, we considered three widely-used features of co-occurrence
matrices: homogeneity ($h$), contrast ($c$) and energy ($e$). In order to
achieve the best performance, we need to explore the best combination of these
statistical features. We have applied our four meta-measures to test the
performance of our measure using each single feature, each feature pair and
the combination of all three features.

The results are shown in Tab. 1. All possibilities ($h,e,c,c+e,h+e,c+h,h+e+c$)
perform well in MM4 (human judgment). $h$ and $c$ are insensitive to resizing
(MM1) and rotation (MM2), while they are not good at content capture (MM3).
$e$ is the opposite compared to $h$ and $c$. Thus, using a single feature is
not good. The results of combining two features show that if $h$ is combined
with $e$, the sensitivity to resizing and rotating will still be high, while
partially overcoming the weakness of $e$. The performance of $h+e+c$ shows no
improvement compared to the combination of $c+e$. Previous work in (Baraldi
and Parmiggiani, 1995) also found energy and contrast to be the most efficient
features for discriminating textural patterns. Therefore, we choose contrast
and energy as our final combination “CE” feature to extract the style
features.

Table 2. Quantitative comparison of different evaluation measures on our 4 meta-measures. MM: meta-measure. The top measure for each MM are shown in boldface. These differences are all statistically significant at the $\alpha<0.05$ level. | CUFS (3380 synthesized images) | CUFSF (9440 synthesized images)
---|---|---
Measure | MM1 | MM2 | MM3 | MM4 | MM1 | MM2 | MM3 | MM4
SSIM (Wang et al., 2004) | 0.162 | 0.086 | 81.4% | 37.3% | 0.073 | 0.074 | 97.4% | 36.8%
FSIM (Zhang et al., 2011b) | 0.268 | 0.123 | 14.2% | 50.0% | 0.151 | 0.058 | 32.4% | 37.5%
VIF (Sheikh and Bovik, 2006) | 0.322 | 0.236 | 43.5% | 44.1% | 0.111 | 0.150 | 22.2% | 52.8%
GMSD (Xue et al., 2014) | 0.417 | 0.210 | 21.9% | 42.6% | 0.259 | 0.132 | 63.6% | 58.6%
Scoot(CE) | 0.037 | 0.025 | 95.9% | 76.3% | 0.012 | 0.008 | 97.5% | 78.8%

###  5.2. Alternative Features

As described in Sec. 3, sketches are quite close to textures. There are many
other texture & edge-based features (_e.g.,_ GLRLM (Galloway, 1974), Gabor
(Gabor, 1946), Canny (Canny, 1987), Sobel (Sobel, 1990)). Here, we select the
most wide-used features as candidate alternatives to replace our “CE” feature.
For GLRLM, we select all five statistics mentioned in the original version.
Results are shown in Tab. 3. Gabor and GLRLM are texture features, while the
other two are edge-based. With all the texture features our measure provides a
high consistency with human ranking (MM4). However, GLRLM does perform well
according to MM1 & 2 & 3\. Gabor is reasonable in terms of MM1 & 2, but fails
in MM3. For edge-based features, Canny fails according to all meta-measures.
Sobel is very stable to slight resizing (MM1) or rotating (MM2), but fails to
capture content (MM3) and is not consistent with human judgment (MM4).

New insight. Note that the alternative features do not consider the pair-wise
“co-occurrence” texture properties (see Fig. 15), only our measure considers
this cue and provides the best performance compared to the other features.
Tab. 3 indicates that the pair-wise co-occurrence texture is suitable for FSS
evaluation.

Table 3. Quantitative comparison of different evaluation measures on our 4 meta-measures. MM: meta-measure. Scoot(CE)† indicates the non-quantized setting. The top measure for each MM are shown in boldface. These differences are all statistically significant at the $\alpha<0.05$ level. | CUFS | CUFSF
---|---|---
Measure | MM1 | MM2 | MM3 | MM4 | MM1 | MM2 | MM3 | MM4
Scoot(Canny) | 0.086 | 0.078 | 33.7% | 27.8% | 0.138 | 0.146 | 0.0% | 0.1%
Scoot(Sobel) | 0.040 | 0.037 | 0.0% | 32.8% | 0.048 | 0.044 | 0.0% | 52.6%
Scoot(GLRLM) | 0.111 | 0.111 | 18.6% | 73.7% | 0.125 | 0.079 | 64.6% | 68.0%
Scoot(Gabor) | 0.062 | 0.055 | 0.0% | 72.2% | 0.089 | 0.043 | 19.3% | 80.9%
Scoot(CE)† | 0.022 | 0.014 | 48.8% | 73.1% | 0.012 | 0.007 | 92.5% | 79.4%
Scoot(CE) | 0.037 | 0.025 | 95.9% | 76.3% | 0.012 | 0.008 | 97.5% | 78.8%

##  6\. Conclusion

In this paper, we summarized the widely-used image quality measures for face
sketch synthesis evaluation based on “pixel-level” degradation, and enumerated
their limitations. Then we focused on the basic principles of sketching and
proposed the use of structure co-occurrence texture as an alternative
motivating principle for the design of face sketch style similarity measures.
To amend the flaws of existing measures, we proposed the novel Scoot-measure
which simultaneously captures both “block-level” spatial structure and co-
occurrence texture. Also, we concluded that the pair-wise co-occurrence
properties are more important than pixel-level properties for style similarity
evaluation of face sketch synthesis images.

In addition, extensive experiments with 4 new meta-measures show that the
proposed measure provides a reliable evaluation and achieves the best
performance. Finally, we created two new human ranked face sketch datasets
(containing 776 images & 1888 images) to examine the correlation between
evaluation measures and human judgment. The study indicated a higher degree
(78.8%) of correlation between our measure and human judgment. The proposed
Scoot-measure is motivated from substantially different design principles, and
we see it as complementary to the existing approaches.

All in all, our measure provides new insights into face sketch synthesis
evaluation where current measures fail to truly examine the pros and cons of
synthesis algorithms. We encourage the community to consider this measure in
future model evaluations and comparisons.

\begin{overpic}[width=433.62pt]{Difference.pdf} \end{overpic} Figure 15. A toy
example illustrating the difference between our pair-wise measure (right) and
a pixel-wise measure (left).

Future works. Though our measure is only tested on face sketch datasets, it is
not limited to evaluating face sketch synthesis. We will also investigate the
potential to use the measure to describe styles in sketches and investigate
new face sketch synthesis models. To promote the development of this field,
our code and dataset will be made publicly available.

