# Preserve Your Own Correlation: A Noise Prior for Video Diffusion Models

Songwei Ge
University of Maryland
Work done during the internship at NVIDIA. Seungjun Nah
NVIDIA
Guilin Liu
NVIDIA
Tyler Poon
University of Chicago
Andrew Tao
NVIDIA
Bryan Catanzaro
NVIDIA
David Jacobs
University of Maryland
Jia-Bin Huang
University of Maryland
Ming-Yu Liu
NVIDIA
Yogesh Balaji
NVIDIA

###### Abstract

Despite tremendous progress in generating high-quality images using diffusion
models, synthesizing a sequence of animated frames that are both
photorealistic and temporally coherent is still in its infancy. While off-the-
shelf billion-scale datasets for image generation are available, collecting
similar video data of the same scale is still challenging. Also, training a
video diffusion model is computationally much more expensive than its image
counterpart. In this work, we explore finetuning a pretrained image diffusion
model with video data as a practical solution for the video synthesis task. We
find that naively extending the image noise prior to video noise prior in
video diffusion leads to sub-optimal performance. Our carefully designed video
noise prior leads to substantially better performance. Extensive experimental
validation shows that our model, Preserve Your Own COrrelation (PYoCo),
attains SOTA zero-shot text-to-video results on the UCF-101 and MSR-VTT
benchmarks. It also achieves SOTA video generation quality on the small-scale
UCF-101 benchmark with a $10\times$ smaller model using significantly less
computation than the prior art. The project page is available at
https://research.nvidia.com/labs/dir/pyoco/.

[width=0.33]16videos/teaser_512/panda/00010076 |  [width=0.33]16videos/teaser_512/supernova/00010076 |  [width=0.33]16videos/teaser_512/bear/00010076
---|---|---
_A very happy fuzzy panda dressed as a chef eating pizza in the New York street food truck._ |  |  _The supernova explosion of a white dwarf in the universe, photo realistic._ |  |  _A high-quality 3D render of hyperrealist, super strong, multicolor stripped, and fluffy bear with wings, highly detailed._
---|---|---|---|---

Figure 1: Given a text description, our approach can faithfully generate
videos that are consistent with the input text while being photorealistic and
temporally consistent. _Best viewed with Acrobat Reader. Click the images to
play the video clips._ 11footnotetext: Work done during an internship at
NVIDIA.

##  1 Introduction

(a)

(b)

Figure 2: Visualizing the noise map correlations. (a) visualizes the t-SNE
plot of the noise maps corresponding to input frames randomly sampled from
videos. These noise maps are obtained by running a diffusion ODE [49, 48] on
the input frames using a trained text-to-image model, but in the opposite
direction of image synthesis ($\sigma:0\rightarrow\sigma_{\text{max}}$). The
green dots in the background denote the reference noise maps sampled from an
i.i.d. Gaussian distribution. The red dots and yellow dots are noise maps
corresponding to input frames coming from different videos. We found they are
spread out and share no correlation. On the other hand, the noise maps
corresponding to the frames coming from the same video (shown in blue dots)
are clustered together. (b) Using an i.i.d. noise model (orange dots) for
finetuning text-to-image models for video synthesis is not ideal since
temporal correlations between frames are not modeled. To remedy this, we
propose a progressive noise model in which the correlation between different
noise maps is injected along the temporal axis. Our progressive noise model
(blue dots) aptly models the correlations present in the video noise maps.

Large-scale diffusion-based text-to-image models [38, 42, 2] have demonstrated
impressive capabilities in turning complex text descriptions into
photorealistic images. They can generate images with novel concepts unseen
during training. Sophisticated image editing and processing tasks can easily
be accomplished through guidance control and embedding techniques. Due to the
immense success in several applications [30, 68, 5], these models are
established as powerful image synthesis tools for content generation. As image
synthesis is largely democratized with the success of these text-to-image
models, it is natural to ask whether we can repeat the same success in video
synthesis with large-scale diffusion-based text-to-video models.

Multiple attempts have been made to build large-scale video diffusion models.
Ho  _et al_. [17] proposed a UNet-based architecture for the video synthesis
task that is trained using joint image-video denoising losses. Imagen video
[14] extends the cascaded text-to-image generation architecture of Imagen [42]
for video generation. In both works, the authors directly train a video
generation model from scratch. While these approaches achieve great success
and produce high-quality videos, they are inherently expensive to train,
requiring hundreds of high-end GPUs or TPUs and several weeks of training.
After all, video generators not only need to learn to form individual images
but should also learn to synthesize coherent temporal dynamics, which makes
the video generation task much more challenging. While the formation of
individual frames is a shared component in an image and video synthesis, these
works disregard the existence of powerful pretrained text-to-image diffusion
models and train their video generators from scratch.

We explore a different avenue for building large-scale text-to-video diffusion
models by starting with a pretrained text-to-image diffusion model. Our
motivation is that most of the components learned for the image synthesis task
can effectively be reused for video generation, leading to knowledge transfer
and efficient training. A similar idea is adopted by several recent works [46,
70, 4]. Without exception, when finetuning, they naively extend the image
diffusion noise prior (i.i.d. noise) used in the text-to-image model to a
video diffusion noise prior by adding an extra dimension to the 2D noise map.
We argue that this approach is not ideal as it does not utilize the natural
correlations in videos that are already learned by the image models. This is
illustrated in Figure 2, where we visualize the t-SNE plot of noise maps
corresponding to different input frames as obtained from a pretrained text-to-
image diffusion model. The noise maps corresponding to different frames coming
from the same video (blue dots in Figure LABEL:fig:tsne_a are clustered
together, exhibiting a high degree of correlation. The use of i.i.d. noise
prior does not model this correlation, which would impede the finetuning
process. Our careful analysis of the video diffusion noise prior leads us to a
noise prior that is better tailored for finetuning an image synthesis model to
the video generation task. As illustrated in Figure LABEL:fig:tsne_b, our
proposed noise prior (shown in blue dots) aptly captures the correlations in
noise maps corresponding to video frames.

We then proceed to build a large-scale diffusion-based text-to-video model. We
leverage several design choices from the prior works, including the use of
temporal attention [17], joint image-video finetuning [17], a cascaded
generation architecture [14], and an ensemble of expert denoisers [2].
Together with these techniques and the proposed video noise prior, our model
establishes a new state-of-the-art for video generation outperforming
competing methods on several benchmark datasets. Figure 1 shows our model can
achieve high-quality zero-shot video synthesis capability with SOTA
photorealism and temporal consistency.

In short, our work makes the following key contributions.

  1. 1.

We propose a video diffusion noise tailored for finetuning text-to-image
diffusion models for text-to-video.

  2. 2.

We conduct extensive experimental validation and verify the effectiveness of
the proposed noise prior.

  3. 3.

We build a large-scale text-to-video diffusion model by finetuning a
pretrained eDiff-I model with our noise prior and achieve state-of-the-art
results on several benchmarks.

##  2 Related Work

Diffusion-based text-to-image models:  Diffusion models have significantly
advanced the progress of text-based photorealistic, compositional image
generation [38, 42]. Given the nature of the iterative denoising process that
requires massive numbers of score function evaluations, earlier diffusion
models focused on generating low-resolution images, e.g., $64\times 64$ [15,
48]. To generate high-resolution images, two common approaches have been used.
The first approach applies cascaded super-resolution models in the RGB space
[32, 16, 42, 38], while the second approach leverages a decoder to exploit
latent space [40, 11]. Based on these models, advanced image and video editing
have been achieved through finetuning the model [41, 68, 5, 23, 61, 29] or
controlling the inference process [30, 13, 34, 10, 35, 7, 31, 3]. Here, we
study the problem of using large-scale diffusion models for text-to-video
generation.

Video generation models:  Generating realistic and novel videos have long been
an attractive and essential research direction [58, 39, 66]. Previously
studies have resorted to different types of generative models such as GANs
[58, 43, 54, 52, 45], Autoregressive models [51, 64, 25, 9, 18], and implicit
neural representations [47, 67]. Recently, driven by the tremendous success of
applying the diffusion model to image synthesis, multiple works have proposed
to explore diffusion models for conditional and unconditional video synthesis
[57, 12, 70, 61, 4, 22, 19, 57, 65, 33, 28, 1, 59]. For example, Singer  _et
al_. extend the unCLIP framework [38] to text-to-video generation, which
allows training without video captions [46]. Ho  _et al_. [17] extend the
Imagen framework [42] by repeatedly up-scaling low-resolution small-fps videos
in both spatial and temporal directions with multiple models [14]. Our work
also falls into this line of work which uses a diffusion model. We focus on
augmenting an image diffusion model for video and study the design choice of
the diffusion noise priors for such an image-to-video finetuning task.

