figurec

2D

    two-dimensional
3D

    three-dimensional
AHRS

    attitude and heading reference system
AUV

    autonomous underwater vehicle
CPP

    Chinese Postman Problem
DoF

    degree of freedom
DVL

    Doppler velocity log
FSM

    finite state machine
IMU

    inertial measurement unit
LBL

    Long Baseline
MCM

    mine countermeasures
MDP

    Markov decision process
POMDP

    Partially Observable Markov Decision Process
PRM

    Probabilistic Roadmap
ROI

    region of interest
ROS

    Robot Operating System
ROV

    remotely operated vehicle
RRT

    Rapidly-exploring Random Tree
SLAM

    Simultaneous Localisation and Mapping
SSE

    sum of squared errors
STOMP

    Stochastic Trajectory Optimization for Motion Planning
TRN

    Terrain-Relative Navigation
UAV

    unmanned aerial vehicle
USBL

    Ultra-Short Baseline
IPP

    informative path planning
FoV

    field of view
CDF

    cumulative distribution function
ML

    maximum likelihood
RMSE

    Root Mean Squared Error
MLL

    Mean Log Loss
GP

    Gaussian Process
KF

    Kalman Filter
IP

    Interior Point
BO

    Bayesian Optimization
SE

    squared exponential
UI

    uncertain input
MCL

    Monte Carlo Localisation
AMCL

    Adaptive Monte Carlo Localisation
SSIM

    Structural Similarity Index
MAE

    Mean Absolute Error
RMSE

    Root Mean Squared Error
AUSE

    Area Under the Sparsification Error curve
AL

    active learning
DL

    deep learning
CNN

    convolutional neural network
MC

    Monte-Carlo
GSD

    ground sample distance
BALD

    Bayesian active learning by disagreement
fCNN

    fully convolutional neural network
FCN

    fully convolutional neural network
CMA-ES

    Covariance Matrix Adaptation Evolution Strategy
FoV

    field of view
mIoU

    mean Intersection-over-Union
ECE

    expected calibration error

# NeU-NBV: Next Best View Planning Using Uncertainty Estimation
in Image-Based Neural Rendering

Liren Jin, Xieyuanli Chen, Julius Rückin, Marija Popović This work has been
fully funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy, EXC-2070 – 390732324
(PhenoRob) and supported by the NVIDIA Academic Hardware Grant Program. All
authors are with the Institute of Geodesy and Geoinformation, University of
Bonn. Corresponding<EMAIL_ADDRESS>
###### Abstract

Autonomous robotic tasks require actively perceiving the environment to
achieve application-specific goals. In this paper, we address the problem of
positioning an RGB camera to collect the most informative images to represent
an unknown scene, given a limited measurement budget. We propose a novel
mapless planning framework to iteratively plan the next best camera view based
on collected image measurements. A key aspect of our approach is a new
technique for uncertainty estimation in image-based neural rendering, which
guides measurement acquisition at the most uncertain view among view
candidates, thus maximising the information value during data collection. By
incrementally adding new measurements into our image collection, our approach
efficiently explores an unknown scene in a mapless manner. We show that our
uncertainty estimation is generalisable and valuable for view planning in
unknown scenes. Our planning experiments using synthetic and real-world data
verify that our uncertainty-guided approach finds informative images leading
to more accurate scene representations when compared against baselines.

##  I Introduction

Active perception and exploration is a core prerequisite for embodied robotic
intelligence. In many applications, including robotic manipulation,
inspection, and vision-based navigation, the ability to autonomously collect
data is crucial for scene understanding and further downstream tasks [1]. A
key challenge in this procedure is planning a view sequence for sensors to
obtain the most useful information given platform-specific constraints [2].

In this work, we present a new framework for iteratively planning the next
best view (NBV) for an RGB camera to explore an unknown scene. Given a limited
measurement budget, our goal is to actively position the sensor to gather the
most informative data of the scene online, i.e. during a robotic mission. To
address this problem, traditional NBV planning methods rely on explicit global
map representations of the scene, such as point clouds [3], volumes [4, 5, 6],
or surface meshes [7, 8], as a basis for planning. However, due to map
discretisation, these approaches scale poorly to larger scenes and have
limited representation ability [9].

Figure 1: Our novel NBV planning framework exploits uncertainty estimation in
image-based neural rendering to guide measurement acquisition. Given reference
images from the current image collection of the scene (black frustums), our
network outputs per-pixel uncertainty estimates at sampled view candidates
(coloured frustums). Brighter frustums indicate higher average uncertainty
from the view. Zoom-in boxes illustrate per-pixel uncertainty estimates at the
most certain and uncertain views. By selecting the most informative, i.e. most
uncertain, view candidate at which to take the next measurement, our approach
efficiently explores the unknown scene in a mapless manner.

Implicit neural representations, such as neural radiance fields (NeRFs) [10],
are drawing immense interest as an alternative approach for complex scene
understanding. Given posed 2D images, NeRFs synthesise novel views by
optimising an underlying continuous function encoding the scene appearance and
geometry. In the context of active perception, emerging works [11, 12, 13, 14,
15] incorporate uncertainty estimation into NeRFs and exploit it to guide NBV
planning. While showing promising results, these studies follow an active
learning [16] paradigm to collect the most informative, i.e. most uncertain,
images for periodically re-training a NeRF to improve the scene representation
with minimal data. Since re-training NeRF models is computationally expensive,
such methods are impractical for online robotic applications.

To overcome the inefficiency and per-scene optimisation requirements of NeRF
models, another line of work focuses on image-based neural rendering [17, 18,
19, 20]. Image-based approaches exploit a shared encoder to map nearby 2D
reference images into latent feature space, upon which the local implicit
representation is conditioned. This allows training the network across
multiple scenes to learn scene priors, enabling it to generalise to new scenes
without test time optimisation. Previous works in image-based neural rendering
[17, 18, 19, 20] mainly study improving network performance assuming pre-
recorded image data. However, exploiting the benefits of image-based neural
rendering for active view planning in robotics has not yet been considered.

The main contribution of this paper is a novel NBV planning framework bridging
the gap between active perception and image-based neural rendering for online
robotic applications. A key aspect of our framework is a new technique for
uncertainty estimation in image-based neural rendering, which enables us to
quantify the informativeness of view candidates without relying on ground
truth images or global scene representations. Intuitively, high uncertainty
indicates where scene information provided by the closest reference images is
insufficient to render the novel view, due to under-sampling or more complex
scene details in these areas. Therefore, we utilise view uncertainty as an
informative exploration objective. As shown in Fig. 1, based on the predicted
uncertainty, we actively select the most uncertain view candidate to maximise
the information acquired during a data collection process in an unknown scene.

To the best of our knowledge, our work is the first to address active
perception using image-based neural rendering. We make the following three
claims: (i) our uncertainty estimation technique generalises to unknown scenes
and provides an informative proxy for rendering quality at novel views; (ii)
our uncertainty-guided NBV planning strategy outperforms baseline approaches
in finding more informative images to represent an unknown scene given a
limited measurement budget; (iii) the informative images collected using our
approach also improve the offline training quality of NeRF models. To support
reproducibility, our implementation and simulation dataset will be released
at: https://github.com/dmar-bonn/neu-nbv.

##  II Related Work

###  II-A Next Best View Planning

View planning for robot active perception is an area of active research [2].
In initially unknown scenes, a common approach is to iteratively select the
NBV from a set of view candidates using an acquisition function capturing
their expected utility based on the current map state.

Isler et al. [4] build a probabilistic volumetric map and select the NBV by
calculating the information gain composed of visibility and the likelihood of
seeing new parts of an object. Bircher et al. [6] find the NBV in a receding-
horizon fashion by generating a random tree and selecting the branch
maximising the amount of unmapped space in a volumetric map from view
candidates. Zaenker et al. [5] maintain a voxel map of the scene and select
the NBV among candidates obtained by targeted region-of-interest sampling and
frontier-based exploration sampling. Zeng et al. [3] propose a point cloud-
based deep neural network to directly predict the information gain of view
candidates from the current raw point cloud of the scene. Song and Jo [8]
evaluate the completeness of reconstructed surfaces and extract low-confidence
surfaces to guide NBV planning. All these approaches require explicit
discretised 3D map representations to maintain current information about the
scene, which limits their scalability and representation ability. In contrast,
our approach utilises a compact implicit neural representation conditioned
only on 2D image inputs for NBV planning.

###  II-B Implicit Neural Representations

Implicit neural representations parameterise a continuous differentiable
signal with a neural network [9]. For example, NeRFs [10] learn a density and
radiance field supervised only by 2D images. To render a novel view, NeRFs
sample points densely along a camera ray, then predict radiance and density
from the position and view direction of each point. The final RGB and depth
estimate of the ray is calculated by differentiable volume rendering. As the
scene information is encoded in the network parameters, NeRFs overfit to a
single scene and require significant training time.

Instead of memorising a specific scene, image-based neural rendering, e.g.
PixelNeRF [17], leverages an encoder to map nearby reference images into
latent feature space. After aggregating features from reference images, a
multilayer perceptron (MLP) is trained to interpret the aggregated feature
into appearance and geometry information at a novel view. By training across
different scenes, image-based approaches generalise well to new scenes without
test time optimisation. We exploit the generalisation ability of image-based
neural rendering to achieve online NBV planning for efficient data collection
in an unknown scene.

###  II-C Uncertainty Estimation in Neural Representations

Estimating uncertainty in learning-based computer vision tasks is a long-
standing problem [21]. Several recent works address uncertainty quantification
in NeRF models. S-NeRF [22] proposes learning a probability distribution over
all possible radiance fields modelling the scene. To this end, it treats
radiance and density as stochastic variables and uses variational inference to
approximate their posterior distribution after training. W-NeRF [23] directly
learns to predict RGB variance as an uncertainty measure in rendering
transient objects in the scene. For image-based neural rendering, Rosu and
Behnke [18] introduce a loss function to learn confidence estimation in the
rendered images. However, they only consider a fixed number of reference
images with small view changes as inputs, which limits the applicability of
their approach in robotics.

Emerging works use uncertainty-guided NBV selection to address NeRF training
with a constrained measurement budget. Pan et al. [11] and Ran et al. [13]
model the emitted radiance as Gaussian distribution and learn to predict the
variance by minimising negative log-likelihood during training. These works
add the view candidate with the highest information gain, i.e. the highest
uncertainty reduction, to the existing training data. Instead of learning
uncertainty in parallel to radiance and density, Lee et al. [15] and Zhan et
al. [12] propose calculating the entropy of the density prediction along the
ray as an uncertainty measure with respect to the scene geometry. The entropy
is used to guide measurement acquisition towards less precise parts.
Sünderhauf et al. [14] exploit the recent development of fast rendering of
Instant-NGP [24] to train an ensemble of NeRF models for a single scene, and
measure uncertainty using the variance of the ensemble’s prediction, which is
utilised for NBV selection.