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# Anisotropic Thermal Transport in Superconductors with Coexisting Spin
Density Waves

Sean F Peterson  Sourav Sen Choudhury  Yves Idzerda  Department of Physics,
Montana State University, Bozeman, Montana 59717, USA

###### Abstract

Thermal conductivity measurements can provide key and experimentally
verifiable insight into the electronic transport of unconventional
superconductors. In this work, electronic thermal transport of two-dimensional
tight-binding metallic systems with coexisting $d$-wave superconducting (SC)
and antiferromagnetic spin density wave (SDW) orders with nesting vector
$\mathbf{Q}=(\pi/2,\pi/2)$ or $(\pi,0)$ are considered. The coexisting SC and
SDW orders are modelled at the mean-field level. Thermal conductivities are
numerically calculated within Boltzmann kinetic theory in the weak impurity
scattering (Born) limit. These SDW nesting vectors are chosen for their unique
property of reconstructing the Fermi surface (FS) parallel to $\mathbf{Q}$ and
preserving the metallic FS perpendicular to $\mathbf{Q}$. This leads to
anisotropic electronic thermal conductivities parallel and perpendicular to
$\mathbf{Q}$, which also depend on the presence or absence of additional
gapless excitations exclusive to the coexistence phase. It was found that the
$\mathbf{Q}=(\pi/2,\pi/2)$ and $(\pi,0)$ SDW systems exhibit equivalent
electron transport relative to $\mathbf{Q}$. These systems also had equivalent
electron transport when coexisting with a $d$-wave SC gap when
$\Delta_{\mathbf{k}}$ had the same symmetry class under translations of
$\mathbf{Q}$.

##  I Introduction

Among the most studied candidates for high temperature superconductors (SCs)
are the cupratesTaillefer (2010), iron pnictidesWen and Li (2011), and iron
chalcogenidesHsu et al. (2008). A common feature for these families of
materials is that they have quasi-two-dimensional sheets of transition metal
atoms (either Cu or Fe) in a square-lattice resulting in cylindrical Fermi
surfacesLebegue (2007),Singh and Du (2008) (FSs) that can be treated as two-
dimensional systems (since they are largely $k_{z}$-independent). Due to the
layered structure of these quasi-two-dimensional sheets, it is possible to
grow single superconducting layers on a substrate and study superconductivity
strictly in two-dimensionsYu et al. (2019),Ge et al. (2015),Bollinger and
Bozovic (2016). It is important to note that many of these high-$T_{c}$ SCs
are unconventional in nature.

Unconventional superconductors often have phase diagrams with multiple broken
symmetry phases which depend on material properties such as electron or hole
doping concentrationBasov and Chubukov (2011),Sachdev (2010). One of the more
common broken symmetry states that superconductivity can coexist with is an
antiferromagnetic (AF) state which couples quasiparticle states in different
parts of the Brillouin Zone by a nesting vector, ${\bf Q}$, forming a spin
density wave (SDW) stateLake et al. (2001),Demler et al. (2001),Harrison and
Sebastion (2009),Tan et al. (2013). While in this work only the interplay
between SDW and SC orders will be investigated, the SDW state will often be
preceded by a structural transition from a tetragonal to an
orthorhombicMcGuire et al. (2008) or a monoclinicde la Cruz et al. (2008)
lattice thus breaking the fourfold rotational symmetry ($C(4)$) of the
crystal. This structural transition can result in an Ising nematic phaseNei et
al. (2017) and its effects on electronic thermal transport in SC systems has
been previously discussedSen Choudhury et al. (2022). Including effects of a
structural transition with SDW and SC ordering is beyond the scope of this
work. While it is assumed here that the $C(4)$ symmetry of the underlying
structural square lattice is preserved, the magnetic structure imposed on the
lattice by the existence of striped AF ordering reduces the $C(4)$ symmetry of
the unit cellTaillefer (2010) to that of a twofold rotational symmetry
($C(2)$) in the magnetic cell. This broken symmetry is often reflected in the
transport properties of such materialsRan et al. (2009).

To better understand these unconventional SCs, thermal conductivity
measurements are an invaluable tool for probing the transport properties of
materialsShakeripour et al. (2021),Hashimoto et al. (2010). In normal metals,
the electronic thermal conductivity at low temperatures is dominated by
electron scattering off impurities, and results in a linear temperature
dependence which is well understood within the framework of semiclassical
transport theory based on the Boltzmann kinetic equationZiman (1960). In
conventional SCs the entire FS is gapped and the thermal conductivity is known
to decrease exponentiallyBardeen et al. (1959) as $T\rightarrow 0$. However in
unconventional SCs (such as $d$-wave SCs) the thermal conductivity is known to
have a linear $T$-dependence at low-$T$ in the limit of weak impurity
scattering, similar to a normal metal due to the existence of zero-energy
quasiparticle excitations (nodes) on the FSArfi and Pethick (1988),Graf et al.
(1996). The band topology in the vicinity of these nodes is of utmost
importance as it determines the quasiparticle velocities, which can
drastically change the transport properties of a material. For example, it has
been shownSen Choudhury and Vorontsov (2021) that two types of $d$-wave SC
($d_{xy}$ vs. $d_{x^{2}-y^{2}}$) have very different thermal conductivities on
tight-binding FSs due to different Fermi velocities and local densities of
states at the nodes. Electron transport within SDW materials was observedKim
et al. (2011) to follow suppressed Fermi liquid behavior, and as such has an
electronic thermal conductivity that is linear in $T$, but diminished from the
normal metallic state thermal conductivitySteckel et al. (2014),Chatterjee et
al. (2017).

In the cuprates, the superconducting gap is known to have $d$-wave
symmetryDamascelli et al. (2003). Superconductivity can be preceded by a
commensurate SDW order with nesting vector ${\bf Q}=(\pi,\pi)$Lake et al.
(2001),Demler et al. (2001). This reconstructs the metallic FS with
quasiparticle pockets located at the $M$ points in the BZMoon and Sachdev
(2009) while preserving the $d$-wave SC symmetry nodes which are the main
contributors to the transport properties in the clean limitChatterjee et al.
(2017). The $d$-wave SC state was also found to coexist with the ${\bf
Q}=(\pi,0)$ SDW state in the underdoped region of two-dimensional Hubbard
modelZheng et al. (2017), which is often used to model the cuprates. In thin
film cupratesYu et al. (2019),Bollinger and Bozovic (2016) the behavior of the
bulk phase was preserved in monolayers, including the high-$T_{c}$ value at
optimal doping, indicating that SC in the cuprates is inherently a two-
dimensional phenomena.

In FeAs-based iron pnictides, an unconventional superconducting gap may emerge
out of a commensurate SDW state with nesting vector ${\bf Q}=(\pi,0)$ (also
known as the AF2 state). This results in a striped AF which reduces the $C(4)$
symmetry of the crystal lattice to a magnetic cell with $C(2)$ symmetryZhao et
al. (2008). In such materials it has been shown that the DC electric
conductivity within the Drude model is highly anisotropic between the
conductivity parallel and perpendicular to ${\bf Q}$ (i.e.
$\sigma_{xx}\neq\sigma_{yy}$)Ran et al. (2009).