Leverage knowledge from images for text-to-video generation:  Like text-to-
image models, text-to-video models require massive amounts of data to learn
caption-relatedness, frame photorealism, and temporal dynamics. But in
contrast to the abundant image data resource, video data are more limited in
style, volume, and quality. To resolve such scarcity issue of text-video data,
previous works have resorted to different strategies to leverage knowledge
from image data for text-to-video generation, including joint training on the
text-image data from scratch [17, 14, 56, 60], first training a text-to-image
model and then finetuning partially [18, 4, 61, 29] or entirely [46, 8] on the
video dataset, and using CLIP image features as the conditional information
[46, 70]. In this paper, we propose a new video diffusion noise prior that is
tailored for finetuning a pretrained diffusion-based image generation model
for the video generation task. We reuse several design choices in the prior
work by finetuning jointly on text-image and text-video datasets. As a result,
we can build a text-to-video generation system that achieves state-of-the-art
zero-shot performances.

##  3 Preliminaries

Diffusion models generate data by iteratively denoising samples drawn from a
noise distribution. In the case of text-to-video models, text embeddings
obtained from a pre-trained text encoder are used as additional inputs in the
denoising process. Formally, let $D(\mathbf{x},\mathbf{e},\sigma)$ denote a
denoising network that operates on the noisy input video
$\mathbf{x}\in\mathbb{R}^{b\times n_{s}\times 3\times h\times w}$ where
$\mathbf{e}$ is the text embedding, and $\sigma$ is the noise level. Here
$n_{s}$ is the sequence length of the input video, $b$ is the batch size, and
$h\times w$ is the spatial resolution. The model $D$ is trained to denoise the
input $\mathbf{x}$.

#### Training

We follow the EDM formulation of Karras  _et al_. [21] to optimize the
denoiser $D$ using the following objective

$\displaystyle\mathbb{E}_{p_{\text{data}}(\mathbf{x}_{\text{clean}},\mathbf{e}),p(\mathbf{\epsilon}),p(\sigma)}$
$\displaystyle\left[\lambda(\sigma)\|D(\mathbf{x}_{\text{noise}};\mathbf{e},\sigma)-\mathbf{x}_{\text{clean}}\|^{2}_{2}\right]$
(1) $\displaystyle\text{where
}~{}~{}\mathbf{x}_{\text{noise}}=\mathbf{x}_{\text{clean}}+\sigma\mathbf{\epsilon}$

Here, $\mathbf{x}_{\text{noise}}$ is the noisy sample obtained by corrupting
the clean video $\mathbf{x}$ with noise $\sigma\mathbf{\epsilon}$, where
$p(\mathbf{\epsilon})=\mathcal{N}(\mathbf{0},\mathbf{I})$ and $\sigma$ is a
scalar for the noise level drawn from $p(\sigma)$. The loss weight,
$\lambda(\sigma)$, is a function of $\sigma$ given by
$\lambda(\sigma)=(\sigma^{2}+\sigma_{\text{data}}^{2})/(\sigma\cdot\sigma_{\text{data}})^{2}$.
Eq. (1) is a simple denoising objective in which the denoiser $D$ is trained
to estimate the clean video $\mathbf{x}_{\text{clean}}$ from the noisy input
$\mathbf{x}_{\text{noise}}$. Following EDM, we use a log-normal distribution
for $\sigma$ i.e.,
$\ln(p(\sigma))=\mathcal{N}(P_{\text{mean}},P_{\text{std}}^{2})$ with
$P_{\text{mean}}=-1.2$ and $P_{\text{std}}=1.2$.

To train the denoising model, EDM uses preconditioning terms in its objective
function to properly scale the inputs and output of the denoiser model $D$.
More specifically, the denoising model $D$ is written as

$\displaystyle
D(\mathbf{x};\mathbf{e},\\!\sigma)\\!:=\\!\Big{(}\frac{\sigma_{\text{data}}}{\sigma^{*}}\Big{)}^{2}\mathbf{x}+\frac{\sigma\cdot\sigma_{\text{data}}}{\sigma^{*}}F_{\theta}\Big{(}\frac{\mathbf{x}}{\sigma^{*}}\,;\mathbf{e},\\!\frac{\ln(\sigma)}{4}\Big{)}$

Here, $F_{\theta}$ is a neural network with parameters $\theta$ and
$\sigma^{*}=\sqrt{\sigma^{2}+\sigma_{\text{data}}^{2}}$. We use
$\sigma_{\text{data}}=0.5$.

#### Sampling

Once the denoising model is trained, sampling can be performed by solving the
following ODE [21]

$\displaystyle\frac{d\mathbf{x}}{d\sigma}=-\sigma\nabla_{\mathbf{x}}\log
p(\mathbf{x}|\mathbf{e},\sigma)=\frac{\mathbf{x}-D(\mathbf{x};\mathbf{e},\sigma)}{\sigma}$
(2)

for $\sigma$ flowing backwards from $\sigma=\sigma_{\text{max}}$ to
$\sigma=0$. The initial value for $\mathbf{x}$ is obtained by sampling from
the prior distribution
$\mathbf{x}\sim\mathcal{N}(\mathbf{0},\sigma_{\max}^{2}\mathbf{I})$. Over the
recent years, several samplers have been proposed for sampling from the
trained diffusion models [69, 48, 26, 27, 15]. In this paper, we use DEIS [69]
and its stochastic variant [21] for synthesizing samples from our model.

##  4 Method

Training text-to-video models is much more challenging than training text-to-
image diffusion models due to practical difficulties in collecting billion-
scale video datasets and securing enough computational resources.
Additionally, generating videos is much more challenging since individual
frames need to be both photorealistic and temporally coherent. Prior works
leverage large-scale image datasets to mitigate these difficulties by either
joint training on the image datasets [60, 17, 14] or finetuning a text-to-
image model on the video datasets [18, 46]. Here, we are interested in
finetuning text-to-image diffusion models jointly on image and video datasets.
We postulate that naively extending the image noise prior to video diffusion
is not ideal. We carefully explore the design space of noise priors and
propose one that is well suited for our video finetuning task, which leads to
significant performance gains.

Table 1: Cosine similarity of the reversed noise. The noise maps corresponding to the frames sampled from the same videos have a higher similarity than those sampled from different videos. | Cosine Similarity
---|---
(a) Same video noise |  $0.206$$\pm 0.156$
(b) Different video noise |  $0.001$$\pm 0.009$
Figure 3: Model architecture. Our pipeline consists of a cascade of four
networks — a base model and three upsampling models. All four models take
inputs as the text embeddings obtained from the T5 encoder and the CLIP text
encoder. The base model produces $16$ video frames of spatial resolution
$64\times 64$ with a frameskip of $5$. The first upsampling model performs a
temporal interpolation, resulting in videos of size $76\times 64\times 64$
while the subsequent two super-resolution models perform spatial super-
resolution to produce videos of sizes $76\times 256\times 256$ and $76\times
1024\times 1024$.

#### Correlated noise model

An image diffusion model is trained to denoise independent noise from a
perturbed image. The noise vector $\mathbf{\epsilon}$ in the denoising
objective (1) is sampled from an i.i.d. Gaussian distribution
$\mathbf{\epsilon}\sim\mathcal{N}(\mathbf{0},\mathbf{I})$. However, after
training the image diffusion model and applying it to reverse real frames from
a video into the noise space in a per-frame manner, we find that the noise
maps corresponding to different frames are highly correlated. This is
illustrated in Figure 2, where the t-SNE plot of noise maps corresponding to
different video frames are plotted. When the input frames come from the same
video (shown in blue dots in Figure LABEL:fig:tsne_a, noise maps are
clustered. The use of i.i.d. sampling (shown in orange dots in Figure
LABEL:fig:tsne_b does not capture these correlations. This is also depicted
quantitatively in Table 1 where we compute the average pairwise cosine
similarity between noise corresponding to (a) same video and (b) different
video. (a) is much higher than (b). As a result, the video diffusion model
trained with i.i.d. noise is coerced to forget such correlation among the
noise between different frames, making it difficult to preserve knowledge from
the image diffusion model. Motivated by this observation, we propose to modify
the noise process to preserve the correlation between different frames. To
this end, we investigate two noising strategies - mixed and progressive
noising.

Mixed noise model:  Let
$\mathbf{\epsilon}^{1},\mathbf{\epsilon}^{2},\ldots\mathbf{\epsilon}^{n_{s}}$
denote the noise corresponding to individual video frames i.e.,
$\mathbf{\epsilon}^{i}$ corresponds to the $i^{th}$ element of the noise
tensor $\mathbf{\epsilon}$. In the mixed noise model, we generate two noise
vectors $\mathbf{\epsilon}_{\text{shared}}$ and
$\mathbf{\epsilon}_{\text{ind}}$. $\mathbf{\epsilon}_{\text{shared}}$ is a
common noise vector shared among all video frames, while
$\mathbf{\epsilon}_{\text{ind}}$ is the individual noise per frame. The linear
combination of both these vectors is used as the final noise.