The above-mentioned approaches address uncertainty-guided NBV selection based
on NeRFs. Although these approaches show NeRF model refinement with limited
input data, deploying such methods in robotic applications is not
straightforward. As the scene information is entirely encoded in the network
weights, after each planning step, the uncertainty estimation must be re-
optimised to account for newly added measurements, which is time- and compute-
consuming. In contrast, our novel approach incorporates uncertainty estimation
in image-based neural rendering to actively select informative images, which
are incrementally added to our image collection. This way, we explore an
unknown scene without the need to maintain an explicit map representation or
re-train an implicit neural representation.

##  III Our Approach

Figure 2: Overview of our mapless NBV planning framework. We leverage
uncertainty estimation in image-based neural rendering to actively guide
measurement acquisition in unknown scenes.

We propose a novel mapless NBV planning framework for robotic exploration
tasks. An overview of our framework is shown in Fig. 2. We first sample view
candidates and query their corresponding closest reference images from the
current image collection. Based on the scene information provided by the
reference images, our image-based neural rendering network predicts per-pixel
uncertainty at these view candidates. The NBV planning strategy selects the
most uncertain view candidate corresponding to the next measurement, which we
add to the image collection. Our image-based neural rendering network
retrieves scene information in a purely image-based manner. This enables us to
achieve efficient autonomous exploration without maintaining an explicit map
or iteratively re-training an implicit neural representation. In the following
subsections, we describe our network architecture, training procedure for
uncertainty estimation, and NBV planning scheme.

###  III-A Network Architecture

Figure 3: Our network architecture. Different colours indicate features from
different reference images. Note that the encoder is not explicitly shown. We
use an LSTM module to predict jumping distance $\Delta t$ to the next sampling
point given the aggregated feature from all reference images acquired at the
current sampling point. After a fixed number of iterations, the aggregated
feature at the final point is interpreted to colour and uncertainty
information. Arrows with dashed lines show the forward pass happening only in
the last iteration.

Our network follows the design choices of PixelNeRF [17] regarding the
architecture of encoder and MLP. However, PixelNeRF uses a volume rendering
technique requiring dense sampling along the ray at predefined intervals,
which is inefficient and limits its online applicability. Inspired by Rosu and
Behnke [18] and Sitzmann et al. [25], we adopt a long short-term memory (LSTM)
module [26] to adaptively predict the jumping distance to the next sampling
point, therefore speeding up the inference of neural rendering. The network is
illustrated in Fig. 3.

Given a novel view, we query our current image collection to find the $N$
closest reference images $\mathbf{I}_{n\in\\{1,2,\ldots,N\\}}$. We use a
shared convolutional-based encoder $E$ to extract latent feature volume
$\mathbf{F}_{n}=E(\mathbf{I}_{n})\in\mathbb{R}^{H\times W\times L}$ from each
reference image. $H$ and $W$ are feature volume’s spatial resolution and $L$
is the channel dimension. We parameterise a ray emitted from the novel view as
$r(t)=\mathbf{o}+t\mathbf{d}$, where $\mathbf{o}\in\mathbb{R}^{3}$ is the
camera centre position and $t$ is the distance along view direction
$\mathbf{d}\in\mathbb{R}^{3}$. Starting from the close end of the ray
$t=t_{s}$, we transform the sampling point’s position $\mathbf{x}=r(t)$ and
view direction $\mathbf{d}$ into each reference view coordinate using known
relative camera poses to get $\mathbf{x}_{n}$ and $\mathbf{d}_{n}$,
respectively. To recover high-frequency details of the scene, the point
position $\mathbf{x}_{n}$ is mapped into higher-dimensional space by the
positional encoding operation $\gamma$ proposed by Mildenhall et al. [10]. By
combining it with its view direction, we compose pose feature
$\mathbf{p}_{n}=(\gamma(\mathbf{x}_{n}),\mathbf{d}_{n})$ for sampling point
expressed in $n^{th}$ reference view coordinate. To retrieve the latent image
feature from reference images, we project $\mathbf{x}_{n}$ onto the
corresponding reference image plane using known camera intrinsics to get image
coordinates $\phi(\mathbf{x}_{n})$, which we use to query the image feature
$\mathbf{f}_{n}=\mathbf{F}_{n}(\phi(\mathbf{x}_{n}))\in\mathbb{R}^{L}$ by grid
sampling with bilinear interpolation [17].

The acquired pose feature $\mathbf{p}_{n}$ and image feature $\mathbf{f}_{n}$
from each reference image are processed individually by MLPfeat. For
aggregating features from all reference images, we use the predicted weight
$w_{n}\in\left[0\,,1\right]$ and processed feature $\mathbf{f}_{n}^{\prime}$
to calculate the weighted mean $\mathbf{f}_{\mathrm{mean}}$ and variance
$\mathbf{f}_{\mathrm{var}}$. This operation downweights the feature from less
informative reference images. Conditioning on the aggregated feature
$(\mathbf{f}_{\mathrm{mean}},\mathbf{f}_{\mathrm{var}})$, our LSTM module
adaptively predicts the jumping distance $\Delta t$ to the next sampling point
$\mathbf{x}=r(t+\Delta t)$, thus mitigating the sampling inefficiency commonly
seen in volume rendering [17, 10]. We iterate this process a fixed number of
times to let the sampling point approach the surface in the scene and acquire
depth prediction. We then use MLPout to interpret the aggregated feature
queried at the final sampling point into colour and uncertainty information,
as detailed in the following subsection.

###  III-B Uncertainty Estimation in Image-based Neural Rendering

Our uncertainty estimation quantifies the uncertainty inherited from the input
data, due to the varying quality of the information provided by the reference
images. For example, we expect reference images with large view differences
and self-occlusions with respect to the novel view to lead to blurry rendering
and thus high uncertainty. An illustration of input-dependent uncertainty
estimated using our new approach is shown in Fig. 4.

Given supervision using only posed 2D images, we incorporate input-dependent
uncertainty estimation in the image-based neural rendering training process.
Considering that the predicted RGB value is normalised between
$\left[0\,,1\right]$, we model each channel value of the RGB prediction
$c_{i}\in\left[0\,,1\right]$, where $i\in\\{1,2,3\\}$, as an independent
logistic normal distribution described by:

$\displaystyle
p(c_{i};\mu_{i},\sigma_{i})=\frac{1}{\sigma_{i}\sqrt{2\pi}}\,\frac{1}{c_{i}(1-c_{i})}\,e^{-\frac{(\mathrm{logit}(c_{i})-\mu_{i})^{2}}{2\sigma_{i}^{2}}}\,,$
(1)

where
$\mathrm{logit}(c_{i})=\mathrm{ln}(\frac{c_{i}}{1-c_{i}})\sim\mathcal{N}(\mu_{i},\,\sigma_{i}^{2})$
follows a normal distribution, with the mean $\mu_{i}$ and variance
$\sigma_{i}^{2}$ predicted by our network. To train the network, following
Kendall and Gal [21], we minimise the negative log-likelihood $-\log
p\left(c_{i}=y_{i}\mid\mu_{i},\sigma_{i}\right)$ given ground truth RGB
channel values $y_{i}\in\left[0\,,1\right]$. For a single pixel RGB
prediction, this leads to our photometric loss function formulated as:

$\small{\mathcal{L}=\sum_{i=1}^{3}\frac{1}{2}\log(\sigma_{i}^{2})+\log(y_{i}(1-y_{i}))+\frac{(\mathrm{logit}(y_{i})-\mu_{i})^{2}}{2\sigma_{i}^{2}}\,.}$
(2)

For calculating the loss, the ground truth RGB channel value is mapped into
logit space by $\mathrm{logit}(y_{i})$. We clamp $y_{i}$ at
$\left[0.001\,,0.999\right]$ to ensure numerical stability.

During deployment in unknown scenes, given a novel view and its reference
images, our network predicts mean $\mu_{i}$ and variance $\sigma_{i}^{2}$
assuming each RGB channel of a pixel is normally distributed in logit space.
We then sample $100$ times from the normal distribution and pass all samples
through a sigmoid function to acquire a valid RGB channel value. We treat the
mean and variance of the $100$ channel values as our final channel-wise RGB
prediction $c_{i}\in\left[0\,,1\right]$ and uncertainty estimate
$u_{i}\in\left[0\,,0.25\right]$ of the respective pixel.

###  III-C Uncertainty Guided Next Best View Planning

Our novel NBV planning framework exploits uncertainty estimation in image-
based neural rendering to guide efficient data collection. Given a limited
measurement budget, our uncertainty-guided approach is effective at finding
more informative images to better represent an unknown scene.

For view planning, we consider a scene-centric hemisphere action space. First,
our planning procedure initialises the image collection with image
measurements at two random views. For planning the next camera view, we
uniformly sample a fixed number of view candidates $k\in\\{1,2,\ldots,K\\}$
within allowable view changes around the current camera view. For each view
candidate, we find at maximum $N$ closest reference images in our current
image collection. Given the novel view and corresponding reference images, our
network renders per-pixel uncertainty estimate
$\mathbf{U}_{k}\in\left[0\,,0.25\right]^{H_{r}\times W_{r}\times 3}$ following
the approach in Sec. IV-B, where $H_{r}$ and $W_{r}$ are the desired rendering
resolution. In this setup, we propose a simple utility function defined as:

$g(k)=\frac{1}{H_{r}\times W_{r}\times 3}\,\left\|\mathbf{U}_{k}\right\|_{1}.$
(3)

The view candidate $k^{*}$ with the highest utility $g(k^{*})$ is selected as
our NBV. High uncertainty indicates that the view candidate cannot be well-
rendered by our network given the current image collection, due to under-
sampling around the view, i.e. the closest reference images are far away, or
the scene is generally complex when observed from the view. Therefore, a new
measurement at the most uncertain view potentially yields the highest
information value for scene representation. The newly-captured image at the
NBV is then added to our image collection. We iterate this planning procedure
until a given measurement budget is exhausted. Note that our framework is
agnostic to sampling strategies and can be easily adapted to other specific
scenarios.

