# Graph Representation Learning for Wireless Communications

Maryam Mohsenivatani, Samad Ali, Vismika Ranasinghe, Nandana Rajatheva, and
Matti Latva-Aho

###### Abstract

Wireless networks are inherently graph-structured, which can be utilized in
graph representation learning to solve complex wireless network optimization
problems. In graph representation learning, feature vectors for each entity in
the network are calculated such that they capture spatial and temporal
dependencies in their local and global neighbourhoods. Graph neural networks
(GNNs) are powerful tools to solve these complex problems because of their
expressive representation and reasoning power. In this paper, the potential of
graph representation learning and GNNs in wireless networks is presented. An
overview of graph learning is provided which covers the fundamentals and
concepts such as feature design over graphs, GNNs, and their design
principles. Potential of graph representation learning in wireless networks is
presented via few exemplary use cases and some initial results on the GNN-
based access point selection for cell-free massive MIMO systems.

###### Index Terms:

Machine learning, graph neural networks, graph representation learning.

##  I Introduction

Wireless networks are shifting from providing connectivity for human-centric
end users to internet of connected things and internet of connected
intelligence. In order to support novel emerging services driven by societal
needs, radical innovations across various layers of communications protocol
stack is required. The traditional technologies and methodologies cannot
efficiently respond to the emerging large-scale heterogeneity in wireless
networks in a real-time fashion as these traditional networking strategies
entail complete or quasi-complete awareness of the environment. Hence, novel
tools and frameworks are needed to address the shortcomings of the traditional
methods that mostly rely on the mathematical realization of the wireless
systems. Data-driven approaches such as machine learning algorithms have
recently shown to be very effective to perform real-time decision making in
wireless networks [1]. However, modern deep learning architectures and
solutions such as convolutional neural networks, feedforward networks, and
recurrent neural networks are usually well-suited for grid structured and
sequential data. On the other hand, many of the datasets in wireless networks
have a graph structure and ignoring the inter-relation between the data points
leads to the loss of information that could be highly beneficial in training
ML algorithms. Therefore, it is highly desirable to use approaches that
utilize all the information that exist when modeling the wireless networks as
graphs.

Graphs are ubiquitous non-Euclidean data structures used to characterize
complex relationships in all kinds of networks including wireless systems. To
enable using deep learning methods on graphs, paradigm of graph representation
learning could be used [2]. Graph representation learning is the art of
utilizing machine learning methods to extract feature vectors from graph
structured data. These feature vectors contain temporal and spatial
information of the local and global neighbourhood of each entity in the graph.
The representation vector of a node is produced by iteratively aggregating and
altering the representation vectors of its neighbors in GNNs. GNNs can be used
to only learn the graph feature vector representation or it can be used to
solve a downstream prediction task (classification as well as regression) over
the given graph in an end-to-end manner. In such an end-to-end learning
approach, the feature vectors are learnt as an intermediate step. Moreover,
highly expressive powers of GNNs that stems from capturing the full
information of the graphs enables them to be extremely effective in solving
combinatorial optimization (CO) problems [3].

As mentioned earlier, many wireless network datasets are graph structured
which could benefit from graph representation learning. This has led to a
recent surge of interest in the research community to use GNNs to solve
various wireless problems such as topology design, scheduling, link activation
and power control to name a few. For example, as an instance of an
autoregressive aproach, in [4], authors capitalize on graph embedding and DRL
to propose a sequential topology design approach for IAB networks that
maximizes the network utility function. To find the optimal topology, IAB-
enabled wireless network is modeled as a graph and RL agent state is the
current graph (subgraph or partial graph) including activated edges (i.e.,
links) and the action is the activation of a certain edge that can contribute
to the cumulative reward. The effectiveness of the proposed topology formation
from various standpoints is then thoroughly presented which highlights the
necessity of using the spatio-temporal dependencies through highly expressive
graph embeddings. The authors in [5] put forward a graph embedding based link
scheduling framework for device-to-device (D2D) networks eracted as a fully-
connected directed graphs to maximize the weighted instantaneous sum rate CO
problem without accurate CSI requisite. Likewise, the authors in [6]
capitalize on a variational graph autoencoder (VGAE) for graph embedding of an
interference graph and propose a link activation strategy which is scalable to
various network topologies such as D2D networks. A permutation equivariant
transferable random edge graph neural networks (REGNNs) followed by a model-
free primal-dual learning weight training algorithm for wireless networks is
introduced in [7]. Shen et al. [8] address the generic problem of radio
resource management problem using message passing GNNs (MPGNNs) without
domain-specific knowledge requirement. Distributed machine learning incurs a
substantial amount of overhead among heterogeneous agents with heterogeneous
data limit. Hence, to settle this issue, a joint Bayesian learning and
information-aware graph optimization that enables agents to engage the most
informative agents in their own learning process is employed in [9], which has
less communication overhead and faster convergence. Then a unsupervised
primal-dual counterfactual optimization, that arises where multiple (often
conflicting) tasks and requirements are demanded from the agent, is applied to
train the network for decision making that ensures a minimum rate. Two novel
GNN based detectors relying upon the expectation propagation (EP) and Bayesian
parallel interference cancellation (BPIC) are proposed in [10] to address the
problem of uplink multi-user multiple-input multiple-output (MU-MIMO)
detection when the multi-user interference (MUI) is strong.

The representative works show the importance and the power of GNNs in solving
complex problems in wireless communication networks. However, there are many
other problems in wireless networks that can benefit from graph learning
approaches, many of which are not yet studied in the literature. The aim of
this paper is to provide an overview of graph representation learning, its
potential to solve wireless network problems and provide future research
directions for the field. The contribution of this work can be summarised as
follows,

  * •

Deep learning on graphs as a frontier in deep learning is introduced.

  * •

Design of GNNs as an effective framework for graph representation learning is
provided with practicalities.

  * •

Some applications of GNNs in wireless networks and open challenges and future
directions are discussed.

  * •

Our findings for GNN-based access point selection for cell-free massive MIMO
systems are also presented to embolden the expressive power of GNNs.

The organization of the paper is as follows. In Section 2 an overview of graph
learning is provided. Graph learning applications and relevance to wireless
networks is then further discussed in Section 3. Followed by the concluding
remarks in Section 4.

##  II Preliminaries on Graphs

Graphs are a general language for describing and analyzing entities with
relations. Graph $G$ is characterized by a set of nodes $V$, edges $E$, and
the matrix of initial features of all nodes $\textbf{X}^{|V|\times d}$ with
$\textbf{x}_{v}\in\textbf{R}^{d}$ being the feature vector of a node $v$ (note
that a graph may also have edge features $\textbf{X}^{e}$ ), i.e.
$G=(V,E,\textbf{X})$. These attached attributes can be weight, ranking, type,
sign, or properties depending on the structure of the graph. Let $v_{i}$
represent a node of the graph, then the edge from $v_{i}$ to $v_{j}$ is
denoted by $e_{i,j}=(v_{i},v_{j})\in E$. The neighborhood of a node $v$ is
defined as $N(v)=\\{u\in V|(v,u)\in E\\}$. The adjacency matrix of $G$ is
denoted by A, which indicates whether pairs of nodes are adjacent or not. For
a given undirected graph, A is a symmetric matrix and for a directed graph
symmetry does not hold. $A_{i,j}=1$ if $e_{i,j}\in E$, otherwise $A_{i,j}=0$.
If the graph edges have some weights associated to them the adjacency matrix
entries can be any real numbers. A graph is extremely sparse if the
$|V|\ll|E|$. Edge lists and adjacency lists are also other common ways of
representing graphs.

There exist a lot of problems in various industries that require using machine
learning methods on graph structured data. Many of these problems could be
modeled as node classification, relation prediction, clustering and community
detection as well as graph classification. Graph structured data has
particular properties that distinguish it from other data types such as
images. For instance, in node classification tasks, the nodes in a given graph
are not independent and identically distributed (i.i.d.), which is the
principal assumption of standard supervised classification. Therefore, there
is a need for solutions to handle graph structured data for machine learning
tasks. In the next section we provide a brief background on the traditional
learning methods over graphs prior to the rise of graph representation
learning approaches, which lays the foundation for the modern deep learning
methods over graph structured data [2].

##  III Traditional Learning Methods for Graph Structured Data

In the traditional methods for handling graph structured data, the graph
features or statistics needed for a given task (e.g., node classification)
were derived by the domain knowledge or heuristic methods. Then, these
extracted features are fed to a standard deep learning model (e.g., a neural
network). These feature vectors describe the structural information around a
given entity in the graph (local neighbourhood structure). Here, we briefly
introduce some of these methods to derive graph statistics (features) at
various levels (e.g. node, edge, or graph level) in a given graph [2].

#### Node-level Features and statistics

To obtain nodes features (statistics) of a graph, two general methods can be
used : 1) importance-based features such as node degree or various node
centrality measures, 2) structure-based features such as node degree,
clustering coefficient or graphlet (a small induced subgraph of a large
network) count vector. The former category, captures how important a given
node is and used to identify the influential nodes in the graph (network). The
later, captures structural characteristics of surrounding neighbourhood of a
given node and used to specify the role of a node in a given network.

#### Edge-level Features and statistics

Edge feature extraction is basically the feature design for pairs of nodes. In
other words, edge-level features quantify the relation between pairs of nodes.
Statistical measurements which detect neighborhood overlap are common tools to
quantify pairwise relation (similarity). A proximity score between a pair of
nodes is calculated. This proximity score can be derived from distance-based
methods by only computing the shortest path between two nodes without
considering the neighbourhood information. It also can be calculated using the
global neighbourhood overlap method that uses the graph structure to find a
defined proximity score. Edge-level features are used mostly in link
prediction tasks in which the emerging or missing links based on the existing
links are predicted.

#### Graph-level Features and Statistics

Graph features characterize the entire graph. These features are useful for
graph-level classification and comparing two graphs. One way to find graph $G$
feature $\phi(G)$ is by counting different graphlets in the given graph. In
other words, given a set of graphlets, we count how many of them exist in a
given graph. Instead of extracting explicit feature vectors, implicit kernel
methods can be used instead. We do not wish to go into the details of it as it
has a rich literature.

Theese methods, in particular node and graph-level features discussed in this
section are task dependent and require hand-crafted solutions for building
feature vectors. Such a feature extraction approaches are computationally
prohibitive as they need to be re-calculated as the task changes. Using such
methods is not feasible in problems with large input graphs or problems with
combinatorial nature with a very large number of possible outputs. Therefore,
an alternative method, graph representation learning is required to tackle
such problems. In the following section, graph representation learning
(automatically learns the structural features of the graph) as an alternative
for traditional learning methods over graphs is introduced in sufficient
details.