While SC often arises out of an AF state in the cuprates and iron pnictides,
iron chalcogenides lack AF orderingMizuguchi and Takano (2010) in bulk.
However, FeSe monolayers can exhibit SDW ordering when grown on substrates
that increase the spacing between Fe atoms due to epitaxial strainTan et al.
(2013). These strained FeSe monolayers have been measuredGe et al. (2015) to
have greatly enhanced transition temperatures ($T_{c}$), when compared to
those measured in bulkHsu et al. (2008). High-$T_{c}$ superconductivity in
these strained FeSe monolayers is likely due to the presence of SDWs in the
material enhancing the SC stateTan et al. (2013). In some iron chalcogenides,
the SDW nesting vector was found to be the commensurate nesting vector ${\bf
Q}=(\pi/2,\pi/2)$Qiu et al. (2009) (also known as the AF3 state).

In this work, the single band electronic transport properties of two cases are
considered. The first being a collinear commensurate SDW state with nesting
vector ${\bf Q}=(\pi/2,\pi/2)$ (AF3) coexisting with the $d$-wave singlet
pairing SC states: $d_{x^{2}-y^{2}}$ and $d_{xy}$. The second case is a
similar collinear commensurate SDW state, but with a nesting vector of ${\bf
Q}=(\pi,0)$ (AF2) coexisting with the same $d$-wave singlet SC pairing states.
The effects arising from the multiplicity of bands were not considered and
therefore orbital degrees of freedom were neglected. Thus the present analysis
is not directly applicable to iron-based SCs (where orbital mixing plays a
significant role), but is more relevant for cuprate SCs which can be
accurately modeled with a single band modelFoley et al. (2019),Xu et al.
(2022). Nevertheless, the analysis given below provides important insights
regarding the interplay of SC and SDW orders and their impact on thermal
transport properties of such systems, particularly in the coexistence phase.

##  II Model and Formalism

###  A  Hamiltonian

In this work the normal state metallic tight-binding Hamiltonian will be
considered

$H_{0}=\sum_{{\bf k},\sigma}\xi_{{\bf k}}c^{\dagger}_{{\bf k}\sigma}c_{{\bf
k}\sigma}$ (1)

Figure 1: Fermi surface nesting between tight-binding FSs (black curves) and
the FSs translated by ${\bf Q}$ (red curves) for SDW orders with the nesting
vector: (a) ${\bf Q}=(\pi/2,\pi/2)$ (b) and ${\bf Q}=(\pi,0)$ where the FS
reconstruction due to SDW ordering occurs in the gray regions and the normal
state FS is preserved in the white regions. These SDW states also result in
magnetic cells which are larger than the unit cells of the underlying lattice,
this results in the periodicity in ${\bf k}$-space being reduced from the
square FBZ to the reduced Brillouin Zone seen as the blue dashed curves.

with the 2D inversion-symmetric ($\xi_{{\bf k}}=\xi_{-{\bf k}}$) dispersion
relations

$\begin{split}\xi^{(1)}_{{\bf k}}&=\mu-t_{1}(\cos 2k_{x}+\cos
2k_{y})-t_{2}\cos 2k_{x}\cos 2k_{y}\\\ \xi^{(2)}_{{\bf k}}&=\mu-
t_{1}(\cos(k_{x}-k_{y})+\cos(k_{x}+k_{y}))\\\
&-t_{2}\cos(k_{x}-k_{y})\cos(k_{x}+k_{y})\end{split}$ (2)

where $\mu$ is the chemical potential, $t_{1}$ is the nearest neighbor
hopping, and $t_{2}$ is the next-nearest neighbor hopping on a two-dimensional
square lattice of spacing one (a = 1), all of which are in units of the Néel
temperature ($T_{N}$). The chemical potential was set to zero ($\mu=0$) and
the hopping parameters were set to $t_{1}=100T_{N}$ and $t_{2}=10T_{N}$ for
both dispersion relations. For $\xi^{(1)}_{{\bf k}}$ this results in the weak
metallic FS ($\xi_{\bf k}^{(1)}=0$) with quasiparticle pockets centered at the
$\Gamma$ and $M$ points in the first Brillouin Zone (FBZ) which can be seen as
the black curves in FIG. 1(a). For $\xi^{(2)}_{{\bf k}}$, this results in a
metallic FS ($\xi_{\bf k}^{(2)}=0$) with a quasiparticle pocket centered
around the $\Gamma$ point in the FBZ which can be seen as the black curve in
FIG. 1(b). The first dispersion relation, $\xi_{\bf k}^{(1)}$, represents a
system where the FS is translated by the SDW nesting vector ${\bf
Q}=(\pi/2,\pi/2)$ and overlaps with the original FS at the edge of the reduced
Brillouin Zone (RBZ) as can be seen in FIG. 1(a). The second dispersion
relation, $\xi_{\bf k}^{(2)}$, represents a SDW nesting vector ${\bf
Q}=(\pi,0)$ as can be seen in FIG. 1(b).

Figure 2: FSs reconstructed by SDW ordering (solid black curves) when
$M=2T_{N}$ for the nesting vectors (a) ${\bf Q}=(\pi/2,\pi/2)$ and (b) ${\bf
Q}=(\pi,0)$. The $d$-wave nodal lines are represented with dotted cyan and
orange lines for $d_{x^{2}-y^{2}}$ and $d_{xy}$ respectively. Cyan and orange
points represent the locations of the $d_{x^{2}-y^{2}}$ and $d_{xy}$ nodes
respectively when their nodal lines cross the reconstructed FS. Red X’s show
the locations of additional mixing nodes that occur only when the SC gap is
even under translations of ${\bf Q}$ ($\Delta_{{\bf k}+{\bf Q}}=\Delta_{\bf
k}$).

The emergence of SDW ordering in these metallic systems occurs below the Néel
temperature and is the result of a striped antiferromagnetic (AF) system. This
broken symmetry is also reflected in the shape of the RBZ which can be seen as
the blue dashed lines in FIG. 1. Due to this reduced rotational symmetry, the
$\xi_{\bf k}$ and $\xi_{{\bf k}+{\bf Q}}$ FSs only overlap in the direction
parallel to the nesting vector ${\bf Q}$, leading to a FS reconstruction
parallel to ${\bf Q}$ while preserving the normal state tight-binding FS
perpendicular to ${\bf Q}$. This reconstructed FS can be seen by the black
curve in FIG. 2.

To begin studying thermal transport in a system with coexisting SDW and SC
orders, the Hamiltonian is modelled at the mean-field level. The order
parameters were found self-consistently by a method outlined in a previous
workSen Choudhury and Vorontsov (2021) and superconductivity was assumed to
arise out of the SDW ordering ($T_{N}>T_{c}$). Here a collinear sinusoidal SDW
system was considered with a spatial magnetization of $\mathbf{m}({\bf
r})=2M\hat{z}\cos{{\bf Q}\cdot{\bf r}}$. SDW ordering couples electron states
of parallel spins whose momenta differ by the nesting vector ${\bf Q}$. For
the SC order, only singlet electron pairing is considered and $\Delta_{\bf
k}=\Delta\mathcal{Y}_{\bf k}$, where $\mathcal{Y}_{{\bf k}}$ is a basis
function compatible with the square symmetry inherent to the lattice. The
basis functions considered were: $d_{x^{2}-y^{2}}$ ($\mathcal{Y}_{\bf k}=\cos
k_{x}-\cos k_{y}$) and $d_{xy}$ ($\mathcal{Y}_{\bf k}=\sin k_{x}\sin k_{y}$)
and qualitative illustrations of these SC gap structures can be seen in FIG. 3
on the two metallic FSs.

