# Reconfigurable Intelligent Surface Identification in Mobile Networks:
Opportunities and Challenges

Emre Arslan  Ali Tugberk Dogukan  Fatih Kilinc  Ahmet Faruk Coskun
and Ertugrul Basar  Emre Arslan and Ahmet Faruk Coskun are with 6GEN. Lab of
Turkcell Iletisim Hizmetleri Inc., Maltepe 34854, Istanbul, Turkiye. Ali
Tugberk Dogukan and Fatih Kilinc are with ULAK Communications, Pendik 34906,
Istanbul, Turkiye. Ertugrul Basar is with Communications Research and
Innovation Laboratory (CoreLab), Department of Electrical and Electronics
Engineering, Koc¸ University, Sariyer 34450, Istanbul, Turkiye. Emre Arslan,
Fatih Kilinc and Ali Tugberk Dogukan are also members of CoreLab. Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

The advent of the sixth generation (6G) wireless networks heralds a
transformative era for mobile communication, where the integration of cutting-
edge technologies like Reconfigurable Intelligent Surfaces (RISs) is paramount
in addressing the burgeoning demands for energy efficiency, high data rates,
reliable connectivity, and enhanced coverage in densely populated areas. RISs
have attracted the attention of academia as well as the industry and emerged
as a beacon of innovation, offering a novel paradigm to reconfigure the
wireless propagation environment beneficially, thereby enhancing the overall
network performance. It is envisioned that the deployment of numerous RISs in
a mobile network will serve user equipments (UEs) to boost various key
performance indicators (KPIs), with lower energy consumption compared to
alternative solutions (e.g., relays, and network-controlled repeaters).
However, the knowledge of whether or not a UE is being served through an RIS
and if so, which RIS it is being served by is crucial and beneficial for
various network planning and operational reasons. In this paper, we address
the importance and the benefits of RIS identification in a mobile network.
Additionally, we guide the readers and researchers by introducing alternative
methods to enable RIS identification. Through the lens of this research, we
unveil and shed light to various benefits, the challenges, and future
opportunities in the identification of RISs serving mobile users in complex
network environments, highlighting the necessity for advanced identification
strategies to fully harness the potential of RIS technology in next-generation
wireless systems.

###### Index Terms:

6G, reconfigurable intelligent surfaces, identification, security, mobile
networks.

##  I Introduction

Advancing into the future, sixth-generation (6G) wireless networks are poised
to revolutionize communication technologies with their promise of terabit-per-
second speeds, enhanced privacy, virtually zero latency and ultra-reliability
even in densely populated areas. Numerous unique technologies and their
coexistence such as novel waveform designs, multiple access techniques,
reflective metasurfaces, machine learning (ML), and much more are investigated
and developed to realize these 6G goals. One critical technology for future
wireless networks is the integration and deployment of reconfigurable
intelligent surfaces (RISs) [1]. Gaining popularity for their ability to
control and optimize the wireless propagation environment, RISs consist of
programmable elements that can dynamically alter the phase, amplitude, and
polarization of the incoming electromagnetic waves [2]. RIS technology offers
substantial benefits in terms of network coverage, signal strength, and data
secrecy, improving the overall user experience (UX), as well as system
capacity and energy efficiency. On the other hand, there are many practical
concerns awaiting solutions regarding the installation, integration, and
management of RIS equipment [3]. Among these, the main challenges might be
listed as

  * •

Clarifying practical deployment benefits and drawbacks compared to previous
network equipment like NCRs and RF repeaters,[4],

  * •

Ensuring effective coordination during operation (control by base stations
(BS, RIS itself, or user equipment (UE)),

  * •

Developing optimization algorithms for RIS configuration with or without
cascaded channel state information (CSI).

Despite these hurdles, RIS stands as a cornerstone technology for 6G,
promising to transform wireless communications by enabling smarter and more
efficient network behaviors [5, 6].

RIS is envisioned for extensive deployment by operators in various
environments, from urban centers to rural areas, enabling networks to overcome
physical obstructions and distance limitations. RIS will be useful in many
indoor/outdoor applications, such as optimizing signal distribution in large
public venues and enhancing communication reliability in emergency response
scenarios. This strategic deployment enhances 6G network capacity and
efficiency, paving the way for innovative services leveraging robust
connectivity [7]. Additionally, RIS serves as an anchor for ISAC applications
[8], enhancing their diversity and accuracy, making RIS a versatile asset for
6G technology.

With the increasing deployment of RIS, identifying which RIS a UE interacts
with is critical [9]. RIS identification, determining which RIS is affecting a
user’s signal, is vital for optimizing real-time communication, enhancing
signal quality, network coverage, capacity, security, and localization
accuracy. It also plays a crucial role in network planning, interference
management, handover, and roaming. Accurate RIS identification can
significantly improve KPIs such as throughput, security, reliability, energy
efficiency, and localization in 6G networks. However, this requires
sophisticated detection and communication protocols to ensure accurate RIS
recognition, especially in environments with multiple RISs.

Despite its importance, the domain of RIS identification remains mainly
underexplored. Most existing research focuses predominantly on the design and
optimization of RISs without adequately addressing the challenges associated
with their discovery and (mis)identification in an operational network.
Identifying an RIS involves determining not just its presence but also its
specific influence on the signal path, a task complicated by the passive
nature of most RIS implementations and the subtle variations in the signal
characteristics they induce. As of now, the identification of RISs in wireless
networks remains largely underdeveloped, primarily due to their passive
nature. Unlike active network components such as BSs and repeaters, which
identify themselves through broadcasts like SSIDs, RISs do not inherently
transmit any identification signals. Current approaches proposed for active
network components might leverage signal strength measurements, timing
advances, or complex CSI, which can be cumbersome and imprecise, particularly
in dynamic and cluttered environments. This challenge introduced by the
passive nature of RIS necessitates innovative approaches to it’s
identification, borrowing concepts from technologies like RF identification
(RFID) and near-field communication (NFC), which rely on distinct signals to
recognize tags or devices.

To adapt these strategies for RISs, potential methods include embedding unique
signatures within the phase shifts or amplitude variations imparted on
reflected signals, detectable by receiving devices to identify the specific
RIS. Alternatively, ML algorithms could analyze and classify subtle
modifications made by different RISs based on extensive training with known
configurations. Developing these identification techniques is crucial,
requiring solutions that integrate effectively with the dynamic nature of 6G
networks while maintaining scalability and efficiency.

For RIS identification, approaches can be broadly categorized based on whether
they are implemented on the BS side or directly over-the-air (OTA) by the RIS.
On the BS side, techniques may focus on marking signal patterns transmitted
over the RIS prior to transmission to indicate which surface is in use. This
method offers centralized control, which can simplify management and
optimization but may struggle with scalability in environments dense with
multiple RIS. Alternatively, OTA methods involve receivers detecting unique
modifications in the signal, such as subtle changes in phase or amplitude,
that indicate specific RIS engagement. These techniques allow for more
distributed processing and can scale effectively as network density increases,
although they place higher demands on receiver sensitivity and complexity.
Both approaches aim to balance the identification accuracy with network
performance, each offering distinct advantages and challenges that need
consideration in diverse operational contexts.