##  IV Graph Representation Learning

Graph representation learning is automatically extracting task-independent
structural features of graphs. In the sequel, we briefly introduce shallow
graph representation learning and deep learning on graph data (GNN) for graph
representation learning at node level and graph level.

###  IV-A Shallow Graph Representation Learning

Shallow representing learning could be categorized into two groups as follows:

Figure 1: GNN training pipeline illustration

#### Embedding Nodes

Here we aim at automatically learning the feature vectors of the nodes such
that the properties of the nodes are preserved throughout the whole process of
learning (embedding). First approach is Encoder-Decoder approach in which
learning node embedding is done with respect to an encoder, a node similarity
function, and a decoder design choices. Encoder maps each node to a low-
dimensional vector. To do this, a similarity function specifies how the
relation in the vector space be mapped to the relations in the original
network. Finally, the decoder, maps from embeddings to the similarity score.
The goal is to optimize the parameters of the encoder such that the similarity
in the embedding space is close to the similarity of the nodes in the original
network. Embedding look-up or shallow encoding is the simplest encoding method
that directly optimizes an embedding matrix that contains the embedding
vectors for all the nodes in the network. Shallow encoder maximizes a
similarity score (e.g., the dot product of the feature vectors of two nodes)
for a pair of nodes that are similar. DeepWalk and node2vec are two commonly
used approaches for shallow encoding which differ in the way that the
similarity function is defined. For instance, in DeepWalk approach, the
similarity function is the probability that a pair of node co-occur on a
random walk on the graph. The objective of random walk is to embed the nodes
that appeared on the same random walk together. In a similar fashion, node2vec
uses biased random walks to yield the embeddings that can either be biased on
the local neighbourhood or the global neighbourhood views in the networks.
These methods are unsupervised embedding learning as they do not use any node
labels or nodes own features [2].

#### Embedding Graphs

The embedding of the entire graph can be derived by averaging or summation of
the embeddings of all the nodes. Graph embedding can also be extracted by
making a sub-graph a virtual super-node and finding the embedding of this
virtual node and then embedding of all the virtual nodes. Anonymous walk
embeddings are similarly a viable candidate to extract the graph embedding.

All the above-mentioned algorithms view the graph as a matrix and heavily rely
on the matrix algebraic operations. No parameter sharing among the nodes is
used to reduce the computational complexity. They are transductive methods and
do not use the nodes own features. In the next section, we focus on deep graph
representation learning (GNN) that address the shortcoming of the methods that
were introduced earlier.

###  IV-B Deep Representation Learning: Graph Neural Networks

Conventional deep learning models (e.g., CNNs, RNNs, and feedforward netwoks)
cannot be directly used on graph structured data. For example, CNN and RNN are
defined for grid and sequence data types, respectively. Since for each node in
the given graph a computational graph needs to be used that captures the local
neighbourhood structure of that certain node. These computational graphs for
all the nodes in the input graph vary in size and topology. One neural network
model cannot be applied to all the nodes in a given graph. Therefore, to
enable ML learning on graphs, one must design a new deep learning architecture
[2].

A class of deep learning techniques called graph neural networks (GNNs) has
been recently introduced to perform deep learning directly on graph structured
data. They can be applied directly to a graph and make predictions and
overcome the shortcomings of other approaches discussed in previous sections.

In order to understand how GNNs work, some concepts (design parameters) are
defined in the following format. First a computational graph is created for
every node that defines the neural structure for that node (it also defines
the receptive field of a GNN similar to the receptive field notion in CNNs).
One layer of GNN computes the nodes very own message or feature on the created
computation graph. It also aggregates the messages or features of the
neighboring nodes and creates node embeddings. Nonlinearity might also be
added to messages or aggregations to increment the expressiveness of the
model. A GNN layer compresses multiple vectors into a single vector. Then
multiple GNN layers are stacked sequentially to create a GNN. Graph feature
and structure augmentation are performed if the original graph is not used
directly as the computational graph to trade off computational complexity
against accuracy. Finally, learning objective is defined to specify the
training of the network (supervised/unsupervised objectives). Different GNNs
basically differ in how these operations are done. Classical Graph
Convolutional Networks (GCNs) output feature vector for node $v$ at $l$-th
layer is calculated by summing the messages of the neighbouring nodes given by

$\mathbf{h}_{v}^{(l)}=\sigma\left(\sum_{u\in
N(v)}\mathbf{W}^{(l)}\frac{\mathbf{h}_{u}^{(l-1)}}{|N(v)|}\right)$ (1)

where neighbouring node $u$’s message is transformed by
$\frac{1}{|N(v)|}\mathbf{W}^{(l)}\mathbf{h}_{u}^{(l-1)}$ where $|N(v)|$ is the
cardinality of the neighbouring nodes set, $\mathbf{W}^{(l)}$ is the learnable
parameter, and $\mathbf{h}_{u}^{(l-1)}$ is the messages of the neighbouring
nodes. $\sigma$ is the activation function applied to boost the expressiveness
of the network. GraphSAGE is also another common GNN that differs from GCN
layer in the fact the GrappgSAGE aggregates node $v$s own message as well by
concatenating it with the aggregated messages of the neighbouring nodes. It
can be computed as

$\mathbf{h}_{v}^{(l)}=\sigma\left(\mathbf{W}^{(l)}\left(\mathbf{h}_{v}^{(l-1)}\mathbin{\|}\operatorname{AGG}\left\\{\mathbf{h}_{u}^{(l-1)},\forall
u\in N(v)\right\\}\right)\right)$ (2)

$\mathbin{\|}$ represents concatenation operation. $AGG$ can be mean, pool, or
long short-term memory (LSTM) functions. To further improve the performance
GraphSAGE $l_{2}$ normalizes the embedding at the end of each layer. Another
common and powerful GNN structure is Graph attention networks (GATs). A GAT
with $L$ layer returns the embedding

$\mathbf{h}_{v}^{(l)}=\sigma\left(\sum_{u\in
N(v)}\alpha_{vu}\mathbf{W}^{(l)}\mathbf{h}_{u}^{(l-1)}\right)$ (3)

$\alpha_{vu}$ is called the attention coefficient reflecting on the importance
of the neighboring nodes. Both in GCNs and GraphSAGE all the neighboring node
are given the same importance that is not computationally efficient as GNN
should allocate more computation power to the more important part of the
neighborhood. $\alpha_{vu}$ is calculated using Attention mechanism. The
attention coefficients can be jointly learnt with GNN weight matrices
(matrices used for transforming nodes own messages and the one used for
neighborhood transformation). GAT is also storage efficient and has inductive
capabilities. $\mathbf{W}^{(l)}$s are learnable parameters that are not
regulated by the number of nodes and enable GNN scalability. In other words,
as the number of nodes increases $\mathbf{W}^{(l)}$ can be shared and utilized
and that is how the problem of different computation graphs with various
shapes are handled and the network scales as the new nodes join.

So far we discussed what a GNN layer is. Depending on the objective of the
task at hand, GNN layers are stacked to create a GNN model. Stacking too many
layers leads to the over-smoothing problem that means that all the nodes
embeddings converge to the same value. Therefore, the embeddings are
indistinguishable. This happens due to the fact that the number of layers in
GNN indicates receptive field of the nodes neighborhood (in $L$\- layer GNN,
every single node has a receptive field of $L$-hop neighborhood). The problem
can be tackled by keeping the number of layers small and add to the
expressiveness of GNN more and in case large number of layers are required
skip connections can be incorporated.

One of the major issues in GNNs that must be handled is that as networks grow
large, using original input graph as the computation graph is going to be
computation inefficient and inaccurate. The given input graph cannot be the
optimal computation graph since it might lack features or it might be too
sparse, dense, or large. Hence, feature augmentation or structure augmentation
is required depending on the given input graph and the task at hand.

GNN models expressiveness is measured in terms of their capabilities to
distinguish different graph structures. Since GNNs are based on the
computational graphs, then the nodes with similar computation graphs will have
the same embeddings though they are distinct nodes. To ensure that the GNN is
powerful enough to capture the local neighborhood structure, GNN should use an
injective function (every distinct point is mapped to a distinct point) for
neighbor aggregation (as it functions over multi-sets) in various layers.
Using multi-layer perceptron (MLP) which is an injective multi-set function
results in the most expressive GNN among the message-passing GNN category,
i.e. Graph Isomorphism Network (GIN).

A GNN training pipeline for node classification is illustrated in Fig. 1.
Depending on the task level prediction head harnesses the node embeddings to
make predictions. The training is done in either supervised or
unsupervised/self-supervised manner. If the labels are given from an external
source the training is supervised learning. Otherwise, if the labels/signals
are provided by the graph itself, the training is unsupervised. For the
classification and regression tasks common loss functions cross entropy and
mean squared error (MSE) can be used, respectively. Last but not least, to
measure the success of the trained GNN for classification or regression,
metrics such as binary classification (accuracy, precision/recall, ROC curves)
and root mean square error (RMSE) could be utilized.

To train GNNs, similar to other ML training approaches, the dataset is split
into train/validation/test set. The partitioning strategy can span from fixed
to random partitioning. As in modern deep learning training pipelines, the
training set is used for GNN parameter optimization. Similarly, the validation
set is used to to tune the model. The test set is held back and isolated until
the training and validation is done. However, the problem with graph-
structured data type, as mentioned earlier, is that data leakage happens since
the data points are not i.i.d.. The nodes that have been considered for the
test phase participate in the node embedding of node that is used for
training. To tackle this problem,two different settings could be used. The
first approach is that the whole graph can be used for all splits subject to
the partitioning of the labels (not applicable to graph-level tasks). The
second approach is to consider multiple sub-graphs over the input graph in the
dataset for train/validation/test sets. The later is an inductive solution
that can generalize over unobserved scenarios (graphs). The dataset is
partitioned in a way to fit the task at hand [11].

##  V Graph Representation Learning in Wireless Communications

Autonomy for robust heterogeneous 6G and beyond wireless networks is
conspicuous. Up until now, many machine learning algorithms have been used to
solve wireless network problems. However, the majority of ML models (e.g.
CNNs, MLPs) do not provide permutation equivariance property which can easily
result in intensive training complexity in dense, heterogeneous, and unseen
scenarios. This permutation equivariance property of GNNs guarantees that
signal processing with GNNs is independent of labeling of graph signals that
implies that relabeling (i.e., applying any permutation and shift) the input
signals results in consistent relabelling of the output signal. In other
words, the solution to a particular permuted problem is the permuted solution
of the original problem. With the rise in agents heterogeneity and magnitude
in such networks, having local overview of agents’ neighbourhood (i.e.,
identifying the interactions among agents) becomes essential for each agent in
order to contribute to better network planning and management which is
delivered by GNNs. Considering the mentioned properties along with their
scalability and generalization power, graph learning-based methods spanning
from graph filters to GNNs are advantageous to wireless communication
networks. In the sequel, we will examine the possible applications of these
methods for wireless systems.