$\displaystyle\mathbf{\epsilon}_{\text{shared}}\sim\mathcal{N}\left(\mathbf{0},\frac{\alpha^{2}}{1+\alpha^{2}}\mathbf{I}\right),$
$\displaystyle\,\mathbf{\epsilon}_{\text{ind}}^{i}\sim\mathcal{N}\left(\mathbf{0},\frac{1}{1+\alpha^{2}}\mathbf{I}\right)$
(3) $\displaystyle\mathbf{\epsilon}^{i}=\mathbf{\epsilon}_{\text{shared}}$
$\displaystyle+\mathbf{\epsilon}_{\text{ind}}^{i}$

Progressive noise model:  In the progressive noise model, the noise for each
frame is generated in an autoregressive fashion in which the noise at frame
$i$ is generated by perturbing the noise at frame $i-1$. Let
$\mathbf{\epsilon}_{\text{ind}}^{i}$ denote the independent noise generated
for frame $i$. Then, progressive noising can be formulated as

$\displaystyle\mathbf{\epsilon}^{0}\sim\mathcal{N}(\mathbf{0},$
$\displaystyle\mathbf{I})\quad\mathbf{\epsilon}_{\text{ind}}^{i}\sim\mathcal{N}(\mathbf{0},{\frac{1}{1+\alpha^{2}}\mathbf{I})}$
(4) $\displaystyle\mathbf{\epsilon}^{i}$
$\displaystyle={\frac{\alpha}{\sqrt{1+\alpha^{2}}}}\mathbf{\epsilon}^{i-1}+\mathbf{\epsilon}_{\text{ind}}^{i}$

In both these models, $\alpha$ controls how much noise is shared among
different video frames. The higher the value of $\alpha$, the more correlation
exists among the noise maps corresponding to different frames. As
$\alpha\to\infty$, all frames would have the same noise which results in
generating a frozen video. On the other hand, $\alpha=0$ corresponds to i.i.d.
noise.

As shown in Figure LABEL:fig:tsne_b, the use of progressive noise sampling
(blue dots) better models the correlations between different noise maps by
obtaining similar clustering patterns to the noise maps of real video frames
embedded by a pretrained text-to-image model in Figure LABEL:fig:tsne_a (blue
dots).

#### Model architecture

As visualized in Figure 3, our model consists of a cascade of four networks —
a base network and three upsampling stacks. The base network generates an
output video of dimension $16\times 64\times 64$ with a frameskip of $5$. It
generates the frames $\\{1,6,11,\ldots 76\\}$. The first upsampling network
performs a temporal interpolation to produce a video of size $76\times
64\times 64$. The second and the third super-resolution network performs
spatial upsampling to produce the outputs of sizes $76\times 256\times 256$
and $76\times 1024\times 1024$. We utilize eDiff-I [2], a state-of-the-art
text-to-image diffusion model, to initialize our base and spatial super-
resolution models. Similar to prior works [17, 46], we adapt the image-based
U-Net model for the video synthesis task by making the following changes: (1)
Transforming 2D convolutions to 3D by adding a dimension $1$ to temporal axis
and (2) Adding temporal attention layers. Please refer to the supplementary
material for more details.

Similar to Ho _et al_. [17], we jointly finetune the model on video and image
datasets by concatenating videos and images in the temporal axis and applying
our temporal modules only on the video part. Similarly to eDiff-I, our model
uses both T5 text embeddings [37] and CLIP text embeddings [36]. We drop each
of the embeddings independently at random during training, as in eDiff-I.

##  5 Experiments

In this section, we evaluate our proposed strategy of training diffusion
models for video synthesis on two sets of experiments. We first
comprehensively analyze our proposed noise model on the small-scale UCF-101
dataset. We then scale up our experiments to the challenging large-scale text-
to-video synthesis task.

###  5.1 Experimental Setups

We conduct ablation experiments in a small-scale unconditional video
generation setting and pick the best configuration for our large-scale text-
to-video generation run.

#### Datasets

We train our model on the UCF-101 dataset [50] for the small-scale
experiments, where we follow the protocol defined in Ho  _et al_. [17] to
generate videos of size $16\times 64\times 64$. UCF-101 dataset contains
$13,320$ videos. We randomly sample frames from these videos to train our
image synthesis model. For our large-scale experiments, we use a combination
of public and proprietary datasets for text-to-image and text-to-video
finetuning. Most of the videos are of 2K resolution with 16:9 aspect ratio.
All data was filtered using a preset CLIP and aesthetic
scores***https://github.com/christophschuhmann/improved-aesthetic-predictor to
ensure high quality. Our final image dataset contains around $1.2$ billion
text-image pairs and $22.5$ million text-video pairs.

#### Training details

In the unconditional generation experiment on the UCF-101 dataset, to do an
ablation study on the model size, we design 3 models where each model has
$69$M, $112$M, and $253$M parameters, respectively. As a comparison, the
baseline Video Diffusion Model (VDM) [17] contains $1.2$B parameters. In the
large-scale text-to-video experiment, our base and temporal interpolation
models contain $1.08$B parameters. Our super-resolution model adapted from the
efficient U-Net [42] architecture with temporal convolution layers [14, 46]
contains $313$M parameters. Please refer to the supplementary material for
more training details.

|  |  |
---|---|---|---

_A cute corgi wearing a red robe holding a sign that says ”Merry Christmas”. There is a Christmas tree in the background._ |  |  _An epic tornado attacking above a glowing city at night, the tornado is made of smoke, highly detailed._ |  |  _Small boat sailing in the ocean, giant Cthulhu monster coming out a dense mist in the background, giant waves attacking._ |  |  _A golden retriever puppy holding a green sign that says ”NVIDIA ROCKS”. Background is a classroom._
---|---|---|---|---|---|---

|  |  |
---|---|---|---

_A cute funny robot dancing, centered, award winning watercolor pen illustration._ |  |  _A cartoon white wolf is giving puppy-dog eyes, detailed fur, very cute kid’s film character._ |  |  _A lightning striking atop of eiffel tower, dark clouds in the sky, slow motion._ |  |  _An anime girl looks at the beautiful nature through the window of a moving train, well rendered._
---|---|---|---|---|---|---

|  |  |
---|---|---|---

_A skull burning while being held up by a skeletal hand._ |  |  _A huge dinosaur skeleton walking in a golden wheat field on a bright sunny day._ |  |  _A cute rabbit is eating grass, wildlife photography._ |  |  _Tomato sauce pouring over fries._
---|---|---|---|---|---|---

Figure 4: Sample generations. _Please check ourproject website to view the videos._ Table 2: Zero-shot text to video generation on UCF-101. Our approach gives significant performance gains compared to the prior baselines both in inception score and FVD metrics. Method | IS $(\uparrow)$ | FVD $(\downarrow)$
---|---|---
CogVideo [18] (Chinese) | 23.55 | 751.34
CogVideo [18] (English) | 25.27 | 701.59
Make-A-Video [46] | 33.00 | 367.23
MagicVideo [70] | - | 655.00
Video LDM [4] | 33.45 | 550.61
VideoFactory [59] | - | 410.00
PYoCo | 47.76 | 355.19
Table 3: Text conditional zero-shot generation on MSRVTT. Our approach with the base config achieves the best results, and using an ensemble further improves the FIDs. Method | CLIP-FID $(\downarrow)$ | FID $(\downarrow)$
---|---|---
NUWA [60] (Chinese) | 47.68 | -
CogVideo [18] (Chinese) | 24.78 | -
CogVideo [18] (English) | 23.59 | -
Make-A-Video [46] | 13.17 | -
MagicVideo [70] | - | 36.50
Latent-Shift [1] | 15.23 | -
PYoCo (Config-A) | 10.21 | 25.39
PYoCo (Config-B) | 9.95 | 24.28
PYoCo (Config-C) | 9.91 | 24.54
PYoCo (Config-D) | 9.73 | 22.14

#### Evaluation

For the small-scale experiments on UCF-101 dataset, we follow the protocol
defined in the prior approaches [52, 47, 17] and report the Inception Score
(IS) [44] calculated by a trained C3D model [53] and Fréchet Video Distance
(FVD) [55] by a trained I3D model [6]. For the large-scale text-to-video
experiments, we perform the zero-shot evaluation of the video generation
quality on the UCF-101 and MSR-VTT datasets following Make-A-Video [46]. We
carefully discuss the evaluation process below.

#### UCF-101 experiment

We use IS and FVD for evaluation in our small-scale experiments. UCF-101 is a
categorical video dataset designed for action recognition. When sampling from
the text-to-video model, we devise a set of prompts for each class name to be
used as the conditional input. This is necessary as some class names (such as
jump rope) are not descriptive. We list all the prompts we use in the
supplementary material. We sample $20$ videos for each prompt to compute the
IS metric. For FVD, we follow the prior work [25, 52] and sample $2,048$
videos for evaluation.