Inspired by the potential and popularity of RIS-empowered systems resonated in
academia and industry to realize smart wireless environments, this study
provides a unified RIS identification framework and a thorough exploration of
the identification process, presenting its fundamental principles and
explaining the substantial advantages it offers to wireless communication
networks. Our comprehensive discussion aims to clarify the operational impact
and strategic importance of accurate RIS identification. We introduce three
promising research directions and preliminary techniques for realizing RIS
identification in wireless networks in different domains each with various
issues and considerations. Our discussion is extended with a critical
examination of the prevailing issues and challenges inherent in RIS
identification. Furthermore, we propose potential strategies and future
research avenues, which are crucial for providing a deeper and insightful
perspective on the practical evaluation of RIS-assisted wireless networks.

The remainder of this article is organized as follows. Section II discusses
RIS identification in a mobile network and the benefits it brings. Section III
then introduces some preliminary RIS identification techniques considering
different domains to provide insight and trigger future alternative solutions.
Section IV highlights and outlines some of the research challenges which may
need to also be considered when performing RIS identification. Finally, the
article is concluded in Section V.

Figure 1: Representative sketch for a densely populated cellular environment
utilizing multiple RISs

##  II  RIS Identification

In the intricate urban tapestry where skyscrapers and concrete establishments
disrupt direct communication pathways, RIS-empowered communication emerges as
an enabling technology for enhanced connectivity. The urban scenario depicted
in the Fig. 1 is representative of modern cities where dead zones with poor
signal quality are commonplace. In such zones, the direct line-of-sight (LoS)
to the BS is often obstructed, leading to deteriorated communication
performance, especially for higher frequency bands established by 5G (e.g.,
FR2) and beyond (e.g., FR3 and D-band). Deploying RISs within these areas is a
strategic move to mitigate such connectivity challenges, with each RIS
assigned a unique identification code, enabling not just signal reflection and
refraction but also intelligent environment-aware signal enhancement.

RIS identification becomes paramount when a user’s device enters such a zone
and should first detect if it is being served by an RIS or not. Beyond mere
detection, the device must ascertain the specific RIS from which it is
receiving the signal. This identification process is crucial as it provides
many benefits with the identification information and determines the quality
of service the UEs experiences, which will be discussed in the following
subsections. However, there could be scenarios where the UE is in the coverage
of multiple RISs so called multi-RIS zones. These are regions where the
coverage zones of multiple RISs overlap. In these areas, the signals from
different RISs might intermingle, similar to inter-cell interference, leading
to a scenario where the UE device may struggle to discern which RIS it is
interacting with. This confusion may cause the device to either oscillate
between RISs or, worse, connect to an RIS with a weaker signal or poorer
performance relative to its position. Hence, RIS identification can aid in
detecting whether a UE is served by a RIS and also in distinguishing which RIS
it is served by when in a multi-RIS zone.

RIS identification thus entails an approach where the device must employ
algorithms capable of distinguishing signals and RIS IDs based on their unique
BS or RIS-induced modifications. This may involve detecting subtle differences
in signal phase, power, polarization, or timing introduced by the BS or RIS’s
reflective properties. For 5G and beyond, the integration of RISs is
anticipated to adhere to stringent standards that ensure seamless operability,
optimally-mitigated interference, and maximal efficacy within the network. The
RIS identification process is thereby not only a question of signal
discernment but also one of compliance with the protocols and performance
benchmarks set forth by these advanced communication standards.

RIS identification techniques may assist in detecting multiple RISs and in
such a scenario choosing the RIS that provides optimal performance. In the
context of 5G networks, by identifying the correct RIS, a UE can optimize its
connection to exploit a wider range of 5G’s capabilities, such as high data
rates, higher efficiency, improved reliability, and reduced latency by
directing its beam towards the dedicated RIS.

In dense urban environments, RIS technology enables networks to proactively
address the diverse demands of modern wireless communication. The convergence
of RIS with 5G standards represents the future, where devices actively
participate in a coordinated flow of data, guided by RISs. Detailed
information on beneficial use cases for RIS identification in beyond 5G
wireless communications is provided in the following subsections.

###  II-A Signal Quality Enhancement

RIS identification has a direct impact on enhancing signal quality by allowing
precise control and optimization of the communication path between the UE and
BS. Specifically, when a UE can identify the RIS it is connected to, it can
engage in more effective uplink communication by directing its transmissions
towards the RIS, thus improving the connection quality and efficiency with the
BS. This targeted approach reduces signal degradation and optimizes bandwidth
usage, crucial for high-demand applications. Additionally, UEs can fine-tune
their device settings, such as transmission power and beam
selection/direction, based on the characteristics of the identified RIS,
addressing factors like signal strength and environmental interference.

Moreover, RIS identification may aid in network-wide signal quality
improvement by informing the BS about which RISs are actively serving UEs.
This information allows the network to prioritize resources and adjust
configurations, ensuring that users connected to a particular RIS receive the
best possible signal. In congested network environments, this can mean
dynamically redistributing network load or enhancing signal fidelity for users
in high-density areas, leading to a more balanced and effective network
performance.

###  II-B Optimized Security and Secrecy

In the wireless communication field, RISs have revolutionized not only signal
optimization but also communication security and secrecy [10]. Identifying the
specific RIS serving a UE can enhance security in wireless networks, allowing
for dynamic adaptation to threats and enabling secure operations for sensitive
services like e-commerce and data exchange. For example, in high-security
zones, devices can activate enhanced security measures, such as advanced
encryption and authentication, based on their RIS connectivity, thus
fortifying the network against unauthorized access and eavesdropping.
Furthermore, UEs may direct signals to the identified RIS to reduce
interception risks, safeguarding the legitimate data path. Moreover, the
unique ability of RIS to manipulate signal phases and amplitudes adds a layer
of physical layer security, concealing communications from eavesdroppers and
representing a significant step toward proactive security measures in wireless
networks [11, 12].

###  II-C Energy Efficiency

The integration of RIS in mobile networks could significantly enhance energy
efficiency by enabling precise RIS identification and interaction. As
illustrated in Fig. 1, when the UE identifies which RIS it is served through
the UE can perform directed beamforming hence, focusing energy use on
necessary paths toward the RIS rather than less efficient non-line-of-sight
(NLOS) or omnidirectional transmissions, thus reducing power wastage. Devices
can adjust their power output based on the RIS coverage, leading to battery
conservation and sustained service quality, especially crucial in mobile
environments. RIS identification also aids in network-wide energy management,
aiding operators to determine the frequently used RISs, allocate resources
intelligently and activate RIS panels as needed hence, enhancing network
efficiency and contributing to environmental sustainability.

###  II-D Localization and Sensing

One of the most prominent benefits of RIS identification complementary to the
6G vision is the enhancement of localization in wireless networks by enabling
UEs to determine their rough location based on the specific RIS connected to.
This process allows for a granular level of location awareness, beneficial in
environments where traditional GPS struggles, such as densely built-up urban
areas or large and complex indoor spaces like malls and airports.

