# Evaluation of Sketch-Based and Semantic-Based Modalities for Mockup
Generation

Tommaso Calò  0000-0002-3200-2348 Politecnico di TorinoCorso Duca degli
Abruzzi, 24Torino10129Italy<EMAIL_ADDRESS>and  Luigi De Russis
0000-0001-7647-6652 Politecnico di TorinoCorso Duca degli Abruzzi,
24Torino10129Italy<EMAIL_ADDRESS>
(2023)

###### Abstract.

Design mockups are essential instruments for visualizing and testing design
ideas. However, the process of generating mockups can be time-consuming and
challenging for designers. In this article, we present and evaluate two
different modalities for generating mockup ideas to support designers in their
work: (1) a sketch-based approach to generate mockups based on hand-drawn
sketches, and (2) a semantic-based approach to generate interfaces based on a
set of predefined design elements. To evaluate the effectiveness of these two
approaches, we conducted a series of experiments with 13 participants in which
we asked them to generate mockups using each modality. Our results show that
sketch-based generation was more intuitive and expressive, while semantic-
based generative AI obtained better results in terms of quality and fidelity.
Both methods can be valuable tools for UI designers looking to increase their
creativity and efficiency.

machine learning, web elements, user interface, convolutional neural network

††journalyear: 2023††copyright: acmcopyright††conference: ; ††booktitle:
;††price: 15.00††ccs: Human-centered computing Graphical user interfaces††ccs:
Human-centered computing Interface design prototyping††ccs: Computing
methodologies Machine learning††ccs: Computing methodologies Computer vision

##  1\. Introduction And Motivation

User interface designers face daily challenges in creating designs that are
effective, usable, and innovative. They often draw inspiration from existing
design samples to come up with new ideas (Gonçalves et al., 2013). There are
two main types of resources that support this process of finding inspiration
(Herring et al., 2009a). The first type are design gallery platforms, such as
Dribbble (dri, 2022) and Behance (Beh, 2022), which allow designers to browse
through a collection of designs and find examples that are interesting or
useful for their work. The second type are design inspirational tools that
suggest examples based on certain types of design input, such as a sketch or
an existing design, using algorithms to determine image similarity (Behrang et
al., 2018; Lee et al., 2010; Ritchie et al., 2011; Swearngin et al., 2018).
While these approaches can be helpful in finding inspiration, they have
limitations. Browsing through design galleries can be overwhelming and lead to
a shift in design ideas away from the original focus, while relying too much
on examples with similar styles can lead to design fixation and hinder the
originality of the work (Jansson and Smith, 1991; Marsh et al., 1996). To find
a balance between targeted and serendipitous inspiration, Mozaffari et al.
(Mozaffari et al., 2022) proposed a style-based generative adversarial network
(StyleGAN) trained on a large dataset of existing interface designs. It
generates a diverse yet focused set of examples based on a preliminary design
input. While it can generate a diverse range of interface mockups, the user
control is limited to injecting style into the latent space of the model. In
other words, the user can specify certain style attributes (such as color and
texture) for the model to incorporate, but has little control over the
specific layout or design elements of the mockup. A better approach would
allow designers to explicitly specify the desired layout and other design
elements, rather than relying on the model to randomly generate these elements
based on style injection. This would enable a more targeted and precise design
process, as designers can focus on specific aspects of the mockup rather than
being limited to injecting style into the model. In this paper, we aim to
achieve this goal by exploring two distinct AI-assisted modalities for mockup
generation: one based on sketches, and the other based on semantic-colored
drawings. Sketch-based mockup generation involves the use of hand-drawn
sketches to represent the desired design. Semantic-colored drawing-based
mockup generation, instead, involves the use of detailed, colored drawings
that convey specific design elements. The choice of these two modalities was
based on a careful review of the literature, indicating that sketching and
semantic drawing are commonly used in the design process due to their ability
to balance precision and speed (Li et al., 2023; Park et al., 2019a; Calò and
De Russis, 2022). Sketch-based mockup generation allows for quick exploration
of ideas while semantic-colored drawing-based mockup generation provides more
accuracy in representing the final product. To achieve this, we used the
Pix2Pix (Wang et al., 2017) model for sketch-based mockup generation and the
SPADE (Park et al., 2019b) model for semantic-colored drawing-based mockup
generation. The Pix2Pix model is well suited for sketch-based mockup
generation because it is a conditional GAN that has been proven to be
effective in generating high-quality images from sketches (Tian et al., 2021).
Similarly, the SPADE model is well suited for semantic colored drawing-based
mockup generation as it is a GAN that is trained to generate images from
semantic maps, it allows for more control over specific design elements, which
is crucial for conveying accurate representations of the final product. We
evaluate the two modalities of mockup generation in terms of their
expressivity, time demand, ease, and intuitiveness. By analyzing the
advantages and disadvantages of both sketch-based and semantic-colored
drawing-based mockup generation, we aim to empower designers to select the
most appropriate modality for their particular design project and boost their
creative potential.

##  2\. Related Works

Inspiration has long been of interest to researchers in the field of
psychology. Thrash et al. (Thrash and Elliot, 2003; Thrash et al., 2014) were
among the first to systematically study inspiration, identifying three
categories: motivation (i.e., goal-oriented self-initiation), evocation (i.e.,
impulsive reactions to stimuli), and transcendence (i.e., feeling of gaining
superior, more elegant or novel ideas than those generated willfully). These
studies have generally focused on how designers access and utilize existing
design artifacts. For instance, Eckert and Stacey (Eckert and Stacey, 2000)
found that knitwear designers used a variety of sources for inspiration,
including artifacts with intriguing shapes, patterns, and colors, as well as
their own previous work. Researchers have also explored how industrial and
user interaction designers are inspired by existing design artifacts.
Bonnardel (Bonnardel, 1999) found that in the context of product design, “the
emergence of new ideas results from analogy-making.” Herring et al. (Herring
et al., 2009b) conducted in-depth interviews with web, graphic, and product
designers and identified common approaches and challenges they faced in
retrieving, storing, and disseminating design examples. Gonçalves et al.
(Gonçalves et al., 2014) surveyed students and industrial designers to
understand their sources and methods of inspiration and found that
professional industrial designers used a wider variety of approaches compared
to students. The literature has also identified several issues with common
inspirational methods, including the risk of “design fixation” from exposure
to a homogenous set of examples and the impact of timing on the quantity and
quality of ideas.

In recent years, Artificial Intelligence (AI) has emerged as a potential tool
for supporting and enhancing the creative inspiration process (Shneiderman,
2022). AI systems can be trained to generate ideas and outputs based on a set
of rules or guidelines, enabling the efficient production of a wide range of
options, and providing a variety of applications and systems to support
professionals in various visual art fields, such as graphic design (Ueno and
Satoh, 2021), UI design (Wang et al., 2021), webtoon (Ko et al., 2022),
digital art (Yurman and Reddy, 2022), and new media art (Qiao et al., 2022).
To ensure a balance between human control and computer automation
(Shneiderman, 2020), some AI systems have employed a “human-in-the-loop”
approach, which allows for collaboration between humans and AI. Overall, AI
has been seen as a helpful tool in the design process, aiding in ideation and
generating a variety of output efficiently.

Our work compares two modalities, sketches and semantic-colored drawings, to
generate user interface mockups and build upon previous knowledge in several
ways. First, the use of sketches as a modality builds upon previous research
on the use of visual representations in the design process (Landay and Myers,
1995). Studies have shown that sketches can be used to quickly generate and
communicate design ideas, allowing designers to explore a wide range of
possibilities in a relatively short amount of time. Additionally, sketches are
often used as a way to capture informal, early-stage design ideas, which can
then be refined and developed further (Landay and Myers, 2001). Second, the
use of semantic-colored drawings (Hong et al., 2018; Zhu et al., 2020), where
each color represents a UI element category, allows for a clear representation
of the design elements and their relationships. The two modalities work
together towards the goal of allowing designers to quickly generate and
communicate ideas using sketches, while also leveraging the power of color
coding to represent the design elements and their relationships. Both
modalities combine the benefits of computer automation and human control
(Shneiderman, 2022) enabling the efficient production of a wide range of
options while maintaining high fidelity to the original design idea.

Figure 1. An example of the same user interface represented in the sketch-
based and semantic-based modalities. On the left, we can see a hand-drawn
sketch of the user interface (A), with various elements such as buttons, text
fields, and icons depicted in a freeform manner. On the right, we can see the
same user interface represented in the semantic-based modality (B). These
elements are filled with different colors, with each color representing a
specific design element such as buttons, text fields, and icons.

The figure shows the skeleton of a mobile user interface with titles,
navigation, and a list of items as both a hand-drawn sketch and a semantic-
colored representation, where each design element is represented by a
different color.

##  3\. Study: Evaluating Generation Modalities

This study aimed to evaluate the semantic drawing-to-mockup and sketch-to-
mockup modalities in terms of time demand, ease, creative expressivity, and
intuitiveness. The study was conducted with 13 human-computer interaction
students from the first-year master’s program in computer engineering. The
participants were asked to use both modalities to create a mockup of a mobile
application and then rate their experience with each modality on various
dimensions. The results of the study provide insights into the strengths and
weaknesses of each modality and can inform the design and development of
mobile applications in the future.

###  3.1. Participants

The study was conducted with 13 participants who gave their consent to
participate in the study. All participants were studying Human-Computer
Interaction (HCI) as part of their curriculum. It is important to note that
all participants in the study had received training in UI design and
programming as part of their curriculum, indicating that they had at least
some level of experience in this area. The sample consisted of 2 female and 11
male participants, with an age range of 23 to 25 years old. Given that the
sample is relatively small, it can be considered an exploratory study. Future
studies should replicate and expand the study on a larger sample to generalize
the findings. The study aimed to be a first step to understanding the
differences between the two modalities.

###  3.2. Procedure

Participants were asked to complete a two-phase task requiring them to design
a mobile user interface by sketching and semantically drawing it. The task was
designed to evaluate the participants’ abilities in terms of time demand (time
to complete), expressiveness (how well the modality conveyed their idea),
intuitiveness (in understanding the modality), and ease of use (of the
modality).

In the first phase, participants were sent a document containing instructions
on how to complete the task. They were asked to design a mobile user interface
and sketch in a given frame. While participants were allowed to use their
preferred digital drawing tool in the study to ensure a more natural and
intuitive design process, future studies may consider restricting the use of
digital drawing tools to a single option to better control for potential
variations in user experience. After sketching, participants were instructed
to semantically draw the user interface. The order of the tasks was chosen
based on the natural progression of a typical design process in which initial,
unstructured ideas and concepts are often followed by more concrete
iterations. This allowed us to evaluate the strengths and weaknesses of each
modality in facilitating the design process, rather than comparing the
effectiveness of one modality over the other. An example from the experiment,
showcasing the same user interface represented in both sketch and semantic-
drawing form, can be seen in Figure 1. Once the participants had completed the
task, they were asked to rate their experience in terms of the above
dimensions on a Likert scale. Participants were also asked to provide
motivation for their ratings.