#### MSR-VTT experiment

MSR-VTT [63] test set contains $2,990$ videos as well as $59,794$ captions.
All the videos have the same resolution of $320\times 240$. We generate a
$76\times 256\times 256$ video for each $59,794$ caption and save the videos
in an mp4 format with a high bit rate. To compare with Make-A-Video, we
compute FID using a ViT-B/32 model [24]. We also report a more common FID
metric computed by an Inception-V3 model. We also examine the idea of ensemble
denoiser [2] by finetuning the level-1 experts of each model. We denote
Config-A as the configuration of using only baseline models and Config-B to
Config-D as incrementally changing super-resolution model, temporal
interpolation model, and base model with the corresponding ensemble models.

###  5.2 Main Results

#### Large-scale text-to-video synthesis

We quantitatively compare our method against Make-A-Video [46], NUWA [60],
CogVideo [18], and several concurrent works [4, 70, 4, 59, 1]. Table 2 shows
that our method outperforms all the baselines on the UCF-101 dataset and
improves the zero-shot Inception Score from $33.45$ to $47.76$. In Table 3, we
show that our baseline model achieves a new state-of-the-art CLIP-FID score
[24] of $10.21$, while using ensemble models further improves both CLIP-FID
and FID scores. In Figure 4, we qualitatively visualize the synthesis
capability of our approach. Our model achieves high-quality zero-shot video
synthesis capability with good photorealism and temporal coherency. We also
provide a qualitative comparison with Make-A-Video [46] and Imagen Video [14]
in Figure 5. We observe that our model is able to produce videos with better
details than both approaches, as shown in the animal videos. We also produce
better-stylized videos than Imagen Video.

#### Small-scale unconditional video synthesis

Table 4: Unconditional UCF-101 generation results. Our approach achieves the state-of-the-art inception score and FVD, while having considerably smaller parameter count compared to other diffusion-based approaches such as VDM (1B parameters). Method | IS $(\uparrow)$ | FVD $(\downarrow)$
---|---|---
TGAN [43] | 15.83$\pm.18$ | -
LDVD-GAN [20] | 22.91$\pm.19$ | -
VideoGPT [64] | 24.69$\pm.30$ | -
MoCoGAN-HD [52] | 32.36$\pm.00$ | 838
DIGAN [67] | 29.71$\pm.53$ | 655$\pm 22$
CCVS [25] | 24.47$\pm.13$ | 386$\pm 15$
StyleGAN-V [47] | 23.94$\pm.73$ | -
VDM [17] | 57.00$\pm.62$ | -
TATS [9] | 57.63$\pm.73$ | 430 $\pm 18$
PYoCo ($112$M) | 57.93$\pm.24$ | 332 $\pm 13$
PYoCo ($253$M) | 60.01$\pm.51$ | 310 $\pm 13$

We report IS and FVD scores on UCF-101 dataset in Table 4 and compare our
model with multiple unconditional video generation baselines. Note that using
class labels as conditional information could lead to sizeable improvement in
IS and FVD scores [9], which we do not consider as the comparison. Our method
attains state-of-the-art unconditional video generation quality. Compared with
previous diffusion-based unconditional generation model [17], our model is
$\sim 10\times$ smaller and has $\sim 14\times$ less training time ($75$ GPU-
days vs. $925$ GPU-days).

Make-A-Video [46] |  |  PYoCo |  |  Imagen Video [14] |  |  PYoCo
---|---|---|---|---|---|---

|  |  |  |
---|---|---|---|---

_A confused grizzly bear in calculus class._ |  |  _A sheep to the right of a wineglass._
---|---|---

|  |  |  |
---|---|---|---|---

_Sailboat sailing on a sunny day in a mountainlake._ |  |  _A cat eating food out of a bowl, in style of Van Gogh._
---|---|---

Figure 5: Qualitative comparison with baseline approaches. The two panels on the left show the comparison of our approach with Make-A-Video [46], while those on the right show the comparison with Imagen Video [14]. PYoCo achieves better photorealism compared to the two approaches. Table 5: Quantitative results of different training strategies on UCF-101 dataset. | IS$(\uparrow)$ | FVD $(\downarrow)$ | FID $(\downarrow)$
---|---|---|---
Image Diffusion (ID) | - | - | 30.05
Training from scratch | 28.25 | 903.37 | 124.75
Finetuning from ID | 41.25 | 566.67 | 56.43
\+ Mixed Noise | 52.71 | 337.40 | 31.57
\+ Progressive Noise | 53.52 | 339.67 | 31.88

###  5.3 Ablation Study

We quantitatively compare several training strategies for video diffusion
models. Then, we perform ablation on the correlation ratio in the Equations 3
and 4, a key hyper-parameter in our approach.

#### Training strategies

We compare training from scratch, a simple finetuning baseline, finetuning
with mixed noising, and progressive noising using IS, FVD, and averaged frame
FID metrics on the UCF-101 dataset in Table 5. We first find that finetuning
from an image diffusion model is much more effective than training from
scratch. For finetuning from the image model, the correlated noise model
produces better video generation quality than the independent noise model. In
addition, we notice that the correlated noise better preserves the image
quality learned by the pretrained image model and produces a lower frame FID.
This is particularly desired in large-scale text-to-video training to fulfill
the goal of inheriting the knowledge from the image model missing in the video
datasets. Specifically, most videos contain realistic scenes captured by
cameras and have infrequent media types like paintings, illustrations,
sketches, etc. Moreover, the video data is much smaller in volume, and the
scenes are less diverse than image datasets. As shown in Figure 4, our model
can preserve properties learned from image datasets that are not presented in
our video dataset, such as the artistic styles, and generate faithful motion
on them.

|  |
---|---|---

$\alpha=0$ |  |  $\alpha=1$ |  |  $\alpha=10$
---|---|---|---|---

Figure 6: Visual ablation on $\alpha$. Small $\alpha=0$ reduces video quality
and diversity and large $\alpha=10$ yields motion artifacts.
$0$$0.1$$0.2$$0.5$$1.0$$2.0$$5.0$$10.0$$\infty$$500$$1000$FVD$\alpha$MixedProgressiveScratch
Figure 7: Quantitative ablation on hyperparameter $\alpha$. Finetuning with
temporally correlated prior improves over training from scratch. Using a too-
large or too-small $\alpha$ leads to inferior results. $\alpha=1$, $\alpha=2$
each works the best for mixed and progressive noising, respectively.

#### Correlation ratio

The hyperparameter $\alpha$ in the Equations 3 and 4 controls the correlation
between the noise of different frames. A larger $\alpha$ injects more
correlation into the noise. The correlation disappears when $\alpha\rightarrow
0$, and the mixed and progressive noise models reproduce the vanilla noise
model. To find optimal $\alpha$, we train our UCF-small model ($69$M
parameters) using $\alpha\in\\{0,0.1,0.2,0.5,1,1,2,5,10,\infty\\}$ and report
FVD in Figure 7. For each $\alpha$ value, we repeat the experiment 3 times and
report the mean. Note that $\alpha=0$ indicates finetuning with the
independent frame noise, and $\alpha=\infty$ indicates using identical noise
maps for all the frames, which produces frozen videos during the inference
time. Finetuning an image diffusion model almost consistently outperforms the
training-from-scratch baseline with different $\alpha$s. Using $\alpha=1$ for
mixed noising and $\alpha=2$ for progressive noising produces similar best
results. We also show qualitiative results for models trained with
$\alpha=0,1,10$ in Figure 6. When $\alpha$ is too small, we notice a
degradation in visual quality in the generated video frames and a reduced
video diversity. For example, we notice many repeated samples and black
borders in almost every video generated with $\alpha=0$. On the other hand,
when $\alpha$ is too large, the model has difficulty generating proper
motions.

#### Model size

We pick the best $\alpha$ for the mixed and progressive noise models and
compare them with the model trained from scratch on models with different
numbers of parameters, $69$M, $112$M, and $253$M. Figure 8 shows that our
mixed and progressive models outperform the baseline consistently by a large
margin in terms of FVD. Overall, mixed and progressive noising provide similar
performance. In our large large-scale experiments, we choose progressive
noising with $\alpha=2$ due to its autoregressive nature.

##  6 Conclusion

We proposed a new efficient way of training text-to-video generation models.
By observing that the noise maps generating the frames of a video are
clustered together, we study mixed and progressive noise priors well-suited
for sequential video frame generation. We apply our progressive noise prior to
finetuning a state-of-the-art diffusion-based text-to-image model to achieve a
state-of-the-art large-scale text-to-video model. The high quality of the
generated videos and the state-of-the-art Inception and FID scores demonstrate
the strength of our approach.

$50$$100$$150$$200$$250$$400$$600$$800$FVDNumber of Parameters (M)Mixed
($\alpha=1$)Progressive ($\alpha=2$)Scratch Figure 8: Ablation on model size.
Larger models consistently improve the performance of both finetuning and
training from scratch. Finetuning from image model consistently outperforms
training from scratch.