###  V-A Cooperative Caching in Mobile Edge Computing(MEC)

With the rise in content-based applications such as video sharing, e-commerce,
and social media networking, content caching in networks has already become a
key component.This calls for some efficient algorithms that can fulfill low-
delay requirement for data fetching. MEC geographic topological structure can
be modeled as a graph with the servers as nodes and the relation between them
as an edge. Spatio-temporal GNNs are enablers of cooperative caching. Unlike
the methods that forecast the popularity of the content and pre-cache the
popular content that only consider the temporal features of content
popularity, spatio-temporal GNNs also consider the spatial features of content
popularity. Another approach could be using graph attention network based
self-attention to find representation vectors of the agents in MEC structure.
Then these calculated feature vectors are used as the state of a fully
decentralized multi-agent deep reinforcement learning system for cooperative
caching [12]. The use of GNNs for balancing out between the power consumption
and computation power is also an interesting problem in cooperative caching in
MEC yet to be addressed.

###  V-B mmWave Beam Selection

For transmitter and receiver pair operating at mmWave frequency, the number of
antennas at each side is huge so choosing an appropriate pair that fulfills a
certain objective becomes an intractable problem. In recent years, ML
algorithms proved to be competent enough to address this issue with lower
complexity and in a more efficient manner. Considering the fact that the
nature of such problems can be further be improved incorporating the spatial
topology information into the network, using GNNs is a potential solution to
reach an efficient solution within more affordable complexity. For example,
[13] thoroughly analyzes the problem of joint beam selection and link
activation in D2D networks in pursuit of maximizing the total throughput using
a GNN-based framework called GBLinks. It is thereby shown that the introduced
framework reaches a sub-optimal solution compared to exhaustive search based
solutions and also outperforms beam selection strategy with all links active.

###  V-C Packet Duplication

PDCP packet duplication is an approach that is used to improve the reliability
and the latency of the wireless networks. ML methods that are used for packet
duplication decisions use various elements such as RSRP and CQI from two or
more base stations to the users [14]. When such information is fed to a neural
network as an input vector, the neural network will not be able to understand
the relationship between the different elements of the feature vector. Using
graph representation for such data will solve this problem. If the data is
formatted such that each base station is one node and connections to the UEs
are edges, then the full information about the network architecture could be
embedded and used as the input for the neural network that is making PDCP
packet duplication decisions.

###  V-D Integrated access and backhaul

As a result of more available bandwidth in the 5th generation mobile network
(5G), a new backhauling technology Integrated Access and Backhaul (IAB), was
proposed. IAB enables cell densification and coverage extension by surmounting
the need for fibre connections between the small cells and core network which
is economically and physically prohibitive. IAB technology uses the same
standardized technology for the access and backhaul connection and the
resources are also flexibly shared between the access and backhaul over-the-
air instead of a prior fixed split of the resources. An IAB network can be
modeled as a graph. One possible case is a graph with IAB nodes or donors as
the nodes and the relations between them as edges. IAB networks can have
either spanning tree, mesh network, directed acyclic graph(DAG), and etc
topologies. topologies. To fully take advantage of the flexibility of IAB
networks, efficient topology formation and accordingly dynamic resource
allocation are needed as they have major impacts on the system performance.
For instance, the authors in [4] investigate a graph-embedding based DRL
solution to incorporate the structural information and reduce the complexity
of IAB topology formation combinatorial optimization problem. Capitalizing on
the aggregated information of a certain node’ (either IAB donor or node)
neighbours provides insights into latent interactions between that certain
node and others as well as indicates the impact intensity of others on a
certain node which can bring significant improvements in manifold aspects such
as load balancing, anomaly detection, and failure robustness in IAB networks.
The above areas are open problems and yet to be explored by the interested
researches.

###  V-E Cell-free massive MIMO networks

Cell-free is a user centric approach where, large number of APs serve a single
UE on the same resource block instead of a dedicated AP as in the case of
conventional cellular network. Furthermore, these APs a connected to central
processing unit (CPU) where the signal processing tasks are performed. It has
been shown that cell-free systems outperform small cell networks due to macro
diversity achieved by geographically distributed APs serving the same UE. Even
though, all the APs in a system can be used to serve a single user, in
practice to achieve scalability best available APs to serve a certain UE must
be selected. This selection may depend on the signal strength between UE and
APs, front-haul network load which connect APs to the CPU, and interference
from other UEs. Since UEs are served non-orthogonally, interference due to use
of same pilot sequence for channel estimation, known as pilot contamination,
is another key issue which needs to be resolved. GNNs can be used to provide
scalable solutions for key problems in cell-free systems like AP selection and
Pilot assignment. In [15] we have formulated a AP selection problem as two
stage classification problem where in the initial stage APs classified based
on some metric and then second stage stage classification is formulated as an
link prediction task. The main advantage of the proposed GNN based AP
selection algorithm is CPU being able to select the potential APs for an UE
trying to establish a connection using RSRP measurements of only few APs.
Here, as illustrated in Fig. 1, spatial convolutions based on GraphSAGE
inductive framework are performed on two graphs [2]. In order to make the
embeddings independent from features of dynamic UE nodes, UE nodes are
connected to APs using only directional edges. In Fig. [13], precision and
recall curves of the proposed GNN based AP selection algorithm for a cell-free
system with 25 APs evenly distributed in a $500\times 500\text{m}^{2}$ are
presented.

Figure 2: Precision and recall of AP selection in a cell-free system with 25
APs

##  VI Conclusion

In this article, we have studied the potential of incorporating graph
representation learning in wireless communication systems. We have reviewed
the challenges faced by conventional ML models for graph-structured wireless
networks and discussed the merits of graph representation learning as an
enabling mechanism in tackling them. Then, we have provided an overview of
graph representation learning, specifically GNNs, thereafter we presented
several problems in wireless communications that can benefit from these
methods. In particular, we have used GNN in access point selection in cell-
free massive MIMO and we have analyzed the performance. By way of conclusion,
the pivotal role of graph representation learning in wireless communications
is undeniable as in other fields. Structural information incorporation and
exploitation can significantly contribute to more efficient and accurate
predictions. This work can be considered as an essential guide toward solving
graph-structured open problems in wireless systems identified earlier for
interested researchers.

## References

  * [1] S. Ali, W. Saad, N. Rajatheva _et al._ , “6G White Paper on Machine Learning in Wireless Communication Networks,” pp. 1–29, apr 2020. [Online]. Available: http://arxiv.org/abs/2004.13875
  * [2] W. L. Hamilton, “Graph Representation Learning,” _Synthesis Lectures on Artificial Intelligence and Machine Learning_ , vol. 14, no. 3, pp. 1–159, 2020\.
  * [3] Q. Cappart, D. Chételat, E. Khalil, A. Lodi, C. Morris, and P. Veličković, “Combinatorial Optimization and Reasoning With Graph Neural Networks,” 2021.
  * [4] M. Simsek, O. Orhan, M. Nassar, O. Elibol, and H. Nikopour, “IAB Topology Design: A Graph Embedding and Deep Reinforcement Learning Approach,” _IEEE Communications Letters_ , vol. 25, no. 2, pp. 489–493, 2021.
  * [5] M. Lee, G. Yu, and G. Y. Li, “Graph Embedding based Wireless Link Scheduling with Few Training Samples,” _IEEE Transactions on Wireless Communications_ , pp. 1–13, 2020.
  * [6] S. Jamshidiha, V. Pourahmadi, A. Mohammadi, and M. Bennis, “Link Activation using Variational Graph Autoencoders,” _IEEE Communications Letters_ , pp. 1–1, 2021.
  * [7] M. Eisen and A. Ribeiro, “Optimal Wireless Resource Allocation With Random Edge Graph Neural Networks,” _IEEE Transactions on Signal Processing_ , vol. 68, pp. 2977–2991, 2020.
  * [8] Y. Shen, Y. Shi, J. Zhang, and K. B. Letaief, “Graph Neural Networks for Scalable Radio Resource Management: Architecture Design and Theoretical Analysis,” _IEEE Journal on Selected Areas in Communications_ , vol. 39, no. 1, pp. 101–115, 2021.
  * [9] T. Alshammari, S. Samarakoon, A. Elgabli, and M. Bennis, “BayGo: Joint Bayesian Learning and Information-Aware Graph Optimization,” nov 2020.
  * [10] A. Kosasih, V. Onasis, V. Miloslavskaya, W. Hardjawana, V. Andrean, and B. Vucetic, “Graph Neural Network Aided MU-MIMO Detectors,” _IEEE Journal on Selected Areas in Communications_ , no. 9, pp. 2540–2555, 2022.
  * [11] Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and P. S. Yu, “A Comprehensive Survey on Graph Neural Networks,” _IEEE Transactions on Neural Networks and Learning Systems_ , vol. 32, no. 1, pp. 4–24, jan 2021.
  * [12] Y. Yan, B. Zhang, C. Li, and C. Su, “Cooperative Caching and Fetching in D2D Communications - A Fully Decentralized Multi-Agent Reinforcement Learning Approach,” _IEEE Transactions on Vehicular Technology_ , vol. 69, no. 12, pp. 16 095–16 109, 2020.
  * [13] S. He, S. Xiong, W. Zhang, Y. Yang, J. Ren, and Y. Huang, “GBLinks: GNN-Based Beam Selection and Link Activation for Ultra-Dense D2D mmWave Networks,” _IEEE Transactions on Communications_ , vol. 70, no. 5, pp. 3451–3466, may 2022.
  * [14] Q. Zhao, S. Paris, T. Veijalainen, and S. Ali, “Hierarchical Multi-Objective Deep Reinforcement Learning for Packet Duplication in Multi-Connectivity for URLLC,” in _2021 Joint European Conference on Networks and Communications & 6G Summit (EuCNC/6G Summit)_, 2021, pp. 142–147.
  * [15] V. Ranasinghe, N. Rajatheva, and M. Latva-aho, “Graph neural network based access point selection for cell-free massive mimo systems,” in _2021 IEEE Global Communications Conference (GLOBECOM)_ , 2021, pp. 01–06.