In the second phase, the participants’ drawings were processed by a network
(see “implementation” below), and users were asked two additional questions
about the quality of the obtained results and how the results respected the
imagined user interface. The participants were asked to rate the quality of
the results obtained from the network on a Likert scale, and to provide
feedback on how well the results corresponded to their imagined user
interface.

Figure 2. Illustration of how Pix2pix (Wang et al., 2017) translates between
sketches and real mockups domains. A generator is trained to generate mockups
from sketches (1) and a discriminator is trained simultaneously to distinguish
between the real and generated mockups (2). A Reconstruction loss, in
addition, measures how close the generated mockup is to the real target mockup
(3). The only difference with SPADE (Park et al., 2019b) is that in the latter
a SPatially-Adaptive Normalization module is added to optimize the translation
of semantic maps.

An architectural diagram: the Pix2pix algorithm starts from the input sketch,
pass it through a generator to generate a mockup. The generated mockup and the
corresponding real user interface go to a discriminator that provides
optimization parameters to the generator, to improve the classification
results.

###  3.3. Implementation

To translate sketches and semantic colored drawing to mockups, we use two deep
learning models: Pix2pix (Wang et al., 2017) and SPADE (Park et al., 2019b)
trained on RICO (Deka et al., 2017) dataset. Pix2pix is the model we used to
generate realistic mockups from sketches, while SPADE is the model we used to
generate mockups from semantic-colored drawings, an illustration of the
framework used to generate mockups can be found in Figure 2. The trained
models are available for download at HIDDENLINK111Temporary link for
preserving the anonimity during the peer-review phase. for the reproducibility
of results.

####  3.3.1. Dataset

In this study, we used the RICO dataset (Deka et al., 2017) for both semantic
drawing-to-mockup and sketch-to-mockup modalities. The RICO dataset is a
publicly available dataset of hand-drawn sketches and corresponding semantic
representations of mobile app user interfaces. It is composed of a total of
approximately 20k sketches and 60k semantic representations. The dataset is
designed to be a benchmark for evaluating the performance of deep learning
models for mobile app user interface generation. To ensure the quality of the
data, we corrected the colors in the semantic representation that were
corrupted and released the corrected version of the dataset on the same GitHub
repository. This was done to ensure that the results of the study were not
affected by any errors or inconsistencies in the data. We found the RICO
dataset to be particularly useful for our study as it provided a large and
diverse set of sketches and semantic representations of mobile app user
interfaces.

####  3.3.2. Sketch-to-mockup

Pix2pix is a model that uses a conditional generative adversarial network
(cGAN) to generate images. It consists of two networks: a generator and a
discriminator. In our case, the generator takes as input a sketch and
generates the corresponding mockup, while the discriminator receives both the
generated image and the target image and tries to distinguish between the two.
The two networks are trained simultaneously, with the generator trying to fool
the discriminator and the discriminator trying to classify the images
correctly.

####  3.3.3. Semantic drawing-to-mockup

SPADE (Semantic Image Synthesis with SPatially-Adaptive Normalization) is a
model that uses a style-based generator architecture to generate images from
semantic colored drawings. It consists of a generator network that takes as
input a semantic-colored drawing and generates a corresponding image, and a
discriminator network that receives both the generated image and the target
image and tries to distinguish between the two. The generator network includes
SPatially-Adaptive Normalization (SPADE) layers that adapt the normalization
parameters of the generator based on the input semantic map, optimizing the
image synthesis for semantic map inputs. This is achieved through the use of a
separate normalization network, which takes the input image and a set of
spatial locations as input and produces the normalization parameters for each
location.

##  4\. Preliminary Results

Table 1. Comparison of sketch-to-mockup and semantic drawing-to-mockup modalities on intuitiveness, ease, time demand, and expressiveness, as well as the quality and fidelity of the generated mockups, on a 1 to 5 Likert scale. | Sketch-to-mockup | Semantic drawing-to-mockup
---|---|---
Intuitiveness |  4.54 $\pm$ 0.78 |  3.15 $\pm$ 1.28
Ease of use | 3.38 $\pm$ 1.04 | 3.69 $\pm$ 1.25
Time Demand |  3.54 $\pm$ 1.13 |  3.77 $\pm$ 1.09
Expressiveness | 4.15 $\pm$ 0.55 | 3.38 $\pm$ 1.12
Quality |  1.50 $\pm$ 0.58 |  2.75 $\pm$ 0.50
Fidelity | 3.38 $\pm$ 2.20 | 3.60 $\pm$ 0.55

The study compares the sketch and semantic drawing modalities by gathering
participant feedback on their level of intuitiveness, ease of use, and time
demand during the generation process, as well as the quality and fidelity of
the generated mockups. The results of the survey provide insight into how
participants perceive these modalities and their relative strengths and
weaknesses. Numerical results are reported in Table 1.

##### Intuitiveness:

The results of the comparison between the sketch and semantic drawing
modalities to input an AI for generating mockups show that the sketch
generation score for intuitiveness is generally higher than the semantic
generation score. Most participants (11 out of 13) gave a score of 5 for the
sketch generation, indicating that they found it fully intuitive. In contrast,
only two participants gave a score of 5 for the semantic generation, with the
majority of participants (8 out of 13) giving a score of 3 or lower. One of
the main motivations for participants finding the sketch generation more
intuitive is that it allows them to “show others what you have in mind” and
“represent at a higher level each part of the layout” (participant 1, 6).
Another motivation is that it is easier to “imagine the layout” (participant
9) and “pour the idea in mind into drawings” (participant 10) with the sketch
generation. However, some participants had issues with the semantic
generation, finding it “difficult due to the rules about the colors”
(participant 4) and “not so clear how it can be useful” (participant 1).
Participant 12 also found semantic generation less intuitive stating that it
is not simple for everyone to understand. Conversely, some participants found
the semantic generation to be more intuitive, with participant 10 stating that
it is more intuitive as it has a “map (indication) about element types” which
makes things “much easier to demonstrate graphically”.

##### Ease of use:

The results of the comparison between the sketch and semantic drawing
modalities for ease of use indicate that there is no statistical significance
between the mean ease of use of the two methods. Most participants (7 out of
13) gave a score of 3 or lower for the sketch generation, indicating that they
found it to be difficult to use. One of the main motivations for participants
finding sketch generation difficult to use is that it requires “drawing skills
and tools” (participant 9) and “some drawing skills are needed to convey
information in a reasonable and clear way” (participant 1). Additionally, some
participants found it difficult to use the mouse to draw (participants 3, 4,
11) and some participants found it difficult to use because they had to use
different devices to complete the task (participant 13). Participants found
the semantic modality difficult to use because it ”needs to remember the
colors according to the legend” (participant 9), while on the other hand, some
participants found it easy to use because it “only requires squares and
rectangles” (participant 4) and is “just a mechanical exercise” (participant
3). Participant 10 also found the semantic generation easy to use because it
is “much more descriptive” and “easier regarding positioning the items” as
compared to sketch generation.

##### Time demands:

When it comes to time demand, participants had mixed opinions on the time
efficiency of sketch and semantic drawing modalities. The main motivation for
participants finding the sketch generation quicker is that it allows them to
represent the layout at a higher level and it is easier to imagine the layout
and that “only took me some minutes to complete” (participant 5). However,
some participants experienced challenges with the sketch generation, citing
issues with tools such as “difficulty drawing with a mouse” (participant 10)
and “difficulty in getting a decent result” (participant 3). On the other
hand, the semantic generation was perceived as less time demanding by some
participants, with participant 8 stating that it is “the quickest one because
it’s similar to sketch generation but with even fewer details to represent”
and participant 3 finding it “fairly quick to draw what I was thinking in the
semantic way”. However, other participants found it to be more time demanding,
with participant 1 stating that they “spent very much time in finding a
software to draw colored squares” and participant 12 stating that “it can take
a long time because there is often a difference between ideas and reality.”

##### Expressivity:

The results of the comparison between the sketch and semantic drawing
modalities for expressivity show that the expressivity score for the sketch
generation is generally higher than the semantic generation score. Most
participants (10 out of 13) gave a score of 4 or 5 for the sketch generation,
indicating that they found it to be fully expressive. The majority of
participants (7 out of 13) gave a score of 3 or lower for the semantic
generation, indicating that they found it to be less expressive. Some of the
main motivations for participants finding the sketch generation to be more
expressive is that it “makes it easiest to describe how you want the final
result to look like” (participant 3) and “irons out some ambiguity”
(participant 5). Participants also found it to be expressive because it allows
them to “draw almost everything” (participant 5) and “the alignments and the
position of the items are easier to express” (participant 6). On the other
hand, some participants found the semantic generation to be less expressive
because it “just conveys information with colored and filled squares that
don’t reflect very well the actual elements” (participant 1) and it is “too
restrictive” (participant 4). Participants also found it difficult to “define
the role of each component” (participant 7). Overall, the results suggest that
participants generally find the sketch generation to be more expressive for
inputting mockups into an AI, but some participants also found the semantic
generation to be useful, particularly for its ability to convey information
through color-name association.

Figure 3. An example of two mockups generated through semantic-drawing
modality (left) and sketch modality (right). The image demonstrates the
improved resulting quality and reduced presence of artifacts achieved through
the use of semantic-drawing modality.

The generated mobile mockups along with their input drawings in the two
modalities.

##### Quality and fidelity of the generated mockups:

Results indicate that the quality of the generated mockup for the sketch-to-
mockup modality is higher than for the semantic drawing-to-mockup modality
with a mean of 2.75 and 1.50 respectively. In terms of fidelity to the
original idea, the sketch-to-mockup modality also scores higher with a mean of
3.60 compared to 3.38 for the semantic drawing-to-mockup modality. However,
this difference is not as large as for the quality of the generated mockup but
still statistically significant. In summary, the results indicate that the
sketch-to-mockup modality generally performs better in terms of quality of the
generated mockup and fidelity to the original idea compared to the semantic
drawing-to-mockup modality, an example is reported in Figure 3. However, it is
worth noting that these results should be taken with caution as it is based on
a small sample size, and it would be beneficial to have more data and a larger
sample size to make more robust conclusion.