##  IV Experimental Evaluation

Reference image set 1

Reference image set 2

Rendered RGB

Uncertainty

Error

Figure 4: Examples of our input-dependent uncertainty estimation in image-based neural rendering. For rendering the same novel view, we select two sets of reference images. The comparison clearly shows how rendering quality depends on scene information provided by reference images. Reference images with low information value lead to blurry rendering and correspondingly high uncertainty prediction (yellower) from our network. The error map shows the mean squared error between ground truth and rendered RGB (whiter areas indicate higher errors). Our uncertainty prediction is strongly correlated with this error, thus serving as a good proxy for view planning.  TABLE I: Evaluation of uncertainty estimation strategies on the 15 test scenes of the DTU dataset. Best results in bold. Scene No. |  | 8 | 21 | 30 | 31 | 34 | 38 | 40 | 41 | 45 | 55 | 63 | 82 | 103 | 110 | 114
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
SRCC $\uparrow$ | Entropy | $0.16$ | $0.52$ | $0.37$ | $0.29$ | $0.21$ | $0.60$ | $0.39$ | $0.52$ | $0.17$ | $0.47$ | $0.53$ | $0.32$ | $0.42$ | $0.33$ | $0.60$
Confidence | $0.83$ | $0.83$ | $0.90$ | $0.80$ | $0.66$ | $0.76$ | $0.81$ | $0.80$ | $0.83$ | $0.78$ | $0.82$ | $0.88$ | $0.48$ | $0.53$ | $0.79$
Ours | $\mathbf{0.84}$ | $\mathbf{0.89}$ | $\mathbf{0.93}$ | $\mathbf{0.88}$ | $\mathbf{0.86}$ | $\mathbf{0.87}$ | $\mathbf{0.83}$ | $\mathbf{0.86}$ | $\mathbf{0.89}$ | $\mathbf{0.91}$ | $\mathbf{0.91}$ | $\mathbf{0.93}$ | $\mathbf{0.73}$ | $\mathbf{0.83}$ | $\mathbf{0.89}$
AUSE $\downarrow$ | Entropy | $0.50$ | $0.48$ | $0.34$ | $0.42$ | $0.55$ | $0.48$ | $0.50$ | $0.51$ | $0.51$ | $0.41$ | $0.38$ | $0.34$ | $0.47$ | $0.36$ | $0.45$
Confidence | $0.25$ | $0.26$ | $0.14$ | $0.18$ | $0.21$ | $0.28$ | $0.27$ | $0.22$ | $0.19$ | $0.23$ | $0.14$ | $0.16$ | $0.23$ | $0.20$ | $0.16$
Ours | $\mathbf{0.17}$ | $\mathbf{0.18}$ | $\mathbf{0.05}$ | $\mathbf{0.11}$ | $\mathbf{0.12}$ | $\mathbf{0.19}$ | $\mathbf{0.14}$ | $\mathbf{0.13}$ | $\mathbf{0.11}$ | $\mathbf{0.15}$ | $\mathbf{0.08}$ | $\mathbf{0.08}$ | $\mathbf{0.18}$ | $\mathbf{0.12}$ | $\mathbf{0.11}$

Our experimental results support our three claims: (i) we show that our
uncertainty estimation in image-based neural rendering is informative to
rendering quality and generalises to new scenes; (ii) we show that our
uncertainty-guided NBV planning strategy collects informative images using a
publicly available real-world dataset and in a simulated environment. To
measure the quality of collected images, we evaluate their influence on image-
based neural rendering performance at test views; and (iii) we show the
benefit of using our online collected images to train NeRF models.
Experimental results indicate that images collected using our planning
framework lead to more accurate implicit representations in both cases when
compared against baselines.

###  IV-A Training Procedure

Datasets. We train our network separately on two datasets for the
corresponding planning experiments. We first use real-world images with a
resolution of $400\times 300$ pixels from the DTU dataset [27]. We follow the
data split proposed by PixelNeRF [17] with $88$ training scenes and $15$ test
scenes, in which no shared or similar scenes exist. For each scene, $49$
images are collected following a fixed path pattern on a section of a scene-
centric hemisphere. We also record our own synthetic dataset, considering $50$
ShapeNet [28] models from $4$ representative categories: car, motorcycle,
camera, and ship. For each model, we record $100$ images with a resolution of
$200\times 200$ pixels from views uniformly distributed on the hemisphere
covering the scene.

Training Setup. We use the Adam optimiser with a learning rate of $10^{-5}$
and exponential decay of $0.999$. LSTM iteration number during a forward pass
is set to $16$. The network is implemented in PyTorch and trained with a
single NVIDIA RTX A5000 GPU for $\sim 2$ days until convergence. Rendering a
novel view with the same resolution as the two dataset images takes $0.6$ s
and $0.3$ s, respectively, which is $60$ times faster than PixelNeRF [17]. Our
network design is agnostic to the number of input reference images. For both
training processes, we randomly select $3$, $4$, or $5$ reference images for
novel view rendering in the scene to restrict memory consumption.

###  IV-B Evaluation of Uncertainty Estimation

Our first experiment is designed to show that our uncertainty estimation
strongly correlates with rendering error in image-based neural rendering in
unknown scenes. To evaluate the quality of uncertainty prediction, we consider
two metrics. We use Spearman’s Rank Correlation Coefficient (SRCC) [29] to
asses the monotonic relationship between averaged uncertainty estimate and
rendering error over a test view. As SRCC only captures the informativeness of
averaged uncertainty prediction, the quality with respect to the structural
similarity between the per-pixel uncertainty estimate and error is not
considered. To evaluate the structural similarity, we report the Area Under
the Sparsification Error (AUSE) curve [30], which reveals how strongly the
uncertainty coincides with the rendering error pixel-wise.

For each test scene in the DTU dataset, we create $100$ test sets. Each test
set consists of four images randomly selected from the scene, from which we
use three as reference images and the remaining one as the test view. We
average the predicted uncertainty and mean squared error (MSE) of each test
view. We then calculate SRCC values with respect to the $100$ pairs of
averaged uncertainty and MSE. Empirically, SRCC values higher than $0.8$
indicate strong monotonicity (high average uncertainty prediction is
consistent with high average rendering error). We also report the average AUSE
over $100$ test views for each scene. AUSE of $0$ means that the order of
pixel-wise uncertainty magnitude perfectly aligns with the order of the MSE
value (uncertain areas at the rendered test view overlap with erroneous
predictions).

We compare our approach against two alternative uncertainty estimation methods
that can be incorporated into image-based neural rendering frameworks. Lee et
al. [15] propose calculating the entropy of the density distribution of the
samples along each ray as uncertainty quantification in NeRF models. We re-
implement this entropy calculation in PixelNeRF, which we denote Entropy in
the experiments. Rosu and Behnke [18] proposes learning to predict RGB
rendering confidence in image-based neural rendering by defining the loss as a
linear combination of the predicted and the ground truth images. As their
approach only handles a fixed number of reference images with small view
changes, we adapt it by replacing our loss function Eq. 2 with their
confidence loss and train the network under the same conditions. We denote
this method Confidence.

Table I summarises the results. Our uncertainty prediction is more informative
with respect to rendering error compared to the other two methods. The poor
performance of the Entropy approach is likely due to the fact that the entropy
of the density distribution mainly captures uncertainty over scene geometry.
As neural rendering can recover colour information under inaccurate depth
prediction [9], naïvely incorporating Entropy as uncertainty estimation in
image-based neural rendering fails to provide useful information about
rendering quality. The superior performance of our approach compared to
Confidence indicates that our probabilistic interpretation of RGB prediction
leads to more consistent uncertainty estimates. A qualitative illustration of
our uncertainty prediction results is exemplified in Fig. 4.

###  IV-C Comparison of Next Best View Planning Strategies

We show that our uncertainty-guided NBV planning collects the most informative
images to better represent an unknown scene. For evaluating planning
performance, we use collected images and our image-based neural rendering
network to render test views. The rendering quality is measured by the peak
signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM)
[10]. Note that, since the image-based neural rendering network is fixed for
test view rendering in all experiments, performance differences arise purely
as the consequence of different NBV planning strategies. We compare our
uncertainty-guided approach against two heuristic baselines:

  * •

Ours: selects the most uncertain view candidate via our uncertainty prediction
as illustrated in Fig. 1;

  * •

Max. View Distance: selects the view candidate that maximises the view
distance with respect to previously collected images;

  * •

Random: selects a view candidate uniformly at random.

We conduct experiments on the DTU dataset and in our simulator with
corresponding pre-trained networks, respectively. For all planning
experiments, we initialise the image collection with two randomly selected
images and use different planning approaches to take the next images until a
given maximum of measurements is reached.

DTU dataset. We set the measurement budget to $9$ images including the $2$
images for initialisation. As the DTU dataset has limited views for each
scene, we treat all unselected views as view candidates. We adopt the three
planning strategies to select the next view from the view candidates and add
it to our image collection. After each view selection step, we use the current
image collection to render all unselected views. We calculate the average PSNR
and SSIM with standard deviations. We repeat the experiment $10$ times for all
$15$ test scenes and report the results in Fig. 5. As shown, NBV planning
guided by our uncertainty estimation selects the most informative view
candidate in each step reflected by better image-based neural rendering
quality.

Figure 5: Comparison of NBV planners on the DTU dataset. For each test scene,
we use our image-based neural rendering network and collected images to render
unselected views. To evaluate planning performance, we report the average PSNR
and SSIM with standard deviations over all test scenes and runs. Note that the
large standard deviations are due to varying rendering difficulty of each
scene. Our uncertainty-guided approach finds informative images in the scene,
improving scene representations via image-based neural rendering.

(a) Car model

(b) Indoor model

Figure 6: Comparison of NBV planners in a ShapeNet-based simulation
environment. We conduct the experiments on (a) car and (b) indoor models,
respectively. We fix $100$ test views for evaluation purposes. For each test
view, we query a maximum of 5 closest reference images from currently
collected images and use our image-based neural rendering network to render
the test view. We report the average PSNR and SSIM with standard deviations
over all test views and experiment runs. Our uncertainty-guided NBV planning
outperforms heuristic baselines in finding more informative images, resulting
in higher rendering quality given a limited measurement budget.

Simulator. To demonstrate the advantages of our NBV planning framework in a
more realistic robotic application scenario, we show the planning experiment
in a simulation environment with a continuous action space. We import two
different ShapeNet 3D models into the simulator. First, we consider a car
model, which belongs to the training category but is not seen during training.
Second, to show the generalisation ability of our approach, we test our
planning framework on an indoor model consisting of a sofa and table. Note
that the sofa and table are not in our training data categories. We configure
our action space as a scene-centric hemisphere and set the measurement budget
to $20$ images including $2$ initialisation images, as in the DTU experiment.
At each planning step, we uniformly sample $50$ view candidates within the
interval of maximal $60^{\circ}$ view angle change with respect to the current
camera view. The three planners select the next view among the sampled view
candidates. For our approach, we predict per-pixel uncertainty at $60\times
60$ pixels resolution for each view candidate using a maximum of $5$ closest
reference images. One planning step takes $1.5$ s in this setting. To evaluate
the quality of collected images during online missions, we fix $100$ random
test views of the scene. After every $2$ measurements, we use our network to
render all test views given a maximum of $5$ closest reference images from the
current image collection and report average PSNR and SSIM with standard
deviations to evaluate implicit scene reconstruction quality. We repeat each
planning experiment $10$ times on the two models, respectively.