# Active actions: effective field theory for active nematics

Michael J. Landry  Center for Theoretical Physics, MIT, Cambridge, MA 02139,
USA

###### Abstract

Active matter consumes energy from the environment and transforms it into
mechanical work. Notable examples from biology include cell division,
bacterial swarms, and muscle contraction. In this work, we investigate the
nature of active matter systems using the powerful effective field theory
toolbox. This allows us to construct the most general theory without ambiguity
up to a given order in the derivative expansion. Our primary focus is active
nematics—liquid crystal systems that spontaneously break rotational but not
translational symmetry—in two spatial dimensions. (Such spontaneous symmetry
breaking is allowed if the nematic is embedded in a higher dimensional space.)
While we focus on this one particular class of physical system, the tools
developed here can in principle be applied to any active matter system. Our
theories give unambiguous predictions for the relationship between
fluctuations and equations of motion in the presence of activity, generalizing
the standard fluctuation-dissipation relations characteristic of passive
systems.

###### Contents

  1. I Introduction
  2. II Passive dynamics
    1. II.1 Probe limit
    2. II.2 The non-Stückelberg trick
  3. III Active dynamics with fuel
  4. IV Modified dynamical KMS symmetry
  5. V Discussion

##  I Introduction

Active matter is a broad area of study that encompasses biological and
synthetic systems that exist far from thermodynamic equilibrium Marchetti  _et
al._ (2013). Such systems consume energy and convert it into mechanical work,
which allows for self-propulsion. In this way, all biological systems fall
under the active matter umbrella Needleman and Dogic (2017). The nature of
this energy source can vary from system to system. In many biological
examples, energy is stored in chemical bonds and is then released to perform
work—this is how we are able to move the muscles in our bodies. But the energy
source could be something entirely different, like a light that shines on an
active sample Bechinger  _et al._ (2016).

Apart from being the fundamental physical principle that undergirds all of
life, active matter systems offer an intriguing space for exploring non-
equilibrium physics in a broader context Ramaswamy (2010). There are various
theoretical questions that accompany the study of active systems. When
constructing a theory of a passive system, principles of local equilibrium can
be invoked that constrain both the equations of motion and the statistical
fluctuations Kubo (1966). When studying active systems, however, it is not so
clear how principles of local equilibrium are relevant. On the one hand,
active systems are very far from equilibrium—they continually consume energy
and use it to move. On the other hand, notions like temperature or chemical
potential—hydrodynamic quantities that are only well-defined in equilibrium
Kovtun (2012)—are often relevant. So while in the most general case in which
we are totally agnostic about the origin of activity, no notion of local
equilibrium can be universally defined, in many real-world applications, some
notion of local equilibrium must persist.

The particular system we will focus on is active nematics in two spatial
dimensions. These are apolar liquid crystal systems consisting of elongated
molecules that spontaneously break spatial rotation symmetry de Gennes and
Prost (1993). They consume energy from their surroundings and convert it into
mechanical work Simha and Ramaswamy (2002); Doostmohammadi  _et al._ (2018).
Active two-dimensional nematic order has been identified in numerous
biological systems. These systems encompass a variety of phenomena, such as
epithelial monolayers Saw  _et al._ (2017); Kawaguchi  _et al._ (2017);
Blanch-Mercader  _et al._ (2018) and suspensions of cytoskeletal filaments
Sanchez  _et al._ (2012); Keber  _et al._ (2014). On the theoretical front,
much attention has been paid to topological defects Vafa  _et al._ (2020,
2022); Vafa (2022); Vafa  _et al._ (2023); Vafa and Mahadevan (2022) and
resulting nematic “turbulence,” which occurs when the complex dynamics of
topological defects are driven by strong active elastic distortions of the
nematic substrate Tan  _et al._ (2019); Alert  _et al._ (2022); Giomi  _et
al._ (2013); Ellis  _et al._ (2018); Shankar and Marchetti (2019); Putzig  _et
al._ (2016); Srivastava  _et al._ (2016); Doostmohammadi  _et al._ (2016);
Pearce  _et al._ (2021); Thijssen  _et al._ (2020); Thampi and Yeomans (2016);
Giomi (2015).

The aim of this paper is to extend the effective field theory (EFT) paradigm
to account for activity, focusing on nematics in two dimensions. In the
passive case, techniques for constructing actions on the Schwinger-Keldysh
(SK) contour that account for the non-equilibrium dynamics of many body
systems—including nematics—have been well-studied Landry (2020); Baggioli and
Landry (2020); Landry (2021a). An important element of passive dynamics is the
dynamical KMS symmetry Crossley  _et al._ (2017); Glorioso and Liu (2016);
Glorioso  _et al._ (2017); Glorioso and Liu (2018). This is a symmetry of the
action that implements both local thermal equilibrium and microscopic time-
reversibility. It ensures that the correlation functions of observables
satisfy the KMS conditions, which are real-time constraints that define
thermal equilibrium. The primary obstacle to constructing active EFTs is that
dynamical KMS symmetry appears to automatically ensure passive dynamics. So
how can any notion of local equilibrium be implemented if activity is present?
To answer this question we we will investigate two approaches:

  1. 1.

The most straight-forward approach is to maintain the usual dynamical KMS
symmetry, but couple the passive system to a far-from-equilibrium sector. For
example, we could introduce a $U(1)$ charge that very slowly decays. This
might represent e.g. a slow chemical reaction like the consumption of ATP that
drives activity in a cell. Such an approach gives a clear account of the
origin of activity at the price of being model-dependent.

  2. 2.

To construct a more model-independent active EFT, we would like to modify the
dynamical KMS relations in such a way that active dynamics emerge. Note that
we could simply do away with dynamical KMS symmetry entirely, but this would
remove all notion of local equilibrium, which as discussed previously, is
undesirable. We will therefore take inspiration from the above approach and
propose such a modified dynamical KMS symmetry.

The paper is organized as follows. In Section II we review how to construct
non-equilibrium EFTs on the SK contour for passive nematics. Working first in
the probe limit, we show how the EFT can be modified to include passive
velocities in open systems. Next, in Section III we couple the passive action
to a far-from-equilibrium sector that drives activity in the system. We find a
leading-order correction to the usual active constitutive relation for the
velocity. Then, taking inspiration from this EFT, we propose a modification to
dynamical KMS symmetry in Section IV. And finally in Section V we discuss the
implications and generalizations for such active EFTs.

##  II Passive dynamics

Before investigating the active case, we begin by considering passive dynamics
for nematics in two spatial dimensions. As a particularly simple example, we
begin with the probe limit in which the only degree of freedom is the
Goldstone corresponding to spontaneously broken rotations.111In two
dimensions, passive nematic phase should be impossible as SSB cannot occur.
The correct way to address this issue is account for the embedding of the two
dimensional nematic into three dimensions. For simplicity, however, we will
merely circumvent this issue by refraining from computing loop diagrams.
Unfortunately, such an EFT has no notion of local velocity, so endowing it
with activity directly would be tricky. Moreover, introducing local velocity
is typically accomplished by considering the conservation of the stress-energy
tensor. But in active systems, energy and momentum can be exchanged with the
environment, so it ought not be conserved. To fix these problems, we propose a
way to introduce dynamical velocity without conserving the stress-energy
tensor, which will ultimately make including activity quite easy.

###  II.1 Probe limit

A nematic liquid crystal consists of oblong molecules that spontaneously break
spatial rotation symmetry, but leave space and time translations unbroken. As
a result, in the infrared, the only relevant degrees of freedom are the
rotation Goldstone $\theta$ and its $a$-type partner $\theta_{a}$. It is
convenient to define the vector pointing parallel to the oblong molecules by
$\vec{n}=(\cos\theta,\sin\theta)$. In nematic phase, flipping the oblong
molecules by 180 degrees leaves the system unchanged, so we postulate the
$Z_{2}$ symmetry $\vec{n}\to-\vec{n}$. It is sometimes convenient to work with
the symmetric, traceless tensor
$Q^{ij}=n^{i}n^{j}-{{\frac{1}{2}}}\delta^{ij}$, which is invariant under this
$Z_{2}$ symmetry.

Suppose the system exists at fixed finite temperature $T_{0}=1/\beta_{0}$. The
effective action $I_{\rm EFT}[\theta,\theta_{a}]$ for the rotation Goldstones
can be constructed using the framework of Crossley  _et al._ (2017); Glorioso
and Liu (2016); Glorioso  _et al._ (2017); Glorioso and Liu (2018); Landry
(2020). In particular, it must satisfy the following conditions:

  1. 1.

It enjoys translational invariance in $x^{\mu}=(t,x^{i})$ and rotational
invariance on $x^{i}$ coordinates. The latter symmetry acts non-trivially on
the Goldstone by

$x^{i}\to R^{ij}(\phi)x^{j},\quad\theta\to\theta+\phi,\quad R(\phi)\in SO(2).$
(1)

  2. 2.

There are various unitarity constraints given by

$\displaystyle I^{*}_{\rm EFT}[\theta,\theta_{a}]=-I_{\rm
EFT}[\theta,-\theta_{a}],$ (2) $\displaystyle\mathop{\rm Im}\,I_{\rm EFT}\geq
0,$ (3) $\displaystyle I_{\rm EFT}[\theta,\theta_{a}=0]=0.$ (4)

These constraints will be straightforwardly generalized when other fields are
included.

  3. 3.

It is invariant under dynamical KMS transformations, that is $I_{\rm
EFT}[\theta,\theta_{a}]=I_{\rm EFT}[\tilde{\theta},\tilde{\theta}_{a}]$, where
dynamical KMS symmetry is a $Z_{2}$ transformation of the form

$\tilde{\theta}={\Theta}\theta,\quad\tilde{\theta}_{a}={\Theta}\theta_{a}+i\beta_{0}{\Theta}\partial_{t}\theta.$
(5)

Here ${\Theta}$ is a discrete anti-unitary, time-reversing symmetry of the
microscopic theory. For example it could be ${\Theta}={\mathcal{T}}$ or
${\Theta}={{\mathcal{C}}}{{\mathcal{P}}}{\mathcal{T}}$. Dynamical KMS symmetry
in conjunction with the unitarity constraints (2)-(4) can be used to deduce an
entropy current $s^{\mu}$ with non-negative divegence
$\partial_{\mu}s^{\mu}\geq 0$. That is, the second law of thermodynamics
follows automatically Glorioso and Liu (2016).

  4. 4.

Impose whatever other discrete symmetries are formed from charge conjugation
${{\mathcal{C}}}$ and parity ${{\mathcal{P}}}$ that are preserved by the
underlying microscopic system. For the purposes of this paper,
${{\mathcal{C}}}$ symmetry is irrelevant as there are no objects that carry
fundamental charge and we will always impose ${{\mathcal{P}}}$ symmetry.