The mean-field Hamiltonian for $T<T_{c}$ and $\Delta\neq 0$ can be written in
matrix form in the Nambu basis as:

$H^{(\sigma)}=\frac{1}{2}\sum_{\vec{k}}\hat{\Psi}_{\vec{k},\sigma}^{{\dagger}}\hat{\mathcal{H}}^{(\sigma)}_{\vec{k}}\hat{\Psi}_{\vec{k},\sigma}$
(3)

the spin-dependent Hamiltonian matrix in the regions of the RBZ where the FS
reconstructs due to SDW ordering (see gray shaded regions in FIG. 1) can be
written as:

$\hat{\mathcal{H}}_{{\bf k}}^{(\sigma)}=\begin{pmatrix}\xi_{{\bf
k}}&\sigma\Delta_{{\bf k}}&\sigma M&0\\\ \sigma\Delta_{{\bf k}}&-\xi_{{\bf
k}}&0&\sigma M\\\ \sigma M&0&\xi_{{\bf k}+{\bf Q}}&\sigma\Delta_{{\bf k}+{\bf
Q}}\\\ 0&\sigma M&\sigma\Delta_{{\bf k}+{\bf Q}}&-\xi_{{\bf k}+{\bf Q}}\\\
\end{pmatrix}$ (4)

and reduces to a pure SC Hamiltonian in regions where the normal state FS is
preserved (see unshaded regions in FIG. 1):

$\hat{\mathcal{H}}_{{\bf k}}^{(\sigma)}=\begin{pmatrix}\xi_{{\bf
k}}&\sigma\Delta_{{\bf k}}&0&0\\\ \sigma\Delta_{{\bf k}}&-\xi_{{\bf k}}&0&0\\\
0&0&\xi_{{\bf k}+{\bf Q}}&\sigma\Delta_{{\bf k}+{\bf Q}}\\\
0&0&\sigma\Delta_{{\bf k}+{\bf Q}}&-\xi_{{\bf k}+{\bf Q}}\\\ \end{pmatrix}$
(5)

where $\hat{\Psi}_{{\bf k},\sigma}^{{\dagger}}=(c_{{\bf
k}\sigma}^{{\dagger}},c_{-{\bf k}-\sigma},c_{{\bf k}+{\bf
Q}\sigma}^{{\dagger}},c_{-{\bf k}-{\bf Q}-\sigma})$ is the Nambu vector. The
eigenvalues of $\hat{\mathcal{H}}_{{\bf k}}$ in the shaded regions when SC and
SDW orders coexist are the quasiparticle energies $\pm E^{(1,2)}_{{\bf k}}$

$E^{(1)}_{{\bf k}}=\sqrt{\Gamma_{{\bf k}}+2\Lambda_{{\bf k}}},\hskip
18.06749ptE^{(2)}_{{\bf k}}=\sqrt{\Gamma_{{\bf k}}-2\Lambda_{{\bf k}}}$ (6)

$\displaystyle\Gamma_{{\bf k}}$ $\displaystyle=(\xi_{{\bf
k}}^{+})^{2}+(\xi_{{\bf k}}^{-})^{2}+(\Delta_{{\bf k}}^{+})^{2}+(\Delta_{{\bf
k}}^{-})^{2}+M^{2}$ $\displaystyle\Lambda_{{\bf k}}$
$\displaystyle=\sqrt{(\xi_{{\bf k}}^{+}\xi_{{\bf k}}^{-}+\Delta_{{\bf
k}}^{+}\Delta_{{\bf k}}^{-})^{2}+M^{2}((\xi_{{\bf k}}^{+})^{2}+(\Delta_{{\bf
k}}^{+})^{2})}$

where $\xi_{\bf k}^{\pm}=(\xi_{\bf k}\pm\xi_{{\bf k}+{\bf Q}})/2$ and
$\Delta_{\bf k}^{\pm}=(\Delta_{\bf k}\pm\Delta_{{\bf k}+{\bf Q}})/2$. In the
regions where the normal state FS is preserved perpendicular to ${\bf Q}$, the
eigenvalues reduce to the typical pure SC eigenvalues:

$E_{\bf k}^{(1)}=\sqrt{\xi_{\bf k}^{2}+\Delta_{\bf k}^{2}},\hskip
18.06749ptE_{\bf k}^{(2)}=\sqrt{\xi_{{\bf k}+{\bf Q}}^{2}+\Delta_{{\bf k}+{\bf
Q}}^{2}}.$ (7)

When $T>T_{c}$ and $\Delta=0$ the eigenvalues in the regions where the
$\xi_{\bf k}$ and $\xi_{{\bf k}+{\bf Q}}$ FSs overlap and the FS reconstructs
reduce to the pure SDW eigenvalues:

$\begin{split}E_{\bf k}^{(1)}=E_{\bf k}^{(\alpha)}=\xi_{\bf
k}^{+}+\sqrt{(\xi_{\bf k}^{-})^{2}+M^{2}}\\\ E_{\bf k}^{(2)}=E_{\bf
k}^{(\beta)}=\xi_{\bf k}^{+}-\sqrt{(\xi_{\bf k}^{-})^{2}+M^{2}}.\end{split}$
(8)

The eigenvalues in the region where the normal state FS is preserved reduce to
$E^{(1)}_{\bf k}=\xi_{\bf k}$ and $E^{(2)}_{\bf k}=\xi_{{\bf k}+{\bf Q}}$.
Furthermore, the FS of this system is reconstructed from the black curves in
FIG. 1 to the black curves in FIG. 2 when $T<T_{N}$ where it can be seen that
sections of the FS parallel to ${\bf Q}$ become gapped by the SDW order.

###  B  Symmetry classes of the SC order parameters

The coexistence of the SDW and SC order parameters $M$ and $\Delta$ depends on
the symmetry of the SC order parameter translated by the SDW nesting vector.
If translations of the SC order parameter by ${\bf Q}$ are even ($\Delta_{{\bf
k}+{\bf Q}}=\Delta_{{\bf k}}$) then the order parameters are competitive with
each other and the existence of SC order suppresses the SDW order and the SC
transition temperature ($T_{c}$)Kato and Machida (1988). Whereas if the SC
order parameter is odd under translations of the nesting vector ($\Delta_{{\bf
k}+{\bf Q}}=-\Delta_{{\bf k}}$) then the order parameters are cooperative with
each other and the existence of SC order enhances the SDW on-site
magnetization and $T_{c}$Kato and Machida (1988).