Fig. 6 summarises the planning results. Our findings confirm that images
collected using our uncertainty-guided approach lead to better image-based
neural rendering quality in both scenes. Non-adaptive heuristic approaches
cannot efficiently utilise the measurement budget, thus limiting their view
planning performance. In contrast, our uncertainty-guided approach collects
informative images in a targeted way, resulting in higher test view rendering
quality.

###  IV-D Data Collection for Offline Modelling

In this experiment, we further show that the images collected by our approach
improve NeRF training using limited data. Note that different from
uncertainty-guided NBV planning based on NeRFs [11, 13, 14, 15], our
uncertainty estimation generalises to unknown scenes, thus the data collection
process and NeRF training can be decoupled in our framework. This avoids
computationally expensive network re-training during online missions.

After online NBV planning experiments in our simulator, described in Sec.
IV-C, we use Instant-NGP [24] to train NeRF models using images collected by
the three planning approaches, respectively, under the same training
conditions. To evaluate the training results, we render $100$ test views using
the trained NeRF models. We report the rendering metrics averaged over all
experiment runs in Table II and show examples of rendering results at complex
views from the scene in Fig. 7. Both quantitative and qualitative results
verify that our planning strategy for collecting informative images boosts
NeRF performance with limited training data. This indicates the benefits of
using our approach to efficiently explore an unknown scene and collect
informative images online. The 3D modelling of the scene can be done by
training NeRFs offline, after a robotic mission, when computational resources
are less constrained.

TABLE II: NeRF training results using images collected from our planning experiments in the simulator. Best results in bold. |  | Car | Indoor
---|---|---|---
PSNR $\uparrow$ | Max. View Distance | $27.37\pm 0.65$ | $30.02\pm 0.55$
Random | $25.73\pm 0.83$ | $28.46\pm 0.92$
Ours | $\mathbf{28.35\pm 0.53}$ | $\mathbf{30.46\pm 0.24}$
SSIM $\uparrow$ | Max. View Distance | $0.925\pm 0.004$ | $0.937\pm 0.003$
Random | $0.908\pm 0.012$ | $0.920\pm 0.007$
Ours | $\mathbf{0.934\pm 0.004}$ | $\mathbf{0.941\pm 0.003}$

Ground Truth

Max. View Distance

Random

Ours

Figure 7: Qualitative rendering results of trained NeRF models using image
collections from three different NBV planners. We select a challenging view
from each scene to show the rendering quality differences. Our uncertainty-
guided planner adapts the acquisition of new measurements according to the
scene structure in contrast to the heuristic strategies. For example, our
approach takes more images of the car bonnet, which has more structural
details compared to the car sides and is thus difficult to model using sparse
views.

##  V Conclusions and Future Work

In this work, we propose a novel mapless NBV planning framework for online
robotic applications. We integrate uncertainty estimation in image-based
neural rendering and exploit the predicted uncertainty to guide our
measurement acquisition. We show that our uncertainty estimation is
informative to the rendering quality at novel views and generalises to new
scenes. Our planning experiments prove that our uncertainty-guided NBV
planning scheme effectively finds informative views in an unknown scene. Image
collection using our approach leads to more accurate scene representations via
online image-based neural rendering and offline implicit reconstruction using
NeRFs.

One limitation of our current framework is that rendering a high-resolution
per-pixel uncertainty or RGB is inefficient for applications that require fast
robot motion. To address this, future work will consider exploiting depth
measurements to achieve more efficient sampling, thus speeding up the
inference of neural rendering. To extend our framework to complex and
cluttered environments, we plan to incorporate geometric uncertainty
estimation for planning in unconstrained action spaces. Finally, we will
investigate integrating semantic prediction with uncertainty estimation to
enable exploring regions of interest in an unknown scene in applications where
targeted inspection is necessary.

## Acknowledgement

We would like to thank Matteo Sodano for proofreading.

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# Angle-dependent Gap state in Asymmetric Nuclear Matter

Xin-le Shang<EMAIL_ADDRESS>Institute of Modern Physics, Chinese
Academy of Sciences, Lanzhou 730000, China  Wei Zuo  Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  State Key
Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese
Academy of Sciences, Beijing 100190, China

###### Abstract

We propose an axi-symmetric angle-dependent gap (ADG) state with the broken
rotational symmetry in isospin-asymmetric nuclear matter. In this state, the
deformed Fermi spheres of neutron and proton increase the pairing
probabilities along the axis of symmetry breaking near the average Fermi
surface. We find the state possesses lower free energy and larger gap value
than the angle-averaged gap state at large isospin asymmetries. These
properties are mainly caused by the coupling of different $m_{j}$ components
of the pairing gap. Furthermore, we find the transition from the ADG state to
normal state is of second order and the ADG state vanishes at the critical
isospin asymmetry $\alpha_{c}$ where the angle-averaged gap vanishes.

###### pacs:

21.65.Cd, 26.60.-c, 74.20.Fg, 74.25.-q

##  I Introduction

The neutron-proton (n-p) pairing properties play an important role in the
description of superfluidity of finite nuclei with $N\simeq Z$ Fn ; Fn2  and
symmetric nuclear matterSnm1 ; Snm2 ; Snm3 . In general, the n-p pair
correlations are considered in different dominant partial-wave channels,
depending on the relevant density and temperature. For weakly isospin-
asymmetric systems, the isospin singlet attractive ${}^{3}S_{1}-^{3}D_{1}$
(${}^{3}SD_{1}$) channel dominates the pairing interaction at relatively low
densities around the nuclear saturation density due to the tensor component of
the nuclear forceSnm1 ; Sdd ; Sdd2 ; Sdd3 ; Sdd4 ; Sdd5 , and the
${}^{3}D_{{}_{2}}$ channel dominates at high densities well above the
saturation densityD1 ; D2 . In neutron star matter, the n-p pair correlations
are strongly suppressed by the isospin-asymmetry. However, the dilute nuclear
matter at sub-saturation densities in supernovas and hot proto-neutron stars
can support ${}^{3}SD_{1}$ channel pairingSdd3 ; Sh1 ; Sh2 ; Sh3 .

Since n-p pair correlations depend crucially on the overlap between the
neutron and proton Fermi surfaces, the pairing gap is suppressed rapidly as
the system is driven out of the isospin-symmetric state. At zero temperature,
a small isospin-asymmetry is enough to prevent the formation of the Cooper
pairs between neutrons and protons with momenta $\overrightarrow{k}$ and
$-\overrightarrow{k}$ around their average Fermi surface where the
contribution to superfluidity is dominant. Near zero temperature, thermal
excitations can reduce the suppression by smearing out the two Fermi surfaces,
however, it is ineffective when the separation between the two Fermi surfaces
is large compared to the temperature. In isospin-asymmetric nuclear matter,
the FFLO ff ; lo  state and the DFS (deformed Fermi surfaces) dfs  state have
been studied in Refs.ffn ; dfsn . In a FFLO state, the shift of the two Fermi
spheres with respect to each other, resulting form the collective motion of
the Cooper pairs with a finite momentum, enhances the overlap between the
neutron and proton Fermi surfaces. The overlap regions then provide the
kinematical phase space for n-p pairing phenomena to occur. And in a DFS
state, the deformation of the neutron and proton Fermi surfaces may increase
the phase-space overlap between the two Fermi surfaces. Both in these two
kinds of possible superfluid states, the quasiparticle excitation spectra are
no longer isotropic, since the anisotropic overlapping configurations could
increase the pairing energy. On the other hand, the usually adopted angle-
averaging procedure in the previous calculationsSdd2 ; ffn , which has been
proved to be a quite good approximation in symmetry nuclear matter aap ,
considers the gap as an isotopic gap by ignoring the angle dependence. As the
true ground state corresponds to the anisotropic overlapping configuration,
the angle-averaging procedure may be an insufficient approximation in isospin-
asymmetric nuclear matter.

In this paper we consider an axi-symmetric angle dependent gap (ADG) state,
and give a general and systematic comparison between the ADG state and the
angle-averaged gap (AAG) state in isospin-asymmetric nuclear matter. The paper
is organized as follows: In Sec. II we briefly review the formalism for the
isotropic AAG state, and derive the angle dependent gap equations from the
Gorkov equations. The numerical solutions of these equations are shown and
discussed in Sec. III, where we compare the AAG state with the ADG state at
finite temperature. Finally, a summary and a conclusion are given in Sec. IV.

##  II Formalism

For isospin-asymmetric nuclear matter, the isospin singlet ${}^{3}SD_{1}$
pairing channel dominates the attractive pairing force at low densities. In
this case we can consider ${}^{3}SD_{1}$ channel only, the gap function is
thus expanded according to

$\displaystyle\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})=\sum_{l,m_{j}}\Delta_{l}^{m_{j}}(k)[G_{l}^{m_{j}}(\textbf{\^{k}})]_{\sigma_{1},\sigma_{2}},$
(1)

with the elements of the spin-angle matrices

$\displaystyle[G_{l}^{m_{j}}(\textbf{\^{k}})]_{\sigma_{1},\sigma_{2}}\equiv\langle\frac{1}{2}\sigma_{1},\frac{1}{2}\sigma_{2}\mid
1\sigma_{1}+\sigma_{2}\rangle\langle 1\sigma_{1}+\sigma_{2},lm_{l}\mid
1m_{j}\rangle Y_{l}^{m_{l}}(\textbf{\^{k}}),$

where $m_{j}$ and $m_{l}$ are the projections of the total angular momentum
$j=1$ and the orbit angular momentum $l=0,2$ of the pair, respectively. The
$Y_{l}^{m_{l}}(\textbf{\^{k}})$ denotes the spherical harmonic with
$\textbf{\^{k}}\equiv\textbf{k}/k$. The anomalous density matrix follows the
same expansion. Moreover the time-reversal invariance implies that

$\displaystyle\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})=(-1)^{1+\sigma_{1}+\sigma_{2}}\Delta_{-\sigma_{1},-\sigma_{2}}^{*}(\textbf{k}).$
(3)

Namely, the pairing gap matrix $\Delta(\textbf{k})$ in spin space possesses
the property

$\displaystyle\Delta(\textbf{k})\Delta^{{\dagger}}(\textbf{k})=ID^{2}(\textbf{k}),$
(4)

i.e., the gap function has the structure of a ``unitary triplet" state aap .
$I$ is the identity matrix and $D(\textbf{k})$ is a scalar quantity in spin
space.