For example, by identifying the RIS that assists the nearby region, a UE can
ascertain that it is within a particular zone of a shopping center, enabling
tailored navigation services and location-specific information such as store
promotions or directory assistance. This identification can also streamline
emergency responses; when a user can pinpoint and communicate their location
via the connected RIS, emergency services can rapidly locate and reach the
individual, reducing response times and potentially saving lives in case of
disasters and possible casualties. As illustrated in Fig. 1, when the user
served by RIS 3 identifies the RIS, it can obtain rough information that it is
in the orange coverage area and have location awareness. Additionally, from a
time difference of arrival localization perspective, identifying all
surrounding RISs also might be utilized for triangulation purposes in case of
coordinated RIS deployments. This level of service personalization not only
improves user engagement but also opens avenues for innovative service
offerings in various sectors, making RIS identification a cornerstone of next-
generation wireless communication systems that aim to be more immersive,
efficient, and user-focused.

###  II-E Smarter Interference Management

Intelligent interference management, enabled by accurate RIS identification,
allows users to actively participate in optimizing network performance. By
recognizing the serving RIS, devices can dynamically adjust their transmission
parameters, such as power levels and beam directions, to minimize inter-cell
interference. This real-time adaptation is crucial in environments with
multiple RISs, where the potential for signal conflict is high. Targeted
communication towards the identified RIS reduces unnecessary signal spread,
avoiding interference with other users in the vicinity. This approach not only
improves the individual user experience by ensuring more reliable connections
but also enhances the overall network efficiency by reducing cross-
interference among users connected to adjacent RISs.

##  III RIS Identification Techniques

This section briefly introduces preliminary and pioneering identification
approaches for RISs in wireless networks in successive subsections. The
following proposed identification methods are briefly summarized in Table I.

TABLE I: Comparison of Proposed Identification Techniques Method | Domain | Computation Complexity | Precision/Robustness | Waveform
---|---|---|---|---
Amplitude Modulation Sequence | Time | Low | Low | Single-/Multi-carrier
Spectral Fingerprinting | Frequency | Moderate | Moderate | Multi-carrier
BS Side Watermarking | Code and Spatial | High | High | Single-/Multi-carrier

###  III-A Amplitude Modulation Sequence

The principle behind this technique involves the alteration of the impinging
signal’s amplitude by the RIS to embed a recognizable pattern. Specifically,
the RIS can selectively boost or attenuate the signal’s strength by either
coherently and destructively manipulating the phases of its elements or
switching off a subset of its elements at predetermined intervals, thereby
reducing the signal reflection or redirection during those periods. This
creates a signature in the signal’s amplitude that can be detected and decoded
by the receiver. This can be applied to both single-carrier and multi-carrier
systems.

Fig. 2 illustrates this concept by depicting multiple RIS units, each
modulating the signal’s amplitude in a distinct manner, creating identifiable
power variations at UEs’ receivers. These variations are encoded into the
signal as unique identifiers, enabling each UE to discern and authenticate the
RIS in communication, thereby ensuring a secure and efficient network
operation.

Figure 2: Illustration of amplitude modulation sequence technique for RIS
identification.

###  III-B Spectral Fingerprinting

The spectral fingerprinting method provides a refined approach that leverages
the frequency domain characteristics of a multi-carrier signal. The central
premise of this method is the strategic enhancement of specific subcarriers or
subsets by a specific RIS for identification. The RISs can adjust themselves
according to the channels of certain subcarriers and boost them. By doing so,
each RIS imprints a distinct spectral signature on the signal that can be
easily recognized and distinguished at the receiver end with subcarriers’
orthogonality feature. This is depicted in Fig. 3, where the power
distribution across subcarriers is noticeably altered at each UE after the
reception of all existing RIS-assisted signals.

In the scenario sketched in Fig. 3, based on the the instant location of the
UEs and the existing RISs, the frequency spectrum given for each UE includes
information about the existing RISs’ dominance. The peaks corresponding to the
intensity of RIS signatures help determine and manage the supporting RISs for
each UE. The presence of distinct peaks correlates to the respective RISs
active during transmission. Through careful analysis of these power
distributions, the UE can discern which RISs have been engaged and, by
extension, infer their relative efficacy and proximity. This RIS
identification approach might momentarily face misidentifications due to
spectral notches that can occur in frequency-selective propagation
environments, hence parameters such as number of symbols to consider for the
identification process, bandwidth and size of subcarrier grouping for spectral
fingerprinting may be considered to increase robustness of the system.

Figure 3: Illustration of spectral fingerprinting technique for RIS
identification

###  III-C BS Watermarking

This core concept revolves around the BS embedding a unique identification
sequence within the single or multi-carrier signals directed towards the
various RISs, as depicted in the Fig. 4. The BS crafts these signals,
$x_{1},x_{2},$ and $x_{3}$, with distinct characteristics, akin to a digital
watermark. This could be a set of orthogonal codes, cyclic shifts, or specific
phase shifts, each serving as a beacon for the UE to recognize the signal’s
origin and path. Such distinct modifications are not merely arbitrary; they
are meticulously designed sequences with exceptional auto-correlation and poor
cross-correlation properties, ensuring minimal interference and optimal path
discernment by the UE. When the BS broadcasts these signals, the UE’s task is
to correlate the received sequences and pinpoint the exact RIS through which
the signal was reflected. This is similar to a detective matching fingerprints
at a crime scene, each unique sequence leads back to a specific RIS, allowing
the UE to ’localize’ its communication pathway.

Fig. 4 illustrates this concept where the BS is the signal origin, with each
RIS reflecting these signals to the UE. The unique identification sequences
encoded within the signals are the critical factor enabling each UE to discern
and trace the signal path back to its RIS of origin. The type and length of
the sequences are important when designing such a system. This not only
enhances the accuracy of localization systems but also adds a layer of
security and efficiency in network communication, as each RIS can be
distinctly and accurately identified by its ’watermarked’ signal.

In Fig. 5 we provide computer simulation results to demonstrate the effects
the parameters in the system has on the identification performance. For this
simulation, there are three RISs with varying number of elements $N$ where the
BS communicates with the UE through only the RISs under a single carrier
system. Considering a coordinate plane the BS, RIS 1, RIS 2, RIS 3, and UE are
located at (0,0), (15,0), (0,20), (20,17), (8,17), respectively. Using Gold
sequences with varying lengths and QPSK modulation, we assume that there is
Rayleigh fading channel and the UE receives the signal from all RISs while RIS
2 is the primary serving RIS. It can be clearly observed that as the code
length and number of RIS elements ($N$) increases, the misidentification rate
decreases hence, a better identification performance.

Figure 4: Illustration of BS watermarking technique for RIS identification.