Effective actions are organized as derivative expansions, so we must assign
weights to derivatives and fields. In the probe limit, boosts are explicitly
broken, so spatial and temporal derivatives need not be on equal footing. It
will turn out that the leading-order dynamics of passive nematics are
diffusive, suggesting the weight-assignments
$[\partial_{t}]=2[\partial_{i}]=2$. Additionally, as $\theta\in[0,2\pi)$ it
ought to have a weight of zero. Finally, as dynamical KMS symmetry relates
$\theta_{a}$ to $\partial_{t}\theta$, we assign $\theta_{a}$ a weight of two
so that this symmetry does not mix terms of differing weights.

The symmetry-invariant building-blocks are as follows. Notice that while
$\theta$ shifts under rotations, $\theta_{a}$ does not. As a result,
$\theta_{a}$ can appear on its own without derivatives. Next, the temporal
derivative $\partial_{t}\theta$ is manifestly rotation-invariant. To take
spatial derivatives, define the covariant derivatives

$\displaystyle\nabla_{1}\theta=\vec{n}\cdot\vec{\nabla}\theta,\quad\nabla_{1}\theta_{a}=-\theta_{a}\vec{n}\times\vec{\nabla}\theta+\vec{n}\cdot\vec{\nabla}\theta_{a},$
(6)
$\displaystyle\nabla_{2}\theta=\vec{n}\times\vec{\nabla}\theta,,\quad\nabla_{2}\theta_{a}=\theta_{a}\vec{n}\cdot\vec{\nabla}\theta-\vec{n}\times\vec{\nabla}\theta_{a}.$
(7)

The covariant derivatives’ actions on $\theta_{a}$ are defined in such a way
that dynamical KMS symmetry can be easily implemented Landry (2020, 2021b).
Lastly, it is often convenient to define
$\hat{\theta}_{a}=\theta_{a}+i\beta_{0}\partial_{t}\theta$.

We are now in a position to construct the leading-order action, which includes
only weight-4 terms given by

${{\mathcal{L}}}={i\gamma\over\beta_{0}}\theta_{a}\hat{\theta}_{a}-K_{1}\nabla_{1}\theta\nabla_{1}\theta_{a}-K_{2}\nabla_{2}\theta\nabla_{2}\theta_{a}.$
(8)

Here $\gamma$ and the Frank coefficients $K_{1,2}$ are constants; (3) requires
that $\gamma>0$ and requiring a stable free energy implies $K_{1,2}>0$. In the
literature it is common to make the so-called one-constant approximation,
which means we fix $K\equiv K_{1}=K_{2}$ Chaikin and Lubensky (2000). The
Lagrangian becomes

${{\mathcal{L}}}_{\text{1-const.}}={i\gamma\over\beta_{0}}\theta_{a}\hat{\theta}_{a}-K\partial_{i}\theta\partial_{i}\theta_{a},$
(9)

whose equations of motion yield simple diffusion

$\partial_{t}\theta=D\vec{\nabla}^{2}\theta,\quad D={K\over\gamma}.$ (10)

Working with $K_{1}\neq K_{2}$, the equations of motion become a bit more
complicated

$\gamma\partial_{t}\theta=(K_{1}-K_{2})\nabla_{1}\theta\nabla_{2}\theta+K_{1}\vec{\nabla}\cdot(\vec{n}\,\nabla_{1}\theta)-K_{2}\vec{\nabla}\times(\vec{n}\,\nabla_{2}\theta).$
(11)

###  II.2 The non-Stückelberg trick

The EFT in the previous subsection is sufficient to describe passive dynamics
in the probe limit. In this limit, there is no notion of velocity. When we
generalize to the active case, however, including a velocity will be crucial.
It is the aim of this subsection to propose how to include a velocity even
when the system is open, meaning that the stress-energy tensor is not
conserved.

When the stress-energy tensor is conserved, we typically introduce metric
sources $g_{\mu\nu},g_{a\mu\nu}$ and corresponding Stückelberg fields
$X^{\mu}(\sigma),X_{a}^{\mu}(\sigma)$ defined on the auxiliary coordinates
$\sigma^{M}$. In the classical limit, we can perform a coordinate
transformation so that the action is defined on the physical spacetime
coordinates $x^{\mu}=X^{\mu}$ at the price of promoting $\sigma^{M}$ to
dynamical fields. The Stückelberg fields must appear in the packages

$G_{a\mu\nu}=g_{a\mu\nu}+{{\mathcal{L}}}_{X_{a}}g_{\mu\nu},$ (12)

where ${{\mathcal{L}}}_{\xi}$ is the Lie derivative with respect to $\xi$, and
all free Lorentz indices must be contracted with $g_{\mu\nu}$. Finally, when
spacetime translations are unbroken the EFT typically enjoys time-independent
diffs

$\sigma^{0}\to\sigma^{0}+f(\sigma^{i}),\quad\sigma^{i}\to\Sigma^{i}(\sigma^{j}).$
(13)

In the active case, we will want an EFT that has active velocity, but fixed
background temperature $T_{0}=1/\beta_{0}$. As a result, we have no need of
the temporal Stückelberg fields above, so we fix $\sigma^{0}=t$ and
$X_{a}^{t}=0$. The local velocity field is given by

$v^{i}\equiv{J^{i}\over J^{t}},\quad
J^{\mu}=\epsilon^{\mu\nu\lambda\rho}\partial_{\nu}\sigma^{1}\partial_{\lambda}\sigma^{2}\partial_{\rho}\sigma^{3}.$
(14)

Next, active systems are open, so the stress-energy tensor cannot be
conserved. We thus must suppose that $X_{a}^{i},\sigma^{i}$ are not true
Stückelberg fields. Nevertheless, if the non-conservation of the stress-energy
tensor is too severe then the energy given to the system by the fuel-source
will be dissipated rapidly and will not lead to activity. Therefore we must
suppose that the stress-energy tensor is only weakly non-conserved. As a
result, we shall treat $X_{a}^{i},\sigma^{i}$ as approximate Stückelberg
fields Landry (2022, 2021b). To this end, decompose the Lagrangian by

${{\mathcal{L}}}={{\mathcal{L}}}^{\text{(non)}}+{{\mathcal{L}}}^{\text{(Stüc)}},$
(15)

where ${{\mathcal{L}}}^{\text{(non)}}$ contains non-Stückelberg terms, e.g.
$X_{a}^{i}$ without derivatives; while ${{\mathcal{L}}}^{\text{(Stüc)}}$
contains only Stückelberg terms. As $X_{a}^{i},\sigma^{i}$ are approximate
Stückelberg fields, we suppose that ${{\mathcal{L}}}^{\text{(non)}}$ is small
even when it is lower-order in the derivative expansion.

Now that the system has a non-trivial velocity, the dynamical KMS
transformations must be modified. We have in particular

$\tilde{\theta}={\Theta}\theta,\quad\tilde{\sigma}^{i}={\Theta}\sigma^{i},\quad\tilde{\theta}_{a}={\Theta}\theta_{a}+i\beta_{0}{\Theta}D_{t}\theta,\quad\tilde{X}_{a}^{i}={\Theta}X_{a}^{i}+i\beta_{0}{\Theta}v^{i},$
(16)

where $D_{t}\equiv\partial_{t}+\vec{v}\cdot\vec{\nabla}$ is the material
derivative. To assign weights to the fields we follow the power-counting
scheme of the previous subsection
$[\partial_{t}]=2[\partial_{i}]=[\theta_{a}]=2$ and $[\theta]=0$. To ensure
that dynamical KMS symmetry does not mix terms of different weights, it is
natural to suppose $[v^{i}]=[X_{a}^{i}]=1$.

Begin by constructing the non-Stückelberg sector. The leading-order terms are
of weight-2, given by

${{\mathcal{L}}}^{\text{(non)}}={i\Lambda^{-1}_{ij}\over\beta_{0}}X_{a}^{i}\hat{X}_{a}^{j},\quad\hat{X}_{a}^{i}=X_{a}^{i}+i\beta_{0}v^{i},\quad\Lambda^{ij}=\Lambda_{0}\delta^{ij}+\Lambda_{Q}Q^{ij}.$
(17)

Let us now make use of field redefinitions. We may redefine $X_{a}^{i}$ by

$X_{a}^{i}\to X_{a}^{i}+if_{a}^{i},$ (18)

where $f_{a}^{i}$ is an (almost) arbitrary higher-weight vector. Such a field
redefinition must however yield an action that is consistent with (2)–(4). As
a result, $f_{a}^{i}$ must be at least linear in $a$-type fields and each
occurrence of an $a$-type field must be accompanied by a factor of $i$.
Further to keep the representation of dynamical KMS symmetry manifest require
the accompanying transformation Jain (2020)

$v^{i}\to v^{i}+\beta_{0}^{-1}(f_{a}^{i}-{\Theta}\tilde{f}_{a}^{i}).$ (19)

The equations of motion for $X_{a}^{i}$ yield

$\partial_{\mu}T^{\mu
i}=\Gamma^{i},\quad\Gamma^{i}=\Lambda^{-1}_{ij}v^{j}+\cdots$ (20)

where $T^{\mu i}$ is furnished by the Stückelberg action (see below) and
$\Gamma^{i}$ by the non-Stückelberg action. Notice that redefinitions of
$X_{a}^{i}$ and the corresponding induced redefinitions (19) have just enough
degrees of freedom to ensure that $\Gamma^{i}$ has no higher-order
corrections. Thus, we may use field redefinitions to remove all higher-order
corrections to ${{\mathcal{L}}}^{\text{(non)}}$. These redefinitions cannot be
used to remove any terms in $T^{\mu i}$ as ${{\mathcal{L}}}^{\text{(non)}}$ is
small even when it is lower-order in the derivative expansion. Further, we
have no left-over field redefinitions to apply to the Stückelberg sector. As a
result, ${{\mathcal{L}}}^{\text{(Stüc)}}$ cannot be modified by field
redefinitions of $X_{a}^{i},v^{i}$.

