# Towards A Unified Agent with
Foundation Models

Norman Di Palo
Imperial College London
London, UK
<EMAIL_ADDRESS>
&Arunkumar Byravan
Google DeepMind
London, UK
&Leonard Hasenclever
Google DeepMind
London, UK
&Markus Wulfmeier
Google DeepMind
London, UK
&Nicolas Heess
Google DeepMind
London, UK
&Martin Riedmiller
Google DeepMind
London, UK
work done during an internship at Google DeepMind.

###### Abstract

Language Models and Vision Language Models have recently demonstrated
unprecedented capabilities in terms of understanding human intentions,
reasoning, scene understanding, and planning-like behaviour, in text form,
among many others. In this work, we investigate how to embed and leverage such
abilities in Reinforcement Learning (RL) agents. We design a framework that
uses language as the core reasoning tool, exploring how this enables an agent
to tackle a series of fundamental RL challenges, such as efficient
exploration, reusing experience data, scheduling skills, and learning from
observations, which traditionally require separate, vertically designed
algorithms. We test our method on a sparse-reward simulated robotic
manipulation environment, where a robot needs to stack a set of objects. We
demonstrate substantial performance improvements over baselines in exploration
efficiency and ability to reuse data from offline datasets, and illustrate how
to reuse learned skills to solve novel tasks or imitate videos of human
experts.

##  1 Introduction

In recent years, the literature has seen a series of remarkable Deep Learning
(DL) success stories (3), with breakthroughs particularly in the fields of
Natural Language Processing (4; 19; 8; 29) and Computer Vision (2; 25; 36;
37). Albeit different in modalities, these results share a common structure:
large neural networks, often Transformers (46), trained on enormous web-scale
datasets (39; 19) using self-supervised learning methods (19; 6). While simple
in structure, this recipe led to the development of surprisingly effective
Large Language Models (LLMs) (4), able to process and generate text with
outstanding human-like capability, Vision Transformers (ViTs) (25; 13) able to
extract meaningful representations from images and videos with no supervision
(6; 18), and Vision-Language Models (VLMs) (2; 36; 28), that can bridge those
data modalities describing visual inputs in language, or transforming language
descriptions into visual outputs. The size and abilities of these models led
the community to coin the term Foundation Models (3), suggesting how these
models can be used as the backbone for downstream applications involving a
variety of input modalities.

This led us to the following question: can we leverage the performance and
capabilities of (Vision) Language Models to design more efficient and general
reinforcement learning agents? After being trained on web-scaled textual and
visual data, the literature has observed the emergence of common sense
reasoning, proposing and sequencing sub-goals, visual understanding, and other
properties in these models (19; 4; 8; 29). These are all fundamental
characteristics for agents that need to interact with and learn from
environments, but that can take an impractical amount of time to emerge tabula
rasa from trial and error. Exploiting the knowledge stored into Foundation
Models, can bootstrap this process tremendously.

Motivated by this idea, we design a framework that puts language at the core
of an RL robotic agent, particularly in the context of learning from scratch.
Our core contribution and finding is the following: we show that this
framework, which leverages LLMs and VLMs, can tackle a series of fundamental
problems in RL settings, such as 1) efficiently exploring sparse-reward
environments, 2) re-using collected data to bootstrap the learning of new
tasks sequentially, 3) scheduling learned skills to solve novel tasks and 4)
learning from observation of expert agents. In the recent literature, these
tasks need different, specifically designed algorithms to be tackled
individually, while we demonstrate that the capabilities of Foundation Models
unlock the possibility of developing a more unified approach.

Figure 1: A high-level illustration of our framework.

##  2 Related Work

Over the past few years, scaling the parameter count of models and the size
and diversity of training datasets led to unprecedented capabilities in
(Vision) Language Models (4; 19; 2; 19; 8). This in turn led to several
applications leveraging these models within agents that interact with the
world. Prior work has used LLMs and VLMs together with RL agents in simulated
environments (12; 44), but they rely on collecting large amounts of
demonstrations for training agents. Instead, we focus on the problem of
learning RL agents from scratch and leverage LLMs and VLMs to accelerate
progress.

Prior work has also looked at leveraging LLMs and VLMs for robotics
applications; particularly (1; 21; 50; 20) leveraged LLMs for planning sub-
goals in the context of long-horizon tasks together with VLMs for scene
understanding and summarization. These sub-goals can then be grounded into
actions through language-conditioned policies (22; 30). While most of these
works focus on deploying and scheduling already learned skills through LLMs,
albeit in the real world, our work focuses on an RL system that learns such
behaviours from scratch, highlighting the benefits that these models bring to
exploration, transfer and experience reuse.

Several methods have been proposed to tackle sparse-reward tasks, either
through curriculum learning (43; 51; 31; 16), intrinsic motivation (17; 35),
or hierarchical decomposition (32; 27). We demonstrate how LLMs can generate
learning curriculums zero-shot, without any additional learning or finetuning,
and VLMs can automatically provide rewards for these sub-goals, greatly
improving learning speed.

Related work has also looked at reusing large datasets of robotic experience
by learning a reward model for the new tasks at hand (5). However, numerous
human annotations of desired rewards need to be gathered for each new task.
Instead, as reported in concurrent related work (48), we show successful
relabeling of past experience leveraging VLMs which can be finetuned with
small amounts of data from the target domain.

(15) is the most similar method to our work: they propose an interplay between
LLMs and VLMs to learn sparse-reward tasks in Minecraft (23; 24). However,
there are some notable differences: they use a vast internet dataset of
videos, posts and tutorials to finetune their models, while we demonstrate
that it is possible to effectively finetune a VLM with as few as 1000
datapoints, and use off-the-shelf LLMs; additionally, we also investigate and
experiment how this framework can be used for data reuse and transfer, and
learning from observation, besides exploration and skills scheduling,
proposing a more unified approach to some core challenges in reinforcement
learning.

##  3 Preliminaries

We use the simulated robotic environment from Lee et al. (26) modelled with
the MuJoCo physics simulator (45) for our experiments: a robot arm interacts
with an environment composed of a red, a blue and a green object in a basket.
We formalise it as a Markov Decision Process (MDP): the state space
$\mathcal{S}$ represents the 3D position of the objects and the end-effector.
The robot is controlled through position control: the action space
$\mathcal{A}$ is composed of an $x,y$ position, that we reach using the known
inverse kinematics of the robot, where the robot arm can either pick or place
an object, inspired by (49; 40). The observation space $\mathcal{O}$ is
composed of $128\times 128\times 3$ RGB images coming from two cameras fixed
to the edges of the basket. The agent receives a language description of the
task $\mathcal{T}$ to solve, which can have two forms: either "Stack X on top
of Y", where X and Y are taken from {"the red object", "the green object",
"the blue object" } without replacement, or "Stack all three objects", that we
also call Triple Stack.

A positive reward of $+1$ is provided if the episode is successful, while a
reward of $0$ is given in any other case. We define the sparseness of a task
as the average number of environment steps needed, when executing random
actions sampled from a uniform distribution, to solve the task and receive a
single reward. With the MDP design we adopt, stacking two objects has a
sparseness of $10^{3}$, while an optimal policy could solve the task with 2
pick-and-place actions/steps (49; 40). Stacking all three objects has a
sparseness of more than $10^{6}$ as measured by evaluating trajectories from a
random policy, while an optimal policy could solve the task in 4 steps.

##  4 A Framework for Language-Centric Agents

The goal of this work is to investigate the use of Foundation Models (3), pre-
trained on vast image and text datasets, to design a more general and unified
RL robotic agent. We propose a framework that augments from-scratch RL agents
with the ability to use the outstanding abilities of LLMs and VLMs to reason
about their environment, their task, and the actions to take entirely through
language.

To do so, the agent first needs to map visual inputs to text descriptions.
Secondly, we need to prompt an LLM with such textual descriptions and a
description of the task to provide language instructions to the agent.
Finally, the agent needs to ground the output of the LLM into actions.

Figure 2: An illustration of CLIP computing the similarity, as dot product,
between observations and text descriptions.

Bridging Vision and Language using VLMs: To describe the visual inputs taken
from the RGB cameras (Sec. 3) in language form, we use CLIP, a large,
contrastive visual-language model (36). CLIP is composed of an image-encoder
$\phi_{I}$ and a text-encoder $\phi_{T}$, trained on a vast dataset of noisily
paired images and text descriptions, that we also refer to as captions. Each
encoder outputs a 128-dimensional embedding vector: embeddings of images and
matching text descriptions are optimised to have large cosine similarity. To
produce a language description of an image from the environment, the agent
feeds an observation $o_{t}$ to $\phi_{I}$ and a possible caption $l_{n}$ to
$\phi_{T}$ (Fig. 2). We compute the dot product between the embedding vectors
and considers the description correct if the result is larger than $\gamma$, a
hyperparameter ($\gamma=0.8$ in our experiments, see Appendix for more
details). As we focus on robotic stacking tasks, the descriptions are in the
form "The robot is grasping X" or "The X is on top of Y", where X and Y are
taken from {"the red object", "the green object", "the blue object" } without
replacement. We finetune CLIP on a small amount of data from the simulated
stacking domain; more details on how this works and analysis on data needs for
finetuning are provided in the appendix.