###  III-D Further possible methods

Further RIS identification methods can use the effects of element spacing in
RIS hardware. The angular width of the main lobe, grating lobes, and sidelobe
levels can create unique patterns detectable at the receiver. The frequency
response, influenced by element spacing, may also show distinct patterns
across frequencies. Additionally, frequency offsets or harmonics from temporal
modulation can serve as unique watermarks for secure communications.
Implementing these methods requires extensive data collection and complex
algorithms but offers promising avenues for precise RIS identification.
However, these attributes present implementation challenges compared to
previous methods. Obtaining these attributes requires comprehensive data
collection and a wide range of equipment. Utilizing them necessitates
cooperation between UEs and more complex identification algorithms.

Figure 5: Performance of BS watermarking technique for varying parameters.

##  IV Future Directions and Open Challenges

As wireless networks evolve, the expected density and scalability of RIS
deployment are expected to significantly increase [13]. Managing and
identifying a large number of RIS panels, especially in dense urban
environments or large indoor spaces, poses a substantial challenge. Ensuring
that each RIS can be uniquely identified without creating an overly complex or
resource-intensive process is crucial. The scalability challenge also extends
to the dynamic nature of wireless environments, where the configuration of RIS
panels may need to change frequently to adapt to varying network conditions
and user demands. It is imperative to address the mechanics of the
identification process, its timing, and the strategic implementation to ensure
seamless network integration and optimal performance. The following
subsections introduce some prominent future directions and open challenges
regarding the identification of RISs.

###  IV-A Identification Period

Determining an optimal RIS identification period presents technical challenges
and opportunities for innovation in wireless networks. The identification
period, or time allocated for a device to identify the RIS, is crucial for
supporting various operational requirements and balancing identification
reliability with overhead. The decision to implement a distinct identification
period requires careful consideration of its benefits and drawbacks, as well
as exploring alternatives for seamless integration without sacrificing
performance.

RIS identification can be done with a dedicated period or simultaneously with
data transmission. A dedicated period allows for focused assessment of signal
characteristics and the RIS’s impact, leading to better decisions on
beamforming, power allocation, and adaptive mechanisms. However, it consumes
valuable time that could be used for data transmission, especially during
high-demand scenarios, potentially increasing latency and reducing throughput.
The dynamic nature of mobile environments means the optimal identification
period may vary, requiring adaptive strategies that complicate network
management.

###  IV-B Misidentification

When a device erroneously identifies a non-serving RIS as its source of signal
enhancement, the primary concern is the potential for reduced user and network
performance. This is particularly problematic in border areas between RIS
zones called multi-RIS zones, where signals from adjacent RISs might overlap.
In such cases, a device might receive conflicting signal enhancements, leading
to a degradation in signal quality and overall communication reliability.
Moreover, false RIS detection can result in suboptimal resource allocation,
where network resources are wasted on attempting to optimize a connection with
an incorrect RIS, thereby reducing the network’s overall efficiency. Some
strategies such as enhanced signal processing techniques to distinguish
reflected signals, strategically planning the deployment of RISs in a network
to mitigate the risk of false detection and/or misidentification, ML-based
solutions, and user feedback mechanisms may aid and are future directions in
the misidentification problem.

###  IV-C Optimal Identification Domain and Technique

The choice of domain for RIS identification (time, power, frequency, or
spatial) significantly influences the system’s efficiency and reliability.
Time-domain identification, for instance, may involve specific time slots
dedicated to transmitting RIS IDs, potentially impacting data throughput due
to the allocation of resources for identification purposes. Power-domain
strategies could involve varying the power levels to encode RIS IDs, posing
challenges in dynamic environments where signal strength can fluctuate due to
mobility or obstacles. Frequency-domain identification offers the advantage of
embedding IDs within certain frequency bands, though this requires careful
spectrum management to avoid interference with data transmission. Spatial
domain identification, leveraging the unique spatial signatures that RIS can
impart on signals, promises a less intrusive method, though it demands
sophisticated signal processing techniques.

As we advance, the exploration of multi-domain strategies that combine time,
power, frequency, and spatial characteristics for RIS identification appears
promising. Such an approach could leverage the strengths of each domain while
mitigating their individual limitations. It should also be noted that the
optimal identification domain and technique may vary in single and multi-RIS
zones.

###  IV-D Other Challenges

Developing RIS identification techniques that seamlessly operate within
current and future wireless standards without extensive modifications is
critical. These approaches should be scalable and backward compatible to meet
the short-term and long-term needs of various wireless networks in real-time,
reducing the integration burden on beyond 5G networks. Key considerations
include:

  1. 1.

Environmental and Physical Limitations: Factors such as multipath propagation,
physical obstructions, and varying atmospheric conditions can affect the
reliability and accuracy of RIS identification. Developing robust
identification techniques that can adapt to or overcome these environmental
challenges is essential for ensuring consistent network performance.

  2. 2.

Security and Privacy Concerns: It is of great importance to ensure that the
process of identifying RIS panels does not become a vector for malicious
activities or privacy breaches. Techniques for secure RIS identification and
data transmission must be developed, alongside measures to protect against
spoofing attacks, where an adversary might mimic an RIS ID to disrupt network
operations or intercept communications.

  3. 3.

Interference Management and Network Coexistence: Developing strategies for
interference mitigation and ensuring coexistence among multiple RISs, while
maintaining optimal network performance, requires innovative approaches to
signal processing and network management.

  4. 4.

Compatibility for Network Automation: These techniques must be capable of
dynamically adjusting to ensure optimal performance without necessitating
constant manual intervention or recalibration.

  5. 5.

Minimal Overhead Footprint: Ensuring that RIS identification and operation do
not adversely affect the user experience or degrade service quality is a
significant challenge. Techniques for RIS identification must be efficient and
unobtrusive, minimizing any potential disruption to data transmission or
delays in service delivery.

Addressing these challenges requires a multidisciplinary approach, leveraging
advancements in signal processing, ML, network theory, and cybersecurity.
Collaboration among academia, industry, and standardization bodies will be the
key for in developing innovative solutions that overcome these hurdles, paving
the way for the successful integration of RIS technology into future wireless
networks.

##  V Conclusion

In this article, we have provided comprehensive discussions and a framework
for RIS identification for the first time in the literature. Fundamental
principles and advantages in wireless networks and the operational impact and
strategic importance of RIS identification have been discussed. In addition,
we have discussed the issues and challenges inherent in RIS identification and
propose potential strategies and future research avenues for realizing RIS
identification in different domains, each with various issues and
considerations.