Once the the isospin singlet ${}^{3}SD_{1}$ channel has been selected, the
pairing gap is an isoscalar and the isospin indices can be dropped off. The
proton/neutron propagators follow from the solution of the Gorkov equations,
and can be present in the form ($\hbar=1$)

$\displaystyle\textbf{G}_{\sigma,\sigma^{{}^{\prime}}}^{(p/n)}(\textbf{k},\omega_{m})=-\delta_{\sigma,\sigma^{{}^{\prime}}}\frac{i\omega_{m}+\xi_{\textbf{k}}\mp\delta\varepsilon_{\textbf{k}}}{(i\omega_{m}+E_{\textbf{k}}^{+})(i\omega_{m}-E_{\textbf{k}}^{-})}.$
(5)

The neutron-proton anomalous propagator matrix in spin space has the form

$\displaystyle\textbf{F}^{{\dagger}}(\textbf{k},\omega_{m})=-\frac{\Delta^{{\dagger}}(\textbf{k})}{(i\omega_{m}+E_{\textbf{k}}^{+})(i\omega_{m}-E_{\textbf{k}}^{-})},$
(6)

where $\omega_{m}$ are the Matsubara frequencies, the uper sign in
$\textbf{G}_{\sigma,\sigma^{{}^{\prime}}}^{(p/n)}$ corresponds to protons, and
the lower to neutrons. The quasiparticle excitation spectra are determined by
finding the poles of the propagators in Gorkov equations,

$\displaystyle
E_{\textbf{k}}^{\pm}=\sqrt{\xi_{\textbf{k}}^{2}+\frac{1}{2}Tr(\Delta\Delta^{{\dagger}})\pm\frac{1}{2}\sqrt{[Tr(\Delta\Delta^{{\dagger}})]^{2}-4\det(\Delta\Delta^{{\dagger}})}}\pm\delta\varepsilon_{\textbf{k}},$
(7)

where

$\displaystyle\xi_{\textbf{k}}\equiv\frac{1}{2}(\varepsilon_{\textbf{k}}^{p}+\varepsilon_{\textbf{k}}^{n}),\delta\varepsilon_{\textbf{k}}\equiv\frac{1}{2}(\varepsilon_{\textbf{k}}^{p}-\varepsilon_{\textbf{k}}^{n}),$

and $\varepsilon_{\textbf{k}}^{(n,p)}$ are the single particle energies of
neutrons and protons. Using the ``unitary" property in Eq. (4), the
quasiparticle spectra are simplified to

$\displaystyle
E_{\textbf{k}}^{\pm}=\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}\pm\delta\varepsilon_{\textbf{k}},$
(8)

which are separated into two branches due to the isospin-asymmetry.

In the present ``unitary triplet" case, the gap equation at finite temperature
can be written in the standard form

$\displaystyle\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})=-\sum_{\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>$
$\displaystyle\times\frac{\Delta_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})}{2\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}}[1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})],$
(9)

where $f(E)=[1+\exp(\beta E)]^{-1}$ is the Fermi distribution at finite
temperature and $V$ is the interaction in the ${}^{3}SD_{1}$ channel.
$\beta^{-1}=k_{B}T$, where $k_{B}$ is the Boltzmann constant and $T$ is the
temperature. Substituting the expansion Eq.(1) into Eqs.(9) and (4), one gets
a set of coupled equations for the quantities $\Delta_{l}^{m_{j}}(k)$

$\displaystyle\Delta_{l}^{m_{j}}(k)=\frac{-1}{\pi}\int_{0}^{\infty}dk^{{}^{\prime}}k^{{}^{\prime}2}\sum_{l^{{}^{\prime}}=0,2}i^{l^{{}^{\prime}}-l}V^{l^{{}^{\prime}}l}_{\lambda}(k^{{}^{\prime}},k)\sum_{l^{{}^{\prime\prime}}\mu}\Delta_{l^{{}^{\prime\prime}}}^{\mu}(k^{{}^{\prime}})$
$\displaystyle\times\int
d\Omega_{\textbf{k}^{{}^{\prime}}}Tr[G_{l^{{}^{\prime}}}^{m_{j}*}(\textbf{\^{k}}^{{}^{\prime}})G_{l^{{}^{\prime\prime}}}^{\mu}(\textbf{\^{k}}^{{}^{\prime}})]\frac{1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})}{\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}},$
(10)

with

$\displaystyle
D^{2}(\textbf{k})=\frac{1}{2}Tr(\Delta\Delta^{{\dagger}})=\sum_{ll^{{}^{\prime}}=0,2}\sum_{m_{j}m_{j^{{}^{\prime}}}}\Delta_{l}^{m_{j}*}(k)\Delta_{l^{{}^{\prime}}}^{m_{j^{{}^{\prime}}}}(k)Tr[G_{l}^{m_{j}{\dagger}}(\textbf{\^{k}})G_{l^{{}^{\prime}}}^{m_{j^{{}^{\prime}}}}(\textbf{\^{k}})],$

where

$\displaystyle
V^{l^{{}^{\prime}}l}_{\lambda}(k^{{}^{\prime}},k)\equiv<k^{{}^{\prime}}\mid
V^{l^{{}^{\prime}}l}_{\lambda}\mid
k>=\int_{0}^{\infty}r^{2}drj_{l^{{}^{\prime}}}(k^{{}^{\prime}}r)V^{l^{{}^{\prime}}l}_{\lambda}(r)j_{l}(kr),$
(12)

is the matrix elements of the NN interaction in different partial wave
($\lambda=T,S,l,l^{{}^{\prime}}$) channels. Here $\lambda$ corresponds to the
coupled ${}^{3}SD_{1}$ channel. Following from Eq.(5), we can get the
densities of neutrons and protons

$\displaystyle\rho^{(p/n)}=\sum_{\textbf{k},\sigma}n_{\sigma}^{(p/n)}(\textbf{k}),$
(13)

with the distributions

$\displaystyle
n_{\sigma}^{(p/n)}(\textbf{k})=\\{\frac{1}{2}(1+\frac{\xi_{\textbf{k}}}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}})f(E_{\textbf{k}}^{\pm})$
$\displaystyle+\frac{1}{2}(1-\frac{\xi_{\textbf{k}}}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}})[1-f(E_{\textbf{k}}^{\mp})]\\}.$
(14)

Summation over frequencies in Eq.(6) leads to the density matrix of the
particles in the condensate,

$\displaystyle\nu(\textbf{k})=\frac{\Delta(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
(15)

It is essential that the coupled Eqs.(10) and (13) should be solved self-
consistently.

The six components $\Delta_{l}^{m_{j}}(k)$ of $\Delta(\textbf{k})$ are
strongly coupled due to the angle dependent energy denominator
$\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}$ in Eqs.(10) and (13). The
equations are thus complicated to be solved accurately, and approximation has
been employed. Before introducing the angle-averaging procedure and ADG, we
need to substitute $\Delta_{l}^{m_{j}}(k)$ with real variables. From Eq.(3) we
can find the relation

$\displaystyle\Delta_{l}^{m_{j}*}(k)=-(-1)^{m_{j}}\Delta_{l}^{-m_{j}}(k).$
(16)

Therefore, we have four independent components $\Delta_{0}^{0}(k)$,
$\Delta_{0}^{1}(k)$, $\Delta_{2}^{0}(k)$ and $\Delta_{2}^{1}(k)$ for
${}^{3}SD_{1}$ channel , and we can describe $\Delta_{l}^{m_{j}}(k)$ as

$\displaystyle\Delta_{0}^{0}(k)$ $\displaystyle=i\delta_{0}(k),$
$\displaystyle\Delta_{0}^{1}(k)$ $\displaystyle=\delta_{1}(k)+in_{1}(k),$
$\displaystyle\Delta_{2}^{0}(k)$ $\displaystyle=i\delta_{2}(k),$
$\displaystyle\Delta_{2}^{1}(k)$ $\displaystyle=\delta_{3}(k)+in_{3}(k),$ (17)

where the six independent variables $\delta_{0}(k)$, $\delta_{1}(k)$,
$n_{1}(k)$, $\delta_{2}(k)$, $\delta_{3}(k)$ and $n_{3}(k)$ are real
quantities. Inserting Eq.(17) into Eq.(11), we get

$\displaystyle D^{2}(\textbf{k})=$
$\displaystyle\frac{1}{32\pi}\Big{\\{}4\delta_{0}^{2}(k)-4\sqrt{2}\delta_{0}(k)\delta_{2}(k)[3\cos^{2}\theta-1]+2\delta_{2}^{2}(k)[3\cos^{2}\theta-1]$
$\displaystyle+8[\delta_{1}^{2}(k)+n_{1}^{2}(k)]+8[\delta_{3}^{2}(k)+n_{3}^{2}(k)]+6[\delta_{3}^{2}(k)+n_{3}^{2}(k)]\sin^{2}\theta$
$\displaystyle+4\sqrt{2}n_{1}(k)n_{3}(k)[3\cos^{2}\theta-1]+4\sqrt{2}\delta_{1}(k)\delta_{3}(k)[3\cos^{2}\theta-1]$
$\displaystyle+12[2\delta_{0}(k)n_{3}(k)+2\delta_{2}(k)n_{1}(k)-\sqrt{2}\delta_{2}(k)n_{3}(k)]\cos\theta\sin\theta\cos\varphi$
$\displaystyle+12[2\delta_{1}(k)\delta_{2}(k)+2\delta_{0}(k)\delta_{3}(k)-\sqrt{2}\delta_{2}(k)\delta_{3}(k)]\cos\theta\sin\theta\sin\varphi$
$\displaystyle+6[n_{3}^{2}(k)-\delta_{3}^{2}(k)+2\sqrt{2}\delta_{1}(k)\delta_{3}(k)-2\sqrt{2}n_{1}(k)n_{3}(k)]\sin^{2}\theta\cos
2\varphi$
$\displaystyle+12[\delta_{3}(k)n_{3}(k)-\sqrt{2}\delta_{1}(k)n_{3}(k)-\sqrt{2}\delta_{3}(k)n_{1}(k)]\sin^{2}\theta\sin
2\varphi\Big{\\}}.$

###  II.1 The angle-averaging procedure

Supposing the angle dependence of the energy denominator
$\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}$ can be neglected, the gap
equations are simplified by substituting $D^{2}(\textbf{k})$ with its angular
average value,

$\displaystyle D^{2}(\textbf{k})\rightarrow$ $\displaystyle
d^{2}(k)=\frac{1}{4\pi}\int d\Omega_{\textbf{k}}D^{2}(\textbf{k})$ (19)
$\displaystyle=\frac{1}{8\pi}\Big{[}2\delta_{1}^{2}(k)+\delta_{0}^{2}(k)+2n_{1}^{2}(k)+2\delta_{3}^{2}(k)+\delta_{2}^{2}(k)+2n_{3}^{2}(k)\Big{]}.$