Now let us construct ${{\mathcal{L}}}^{\text{(Stüc)}}$. Turning off external
sources, the covariant building-blocks are

$\displaystyle G_{ati}=\partial_{t}X_{ai},\quad
G_{aij}=\partial_{i}X_{aj}+\partial_{j}X_{ai},$ (21)
$\displaystyle\vartheta_{a}=\theta_{a}-{{\frac{1}{2}}}\vec{\nabla}\times\vec{X}_{a},\quad\hat{\vartheta}_{a}=\vartheta_{a}+i\beta_{0}D_{t}\vartheta,$
(22)
$\displaystyle\nabla_{1}\theta=\vec{n}\cdot\vec{\nabla}\theta,\quad\nabla_{1}\theta_{a}=-\theta_{a}\vec{n}\times\vec{\nabla}\theta+\vec{n}\cdot\vec{\nabla}\theta_{a}-\vec{n}\cdot\vec{\nabla}X_{a}^{i}\partial_{i}\theta,$
(23)
$\displaystyle\nabla_{2}\theta=\vec{n}\times\vec{\nabla}\theta,\quad\nabla_{2}\theta_{a}=\theta_{a}\vec{n}\cdot\vec{\nabla}\theta-\vec{n}\times\vec{\nabla}\theta_{a}+\vec{n}\times\vec{\nabla}X_{a}^{i}\partial_{i}\theta,$
(24)

where $D_{t}\vartheta\equiv D_{t}\theta-{{\frac{1}{2}}}\omega$, with
$\omega=\vec{\nabla}\times\vec{v}\,$ the pseudo-scalar vorticity. These
building-blocks can be identified using the coset construction Landry (2020,
2021b); Landry and Sun (2021). We can now construct the Stückelberg sector
Lagrangian

$\begin{split}{{\mathcal{L}}}^{\text{(Stüc)}}={i\gamma\over\beta_{0}}\vartheta_{a}\hat{\vartheta}_{a}-K_{1}\nabla_{1}\theta\nabla_{1}\theta_{a}-K_{2}\nabla_{2}\theta\nabla_{2}\theta_{a}\\\
+{{\frac{1}{2}}}T_{0}^{ij}G_{aij}+T^{ti}G_{ati}+{i\over
4\beta_{0}}W^{ijkl}G_{aij}\hat{G}_{akl},\end{split}$ (25)

for
$T_{0}^{ij}=p_{0}\delta^{ij}+w_{0}v^{i}v^{j}+{{\frac{1}{2}}}w_{0}v^{2}\delta^{ij}$,
$T^{ti}=\tilde{w}_{0}v^{i}$, and

$\begin{split}W^{ijkl}=2\eta_{\perp}\Delta^{i(k}\Delta^{l)j}+\bigg{(}\zeta_{\perp}-\frac{2}{d}\eta_{\perp}\bigg{)}\Delta^{ij}\Delta^{kl}+4\eta_{\parallel}n^{(i}\Delta^{j)(k}n^{l)}\\\
+\bigg{(}\zeta_{\times}-\frac{2}{d}\eta_{\parallel}\bigg{)}(\Delta^{ij}n^{k}n^{l}+\Delta^{kl}n^{i}n^{j})+\zeta_{\parallel}n^{i}n^{j}n^{k}n^{l},\end{split}$
(26)

where $\Delta^{ij}=\delta^{ij}-n^{i}n^{j}$. The coefficients
$\gamma,K_{1,2},p_{0},w_{0},\tilde{w}_{0},\eta_{\parallel,\perp},\zeta_{\parallel,\perp,\times}$
are all constant. Physically, $\eta_{\perp,\parallel}$ are shear viscosities,
and $\zeta_{\perp,\parallel,\times}$ are bulk viscosities, $K_{1,2}$ are the
Frank constants, and $p_{0}$ is the pressure. With boost symmetry restored,
$w_{0},\tilde{w}_{0}$ would coincide and equal the enthalpy density. The
unitarity constraint (3) give positivity conditions on the dissipative
transport coefficients, namely

$\gamma>0,\quad\Lambda_{0}>0,\quad\Lambda_{Q}^{2}<\Lambda_{0}^{2},\quad\zeta_{\parallel,\perp}>0,\quad\eta_{\parallel,\perp}>0,\quad\zeta_{\parallel}\zeta_{\perp}>\zeta_{\times}^{2},$
(27)

and requiring the stability of the free energy implies $K_{1,2}\geq 0.$

The equation of motion for $\theta_{a}$ is

$\gamma
D_{t}\vartheta=(K_{1}-K_{2})\nabla_{1}\theta\nabla_{2}\theta+K_{1}\vec{\nabla}\cdot(\vec{n}\,\nabla_{1}\theta)-K_{2}\vec{\nabla}\times(\vec{n}\,\nabla_{2}\theta),$
(28)

which is simply (11) modified to include a velocity. Next, the equations of
motion for $X_{a}^{i}$ give the non-conservation equation (20) where
$\Gamma^{i}=\Lambda^{-1}_{ij}v^{j}$, $T^{ti}=\tilde{w}_{0}v^{i}$ and

$T^{ij}=p_{0}\delta^{ij}+w_{0}v^{i}v^{j}+{{\frac{1}{2}}}w_{0}v^{2}\delta^{ij}-K_{1}\nabla_{1}\theta
n^{(i}\partial^{j)}-K_{2}\nabla_{2}\theta
n^{k}\epsilon^{k(i}\partial^{j)}\theta-W_{0}^{ijkl}\partial_{k}v_{l}.$ (29)

In the linearized limit, $v^{i}$ exponentially decays to zero. In the deep
infrared, the leading-order equation of motion simply fixes $v^{i}=0$.
Similarly the leading-order equation of motion for $\sigma^{i}$ is
$X_{a}^{i}=0$. Plugging these back into the EFT yields (8).

##  III Active dynamics with fuel

To introduce activity to the nematic system described in the previous section,
we simply couple it to a far-from-equilibrium sector. This sector will be
described by a non-conserved $U(1)$ charge that begins at a large value and
slowly decays, which models the burning of fuel. Active dynamics will take
place on time scales much longer than the collision time, but much shorter
than the decay time for the fuel. To ensure that such an intermediate regime
exists and is large, we suppose that the fuel is described by approximate
Stückelberg fields $\varphi,\varphi_{a}$.

We follow the formalism of Landry (2022). If the fuel were exactly conserved,
then we would introduce background $U(1)$ gauge sources $a_{\mu},a_{a\mu}$
such that $\varphi,\varphi_{a}$ would always appear in the packages

$A_{\mu}=a_{\mu}+\partial_{\mu}\varphi,\quad
A_{a\mu}=a_{a\mu}+{{\mathcal{L}}}_{X_{a}}a_{\mu}+\partial_{\mu}\varphi.$ (30)

To ensure that this $U(1)$ charge exists in normal (i.e. unbroken) phase,
impose the time-independent shift symmetries Dubovsky  _et al._ (2012)

$\varphi\to\varphi+g(\sigma^{i}).$ (31)

It is convenient to define the local chemical potential222Typically a chemical
potential is only well-defined when a charge is exactly conserved. Here
because the non-conservation is very weak, an approximate notion of chemical
potential can make sense on sufficiently short time scales. But eventually it
will decay to zero, which is consistent with the common lore that chemical
potentials for non-conserved quantities must vanish in equilibrium.

$\mu=D_{t}\varphi,\quad D_{t}=\partial_{t}+\vec{v}\cdot\vec{\nabla},$ (32)

which is manifestly invariant under (31).

As this $U(1)$ charge is not actually conserved, and $\varphi,\varphi_{a}$ are
only approximate Stückelberg symmetries, we can decompose the Lagrangian by
(15). Like before, ${{\mathcal{L}}}^{\text{(non)}}$ is considered small.

Dynamical KMS symmetry acts by (16) and

$\tilde{\varphi}={\Theta}\varphi,\quad\tilde{\varphi}_{a}={\Theta}\varphi_{a}+i\beta_{0}{\Theta}\mu.$
(33)

We want the chemical potential to begin at a rather large value, so we suppose
that $\mu$ has a weight of zero, meaning that $\varphi_{a}$ also has a weight
of zero.

Let us begin by constructing the non-Stückelberg Lagrangian. It is given by

${{\mathcal{L}}}^{\text{(non)}}={ib\over\beta_{0}}\varphi_{a}\hat{\varphi}_{a}+{i\Lambda^{-1}_{ij}\over\beta}X_{a}^{i}\hat{X}_{a}^{j},\quad\Lambda^{ij}=\Lambda_{0}\delta^{ij}+\Lambda_{Q}Q^{ij},$
(34)

where $\hat{\varphi}_{a}=\varphi_{a}+i\beta_{0}\mu$. Here $b,\Lambda_{0,Q}$
are functions of $\mu$.333In principle they could also depend on $\varphi_{a}$
but we will truncate after quadratic order in $a$-type fields for simplicity.
Like the non-Stückelberg action discussed in the previous section, field
redefinitions can be used to ensure that there are no higher-order corrections
to this sector. These field redefinitions cannot be used to remove any terms
from ${{\mathcal{L}}}^{\text{(Stüc)}}$ as ${{\mathcal{L}}}^{\text{(non)}}$ is
considered small. Further, all field redefinitions have been used up, so terms
of ${{\mathcal{L}}}^{\text{(Stüc)}}$ that under ordinary circumstances could
be removed via field redefinition, can no longer be removed. This fact will
have important consequences.

Let us now turn attention to the Stückelberg terms. When external sources are
turned off, in addition to the building-blocks (21)–(24), we have

$\displaystyle\mu,\quad{{\mathcal{A}}}_{a}=D_{t}\varphi_{a},\quad
A_{ai},\quad\hat{{\mathcal{A}}}_{a}={{\mathcal{A}}}_{a}+i\beta_{0}D_{t}\mu,\quad\hat{A}_{ai}=A_{ai}+i\beta_{0}\partial_{i}\mu.$
(35)

The Stückelberg Lagrangian up to weight-4 terms is

$\begin{split}{{\mathcal{L}}}^{\text{(Stüc)}}={i\gamma\over\beta_{0}}\vartheta_{a}\hat{\vartheta}_{a}-K_{1}\nabla_{1}\theta\nabla_{1}\theta_{a}-K_{2}\nabla_{2}\theta\nabla_{2}\theta_{a}+T^{ti}G_{ati}+{{\frac{1}{2}}}T_{0}^{ij}G_{aij}+n_{0}{{\mathcal{A}}}_{a}\\\
+{i\sigma^{ij}\over\beta_{0}}A_{ai}\hat{A}_{aj}+{i\kappa^{ij}\over
2\beta_{0}}(G_{aij}\hat{{\mathcal{A}}}_{a}+\hat{G}_{aij}{{\mathcal{A}}}_{a})+{i\over
4\beta_{0}}W^{ijkl}G_{aij}\hat{G}_{akl},\end{split}$ (36)

where
$T_{0}^{ij}=p_{0}\delta^{ij}+w_{0}v^{i}v^{j}+{{\frac{1}{2}}}w_{0}v^{2}\delta^{ij}$,
$T^{ti}=\tilde{w}_{0}v^{i}$, such that

$w_{0}-\tilde{w}_{0}={\rm const.},\quad
n_{0}=-{\partial{{\mathcal{F}}}\over\partial\mu},$ (37)

for free energy444Note that free energy is a sensible quantity as long as some
notion of local equilibrium, which is guaranteed by dynamical KMS symmetry,
exists.