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(S47) $\displaystyle W_{ii\,jk}$
$\displaystyle=W_{ii\,kj}=W_{kj\,ii}=-W_{ik\,ji}=\,\partial_{j}\,\partial_{k}\,{\epsilon}_{ii}+\partial_{i}^{2}\,{\epsilon}_{jk}-\partial_{i}\,\partial_{j}\,{\epsilon}_{ik}-\partial_{i}\,\partial_{k}\,{\epsilon}_{ij}\,,$

which explicitly are $W_{11\,22}$, $W_{22\,33}$, $W_{33\,11}$, $W_{11\,23}$,
$W_{22\,31}$ and $W_{33\,12}$, leading to

$\displaystyle\partial_{2}^{2}\,{\epsilon}_{11}(\mathbf{r})+\partial_{1}^{2}\,{\epsilon}_{22}(\mathbf{r})-2\,\partial_{1}\partial_{2}\,{\epsilon}_{12}(\mathbf{r})$
$\displaystyle=0\,,$ (S48)
$\displaystyle\partial_{3}^{2}\,{\epsilon}_{22}(\mathbf{r})+\partial_{2}^{2}\,{\epsilon}_{33}(\mathbf{r})-2\,\partial_{2}\partial_{3}\,{\epsilon}_{23}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{1}^{2}\,{\epsilon}_{33}(\mathbf{r})+\partial_{3}^{2}\,{\epsilon}_{11}(\mathbf{r})-2\,\partial_{3}\partial_{1}\,{\epsilon}_{31}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{2}\,\partial_{3}\,{\epsilon}_{11}(\mathbf{r})+\partial_{1}^{2}\,{\epsilon}_{23}(\mathbf{r})-\partial_{3}\partial_{1}\,{\epsilon}_{12}(\mathbf{r})-\partial_{1}\,\partial_{2}\,{\epsilon}_{31}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{3}\,\partial_{1}\,{\epsilon}_{22}(\mathbf{r})+\partial_{2}^{2}\,{\epsilon}_{31}(\mathbf{r})-\partial_{2}\partial_{3}\,{\epsilon}_{12}(\mathbf{r})-\partial_{1}\,\partial_{2}\,{\epsilon}_{23}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{1}\,\partial_{2}\,{\epsilon}_{33}(\mathbf{r})+\partial_{3}^{2}\,{\epsilon}_{12}(\mathbf{r})-\partial_{3}\partial_{1}\,{\epsilon}_{23}(\mathbf{r})-\partial_{2}\,\partial_{3}\,{\epsilon}_{31}(\mathbf{r})$
$\displaystyle=0\,.$

In reality the last 3 are automatically satisfied if the first 3 are.
If we assume a C-plane oriented film geometry, the strain is constant in $z$,
so that the derivatives with respect to it vanish, and the compatibility
conditions simplify into

$\displaystyle\partial_{2}^{2}\,{\epsilon}_{11}(\mathbf{r})+\partial_{1}^{2}\,{\epsilon}_{22}(\mathbf{r})-2\,\partial_{1}\partial_{2}\,{\epsilon}_{12}(\mathbf{r})$
$\displaystyle=0\,,$ (S49)
$\displaystyle\partial_{2}^{2}\,{\epsilon}_{33}(\mathbf{r})=\partial_{1}^{2}\,{\epsilon}_{33}(\mathbf{r})=\partial_{1}\,\partial_{2}\,{\epsilon}_{33}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{1}^{2}\,{\epsilon}_{23}(\mathbf{r})-\partial_{1}\,\partial_{2}\,{\epsilon}_{31}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{1}\,\partial_{2}\,{\epsilon}_{23}(\mathbf{r})-\partial_{2}^{2}\,{\epsilon}_{31}(\mathbf{r})$
$\displaystyle=0\,,$

with $\mathbf{r}$ now being the planar coordinate, which, through Eq. (S36),
can be written as

$\displaystyle\Big{(}\partial_{1}^{2}-\partial_{2}^{2}\Big{)}\,\varepsilon_{1,1}(\mathbf{r})+2\,\partial_{1}\,\partial_{2}\,\varepsilon_{1,2}(\mathbf{r})$
$\displaystyle=\nabla^{2}\,{\epsilon}(\mathbf{r})\,,$ (S50)
$\displaystyle\partial_{2}^{2}\,{\epsilon}_{33}(\mathbf{r})=\partial_{1}^{2}\,{\epsilon}_{33}(\mathbf{r})=\partial_{1}\,\partial_{2}\,{\epsilon}_{33}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{1}\,\Big{(}\partial_{1}\,\varepsilon_{2,2}(\mathbf{r})-\partial_{2}\,\varepsilon_{2,1}(\mathbf{r})\Big{)}=\partial_{1}\,{\boldsymbol{\nabla}}\wedge{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$
$\displaystyle=0\,,$
$\displaystyle\partial_{2}\,\Big{(}\partial_{1}\,\varepsilon_{2,2}(\mathbf{r})-\partial_{2}\,\varepsilon_{2,1}(\mathbf{r})\Big{)}=\partial_{2}\,{\boldsymbol{\nabla}}\wedge{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$
$\displaystyle=0\,.$

They are satisfied by a constant
${\epsilon}_{33}(\mathbf{r})={\epsilon}_{33}$, $\forall\,\mathbf{r}$, a curl
free ${\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$, i.e.,

$\displaystyle{\boldsymbol{\nabla}}\wedge{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$
$\displaystyle=0\,,$ (S51)

and, finally, by the constraint

$\displaystyle\Big{(}\partial_{1}^{2}-\partial_{2}^{2}\Big{)}\,\varepsilon_{1,1}(\mathbf{r})+2\,\partial_{1}\,\partial_{2}\,\varepsilon_{1,2}(\mathbf{r})$
$\displaystyle=\nabla^{2}\,{\epsilon}(\mathbf{r})\,,$ (S52)

which can be enforced adding a Lagrange multiplier and the following term to
the elastic energy, written for a generic non-uniform strain

$\displaystyle\delta E_{\text{compatibility}}$ $\displaystyle=\iint
d\mathbf{r}\,d\mathbf{r}^{\prime}\,\lambda(\mathbf{r})\,\bigg{[}\;R_{1}(\mathbf{r}-\mathbf{r}^{\prime})\,\varepsilon_{1,1}(\mathbf{r}^{\prime})+R_{2}(\mathbf{r}-\mathbf{r}^{\prime})\,\varepsilon_{1,2}(\mathbf{r}^{\prime})-\delta\big{(}\mathbf{r}-\mathbf{r}^{\prime}\big{)}\,{\epsilon}(\mathbf{r}^{\prime})\;\bigg{]}$
(S53)
$\displaystyle=\frac{\displaystyle\;1\;}{\displaystyle\;V\;}\sum_{\mathbf{k}}\,\lambda(-\mathbf{k})\,\Bigg{[}\;\frac{\displaystyle\;\;k_{1}^{2}-k_{2}^{2}\;\;}{\displaystyle\;k_{1}^{2}+k_{2}^{2}\;}\;\varepsilon_{1,1}(\mathbf{k})+\frac{\displaystyle\;\;2\,k_{1}\,k_{2}\;\;}{\displaystyle\;\;k_{1}^{2}+k_{2}^{2}\;\;}\;\varepsilon_{1,2}(\mathbf{k})-{\epsilon}(\mathbf{k})\;\Bigg{]}\,.$

Specifically, for a two dimensional coordinate
$\mathbf{r}=r\,\big{(}\cos\phi,\sin\phi\big{)}$, and recalling the following
equalities involving the Bessel functions