Acknowledgment. We would like to thank Amanda Moran, John Dickinson, Sivakumar
Arayandi Thottakara, David Page, Ranjitha Prasanna, Venkata Karri, and others
in NGC and PBSS team for the computing infrastructure support. We also give
thanks to Qinsheng Zhang, Zekun Hao, Tsung-Yi Lin, Ajay Jain, and Chen-Hsuan
Lin for useful discussions and feedback. Thanks also go to Yin Xi and Thomas
Hayes for clarifying the evaluation protocol of Make-A-Video and to Ming Ding
for the details of CogVideo. We thank the ICCV PCs, ACs, and reviewers for
their service and valuable feedback. This work is partly supported by NSF
grants No. IIS-1910132 and IIS-2213335.

##  Appendix A Experimental Setups

In this section, we provide additional details of our experiments in terms of
implementation, dataset, evaluation, model, and training.

###  A.1 Implementation details

Similar to prior works [17, 46], we adapt the image-based U-Net model for the
video synthesis task by making the following changes: (1) We transform the 2D
convolution layers to 3D by adding a dimension of $1$ to the temporal axis.
For instance, we convert a $3\times 3$ convolution layer to $1\times 3\times
3$ layer. (2) We replace the attention layers in the base and temporal
interpolation models with a cascade of spatial and temporal attention layers.
The spatial attention layers are reused from eDiff-I [2], while the temporal
attention layers are initialized randomly with a projection layer at the end
using zero-initialization. We apply temporal attention to the activation maps
obtained by moving the spatial dimension of the feature tensor to the batch
axis. (3) For the temporal interpolation model, we concatenate the input noise
in the channel axis with $16$ frames by infilling $4$ real frames with zero
frames. (4) We add a $3\times 1\times 1$ convolution layer at the end of each
efficient block of the super-resolution model [42]. (5) For all the models, we
apply spatial attention to the reshaped activation maps obtained by moving the
temporal dimension of the feature tensor to the batch axis. We apply the same
operation to the feature maps input the GroupNorm [62] to mimic better the
statistics the image model learned. We use cross-attention layers (between
text and videos) only in the spatial attention block, as adding it to the
temporal attention resulted in significant memory overhead. (6) We utilize
eDiff-I [2] to initialize our base and spatial super-resolution models. We use
a similar model architecture as the base model for our temporal interpolation
model, as they share the same function of hallucinating unseen frames. After
finetuning the base model for some time, we use its checkpoint to initialize
the temporal interpolation model. (7) Similar to Ho _et al_. [17], we jointly
finetune the model on video and image datasets by concatenating videos and
images in the temporal axis and applying our temporal modules only on the
video part. (8) Similarly to eDiff-I, our model uses both T5 [37] text
embeddings and CLIP text embeddings [36]. During training, we drop each of the
embeddings independently at random, as in eDiff-I.

###  A.2 Dataset and evaluation details

#### Caption templates for categorical video datasets

Given the name of the category [class] such as kayaking and yoga, we consider
the following templates to create video captions:

  * •

a man is [class].

  * •

a woman is [class].

  * •

a kid is [class].

  * •

a group of people are [class].

  * •

doing [class].

  * •

a man is doing [class].

  * •

a woman is doing [class].

  * •

a kid is doing [class].

  * •

a group of people are doing [class].

  * •

[class].

#### Prompts used for UCF-101 evaluation

In our initial explorations, we find that the original class labels in the
UCF-101 dataset often cannot describe the video content correctly. For
example, the class jump rope is more likely describing an object rather than a
complete video. Therefore, we write one sentence for each class as the caption
for video generation. We list these prompts for evaluating text-to-video
generation models on the standard UCF-101 benchmark below †††A copy-paste
friendly version is available in the Google Spreadsheet at
https://docs.google.com/spreadsheets/d/1teEGth-
Iy1be4Tx7xfXUKBA3aGZ9Hhr2gueTpuuwv94/edit?usp=sharing.

applying eye makeup, applying lipstick, archery, baby crawling, gymnast
performing on a balance beam, band marching, baseball pitcher throwing
baseball, a basketball player shooting basketball, dunking basketball in a
basketball match, bench press, biking, billiards, blow dry hair, blowing
candles, body weight squats, a person bowling on bowling alley, boxing
punching bag, boxing speed bag, swimmer doing breast stroke, brushing teeth,
clean and jerk, cliff diving, bowling in cricket gameplay, batting in cricket
gameplay, cutting in kitchen, diver diving into a swimming pool from a
springboard, drumming, two fencers have fencing match indoors, field hockey
match, gymnast performing on the floor, group of people playing frisbee on the
playground, swimmer doing front crawl, golfer swings and strikes the ball,
haircuting, a person hammering a nail, an athlete performing the hammer throw,
an athlete doing handstand push up, an athlete doing handstand walking,
massagist doing head massage to man, an athlete doing high jump, group of
people racing horse, person riding a horse, a woman doing hula hoop, man and
woman dancing on the ice, athlete practicing javelin throw, a person juggling
with balls, a young person doing jumping jacks, a person skipping with jump
rope, a person kayaking in rapid water, knitting, an athlete doing long jump,
a person doing lunges with barbell, military parade, mixing in the kitchen,
mopping floor, a person practicing nunchuck, gymnast performing on parallel
bars, a person tossing pizza dough, a musician playing the cello in a room, a
musician playing the daf, a musician playing the indian dhol, a musician
playing the flute, a musician playing the guitar, a musician playing the
piano, a musician playing the sitar, a musician playing the tabla, a musician
playing the violin, an athlete jumps over the bar, gymnast performing pommel
horse exercise, a person doing pull ups on bar, boxing match, push ups, group
of people rafting on fast moving river, rock climbing indoor, rope climbing,
several people rowing a boat on the river, couple salsa dancing, young man
shaving beard with razor, an athlete practicing shot put throw, a teenager
skateboarding, skier skiing down, jet ski on the water, sky diving, soccer
player juggling football, soccer player doing penalty kick in a soccer match,
gymnast performing on still rings, sumo wrestling, surfing, kids swing at the
park, a person playing table tennis, a person doing TaiChi, a person playing
tennis, an athlete practicing discus throw, trampoline jumping, typing on
computer keyboard, a gymnast performing on the uneven bars, people playing
volleyball, walking with dog, a person standing and doing pushups on the wall,
a person writing on the blackboard, a kid playing Yo-Yo

###  A.3 Training details

#### UCF-101 experiments.

For image pretraining phase on the UCF-101 frames, we use an ADAM optimizer
with a base learning rate of $2e-4$. For video finetuning phase, we adopt an
ADAM optimizer with a base learning rate of $1e-4$. We use a linear warm up of
$5,000$ steps for both phases. For sampling, we use stochastic DEIS sampler
[zhang2022fast, 21] with 3kutta, order $6$ and $25$ steps.

#### Large-scale experiments.

The hyper-parameters we use for the large-scale text-to-video experiments are
provided in Table F.

Table F: Hyperparameters Hyperparameters for large-scale experiments
---
Optimizer | AdamW
Learning rate | $0.0001$
Weight decay | $0.01$
Betas | (0.9, 0.999)
EMA | $0.9999$
CLIP text embedding | $0.2$
dropout rate
T5 text embedding | $0.25$
dropout rate
Gradient checkpointing | Enabled
# iterations for base model | $150K$
# iterations for super-res model | $220K$
Sampler for base model | Stochastic DEIS [69, 21],
3kutta, Order 3, 60 steps
Sampler for super-res models | DEIS, 3kutta
Order 3, 20 steps

###  A.4 Architecture details

The architectures used for the small-scale UCF experiments are provided in
Tables G, H and I. For the large-scale experiment, the architectures used for
base model, temporal interpolation model, and the two spatial super-resolution
stacks are provided in tables J, K, L and M respectively.