##### Additional Comments

In analyzing the additional comments, it is clear that there are different
preferences for the best method of inputting AI to generate mockups.
Participant 7 suggests that the sketch generation modality is the best, but
that the problem of describing components’ function can be solved by adding a
color convention to the sketch generation. This suggests that the participant
values the intuitiveness and expressivity of the sketch generation modality,
but also recognizes the benefits of using a semantic approach for describing
the function of components. Participant 8 also prefers the sketch generation
modality, citing its ability to describe the interface in more detail than the
semantic method. Participant 11 suggests that an hybrid solution between the
sketch and semantic modality may be a good option. Participant 11 recognizes
the limitations of both modalities, specifically the lack of semantic meaning
in the sketch generation and the difficulty of representing specific shapes in
the semantic method, and suggests that combining them may provide the best
solution. Participant 12 believes that the best method depends on the context.
They also believe that there is no one method that is better than others, but
they are complementary and useful to provide a complete description.

##  5\. Conclusions and Future Work

The paper evaluated the sketch and semantic drawing modalities by gathering
user feedback on their level of intuitiveness, ease of use, and time demand
during the generation process, as well as the quality and fidelity of the
generated mockups.

The results of the study indicate that users generally found the sketch
generation to be more intuitive and expressive compared to the semantic
generation. The study provides insight into how users perceive these
modalities and their relative strengths and weaknesses and can help inform
future design decisions for AI mockup generation tools. Many users have
suggested that a possible way to optimize the interaction between the user and
the system would be to combine the two approaches, taking advantage of the
strengths of each modality while minimizing their weaknesses. For example, by
using sketching to quickly and intuitively create a rough prototype, and
semantic drawing to add more detailed and expressive design elements,
designers would have a more flexible and powerful tool for creating their
mockups. Additionally, this approach could help designers to express their
ideas and designs more easily and accurately, which may result in a better
final product.

In conclusion, a mixed approach that combines sketching and semantic drawing
could be a promising solution for generating mockups of mobile applications
and additional research should explore this possibility.

## References

  * (1)
  * Beh (2022) 2022\.  Behance - Search Projects: Photos, videos, logos, illustrations and branding.  https://www.behance.net https://www.behance.net.
  * dri (2022) 2022\.  Dribbble - Discover the World’s Top Designers and Creative.  https://dribbble.com/ https://dribbble.com/.
  * Behrang et al. (2018) Farnaz Behrang, Steven P. Reiss, and Alessandro Orso. 2018\.  GUIfetch: Supporting App Design and Development through GUI Search. In _Proceedings of the 5th International Conference on Mobile Software Engineering and Systems_ (Gothenburg, Sweden) _(MOBILESoft ’18)_. Association for Computing Machinery, New York, NY, USA, 236–246.  https://doi.org/10.1145/3197231.3197244
  * Bonnardel (1999) Nathalie Bonnardel. 1999\.  Creativity in design activities: The role of analogies in a constrained cognitive environment. In _Proceedings of the 3rd conference on Creativity & cognition_. 158–165.
  * Calò and De Russis (2022) Tommaso Calò and Luigi De Russis. 2022.  Style-Aware Sketch-to-Code Conversion for the Web. In _Companion of the 2022 ACM SIGCHI Symposium on Engineering Interactive Computing Systems_ (Sophia Antipolis, France) _(EICS ’22 Companion)_. Association for Computing Machinery, New York, NY, USA, 44–47.  https://doi.org/10.1145/3531706.3536462
  * Deka et al. (2017) Biplab Deka, Zifeng Huang, Chad Franzen, Joshua Hibschman, Daniel Afergan, Yang Li, Jeffrey Nichols, and Ranjitha Kumar. 2017\.  Rico: A Mobile App Dataset for Building Data-Driven Design Applications. In _Proceedings of the 30th Annual Symposium on User Interface Software and Technology_ _(UIST ’17)_.
  * Eckert and Stacey (2000) Claudia Eckert and Martin Stacey. 2000.  Sources of inspiration: a language of design.  _Design Studies_ 21, 5 (2000), 523–538.  https://doi.org/10.1016/S0142-694X(00)00022-3
  * Gonçalves et al. (2014) M Gonçalves, C Cardoso, and P Badke-Schaub. 2014\.  What inspires designers? Preferences on inspirational approaches during idea generation. Des Stud 35 (1): 29–53.
  * Gonçalves et al. (2013) Milene Gonçalves, Carlos Cardoso, and Petra Badke-Schaub. 2013\.  What inspires designers? Preferences on inspirational approaches during idea generation.  _Design Studies_ 35 (01 2013).  https://doi.org/10.1016/j.destud.2013.09.001
  * Herring et al. (2009a) Scarlett R. Herring, Chia-Chen Chang, Jesse Krantzler, and Brian P. Bailey. 2009a.  Getting Inspired! Understanding How and Why Examples Are Used in Creative Design Practice. In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ (Boston, MA, USA) _(CHI ’09)_. Association for Computing Machinery, New York, NY, USA, 87–96.  https://doi.org/10.1145/1518701.1518717
  * Herring et al. (2009b) Scarlett R Herring, Chia-Chen Chang, Jesse Krantzler, and Brian P Bailey. 2009b.  Getting inspired! Understanding how and why examples are used in creative design practice. In _Proceedings of the SIGCHI conference on human factors in computing systems_. 87–96.
  * Hong et al. (2018) Seunghoon Hong, Dingdong Yang, Jongwook Choi, and Honglak Lee. 2018\.  Inferring semantic layout for hierarchical text-to-image synthesis. In _Proceedings of the IEEE conference on computer vision and pattern recognition_. 7986–7994.
  * Jansson and Smith (1991) David Jansson and Steven Smith. 1991.  Design fixation.  _Design Studies_ 12 (01 1991).  https://doi.org/10.1016/0142-694X(91)90003-F
  * Ko et al. (2022) Hyung-Kwon Ko, Subin An, Gwanmo Park, Seung Kwon Kim, Daesik Kim, Bohyoung Kim, Jaemin Jo, and Jinwook Seo. 2022\.  We-Toon: A Communication Support System between Writers and Artists in Collaborative Webtoon Sketch Revision. In _Proceedings of the 35th Annual ACM Symposium on User Interface Software and Technology_ (Bend, OR, USA) _(UIST ’22)_. Association for Computing Machinery, New York, NY, USA, Article 76, 14 pages.  https://doi.org/10.1145/3526113.3545612
  * Landay and Myers (2001) J.A. Landay and B.A. Myers. 2001.  Sketching interfaces: toward more human interface design.  _Computer_ 34, 3 (2001), 56–64.  https://doi.org/10.1109/2.910894
  * Landay and Myers (1995) James A. Landay and Brad A. Myers. 1995.  Interactive Sketching for the Early Stages of User Interface Design. In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ (Denver, Colorado, USA) _(CHI ’95)_. ACM Press/Addison-Wesley Publishing Co., USA, 43–50.  https://doi.org/10.1145/223904.223910
  * Lee et al. (2010) Brian Lee, Savil Srivastava, Ranjitha Kumar, Ronen Brafman, and Scott R. Klemmer. 2010\.  Designing with Interactive Example Galleries. In _Proceedings of the SIGCHI Conference on Human Factors in Computing Systems_ (Atlanta, Georgia, USA). Association for Computing Machinery, New York, NY, USA, 2257–2266.  https://doi.org/10.1145/1753326.1753667
  * Li et al. (2023) Yuheng Li, Haotian Liu, Qingyang Wu, Fangzhou Mu, Jianwei Yang, Jianfeng Gao, Chunyuan Li, and Yong Jae Lee. 2023\.  GLIGEN: Open-Set Grounded Text-to-Image Generation.  _CVPR_ (2023).
  * Marsh et al. (1996) Richard Marsh, Joshua Landau, and Jason Hicks. 1996\.  How Examples May (and May Not) Constrain Creativity.  _Memory and Cognition_ 24 (09 1996), 669–680.  https://doi.org/10.3758/BF03201091
  * Mozaffari et al. (2022) Mohammad Amin Mozaffari, Xinyuan Zhang, Jinghui Cheng, and Jin L.C. Guo. 2022.  GANSpiration: Balancing Targeted and Serendipitous Inspiration in User Interface Design with Style-Based Generative Adversarial Network. In _Proceedings of the 2022 CHI Conference on Human Factors in Computing Systems_ (New Orleans, LA, USA) _(CHI ’22)_. Association for Computing Machinery, New York, NY, USA, Article 537, 15 pages.  https://doi.org/10.1145/3491102.3517511
  * Park et al. (2019a) Taesung Park, Ming-Yu Liu, Ting-Chun Wang, and Jun-Yan Zhu. 2019a.  Semantic Image Synthesis with Spatially-Adaptive Normalization. In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_.
  * Park et al. (2019b) Taesung Park, Ming-Yu Liu, Ting-Chun Wang, and Jun-Yan Zhu. 2019b.  Semantic Image Synthesis With Spatially-Adaptive Normalization.  _2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ (2019), 2332–2341.
  * Qiao et al. (2022) Han Qiao, Vivian Liu, and Lydia Chilton. 2022.  Initial Images: Using Image Prompts to Improve Subject Representation in Multimodal AI Generated Art. In _Creativity and Cognition_ (Venice, Italy). Association for Computing Machinery, New York, NY, USA, 15–28.  https://doi.org/10.1145/3527927.3532792
  * Ritchie et al. (2011) Daniel Ritchie, Ankita Arvind Kejriwal, and Scott R. Klemmer. 2011.  D.Tour: Style-Based Exploration of Design Example Galleries. In _Proceedings of the 24th Annual ACM Symposium on User Interface Software and Technology_ (Santa Barbara, California, USA). Association for Computing Machinery, New York, NY, USA, 165–174.  https://doi.org/10.1145/2047196.2047216
  * Shneiderman (2020) Ben Shneiderman. 2020\.  Human-Centered Artificial Intelligence: Reliable, Safe & Trustworthy.  _International Journal of Human–Computer Interaction_ 36, 6 (2020), 495–504.  https://doi.org/10.1080/10447318.2020.1741118 arXiv:https://doi.org/10.1080/10447318.2020.1741118
  * Shneiderman (2022) B. Shneiderman. 2022\.  _Human-Centered AI_.  Oxford University Press.
  * Swearngin et al. (2018) Amanda Swearngin, Mira Dontcheva, Wilmot Li, Joel Brandt, Morgan Dixon, and Amy J. Ko. 2018\.  Rewire: Interface Design Assistance from Examples. In _Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems_ (Montreal QC, Canada). Association for Computing Machinery, New York, NY, USA, 1–12.  https://doi.org/10.1145/3173574.3174078
  * Thrash and Elliot (2003) Todd Thrash and Andrew Elliot. 2003.  Inspiration as a Psychological Construct.  _Journal of personality and social psychology_ 84 (05 2003), 871–89.  https://doi.org/10.1037/0022-3514.84.4.871
  * Thrash et al. (2014) Todd M Thrash, Emil G Moldovan, Victoria C Oleynick, and Laura A Maruskin. 2014.  The psychology of inspiration.  _Social and personality psychology compass_ 8, 9 (2014), 495–510.  https://doi.org/10.1111/spc3.12127
  * Tian et al. (2021) Nannan Tian, Yuan Liu, Bo Wu, and Xiaofeng Li. 2021\.  Colorization of Logo Sketch Based on Conditional Generative Adversarial Networks.  _Electronics_ 10, 4 (2021).  https://doi.org/10.3390/electronics10040497
  * Ueno and Satoh (2021) Michihiko Ueno and Shin’ichi Satoh. 2021.  Continuous and Gradual Style Changes of Graphic Designs with Generative Model. Association for Computing Machinery, New York, NY, USA, 280–289.  https://doi.org/10.1145/3397481.3450666
  * Wang et al. (2021) Bryan Wang, Gang Li, Xin Zhou, Zhourong Chen, Tovi Grossman, and Yang Li. 2021\.  Screen2Words: Automatic Mobile UI Summarization with Multimodal Learning. 498–510.  https://doi.org/10.1145/3472749.3474765
  * Wang et al. (2017) Ting-Chun Wang, Ming-Yu Liu, Jun-Yan Zhu, Andrew Tao, Jan Kautz, and Bryan Catanzaro. 2017\.  High-Resolution Image Synthesis and Semantic Manipulation with Conditional GANs.  _2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition_ (2017), 8798–8807.
  * Yurman and Reddy (2022) Paulina Yurman and Anuradha Venugopal Reddy. 2022.  Drawing Conversations Mediated by AI. In _Creativity and Cognition_ (Venice, Italy) _(C &C ’22)_. Association for Computing Machinery, New York, NY, USA, 56–70.  https://doi.org/10.1145/3527927.3531448
  * Zhu et al. (2020) Peihao Zhu, Rameen Abdal, Yipeng Qin, and Peter Wonka. 2020\.  Sean: Image synthesis with semantic region-adaptive normalization. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_. 5104–5113.