$\displaystyle\text{e}^{ix\cos\theta}$ $\displaystyle=J_{0}(x)+2\,\sum_{n\geq
1}\,i^{n}\,J_{n}(x)\,\cos n\theta\,,$
$\displaystyle\int_{0}^{\infty}dx\,J_{n}(x)$ $\displaystyle=1\,,$ (S54)
$\displaystyle 2n\,J_{n}(x)$ $\displaystyle=x\,J_{n-1}(x)+x\,J_{n+1}(x)\,,$
$\displaystyle x\,J_{0}(x)$
$\displaystyle=\frac{\displaystyle\;d\;}{\displaystyle\;dx\;}\,\Big{[}x\,J_{1}(x)\Big{]}\,,$

we find

$\displaystyle R_{1}(\mathbf{r})$
$\displaystyle=\int_{0}^{\infty}\frac{\displaystyle\;k\,dk\;}{\displaystyle\;4\pi^{2}\;}\int_{0}^{2\pi}\\!d\theta\,\text{e}^{ikr\,\cos(\theta-\phi)}\;\cos
2\theta=\int_{0}^{\infty}\frac{\displaystyle\;k\,dk\;}{\displaystyle\;4\pi^{2}\;}\int_{0}^{2\pi}\\!d\theta\,\text{e}^{ikr\,\cos\theta}\;\cos
2\big{(}\theta+\phi\big{)}$ (S55)
$\displaystyle=\int_{0}^{\infty}\frac{\displaystyle\;k\,dk\;}{\displaystyle\;4\pi^{2}\;}\int_{0}^{2\pi}\\!d\theta\,\text{e}^{ikr\,\cos\theta}\;\cos
2\theta\,\cos 2\phi$ $\displaystyle=-2\cos
2\phi\,\int_{0}^{\infty}\frac{\displaystyle\;k\,dk\;}{\displaystyle\;4\pi^{2}\;}\,J_{2}(kr)\,\int_{0}^{2\pi}d\theta\,\cos^{2}2\theta=-\frac{\displaystyle\;\cos
2\phi\;}{\displaystyle\;\;2\pi\,r^{2}\;\;}\,\int_{0}^{\infty}\\!dx\,x\,J_{2}(x)$
$\displaystyle=-\frac{\displaystyle\;\cos
2\phi\;}{\displaystyle\;\;2\pi\,r^{2}\;\;}\,\int_{0}^{\infty}\\!dx\,\Big{(}\,2J_{1}(x)-x\,J_{0}(x)\Big{)}=-\frac{\displaystyle\;\cos
2\phi\;}{\displaystyle\;\;\pi\,r^{2}\;\;}\;,$ $\displaystyle
R_{2}(\mathbf{r})$
$\displaystyle=\int_{0}^{\infty}\frac{\displaystyle\;k\,dk\;}{\displaystyle\;4\pi^{2}\;}\int_{0}^{2\pi}\\!d\theta\,\text{e}^{ikr\,\cos(\theta-\phi)}\;\sin
2\theta=\int_{0}^{\infty}\frac{\displaystyle\;k\,dk\;}{\displaystyle\;4\pi^{2}\;}\int_{0}^{2\pi}\\!d\theta\,\text{e}^{ikr\,\cos\theta}\;\sin
2\big{(}\theta+\phi\big{)}$ $\displaystyle=-\frac{\displaystyle\;\sin
2\phi\;}{\displaystyle\;\;\pi\,r^{2}\;\;}\;,$

Figure S3: Possible interfaces

Equation (S51) implies instead that, if
${\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$ changes abruptly across an
interface, its component tangential to it must be continuous, which leads to
the allowed interfaces in Fig. S3.

###  S5.3 Landau-Ginzburg functional

Since ${\epsilon}_{33}={\epsilon}_{3}$ is constant in $\mathbf{r}$, we shall
discard it in the following analysis. We then add to the elastic energy Eq.
(S44) the term (S53) that enforces the compatibility equations, and integrate
out all strain components but the order parameter
${\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$.

The energy functionals of ${\epsilon}(\mathbf{r})$ and
${\boldsymbol{{\epsilon}}}_{1}(\mathbf{r})$ can be written as

$\displaystyle E\big{[}{\epsilon}\big{]}$ $\displaystyle=\int
d\mathbf{r}\Bigg{\\{}\;\big{(}c_{11}+c_{12}\big{)}\,{\epsilon}(\mathbf{r})^{2}-\Lambda(\mathbf{r})\,{\epsilon}(\mathbf{r})\;\Bigg{\\}}\,,$
(S56) $\displaystyle E\Big{[}{\boldsymbol{{\epsilon}}}_{1}\Big{]}$
$\displaystyle=\int
d\mathbf{r}\Bigg{\\{}\;\big{(}c_{11}-c_{12}\big{)}\,{\boldsymbol{{\epsilon}}}_{1}(\mathbf{r})\cdot{\boldsymbol{{\epsilon}}}_{1}(\mathbf{r})+\Big{(}\Lambda_{1}(\mathbf{r})\,\varepsilon_{1,1}(\mathbf{r})+\Lambda_{2}(\mathbf{r})\,\varepsilon_{1,2}(\mathbf{r})\Big{)}\;\Bigg{\\}}\,,$

where

$\displaystyle\Lambda(\mathbf{r})$
$\displaystyle=\lambda(\mathbf{r})+2g\,{\epsilon}_{2}(\mathbf{r})^{2}\,,$
(S57) $\displaystyle\Lambda_{1}(\mathbf{r})$ $\displaystyle=\int
d\mathbf{r}^{\prime}\,\lambda(\mathbf{r}^{\prime})\,R_{1}(\mathbf{r}^{\prime}-\mathbf{r})+4c_{14}\,\varepsilon_{2,2}(\mathbf{r})+g\,\Big{(}\varepsilon_{2,1}(\mathbf{r})^{2}-\varepsilon_{2,2}(\mathbf{r})^{2}\Big{)}\,,$
$\displaystyle\Lambda_{2}(\mathbf{r})$ $\displaystyle=\int
d\mathbf{r}^{\prime}\,\lambda(\mathbf{r}^{\prime})\,R_{2}(\mathbf{r}^{\prime}-\mathbf{r})+4c_{14}\,\varepsilon_{2,1}(\mathbf{r})+2g\,\varepsilon_{2,1}(\mathbf{r})\,\varepsilon_{2,2}(\mathbf{r})\,,$