Figure 3: Qualitative illustration of the amplitude and sign of the superconducting gap along the normal state tight-binding FSs (band parameters $t_{1}=100T_{N}$ and $t_{2}=10T_{N}$) to show the symmetry of the SC gap under translations of the ${\bf Q}$-vector. (a) $d_{x^{2}-y^{2}}$ SC pairing on the $\xi_{{\bf k}}^{(1)}$ FS (b) $d_{xy}$ pairing on the $\xi_{{\bf k}}^{(1)}$ FS (c) $d_{x^{2}-y^{2}}$ SC pairing on the $\xi_{{\bf k}}^{(2)}$ FS (d) $d_{xy}$ pairing on the $\xi_{{\bf k}}^{(2)}$ FS. Table 1:  Symmetry class of the SC basis function, $\mathcal{Y}_{{\bf k}}$, under translations of the SDW nesting vector ${\bf Q}$. ${\bf Q}$ |  $d^{(O)}$-wave |  $d^{(E)}$-wave
---|---|---
$(\pi/2,\pi/2)$ | $d_{x^{2}-y^{2}}$ | $d_{xy}$
$(\pi,0)$ | $d_{xy}$ | $d_{x^{2}-y^{2}}$

As can be seen from FIG. 3, for the SDW state with nesting vector ${\bf
Q}=(\pi/2,\pi/2)$, the SC gap is even under translations of ${\bf Q}$ for
$d_{xy}$ and odd for $d_{x^{2}-y^{2}}$. However for the SDW state with ${\bf
Q}=(\pi,0)$ the $d_{x^{2}-y^{2}}$ SC pairing state is even under translations
of ${\bf Q}$ and $d_{xy}$ is odd under these same translations (these symmetry
classifications are listed in Table 1). This switching between the $d$-wave
symmetry classes under translations of ${\bf Q}$ is a direct result of both
the ${\bf Q}$ vectors and the $d_{x^{2}-y^{2}}$ and $d_{xy}$ SC basis
functions ($\mathcal{Y}_{\bf k}$) being rotated by $\theta_{\bf k}=\pi/4$
relative to each other. Therefore to maintain the same symmetry in
$\Delta_{\bf k}$ under translations of ${\bf Q}$, the $d$-wave basis functions
are switched when ${\bf Q}$ is rotated.

Figure 4: (a) Band paths relative to the $\xi_{\bf k}^{(1)}$ FS for the ${\bf
Q}=(\pi/2,\pi/2)$ SDW systems both parallel to ${\bf Q}$ (green) and
perpendicular to ${\bf Q}$ (magenta). (b) Band paths relative to the $\xi_{\bf
k}^{(2)}$ FS for the ${\bf Q}=(\pi,0)$ SDW systems both parallel to ${\bf Q}$
(green) and perpendicular to ${\bf Q}$ (magenta). (c) Band structure for the
$d$-wave SC gap which is odd under translations of ${\bf Q}$ plotted along the
band paths parallel to ${\bf Q}$ for both nesting vectors (green paths). (d)
Band structure for the $d$-wave SC gap which is odd under translations of
${\bf Q}$ plotted along the band paths perpendicular to ${\bf Q}$ for both
nesting vectors (magenta paths). (e) Band structure for the $d$-wave SC gap
which is even under translations of ${\bf Q}$ plotted along the band paths
parallel to ${\bf Q}$ for both nesting vectors (green paths). (f) Band
structure for the $d$-wave SC gap which is even under translations of ${\bf
Q}$ plotted along the band paths perpendicular to ${\bf Q}$ for both nesting
vectors (magenta paths).

The nature of the zero-energy excitations critically depends on the symmetry
of the SC gap under translations of the nesting vectorSen Choudhury and
Vorontsov (2021). When this symmetry is even, $\Delta_{{\bf k}+{\bf
Q}}=\Delta_{{\bf k}}$ so $\Delta_{\bf k}^{+}=\Delta_{\bf k}$ and $\Delta_{\bf
k}^{-}=0$. Similarly, when this symmetry is odd, $\Delta_{{\bf k}+{\bf
Q}}=-\Delta_{{\bf k}}$ so $\Delta_{\bf k}^{+}=0$ and $\Delta_{\bf
k}^{-}=\Delta_{\bf k}$. This simplifies the eigenvalues when both $\Delta$ and
$M$ are nonzero in Equation (6) to

$E_{\bf k}^{(1,2;E,O)}=\sqrt{\Gamma_{\bf k}\pm 2\Lambda_{\bf k}^{(E,O)}}$ (9)
$\displaystyle\Gamma_{\bf k}$ $\displaystyle=(\xi_{\bf k}^{+})^{2}+(\xi_{\bf
k}^{-})^{2}+\Delta_{\bf k}^{2}+M^{2}$ $\displaystyle\Lambda_{\bf k}^{(E)}$
$\displaystyle=\sqrt{(\xi_{\bf k}^{+}\xi_{\bf k}^{-})^{2}+M^{2}((\xi_{\bf
k}^{+})^{2}+\Delta_{\bf k}^{2})}$ $\displaystyle\Lambda_{\bf k}^{(O)}$
$\displaystyle=\sqrt{(\xi_{\bf k}^{+}\xi_{\bf k}^{-})^{2}+M^{2}(\xi_{\bf
k}^{+})^{2}}$

where $(E)$ identifies even symmetry and $(O)$ identifies odd symmetry. For
even symmetry, nodal points exist in addition to the SC symmetry nodes near
the edge of the RBZ where $\xi_{\bf k}^{-}=0$ and when $E_{\bf
k}^{(2;E)}=M-\sqrt{(\xi_{\bf k}^{+})^{2}+\Delta_{\bf k}^{2}}=0$. The locations
of these nodes can be seen as the red crosses in FIG. 2. In the odd symmetry
cases, these additional nodes are absent. Furthermore, two of the $d$-wave
symmetry nodes with odd translational symmetry are gapped by the SDW order due
to their nodal lines aligning with ${\bf Q}$. This can be seen from the cyan
$d_{x^{2}-y^{2}}$ and orange $d_{xy}$ nodal lines parallel to ${\bf Q}$ in
FIG. 2 (a) and (b) respectively.

The resulting band structures are shown in FIG. 4 where the first two panels,
FIG. 4 (a) and (b), illustrate the band paths relative to the normal state
FSs. The middle two panels, FIG. 4 (c) and (d), display the calculated band
structure for both SDW states coexisting with the $d$-wave SC gaps which are
odd under translations of ${\bf Q}$ along the indicated paths. The final two
panels, FIG. 4 (e) and (f), display the calculated band structure for both SDW
states coexisting with the $d$-wave SC gaps which are even under translations
of ${\bf Q}$ again along the indicated paths. For ${\bf Q}=(\pi/2,\pi/2)$ the
odd $d$-wave state is the $d_{x^{2}-y^{2}}$ pairing and for ${\bf Q}=(\pi,0)$
the odd state is the $d_{xy}$ pairing state. In FIG. 4(c), the bands shown are
those in the region where SDW ordering reconstructs the FS parallel to ${\bf
Q}$ along the green paths. The SC node typically present in the vicinity of
${\bf k}=(\pi/4,\pi/4)$ [for ${\bf Q}=(\pi/2,\pi/2)$] or near ${\bf
k}=(\pi/2,0)$ [for ${\bf Q}=(\pi,0)$] becomes gapped by the SDW. This again
corresponds to the cyan $d_{x^{2}-y^{2}}$ and orange $d_{xy}$ nodal lines in
FIG. 2 (a) and (b), respectively. In FIG. 4(d) the bands determined along the
magenta paths in FIG. 4 (a) and (b) are shown where the normal state FS is
preserved perpendicular to ${\bf Q}$, resulting in the pure SC bands and the
$d$-wave symmetry nodes in the vicinity of ${\bf k}=(\pi/4,-\pi/4)$ [for ${\bf
Q}=(\pi/2,\pi/2)$] or near ${\bf k}=(0,\pi/2)$ [for ${\bf Q}=(\pi,0)$] are
preserved.