Table G: Small (69M parameters) UCF-101 model architecture. Small (69M
parameters) UCF-101 model
---
Channel multiplier | $[1,2,2,3]$
Dropout | $0.1$
Number of channels | $128$
Number of residual blocks | $2$
Spatial self attention resolutions | $[32,16,8]$
Spatial cross attention resolutions | $[32,16,8]$
Temporal attention resolution | $[32,16,8]$
Number of channels in attention heads | 64
Use scale shift norm | True
Table H: Medium (112M parameters) UCF-101 model architecture. Medium (112M
parameters) UCF-101 model
---
Channel multiplier | $[1,2,3,4]$
Dropout | $0.1$
Number of channels | $128$
Number of residual blocks | $2$
Spatial self attention resolutions | $[32,16,8]$
Spatial cross attention resolutions | $[32,16,8]$
Temporal attention resolution | $[32,16,8]$
Number of channels in attention heads | 64
Use scale shift norm | True
Table I: Large (253M parameters) UCF-101 model architecture. Large (253M
parameters) UCF-101 model
---
Channel multiplier | $[1,2,3,4]$
Dropout | $0.1$
Number of channels | $192$
Number of residual blocks | $2$
Spatial self attention resolutions | $[32,16,8]$
Spatial cross attention resolutions | $[32,16,8]$
Temporal attention resolution | $[32,16,8]$
Number of channels in attention heads | 64
Use scale shift norm | True

Table J: Architecture for the base model in text-to-video experiments. Text-
to-video base model (1.08B parameters)
---
Channel multiplier | $[1,2,4,4]$
Dropout | $0$
Number of channels | $256$
Number of residual blocks | $3$
Spatial self attention resolutions | $[32,16,8]$
Spatial cross attention resolutions | $[32,16,8]$
Temporal attention resolution | $[32,16,8]$
Number of channels in attention heads | 64
Use scale shift norm | True
Table K: Architecture for the temporal interpolation model in text-to-video
experiments. Temporal interpolation model (1.08B parameters)
---
Channel multiplier | $[1,2,4,4]$
Dropout | $0$
Number of channels | $256$
Number of residual blocks | $3$
Spatial self attention resolutions | $[32,16,8]$
Spatial cross attention resolutions | $[32,16,8]$
Temporal attention resolution | $[32,16,8]$
Number of channels in attention heads | 64
Use scale shift norm | True
Table L: Architecture for the spatial super-resolution model in text-to-video
experiments. Spatial super-resolution 256 (300M parameters)
---
Channel multiplier | $[1,2,4,8]$
Block multiplier | $[1,2,4,4]$
Dropout | $0$
Number of channels | $128$
Number of residual blocks | $2$
Spatial self attention resolutions | $[32]$
Spatial cross attention resolutions | $[32]$
Number of channels in attention heads | 64
Use scale shift norm | True
Table M: Architecture for the spatial super-resolution model in text-to-video
experiments. Spatial super-resolution 1024 (170M parameters)
---
Patch size | $256\times 256$
Channel multiplier | $[1,2,4,4]$
Block multiplier | $[1,2,4,4]$
Number of channels | $128$
Number of residual blocks | $2$
Spatial cross attention resolutions | $[32]$
Use scale shift norm | True

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# Differences in the Resistive and Thermodynamic Properties of the Single
Crystalline Chiral Superconductor Candidate SrPtAs

A. Weiland, F. B. Santos, J. D. Thompson, E. D. Bauer, S. M. Thomas, and P. F.
S. Rosa  MPA-Q, Los Alamos National Laboratory, Los Alamos, New Mexico 87545,
U.S.A.

###### Abstract

The locally non-centrosymmetric superconductor SrPtAs is proposed to host a
topological chiral d-wave state, but experimental reports have been limited to
polycrystalline samples. Here we report the synthesis of single crystalline
SrPtAs grown from Pb flux. SrPtAs crystallizes in the hexagonal space group
P63/mmc with lattice parameters a = 4.2445(4) Å and c = 8.9513(18) Å. Magnetic
susceptibility and electrical resistivity measurements reveal a
superconducting transition at Tc $\sim$2.2 K, in agreement with previous
reports on polycrystalline samples. Surprisingly, heat capacity data show only
a small bulk transition at 0.7 K. We discuss the possible origins of the
discrepancy between the various measurements.

††preprint: APS/123-QED

SrPtAs, the first hexagonal pnictide-based superconductor [1], is a potential
chiral d-wave superconductor due to its hexagonal symmetry and quasi-two-
dimensional multiband Fermi surface [2, 3]. Chiral superconductors, whose
complex superconducting gap function breaks time reversal symmetry (TRS), are
of interest due to their non-trivial topology and the potential for quantum
computing applications [4]. In non $s$-wave pairing, the gap function contains
nodes which may be detrimental for superconductivity, but chiral
superconducting order parameters may naturally reduce the number of nodes.

In high-symmetry crystal systems, multiple gap functions can be degenerate.
For example, in hexagonal systems, $d_{x^{2}-y^{2}}$ and $d_{xy}$
superconducting channels are degenerate, and a linear combination of the two
results in a chiral $d_{x^{2}-y^{2}}$ $\pm$ i$d_{xy}$ state that would
maximize the condensation energy of the superconducting state. Superconductors
that contain a high-symmetry crystal structure, strong spin-orbit coupling,
magnetic interactions, noncentrosymmetry, TRS breaking, and a quasi-two-
dimensional Fermi surface are suitable systems to search for chiral
superconductivity. Few known materials are considered candidate chiral
superconductors. Examples include Sr2RuO4 [5], UPt3 [6], URu2Si2 [7], and
potentially SrPtAs [8, 2].

SrPtAs crystallizes in the hexagonal non-symmorphic space group P63/mmc (No.
194) [9], wherein Pt and As form a honeycomb lattice. This is in contrast to
other pnictide superconductors which form with a square lattice such as LaFePO
[10], LaFeAsO [11], LiFeAs [12], NaFeAs [13] and ThCr2Si2-type structures like
(Ba1-xKx)Fe2As2 [14]. Although globally centrosymmetric, SrPtAs is locally
noncentrosymmetric due to the As-Pt layer breaking inversion symmetry.

Initial measurements, including nuclear magnetic resonance and nuclear
quadrupole resonance (NMR/NQR) $T^{-1}_{1}$ relaxation rates [15] argue that
SrPtAs is a conventional $s$-wave superconductor because the spin-lattice
relaxation rate shows a coherence peak and the Knight shift decreases below
Tc. Magnetic penetration depth data may be fit to an exponential model and the
inferred superfluid density agrees with an isotropic Bardeen–Cooper–Schrieffer
(BCS) model [16].

However, muon spin relaxation ($\mu$SR) data showing TRS breaking below the
superconducting transition temperature, Tc, [8] coupled with a recent
reevaluation of the NMR data [3], indicate that these data could also
correspond to $d$-wave pairing. Although SrPtAs is the subject of many
theoretical investigations discussing band structure topology [17], possible
pairing states [18, 19], spin orbit coupling [20], topological properties
[15], local non-centrosymmetry [21], and gap function symmetry analysis [22],
there are very few experimental reports, all of which are limited to
polycrystalline samples [1, 9, 8, 15, 16]. Several previous works mention the
advantages of growing single crystals of SrPtAs to further probe the bulk
properties [17, 8, 19, 2, 16, 15]. To this end we set out to synthesize and
characterize single crystals of SrPtAs.

Figure 1: Crystal structure of SrPtAs (P63/mmc, No. 194), with Sr in purple,
Pt in black, and As in yellow.

This work presents the synthesis and physical properties of single crystalline
SrPtAs. Although magnetic susceptibility and electrical resistivity
measurements reveal superconducting transitions at 2.2 K, specific heat and
thermal expansion measurements show a significantly decreased Tc or no Tc,
respectively. Herein, we discuss the varying superconducting transition
temperatures and draw parallels to other AlB2 derived superconductor MgB2.

Table 1: Crystallographic data, data collection, and refinement parameters of SrPtAs. Formula | SrPt0.97As1.03
---|---
Space Group |  P63/mmc
a (Å) | 4.2445(4)
c (Å) | 8.9513(18)
V (Å3) | 139.66(4)
Z | 2
T (K) | 298
$\theta$ range (degrees) | 4.55-30.40
$\mu$ (absorption coefficient, $mm^{-1}$) | 79.37
measured reflections | 6210
independent reflections | 105
$\Delta\rho_{max}$ (largest peak, eÅ-3) | 2.00
$\Delta\rho_{min}$ (deepest hole, eÅ-3) | -2.34
extinction coefficient | 0.031(7)
R1(F2$>$2$\sigma$(F2)) | 0.029
wR2(F2) | 0.059
Table 2: Fractional atomic coordinates and anisotropic displacement parameters of SrPtAs. Site Label | Wyckoff | x | y | z | Ueq | Occupancy | U11 | U22 | U33 | U12
---|---|---|---|---|---|---|---|---|---|---
Sr1 | 2a | 0 | 0 | 0 | 0.0099(10) | 1 | 0.0091(10) | 0.0091(10) | 0.0117(13) | 0.0045(5)
Pt2/As2 | 2c | $\frac{1}{3}$ | $\frac{2}{3}$ | $\frac{1}{4}$ | 0.0084(4) | 0.85(3)/0.15(3) | 0.0069(5) | 0.0069(5) | 0.0114(6) | 0.0034(2)
As3/Pt3 | 2d | $\frac{2}{3}$ | $\frac{1}{3}$ | $\frac{1}{4}$ | 0.0100(9) | 0.884(10)/0.116(10) | 0.0070(10) | 0.0070(10) | 0.0161(12) | 0.0035(5)

As described in Methods, single crystals of SrPtAs were grown using Pb flux.
SrPtAs, of the KZnAs-type derived from the AlB2 (P6/mmm) structure type,
crystallizes in the hexagonal space group P63/mmc (No. 194) with lattice
parameters a = 4.2445(4) Å and c = 8.9513(18) Å, as shown in Figure 1. In the
AlB2 structure, the B atoms form honeycomb layers separated by Al atoms in the
c direction. In contrast, in SrPtAs the B site is occupied by both Pt and As
sites, which alternate both within the honeycomb lattice and in the c
direction such that above each As atom is a Pt atom and above each Pt atom is
an As atom. This deviation from the AlB2 structure results in the doubling of
the unit cell along the c axis [9] and breaks local centrosymmetry although
the global structure remains centrosymmetric. SrPtAs consists of three
crystallographically unique sites, Sr (2a), Pt (2c), and As (2d), where the Pt
and As sites are disordered. A more accurate description therefore is
Sr(Pt1-xAsx)(As1-yPty) where x = 0.15(3) and $y$ = 0.116(10). Table 1 shows
data collection and refinement parameters, and Table 2 presents fractional
atomic coordinates and displacement parameters.