Thereby, the energy denominator and the quasiparticle spectra are isotropic.
Noting the properties of $G_{l}^{m_{j}}(\textbf{\^{k}})$

$\displaystyle\int
d\Omega_{\textbf{k}}Tr[G_{l}^{m_{j}*}(\textbf{\^{k}})G_{l^{{}^{\prime}}}^{m_{j^{{}^{\prime}}}*}(\textbf{\^{k}})]=\delta_{ll^{{}^{\prime}}}\delta_{m_{j}m_{j^{{}^{\prime}}}},$
(20)

the different $m_{j}$ components $\Delta_{l}^{m_{j}}(k)$ with the same $l$
become uncoupled and all equal to each other. It follows that

$\displaystyle\delta_{1}(k)=n_{1}(k)=\sqrt{\frac{1}{2}}\delta_{0}(k),\delta_{3}(k)=n_{3}(k)=\sqrt{\frac{1}{2}}\delta_{2}(k),$
(21)

and

$\displaystyle d^{2}(k)=\frac{3}{8\pi}[\delta_{0}^{2}(k)+\delta_{2}^{2}(k)].$
(22)

Taking the normalization

$\displaystyle\Delta_{0}(k)=\sqrt{\frac{3}{8\pi}}\delta_{0}(k),\Delta_{2}(k)=-\sqrt{\frac{3}{8\pi}}\delta_{2}(k),$
(23)

the set of equations in Eq.(10) reduces to two coupled equations for the
${}^{3}S_{1}$ and ${}^{3}D_{1}$ gap components $\Delta_{0}(k)$ and
$\Delta_{2}(k)$, respectively. They read

$\displaystyle\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k)=\frac{-1}{\pi}\int
dk^{{}^{\prime}}k^{{}^{\prime}2}\left(\begin{array}[]{ll}V^{00}&V^{02}\\\
V^{20}&V^{22}\end{array}\right)(k,k^{{}^{\prime}})\frac{1-f(E_{k^{{}^{\prime}}}^{+})-f(E_{k^{{}^{\prime}}}^{-})}{\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(k^{{}^{\prime}})}}\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k^{{}^{\prime}})$ (30) , (31)

where $V^{00}$, $V^{02}$, $V^{20}$, $V^{22}$ are given in Eq.(12) with
$l,l^{{}^{\prime}}=0,2$ and

$\displaystyle
E^{\pm}_{k}=\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(k)}\pm\delta\varepsilon_{\textbf{k}},$
$\displaystyle D^{2}(k)\equiv d^{2}(k)=\Delta_{0}^{2}(k)+\Delta_{2}^{2}(k).$
(32)

Eqs.(13), (24) and (25) compose the angle-averaged gap equations and should be
solved simultaneously for isospin-asymmetric nuclear matter. The quasiparticle
spectra here are isotropic and the gapless excitation exists at large
asymmetry ($|\delta\varepsilon_{\textbf{k}_{F}}|\geq D(k_{F})$) near zero
temperature.

###  II.2 The angle dependent gap

As pointed out in the Sec.I, the angle dependence of quasiparticle spectra due
to $D^{2}(\textbf{k})$ may increase the phase-space overlap of neutron and
proton near their average Fermi surface. We consider an axi-symmetric
$D^{2}(\textbf{k})$ solution which corresponds to an axi-symmetric deformation
of the neutron and proton Fermi spheres. From the expression in Eq.(18), the
axi-symmetric solutions are restricted by

$\displaystyle
2\delta_{0}(k)n_{3}(k)+2\delta_{2}(k)n_{1}(k)-\sqrt{2}\delta_{2}(k)n_{3}(k)=0,$
$\displaystyle
2\delta_{1}(k)\delta_{2}(k)+2\delta_{0}(k)\delta_{3}(k)-\sqrt{2}\delta_{2}(k)\delta_{3}(k)=0,$
$\displaystyle
n_{3}^{2}(k)-\delta_{3}^{2}(k)+2\sqrt{2}\delta_{1}(k)\delta_{3}(k)-2\sqrt{2}n_{1}(k)n_{3}(k)=0,$
$\displaystyle\delta_{3}(k)n_{3}(k)-\sqrt{2}\delta_{1}(k)n_{3}(k)-\sqrt{2}\delta_{3}(k)n_{1}(k)=0.$
(33)

There exists only one nontrivial solution

$\displaystyle\delta_{1}(k)=n_{1}(k)=\delta_{3}(k)=n_{3}(k)=0,$ (34)

which corresponds to the $m_{j}=0$ gap components of $\Delta_{l}^{m_{j}}(k)$.
In this case

$\displaystyle D^{2}(\textbf{k})\rightarrow D^{2}(k,\theta)=$
$\displaystyle\frac{1}{8\pi}\Big{[}\delta_{0}^{2}(k)-\sqrt{2}\delta_{0}(k)\delta_{2}(k)(3\cos^{2}\theta-1)$
(35) $\displaystyle+\delta_{2}^{2}(k)\frac{3\cos^{2}\theta+1}{2}\Big{]}.$

Using the normalization

$\displaystyle\Delta_{0}(k)=\sqrt{\frac{1}{8\pi}}\delta_{0}(k),\Delta_{2}(k)=-\sqrt{\frac{1}{8\pi}}\delta_{2}(k),$
(36)

one gets the angle dependent gap equations

$\displaystyle\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k)$ $\displaystyle=\frac{-1}{\pi}\int
dk^{{}^{\prime}}k^{{}^{\prime}2}\left(\begin{array}[]{ll}V^{00}&V^{02}\\\
V^{20}&V^{22}\end{array}\right)(k,k^{{}^{\prime}})$ (46)
$\displaystyle\times\int
d\Omega_{\textbf{k}^{{}^{\prime}}}\frac{1-f(E_{k^{{}^{\prime}}}^{+})-f(E_{k^{{}^{\prime}}}^{-})}{\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(k^{{}^{\prime}},\theta)}}\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)\left(\begin{array}[]{l}\Delta_{0}\\\
\Delta_{2}\end{array}\right)(k^{{}^{\prime}}),$

with the following axi-symmetric quantities,

$\displaystyle D^{2}(k,\theta)$
$\displaystyle=\Delta_{0}^{2}(k)+\sqrt{2}\Delta_{0}(k)\Delta_{2}(k)[3\cos^{2}\theta-1]+\Delta_{2}^{2}(k)[\frac{3\cos^{2}\theta+1}{2}],$
$\displaystyle E^{\pm}_{k}$
$\displaystyle=\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(k,\theta)}\pm\delta\varepsilon_{\textbf{k}}.$
(47)

The angle matrix
$\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)$ comes from the coupling
among the different $m_{j}$ components of $\Delta(\textbf{k})$. The matrix
elements are

$\displaystyle\emph{f}(\theta)$
$\displaystyle=Tr[G_{0}^{0*}(\textbf{\^{k}}^{{}^{\prime}})G_{0}^{0}(\textbf{\^{k}}^{{}^{\prime}})]=\frac{1}{4\pi},$
$\displaystyle\emph{g}(\theta)$
$\displaystyle=-Tr[G_{0}^{0*}(\textbf{\^{k}}^{{}^{\prime}})G_{2}^{0}(\textbf{\^{k}}^{{}^{\prime}})]=\frac{\sqrt{2}}{8\pi}(3\cos^{2}\theta-1),$
$\displaystyle\emph{h}(\theta)$
$\displaystyle=Tr[G_{2}^{0*}(\textbf{\^{k}}^{{}^{\prime}})G_{2}^{0}(\textbf{\^{k}}^{{}^{\prime}})]=\frac{1}{8\pi}(3\cos^{2}\theta+1).$
(48)

As a first inspection, when applying following the substitution [both in the
gap equations (30) and the expression of $E_{k}^{\pm}$ in Eq.(31)]

$\displaystyle\frac{3\cos^{2}\theta}{8\pi}\rightarrow\frac{1}{8\pi},$ (49)

which has been used as the angle-averaging procedure for ${}^{3}PF_{2}$
superfluidity in Ref.3pf2 , Eq.(30) reduces to the form of angle-averaged gap
Eq.(24). At zero temperature, the pairing is suppressed by the gapless
excitation near the average Fermi surface in the AAG state. However, pairing
can exist in the interval $(0,\theta_{1})\bigcup(\pi,\pi-\theta_{1})$ of
$\theta$ near the average Fermi surface in the ADG state, where

$\displaystyle\cos^{2}\theta_{1}=\frac{\delta\mu^{2}-\Delta_{0}^{2}(k_{F})+\sqrt{2}\Delta_{0}(k_{F})\Delta_{2}(k_{F})-\Delta_{2}^{2}(k_{F})/2}{3\sqrt{2}\Delta_{0}(k_{F})\Delta_{2}(k_{F})+3\Delta_{2}^{2}(k_{F})/2}$

and $\delta\mu$ is the difference between the neutron and proton chemical
potentials. This mechanism is consistent with that of the FFLO state.
Furthermore, the influences from the coupling of different $m_{j}$ components
are partially taken into account via the angle matrix
$\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)$ in the ADG state.

###  II.3 Thermodynamics

For isospin-asymmetric nuclear matter at a fixed temperature and given neutron
and proton densities, the essential quantity to describe the thermodynamics of
the system is the free energy defined as

$\displaystyle\emph{F}|_{\rho,\beta}=\emph{U}-\beta^{-1}\emph{S},$ (50)

where U is the internal energy and S is the entropy. In the mean-field
approximation, the entropy of the superfluid state is

$\displaystyle\emph{S}=-2k_{B}\sum_{\textbf{k}}$
$\displaystyle\\{f(E_{\textbf{k}}^{+})\ln
f(E_{\textbf{k}}^{+})+\bar{f}(E_{\textbf{k}}^{+})\ln\bar{f}(E_{\textbf{k}}^{+})$
(51) $\displaystyle+f(E_{\textbf{k}}^{-})\ln
f(E_{\textbf{k}}^{-})+\bar{f}(E_{\textbf{k}}^{-})\ln\bar{f}(E_{\textbf{k}}^{-})\\},$

where $\bar{f}(E_{\textbf{k}}^{\pm})=1-f(E_{\textbf{k}}^{\pm})$. The internal
energy of the superfluid state reads

U
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]$
(52)
$\displaystyle+\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}}).$

The first term in Eq.(36) includes the kinetic energy of the quasiparticles
which is a functional of the pairing gap. In the normal state it reduces to
the kinetic energy of the neutrons and protons. The second term includes the
BCS mean-field interaction among the particles in the condensate and can be
eliminated in terms of the gap equation (9) (shown in Appendix). Finally, the
internal energy is written as

U
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]$
(53)
$\displaystyle-\sum_{\textbf{k}}\frac{D^{2}(\textbf{k})}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$

A thermodynamically stable state minimizes the difference of the free energies
between the superconducting and normal states,
$\delta\emph{F}=\emph{F}_{S}-\emph{F}_{N}$ [the free energy in the normal
state follows from Eqs.(35) and (37) when $\Delta\rightarrow 0$].