Reasoning through Language with LLMs: Language Models take as input a prompt
in the form of language and produce language as output by autoregressively
computing the probability distribution of the next token and sampling from
this distribution. In our setup, the goal of LLMs is to take a text
instruction that represents the task at hand (e.g. "Stack the red object on
the blue object"), and generate a set of sub-goals for the robot to solve. We
use FLAN-T5 (10), an LLM finetuned on datasets of language instructions. A
qualitative analysis we performed showed that it performed slightly better
than LLMs not finetuned on instructions.

Figure 3: The VLM can act as an internal reward model by comparing language
goals proposed by the LLM to the collected observations.

The extraordinary in-context learning capabilities of these LLMs allowed us to
use them off-the-shelf (4; 34), without the need for in-domain finetuning, and
guide their behaviour by providing as few as two examples of task instruction
and desired language outputs: we describe the environment setting, asking the
LLM to find sub-goals that would lead to solving a proposed task, providing
two examples of such tasks and relative sub-goals decomposition. With that,
the LLM was able to emulate the desired behaviour, not only in content, but
also in the formatting of the output language which allowed for efficient
parsing. In the Appendix we provide a more detailed description of the prompts
we use and the behaviour of the LLMs.

Grounding Instructions into Actions: The language goals provided by the LLMs
are then grounded into actions using a language-conditioned policy network.
This network, parameterized as a Transformer (46), takes an embedding of the
language sub-goal and the state of the MDP at timestep $t$, including objects’
and robot end-effector’s positions, as input, each represented as a different
vector, and outputs an action for the robot to execute as timestep $t+1$. This
network is trained from scratch within an RL loop as we describe below.

Collect & Infer Learning Paradigm: Our agent learns from interaction with the
environment through a method inspired by the Collect & Infer paradigm (38).
During the Collect phase, the agent interacts with the environment and
collects data in the form of states, observations, actions and current goal as
$(s_{t},o_{t},a_{t},g_{i})$, predicting actions through its policy network,
$f_{\theta}(s_{t},g_{i})\rightarrow a_{t}$. After each episode, the agent uses
the VLM to infer if any sub-goals have been encountered in the collected data,
extracting additional rewards, as we explain in more detail later. If the
episode ends with a reward, or if any reward is provided by the VLM, the agent
stores the episode data until the reward timestep
$[(s_{0},o_{0},a_{0},g_{i}),\dots,(s_{T_{r}-1},o_{T_{r}-1},a_{T_{r}-1},g_{i})]$
in an experience buffer. We illustrate this pipeline in Fig. 4 (Left). These
steps are executed by $N$ distributed, parallel agents, that collect data into
the same experience buffer ($N=$1000 in our work). During the Infer phase, we
train the policy through Behavioural Cloning on this experience buffer after
each agent has completed an episode, hence every $N$ total episodes,
implementing a form of Self-Imitation on successful episodes (33; 14; 7). The
updated weights of the policy are then shared with all the distributed agents
and the process repeats.

##  5 Applications and Results

We described the building blocks that compose our framework. The use of
language as the core of the agent provides a unified framework to tackle a
series of fundamental challenges in RL. In the following sections, we will
investigate each of those contributions, focusing on exploration, reusing past
experience data, scheduling and reusing skills and learning from observation.
The overall framework is also described in Algorithm 1.

Algorithm 1 Language-Centric Agent

1: Training time:

2: for task in tasks do

3: subgoals = LLM(task) //find text subgoals given task description

4: exp_buffer.append( VLM(offline_buffer, subgoals)) //extract successful eps
from offline buff. collected in past tasks(Sec. 5.2)

5: for $ep$ in episodes do

6: (Sec. 5.1)

7: $E\leftarrow[s_{0:T},o_{0:T},a_{0:T},g_{i}]$ //collect ep. trajectory

8: $r\leftarrow$ collect final reward

9: $r_{internal}\leftarrow$ VLM($E$, subgoals) //extract additional rewards
for subgoals

10: if $r$ or $r_{internal}$ then

11: exp_buffer.append($E_{0:T_{r}}\texttt{)}$ //Add timesteps until reward

12: if ep%$N==0$ then

13: $\theta\leftarrow$ BC(episode_buffer) //train agent with BC every $N$ eps

14: Test time:

15: Receive text_instruction or video_demo

16: if text_instruction then

17: subgoals = LLM(text_instruction) (Sec. 5.3)

18: else if video_demo then

19: subgoals = VLM(video_demo) (Sec. 5.4)

20: execute(subgoals) (Sec. 5.3)

###  5.1 Exploration - Curriculum Generation through Language

RL benefits substantially from carefully crafted, dense rewards (5). However,
the presence of dense rewards is rare in many real-world environments. Robotic
agents need to be able to learn a wide range of tasks in complex environments,
but engineering dense reward functions becomes prohibitively time-consuming as
the number of tasks grows. Efficient and general exploration is therefore
imperative to overcome these challenges and scale RL.

A wide variety of methods have been developed over the years to tackle
exploration of sparse-reward environments (43; 51; 31; 16; 17; 35; 32; 27).
Many propose decomposing a long-horizon task into shorter, easier to learn
tasks, through curriculum generation and learning. Usually, these methods need
to learn to decompose tasks from scratch, hindering overall learning
efficiency. We demonstrate how an RL agent leveraging LLMs can take advantage
of a curriculum of text sub-goals that are generated without any past
environment interaction.

Figure 4: Left: Illustration of our Collect & Infer pipeline. Middle, Right:
Learning curves of our framework and a baseline in the Stack Red on Blue and
Triple Stack tasks.

To guide exploration, the agent provides the task description
$\mathcal{T}_{n}$ to the LLM, instructing it to decompose the task into
shorter-horizon sub-goals, effectively generating a curriculum of goals
$g_{0:G}$ in text form 111For example, the LLM decomposes ”Stack the red
object on the blue object” into the following sub-goals: [”The robot is
grasping the red object”, ”The red object is on top of the blue object”]. The
agent selects actions as $f_{\theta}(s_{t},\mathcal{T}_{n})\rightarrow a_{t}$.
While the environment provides a reward only if $\mathcal{T}_{n}$ is solved,
the VLM is deployed to act as an additional, less sparse reward model: given
the observations $o_{0:T}$ collected during the episode and all the text sub-
goals $g_{0:G}$ proposed by the LLM, it verifies if any of the sub-goals were
solved at any step.

We consider an observation $o_{t}$ to represent a completion state for a sub-
goal $g_{i}$ if $\phi_{T}(g_{i})\cdot\phi_{I}(o_{t})>\gamma$. In that case,
the agent adds
$[(s_{0},o_{0},a_{0},\mathcal{T}_{n}),\dots,(s_{t-1},o_{t-1},a_{t-1},\mathcal{T}_{n})]$
to our experience buffer. The process is illustrated in Fig. 3, 11 (in the
Appendix).

Results on Stack X on Y and Triple Stack. We compare our framework to a
baseline agent that learns only through environment rewards in Fig. 4. The
learning curves clearly illustrate how our method is substantially more
efficient than the baseline on all the tasks. Noticeably, our agent’s learning
curve rapidly grows in the Triple Stack task, while the baseline agent still
has to receive a single reward, due to the sparseness of the task being
$10^{6}$. We provide a visual example of the extracted sub-goals and rewards
in the Appendix.

These results suggest something noteworthy: we can compare the sparseness of
the tasks with the number of steps needed to reach a certain success rate, as
in Fig. 5. We train our method also on the Grasp the Red Object task, the
easiest of the three, with sparseness in the order of $10^{1}$. We can see
that, under our framework, the number of steps needed grows more slowly than
the sparseness of the task. This is a particularly important result, as
generally the opposite is true in Reinforcement Learning (35).

Figure 5: With our framework, the number of steps needed to reach a certain
success rate grows more slowly than the sparseness of the task.

This slower growth, enabled by the increase in the amount of sub-goals
proposed by the LLM as the task becomes sparser, suggests that our framework
can scale to even harder tasks and make them tractable, assuming sub-goals can
be encountered with a uniform-like distribution at any point during
exploration. Additionally, unlike prior approaches that need carefully crafted
intrinsic rewards or other exploration bonuses our approach can directly
leverage prior knowledge from LLMs and VLMs to generate a semantically
meaningful curriculum for exploration, thereby paving the way for general
agents that explore in a self-motivated manner even in sparse-reward
environments.

###  5.2 Extract and Transfer - Efficient Sequential Tasks Learning by Reusing
Offline Data

When interacting with their environments, our agents should be able to learn a
series of tasks over time, reusing the prior collected data to bootstrap
learning on any new task instead of starting tabula rasa. This is a
fundamental ability to scale up RL systems that learn from experience. Recent
work has proposed techniques to adapt task-agnostic offline datasets to new
tasks, but they can require laborious human annotations and learning of reward
models (5; 47; 9).

Figure 6: Our framework can reutilise offline data collected on other tasks,
extracting successful trajectories for the new task at hand, bootstrapping
policy learning.