# Thermodynamic Susceptibilities for a Unitary Fermi Gas

Max Weiner  Department of Physics, University of Colorado, Boulder, Colorado
80309, USA

###### Abstract

The unitary Fermi gas provides a unique window into both cold atom experiments
and neutron star properties. There are major challenges in determining the
physical properties within a neutron star, both experimentally and
theoretically. However, there is a region within the crust of a neutron star
that resembles a gas of fermions that is recreated in a cold atom laboratory.
This is the so-called unitary Fermi gas characterized by a large negative
scattering length and small effective range. We set out to calculate from
first principles certain transport coefficients that appear at second order in
the hydrodynamic expansion. These particular transport coefficients are
obtained from two point correlation functions in flat space that do not
require analytic continuation from Euclidean to Minkowski space. We are
motivated by the potential to learn about neutron star properties from cold
atom experiments.

##  I Introduction

Currently, one of the prominent unresolved challenges in physics is
understanding the properties of matter under conditions of ultra-high density
and low temperature. Neutron stars (NS) are an embodiment of such a physical
regime, yet the conditions within a NS are far removed from terrestrial
experiments and observations of such objects are limited. Since their
discovery in 1967 [1] over 3,000 neutron stars have been cataloged, the vast
majority detected as pulsars [2]. This relatively small set of data, which
mostly relies on a particular alignment of Earth with the axis of the NS
beacon, puts into perspective the challenges experimentalists face.

Theoretical obstacles are no less significant. Calculating properties of
nuclear matter properties at high densities from first principles requires
solving the equations of quantum chromodynamics (QCD), the accepted theory of
nuclear interactions. Unfortunately, while efficient numerical methods exist
for calculating QCD matter properties in equilibrium at small densities, these
so-called lattice gauge theory calculations cannot be applied to high
densities due to the infamous sign problem. In other words, the equations for
calculating neutron star matter from first principles are known, but no one
knows how to solve them.

The purpose of this paper is to calculate NS properties from first principles
that can be measured in the lab. We are motivated by the fact that a region
within the crust of a NS is composed of neutron-rich nuclei surrounded by a
sea of free neutrons. Although free, these neutrons are strongly interacting,
characterized by a large s-wave scattering length $a_{s}\approx-18.5$ fm and a
small effective range $r_{\text{eff}}\approx 2.7$ fm [3, 4]. When the
scattering length is sent to infinity and the effective range to zero, all
scales drop out and the system exhibits universal behavior since it is
characterized only by its temperature, chemical potential, and density. Such a
system is called a unitary Fermi gas (UFG) and the neutron sea we are
interested in is close to this limit. Whether the fermions are atoms,
electrons, quarks or something else, the way these systems interact in bulk
will always be similar. UFGs have been studied extensively in cold atom
laboratories where the s-wave scattering length of atomic fermions can be
varied by applying an external magnetic field [5]. Utilizing Feshbach
resonances, the s-wave scattering length of certain atomic systems can be made
very large such that effectively $k_{F}|a_{s}|\to\infty$, and sufficiently
dilute such that $k_{F}r_{s}\to 0$, where the Fermi momentum is related to the
density of the system via $k_{F}\sim n^{-1/3}$ [6]. With this in mind, it may
be possible to probe NS properties using table-top experiments. In particular,
we are interested in transport properties of the UFG.

When a system is perturbed out of equilibrium, transport coefficients govern
how the system relaxes back to equilibrium. Well-known examples include
conductivities, viscosities, and diffusion constants [7]. Over the past
several years many experimental and theoretical studies have been devoted to
hydrodynamic properties of strongly interacting quantum fluids, such as the
UFG or the quark gluon plasma (QGP). It is interesting that these two systems
are considered the most perfect fluids in the universe [8], meaning their
shear viscosity to entropy density ratio is close to the conjectured quantum
lower bound $\eta/s\lesssim\frac{\hbar}{4\pi k_{B}}$ [9]. Transport
coefficients such as the shear viscosity are, in general, difficult to
calculate. However, in a remarkable paper Kovtun and Shukla [10] showed that a
certain family of transport coefficients, so-called thermodynamic transport
coefficients, are considerably easier to calculate because they do not require
analytic continuation from Euclidean to Minkowski spacetime. These quantities
are classified as “thermodynamic” because they can be obtained by calculating
certain thermodynamic susceptibilities in equilibrium.

Hydrodynamics is an effective low energy theory of systems near thermal
equilibrium [11, 12]. The expressions are valid on length scales much larger
than the characteristic microscopic sizes of the system, such as the mean free
path or correlation length. Hydrodynamics admits a derivative expansion in
terms of the fluid variables such as temperature, chemical potential, and
fluid velocity. At each order in this expansion are various transport
coefficients quantifying the response of the system to external perturbations.
Certain transport coefficients exhibit dissipation, leading to the generation
of entropy in non-equilibrium conditions, while others remain non-dissipative
and persist even in the limit of thermal equilibrium. Thermodynamic transport
coefficients appear at second order in the gradient expansion and are non-
dissipative in nature [13, 14].

These coefficients have been calculated for free uncharged relativistic
scalar, vector, and fermionic quantum field theories [15, 16, 10] as well as
free Dirac fermions at finite density [17]. For interacting theories, results
have been obtained for large $N$, $\mathcal{N}=4$ super Yang-Mills theory at
infinite coupling for both zero and finite density utilizing the AdS/CFT
conjecture [13, 18, 19, 20, 21]. For coupling strengths between the weak and
strong limits, analytic results were obtained in the large $N$ limit for the
$O(N)$ model [22, 23] and for cold non-relativistic fermions at finite density
[24]. In this latter paper, Lawrence and Romatschke calculated the second
order transport coefficient $\kappa$ for a UFG and we set out to calculate one
more.

We work in natural units, $\hbar=c=k_{B}=1$.

##  II Calculation

A system of fermions at unitarity begins with the Hamiltonian density in three
spatial dimensions,

$\displaystyle\mathcal{H}$
$\displaystyle=\sum_{s=\uparrow,\downarrow}\psi_{s}^{\dagger}(\textbf{x})\left(\frac{-\gradient^{2}}{2m}\right)\psi_{s}(\textbf{x})+\frac{4\pi
a_{s}}{m}\psi_{\uparrow}^{\dagger}(\textbf{x})\psi_{\downarrow}^{\dagger}(\textbf{x})\psi_{\downarrow}(\textbf{x})\psi_{\uparrow}(\textbf{x}),$
(1)

where $\psi_{s}^{\dagger}$, $\psi_{s}$ are the fermionic creation and
annhilation operators for states $s\in\\{\uparrow$, $\downarrow\\}$,
respectively, $m$ is the fermion mass, and $a_{s}<0$ is the s-wave scattering
length. Thermodynamic properties of this system are derived from its partition
function at finite temperature, $T$ and chemical potential, $\mu$. The
partition function in the path integral formulation is,

$\displaystyle Z(T,\mu)$
$\displaystyle=\int\mathcal{D}\psi_{s}^{\dagger}\mathcal{D}\psi_{s}e^{-S_{E}},$
(2) $\displaystyle S_{E}$ $\displaystyle=\int_{0}^{\beta}d\tau\int
d^{3}\textbf{x}\left[\psi_{s}^{\dagger}(\textbf{x})\left(\partial_{\tau}-\frac{\gradient^{2}}{2m}-\mu\right)\psi_{s}(\textbf{x})+\frac{4\pi
a_{s}}{m}\psi_{\uparrow}^{\dagger}(\textbf{x})\psi_{\downarrow}^{\dagger}(\textbf{x})\psi_{\downarrow}(\textbf{x})\psi_{\uparrow}(\textbf{x})\right],$

where $\beta\equiv 1/T$ is the inverse temperature and we utilize the Einstein
summation convention to sum over the repeated indices $s$. We can make the
path integral tractable by decoupling the four-fermi interaction with a
Hubbard-Stratonovich transformation and introducing a complex auxiliary field
$\zeta(\textbf{x})$,

$\displaystyle\exp\left(-\int_{\tau,\textbf{x}}\frac{4\pi
a_{s}}{m}\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\psi_{\downarrow}\psi_{\uparrow}\right)=\int\mathcal{D}\zeta
e^{\int_{\tau,\textbf{x}}\left(\frac{m}{4\pi
a_{s}}\zeta\zeta^{*}-i\psi_{\downarrow}\psi_{\uparrow}\zeta^{*}+i\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\zeta\right)}$
(3)

where we ignore an irrelevant normalization constant. Our partition function
then reads,

$\displaystyle Z(T,\mu)$
$\displaystyle=\int\mathcal{D}\psi_{s}^{\dagger}\mathcal{D}\psi_{s}\mathcal{D}\zeta
e^{-S_{E,\text{eff}}},$ (4) $\displaystyle S_{E,\text{eff}}$
$\displaystyle=\int_{\tau,\textbf{x}}\left[\psi_{s}^{\dagger}\left(\partial_{\tau}-\frac{\gradient^{2}}{2m}-\mu\right)\psi_{s}+i\psi_{\downarrow}\psi_{\uparrow}\zeta^{*}-i\psi_{\uparrow}^{\dagger}\psi_{\downarrow}^{\dagger}\zeta-\frac{m}{4\pi
a_{s}}\zeta\zeta^{*}\right].$

where all path integrals are now taken with respect to the effective Euclidean
action $S_{E,\text{eff}}$.