Integrating out ${\epsilon}$ and ${\boldsymbol{{\epsilon}}}_{1}$ leads to

$\displaystyle\delta_{\epsilon}E\Big{[}{\boldsymbol{{\epsilon}}}_{2},\lambda\Big{]}$
$\displaystyle=-\frac{\displaystyle\;1\;}{\displaystyle\;\;4\,\big{(}c_{11}+c_{12}\big{)}\;\;}\,\int
d\mathbf{r}\,\Lambda(\mathbf{r})^{2}\,,$ (S58)
$\displaystyle\delta_{{\boldsymbol{{\epsilon}}}_{1}}E\Big{[}{\boldsymbol{{\epsilon}}}_{2},\lambda\Big{]}$
$\displaystyle=-\frac{\displaystyle\;1\;}{\displaystyle\;\;4\,\big{(}c_{11}-c_{12}\big{)}\;\;}\,\int
d\mathbf{r}\,\bigg{[}\;\Lambda_{1}(\mathbf{r})^{2}+\Lambda_{2}(\mathbf{r})^{2}\;\bigg{]}\,,$

where, explicitly,

$\displaystyle\Lambda(\mathbf{r})^{2}$
$\displaystyle=\lambda(\mathbf{r})^{2}+4g^{2}\,{\epsilon}_{2}(\mathbf{r})^{4}+4g\,\lambda(\mathbf{r})\,{\epsilon}_{2}(\mathbf{r})^{2}\,,$
(S59) $\displaystyle\Lambda_{1}(\mathbf{r})^{2}+\Lambda_{2}(\mathbf{r})^{2}$
$\displaystyle=\lambda(\mathbf{r})^{2}+16\,c_{14}\,{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})+g^{2}\,{\epsilon}_{2}(\mathbf{r})^{4}+8\,c_{14}\,g\,{\epsilon}_{2}(\mathbf{r})^{3}\,\sin
3\phi_{2}(\mathbf{r})$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\int
d\mathbf{r}^{\prime}\,\lambda(\mathbf{r}^{\prime})\,\Bigg{\\{}\;R_{2}(\mathbf{r}^{\prime}-\mathbf{r})\,\bigg{[}8c_{14}\,\varepsilon_{2,1}(\mathbf{r})+4g\,\varepsilon_{2,1}(\mathbf{r})\,\varepsilon_{2,2}(\mathbf{r})\bigg{]}$
$\displaystyle\qquad\qquad\quad+R_{1}(\mathbf{r}^{\prime}-\mathbf{r})\,\bigg{[}8c_{14}\,\varepsilon_{2,2}(\mathbf{r})+2g\,\Big{(}\varepsilon_{2,1}(\mathbf{r})^{2}-\varepsilon_{2,2}(\mathbf{r})^{2}\Big{)}\bigg{]}\;\Bigg{\\}}\,.$

The Lagrange multiplier is determined through the saddle point of the
functional

$\displaystyle F\big{[}\lambda\big{]}$ $\displaystyle=\int
d\mathbf{r}\,\Bigg{\\{}\;-\frac{\displaystyle\;1\;}{\displaystyle\;2\;}\,\frac{\displaystyle\;c_{11}\;}{\displaystyle\;\;c_{11}^{2}-c_{12}^{2}\;\;}\;\lambda(\mathbf{r})^{2}-\Omega(\mathbf{r})\,\lambda(\mathbf{r})\;\Bigg{\\}}\,,$
(S60)

where, by definition,

$\displaystyle\Omega(\mathbf{r})$
$\displaystyle=\frac{\displaystyle\;g\;}{\displaystyle\;\;c_{11}+c_{12}\;\;}\;{\epsilon}_{2}(\mathbf{r})^{2}$
(S61)
$\displaystyle\quad+\frac{\displaystyle\;1\;}{\displaystyle\;\;4\big{(}c_{11}-c_{12}\big{)}\;\;}\,\int
d\mathbf{r}^{\prime}\,\Bigg{\\{}\;R_{2}(\mathbf{r}-\mathbf{r}^{\prime})\,\bigg{[}8c_{14}\,\varepsilon_{2,1}(\mathbf{r}^{\prime})+4g\,\varepsilon_{2,1}(\mathbf{r}^{\prime})\,\varepsilon_{2,2}(\mathbf{r}^{\prime})\bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+R_{1}(\mathbf{r}-\mathbf{r}^{\prime})\,\bigg{[}8c_{14}\,\varepsilon_{2,2}(\mathbf{r}^{\prime})+2g\,\Big{(}\varepsilon_{2,1}(\mathbf{r}^{\prime})^{2}-\varepsilon_{2,2}(\mathbf{r}^{\prime})^{2}\Big{)}\bigg{]}\;\Bigg{\\}}$
$\displaystyle\equiv\frac{\displaystyle\;g\;}{\displaystyle\;\;c_{11}+c_{12}\;\;}\;{\epsilon}_{2}(\mathbf{r})^{2}+\Omega_{R}(\mathbf{r})\;,$

yielding the following contribution to the energy functional of
${\boldsymbol{{\epsilon}}}_{2}$

$\displaystyle\delta E_{\lambda}\big{[}{\boldsymbol{{\epsilon}}}_{2}\big{]}$
$\displaystyle=\frac{\displaystyle\;\;c_{11}^{2}-c_{12}^{2}\;\;}{\displaystyle\;2c_{11}\;}\,\int
d\mathbf{r}\,\Omega(\mathbf{r})^{2}=\frac{\displaystyle\;\;g^{2}\big{(}c_{11}-c_{12}\big{)}\;\;}{\displaystyle\;\;2c_{11}\big{(}c_{11}+c_{12}\big{)}\;\;}\,\int
d\mathbf{r}\,{\epsilon}_{2}(\mathbf{r})^{4}$ (S62)
$\displaystyle\qquad+\frac{\displaystyle\;\;g\big{(}c_{11}-c_{12}\big{)}\;\;}{\displaystyle\;c_{11}\;}\,\int
d\mathbf{r}\,{\epsilon}_{2}(\mathbf{r})^{2}\,\Omega_{R}(\mathbf{r})+\frac{\displaystyle\;\;c_{11}^{2}-c_{12}^{2}\;\;}{\displaystyle\;2c_{11}\;}\,\int
d\mathbf{r}\,\Omega_{R}(\mathbf{r})^{2}\;.$

The last term favours the growth of domains [61]. In its presence, the
amplitude of the strain
${\epsilon}_{2}(\mathbf{r})=\big{|}{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\big{|}$
would be constant everywhere but at the interfaces between domains, where it
would vanish. Therefore, we can safely neglect the term
$\propto\int{\epsilon}_{2}(\mathbf{r})^{2}\,\Omega_{R}(\mathbf{r})$, since
either ${\epsilon}_{2}(\mathbf{r})$ is zero, or, when it is constant, it is
the angular average of the kernels $R_{1}(\mathbf{r})$ and $R_{2}(\mathbf{r})$
that makes the integral to vanish.

Collecting all terms, and further adding a stiffness , we finally obtain the
Landau-Ginzburg functional of the order parameter

$\displaystyle E\Big{[}{\boldsymbol{{\epsilon}}}_{2}\Big{]}$
$\displaystyle=\int
d\mathbf{r}\,\Bigg{\\{}\;-\frac{\displaystyle\;K\;}{\displaystyle\;2\;}\;{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot\nabla^{2}{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})+\tau\,{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})+\frac{\displaystyle\;\;c_{11}^{2}-c_{12}^{2}\;\;}{\displaystyle\;2c_{11}\;}\;\Omega_{R}(\mathbf{r})^{2}$
(S63)
$\displaystyle\qquad\qquad\qquad-\gamma\,{\epsilon}_{2}(\mathbf{r})^{3}\,\sin
3\phi_{2}(\mathbf{r})+\mu\,\Big{(}{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\Big{)}^{2}\;\Bigg{\\}}\,,$

where $\mu>0$ and $\gamma\propto c_{14}\,g$. The quadratic coupling constant
$\tau$ is positive above the transition, and negative below. We mentioned that
$\gamma>0$ below the transition, but negative, and small in absolute value,
quite above it, possibly vanishing at or right before the transition.