We leverage our language based framework to showcase bootstrapping based on
the agent’s past experience. We train three tasks in sequence: Stack the red
object on the blue object, Stack the blue object on the green object, and
Stack the green object on the red object, that we call
$[\mathcal{T}_{R,B},\mathcal{T}_{B,G},\mathcal{T}_{G,R}]$. The intuition is
simple: while exploring to solve, for example, $\mathcal{T}_{R,B}$, it is
likely that the agent had solved other related tasks, like $\mathcal{T}_{B,G}$
or $\mathcal{T}_{G,R}$, either completely or partially. The agent should
therefore be able to extract these examples when trying to solve the new
tasks, in order not to start from scratch, but reuse all the exploration data
gathered for previous tasks.

As discussed in Sec. 4, our agent gathers an experience buffer of interaction
data. We now equip the agent with two different buffers: a lifelong buffer, or
offline buffer, where the agent stores each episode of interaction data, and
continues expanding it task after task. Then, the agent has a new task buffer,
re-initialised at the beginning of each new task, that is filled, as in Sec.
5.1, with trajectories that result in a reward, either external or internally
provided by the VLM using LLM text sub-goals (Fig. 3). The policy network is
optimised using the new task buffer.

Differently from before however, while the first task, $\mathcal{T}_{R,B}$, is
learned from scratch, the agent reuses the data collected during task $n$ to
bootstrap the learning of the next task $n+1$. The LLM decomposes
$\mathcal{T}_{n+1}$ into text sub-goals $[g_{0},\dots,g_{L-1}]$. The agent
then extracts from the lifelong/offline buffer each stored episode
$\mathcal{E}_{n}=[(s_{0:T,n},o_{0:T,n},a_{0:T,n})]$. It then takes each
episode’s observation $o_{t,n}$ and uses the VLM to compute dot-products score
between all image observations and all text sub-goals as
$\phi_{T}(g_{l})\cdot\phi_{I}(o_{t})$. If the score is larger than the
threshold $\gamma$ the agent adds all the episode’s timesteps up to $t$,
$[(s_{0:t,n},o_{0:t,n},a_{0:t,n})]$ to the new task buffer. The process is
illustrated in Fig. 6.

This procedure is repeated for each new task at the beginning of training.
Following this procedure, the agent does not start learning new tasks tabula
rasa: at the beginning of task $\mathcal{T}_{n}$, the current experience
buffer is filled with episodes useful to learn the task extracted from
$\mathcal{T}_{0:n}$. When $n$ increases, the amount of data extracted from
$\mathcal{T}_{0:n}$ increases as well, speeding up learning.

Figure 7: In our experiments, the agent can learn task $n+1$ faster than task
$n$ by reusing past experience data.

Results on Experience Reuse for Sequential Tasks Learning. The agent applies
this method to learn $[\mathcal{T}_{R,B},\mathcal{T}_{B,G},\mathcal{T}_{G,R}]$
in succession. At the beginning of each new task we re-initialise the policy
weights: our goal is to investigate the ability of our framework to extract
and re-use data, therefore we isolate and eliminate effects that could be due
to network generalisation.

We plot how many interaction steps the agent needs to take in the environment
to reach 50% success rate on each new task in Fig. 7. Our experiments clearly
illustrate the effectiveness of our technique in reusing data collected for
previous tasks, improving the learning efficiency of new tasks.

These results suggest that our framework can be employed to unlock lifelong
learning capabilities in robotic agents: the more tasks are learned in
succession, the faster the next one is learned. This can be particularly
beneficial when deploying agents in open-ended environments, particularly in
the real world; by leveraging data across its lifetime the agent has
encountered it should be able to learn novel tasks far faster than learning
purely from scratch.

###  5.3 Scheduling and Reusing Learned Skills

We described how our framework enables the agent with the ability to
efficiently explore and learn to solve sparse-reward tasks, and to reuse and
transfer data for lifelong learning.

Figure 8: Our framework can break down a task into a list of skills using the
LLM, and execute each skill until the VLM predicts that its sub-goal has been
reached.

Using its language-conditioned policy (Sec. 4), the agent can thus learn a
series of $M$ skills, described as a language goal $g_{0:M}$ (e.g. "The green
object is on top of the red object" or "The robot is grasping the blue
object").

Our framework allows the agent to schedule and reuse the $M$ skills it has
learned to solve novel tasks, beyond what the agent encountered during
training. The paradigm follows the same steps we encountered in the previous
sections: a command like Stack the green object on top of the red object or
Stack the red on the blue and then the green on the red is fed to the LLM,
which is prompted to decompose it into a list of shorter-horizon goals,
$g_{0:N}$. The agent can then ground these into actions using the policy
network as $f_{\theta}(s_{t},g_{n})\rightarrow a_{t}$.

When executing the $n$-th skill, the VLM computes at each timestep if
$\phi_{T}(g_{n})\cdot\phi_{I}(o_{t})>\gamma$, thus checking if the goal of the
skill has been reached in the current observation. In that case, the agent
starts executing the $n+1$-th skill, unless the task is solved.

###  5.4 Learning from Observation: Mapping Videos to Skills

Learning from observing an external agent is a desirable ability for general
agents, but this often requires specifically designed algorithms and models
(42; 11; 52). Our agent can be conditioned on a video of an expert performing
the task, enabling one-shot learning from observation. In our tests, the agent
takes a video of a human stacking the objects with their hand. The video is
divided into $F$ frames, $v_{0:F}$. The agent then uses the VLM, paired with
the $M$ textual description of the learned skills, expressed as sub-goals
$g_{0:M}$, to detect what sub-goals the expert trajectory encountered as
follows: (1) the agent embeds each learned skill/sub-goal through
$\phi_{T}(g_{m})$ and each video frame through $\phi_{I}(v_{f})$ and compute
the dot product between each pair. (2) it lists all the sub-goals that obtain
a similarity larger than $\gamma$, collecting the chronological list of sub-
goals the expert encountered during the trajectory. (3) It executes the list
of sub-goals as described in Fig. 8.

Despite being finetuned only on images from the MuJoCo simulation (Sec. 4),
the VLM was able to accurately predict the correct text-image correspondences
on real-world images depicting both a robot or a human arm. Notice also how we
still refer to it as "the robot" in the captions (Fig. 9), but the VLM
generalises to a human hand regardless.

Figure 9: An illustration of the agent learning from observation using our
framework.

##  6 Conclusion

We propose a framework that puts language at the core of an agent. Through a
series of experiments, we demonstrate how this framework, by leveraging the
knowledge and capabilities of Foundation Models, can provide a more unified
approach with respect to the current literature to tackle a series of core RL
challenges, that would normally require separate algorithms and models: 1)
exploring in sparse-reward tasks 2) reusing experience data to bootstrap
learning of new skills 3) scheduling learned skills to solve novel tasks and
4) learning from observing expert agents. These initial results suggest that
leveraging foundation models can lead to general RL algorithms able to tackle
a variety of problems with improved efficiency and generality. By leveraging
the prior knowledge contained within these models we can design better robotic
agents that are capable of solving challenging tasks directly in the real
world. We provide a list of current limitations and future work in the
Appendix.

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##  7 Appendix

###  7.1 Finetuning CLIP on in-domain Data

Figure 10: Captioning precision and recall of finetuned CLIP as a function of
the dataset size. The logarithmic trend suggests that around $10^{3}$ image-
caption pairs unlock sufficient performance. Values obtained with
$\gamma=0.8$.

In our experiments, the dot products between the embeddings of possible
captions and of an RGB observation from our environment
$y=\phi_{I}(o_{t})\cdot\phi_{T}(l_{i})$ were often uninformative: correct and
wrong pairs obtained very similar scores, and varied too little in range. Our
goal is to set a threshold $\gamma$ to recognise correct and wrong
descriptions given an image: therefore we need a larger difference in score.
To tackle this, we collect a dataset of image observations with various
configurations of the objects and the corresponding language descriptions
using an automated annotator based on the MuJoCo state of the simulation to
finetune CLIP with in-domain data. The plot on the right provides an analysis
of our findings: precision and recall tend to increase logarithmically with
the dataset size. The key takeaway message is that, although CLIP is trained
on around $10^{8}$ images, just $10^{3}$ in-domain pairs are enough to improve
its performance on our tasks.

In our case, a high precision is more desirable than high recall: the former
indicates that positive rewards are not noisy, while the opposite may disrupt
the learning process. A lower recall indicates that the model may not be able
to correctly identify all successful trajectories, but this simply translate
in the need for more episodes to learn, and does not disrupt the learning
process. We found a value of $\gamma=0.8$ to be the best performing choice
after finetuning.

Figure 11: Autonomously identifying sub-goals and corresponding rewards
becomes especially important when tasks become prohibitively sparse, like
Triple Stack.

###  7.2 Current Limitations and Future Work

1) In our current implementation, we use a simplified input and output space
for the policies, namely the state space of the MDP - i.e. the positions of
the objects and the end-effector as provided by the MuJoCo simulator - and a
pick and place action space, as described in Sec. 3, where the policy can
output a $x,y$ position for the robot to either pick and place. This choice
was adopted to have faster experiments iteration and therefore be able to
focus our search on the main contribution of the paper: the interplay with the
LLM and the VLM. Nevertheless, the recent literature has demonstrated that a
wide range of robotics tasks can be executed through this action space
formulation Zeng et al. [2021], Shridhar et al. [2022].