${{\mathcal{F}}}=-p_{0}(\mu)-{{\frac{1}{2}}}w_{0}(\mu)v^{2}+{{\frac{1}{2}}}K_{1}(\mu)(\nabla_{1}\theta)^{2}+{{\frac{1}{2}}}K_{2}(\mu)(\nabla_{2}\theta)^{2}.$
(38)

Stability of the free energy requires $K_{1,2}\geq 0$. All coefficients are
functions of $\mu$ and we can decompose various tensor quantities by

$\displaystyle\sigma^{ij}=\sigma_{0}\delta^{ij}+\sigma_{Q}Q^{ij},\quad\kappa^{ij}=\kappa_{0}\delta^{ij}+\kappa_{Q}Q^{ij},$
(39)

and $W_{0}^{ijkl}$ is given by (26). The unitarity constraint (3) places
positivity conditions on dissipative transport coefficients (27) and

$\sigma_{0}>0,\quad\kappa_{0}>0,\quad\sigma_{Q}^{2}<4\sigma_{0}^{2},\quad\kappa_{Q}^{2}<4\kappa_{0}^{2}.$
(40)

Physically, $\sigma_{0,Q}$ are conductivities, $\eta_{\perp,\parallel}$ are
shear viscosities, and $\zeta_{\perp,\parallel,\times}$ are bulk viscosities.
The coefficients $\kappa_{0,Q}$ modify both the stress and the local fuel
density $n_{0}$. Notice that if the $U(1)$ charge were conserved, we could
work in Landau frame, thereby removing such terms. However, we already used up
all field redefinitions for $\varphi,\varphi_{a}$ and $\sigma^{i},X_{a}^{i}$
in the non-Stückelberg sector, making Landau frame impossible.555The fact that
Landau frame is impossible is just an artefact of a choice we made. We found
it convenient to make ${{\mathcal{L}}}^{\text{(non)}}$ as simple as possible.
But we could equally well have imposed Landau frame on
${{\mathcal{L}}}^{\text{(Stüc)}}$ at the price of generating additional terms
in ${{\mathcal{L}}}^{\text{(non)}}$.  As a result, this term is in fact
physical. This point is crucial as the $\kappa_{Q}$-term modifies the stress
by an amount proportional to $Q^{ij}$, which is the hallmark of activity in
nematics Doostmohammadi  _et al._ (2018).

Let us now investigate how this action gives rise to active dynamics. We will
consider the equations of motion in the “slow roll” regime in which $\mu$
begins with a large and spatially homogeneous value and decays slowly. We
further suppose that gradients in $\mu$ remain negligible. In this regime, the
equations of motion for $\varphi_{a}$ are

$\partial_{t}\mu=-{\mu\over\tau}+\cdots,\quad\tau={\chi\over
b},\quad\chi\equiv{\partial n_{0}\over\partial\mu},$ (41)

where $\cdots$ represent higher-order corrections. We therefore see that $\mu$
has instantaneous decay rate $\tau$. As ${{\mathcal{L}}}^{\text{(non)}}$ is
small, we must take $b$ small, meaning that $\tau$ is large, as expected.
Next, the equations of motion for $X_{a}^{i}$ give the non-conservation
equation (20) where $\Gamma^{i}=\Lambda^{-1}_{ij}v^{j}$,
$T^{ti}=\tilde{w}_{0}v^{i}$ and

$T^{ij}=p_{0}\delta^{ij}+w_{0}v^{i}v^{j}+{{\frac{1}{2}}}w_{0}v^{2}\delta^{ij}-K_{1}\nabla_{1}\theta\,n^{(i}\partial^{j)}-K_{2}\nabla_{2}\theta\,n^{k}\epsilon^{k(i}\partial^{j)}\theta-
W_{0}^{ijkl}\partial_{k}v_{l}-\kappa^{ij}\partial_{t}\mu.$ (42)

Expanding $\kappa^{ij}$ according to (39) we find that the $\kappa_{Q}$-term
augments the stress tensor with a term proportional to $Q^{ij}$, which is the
hallmark of activity in nematic systems. Solving for $v^{i}$, we find that

$v^{i}=(\alpha_{0}\delta^{ij}+\alpha_{Q}Q^{ij})\partial_{k}Q^{jk},\quad\alpha_{0,Q}\equiv{\mu\kappa_{Q}\Lambda_{0,Q}\over\tau},$
(43)

where we have dropped higher-order terms. The equations of motion for
$\theta_{a}$ are, in the slow-roll approximation, given by (28). In
conjunction with the active constitutive relation for the velocity (43), we
can compare with the literature. We find that our expression for the velocity
contains the standard activity $\alpha_{0}$ and also accounts for the
anisotropic friction contribution to the activity $\alpha_{Q}$, proposed in
Kawaguchi  _et al._ (2017). We emphasize that both of these terms contribute
at the same order in the derivative expansion. As a result, unless there is
fine-tuning, we cannot neglect either of these active terms. Notice that we
must require $\tau$ to be large for there to be a sufficiently large
separation of scales for interesting active dynamics to take place. So we
might expect the activity $\alpha_{0,Q}$ must be small. Fortunately, this is
not necessary as we may start with an arbitrarily large value of $\mu$. This
is to say that the total amount of energy consumed can be very large even when
the percent of fuel consumed per unit time is small so long as the total
amount of fuel is large.

##  IV Modified dynamical KMS symmetry

Taking inspiration from the action in the previous section, we now attempt to
modify dynamical KMS symmetry to describe active systems in the slow-roll
regime on time-scales over which $\mu$ and $\partial_{t}\mu$ do not change
appreciably. To do this, remove $\varphi,\varphi_{a}$ from the EFT and
introduce non-dynamical sources $M,M_{a}$ that transform under dynamical KMS
symmetry by

$\tilde{M}={\Theta}M,\quad\tilde{M}_{a}={\Theta}M_{a}+i\beta_{0}{\Theta}M.$
(44)

Notice that there is no time-derivative acting on $M$ in the expression for
$\tilde{M}_{a}$. As a result, we choose $M,M_{a}$ to have opposite
transformation properties under ${\Theta}$. In particular, we take
${\Theta}M_{a}=M_{a}(-t,\eta\vec{x})$ and ${\Theta}M=-M(-t,\eta\vec{x})$,
where $\eta=+1$ when ${\Theta}$ contains no factors of ${{\mathcal{P}}}$ and
$\eta=-1$ when ${\Theta}$ contains a factor of ${{\mathcal{P}}}$. It is
convenient to define $\hat{M}_{a}=M_{a}+i\beta_{0}M$.

The dynamical fields are now $X_{a}^{i},\sigma^{i}$ and $\theta_{a},\theta$.
As $X_{a}^{i},\sigma^{i}$ are still only approximate Stückelberg fields, we
will continue to decompose the action according to (15). The non-Stückelberg
terms are

${{\mathcal{L}}}^{\text{(non)}}={i\Lambda^{-1}_{ij}\over\beta}X_{a}^{i}\hat{X}_{a}^{j},\quad\Lambda^{ij}=\Lambda_{0}\delta^{ij}+\Lambda_{Q}Q^{ij},$
(45)

for constants $\Lambda_{0,Q}$. And the Stückelberg terms are

$\begin{split}{{\mathcal{L}}}^{\text{(Stüc)}}={i\gamma\over\beta_{0}}\vartheta_{a}\hat{\vartheta}_{a}-K_{1}\nabla_{1}\theta\nabla_{1}\theta_{a}-K_{2}\nabla_{2}\theta\nabla_{2}\theta_{a}+T^{ti}G_{ati}+{{\frac{1}{2}}}T_{0}^{ij}G_{aij}\\\
+{i\kappa^{ij}\over 2\beta_{0}}(G_{aij}\hat{M}_{a}+\hat{G}_{aij}M_{a})+{i\over
4\beta_{0}}W^{ijkl}G_{aij}\hat{G}_{akl}.\end{split}$ (46)

where $W_{0}^{ijkl}$ is given by (26),
$T_{0}^{ij}=p_{0}\delta^{ij}+w_{0}v^{i}v^{j}+{{\frac{1}{2}}}w_{0}v^{2}\delta^{ij}$,
$T^{ti}=\tilde{w}_{0}v^{i}$, and
$\kappa^{ij}=\kappa_{0}\delta^{ij}+\kappa_{Q}Q^{ij}$ with constants
$\gamma,K_{1,2},p_{0},w_{0},\tilde{w}_{0},\kappa_{0,Q},\eta_{\parallel,\perp},\zeta_{\parallel,\perp,\times}$.
Positivity constraints from (3) are given by (27) and (40). And requiring the
free energy to be stable imposes $K_{1,2}\geq 0$.

Turning off the $a$-type source $M_{a}=0$, setting $M$ constant, and solving
the $X_{a}^{i}$-equations of motion for $v^{i}$, we find

$v^{i}=(\alpha_{0}\delta^{ij}+\alpha_{Q}Q^{ij})\partial_{k}Q^{jk},\quad\alpha_{0,Q}\equiv
M\kappa_{Q}\Lambda_{0,Q},$ (47)

where sub-leading terms have been dropped. We therefore see that $M$ is the
external source for activity; taking $M=0$ recovers the passive Lagrangian
(25). The equations of motion for $\theta_{a}$ are given by (28), which in
conjunction with (47), yields active dynamics. Notice that the
$\alpha_{Q}$-term accounts for the effects of anisotropic friction Kawaguchi
_et al._ (2017).

An important note is that dynamical KMS symmetry ensures the second law of
thermodynamics. That is, it provides a prescription for constructing an
entropy current $s^{\mu}$ whose on-shell divergence is non-negative,
$\partial_{\mu}s^{\mu}\geq 0$. This is, however, only guaranteed when all
external sources are turned off, $M=M_{a}=0$. As a result, the Lagrangian
constructed in this section will not satisfy the second law of thermodynamics.
This finding should not be surprising: by replacing the $U(1)$ non-conserved
fuel charge with external sources, we are no longer able to keep track of the
entropy produced by burning fuel.

Notice that the EFT constructed in the previous section is more complete in
that it gives a detailed account of how activity is generated. While this may
be good for some purposes, it also has the downside of being model-dependent.
By contrast the EFT constructed here is agnostic about the mechanism of
activity and is therefore model-independent. All predictions of this theory
are, however, only valid up to corrections of order $O(1/\tau)$, where $\tau$
is the characteristic decay time of the fuel.