In FIG. 4 (e) and (f) the band structure is shown for the SDW states
coexisting with the $d$-wave SC gaps with even symmetry when translated by
${\bf Q}$. For the ${\bf Q}=(\pi/2,\pi/2)$ SDW state this is the $d_{xy}$
pairing state and for the ${\bf Q}=(\pi,0)$ SDW state this is the
$d_{x^{2}-y^{2}}$ pairing state. The quasiparticle bands parallel to ${\bf Q}$
along the green paths in FIG. 4 (a) and (b) are shown in FIG. 4(e). The
typical $d$-wave nodes occurring in the vicinity of ${\bf k}=(\pi/2,0)$ [for
${\bf Q}=(\pi/2,\pi/2)$] or ${\bf k}=(\pi/2,\pi/2)$ [for ${\bf Q}=(\pi,0)$]
remain intact when both SC and SDW coexist. In addition to the $d$-wave
symmetry nodes, additional nodes appear parallel to ${\bf Q}$ near ${\bf
k}=(\pi/4,\pi/4)$ [when ${\bf Q}=(\pi/2,\pi/2)$] or near ${\bf k}=(\pi/2,0)$
[when ${\bf Q}=(\pi,0)$]. These additional nodes correspond to the red crosses
in FIG. 2. FIG. 4(f) displays the band structure along the magenta band paths
in FIG. 4 (a) and (b) where the normal state FS is preserved, again resulting
in the pure SC bands.

###  C  Kinetic method for heat conductivity

From these band structures, the electronic thermal conductivity was calculated
using the Boltzmann kinetic equation similar to calculations for the thermal
conductivities of both $s$-wave and unconventional superconductors. Within the
Boltzmann kinetic approach the thermal conductivity tensor is given by the
equationMineev and Samokin (1998)

$\kappa_{ij}=-\frac{2}{T}\sum_{n=1}^{2}\int\frac{d^{2}k}{(2\pi)^{2}}(E^{(n)}_{\bf
k})^{2}(\mathbf{v}_{\bf k}^{(n)})_{i}(\mathbf{v}_{\bf
k}^{(n)})_{j}\frac{\partial f^{0}_{\bf k}}{\partial
E}(\tau_{n1}^{-1}+\tau_{n2}^{-1})^{-1}$ (10)

where $f^{0}_{\bf k}=\frac{1}{e^{E_{\bf k}/T}+1}$ is the equilibrium Fermi-
Dirac distribution function, $\mathbf{v}_{\bf k}^{(n)}=\mathbf{\nabla}_{\bf
k}E^{(n)}_{\bf k}$ is the quasiparticle velocity, and $\tau_{nm}$ is the
quasiparticle relaxation time defined as

$\tau_{nm}^{-1}({\bf
k})=N_{imp}V^{2}\frac{2\pi}{\hbar}\int\frac{d^{2}k^{\prime}}{(2\pi)^{2}}|C_{nm}({\bf
k},{\bf k}^{\prime})|^{2}\delta(E_{\bf k}^{(n)}-E_{{\bf k}^{\prime}}^{(m)})$
(11)

where $C_{nm}({\bf k},{\bf k}^{\prime})$ is known as the coherence factor and
is the amplitude for a single impurity to scatter a quasiparticle from the
state with momentum ${\bf k}$ and energy $E^{(n)}_{\bf k}$ to a state with
momentum ${\bf k}^{\prime}$ and energy $E^{(m)}_{{\bf k}^{\prime}}$ within the
Born limitSen Choudhury and Vorontsov (2021). $N_{imp}$ is the density of
impurities, and $V$ is the isotropic scattering amplitude. The quasiparticle
relaxation time integral was calculated numerically with the unknown
$N_{imp}V^{2}$ eliminated in favor of the normal state quasiparticle
relaxation times; $\tau_{n}^{-1}=N_{imp}V^{2}\frac{2\pi}{\hbar}N_{F}$ where
$N_{F}$ is the density of states at the Fermi level in the normal state.

##  III Numerical Results and Discussion

The Cartesian components of the thermal conductivity tensor $\kappa_{xx}(T)$,
$\kappa_{yy}(T)$, and $\kappa_{xy}(T)$ ($\kappa_{yx}(T)=\kappa_{xy}(T)$) were
numerically calculated in the RBZ. However, the frame-of-reference of the
nesting vector ${\bf Q}$ diagonalizes the thermal conductivity tensor, and is
therefore the more natural frame to study thermal transport. This is straight-
forwardly accomplished by rotating the coordinate system by $\theta=\pi/4$ for
the case when ${\bf Q}=(\pi/2,\pi/2)$. The rotated conductivity tensor
$\hat{\kappa}^{\prime}=\hat{R}(\theta=\pi/4)\hat{\kappa}\hat{R}^{T}(\theta=\pi/4)$
is:

$\begin{pmatrix}\kappa_{\perp}&0\\\ 0&\kappa_{\parallel}\\\
\end{pmatrix}=\frac{1}{2}\begin{pmatrix}1&-1\\\ 1&1\\\
\end{pmatrix}\begin{pmatrix}\kappa_{xx}&\kappa_{xy}\\\
\kappa_{xy}&\kappa_{xx}\\\ \end{pmatrix}\begin{pmatrix}1&1\\\ -1&1\\\
\end{pmatrix}$ (12)

which was simplified by using the inherent symmetries
($\kappa_{xx}=\kappa_{yy}$, $\kappa_{yx}=\kappa_{xy}$) in the case when ${\bf
Q}=(\pi/2,\pi/2)$, and leads to the diagonalized thermal conductivity
components $\kappa_{\perp}=\kappa_{xx}-\kappa_{xy}$ and
$\kappa_{\parallel}=\kappa_{xx}+\kappa_{xy}$. In the SDW system with ${\bf
Q}=(\pi,0)$, no rotation is needed and $\kappa_{xy}$ integrates to 0,
resulting in a diagonal thermal conductivity tensor.