Many works from the current literature Lee et al. [2021], Silver et al.
[2017], Cabi et al. [2019], Fan et al. [2022] demonstrate that, in order for
the policy to scale to image observations as input and end-effector velocities
as output, the model only needs more data, and therefore interaction time. As
our goal was demonstrating the relative performance improvements brought by
our method, our choice of MDP design does not reduce the generality of our
findings. Our results will most likely translate also to models that use
images as inputs, albeit with the need for more data.

2) We finetune CLIP on in-domain data, using the same objects we then use for
the tasks. In future work, we plan to perform a larger scale finetuning of
CLIP on more objects, possibly leaving out the object we actually use for the
tasks, therefore also investigating the VLM capabilities to generalise to
inter-class objects. At the moment, this was out of the scope of this work, as
it would have required a considerable additional amount of computation and
time.

3) We train and test our environment only in simulation: we plan to test the
framework also on real-world environments, as our results suggest that 1) we
can finetune CLIP with data from simulation and it generalises to real images
(Sec. 5.4), therefore we can avoid expensive human annotations 2) the
framework allows for efficient learning of even sparse tasks from scratch
(Sec. 5.1), suggesting the applicability of our method to the real-world,
where collecting robot experience is substantially more time expensive.

###  7.3 Prompts and outputs of the LLM

In Fig. 12 we show the prompt we used to allow in-context learning of the
behaviour we expect from the LLM Olsson et al. [2022]. With just two examples
and a general description of the setting and its task, the LLM can generalise
to novel combinations of objects and even novel, less well-defined tasks, like
"Stack all three objects", outputting coherent sub-goals.

Figure 12: An example of the prompt we used to condition the LLM, and its
outputs. Normal text: user inserted text, bold text: LLM outputs.


# Anisotropy of magnetized quark matter

Kangkan Goswami  Dushmanta Sahu  Jayanta Dey  Raghunath Sahoo  Corresponding
Author<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Indore, Simrol, Indore 453552, India  Reinhard Stock  Institute of
Nuclear Physics, University of Frankfurt, Germany

###### Abstract

Strong transient magnetic fields are generated in non-central relativistic
heavy-ion collisions. These fields induce anisotropy within the strongly
interacting medium that, in principle, can affect the thermodynamic properties
of the medium. We use the Polaykov loop extended Nambu Jona-Lasinio (PNJL)
model to study the quark matter subjected to an external magnetic field at
vanishing baryon chemical potential. We have estimated the degree of
anisotropy in the speed of sound and isothermal compressibility within the
magnetized quark matter as a function of temperature and magnetic field. This
study helps us to understand the extent of directionality generated in the
initial stages of non-central collisions while giving us useful information
about the system.

##  I Introduction

One of the primary goals of relativistic heavy-ion collisions is to study the
deconfined state of strongly interacting quarks and gluons in local thermal
equilibrium, known as the quark-gluon plasma (QGP). In the non-central heavy-
ion collision, the charged spectators move past the fireball at a relativistic
speed. According to the Biot-Savart law, these moving charge particles create
a large transient electromagnetic field of the order of $10^{18}$ G at the
Large Hadron Collider (LHC) [1, 2]. Direct experimental evidence for the
strength of the magnetic field ($B$) is yet to be discovered. However, recent
measurements of the directed flow of $D^{0}$ and $\bar{D}^{0}$ at the
Relativistic Heavy Ion Collider (RHIC) [3] and Large Hadron Collider [4],
indicate the creation of a strong magnetic field during the collision.

The first principle non-perturbative theory of strong interaction, lattice
quantum chromodynamics (lQCD), found many interesting phenomena in QGP in the
presence of magnetic field such as chiral magnetic effect [5], magnetic and
inverse magnetic catalysis [6]. However, experimental verification of these
phenomena are yet to be observed. Under an external magnetic field, the energy
levels of the charged particles get quantized following the Landau
quantization, which creates momentum anisotropy affecting various
thermodynamical [7, 8, 9] and dissipative quantities [10, 11, 12]. For
instance, the thermodynamical pressure becomes anisotropic with a longitudinal
component along the magnetic field and a transverse component in the
transverse plane of the field. This leads to anisotropy in many other
thermodynamical quantities, such as speed of sound ($c_{s}$), and isothermal
compressibility ($\kappa_{\rm{T}}$) as we will show here. In the strong field
limit, this leads to a dimensional reduction in the phase space ($3\rightarrow
1$ dimension) at the lowest Landau level (LLL). Moreover, the magnetic field
influences the QCD phase diagram. Different effective QCD models are used to
study the phase diagram in the $B-T$ plane, such as linear sigma model [13],
Nambu Jona-Lasinio (NJL) model, and its extended version Polayakov loop
extended NJL (PNJL) model [14, 15]. Initial prediction of the PNJL model
showed that the transition temperature ($T_{c}$) and its strength increase
with $eB$, leading to a first-order phase transition. However, lQCD
calculation [6] found an opposite trend, which means $T_{c}$ decreases with
$eB$. The same was also observed in the models, such as in NJL and PNJL models
[16, 17] with a magnetic field-dependent coupling constant.

Many indirect probes and observables are suggested to study the microscopic
and bulk features of QGP under the effect of the magnetic field.
Theoretically, one can study the change in the thermodynamic observables to
understand the changes in the deconfined medium in the presence of an external
magnetic field. The behavior of certain thermodynamic observables such as
$\kappa_{\rm T}$ and $c_{\rm s}^{2}$ provide useful information about the
nature of the phase transition of the system. $\kappa_{\rm T}$ represents the
rate of change in the volume of the system concerning pressure at a constant
temperature. Precisely, $\kappa_{\rm T}$ measures the extent to which the
density of quarks and gluons changes in response to changes in external
pressure, which is an essential factor in determining the equation of state of
the medium [18, 19, 20]. Moreover, it can tell us about the degree of
deviation of a system from a perfect fluid. $\kappa_{\rm{T}}$ is expected to
show a sudden jump near the critical end-point (CEP), where the smooth
crossover in the QCD phase diagram meets the first-order phase transition.
Thus, it is an interesting observable to explore in the QCD phase diagram
[21]. In literature, $\kappa_{\rm{T}}$ has been studied as a function of
temperature and charged particle multiplicity [22, 23, 24, 25, 26, 27]. In
Ref. [23], the high-temperature QCD matter has been found to be the closest to
a perfect fluid.

On the other hand, the speed of sound reflects the propagation of small
perturbations produced in the system in its local rest frame. Its dependence
on the environment, i.e. temperature, density, and baryon chemical potential,
means that it is an ideal probe to explore the evolution of the fireball [28].
The studies in Refs. [29, 30, 27, 31] reveal that exploring $c_{\rm s}^{2}$ as
a function of charged particle multiplicity can be utilized to study the
dynamics of heavy-ion collisions. Moreover, it shows minima near the phase
transition. This also makes $c_{\rm s}^{2}$ a suitable observable to study the
QCD phase space. In literature, many phenomenological models are employed to
compute the speed of sound such as the quasiparticle model [32], dynamic
quasiparticle model [33], color string percolation model [34, 35], hadron
resonance gas model [36, 37], NJL model [38, 39], and PNJL model [40, 41, 42].
It has also been estimated using the lQCD calculations [43, 44, 45, 46].

The lattice QCD model is a first principle theory of strongly interacting
matter. Despite its many successes, it fails to explain the high-baryon-rich
environment due to its fermion sign problem [21]. Hence, exploring the CEP
within the lQCD formalism is not feasible. Thus, the NJL and PNJL models are
good alternatives in this regard. The PNJL model is an extended version of the
NJL model, initially developed for nuclear matter. Due to its respect for the
global symmetries of QCD, particularly the chiral symmetry, this model has
been widely utilized to explore some of the non-perturbative features of the
QCD vacuum. The NJL model is not renormalizable due to the point-like
interaction between the quarks [47, 48]. A suitable regularisation method must
be established to deal with the divergent integrals. The three-momentum cutoff
scheme is widely accepted among many others in the literature [49]. The
model’s parameters must be fixed to replicate well-known phenomenological
quantities, such as the pion-decay constant and quark condensate density [49].
The NJL model is based on the idea that the chiral symmetry of quantum
chromodynamics (QCD) is spontaneously broken at low temperatures and
densities, which leads to the appearance of a non-zero quark condensate. The
NJL model incorporates this symmetry breaking by introducing a four-fermion
interaction between quarks. The interaction is attractive in the scalar
channel and repulsive in the pseudoscalar channel, which leads to the
formation of a chiral condensate and the generation of quark masses [50].