###  II.1 Thermodynamics

We may repackage the fermion bilinears into matrix form using Nambu-Gor’kov
spinors, $\Psi=\left(\psi_{\uparrow},\psi_{\downarrow}^{\dagger}\right)$, such
that the effective Euclidean action becomes,

$\displaystyle S_{E,\text{eff}}$
$\displaystyle=\int_{\tau,\textbf{x}}\left[\Psi^{\dagger}\left(\partial_{\tau}\mathbb{1}-\frac{\gradient^{2}}{2m}\sigma_{z}-\mu\sigma_{z}+i\zeta^{*}\sigma_{-}-i\zeta\sigma_{+}\right)\Psi-\frac{m\zeta\zeta^{*}}{4\pi
a_{s}}\right],$ (5)

where $\mathbb{1}$ is the $2\times 2$ identity matrix, $\sigma_{i}$ denote
Pauli matrices, and $\sigma_{\pm}\equiv\frac{1}{2}\left(\sigma_{x}\pm
i\sigma_{y}\right)$. The path integral is now quadratic in fermion fields and
can be integrated out exactly,

$\displaystyle Z(T,\mu)$ $\displaystyle=\int\mathcal{D}\zeta
e^{\int_{\tau,\textbf{x}}\left[\frac{m\zeta\zeta^{*}}{4\pi
a_{s}}+\ln\left(-\det G^{-1}\right)\right]}$ (6)

where
$G^{-1}(\zeta,\zeta^{*})=\begin{pmatrix}\partial_{\tau}-\frac{\gradient^{2}}{2m}-\mu&-i\zeta\\\
i\zeta^{*}&\partial_{\tau}+\frac{\gradient^{2}}{2m}+\mu\end{pmatrix}$ and the
minus sign in front of the determinant comes from the Jacobian
$\int\mathcal{D}\psi_{s}^{\dagger}\mathcal{D}\psi_{s}=-\int\mathcal{D}\Psi^{\dagger}\mathcal{D}\Psi$.

Up to this point we have not made any approximations. To perform the path
integral over the $\zeta$ fields we employ the non-perturbative resummation
method introduced by Romatschke [25] by expanding the auxiliary field around
the global zero mode,
$\zeta(\tau,\textbf{x})=i\Delta+\zeta^{\prime}(\tau,\textbf{x})$. In the $R0$
approximation (or equivalently the leading large $N$ expansion [26, 27]) we
neglect the field fluctuations $\zeta^{\prime}$ and assume the zero mode
$\Delta$ is real,

$\displaystyle Z_{R0}(T,\mu)=\int_{-\infty}^{\infty}d\Delta e^{\beta
V\left[\frac{m\Delta^{2}}{4\pi
a_{s}}+T\sum_{\omega_{n}}\int_{\textbf{p}}\ln\left(-\det\tilde{G}^{-1}\right)\right]}$
(7)

where the sum is over fermionic Matsubara frequencies $\omega_{n}=2\pi
T(n+1/2)$ with $n\in\mathbb{Z}$, $V$ is the volume of the system, and the
inverse propagator in frequency-momentum space is

$\displaystyle\tilde{G}^{-1}(\omega_{n},\textbf{p},\Delta)=\begin{pmatrix}i\omega_{n}+\epsilon_{\textbf{p}}-\mu&\Delta\\\
\Delta&i\omega_{n}-\epsilon_{\textbf{p}}+\mu\end{pmatrix}$ (8)

and $\epsilon_{\textbf{p}}\equiv\frac{\textbf{p}^{2}}{2m}$.

In the large volume limit, $Z_{R0}$ is solved exactly by the saddle point
condition with respect to $\Delta$,

$\displaystyle\frac{m\Delta}{4\pi
a_{s}}+T\sum_{\omega_{n}}\int\frac{d^{3}\textbf{p}}{(2\pi)^{3}}\frac{\Delta}{\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}+\omega_{n}^{2}}=0.$
(9)

We see that there is the trivial solution $\Delta=0$, but we proceed to look
for non-trivial solutions. After performing the sum over fermionic Matsubara
frequencies we obtain the condition

$\displaystyle\frac{-m}{4\pi
a_{s}}=\frac{1}{2}\int\frac{d^{3}\textbf{p}}{(2\pi)^{3}}\frac{\tanh\left(\frac{\sqrt{\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}}}{2T}\right)}{\sqrt{\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}}}.$
(10)

We note here that the poles of the Green’s function, whose inverse is given in
(8), define the spectrum located at
$i\omega_{n}=\sqrt{\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}}$. We see that
excitations require a minimum energy input of $\Delta$ which characterizes a
gapped spectrum. We refer to (10) as the gap equation and such systems are
characteristic of superfluids.

At zero temperature the Matsubara sum becomes an integral,
$T\sum_{\omega_{n}}\to\int\frac{d\omega}{2\pi}$, and the pressure is given by

$\displaystyle p=\frac{\ln Z_{R0}}{\beta V}=\frac{m\Delta^{2}}{4\pi
a_{s}}+\int\frac{d\omega
d^{3}\textbf{p}}{(2\pi)^{4}}\ln\left(\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}+\omega^{2}\right).$
(11)

The argument inside the the logarithm needs to be unitless so we regulate by
subtracting $\int d\omega\ln\omega^{2}$. This ensures the integral over
$\omega$ converges since in dimensional regularization $\int
d\omega\ln\omega^{2}=0$. Thus, in dimensional regularization the pressure is

$\displaystyle p=\frac{m\Delta^{2}}{4\pi
a_{s}}+\int\frac{d^{3}\textbf{p}}{(2\pi)^{3}}\sqrt{\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}}.$
(12)

The integral over momenta is now divergent but this is taken care of using a
well-known technique in dimensional regularization. We can expand the
integrand in (12) as a binomial series

$\displaystyle\sqrt{\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}}=\sum_{n=0}^{\infty}\binom{\frac{1}{2}}{n}\frac{(-2\mu\epsilon_{\textbf{p}})^{n}}{\left(\Delta^{2}+\mu^{2}+\epsilon_{\textbf{p}}^{2}\right)^{n-\frac{1}{2}}}$
(13)

and each integral is solved in dimensional regularization using [28],

$\displaystyle\int\frac{d^{D}\textbf{p}}{(2\pi)^{D}}\frac{\epsilon_{\textbf{p}}^{\alpha}}{(\epsilon_{\textbf{p}}^{2}+A^{2})^{\frac{\beta}{2}}}=\frac{\Gamma\left(\frac{D+2\alpha}{4}\right)\Gamma\left(\frac{2\beta-2\alpha-D}{4}\right)}{2\Gamma\left(\frac{D}{2}\right)\Gamma\left(\frac{\beta}{2}\right)}\left(\frac{mA}{2\pi}\right)^{\frac{D}{2}}A^{\alpha-\beta}.$
(14)

We find that in the limit $D\to 3$,

$\displaystyle\int\frac{d^{3}\textbf{p}}{(2\pi)^{3}}\sqrt{\Delta^{2}+(\epsilon_{\textbf{p}}-\mu)^{2}}=\frac{2}{5}\frac{\mu(2m\mu)^{3/2}}{3\pi^{2}}g(y),$
(15)

where $y\equiv\frac{\mu}{\sqrt{\Delta^{2}+\mu^{2}}}$, the function is
$g(y)=y^{-5/2}\left[(4y^{2}-3)E\left(\frac{1+y}{2}\right)+\frac{3+y-4y^{2}}{2}K\left(\frac{1+y}{2}\right)\right]$,
and $E(x)$, $K(x)$, are the complete elliptic integrals of the first and
second kind, respectively. At zero temperature, the non-trivial solution to
(10) fulfills

$\displaystyle
2E\left(\frac{1+y}{2}\right)-K\left(\frac{1+y}{2}\right)=\frac{\pi
y^{1/2}}{2\sqrt{2m\mu}a_{s}}$ (16)

One can check that the non-trivial solution to (10) has lower free energy than
the trivial solution $\Delta=0$ for all $a_{s}<0$. For the unitary fermi gas
$a_{s}\to-\infty$ and the gap equation becomes
$K\left(\frac{1+y}{2}\right)=2E\left(\frac{1+y}{2}\right)$ corresponding to
the numerical soulution $y\approx 0.652$ or $\Delta\approx 1.162\mu$. It is
noted in [24] that for small scattering lengths the solution is close to
$y\approx 1$ and (16) can be expanded around this point to obtain the analytic
result

$\displaystyle\Delta=\mu\,e^{-\frac{m}{2\pi a_{s}\nu}-2+3\ln
2},\quad|a_{s}|\ll 1$ (17)

Lawrence and Romatschke point out that this analytic solution is within ten
percent of the numerical solution of (16) for all $a_{s}<0$. For the cold
unitary Fermi gas,

$\displaystyle\lim_{a_{s}\to-\infty}\Delta=\mu e^{-2+3\ln 2}\approx 1.083\mu$
(18)

we see that the R0 approximation gives the correct order of magnitude:
$\Delta/\mu=\mathcal{O}(1)$ [29]. At unitarity,
$g(y)=y^{-3/2}E(\frac{1+y}{2})$ and so the pressure is given by,

$\displaystyle p$
$\displaystyle=\frac{2}{5}\frac{\mu(2m\mu)^{3/2}}{3\pi^{2}}\frac{E(\frac{1+y}{2})}{y^{3/2}}\Bigg{|}_{y\equiv\left(1+\frac{\Delta^{2}}{\mu^{2}}\right)^{-1/2}}\approx
2.2p_{\text{free}}$ (19)

where $p_{\text{free}}=\frac{2}{5}\frac{\mu(2m\mu)^{\frac{3}{2}}}{3\pi^{2}}$
is the pressure of a cold _non-interacting_ fermi gas. The universality of a
unitary fermi gas dictates $p=\xi^{-3/2}p_{\text{free}}$ where $\xi$ is a
constant known as the Bertsch parameter. In our approximation we find
$\xi_{R0}\approx 0.59$ (which is what one obtains from a mean-field theory
calculation [26]) which should be compared to the measured value of
$\xi\approx 0.38$ [30, 31].