Appropriately integrating the band structure generates the electronic density
of states (DOS) for both SC + SDW coexistence states. One important aspect of
the nodal structures is the variation in the DOS just above the Fermi level
(which occurs when $E=0$) for the two SC + SDW coexistence states (shown in
FIG. 5). The enhancement occurring just above the Fermi level has important
consequences for the low-$T$ behavior of the thermal conductivity elements.

The electronic thermal conductivity calculated on the normal state tight-
binding FS was found to have a linear dependence on temperature. To accentuate
the deviation from the normal metallic thermal conductivity, the thermal
conductivity elements in FIG. 6(a) had their linear $T$-dependence removed by
plotting $\kappa(T)/T$ and were normalized by $\kappa(T_{N})/T_{N}$. In FIG.
6(b) the conductivity elements were normalized by $\kappa(T_{N})$ on a log-log
scale to emphasize which conductivity elements preserve this linear
$T$-dependence and which ones deviate from it as $T\rightarrow 0$.

Figure 5: Density of states normalized by the normal state DOS at the Fermi
level ($N_{F}$) for SDW + $d$-wave SC order with odd symmetry under
translations of ${\bf Q}$ (cyan) and SDW + $d$-wave SC order with even
symmetry under translations of ${\bf Q}$ (orange) plotted vs. energy
normalized by the SC gap maxima on the FS ($\Delta$).

###  A  Pure SDW Thermal Conductivity

In the pure SDW case the electronic thermal conductivities were calculated
numerically in the directions parallel and perpendicular to the nesting vector
${\bf Q}$ (see the black curves of FIG. 6). Parallel to the nesting vector,
the thermal conductivity, $\kappa_{\parallel}^{SDW}(T)$, is diminished from
its normal state value as the temperature is decreased from $T_{N}$ as a
result of a growing gap in the reconstructed FS (seen in FIG. 2) as
$T\rightarrow 0$. Thus $\kappa_{\parallel}^{SDW}(T)/T$ becomes a constant as
the on-site magnetization, $M(T)$, reaches its maximum value and results in a
diminished metallic state. This result is comparable to the pure SDW thermal
conductivity for the case when ${\bf Q}=(\pi,\pi)$ with a nearly half electron
filling for similar band parametersSen Choudhury and Vorontsov (2021).
Perpendicular to ${\bf Q}$ in these SDW systems the normal state tight-binding
FS is preserved and $\kappa_{\perp}^{SDW}(T)$ is essentially that of the
normal state thermal conductivity. However, as $T\rightarrow 0$ this thermal
conductivity becomes slightly enhanced due to the gap in the FS generated by
the SDW order which reduces the available states at the Fermi level,
increasing the quasiparticle lifetimes resulting in a slightly enhanced
thermal conductivity.

Figure 6: Diagonalized thermal conductivity tensor elements parallel (dashed
curves) and perpendicular (dotted curves) to the SDW nesting vector ${\bf Q}$
in the pure SDW state (black), SDW + $d$-wave SC state with odd translational
symmetry (cyan), and SDW + $d$-wave SC state with even translational symmetry
(orange). (a) $\kappa(T)T_{N}/\kappa(T_{N})T$ plotted to remove linear $T$\-
dependence and emphasize deviations from normal state conductivity. (b)
$\kappa(T)/\kappa(T_{N})$ plotted on a log-log scale to emphasize low-$T$
linearity.

###  B  SDW + $d$-wave SC Thermal Conductivity

When the propagation direction of the SDW is aligned (parallel case) with a
$d$-wave nodal line, the FS reconstruction destroys two of the $d$-wave
symmetry nodes (e.g. this occurs for the ${\bf Q}=(\pi/2,\pi/2)$ SDW and the
$d_{x^{2}-y^{2}}$ SC gap or ${\bf Q}=(\pi,0)$ SDW and the $d_{xy}$ SC gap),
while the symmetry nodes perpendicular to the SDW propagation direction are
unaffected. This can be seen by the cyan lines in FIG. 2(a) and the orange
lines in FIG. 2(b), where the remaining nodes occur on the low local DOS
regions (flat regions) of the FS. These two coexistence phases have SC gaps
which are odd under translations of their respective ${\bf Q}$ vectors (i.e.
$\Delta_{{\bf k}+{\bf Q}}=-\Delta_{\bf k}$) (see FIG. 3 (a) and (d)). More so,
these systems have equivalent band structures along their respective band
paths (see FIG. 4 (c) and (d)). Due to these similarities these coexistence
states have equivalent transport properties relative to their ${\bf Q}$
vectors, which will be referred to as the $d^{(O)}$-wave state (odd symmetry
state).

The ${\bf Q}=(\pi/2,\pi/2)$ SDW state coexisting with the $d_{xy}$ SC gap and
the ${\bf Q}=(\pi,0)$ SDW state coexisting with the $d_{x^{2}-y^{2}}$ SC gap
have symmetry nodes on the high local DOS regions (curved regions) of their
respective normal state FSs, all of which remain unchanged by the FS
reconstruction (note the orange points in FIG. 2(a) and the cyan points in
FIG. 2(b)). These coexistence states both have SC gaps which are even under
translations of ${\bf Q}$ (see FIG. 3 (b) and (c)). Due to this translational
symmetry, additional mixing nodes appear near the FS reconstruction which can
be seen as the red crosses in FIG. 2. Furthermore, these coexistence states
have equivalent band structures along their respective band paths (see FIG. 4
(e) and (f)) which lead to them having equivalent transport properties
relative to ${\bf Q}$. These states will be referred to as the $d^{(E)}$-wave
state (even symmetry state).

####  B.1  SDW + $d^{(O)}$-wave SC Thermal Conductivity

Perpendicular to ${\bf Q}$, this system behaves like a pure $d$-wave SC
system. However, only half the symmetry nodes typically present in a $d$-wave
SC system survive the FS reconstruction, so the DOS just above the Fermi level
($E=0$) is approximately half that of the pure $d$-wave system with the same
band parameters (note the reduction in the DOS just above the Fermi level for
$d^{(O)}$ in FIG. 5). With half the available states to scatter to just above
the Fermi level, there is a doubling in the quasiparticle lifetimes at the
remaining nodes. Since the remaining symmetry nodes contain quasiparticles
with Fermi velocities purely perpendicular to ${\bf Q}$ with lifetimes
approximately twice that of their pure SC counterparts, the residual
$\kappa_{\perp}(T\rightarrow 0)/T$ is therefore roughly twice the residual
$\kappa(T\rightarrow 0)/T$ for a pure $d$-wave SC with nodes located on the
flat parts of a tight-binding FSSen Choudhury and Vorontsov (2021) (see FIG.
6(a)). The thermal conductivity perpendicular to ${\bf Q}$ is linear in $T$,
(see FIG. 6(b)), and therefore behaves like a SC with line nodes in this
directionGraf et al. (1996). The thermal conductivity parallel to ${\bf Q}$
decreases exponentially as $T\rightarrow 0$ since no nodal quasiparticle
states have Fermi velocities perpendicular to ${\bf Q}$, and therefore behaves
like a fully-gapped SC in this directionBardeen et al. (1959).