Albeit being a successful model, one drawback of the NJL model is that it
doesn’t incorporate the gluon dynamics, and as a result, quark confinement is
absent in this model. As suggested in [51], the Polyakov loop characterizes
the effects of confinement, which prevents quarks from existing as free
particles. The trace of the Polyakov loop ($\rm\Phi$) is the order parameter
for the confinement transition in a pure gauge theory [52, 53, 54, 55]. It can
be understood as $\Phi$ is related to $e^{-F/T}$, where $F$ is the free energy
of the static quark. The free energy of a confined single quark is infinite,
which leads to $\Phi=0$, and in the deconfined phase, free energy is finite,
which makes $\Phi\neq 0$ [56]. In the PNJL model, the Polyakov loop is
included as a background temporal gluon field that interacts with the quarks.
The PNJL model has been successful in reproducing lattice results
qualitatively [57, 41, 58, 59, 28]. The PNJL model has also been used to study
a wide range of phenomena in QCD matter, including thermodynamic and transport
properties [60, 61, 62, 63, 64, 33], probing the critical point in the
$(T-\mu)$ plane [65, 66, 67, 68], and exploring the QCD medium under the
effect of an external magnetic field [9, 69, 16, 70, 71].

In this work, we investigate the behavior of the speed of sound and isothermal
compressibility and explore the degree of anisotropy generated in baryon-free
quark matter in the presence of an external magnetic field. This paper is
organized as follows. In section II.1 and II.2, we have discussed the PNJL
model formulation with and without an external magnetic field, respectively.
Section III comprises the results of this work. Finally, section IV summarizes
this work.

##  II Formulation

###  II.1 Quark matter in zero magnetic field

The Polyakov loop extended Nambu-Jona Lasinio (PNJL) model was introduced as
an improvement to the Nambu-Jona Lasinio (NJL) model [72, 73, 51, 57]. The NJL
model effectively takes care of the chiral symmetry breaking. However, due to
the lack of gluonic interaction in the model, it fails to explain the
deconfinement dynamics of quarks. PNJL model takes care of this problem by
adding a Polyakov loop effective potential to the NJL Lagrangian. This model
simplifies the interaction between quarks and gluons by considering chiral
point couplings between quarks and a temporal background gauge field.
Extensive studies have been done using the PNJL model with 2 and 2+1 flavors
[57, 65, 66, 41, 74, 15, 75, 76]. The PNJL lagrangian for 2+1 flavors is given
by [76, 77, 66],

$\begin{split}{\mathcal{L}_{PNJL}}=\bar{q}(i\gamma^{\mu}D_{\mu}-\hat{m})q+G\sum_{a=0}^{8}\left[(\bar{q}\lambda^{a}q)^{2}+(\bar{q}\iota\gamma_{5}\lambda^{a}q)^{2}\right]-K\\{{\rm
det}[\bar{q}(1+\gamma_{5})q]+\mbox{det}[\bar{q}(1-\gamma_{5})q]\\}-\mathcal{U}(\Phi,\overline{\Phi},T)\end{split}$
(1)

where $q$ is the three-flavor quark field, $q=(u,d,s)$, $\lambda^{a}$ are the
Gell-Mann matrices in flavor space, and $\hat{m}$ is the current quark mass,
$\hat{m}$ = diag $(m_{u},m_{d},m_{s})$. The covariant derivative is defined
as,

$D^{\mu}=\partial^{\mu}-\iota A^{\mu},$

where $A^{\mu}=\delta^{\mu}_{0}A^{0}$ and in the Euclidian notation, we can
write $A^{0}=-\textit{i}A_{4}$. The four-point interaction and six-point
interaction of the quark field are incorporated with an effective coupling
strength $G$ and $K$, respectively. $\mathcal{U}(\Phi,\overline{\Phi},T)$ is
the effective Polyakov loop potential. The parameters in the Lagrangian are
the cut-off parameter $\rm\lambda$, the coupling constants $\rm G$ and $\rm
K$, and the current quark masses $\rm m_{u}$ and $\rm m_{s}$, we use the value
set in Ref. [78] as $\lambda=602.3$ MeV, $G\lambda^{2}=1.835$,
$K\lambda^{5}=12.36$, $m_{u}=m_{d}=5.5$ MeV and $m_{s}=140.7$ MeV. These
parameters are obtained by fixing the values of $m_{\pi}=135$ MeV,
$m_{K}=497.7$ MeV, $m_{\eta^{\prime}}=957.8$ MeV and $f_{\pi}=92.4$ MeV.

This model takes the analogy from the BCS theory of superconductor where
pairing between electrons in a spin-singlet state leads to a condensation
which introduces a gap in the energy spectra. Similarly, the condensation
$\langle\bar{q}_{R}q_{L}\rangle$ (or $\langle\bar{q}_{L}q_{R}\rangle$) arises
due to the pairing between the quark-antiquark of the same chirality. Thus,
$\langle\bar{q}{q}\rangle$ =
$\langle\bar{q}_{R}q_{L}+\bar{q}_{L}q_{R}\rangle\sim\sigma$ can be taken as
the order parameter and in the mean field approximation gives rise to a
dynamical mass $M\sim\langle\bar{q}{q}\rangle$ [79]. Thus, the gap equation
for the PNJL model can be expressed as [76],

$M_{i}=m_{i}-4G\sigma_{i}+2K\sigma_{j}\sigma_{k}$ (2)

where $i,j,k=u,d,s$ in cyclic order, $M_{i}$ is the constituent quark mass,
$m_{i}$ is the bare quark mass, and $\sigma_{i,j,k}$ is the quark condensate
for different flavors. The third term exists as a result of introducing six-
fermion interactions and in turn, generating a flavor mixing in the quark
mass.

The thermodynamic potential is given as [76, 80],

$\begin{split}\Omega=\mathcal{U}(\Phi,\overline{\Phi},T)+2{G}\sum_{{f=u,d,s}}\sigma_{f}^{2}-4{K}\sigma_{u}\sigma_{d}\sigma_{s}-2{N_{c}}\sum_{{f=u,d,s}}\int_{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\sqrt{p^{2}+M^{2}_{f}}\\\
-2T\sum_{{f=u,d,s}}\int\frac{d^{3}p}{(2\pi)^{3}}\rm{ln}\left[1+3\Phi
e^{-\beta(E-\mu)}+3\bar{\Phi}e^{-2\beta(E-\mu)}+e^{-3\beta(E-\mu)}\right]\\\
-2T\sum_{{f=u,d,s}}\int\frac{d^{3}p}{(2\pi)^{3}}\rm{ln}\left[1+3\bar{\Phi}e^{-\beta(E+\mu)}+3\Phi
e^{-2\beta(E+\mu)}+e^{-3\beta(E+\mu)}\right]~{},\end{split}$ (3)

where $\beta$ = 1/T, $E=\sqrt{p^{2}+M^{2}}$ is the energy of the quark and
$\mathcal{U}(\Phi,\overline{\Phi},T)$ is the effective Polyakov loop potential
for the $\rm\Phi$ and $\rm\bar{\Phi}$ fields. Different versions of this
potential exist in the literature [65, 81, 82]. For our study, we choose the
following form of the potential, which mathematically limits the value of
$\Phi$ to be unity as $T~{}\rightarrow\infty$ [65],

$\begin{split}\frac{\mathcal{U}(\Phi,\overline{\Phi},T)}{T^{4}}=-\frac{a(T)}{2}\Phi\overline{\Phi}+b(T)\rm{ln}[1-\\\
6\Phi\overline{\Phi}+4(\Phi^{3}+\overline{\Phi}^{3})-3(\Phi\overline{\Phi})^{2}]~{},\end{split}$
(4)

where $a(T)$ and $b(T)$ reads,

$\displaystyle
a(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2}~{},$
$\displaystyle b(T)=b_{3}\left(\frac{T_{0}}{T}\right)^{3}~{},$ (5)

where the parameters $a_{0}$, $a_{1}$, $a_{2}$ and $b_{3}$ are fixed by
performing a simultaneous fit of the thermodynamic observables obtained from
lattice QCD calculation and given as [65],

$\begin{split}a_{0}=3.51,~{}a_{1}=-2.47\\\
a_{2}=15.2,~{}b_{3}=-1.75\end{split}$ (6)

The parameter $T_{0}$ is the critical temperature for the deconfinement phase
transition in a pure-gauge system. In a pure gauge approach, $T_{0}$ is fixed
to be 270 MeV. One can find different values of $T_{0}$ in literature. In Ref.
[83], a $N_{f}$ dependent form of $T_{0}$ is taken. In Ref. [84], the authors
studied the effect of different $T_{0}$ values on the thermodynamic
observables. However, following Ref. [57], for quantitative comparison of
results with lQCD data, we choose $T_{0}$ = 190 MeV.