From the pressure we can obtain the number density,

$\displaystyle\lim_{a_{s}\to-\infty}n(\mu)=\frac{\partial
p}{\partial\mu}=\frac{(2m\mu)^{3/2}}{3\pi^{2}}\xi^{-3/2}$ (20)

which is used to define the Fermi momentum

$\displaystyle k_{F}\equiv(3\pi^{2}n)^{1/3}.$ (21)

###  II.2 Transport Coefficients

Hydrodynamics provides an effective low energy description of physical systems
near thermal equilibrium [11]. Hydrodynamics describes the dynamics of
conserved macroscopic quantities of an underlying microscopic theory. The
conserved hydrodynamic variable $T^{\mu\nu}$ (the energy-momentum tensor) is
related to the symmetries of the microscopic theory via Noether’s theorem.
Further, we assume that these variables can be expressed as functions of a
local temperature $T(x)$, a local chemical potential $\mu(x)$, and a local
fluid four-velocity $u_{\mu}(x)$ normalized to $u^{2}=-1$.

The energy-momentum tensor can be decomposed in the following way [12],

$\displaystyle
T^{\mu\nu}=\mathcal{E}u^{\mu}u^{\nu}+\mathcal{P}\Delta^{\mu\nu}+\mathcal{Q}^{\mu}u^{\nu}+\mathcal{Q}^{\nu}u^{\mu}+\mathcal{T}^{\mu\nu}$
(22)

where $\Delta_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$ is a projector transverse to
the fluid four-velocity (e.g. $u^{\mu}\Delta_{\mu\nu}=0$),
$\mathcal{E}=u_{\mu}u_{\nu}T^{\mu\nu}$ is the energy density,
$\mathcal{P}=\frac{1}{3}\Delta_{\mu\nu}T^{\mu\nu}$ is the pressure, the energy
flux $\mathcal{Q}^{\mu}=-\Delta^{\mu\alpha}T_{\alpha\beta}u^{\beta}$ is
transverse to $u_{\mu}$, and the stress
$\mathcal{T}^{\mu\nu}=T^{\langle\mu\nu\rangle}$ is transverse to $u_{\mu}$,
symmetric, and traceless. The angled brackets denote the symmetric transverse
traceless part of a tensor,
$X_{\langle\mu\nu\rangle}=\frac{1}{2}\left(\Delta_{\mu\alpha}\Delta_{\nu\beta}+\Delta_{\mu\beta}\Delta_{\nu\alpha}-\frac{2}{3}\Delta_{\mu\nu}\Delta_{\alpha\beta}\right)X^{\alpha\beta}$.

Hydrodynamics assumes that the quantities
$\mathcal{E},\mathcal{P},\mathcal{Q}^{\mu}$, and $\mathcal{T}^{\mu\nu}$ can be
expanded in terms of the hydrodynamic variables $u^{\mu}$, $T$, and $\mu$ and
their derivatives. These expressions are called the constitutive relations
[12]. Gradients of hydrodynamic variables correspond to perturbations out of
equilibrium and each term is coupled to a quantity called a transport
coefficient. Transport coefficients dictate the real-time evolution for
equilibrium to be restored, familiar examples being conductivities,
viscosities, and diffusion coefficients. These well-known quantities, however,
control the linear (first-order) response of the system to a perturbation
whereas non-linear contributions (second-order, third-order, etc.) will also
be present.

In general, transport coefficients of a strongly interacting quantum many-body
system are difficult to calculate. However, for a particular subset of second-
order transport coefficients, so-called thermodynamic transport coefficients,
Kovtun & Shukla showed such quantities are considerably easier to calculate
from two-point correlation functions in flat space without the need to
analytically continue from Euclidean to Minkowski spacetime [10]. These
transport coefficients are non-dissipative in nature and exist even in the
limit of thermal equilibrium [17]. As first noted in [32, 14], these non-
dissipative transport coefficients follow from an equilibrium generating
functional which itself admits a derivative expansion, with various
thermodynamic susceptibilities appearing as coefficients of different terms in
the expansion. The variation of the equilibrium generating functional with
respect to external sources produces the constitutive relations for the
conserved currents in the system, which contain various transport coefficients
that do not vanish in the equilibrium limit. Thus, these second-order
thermodynamic transport coefficients are linear combinations of the
thermodynamic susceptibilities. So we proceed to calculate susceptibilities
and use the dictionary provided in [10] to obtain transport coefficients.
Susceptibilities are denoted as $f_{i}$, $i=\\{1,2,3,...\\}$ and the
constitutive relations are,

$\displaystyle\mathcal{E}$
$\displaystyle=\epsilon+\left(f_{1}^{\prime}-f_{1}\right)R+\left(4f_{1}^{\prime}+2f_{1}^{\prime\prime}-f_{2}-f_{2}^{\prime}\right)a^{2}$
$\displaystyle\quad+\left(f_{1}^{\prime}-f_{2}-3f_{3}+f_{3}^{\prime}\right)\Omega^{2}-2\left(f_{1}+f_{1}^{\prime}-f_{2}\right)u^{\alpha}R_{\alpha\beta}u^{\beta},$
(23a) $\displaystyle\mathcal{P}$
$\displaystyle=p+\tfrac{1}{3}f_{1}R-\tfrac{1}{3}\left(2f_{1}^{\prime}+f_{3}\right)\Omega^{2}-\tfrac{1}{3}\left(2f_{1}^{\prime}+4f_{1}^{\prime\prime}-f_{2}\right)a^{2}+\tfrac{2}{3}\left(2f_{1}^{\prime}-f_{1}\right)u^{\alpha}R_{\alpha\beta}u^{\beta},$
(23b) $\displaystyle\mathcal{Q}_{\mu}$
$\displaystyle=\left(f_{1}^{\prime}+2f_{3}^{\prime}\right)\epsilon_{\mu\lambda\rho\sigma}a^{\lambda}u^{\rho}\Omega^{\sigma}+\left(2f_{1}+4f_{3}\right)\Delta^{\rho}_{\mu}R_{\rho\sigma}u^{\sigma},$
(23c) $\displaystyle\mathcal{T}^{\mu\nu}$
$\displaystyle=\left(4f_{1}^{\prime}+2f_{1}^{\prime\prime}-2f_{2}\right)a_{\langle\mu}a_{\nu\rangle}-\tfrac{1}{2}\left(f_{1}^{\prime}-4f_{3}\right)\Omega_{\langle\mu}\Omega_{\nu\rangle}+2f_{1}^{\prime}u^{\alpha}R_{\alpha\langle\mu\nu\rangle\beta}u^{\beta}-2f_{1}R_{\langle\mu\nu\rangle},$
(23d)

where

$\displaystyle f_{n}^{\prime}$ $\displaystyle\equiv Tf_{n,T}+\mu f_{n,\mu}$
$\displaystyle f_{n}^{\prime\prime}$ $\displaystyle\equiv T^{2}f_{n,T,T}+2\mu
Tf_{n,T,\mu}+\mu^{2}f_{n,\mu,\mu}$

and the comma denotes the partial derivative with respect to the argument that
follows. Terms appearing in 23 include the Riemann scalar $R$, Riemann tensor
$R^{\mu\nu}$, acceleration $a^{\mu}\equiv u^{\lambda}\nabla_{\lambda}u^{\mu}$,
and vorticity
$\Omega^{\mu}\equiv\epsilon^{\mu\nu\alpha\beta}u_{\nu}\nabla_{\alpha}u_{\beta}$.111With
the convention
$\epsilon^{\mu\nu\alpha\beta}=\varepsilon^{\mu\nu\alpha\beta}/\sqrt{-g}$ and
$\varepsilon^{0123}=1$.

We set out to calculate $f_{3}$ given by [10],

$\displaystyle f_{3}$ $\displaystyle=\frac{1}{4}\lim_{\textbf{k}\to
0}\frac{\partial^{2}}{\partial k^{2}}\Big{(}-\langle
T_{E}^{01}T_{E}^{01}\rangle(\textbf{k})+\langle
T_{E}^{12}T_{E}^{12}\rangle(\textbf{k})\Big{)}.$ (24)

where $T_{E}^{\mu\nu}$ is the energy-momentum tensor of the underlying quantum
system in Euclidean space. Without loss of generality we assume the external
momentum is oriented in the $z$-direction, $\vec{k}=k\hat{z}$. We wish to
calculate the Euclidean correlator

$\displaystyle C_{E}(\omega,k\hat{\textbf{z}})$ $\displaystyle\equiv-\langle
T_{E}^{01}T_{E}^{01}\rangle(\omega,k\hat{\textbf{z}})+\langle
T_{E}^{12}T_{E}^{12}\rangle(\omega,k\hat{\textbf{z}}).$ (25)

In particular, we will calculate

$\displaystyle f_{3}=\frac{1}{4}\lim_{k\to 0}\frac{\partial^{2}}{\partial
k^{2}}C_{E}(\omega_{n}=0,k)\Bigg{|}_{k=0}$ (26)

###  II.3 Stress-Tensor Correlators in the Cold Superfluid Fermi Gas

The Eucildean stress-tensor two-point correlator in the path integral
formulation is

$\displaystyle C_{E}(x-y)$
$\displaystyle=Z^{-1}\int\mathcal{D}\psi^{\dagger}\mathcal{D}\psi
e^{-S_{E}}\left[-T_{E}^{01}(x)T_{E}^{01}(y)+T_{E}^{12}(x)T_{E}^{12}(y)\right].$
(27)

We obtain $T_{E}^{\mu\nu}(x)$ by starting with the energy-momentum tensor of
relativistic Dirac fermions and taking the non-relativistic limit (e.g. where
the energy $m$ is much higher than any other energy scale in our system). We
find

$\displaystyle T_{E}^{01}(x)$
$\displaystyle=\frac{1}{4}\left[\psi_{\uparrow}^{\dagger}\left(\partial_{1}\psi_{\uparrow}\right)+\psi_{\downarrow}^{\dagger}\left(\partial_{1}\psi_{\downarrow}\right)-\left(\partial_{1}\psi_{\uparrow}^{\dagger}\right)\psi_{\uparrow}-\left(\partial_{1}\psi_{\downarrow}^{\dagger}\right)\psi_{\downarrow}\right]+\mathcal{O}(1/m)$
(28)

and $T_{E}^{12}(x)\propto\frac{1}{m}$, where we ignore all $\mathcal{O}(1/m)$
and higher terms. Expressing the stress-tensor component using Nambu-Gor’kov
spinors, $\Psi=\begin{pmatrix}\psi_{\uparrow}\\\
\psi_{\downarrow}^{\dagger}\end{pmatrix}$, our path integral becomes straight-
forward to calculate. Since a vanishing total divergence of fields implies
$\left(\partial_{1}\psi_{s}^{\dagger}\right)\psi_{s}=-\psi_{s}^{\dagger}\left(\partial_{1}\psi_{s}\right)$,
$T^{01}_{E}$ simplifies to