####  B.2  SDW + $d^{(E)}$-wave SC Thermal Conductivity

Since none of these $d$-wave symmetry nodes become gapped by the FS
reconstruction and additional mixing nodes appear due to the SC gap being even
under translations of ${\bf Q}$, the DOS just above the Fermi level for
$d^{(E)}$ is relatively large (see FIG. 5) and results in short-lived
quasiparticles. The majority of the states just above the Fermi level are
located in the vicinity of the $d$-wave symmetry nodes rather than the mixing
nodes, which results in the symmetry nodes dominating low-$T$ thermal
transport. Quasiparticles occupying states at the symmetry nodes have Fermi
velocities with equal magnitude components perpendicular and parallel to ${\bf
Q}$ and contribute equally to the low-$T$ transport in both directions.
However, quasiparticles occupying states at the mixing nodes have Fermi
velocities parallel to ${\bf Q}$ and only contribute to $\kappa_{\parallel}$,
thus leading to a system with weakly anisotropic thermal transport where
$\kappa_{\parallel}(T\rightarrow 0)>\kappa_{\perp}(T\rightarrow 0)$ (see the
orange curves in Fig 6 (a)).

The residual $\kappa(T\rightarrow 0)/T$ thermal conductivity elements in these
cases are much smaller than $\kappa_{\perp}(T\rightarrow 0)/T$ in the
previously discussed $d$-wave system due to the quasiparticles at these
$d$-wave symmetry nodes having significantly lower Fermi velocities and
lifetimes (see FIG. 6(a)). While the quasiparticles occupying the states at
the mixing nodes have high Fermi velocities parallel to ${\bf Q}$, the
relative dearth of available states means they don’t play a significant role
in thermal transport. This $d$-wave SC state decreases linearly with $T$ at
low $T$ as can be seen in FIG. 6(b) both parallel and perpendicular to ${\bf
Q}$, and therefore behaves like a SC with line nodes in both directionsGraf et
al. (1996).

##  IV Conclusion

While this work ignores the effects of band multiplicity, these results are
still useful in determining the nodal structures of commensurate SDW systems
with nesting vectors ${\bf Q}=(\pi/2,\pi/2)$ and ${\bf Q}=(\pi,0)$ coexisting
with singlet $d$-wave SC pairings. Commensurate SDW systems of type AF3 and
AF2 were considered on two-dimensional tight-binding square-lattices and found
to have equivalent transport properties within Boltzmann kinetic theory in the
weak impurity scattering (Born) limit relative to their nesting vectors.
Parallel to their nesting vectors these systems behave similar to a suppressed
metal, where the electronic thermal conductivity is linear in $T$ but
diminished from the normal state thermal conductivity. However, perpendicular
to their nesting vectors, the transport properties of these systems are almost
identical to that of the normal metallic state, except they are slightly
enhanced as $T\rightarrow 0$ due to the FS reconstruction parallel to ${\bf
Q}$ creating a gap in the FS and reducing $N_{F}$, thus enhancing the
quasiparticle lifetimes.

The $d$-wave SC states coexisting with the ${\bf Q}=(\pi/2,\pi/2)$ and ${\bf
Q}=(\pi,0)$ nesting vectors have equivalent transport properties, with the
$d_{x^{2}-y^{2}}$ and $d_{xy}$ states swapped between the nesting vectors. The
$d$-wave symmetry nodes are located on regions of the tight-binding FS with
the same relatively small local DOS and have equivalent band structures when
the ${\bf Q}=(\pi/2,\pi/2)$ SDW state coexists with the $d_{x^{2}-y^{2}}$ SC
gap and the ${\bf Q}=(\pi,0)$ SDW state coexists with the $d_{xy}$ SC gap
(these are the odd symmetry, $d^{(O)}$-wave state in this work). Similarly,
the $d$-wave nodes occur on regions of the tight-binding FS with the same
relatively large local DOS and have equivalent band structures when the ${\bf
Q}=(\pi/2,\pi/2)$ SDW state coexists with the $d_{xy}$ SC gap and the ${\bf
Q}=(\pi,0)$ SDW state coexists with the $d_{x^{2}-y^{2}}$ SC gap (these are
the even symmetry, $d^{(E)}$-wave state in this work).

The electron transport properties of these commensurate SDW systems were also
studied when SC singlet pairing arises ($T_{N}>T_{c}$) out of it. The
electronic thermal conductivity for a $d^{(O)}$-wave SC gap measured parallel
to ${\bf Q}$ was found to decrease exponentially with $T$, consistent with
results for fully-gapped SCsBardeen et al. (1959). Perpendicular to ${\bf Q}$,
the conductivity was found to decrease linearly with $T$, consistent with SCs
with line nodesGraf et al. (1996). Furthermore, the residual
$\kappa_{\perp}(T\rightarrow 0)/T$ value was found to be roughly twice the
reported valueSen Choudhury and Vorontsov (2021) for pure $d$-wave SCs
containing nodal quasiparticle states situated on the flat regions of a tight-
binding FS. This results in a system with highly anisotropic electron
transport where fast long-lived quasiparticles tend to travel perpendicular to
${\bf Q}$.

When the $d$-wave SC gap is even under translations of ${\bf Q}$
($d^{(E)}$-wave), the SC symmetry nodes are preserved as none of them appear
in the region of the FS which is reconstructed by the SDW order. Due to the
translational symmetry of the SC gap, additional mixing nodes appear in the
vicinity of the FS reconstruction parallel to ${\bf Q}$. The electronic
thermal conductivity both parallel and perpendicular to ${\bf Q}$ were found
to decrease linearly with $T$. In fact, $\kappa_{\parallel}(T\rightarrow 0)/T$
and $\kappa_{\perp}(T\rightarrow 0)/T$ were nearly identical due to
quasiparticles occupying states at the $d$-wave symmetry nodes contributing
equally to thermal transport in both directions. However, thermal transport
parallel to ${\bf Q}$ was slightly enahnced since quasiparticles occupying
states at the mixing nodes enhanced transport in that direction. Therefore,
this results in a system with weakly anisotropic thermal transport, where slow
short-lived quasiparticles travel both parallel and perpendicular to ${\bf
Q}$, but slightly prefer to travel parallel to ${\bf Q}$.

These results could be relevant in determining the nature of the $d$-wave gap
in cuprates with commensurate SDW orders of nesting vectors: ${\bf
Q}=(\pi/2,\pi/2)$ or ${\bf Q}=(\pi,0)$. If the thermal conductivity is
measured both parallel and perpendicular to ${\bf Q}$ and the nesting vector
is known, weakly or strongly anisotropic thermal transport at low-$T$ could be
used to determine whether the SC gap is $d_{x^{2}-y^{2}}$ or $d_{xy}$ in
nature.

## Acknowledgements

This work supported by NSF Award No. 1809846 and through the NSF MonArk
Quantum Foundry supported under Award No. 1906383. The authors would like to
thank Anton Vorontsov for the initial suggestion of the problem.

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