In order to estimate $\Phi$, $\bar{\Phi}$ and $\sigma_{f}$, one needs to
minimize $\Omega$ with respect to the above-mentioned quantities [76, 85],

$\frac{\partial\Omega}{\partial\sigma_{f}}=0,~{}\frac{\partial\Omega}{\partial\Phi}=0,~{}\frac{\partial\Omega}{\partial\bar{\Phi}}=0$

Solving $\frac{\partial\Omega}{\partial\sigma_{f}}=0$, we get the condensates
as [85],

$\sigma_{f}=-2N_{c}\int_{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\frac{M_{f}}{E_{f}}(1-f_{\Phi}^{+}-f_{\Phi}^{-})$
(7)

where $f_{\Phi}^{+}$ and $f_{\Phi}^{-}$ are the quark and antiquark
distribution functions respectively [74, 80],

$f_{\Phi}^{+}=\frac{(\Phi+2\overline{\Phi}e^{-\beta(E_{f}-\mu_{f})})e^{-\beta(E_{f}-\mu_{f})}+e^{-\beta(E_{f}-\mu_{f})}}{1+3\left(\Phi+\overline{\Phi}e^{-\beta(E_{f}-\mu_{f})}\right)e^{-\beta(E_{f}-\mu_{f})}+e^{-3\beta(E_{f}-\mu_{f})}},$
$f_{\Phi}^{-}=\frac{(\overline{\Phi}+2\Phi
e^{-\beta(E_{f}+\mu_{f})})e^{-\beta(E_{f}+\mu_{f})}+e^{-\beta(E_{f}+\mu_{f})}}{1+3\left(\overline{\Phi}+\Phi
e^{-\beta(E_{f}+\mu_{f})}\right)e^{-\beta(E_{f}+\mu_{f})}+e^{-3\beta(E_{f}+\mu_{f})}}.$

Finally, pressure can be estimated as $P=-\Omega$. We take care of the vacuum
pressure, taken numerically at $T=0.001$ and $\mu=0$, by subtracting its
contribution from the total pressure given as [86],

$P=-\Big{(}\Omega(T,\mu)-\Omega(0.001,0)\Big{)}~{}.$ (8)

Now, the entropy density ($s$) and the energy density ($\epsilon$) are
estimated as,

$\begin{split}s=-\frac{\partial\Omega}{\partial T}=\frac{\partial P}{\partial
T}~{},\end{split}$ $\epsilon=-P+Ts~{}.$

Furthermore, the speed of sound, $c_{s}$ and the isothermal compressibility,
$\kappa_{\rm{T}}$ can be obtain as,

$c_{s}^{2}=\frac{\partial P}{\partial\epsilon}=\frac{\frac{\partial
P}{\partial T}}{\frac{\partial\epsilon}{\partial T}}~{},$ (9)
$\kappa_{T}=-\frac{1}{V}\frac{\partial V}{\partial P}~{},$ (10)

where volume, $V=N/n$ with the total number of partons $N$ and the number
density $n$. Plugging this into the above equation, we get

$\kappa_{T}=\frac{1}{n}\frac{\partial n}{\partial
P}=\frac{1}{n}\frac{\frac{\partial n}{\partial\mu}}{\frac{\partial
P}{\partial\mu}}~{}.$ (11)

###  II.2 Quark matter in finite magnetic field

Under the effect of an external magnetic field, the PNJL lagrangian changes as
[87, 16],

$\begin{split}{\mathcal{L}_{PNJL}^{B}}=\bar{q}(i\gamma^{\mu}D_{\mu}-\hat{m})q+G_{B}(eB)\sum_{a=0}^{8}\left[(\bar{q}\lambda^{a}q)^{2}+(\bar{q}\iota\gamma_{5}\lambda^{a}q)^{2}\right]-K\\{{\rm
det}[\bar{q}(1+\gamma_{5})q]+\mbox{det}[\bar{q}(1-\gamma_{5})q]\\}\\\
-\mathcal{U}(\Phi,\overline{\Phi},T)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\end{split}$
(12)

The coupling of the quarks and the temporal effective gluon field with the
external magnetic field is incorporated by adding the
$\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ term, where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}^{EM}-\partial_{\nu}A_{mu}^{EM}$, and
altering the covariant derivative as $D^{\mu}=\partial^{\mu}-\iota
q_{f}A_{EM}^{\mu}-\iota A^{\mu}$, where $e_{f}$ is the electric charge of the
quark of flavor $f$. The coupling constant $G$ is taken as, $G_{B}(eB)$, a
function of $eB$ to incorporate the effect of an external magnetic field.
Various forms of $G_{B}(eB)$ exist in literature [88, 89, 90, 91, 69, 92, 16].
For our study, we use the form given in [16],

$G_{B}(eB)=G\left(\frac{1+a\zeta^{2}+b\zeta^{3}}{1+c\zeta^{2}+d\zeta^{4}}\right)$
(13)

This form is obtained by reproducing the chiral transition temperature,
$T_{C}^{\chi}(eB)$, obtained in lQCD [6]. This form of coupling constant is
able to yield qualitatively precise results for quark condensate values at a
finite magnetic field. where $a=0.0108805$, $b=-1.0133\times 10^{-4}$,
$c=0.02228$, $d=1.84558\times 10^{-4}$, and $\zeta=eB/\Lambda^{2}_{QCD}$ with
$\Lambda_{QCD}=300$ MeV. Finally, the thermodynamic potential due to an
external magnetic field changes as [87, 16],

$\begin{split}\Omega(T,\mu,eB)=\mathcal{U}(\Phi,\overline{\Phi},T)+2G_{B}(eB)\sum_{{f=u,d,s}}\sigma_{f}^{2}-4K\sigma_{u}\sigma_{d}\sigma_{s}+\left(\sum_{{f=u,d,s}}\Omega_{f}^{vac}+\Omega_{f}^{med}+\Omega_{f}^{mag}\right)\end{split}$
(14)

where contributions from the vacuum, $\Omega^{vac}_{f}$, medium,
$\Omega^{med}_{f}$, and magnetic field, $\Omega^{mag}_{f}$ which are given as
[93, 94, 16],

$\displaystyle\Omega_{f}^{vac}=-6\int_{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\sqrt{p^{2}+M_{f}^{2}},$
$\displaystyle\Omega_{f}^{med}=-T\frac{|e_{f}B|}{4\pi^{2}}\sum_{k}a(k)\int_{-\infty}^{\infty}dp_{z}(\mathcal{Z}_{\Phi}^{+}+\mathcal{Z}_{\Phi}^{-}),$
$\displaystyle\Omega_{f}^{mag}=-\frac{3|e_{f}B|^{2}}{2\pi^{2}}\left[\zeta^{\prime}(-1,x_{f})-\frac{1}{2}(x_{f}^{2}-x_{f}){\rm
ln}(x_{f})+\frac{x_{f}^{2}}{4}\right].$ (15)

Here, ${Z}_{\Phi}^{+}$ and ${Z}_{\Phi}^{-}$ are the partition function density
given as[74, 80],

$\begin{split}{Z}_{\Phi}^{+}={\rm
ln}\Big{[}1+3\left(\Phi+\overline{\Phi}e^{-\beta(E_{f}-\mu_{f})}\right)e^{-\beta(E_{f}-\mu_{f})}\\\
+e^{-3\beta(E_{f}-\mu_{f})}\Big{]}\\\ {Z}_{\Phi}^{-}={\rm
ln}\Big{[}1+3\left(\overline{\Phi}+\Phi
e^{-\beta(E_{f}+\mu_{f})}\right)e^{-\beta(E_{f}+\mu_{f})}\\\
+e^{-3\beta(E_{f}+\mu_{f})}\Big{]}~{}.\end{split}$

$x_{f}=M_{f}^{2}/(2|e_{f}|B)$, $\zeta(z,x)$ is the Riemann-Hurwitz zeta
function, and $\zeta^{\prime}(-1,x_{f})=d\zeta(z,x_{f})/dz\mid_{z=-1}$ can be
expressed as [95, 96],

$\begin{split}\zeta^{\prime}(-1,x_{f})=\frac{1}{12}-\frac{x^{2}_{f}}{4}+(\frac{1}{12}-\frac{x_{f}}{2}+\frac{x^{2}_{f}}{2}){\rm
log}(x_{f})+\mathcal{O}(x^{-2}_{f})\end{split}$

In the presence of a magnetic field, the dispersion relation of quarks will be
modified due to the Landau quantization as,

$E_{f}=\sqrt{p^{2}+M_{f}^{2}+2k|e_{f}|B}~{}.$ (16)

Minimizing the thermodynamic potential with respect to $\Phi$, $\bar{\Phi}$
and $\sigma_{f}$, we get a set of coupled equations to be solved
simultaneously [76, 85],

$\frac{\partial\Omega}{\partial\sigma_{f}}=0,~{}\frac{\partial\Omega}{\partial\Phi}=0,~{}\frac{\partial\Omega}{\partial\bar{\Phi}}=0$

Further, the condensates split under the effect of an external magnetic field
as,

$\sigma_{f}=\sigma_{f}^{vac}+\sigma_{f}^{med}+\sigma_{f}^{mag}$

where $\sigma_{f}^{vac}$, $\sigma_{f}^{med}$, and $\sigma_{f}^{mag}$ are the
contributions from vacuum, medium and the magnetic field respectively, which
are given as [9, 71, 93, 89, 80],

$\displaystyle\sigma_{f}^{vac}=-6\int_{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\frac{M_{f}}{E_{f}}~{},$
$\displaystyle\sigma_{f}^{med}=\frac{3(|e_{f}B|)^{2}}{4\pi^{2}}\sum_{k}a(k)\int_{-\infty}^{\infty}\frac{dp_{z}}{E_{f}}(f_{\Phi}^{+}+f_{\Phi}^{-})~{},$
$\displaystyle\sigma_{f}^{mag}=-\frac{M|e_{f}|BN_{c}}{2\pi^{2}}\Big{[}{\rm
ln\Gamma}(x_{f})-\frac{1}{2}{\rm ln}(2\pi)$
$\displaystyle+x_{f}-\frac{1}{2}(2x_{f}-1){\rm ln}(x_{f})\Big{]}~{},$ (17)