$\displaystyle
T_{E}^{01}(x)=\frac{1}{2}\Psi^{\dagger}(x)\partial_{1}\Psi(x)+\mathcal{O}(1/m).$
(29)

Using the same auxiliary fields as in (4) and to leading order in $1/m$, our
path integral is

$\displaystyle C_{E}(x-y)$
$\displaystyle=\frac{1}{4Z}\int\mathcal{D}\Psi^{\dagger}\mathcal{D}\Psi\mathcal{D}\zeta\Tr\left[\partial_{1}^{y}G(y-x)\partial_{1}^{x}G(x-y)\right]$
(30)

where the trace is over spinor indices and

$\displaystyle\left<\Psi(x)\Psi^{\dagger}(y)\right>=G(x-y).$ (31)

In the R0 approximation we restrict the auxiliary fields to only the global
zero mode. The R0-propagator from (8) is

$\displaystyle
G(\omega_{n},\textbf{p},\Delta)=\frac{1}{(\epsilon_{\textbf{p}}-\mu)^{2}+\omega_{n}^{2}+\Delta^{2}}\begin{pmatrix}i\omega_{n}+\epsilon_{\textbf{p}}-\mu&\Delta\\\
\Delta&i\omega_{n}-\epsilon_{\textbf{p}}+\mu\end{pmatrix}$ (32)

and the stress-tensor correlator in frequency-momentum space
$C_{E}(k_{E},\textbf{k})$ at zero temperature is

$\displaystyle
C_{E}(k_{E},\textbf{k})=-\int_{-\infty}^{\infty}d\Delta\frac{e^{\beta
Vp}}{4Z}\int\frac{d^{4}p}{(2\pi)^{4}}p_{1}^{2}\Tr\left[G(k_{E}+\omega,\textbf{k}+\textbf{p})G(\omega,\textbf{p})\right]$
(33)

where $\int d\Delta e^{\beta Vp}=Z_{R0}$ and the integral over $\Delta$ will
once again restrict its value to the saddle point condition approximated in
(18). Upon performing the trace over spinor components and restricting to zero
external frequency $k_{E}=0$, our correlator is

$\displaystyle C_{E}(k_{E}=0,\textbf{k})$
$\displaystyle=-\frac{1}{2}\int\frac{d\omega}{2\pi}\frac{d^{3}\textbf{p}}{(2\pi)^{3}}p_{1}^{2}\frac{(\epsilon_{\textbf{k}+\textbf{p}}-\mu)(\epsilon_{\textbf{p}}-\mu)+\Delta^{2}-\omega^{2}}{[(\epsilon_{\textbf{k}+\textbf{p}}-\mu)^{2}+\Delta^{2}+\omega^{2}][(\epsilon_{\textbf{p}}-\mu)^{2}+\Delta^{2}+\omega^{2}]}$
(34)

After $\lim_{\textbf{k}\to 0}\frac{\partial^{2}}{\partial k^{2}}C_{E}$ and
averaging over angles,

$\displaystyle C_{E}^{\prime\prime}(K=0)$
$\displaystyle=-\int_{P}\left\\{\tfrac{32\epsilon_{\textbf{p}}^{2}\left(\epsilon_{\textbf{p}}-\mu\right){}^{4}}{15\left[\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}+\omega^{2}\right]^{4}}-\tfrac{4\epsilon_{\textbf{p}}\left(13\epsilon_{\textbf{p}}-5\mu\right)\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}}{15\left[\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}+\omega^{2}\right]^{3}}+\tfrac{\epsilon_{\textbf{p}}\left(19\epsilon_{\textbf{p}}-15\mu\right)}{15\left[\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}+\omega^{2}\right]^{2}}\right\\}$
$\displaystyle\quad-\Delta^{2}\int_{P}\left\\{\tfrac{32\epsilon_{\textbf{p}}^{2}\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}}{15\left[\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}+\omega^{2}\right]^{4}}-\tfrac{4\epsilon_{\textbf{p}}\left(7\epsilon_{\textbf{p}}-5\mu\right)}{15\left[\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}+\omega^{2}\right]^{3}}\right\\}$
(35) $\displaystyle\equiv C_{E}^{a^{\prime\prime}}+C_{E}^{b^{\prime\prime}}$
(36)

where the top line of (35) is $C_{E}^{a^{\prime\prime}}$ and the bottom line
proportional to $\Delta^{2}$ is $C_{E}^{b^{\prime\prime}}$. Note that the
integral in the bottom line of (35) is finite so that
$C_{E}^{b^{\prime\prime}}$ vanishes in the limit $\Delta\to 0$. We can rewrite
the denominator of each term as

$\displaystyle\frac{1}{\left[\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}+\omega^{2}\right]^{n}}=\frac{(-1)^{n-1}}{\left(n-1\right)!}\frac{\partial^{n-1}}{\partial(\Delta^{2})^{n-1}}\frac{1}{\left[\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right){}^{2}+\omega^{2}\right]}$
(37)

for integers $n\geq 1$ so that we are left with an integral over $\omega$ with
simple poles at $\pm
i\sqrt{\Delta^{2}+\left(\epsilon_{\textbf{p}}-\mu\right)^{2}}$. The next step
is to integrate over momenta utilizing equations (13) and (14) which, in the
unitary limit $a_{s}\to-\infty$, leads to

$\displaystyle C_{E}^{a^{\prime\prime}}(K=0)$
$\displaystyle=-\frac{(2m\mu)^{3/2}}{48\pi^{2}\mu}\frac{E\left(\frac{1+y}{2}\right)}{y^{3/2}}y^{2}=-\frac{n(\mu)}{16\mu}y^{2}$
(38) $\displaystyle C_{E}^{b^{\prime\prime}}(K=0)$
$\displaystyle=\frac{(2m\mu)^{3/2}}{720\pi^{2}\mu}\frac{E\left(\frac{1+y}{2}\right)}{y^{3/2}}\frac{8\mu^{2}+23\Delta^{2}}{\Delta^{2}}y^{2}=\frac{n(\mu)}{16\mu}\frac{8\mu^{2}+23\Delta^{2}}{15\Delta^{2}}y^{2}$
(39) $\displaystyle C^{\prime\prime}_{E}(K=0)$
$\displaystyle=\frac{n(\mu)}{30\mu}\left(\frac{\mu^{2}}{\Delta^{2}}\right)$
(40) $\displaystyle f_{3}$
$\displaystyle=\frac{1}{4}C^{\prime\prime}_{E}(K=0)=\frac{n(\mu)}{120\mu}\left(\frac{\mu^{2}}{\Delta^{2}}\right).$
(41)

This is our final result for the thermodynamic susceptibility $f_{3}$. Second
order thermodynamic transport coefficients are linear combinations of second
order thermodynamic susceptibilities and their derivatives. These transport
coefficients, as formulated in [13, 11], and their relation to the
susceptibilities are given in [10]. For example, the second order transport
coefficient $\lambda_{3}$ couples to the vorticity and, to leading order in
the UFG, is given by $\lambda_{3}\approx-8f_{3}$.

##  III Conclusion

The unitarity Fermi gas is an important example of a strongly correlated
quantum fluid. Measurements of both equilibrium and non-equilibrium properties
serves as a crucial reference point for a diverse range of physical systems,
spanning from the study of dilute neutron matter in neutron stars to the
investigation of the quark gluon plasma in relativistic heavy ion collisions.

Neutron star matter cannot be recreated in the lab and cannot be studied using
lattice gauge theory because of the sign problem. Our result, equation (41),
is an analytic solution from first principles and attempts to shed light on
these issues. Because thermodynamic transport coefficients (which are linear
combinations of thermodynamic susceptibilities as illustrated in equations
(23) do not require analytic continuation from Euclidean to Minkowski
spacetime, our calculation is amenable to lattice calculations. Such
calculations could further our understanding of such transport properties in
NS crusts. Ultracold atomic experiments are known for their extraordinarily
precise measurements. Since they are capable of creating UFGs in the lab,
perhaps there is (or will be in the future) a means of measuring second-order
transport coefficients.

It has been suggested that superfluid vortices are relevant to NS glitches.
Since $f_{3}$ couples to the vorticity (at second order) further study as to
its relation to NS glitches might be worthwhile, although the relation between
vorticity and superfluid vortices is unclear.

Our results may be of assistance to those studying the NS equation of state
(EoS). The authors of [33] propose a “unitary gas conjecture” which states
that the energy of a UFG serves as a lower bound for pure neutron matter. This
conjecture imposes constraints on EoS parameters.

There are six other thermodynamic susceptibilities that can be calculated for
the unitary fermi gas [17], one of which has already been calculated [24].
Calculating the rest of these transport coefficients would be the logical next
step to this paper. The susceptibilities $f_{1}$, $f_{2}$, and $f_{3}$
characterize the constitutive relations in equations (23), where $f_{2}$ has
not been calculated for the UFG. The reason for this is that the two-point
correlation functions used to determine $f_{2}$ consist of diagonal components
of $T^{\mu\nu}$ which make it much more complicated. We leave this for future
work.

##  IV Acknowledgments

This work was supported by the Department of Energy, DOE award No DE-
SC0017905. I would like to thank Paul Romatschke for suggesting such an
exciting and rich research topic as well as the enormous amount of help to
make this work possible.

##  Appendix A Free and Non-Relativistic Limit

We compare our result in the free limit ($\Delta\to 0$) with the non-
relativistic result in [17]. In that paper, Shukla calculated all
thermodynamic transport coefficients for a relativistic system of free
fermions. There he found,

$\displaystyle f_{3}$
$\displaystyle=-\frac{|\mu_{r}|}{96\pi^{2}}\sqrt{\mu_{r}^{2}-m^{2}}$ (42)

where we distinguish between the relativistic chemical potential $\mu_{r}$ and
the non-relativistic chemical potential used in this paper, $\mu$. In the non-
relativistic limit, the chemical potential is only slightly larger than the
mass,

$\displaystyle\mu_{r}\approx m+\mu$ (43)

where $\mu\ll m$. We can then expand $f_{3}$ around $\mu=0$,

$\displaystyle f_{3}$
$\displaystyle\approx-\frac{m^{3/2}\sqrt{\mu}}{48\sqrt{2}\pi^{2}}=-\frac{(2m\mu)^{3/2}}{192\pi^{2}\mu}=-\frac{n_{\text{free}}}{64\mu}$
(44)

where $n_{\text{free}}=\frac{(2m\mu)^{3/2}}{3\pi^{2}}$ is the density of a
degenerate Fermi gas. Looking at Eqns (38) & (41) and noting that
$\lim_{\Delta\to 0}$ is equivalent to $\lim_{y\to 1}$,

$\displaystyle f_{3}=\frac{1}{4}\lim_{\Delta\to
0}C_{E}^{a^{\prime\prime}}(K=0)$ $\displaystyle=\lim_{y\to
1}-\frac{(2m\mu)^{3/2}}{192\pi^{2}\mu}\frac{E\left(\frac{1+y}{2}\right)}{y^{3/2}}=-\frac{(2m\mu)^{3/2}}{192\pi^{2}\mu}$
(45)

which agrees with the non-relativistic limit obtained in [17]. In order that
our results agree with free fermions in the non-relativistic limit, we must
also show that $\lim_{\Delta\to 0}C_{E}^{b^{\prime\prime}}(K=0)=0$.