##  III Results

The numerical calculations here focus on the effects of the angle dependence
of the quasiparticle spectra and the emergence of the ADG phase in isospin-
asymmetric nuclear matter. To simplify the calculations, several assumptions
have been adopted. Firstly, the pairing interaction is approximated by the
bare interaction; i.e., the effects of the screening of the pairing
interaction are ignored. Secondly, we adopt the free single particle (s.p.)
spectrum, which may affect the density of the states at the Fermi surface.
Previous calculations bhf1 ; bhf2  show that using a more realistic s.p.
spectrum obtained from the BHF approach (the BHF spectrum) may reduce the
${}^{3}SD_{1}$ channel pairing gap as compared with the free spectrum. As for
the pairing interaction, the screening potential (i.e., the higher-order
contribution in the pairing interaction) for the ${}^{3}S_{1}$ pairing channel
in nuclear matter under different approximations has been discussed in
Refs.scr1 . It has been shown the screening potential is repulsive at low
densities in the one-bubble approximation, whereas it is slightly attractive
in the full RPA (suitably renormalized to cure the low density mechanical
instability of nuclear matter scr1 ; scr2 ). Up to now, the screening effect
on the pairing gap remains an open problem. Finally, we ignore the isospin
triplet states, which is valid when the pairing in the isospin singlet channel
is much larger than that in the isospin triplet channel. However, the argument
could be questionable when the first two approximations are abandoned. In the
present calculations, the net density is fixed at the empirical saturation
density of nuclear matter $\rho=\rho_{0}=0.17fm^{-3}$ except for Fig.7, and
the Argonne $V_{18}$ potential is adopted as the pairing interaction.

Figure 1: (Color online). The upper and lower curves in the figures are
related to the values of $\Delta_{0}(k_{F})$ and $\Delta_{2}(k_{F})$ vs
isospin-asymmetry $\alpha$. The blue solid and red dashed lines correspond to
the ADG and angle-averaged gap, respectively.

Fig.1 shows the angle-averaged and angle dependent gaps $\Delta_{0}(k_{F})$
and $\Delta_{2}(k_{F})$ in the ${}^{3}SD_{1}$ partial-wave channel as a
function of isospin-asymmetry $\alpha$, defined as
$\alpha=(\rho_{n}-\rho_{p})/\rho$. The temperatures are set at low-temperature
regime $\beta^{-1}=$ $0.5$ MeV, $1.0$ MeV, $2.0$ MeV, $3.0$ MeV (the critical
temperature $\beta^{-1}_{c}$ where the superfluid vanishes is about $7.5$ MeV
for isospin-symmetric case). At temperature $\beta^{-1}=0.5$ MeV, the value of
$\Delta_{0}(k_{F})$ in the ADG state becomes larger than that of the angle-
averaged gap state for $\alpha\geq 0.07$, and the difference of
$\Delta_{0}(k_{F})$ between the ADG and angle-averaged gap states reaches 22
percent at $\alpha=0.23$. With increasing temperature, the difference of
$\Delta_{0}(k_{F})$ between the two kinds of states decreases rapidly. The
critical isospin-asymmetries $\alpha_{c}$ at which the gaps vanish are the
same in the two states, and their values are $0.267$, $0.275$, $0.30$ and
$0.315$ for the temperatures $0.5$ MeV, $1.0$ MeV, $2.0$ MeV and $3.0$ MeV,
respectively. It implies that the thermal excitation can promote pairing in
large isospin-asymmetry nuclear matter at low temperature regime.

In order to have an entire inspection of the difference between the pairing
gaps of the ADG state and the angle-averaged gap state, we exhibit the gap
functions in Fig.2.

Figure 2: (Color online). The curves marked with symbols ${}^{3}S_{1}$ and
${}^{3}D_{1}$ are related to the gap functions $\Delta_{0}(k)$ and
$\Delta_{2}(k)$ in Eqs.(24) and (30). The blue solid and red dashed lines
correspond to the ADG and angle-averaged gap, respectively.

At temperature $\beta^{-1}=0.5$ MeV, the gap functions of the two different
kinds of states are almost the same except a little difference of
$\Delta_{0}(k)$ near the zero momentum for the asymmetry $\alpha=0.02$ [in
Fig.2.(a)]. When the system becomes more asymmetric, the difference gets
larger [in Fig.2.(b)]. However, the curves of the ADG coincide with these of
the angle-averaged gap for $\beta^{-1}=3.0$ MeV with $\alpha=0.16$ [in
Fig.2.(d)]. That implies the angle-averaging procedure is a satisfactory
approximation for asymmetric nuclear matter at high temperatures.

Figure 3: (Color online). The difference of the free energy between the
superconducting and normal states as a function of the isospin-symmetry
$\alpha$ for diffderent temperatures. The blue solid and red dashed lines
correspond to the ADG and angle-averaged gap, respectively.

A larger gap value in ADG state may result in a larger pairing energy in the
condensate [second term in Eqs.(36) and (37)], which has important influence
on the free energy of the superconducting state. Thus we calculate the free-
energy difference $\delta\emph{F}$ between the normal and superconducting
states. The results are shown in Fig.3, where the parameters are set as the
same as those in Fig.1. At temperature $\beta^{-1}=0.5$ MeV, $\delta F$ in the
ADG state gets smaller than that of the angle-averaged gap state when
$\alpha\geq 0.06$, especially, the former is about 35 percent lower than the
latter in the regime $\alpha>0.17$. We can conclude that the ADG state is more
favored than the angle-averaged gap state for large asymmetry at low
temperature, since the angle dependence of the pairing gap enhances the
pairing energy and has little effect on the kinetic energy. However, the
thermal excitation can reduce the effects of angle dependence of the pairing
gap [comparing the Fig.3.(a) with Fig.3.(d)]. It is also shown in Fig.3 that
the values of $\delta F$ tend to zero gently when
$\alpha\rightarrow\alpha_{c}$ at different temperatures.

Figure 4: (Color online). The higher and lower curves in the upper two figures
are related to the neutron and proton occupation probabilities, respectively.
The curves in the lower two figures are the pairing probabilities. The blue
solid and red dashed lines correspond to the ADG and angle-averaged gap,
respectively.

One straightforward way to understand the effects of angle dependence of the
pairing gap is to investigate the normal and superconducting occupation
probabilities [obtained from Eqs.(14) and (15)] near the average Fermi surface
(related to the average chemical potential of neutron and proton). The results
are depicted in Fig.4, where the spin summation has been carried out. In this
figure, the neutron/proton and pairing particle occupation probabilities at
the average Fermi surface for a fixed asymmetry $\alpha=0.16$ at temperature
$\beta^{-1}=$ $0.5$ MeV has been compared with those at $3.0$ MeV. In isospin-
asymmetric nuclear matter, the large splitting between the neutron and proton
occupation probabilities prevents the pairing around the average Fermi surface
in the angle-averaging procedure. However, in the ADG state, the splitting is
reduced by the angle dependence of the pairing gap in partial area around the
average Fermi surface, i.e., in the regime
$\theta\subset(0,\frac{\pi}{5})\cup(\frac{4\pi}{5},\pi)$ as shown in
Fig.4.(a). In Fig.4.(c), as compared with the angle-averaged gap, although the
pairing in the ADG state is almost fully suppressed in the regime
$\theta\subset(\frac{\pi}{5},\frac{4\pi}{5})$ in ADG state, it is obviously
enhanced at $\theta$ smaller than $\frac{\pi}{5}$ and greater than
$\frac{4\pi}{5}$.

Substituting the expression of $D^{2}(\textbf{k})$ in Eq.(31) into Eq.(14), we
can find that the Fermi spheres of neutron and proton are no longer isotropic
in the ADG state. Since we assume an axi-symmetric quasiparticle spectrum in
the ADG state, the rotational symmetry is spontaneously broken [in terms of
group theory the $O(3)$ symmetry breaks down to $O(2)$] and there exists one
favored direction. The neutron Fermi sphere possesses an oblate deformation
perpendicular to the favored direction, whereas the proton Fermi sphere has a
prolate deformation along the favored direction. The two different
deformations enhance the correlation between neutrons and protons near their
average Fermi surface. However, at high temperature the neutron/proton
occupation probability in the ADG becomes almost isotropic as shown in
Fig.4.(b), namely, the thermal excitation reduces the angle dependence of
quasiparticle spectra. In this case, the deformation of the neutron/proton
Fermi sphere fails to increase the phase-space overlap of neutron and proton
near their average Fermi surface effectively. Thus the results of the ADG
state are nearly the same as that of the angle-averaged gap state, i.e., the
angle-averaging procedure becomes an adequate approximation at high
temperatures $\beta^{-1}\geq 3$ MeV.

Figure 5: (Color online). The entropy (scaled by $\beta^{-1}$) as a function
of the isospin-symmetry $\alpha$ for different temperature. The blue solid,
red dashed and black dash-dotted lines correspond to the ADG, angle-averaged
gap and normal state, respectively.

Fig.5 displays the entropy ($\beta^{-1}S$) as a function of isospin-asymmetry
$\alpha$ for different temperatures $\beta^{-1}=$ $0.5$ MeV, $1.0$ MeV, $2.0$
MeV and $3.0$ MeV. The entropy in the superconducting state is smaller than
that in the normal state near $\alpha=0$, and gets larger than that in the
normal state at sufficiently large asymmetry. However, around the transition
point $\alpha_{c}$ from the superconducting state to the normal state, the
entropies of the superconducting states (both of the ADG state and the angle-
averaged gap state) approach to the value of the normal state, i.e., the
latent heats $Q=\beta^{-1}(S_{s}-S_{n})\rightarrow 0$ when
$\alpha\rightarrow\alpha_{c}$. Hence the transitions are of second order. At
temperature $\beta^{-1}=$ $0.5$ MeV, the entropy in the ADG state is nearly a
linear function of the isospin-asymmetry when $0.02<\alpha<0.22$. With
increasing temperature, the linear property of the entropy curve disappears
and the difference between the ADG state and the angle-averaged gap state gets
smaller.