Any thermodynamic quantity can be derived from the thermodynamic potential
$\Omega$. In the presence of a magnetic field, pressure becomes anisotropic
due to the Landau quantization. Pressure along the direction of the magnetic
field is longitudinal pressure ($P^{\parallel}$) and in the direction
perpendicular to the magnetic field it is transverse pressure ($P^{\perp}$),
and can be expressed as[97, 98],

$\displaystyle P^{\parallel}=-\Omega-\frac{(eB)^{2}}{2}~{},$ $\displaystyle
P^{\perp}=-\Omega-eB\mathcal{M}+\frac{(eB)^{2}}{2}~{},$ (18)

where $\mathcal{M}$ is the magnetization,

$\mathcal{M}=-\Bigg{(}\frac{\partial\Omega}{\partial eB}\Bigg{)}_{\mu}~{}.$
(19)

The normalized pressure is taken in a way that it vanishes at vacuum [93, 9,
91, 89],

$P^{\parallel}_{N}=P^{\parallel}(T,\mu,B)-P^{\parallel}(0.001,0,B)$
$P^{\perp}_{N}=P^{\perp}(T,\mu,eB)-P^{\perp}(0.001,0,eB)~{}.$

In the weak magnetic field limit, for simplicity, we can take
$P^{\perp}_{N}\simeq P^{\parallel}_{N}=P_{N}$. Using the Euler thermodynamic
relation, energy density can be written as,

$\epsilon=-P_{N}+Ts+\mathcal{M}eB$ (20)

The speed of sound also becomes anisotropic in the presence of a magnetic
field due to changes in pressure in the longitudinal and transverse
directions. The anisotropic squared speed of sound reads as [99, 100],

$\displaystyle
c_{s}^{2\parallel(\perp)}=\Bigg{(}\frac{dP^{\parallel(\perp)}_{N}}{d\epsilon}\Bigg{)}_{s/n}~{},$
(21) $\displaystyle c_{s}^{2\parallel(\perp)}=\frac{\frac{\partial
P^{\parallel(\perp)}_{N}}{\partial T}+\frac{\partial
P^{\parallel(\perp)}_{N}}{\partial
eB}\frac{deB}{dT}}{\frac{\partial\epsilon}{\partial
T}+\frac{\partial\epsilon}{\partial eB}\frac{deB}{dT}}~{},$ (22)

where

$\frac{deB}{dT}=\frac{s\frac{\partial n}{\partial T}-n\frac{\partial
s}{\partial T}}{n\frac{\partial s}{\partial eB}-s\frac{\partial n}{\partial
eB}}.$

Isothermal compressibility can be expressed as in Eq. (11) for the vanishing
magnetic field. For a system with a finite magnetic field, the pressure term
in the denominator splits isothermal compressibility into longitudinal and
transverse components, which can be expressed as [98],

$\displaystyle\kappa_{T}^{\parallel}=\frac{1}{n}\frac{\frac{\partial
n}{\partial\mu}}{\frac{\partial
P^{\parallel}}{\partial\mu}}=\frac{1}{n^{2}}{\frac{\partial
n}{\partial\mu}}~{}.$
$\displaystyle\kappa_{T}^{\perp}=\frac{1}{n}\frac{\frac{\partial
n}{\partial\mu}}{\frac{\partial
P^{\perp}}{\partial\mu}}=\frac{1}{n}{\frac{\partial
n}{\partial\mu}}\Bigg{(}\frac{1}{n-\frac{\partial\mathcal{M}}{\partial\mu}B}\Bigg{)}~{}.$
(23)

##  III Results and Discussion

Figure 1: Squared speed of sound’s longitudinal ($c_{s}^{2\parallel}$) (left)
and transverse ($c_{s}^{2\perp}$) (right) components against temperature ($T$)

Figure 2: Squared speed of sound ($c_{s}^{2\parallel}$ and $c_{s}^{2\perp}$)
against magnetic field ($eB$) for different temperature; $T=0.12$ GeV (left),
$T=0.155$ GeV (middle), and $T=0.3$ GeV (right)

.

Figure 3: Anisotropy in squared speed of sound ($\Delta
c_{s}^{2}/c_{s}^{2\parallel}$) against $T$ (left) and $eB$ (right). Here,
$\Delta c_{s}^{2}=c_{s}^{2\parallel}-c_{s}^{2\perp}$.

In the presence of a magnetic field, $c_{s}^{2}$ becomes anisotropic. It
signifies that any disturbance in the medium will not travel at the same speed
in all directions in the presence of an external magnetic field. In Fig. 1,
longitudinal ($\parallel$) (left) and transverse ($\perp$) (right) components
of $c_{s}^{2}$ are plotted against temperature for different values of
magnetic fields; $eB=0$ (black dotted line), $eB=0.2~{}\rm GeV^{2}$ (red
dashed line), and $eB=0.4~{}\rm GeV^{2}$ (solid blue line). Lattice estimation
at the zero magnetic fields is also presented here from WB group [101]. At
zero magnetic field, the PNJL result agrees with lattice estimation at the
high temperature. The dip around transition temperature ($T_{c}$) in the PNJL
result is the consequence of deconfinement phase transition which is absent in
NJL estimations [38]. With the increasing magnetic field, the dip shifts
towards lower temperatures. This signifies that with the increasing magnetic
field, the deconfinement temperature decreases, which agrees with the lattice
results obtained in Ref. [102].

In a massless non-interacting (ideal) gas, the speed of sound is estimated to
be $1/3$ [103]. In Fig. 1, we observe that as the temperature increases,
chiral symmetry is restored, and $c_{s}^{2}$ approaches the ideal gas limit.
We observe an interesting trend of $c^{2}_{s}$ near the transition
temperature. Below $T_{c}$, $c_{s}^{2}$ decreases with increasing magnetic
field, whereas slightly above $T_{c}$, the speed of sound is higher at a
larger magnetic field. This shows that $c_{s}^{2}$ is very sensitive to mass
ordering. This behavior could be a consequence of magnetic catalysis (MC) and
inverse magnetic catalysis (IMC) phenomena. Due to magnetic catalysis, at low
temperatures, the quark condensate is enhanced in the presence of a magnetic
field, and as a result, the dynamical mass of the quarks is directly
proportional to the magnetic field strength. However, due to inverse magnetic
catalysis, beyond $T_{c}$, the quark condensate is reduced with increasing
magnetic field and the mass is inversely proportional to the external magnetic
field. The same behaviour can be seen for the speed of sound near $T_{c}$.
These phenomena have been widely explored in literature [104, 16, 89].

In Fig. 2, $c_{s}^{2}$ is plotted against $eB$ for three different values of
temperatures, such that $T=0.12$ GeV (left) represents the confined or
hadronic medium, $T=0.155$ GeV (middle) is around $T_{c}$, and $T=0.3$ GeV
(right) represents the deconfined or partonic medium. $\parallel$ and $\perp$
components are represented by a solid green and red dotted line, respectively.
$c_{s}^{2}$ shows a non-monotonic behavior with respect to $eB$. $\parallel$
component is almost independent of the magnetic field because, along the
magnetic field, direct contribution from Lorentz force vanishes. In the
confinement zone, the effect of the magnetic field is large in the $\perp$
component. We observe a decrease in the transverse component of the speed of
sound with an increase in magnetic field strength. Around the transition
temperature, the effect of the magnetic field on both of the components is not
significant. However, there is a steady rise for the longitudinal component at
a strong magnetic field. On the right, for the partonic phase, we have both
weak and strong field regions. In the weak field region, we observe an
increasing and then decreasing behavior that could be attributed to the
interplay of decreasing coupling constant with an increase in $eB$ and
dependence of dynamical mass on the magnetic field strength.

To better understand the anisotropy in the components of $c_{s}^{2}$ due to
magnetic field, we have plotted $\Delta c_{s}^{2}/c_{s}^{2\parallel}$ with $T$
for three different values of magnetic field
($eB=0.1,~{}0.2,~{}0.4~{}\rm{GeV}^{2}$) in Fig. (3) (left). On the right
panel, we have plotted the same against $eB$ for different temperatures
($T=0.12,~{}0.155,~{}0.3$ GeV). Here, $\Delta
c_{s}^{2}=c_{s}^{2\parallel}-c_{s}^{2\perp}$ represents normalized anisotropy
in the components of $c_{s}^{2}$. In other words, Fig. (3) represents the
deviation of $\perp$ component from $\parallel$ component in the presence of
magnetic fields. In the left panel, anisotropy is larger at a low temperature
and decreases as temperature increases. For $eB=0.2,~{}0.4~{}\rm{GeV}^{2}$,
anisotropy is around $20\%$ to $60\%$, respectively, at $T=0.1$ GeV. The
degree of anisotropy decreases with temperature and obtains minima near the
transition temperature. The position of the minima changes with magnetic field
strength. With an increase in temperature, the deviation increases as well.
The figure on the right shows that anisotropy increases as the magnetic field
increases. Anisotropy is smaller at low $eB$ for all three temperatures and
increases gradually with increasing $eB$. Therefore, from both figures, we can
conclude that anisotropy is higher in the strong field zone, i.e., low $T$ and
high $eB$. With decreasing magnetic field, anisotropy diminishes, and
$\parallel$ and $\perp$ components merge to the isotropic component of
$c_{s}^{2}$ as soon as the magnetic field vanishes.