Finally, it is important to note that setting $M$ constant and $M_{a}=0$ is
not the only possibility for generating active dynamics. This is the correct
prescription if our source of activity is a slowly decaying fuel, but in
principle, activity could be generated from an inherently noisy process. In
such cases, if the noise profile of the activity is known, then $M,M_{a}$ can
be treated as stochastic variables. This stochastic behavior could lead to
different relations between the equations of motion and noise profiles of the
active system. In this way, such a theory will give an unambiguous
generalization of the fluctuation-dissipation relations.

##  V Discussion

In this work, we constructed non-equilibrium EFTs defined on the SK contour
for active nematics. We took two approaches, one that describes a particular
mechanism for activity and the other, which is more model-independent.666It is
not absolutely model-independent as we still suppose that some notion of local
thermal equilibrium exists. To account for the case in which no notion of
local equilibrium exists in trivial: simply discard dynamical KMS symmetry.
Our EFTs are based entirely on symmetry principles organized in a derivative
expansion that account for both classical equations of motion and statistical
fluctuations about the mean. As a result, they give unambiguous predictions
regarding the relationship between statistical fluctuations and classical
equations of motion. That is, these EFTs generalize the famous fluctuation-
dissipation relations to a wide class of active system. A potentially fruitful
further direction is to study defects Vafa  _et al._ (2020, 2022); Vafa
(2022); Vafa  _et al._ (2023); Vafa and Mahadevan (2022) using EFT. In
particular, the language of field theory is well-suited to studying phase
transitions, so EFT may shed light on the activity-induced BKT-type phase
transition Shankar  _et al._ (2018).

While we focused on constructing EFTs for active nematics, the principles
exploited in this paper can be easily generalized to active systems more
broadly. We propose the following prescription for constructing model-
independent active actions:

  1. 1.

Identify the degrees of freedom associated with the passive system in the
probe limit.

  2. 2.

Introduce non-Stückelberg fields $\sigma^{i},X_{a}^{i}$, which describe
velocity. Treat them as approximate Stückelberg fields and decompose the
Lagrangain by

${{\mathcal{L}}}={{\mathcal{L}}}^{\text{(non)}}+{{\mathcal{L}}}^{\text{(Stüc)}},$
(48)

where ${{\mathcal{L}}}^{\text{(non)}}$ is small and contains non-Stückelberg
terms, while ${{\mathcal{L}}}^{\text{(Stüc)}}$ contains Stückelberg terms.
Suppose all fields transform under dynamical KMS symmetry in the usual way.

  3. 3.

Couple the action to active sources $M,M_{a}$ that transform under the
dynamical KMS symmetry by

$\tilde{M}={\Theta}M,\quad\tilde{M}_{a}={\Theta}M_{a}+i\beta_{0}{\Theta}M,$
(49)

where $M,M_{a}$ transform with opposite signs under ${\Theta}$.

  4. 4.

Identify a derivative power-counting scheme and construct the most general
action consistent with symmetries up to a given order.

For specific models of activity, instead of introducing sources $M,M_{a}$, one
could couple the system to a far-from-equilibrium sector, like the non-
conserved $U(1)$ fuel charge discussed in this work.

ACKNOWLEDGMENTS: I would like to thank Farzan Vafa and Matteo Baggioli for
many insightful discussions and comments on the manuscript. This work was
supported by the ALFA Foundation.

## References

  * Marchetti  _et al._ (2013) M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, M. Rao, and R. A. Simha, Rev. Mod. Phys. 85, 1143 (2013).
  * Needleman and Dogic (2017) D. Needleman and Z. Dogic, Nature reviews materials 2, 1 (2017).
  * Bechinger  _et al._ (2016) C. Bechinger, R. Di Leonardo, H. Löwen, C. Reichhardt, G. Volpe, and G. Volpe, Rev. Mod. Phys. 88, 045006 (2016).
  * Ramaswamy (2010) S. Ramaswamy, Annual Review of Condensed Matter Physics 1, 323 (2010).
  * Kubo (1966) R. Kubo, Reports on Progress in Physics 29, 255 (1966).
  * Kovtun (2012) P. Kovtun, J. Phys. A 45, 473001 (2012), arXiv:1205.5040 [hep-th] .
  * de Gennes and Prost (1993) P. G. de Gennes and J. Prost,  _The physics of liquid crystals_ (Clarendon Press, Oxford, 1993).
  * Simha and Ramaswamy (2002) R. A. Simha and S. Ramaswamy, Physical review letters 89, 058101 (2002).
  * Doostmohammadi  _et al._ (2018) A. Doostmohammadi, J. Ignés-Mullol, J. M. Yeomans, and F. Sagués, Nature Communications 9, 3246 (2018).
  * Saw  _et al._ (2017) T. B. Saw, A. Doostmohammadi, V. Nier, L. Kocgozlu, S. Thampi, Y. Toyama, P. Marcq, C. T. Lim, J. M. Yeomans, and B. Ladoux, Nature 544, 212–216 (2017).
  * Kawaguchi  _et al._ (2017) K. Kawaguchi, R. Kageyama, and M. Sano, Nature 545, 327–331 (2017).
  * Blanch-Mercader  _et al._ (2018) C. Blanch-Mercader, V. Yashunsky, S. Garcia, G. Duclos, L. Giomi, and P. Silberzan, Phys. Rev. Lett. 120, 208101 (2018).
  * Sanchez  _et al._ (2012) T. Sanchez, D. T. N. Chen, S. J. DeCamp, M. Heymann, and Z. Dogic, Nature 491, 431 (2012).
  * Keber  _et al._ (2014) F. C. Keber, E. Loiseau, T. Sanchez, S. J. DeCamp, L. Giomi, M. J. Bowick, M. C. Marchetti, Z. Dogic, and A. R. Bausch, Science 345, 1135 (2014).
  * Vafa  _et al._ (2020) F. Vafa, M. J. Bowick, M. C. Marchetti, and B. I. Shraiman, “Multi-defect dynamics in active nematics,” (2020), arXiv:2007.02947 [cond-mat.soft] .
  * Vafa  _et al._ (2022) F. Vafa, G. H. Zhang, and D. R. Nelson, Phys. Rev. E 106, 024704 (2022).
  * Vafa (2022) F. Vafa, Soft Matter 18, 8087 (2022).
  * Vafa  _et al._ (2023) F. Vafa, D. R. Nelson, and A. Doostmohammadi, Journal of Physics: Condensed Matter 35, 425101 (2023).
  * Vafa and Mahadevan (2022) F. Vafa and L. Mahadevan, Phys. Rev. Lett. 129, 098102 (2022).
  * Tan  _et al._ (2019) A. J. Tan, E. Roberts, S. A. Smith, U. A. Olvera, J. Arteaga, S. Fortini, K. A. Mitchell, and L. S. Hirst, Nature Physics 15, 1033 (2019).
  * Alert  _et al._ (2022) R. Alert, J. Casademunt, and J.-F. Joanny, Annual Review of Condensed Matter Physics 13, 143 (2022).
  * Giomi  _et al._ (2013) L. Giomi, M. J. Bowick, X. Ma, and M. C. Marchetti, Phys. Rev. Lett. 110, 228101 (2013).
  * Ellis  _et al._ (2018) P. W. Ellis, D. J. Pearce, Y.-W. Chang, G. Goldsztein, L. Giomi, and A. Fernandez-Nieves, Nature Physics 14, 85 (2018).
  * Shankar and Marchetti (2019) S. Shankar and M. C. Marchetti, Phys. Rev. X 9, 041047 (2019).
  * Putzig  _et al._ (2016) E. Putzig, G. S. Redner, A. Baskaran, and A. Baskaran, Soft matter 12, 3854 (2016).
  * Srivastava  _et al._ (2016) P. Srivastava, P. Mishra, and M. C. Marchetti, Soft matter 12, 8214 (2016).
  * Doostmohammadi  _et al._ (2016) A. Doostmohammadi, M. F. Adamer, S. P. Thampi, and J. M. Yeomans, Nature communications 7, 10557 (2016).
  * Pearce  _et al._ (2021) D. Pearce, J. Nambisan, P. Ellis, A. Fernandez-Nieves, and L. Giomi, Physical Review Letters 127 (2021), 10.1103/physrevlett.127.197801.
  * Thijssen  _et al._ (2020) K. Thijssen, M. R. Nejad, and J. M. Yeomans, Physical Review Letters 125 (2020), 10.1103/physrevlett.125.218004.
  * Thampi and Yeomans (2016) S. P. Thampi and J. M. Yeomans, The European Physical Journal Special Topics 225, 651 (2016).
  * Giomi (2015) L. Giomi, Physical Review X 5, 031003 (2015).
  * Landry (2020) M. J. Landry, JHEP 07, 200 (2020), arXiv:1912.12301 [hep-th] .
  * Baggioli and Landry (2020) M. Baggioli and M. Landry, SciPost Phys. 9, 062 (2020), arXiv:2008.05339 [hep-th] .
  * Landry (2021a) M. J. Landry, JHEP 04, 213 (2021a), arXiv:2008.11725 [hep-th] .
  * Crossley  _et al._ (2017) M. Crossley, P. Glorioso, and H. Liu, JHEP 09, 095 (2017), arXiv:1511.03646 [hep-th] .
  * Glorioso and Liu (2016) P. Glorioso and H. Liu, (2016), arXiv:1612.07705 [hep-th] .
  * Glorioso  _et al._ (2017) P. Glorioso, M. Crossley, and H. Liu, JHEP 09, 096 (2017), arXiv:1701.07817 [hep-th] .
  * Glorioso and Liu (2018) P. Glorioso and H. Liu, (2018), arXiv:1805.09331 [hep-th] .
  * Landry (2021b) M. J. Landry, (2021b), arXiv:2101.02210 [hep-th] .
  * Chaikin and Lubensky (2000) P. M. Chaikin and T. C. Lubensky,  _Principles of condensed matter physics_ (Cambridge university press, 2000).
  * Landry (2022) M. J. Landry, J. Stat. Mech. 2207, 073205 (2022), arXiv:2006.13220 [hep-th] .
  * Jain (2020) A. Jain, JHEP 10, 208 (2020), arXiv:2008.03994 [hep-th] .
  * Landry and Sun (2021) M. J. Landry and G. Sun, JHEP 05, 040 (2021), arXiv:2010.11191 [hep-th] .
  * Dubovsky  _et al._ (2012) S. Dubovsky, L. Hui, A. Nicolis, and D. T. Son, Physical Review D 85 (2012), 10.1103/physrevd.85.085029.
  * Shankar  _et al._ (2018) S. Shankar, S. Ramaswamy, M. C. Marchetti, and M. J. Bowick, Phys. Rev. Lett. 121, 108002 (2018).