Figure 6: (Color online). $\Delta_{0}(k_{F})$ and $\Delta_{2}(k_{F})$ as a
function of isospin-asymmetry $\alpha$ for the ADG, angle-averaged gap and
`approximation in ADG' are shown in Fig.(a). Fig.(b) exhibits the gap
functions for the three case. The normal and superconducting occupation
probabilities at the average Fermi surface for the three case are shown in
Figs.(c) and (d), respectively. The blue solid, red dashed and green dash-
dotted lines correspond to the ADG, angle-averaged gap and the `approximation
in ADG', respectively. The temperature is set to be $\beta^{-1}=0.5$ MeV, and
the isospin-asymmetry $\alpha=0.16$ in (b), (c), (d).

Comparing the gap equations (30) for the ADG state with Eq.(24) for the angle-
averaged gap state, two differences appear in the ADG state, i.e., the angle
dependent quasiparticle spectrum and the angle matrix
$\left(\begin{array}[]{ll}\emph{f}(\theta)&\emph{g}(\theta)\\\
\emph{g}(\theta)&\emph{h}(\theta)\end{array}\right)$. The first leads to the
deformation of neutron/proton Fermi sphere, and the second corresponds to the
coupling among different $m_{j}$ gap components. Actually, the angle matrix
modifies the strength of $V^{l^{{}^{\prime}}l}_{\lambda}(k^{{}^{\prime}},k)$
in different directions in momentum space. We replace the angle matrix by
$\frac{1}{4\pi}\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$ to
inspect the influence of the angle matrix. The results are shown in Fig.6 for
asymmetry $\alpha=0.16$ in (b), (c), (d) and the temperature is set to be
$\beta^{-1}=0.5$ MeV. The dash-doted lines denoted by `approximation in ADG'
are obtained by replacing the angle matrix with
$\frac{1}{4\pi}\left(\begin{array}[]{ll}1&0\\\ 0&1\end{array}\right)$.
Figs.6.(c) and (d) exhibit the neutron/proton and pairing particle occupation
probabilities at the average Fermi surface, respectively. The curves of ADG
and `approximation in ADG' are nearly the same in Figs.6.(c) and (d). Whereas
the gap functions in Fig.6.(b) show that the curves of `approximation in ADG'
behave closer to those of the angle-averaged gap state than those of the ADG
state. Fig.6.(a) displays the $\Delta_{0}(k_{F})$ and $\Delta_{2}(k_{F})$ vs
isospin-asymmetry $\alpha$. The gaps of `approximation in ADG' turn out to be
smaller than both the gaps in the ADG state and angle-averaged gap state when
$\alpha>0.07$. Moreover, the curves of `approximation in ADG' are much closer
to that of the angle-averaged gap state. All these results indicate that the
influence of the angle matrix is much more important than that of the angle
dependence of quasiparticle spectrum. Furthermore, the coupling from different
$m_{j}$ gap components may strengthen the pairing interaction for large
isospin-asymmetry at low temperatures.

Figure 7: (Color online). The difference of the free energy between the
superconducting and normal states as a function of the isospin-symmetry
$\alpha$ for different densities at a fixed temperature $\beta^{-1}=0.5$ MeV.
The blue solid and red dashed lines correspond to the ADG and angle-averaged
gap, respectively.

In order to discuss the effect of angle dependence of the pairing gap for
different densities, we show the free energy difference $\delta\emph{F}$
between the superconducting and normal states at temperature $\beta^{-1}=0.5$
MeV vs isospin-asymmetry $\alpha$ in Fig.7. The densities are set to be
$\rho=0.05$, $0.1$, $1.5\rho_{0}$ and $2\rho_{0}$ for (a), (b), (c) and (d),
respectively. At the density $\rho=0.05$ [in Fig.7.(a)], the two curves of
$\delta\emph{F}$ for the ADG and angle-averaged gap states are very close to
each other, indicating the effect of angle dependence of the pairing gap is
quite small at low densities. When the density increases, the difference of
$\delta\emph{F}$ for the ADG and angle-averaged gap states increases rapidly,
implying that the angle dependence of the pairing gap is more important at
higher densities. As the Fermi energy $\emph{E}_{F}\propto\rho^{\frac{2}{3}}$,
the value of $\frac{\Delta}{\emph{E}_{F}}$ is thus small at high densities. In
this case, the summations over ${k}^{{}^{\prime}}$ in the gap equation (9)
concentrate near the average Fermi surface (i.e., the contribution to
superfluidity from the Cooper pairs around the average Fermi surface is
dominant). A little separation of the neutron and proton Fermi surfaces
$\delta\mu$ may suppress the superfluidity strongly. In the ADG configuration,
the angle dependence can reduce the suppression. However, at low densities,
the value of $\frac{\Delta}{\emph{E}_{F}}$ gets large. Thus the contribution
to superfluidity from the Cooper pairs near the average Fermi surface is no
longer as important as that at high densities. Since the angle dependence
mainly increases the pairing probability around the average Fermi surface, the
effect of the angle dependence becomes weak at low densities.

##  IV Summary and Outlook

The fermionic condensation in asymmetric nuclear matter leads to
superconducting states which spontaneously break the spatial symmetries (such
as FFLO and DFS states). The quasiparticle spectrum behaves as an isotropic
one and the angle dependence of the pairing gap should be reconsidered. In
this work we propose an axi-symmetric angle dependent gap state in which the
isotropic symmetry is broken in isospin-asymmetric nuclear matter, and compare
with the angle-averaged gap state. It is shown the ADG state is more favored
than the angle-averaged gap state for large asymmetry at low temperature, and
the differences of both the gap values and the free energies between the two
kinds of states get small with increasing temperature. At temperature
$\beta^{-1}=0.5$ MeV with density $\rho_{0}$, the maximal differences of
$\Delta_{0}(k_{F})$ and $\delta\emph{F}$ between the ADG state and angle-
averaged gap state are about 22 and 35 percent, respectively. The differences
get larger at higher densities for $\beta^{-1}=0.5$ MeV. In the ADG state, the
neutron and proton deformed Fermi spheres increase the pairing probability
along the axis of symmetry breaking near their average Fermi surface. The
effect of the coupling among different $m_{j}$ gap components is also
investigated in this work and we find the coupling dominates the main
contribution to the mechanism of the ADG state.

The ADG state vanishes at the critical value $\alpha_{c}$, where the angle-
averaged gap vanishes. And the phase transition from the ADG state to the
normal state is of the second order. When temperature goes up, $\alpha_{c}$
rises and the effect of angle dependence of pairing gap becomes weak. In a
certain region of $\alpha$ the latent heat has an anomalous negative sign,
which is consistent with the result if Ref.Sdd2 . However, this does not
affect the stability of the ADG state, since its energy budget is dominated by
the pair-condensation energy.

In the ADG state, the symmetry is broken spontaneously. It is different from
that in the FFLO state, where the symmetry is broken by the collective motion
of the cooper pairs (the translation and rotational symmetries are both
broken). The translation symmetry is maintained in the ADG state. The
deformation of the neutron/proton Fermi sphere in the ADG state is similar to
the DFS configuration, however, the mechanisms are different. In the DFS state
the symmetry breaking corresponds to the deformed Fermi surface, while in the
ADG state the symmetry breaking results from the angle dependence of the
pairing gap. As is well known, the continuous symmetry breaking leads to
collective excitations with vanishing minimal frequency (Goldstone's theorem).
The breaking of rotational symmetry, which corresponds to the anisotropic
$D^{2}(\textbf{k})$ in the ADG state, may imply new collective bosonic modes
in asymmetric nuclear matter. However, the true ground state could be a
combination of the ADG state and the FFLO state, we should consider the ADG
state with the cooper pair momentum together which is in progress.

## Acknowledgments

The work is supported by the 973 Program of China under No. 2013CB834405, the
National Natural Science Foundation of China (No. 11175219), and the Knowledge
Innovation Project(No. KJCX2-EW-N01) of the Chinese Academy of Sciences.

## Appendix

We present here the main steps of the elimination of the second term in
Eq.(36) by using the gap equation (9). The elements of the density matrix of
the particles in condensate are,

$\displaystyle\nu_{\sigma_{1},\sigma_{2}}(\textbf{k})=\frac{\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
(54)

The second term of Eq.(36) is written as

$\displaystyle\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})$
$\displaystyle=\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\frac{\Delta^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle\times\frac{\Delta_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})}{2\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}}[1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})]$
$\displaystyle=\sum_{\textbf{k},\sigma_{1},\sigma_{2}}\frac{\Delta^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle\times\sum_{\textbf{k}^{{}^{\prime}},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\frac{\Delta_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})}{2\sqrt{\xi_{\textbf{k}^{{}^{\prime}}}^{2}+D^{2}(\textbf{k}^{{}^{\prime}})}}[1-f(E_{\textbf{k}^{{}^{\prime}}}^{+})-f(E_{\textbf{k}^{{}^{\prime}}}^{-})].$
(55)

Noting that the second summation over
${k}^{{}^{\prime}},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}$ is
$-\Delta_{\sigma_{2},\sigma_{1}}$ (using the gap equation (9)), thus

$\displaystyle\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})$
$\displaystyle=-\sum_{\textbf{k},\sigma_{1},\sigma_{2}}\frac{\Delta^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\Delta_{\sigma_{1},\sigma_{2}}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle=-\sum_{\textbf{k}}\frac{Tr[\Delta(\textbf{k})\Delta^{{\dagger}}(\textbf{k})]}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$
(56)

Using the ``unitary" property Eq.(4),

U
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]$
(57)
$\displaystyle+\sum_{\textbf{k},\textbf{k}^{{}^{\prime}}}\sum_{\sigma_{1},\sigma_{2},\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}<\textbf{k}\sigma_{1},-\textbf{k}\sigma_{2}\mid
V\mid\textbf{k}^{{}^{\prime}}\sigma_{1}^{{}^{\prime}},-\textbf{k}^{{}^{\prime}}\sigma_{2}^{{}^{\prime}}>\nu^{{\dagger}}_{\sigma_{2},\sigma_{1}}(\textbf{k})\nu_{\sigma_{1}^{{}^{\prime}},\sigma_{2}^{{}^{\prime}}}(\textbf{k}^{{}^{\prime}})$
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]-\sum_{\textbf{k}}\frac{2D^{2}(\textbf{k})}{2\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})]$
$\displaystyle=\sum_{\sigma\textbf{k}}[\varepsilon_{\textbf{k}}^{(n)}n_{\sigma}^{(n)}(\textbf{k})+\varepsilon_{\textbf{k}}^{(p)}n_{\sigma}^{(p)}(\textbf{k})]-\sum_{\textbf{k}}\frac{D^{2}(\textbf{k})}{\sqrt{\xi_{\textbf{k}}^{2}+D^{2}(\textbf{k})}}[1-f(E_{\textbf{k}}^{+})-f(E_{\textbf{k}}^{-})].$

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