Figure 4: Isothermal compressibility ($\kappa_{T}$) against temperature ($T$)
at zero magnetic field. A comparison of this work (PNJL) with different models
(HRG, EVHRG [24], and CSPM [23]), and experimental results from ALICE [105].

Figure 5: Isothermal compressibility: longitudinal component
($\kappa_{T}^{\parallel}$) (left) and transverse component
($\kappa_{T}^{\perp}$) (right) against temperature ($T$) for different value
of magnetic field ($eB$).

Figure 6: $\kappa_{T}^{\parallel}$ and $\kappa_{T}^{\perp}$ against $eB$ at
$T=0.12$ GeV (left), $T=0.155$ GeV (middle), and $T=0.3$ GeV (right).

In Fig. 4, $\kappa_{T}$ of QCD matter is plotted against temperature for QGP
using the PNJL model. We compare our results with experimental results and
different models. Experimentally, the isothermal compressibility is estimated
using charged particle multiplicity fluctuations in Pb-Pb collisions at
$\sqrt{s_{NN}}=2.76$ TeV [105]. However, this estimation requires the
information of temperature and volume of the system at chemical freeze-out
which is taken from the statistical hadronization model. Results from hadronic
models HRG (green dotted line), EV-HRG (solid black line) [24], and partonic
models CSPM (solid violet circle) [23], and the PNJL (blue dash line) in the
absence of magnetic field are presented for comparison. $\kappa_{T}$ indicates
stiffness in the equation of state (EoS). The smaller the value of
$\kappa_{T}$, the stiffer the EoS is. Here, $\kappa_{T}$ decreases as $T$
increases, which means at very high $T$, the QCD matter is highly
incompressible. This makes the QGP matter almost perfect fluid. In the absence
of magnetic fields, $\kappa_{T}$ is isotropic. The EoS is the same in all
directions. As we introduce the magnetic field, $\kappa_{T}$ becomes
anisotropic, and we get two independent components. Along the magnetic field,
EoS is defined by $\kappa_{T}^{\parallel}$ component, and in the plane
perpendicular to the magnetic field, it is $\kappa_{T}^{\perp}$.

In Fig. (5), we have plotted $\kappa_{T}^{\parallel}$ (left) and
$\kappa_{T}^{\perp}$ (right) as a function of $T$ for
$eB=0,~{}0.2,~{}0.4~{}\rm{GeV}^{2}$. $\kappa_{T}$ is a proxy to the phase
transition. For the baryon-free ($\mu_{\rm{B}}=0$) QCD matter, our results
show a smooth transition from deconfinement to confinement temperature zone.
Even in the presence of a magnetic field, both the components show smooth
transition or crossover phase transition. However, one recent study [98] in
NJL model discovered a first-order phase transition in
$\kappa_{T}^{\parallel}$ component in the presence of a magnetic field at very
low temperature and finite baryon chemical potential, which may not be
realized in RHIC or LHC energies. Here, we see that the magnetic field reduces
the compressibility, and the effect is nearly independent of $T$.

In Fig. (6), we have plotted $\kappa_{T}^{\parallel}$ (green solid line) and
$\kappa_{T}^{\perp}$ (red dotted line) as a function of $eB$ for $T=0.12$ GeV
(left), $T=0.155$ GeV (middle), and $T=0.3$ GeV (right). At any fixed
temperature, both the components decrease with increasing magnetic field. In
the middle figure, where $T$ is around $T_{c}$, a peak-like structure is seen
in $\kappa_{T}^{\parallel}$. However, we do not recognize it as a first-order
phase transition because the peak is very small, and no other signature is
found in the order parameters, such as the deconfinement parameter and chiral
condensate. In the low-temperature regime, saturation in $\kappa_{T}$ can be
observed for both parallel and perpendicular components. The value of the
$\perp$ component for $\kappa_{T}$ is smaller compared to the $\parallel$
component. This signifies that along the magnetic field, EoS is stiffer.

Figure 7: Anisotropy in isothermal compressibility
$(\Delta\kappa_{T}/\kappa_{T}^{\parallel})$ as a function of temperature ($T$)
(left) and magnetic field ($eB$) (right),
$\Delta\kappa_{T}=\kappa_{T}^{\parallel}-\kappa_{T}^{\perp}$.

The anisotropy between the two components is very prominent. In order to
quantify the anisotropy in $\kappa_{T}$ we have plotted
$\Delta\kappa_{T}/\kappa_{T}^{\parallel}$ as a function of $T$ (left) and $eB$
(right) in Fig. (7), where
$\Delta\kappa_{T}=\kappa_{T}^{\parallel}-\kappa_{T}^{\perp}$. Similar to
$c_{s}^{2}$ in Fig. (3), here also we see that anisotropy decreases with
increasing $T$ at a fixed $eB$, reaching minima. For temperatures higher than
$T_{\rm{c}}$, there is an increase in anisotropic behavior with temperature
for each magnetic field value. On the right, we have studied the degree of
anisotropy as a function of $eB$. Anisotropy increases with $eB$ at a fixed
$T$. Anisotropy is maximum in the strong field zone. Compared to $c_{s}^{2}$,
anisotropy is higher in $\kappa_{T}$. In the low $T$ zone, anisotropy in
$\kappa_{\rm{T}}$ varies from $60\%$ to $80\%$, and in high $T$, it is $30\%$
to $40\%$ in the mentioned magnetic field regime. In the case of $c_{s}^{2}$,
the effect is comparatively $10\%$ to $20\%$ less. This signifies that the
magnetic field effect is more prominent in the anisotropy of $\kappa_{\rm{T}}$
than that of $c_{s}^{2}$.

##  IV Summary and outlook

In summary, we have estimated the squared speed of sound ($c_{s}^{2}$) and
isothermal compressibility ($\kappa_{T}$) of baryon-free QCD matter in the
presence of a magnetic field using the PNJL model. With the increasing
temperature, $c_{s}^{2}$ approaches the conformal limit. It shows a dip around
the $T_{c}$, which shifts to a lower temperature with the increasing magnetic
field. The effect of MC and IMC is also found in the components of $c_{s}^{2}$
near the transition temperature. The isothermal compressibility is estimated
within the PNJL model for the first time. Our results match with other results
in the literature qualitatively, confirming that the QCD matter is the most
perfect fluid found in nature. With the increasing temperature, $\kappa_{T}$
decreases. Smooth transition of longitudinal and transverse components from
confinement to deconfinement phase indicates crossover transition in the
presence of magnetic field at zero chemical potential. Furthermore, we noticed
that anisotropy is higher in $\kappa_{T}$ compared to $c_{s}^{2}$, and in the
strong field limit, the lowest Landau level is a good approximation. We found
that the degree of anisotropy increases with the magnetic field and shows
minima near $T_{c}$ when plotted against temperature for both thermodynamical
quantities.

The anisotropy in the EoS, due to the magnetic field, is not directly
measurable in experiments. However, the observable, directed flow is a
promising probe to the initial magnetic field [3, 4]. Recent studies [106,
107] show that the directed flow is sensitive to the EoS. The EoS is affected
significantly by the external magnetic field. In principle, one should be able
to extract information about the effect of the magnetic field on the hot QCD
matter by studying the splitting of the directed flow of particle and anti-
particle pair. Experimentally, at LHC and RHIC, the splitting of the directed
flow of $D^{0}$ and $\bar{D^{0}}$ has been observed [4, 3].

As a preliminary study, our work concentrates only on baryon-free QCD matter
at a constant magnetic field. This work can be extended to the baryonic matter
with a time-dependent magnetic field, which applies to matter created at RHIC
and energies achievable below RHIC. Moreover, in this study, the upper limit
of the magnetic field is around $1$ GeV owing to the fact that $G(eB)$ is
obtained by fitting lQCD data that is available up to $1~{}\rm{GeV}^{2}$ only.
However, the magnetic field generated at RHIC and LHC is around
$0.2-0.4~{}\rm{GeV}^{2}$. Thus making $1~{}\rm{GeV}^{2}$ a suitable upper
limit. A further theoretical improvement would also enable us to explore the
high $eB$ region. In addition, as mentioned in [51], the dependence of $G$ on
$\mu$ and $\Phi$ has been neglected for simplicity. One can find different
forms of $G$ in use, but for practical purposes, $G$ should be a function of
all the parameters, i.e., $T$, $\mu$, $B$, and $\Phi$, to successfully explain
the hot QCD matter. However, considering that the $B$ dependent coupling
constant contemplated in this work can reproduce lQCD results qualitatively
well, one can proceed with this approximation for further phenomenological
studies to explore the QCD medium.

##  V Acknowledgement

K.G. acknowledges the financial support from the Prime Minister’s Research
Fellowship (PMRF), Government of India. The authors gratefully acknowledge the
DAE-DST, Government of India funding under the mega-science project “Indian
participation in the ALICE experiment at CERN” bearing Project No.
SR/MF/PS-02/2021-IITI(E-37123).

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