To this end, we must look to [34, 35] which conclude calculations (at zero
temperature and finite chemical potential) that involve integrals over momenta
and frequencies do not commute. That is, the physically correct results are
obtained by integrating over momenta first, and not vice versa as is commonly
done. To this point we also note that when one sets $\Delta=0$ at the outset
and goes on to calculate $\langle T^{01}T^{01}\rangle$ to leading order,
integrating over frequencies first produces zero, which disagrees with [17].
It is straightforward from eqn (35) that $\lim_{\Delta\to
0}C_{E}^{b^{\prime\prime}}(K=0)=0$ since it is the product of a finite
integral and $\Delta^{2}$, so it must vanish in this limit.

## References

  * Hewish  _et al._ [1968] A. Hewish, S. J. Bell, J. D. H. Pilkington, P. F. Scott, and R. A. Collins, Observation of a Rapidly Pulsating Radio Source, Nature (London) 217, 709 (1968).
  * Manchester  _et al._ [2005] R. N. Manchester, G. B. Hobbs, A. Teoh, and M. Hobbs, The australia telescope national facility pulsar catalogue, The Astronomical Journal 129, 1993 (2005).
  * Chen  _et al._ [2008] Q. Chen, C. R. Howell, T. S. Carman, W. R. Gibbs, B. F. Gibson, A. Hussein, M. R. Kiser, G. Mertens, C. F. Moore, C. Morris, A. Obst, E. Pasyuk, C. D. Roper, F. Salinas, H. R. Setze, I. Slaus, S. Sterbenz, W. Tornow, R. L. Walter, C. R. Whiteley, and M. Whitton, Measurement of the neutron-neutron scattering length using the $\pi$-d capture reaction, Phys. Rev. C 77, 054002 (2008).
  * González Trotter  _et al._ [1999] D. E. González Trotter, F. Salinas, Q. Chen, A. S. Crowell, W. Glöckle, C. R. Howell, C. D. Roper, D. Schmidt, I. Šlaus, H. Tang, W. Tornow, R. L. Walter, H. Witała, and Z. Zhou, New Measurement of the 1S0 Neutron-Neutron Scattering Length Using the Neutron-Proton Scattering Length as a Standard, Phys. Rev. Lett. 83, 3788 (1999), arXiv:nucl-ex/9904011 [nucl-ex] .
  * Ketterle and Zwierlein [2008] W. Ketterle and M. W. Zwierlein, Making, probing and understanding ultracold Fermi gases, Nuovo Cimento Rivista Serie 31, 247 (2008), arXiv:0801.2500 [cond-mat.other] .
  * Gurarie and Radzihovsky [2007] V. Gurarie and L. Radzihovsky, Resonantly paired fermionic superfluids, Annals of Physics 322, 2 (2007), arXiv:cond-mat/0611022 [cond-mat.supr-con] .
  * Landau and Lifshitz [1959] L. D. Landau and E. M. Lifshitz,  _Fluid mechanics_ (1959).
  * Schäfer and Teaney [2009] T. Schäfer and D. Teaney, Nearly perfect fluidity: from cold atomic gases to hot quark gluon plasmas, Reports on Progress in Physics 72, 126001 (2009), arXiv:0904.3107 [hep-ph] .
  * Kovtun  _et al._ [2005] P. K. Kovtun, D. T. Son, and A. O. Starinets, Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics, Phys. Rev. Lett. 94, 111601 (2005), arXiv:hep-th/0405231 [hep-th] .
  * Kovtun and Shukla [2018] P. Kovtun and A. Shukla, Kubo formulas for thermodynamic transport coefficients, Journal of High Energy Physics 2018, 7 (2018), arXiv:1806.05774 [hep-th] .
  * Romatschke and Romatschke [2019] P. Romatschke and U. Romatschke,  _Relativistic Fluid Dynamics In and Out of Equilibrium_, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2019) arXiv:1712.05815 [nucl-th] .
  * Kovtun [2012] P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, Journal of Physics A Mathematical General 45, 473001 (2012), arXiv:1205.5040 [hep-th] .
  * Baier  _et al._ [2008] R. Baier, P. Romatschke, D. Thanh Son, A. O. Starinets, and M. A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance, and holography, Journal of High Energy Physics 2008, 100 (2008), arXiv:0712.2451 [hep-th] .
  * Jensen  _et al._ [2012] K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz, and A. Yarom, Towards Hydrodynamics without an Entropy Current, Phys. Rev. Lett. 109, 101601 (2012), arXiv:1203.3556 [hep-th] .
  * Romatschke and Son [2009] P. Romatschke and D. T. Son, Spectral sum rules for the quark-gluon plasma, Phys. Rev. D 80, 065021 (2009), arXiv:0903.3946 [hep-ph] .
  * Moore and Sohrabi [2012] G. D. Moore and K. A. Sohrabi, Thermodynamical second-order hydrodynamic coefficients, Journal of High Energy Physics 2012, 148 (2012), arXiv:1210.3340 [hep-ph] .
  * Shukla [2019] A. Shukla, Equilibrium thermodynamic susceptibilities for a dense degenerate Dirac field, Phys. Rev. D 100, 096010 (2019), arXiv:1906.02334 [hep-th] .
  * Bhattacharyya  _et al._ [2008] S. Bhattacharyya, S. Minwalla, V. E. Hubeny, and M. Rangamani, Nonlinear fluid dynamics from gravity, Journal of High Energy Physics 2008, 045 (2008), arXiv:0712.2456 [hep-th] .
  * Finazzo  _et al._ [2015] S. I. Finazzo, R. Rougemont, H. Marrochio, and J. Noronha, Hydrodynamic transport coefficients for the non-conformal quark-gluon plasma from holography, Journal of High Energy Physics 2015, 51 (2015), arXiv:1412.2968 [hep-ph] .
  * Grozdanov and Starinets [2015] S. Grozdanov and A. O. Starinets, On the universal identity in second order hydrodynamics, Journal of High Energy Physics 2015, 7 (2015), arXiv:1412.5685 [hep-th] .
  * Grieninger and Shukla [2021] S. Grieninger and A. Shukla, Second order equilibrium transport in strongly coupled N = 4 supersymmetric SU(Nc) Yang-Mills plasma via holography, Journal of High Energy Physics 2021, 108 (2021), arXiv:2105.08673 [hep-th] .
  * Romatschke [2019a] P. Romatschke, Analytic transport from weak to strong coupling in the O (N ) model, Phys. Rev. D 100, 054029 (2019a), arXiv:1905.09290 [hep-th] .
  * Weiner and Romatschke [2022] M. Weiner and P. Romatschke, Determining all thermodynamic transport coefficients for an interacting large N quantum field theory, (2022), arXiv:2208.10502 .
  * Lawrence and Romatschke [2023] S. Lawrence and P. Romatschke, Gravitational-wave-to-matter coupling of superfluid Fermi gases near unitarity, Phys. Rev. A 107, 033327 (2023), arXiv:2206.04765 [cond-mat.str-el] .
  * Romatschke [2019b] P. Romatschke, Simple non-perturbative resummation schemes beyond mean-field: case study for scalar $\phi$ 4 theory in 1+1 dimensions, Journal of High Energy Physics 2019, 149 (2019b), arXiv:1901.05483 [hep-th] .
  * Veillette  _et al._ [2007] M. Y. Veillette, D. E. Sheehy, and L. Radzihovsky, Large- N expansion for unitary superfluid Fermi gases, Phys. Rev. A 75, 043614 (2007), arXiv:cond-mat/0610798 [cond-mat.other] .
  * Nikolić and Sachdev [2007] P. Nikolić and S. Sachdev, Renormalization-group fixed points, universal phase diagram, and 1/N expansion for quantum liquids with interactions near the unitarity limit, Phys. Rev. A 75, 033608 (2007), arXiv:cond-mat/0609106 [cond-mat.supr-con] .
  * Nishida and Son [2007] Y. Nishida and D. T. Son, Fermi gas near unitarity around four and two spatial dimensions, Phys. Rev. A 75, 063617 (2007), arXiv:cond-mat/0607835 [cond-mat.other] .
  * Gandolfi  _et al._ [2015] S. Gandolfi, A. Gezerlis, and J. Carlson, Neutron Matter from Low to High Density, Annual Review of Nuclear and Particle Science 65, 303 (2015), arXiv:1501.05675 [nucl-th] .
  * Ku  _et al._ [2012] M. J. H. Ku, A. T. Sommer, L. W. Cheuk, and M. W. Zwierlein, Revealing the Superfluid Lambda Transition in the Universal Thermodynamics of a Unitary Fermi Gas, Science 335, 563 (2012), arXiv:1110.3309 [cond-mat.quant-gas] .
  * Endres  _et al._ [2013] M. G. Endres, D. B. Kaplan, J.-W. Lee, and A. N. Nicholson, Lattice Monte Carlo calculations for unitary fermions in a finite box, Phys. Rev. A 87, 023615 (2013), arXiv:1203.3169 [hep-lat] .
  * Banerjee  _et al._ [2012] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Jain, S. Minwalla, and T. Sharma, Constraints on fluid dynamics from equilibrium partition functions, Journal of High Energy Physics 2012, 46 (2012), arXiv:1203.3544 [hep-th] .
  * Tews  _et al._ [2017] I. Tews, J. M. Lattimer, A. Ohnishi, and E. E. Kolomeitsev, Symmetry Parameter Constraints from a Lower Bound on Neutron-matter Energy, Astrophys. J. 848, 105 (2017), arXiv:1611.07133 [nucl-th] .
  * Gorda  _et al._ [2022] T. Gorda, J. Österman, and S. Säppi, Augmenting the residue theorem with boundary terms in finite-density calculations, Phys. Rev. D 106, 105026 (2022), arXiv:2208.14479 [hep-th] .
  * Österman  _et al._ [2023] J. Österman, P. Schicho, and A. Vuorinen, Integrating by parts at finite density, arXiv e-prints , arXiv:2304.05427 (2023), arXiv:2304.05427 [hep-ph] .