Figure 2: A) Temperature dependent magnetic susceptibility ($\chi$(T)) of
SrPtAs with field (H = 0.35 Oe) applied along the c axis. Field cooled (FC)
data are shown with red triangles and zero field cooled (ZFC) data are shown
with black squares. Inset shows field dependent magnetization (M(H) at T = 1.6
K) in blue circles with the initial slope denoted with a red line. B)
Electrical resistivity ($\rho$(T)) of SrPtAs at H = 0. The bottom right inset
shows the low temperature region (below 10 K), highlighting the
superconducting transition of Pb at $\sim$7 K. The top left inset shows
resistivity below 3 K where the superconducting transition of SrPtAs occurs at
2.2 K. Figure 3: A) $\chi_{ac}$(T) with various fields applied perpendicular
to the c-axis where the driving ac field $\sim$0.042 Oe. B) $\chi_{ac}$(T)
with various fields applied along the c-axis where the driving ac field
$\sim$0.042 Oe. C) Temperature dependent resistivity ($\rho$(T)) of SrPtAs
with varying fields applied perpendicular to the c-axis. D) Temperature
dependence of the upper critical field, $H_{c2}$. Data determined from ac
susceptibility (resistivity) measurements are shown in closed black (open
blue) symbols. Data collected with field applied parallel (perpendicular) to
the c axis are shown with triangles (squares).

The temperature dependent magnetic susceptibility ($\chi$) and electrical
resistivity ($\rho$) measurements show a superconducting transition
temperature of about 2.2 K, as shown in Figures 2A and 2B, respectively,
consistent with previously reported works on polycrystalline samples of SrPtAs
[1]. Transitions visible at 7 K are due to excess elemental Pb that was used
as a flux for the crystal growth. The susceptibility of SrPtAs with magnetic
field (H) (0.35 Oe) applied along the c axis under zero-field cooling (ZFC)
and field-cooling (FC) conditions is shown in Figure 2A. A large diamagnetic
repulsion response indicates a Tc of $\sim$2.0 K as determined by the midpoint
of the drop. The ZFC signal, including a demagnetization factor [23], at 1.8 K
corresponds to $\sim$ 110$\%$ perfect diamagnetism, where $\sim$ 5$\%$ comes
from Pb inclusions. A value greater than 100$\%$ could be due to error in the
demagnetization factor estimation and/or the presence of chemical impurities.
The FC curve has a slight drop that begins at $\sim$ 2.2 K. The lack of a
large diamagnetic response could be due to nonsuperconducting impurities,
vortex pinning, or a lack of bulk superconductivity. The inset to Figure 2A
shows the field dependent magnetization (M) wherein the solid red line
indicates the initial slope, 4$\pi\frac{dM}{dH}$ = - 1.142, with a
corresponding superconducting volume of $\sim$110$\%$. The lower critical
field, $H_{c1}$ $\sim$15 Oe at 1.6 K, is estimated from the deviation of the
data compared to the initial slope.

The temperature dependent electrical resistivity, $\rho$(T), for a SrPtAs
crystal is shown in Figure 2B. The Pb superconducting transition at $\sim$7 K
is shown in the bottom right inset. The top left inset shows a superconducting
transition at 2.2 K, attributed previously to the bulk superconductivity of
polycrystalline SrPtAs. Here, Tc is defined as the midpoint of the resistivity
drop. The residual resistivity ratio (RRR),
$\frac{\rho_{300K}-\rho_{0}}{\rho_{0}}$, is $\sim$15, where $\rho_{0}$ is
determined to be to be $\sim$25 $\mu\Omega\cdot$cm by extrapolating $\rho(T)$
from above the Pb transition at 7 K to zero temperature.

The temperature dependent real part of the ac susceptibility ($\chi_{ac}$),
collected with various fields perpendicular and parallel to the c axis, are
shown in Figure 3A and Figure 3B, respectively. The temperature dependence of
the electrical resistivity (I$||$c) at different fields perpendicular to the c
axis is shown in Figure 3C. The upper critical field, Hc2, for both
directions, is shown in Figure 3D. Tcs for $\chi_{ac}$ were determined by the
onset of each transition shown in Figure 3A, and 3B. Tcs for resistivity data
were determined by the point where $\rho$ = 0 in Figure 3C. For fields
parallel (perpendicular) to the c axis, the estimated upper critical field is
1.5 kOe (11 kOe). Similar extrapolations from powder samples of SrPtAs yield
an upper critical field ($H_{c2(0)}$) of 2.2 kOe [1]. Utilizing both the
isotropic single band Eliashberg model [24] and Werthamer–Helfand–Hohenberg
(WHH) theory [25], $H_{c2(0)}$ was previously calculated (based on
polycrystalline SrPtAs) [17]. Both values, 1.4 kOe and 1.58 kOe, respectively,
agree with our linearly extrapolated lower $H_{c2(0)}$, but are significantly
smaller than the $H_{c2(0)}$ perpendicular to the c axis. To calculate the WHH
Hc2 from the anisotropic data of this work, the WHH equation $H_{c2}(0)$ =
-0.69$T_{c}(dH_{c2}/dT)_{T_{c}}$ was utilized, where the resistive $T_{c}$ of
2.2 K was used.

For the ac susceptibility data, the resulting H${}_{c2}(0)$ for field parallel
to $c$ is 0.60 kOe and 4.4 kOe for field perpendicular to $c$. For the
resistivity data, Hc2 for field perpendicular to $c$ is 3.6 kOe. Reasonably,
the isotropic value determined from polycrystalline data is between the values
for fields parallel and perpendicular to $c$. The weak BCS coupling Pauli
limit H${}_{c2}(0)$ = (1.84(T)*Tc(K)) is determined to be 4.0 T (40 kOe) which
far exceeds any estimate or extrapolation of Hc2 for the single crystalline
data, possibly suggesting a singlet state.

First-principles calculations [20] obtain a quasi-two-dimensional Fermi
surface with small corrugations along the $c$ direction. Therefore, the upper
critical field along this direction is significantly smaller than $H_{c2(0)}$
perpendicular to the c axis. This, coupled with the upward curvature of the
upper critical field curves, could indicate that a multiband model may be
applicable.

Multiple aspects of SrPtAs, such as resistivity, ac susceptibility, and the
resulting upper critical field curves, are reminiscent of MgB2. For example,
(i) the highly anisotropic nature of Hc2 is reminiscent of MgB2 where
H${}_{c2}^{c}\sim$ 2.5 T and H${}_{c2}^{ab}\sim$ 16 T resulting in a low
temperature anisotropy value of H${}_{c2}^{ab}$/H${}_{c2}^{c}$ = 6-7 [26],
(ii) the upward curvature of Hc2, is an indicator of multiband
superconductivity [27], and (iii) the ac susceptibility curves exhibit a kink
below the onset of Tc [28], (for example, in Figure 4A, at 5 kOe there are two
kinks, one at $\sim$1 K and one at $\sim$0.8 K) although less pronounced in
the $ab$ plane, which could indicate multiband superconductivity [26], vortex
physics [26], surface superconductivity [29], or imperfect crystallinity [28].
However, the kinks have the same field dependence and anisotropy as Tc. This
could indicate a change in the gap structure or vortex excitations.

Figure 4: Low temperature thermal expansion of SrPtAs along the c-axis from
0.1 K to 7 K (black line). The shaded region indicates the detection threshold
of the experimental setup.

Thermal expansion of the same SrPtAs crystal along the c axis, shown in black
in Figure 4, does not reveal a bulk superconducting transition. Multiple runs
reveal statistical noise with no transition. The gray shaded region in Figure
4 indicates the resolution of the experimental setup. The lack of a feature
indicating a bulk superconducting transition could be because the magnitude of
the transition is below the detection threshold of this experimental setup.
Elemental Nb for example has a thermal expansion on the order of 10-8K-1 which
would not be distinguishable in this setup [30]. Additionally, SrPtAs may not
have much pressure dependence of Tc along the $c$ axis.

Heat capacity measurements were performed on the same crystal. Figure 5A shows
specific heat divided by temperature (Cp/T) as a function of temperature.
Figure 5B shows Cp/T as a function of temperature squared where the red line
denotes the fit to the data by C/T = $\gamma$ \+ $\beta$T2. Here $\gamma$, the
electronic specific heat coefficient, is 4.27 mJ/K2mol and $\beta$, a constant
corresponding to the Debye phonon contribution, is 0.65 mJ/K4mol. The Debye
temperature $\Theta_{D}$ is calculated from: ${\Theta_{D}}$ =
$\sqrt[3]{\frac{12\pi^{4}nR}{5\beta}}$ where R is the ideal gas constant
(8.314 J/molK) and $n$ = 3, the number of atoms per formula unit [31],
yielding $\Theta_{D}$ = 208 K. A small heat capacity jump ($\Delta
C/T_{c}\sim$2 mJ/K2mol) is visible at 0.76 K. Using the value of $\gamma$ from
the fit (= 4.27 mJ/K2mol), the quantity $\Delta$C/$\gamma T_{c}$ = 0.47 is
obtained. Assuming a weak-coupling value of $\Delta$C/$\gamma T_{c}$ = 1.43
[32], a superconducting volume fraction is estimated to be 33%. These values
may be compared to $\gamma$ = 7.31 mJ/K2mol, ${\Theta_{D}}$ = 241 K, and
$\Delta$C/$\gamma T_{c}$ = 1.07 for unpublished SrPtAs data mentioned in [33]
and $\Delta$C/$\gamma T_{c}$ for MgB2 of 1.09 [34].

The discrepancy between the higher Tc values ($\sim$2.0-2.4 K) determined by
non thermodynamic measurements compared to the lower Tc ($\sim$0.76 K) bulk
measurements continues to be a puzzle. Possible explanations include strain
induced filamentary or surface superconductivity and/or the presence of an
impurity phase. For example, in CeIrIn5 resistance goes to zero at $\sim$1 K
whereas specific heat data indicate bulk superconductivity at 0.4 K [35]. This
discrepancy is considered to be due to strain introduced by crystallographic
defects [36, 37]. Other examples of filamentary or surface superconductivity
include CePt3Si [38] and WO2.90 [39]. Additional measurements, such as
comparisons of multiple growth batches from various different research groups,
are required to elucidate the underlying reason for the discrepancy in Tc.
Because the recipe for the growth of single crystals has been determined the
opportunity for additional measurements on single crystals of SrPtAs is now
available.

Figure 5: A) Heat capacity divided by temperature (C/T) shown as a function of
temperature for SrPtAs. Inset shows zoomed low temperature region. B) Cp/T as
a function of T2. The red line denotes a fit to the data of C/T = $\gamma$ \+
$\beta$T2.

In summary, we have presented the synthesis and characterization of single
crystalline SrPtAs. Magnetic susceptibility and electrical resistivity
measurements show superconducting transitions at 2.0-2.4 K, consistent with
previous works [1, 8, 15, 16]. However, bulk specific heat show decreased Tc
of 0.7 K. Multiple similarities between SrPtAs and MgB2 have been discussed
and the possibility of SrPtAs as a multiband superconductor has been
considered. Our results on single crystals provide a new avenue to explore the
underlying mechanisms for the superconductivity of SrPtAs.

##  .1 Methods

Crystals of SrPtAs were grown in a Pb flux with a reaction ratio of 1:1:0.7:20
of Sr:Pt:As:Pb. The elements were weighed and placed into an alumina crucible
set [40]. The crucibles were placed in a fused-silica tube and flame sealed
under vacuum using a hydrogen torch. The reaction vessel was then heated in a
programmable furnace to 1150 ∘C in 72 hours, held for 72 hours, then cooled to
600 ∘C at a rate of 3 ∘C/h. The vessel was removed from the furnace at 600 ∘C
and inverted into a centrifuge to decant the Pb flux. Reaction ratios of
1:1:1:20 of Sr:Pt:As:Pb following the same heating profile resulted in
crystals of non superconducting SrPt0.7As0.9 crystallizing in the
${P}$$\overline{6}$${m}$2 space group. A single crystal $<$ 0.1 mm on each
edge was cut from the larger needle used for property measurements and mounted
to a CryoLoop using vacuum grease. Data were collected with a Bruker D8
Venture single crystal X-ray diffractometer with an Incoatec I$\mu$S
microfocus source (Mo radiation $\lambda$ = 0.71073 Å) and a PHOTON II CPAD
area detector. The raw data frames were processed with the Bruker SAINT
software, and a multi-scan absorption correction was applied with Bruker
SADABS. [41] Starting crystallographic models were obtained in SHELXT [42]
using the intrinsic phasing method, and least-squares refinements were
performed with SHELXL2018. [43] Single crystals of SrPtAs were kept in an
argon glovebox between measurements. Magnetization measurements were obtained
through a commercial Quantum Design MPMS SQUID-based magnetometer. Specific
heat measurements were made using a commercial Quantum Design PPMS calorimeter
that utilizes a quasi-adiabatic thermal relaxation technique. The electrical
resistivity $\rho$ was characterized using a standard four-probe configuration
with an ac resistance bridge. Thermal expansion measurements were performed
using a capacitance dilatometer described in Ref. [44] in an adiabatic
demagnetization cryostat. ac susceptibility was measured using a set of
commercial drive and pickup coils. The ac excitation field was estimated to be
0.042 Oe based on the geometry of the coils.

###### Acknowledgements.

The work at Los Alamos National Laboratory was primarily supported by the U.S.
Department of Energy, Office of Science, National Quantum Information Science
Research Centers, Quantum Science Center. Scanning electron microscope and
energy dispersive X-ray measurements were supported by the Center for
Integrated Nanotechnologies, an Office of Science User Facility operated for
the U.S. Department of Energy Office of Science. EDB acknowledges support from
the U.S. Department of Energy, Office of Basic Energy Sciences, Division of
Materials Sciences and Engineering, under the ”Quantum Fluctuations in Narrow
Band Systems.” AW acknowledges support from the Laboratory Directed Research
and Development program at LANL.

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$\varepsilon\to 0$, we deduce the following uniform bounds and set
$\gamma_{*}=\min\\{\gamma,2\\}$:

$\displaystyle\mathbb{E}\Big{[}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{1}{2}\varrho|{\bf
u}|^{2}\right>(\tau\wedge\tau_{M})\,dx\ \Big{]}\leq
C,\quad\mathbb{E}\Big{[}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{|\varrho-1|^{\gamma_{*}}}{\varepsilon^{2}}\right>(\tau\wedge\tau_{M})\,dx\
\Big{]}\leq C$

Now using (5.7), the continuity of $\nabla\Delta^{-1}\mathrm{div}$, and
regularity of ${\bf v}$, we obtain

$\displaystyle\left|\mathbb{E}\left[\int_{0}^{\tau\wedge\tau_{M}}\int_{\mathbb{T}^{3}}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{\varrho-1}{\varepsilon}\right>\,\nabla_{x}\Pi\cdot{\bf
v}\,dx\ \,{\rm d}t\right]\right|$
$\displaystyle\qquad\leq\mathbb{E}\Bigg{[}\Big{\|}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{\varrho-1}{\varepsilon}\right>\Big{\|}_{L^{\gamma_{*}}(\mathbb{T}^{3})}\Bigg{]}\,\mathbb{E}\Big{[}\|\nabla_{x}\Pi\|_{L^{2\gamma^{{}^{\prime}}_{*}}(\mathbb{T}^{3})}\Big{]}\,\mathbb{E}\Big{[}\|{\bf
v}\|_{L^{2\gamma^{{}^{\prime}}_{*}}(\mathbb{T}^{3})}\Big{]}$
$\displaystyle\qquad\qquad\leq\mathbb{E}\Bigg{[}\Big{\|}\left<{\mathcal{V}^{\omega,\varepsilon}_{t,x}};\frac{\varrho-1}{\varepsilon}\right>\Big{\|}_{L^{\gamma_{*}}(\mathbb{T}^{3})}\Bigg{]}\,\mathbb{E}\Big{[}\|{\bf
v}\otimes{\bf
v}\|_{L^{2\gamma^{{}^{\prime}}_{*}}(\mathbb{T}^{3})}\Big{]}\,\mathbb{E}\Big{[}\|{\bf
v}\|_{L^{2\gamma^{{}^{\prime}}_{*}}(\mathbb{T}^{3})}\Big{]}\leq C,$

uniformly for $\varepsilon\to 0$. In particular, the last two relevant terms
on the right-hand side of (5.15) and (5.16) vanish for $\varepsilon\to 0$. In
view of this result, along with (5.10), (5.11), and (5.12), the proof of the
theorem follows after taking expectation in (5.9), and applying Gronwall’s
lemma.

## Acknowledgements

U.K. acknowledges the support of the Department of Atomic Energy, Government
of India, under project no.$12$-R$\&$D-TFR-$5.01$-$0520$, and India SERB
Matrics grant MTR/$2017/000002$.

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