Let us now analyse in detail the term $\propto\int\Omega_{R}(\mathbf{r})^{2}$.
Through (S61), $\Omega_{R}(\mathbf{r})$ involves the two combinations

$\displaystyle\Phi_{1}(\mathbf{r})$
$\displaystyle=8c_{14}\,\varepsilon_{2,1}(\mathbf{r})+4g\,\varepsilon_{2,1}(\mathbf{r})\,\varepsilon_{2,2}(\mathbf{r})=8c_{14}\,{\epsilon}_{2}(\mathbf{r})\,\cos\phi_{2}(\mathbf{r})+2g\,{\epsilon}_{2}(\mathbf{r})^{2}\,\sin
2\phi_{2}(\mathbf{r})\,,$ (S64) $\displaystyle\Phi_{2}(\mathbf{r})$
$\displaystyle=8c_{14}\,\varepsilon_{2,2}(\mathbf{r})+2g\Big{(}\varepsilon_{2,1}(\mathbf{r})^{2}-\varepsilon_{2,2}(\mathbf{r})^{2}\Big{)}=8c_{14}\,{\epsilon}_{2}(\mathbf{r})\,\sin\phi_{2}(\mathbf{r})+2g\,{\epsilon}_{2}(\mathbf{r})^{2}\,\cos
2\phi_{2}(\mathbf{r})\,.$

At the values of the angle $\phi_{2}$ that minimise the energy, i.e.,
$\phi_{2}=\pi/6,\,5\pi/6,\,3\pi/2$,
$\Phi_{1}(\mathbf{r})=\Phi(\mathbf{r})\,\cos\phi_{2}$ and
$\Phi_{2}(\mathbf{r})=\Phi(\mathbf{r})\,\sin\phi_{2}$. Therefore, without loss
of generality, we can approximate $\Phi_{1}(\mathbf{r})$ and
$\Phi_{2}(\mathbf{r})$ with $\varepsilon_{2,1}(\mathbf{r})$ and
$\varepsilon_{2,2}(\mathbf{r})$, respectively, apart from a proportionality
constant that we will absorb into an effective coupling constant, so that

$\displaystyle\frac{\displaystyle\;\;c_{11}^{2}-c_{12}^{2}\;\;}{\displaystyle\;2c_{11}\;}\,\int
d\mathbf{r}\,\Omega_{R}(\mathbf{r})^{2}$ $\displaystyle\simeq\kappa\,\iint
d\mathbf{r}\,d\mathbf{r}^{\prime}\;{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot\begin{pmatrix}K_{11}(\mathbf{r}-\mathbf{r}^{\prime})&K_{12}(\mathbf{r}-\mathbf{r}^{\prime})\\\
K_{21}(\mathbf{r}-\mathbf{r}^{\prime})&K_{22}(\mathbf{r}-\mathbf{r}^{\prime})\end{pmatrix}\,{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r}^{\prime})\,,$
(S65)

with $\kappa>0$, and

$\displaystyle K_{11}(\mathbf{r})$
$\displaystyle=\frac{\displaystyle\;1\;}{\displaystyle\;V\;}\,\sum_{\mathbf{k}}\,\text{e}^{i\mathbf{k}\cdot\mathbf{r}}\;\left(\frac{\displaystyle\;2k_{1}\,k_{2}\;}{\displaystyle\;k_{1}^{2}+k_{2}^{2}\;}\right)^{2}=-\cos
4\phi\;\frac{\displaystyle\;1\;}{\displaystyle\;\;\pi\,r^{2}\;\;}\;,$ (S66)
$\displaystyle K_{22}(\mathbf{r})$
$\displaystyle=\frac{\displaystyle\;1\;}{\displaystyle\;V\;}\,\sum_{\mathbf{k}}\,\text{e}^{i\mathbf{k}\cdot\mathbf{r}}\;\left(\frac{\displaystyle\;k_{1}^{2}-k_{2}^{2}\;}{\displaystyle\;k_{1}^{2}+k_{2}^{2}\;}\right)^{2}=\cos
4\phi\;\frac{\displaystyle\;1\;}{\displaystyle\;\;\pi\,r^{2}\;\;}=-K_{11}(\mathbf{r})\;,$
$\displaystyle K_{12}(\mathbf{r})$
$\displaystyle=K_{21}(\mathbf{r})=\frac{\displaystyle\;1\;}{\displaystyle\;V\;}\,\sum_{\mathbf{k}}\,\text{e}^{i\mathbf{k}\cdot\mathbf{r}}\;\left(\frac{\displaystyle\;2k_{1}\,k_{2}\;}{\displaystyle\;k_{1}^{2}+k_{2}^{2}\;}\right)\,\left(\frac{\displaystyle\;k_{1}^{2}-k_{2}^{2}\;}{\displaystyle\;k_{1}^{2}+k_{2}^{2}\;}\right)=\sin
4\phi\;\frac{\displaystyle\;1\;}{\displaystyle\;\;\pi\,r^{2}\;\;}\;.$

In conclusion, the Ginzburg-Landau functional reads

$\displaystyle E\Big{[}{\boldsymbol{{\epsilon}}}_{2}\Big{]}$
$\displaystyle\to\int
d\mathbf{r}\,\Bigg{\\{}-\frac{\displaystyle\;K\;}{\displaystyle\;2\;}\;{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot\nabla^{2}{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})+\tau\,{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$
(S67)
$\displaystyle\qquad\qquad\qquad-\gamma\,{\epsilon}_{2}(\mathbf{r})^{3}\,\sin
3\phi_{2}(\mathbf{r})+\mu\,\Big{(}{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\Big{)}^{2}\;\Bigg{\\}}$
$\displaystyle\qquad\qquad\quad+\kappa\,\iint
d\mathbf{r}\,d\mathbf{r}^{\prime}\;{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})\cdot\begin{pmatrix}K_{11}(\mathbf{r}-\mathbf{r}^{\prime})&K_{12}(\mathbf{r}-\mathbf{r}^{\prime})\\\
K_{21}(\mathbf{r}-\mathbf{r}^{\prime})&K_{22}(\mathbf{r}-\mathbf{r}^{\prime})\end{pmatrix}\,{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r}^{\prime})\,,$

with positive $\mu$, $\gamma$ and $\kappa$, supplemented by the condition Eq.
(S51). Eq. (S67) can describe a transition once $\tau$ changes sign from
positive to negative. Since $\gamma$ is likely very small near the structural
transition, the latter might well be continuous or, if discontinuous, not as
sharp as the jump in resistivity, as suggested by Ref. [54].

###  S5.4 Mean-field approximation in the pseudo-spin representation

As discussed in the main text, the dimer stretching, main responsible of the
metal-insulator transition, add a further term to the energy functional (S67)
that depends on a dimensionless field $\eta(\mathbf{r})$ and reads

$\displaystyle\delta E\big{[}{\epsilon}_{2},\eta\big{]}$
$\displaystyle=a\,\int
d\mathbf{r}\,\bigg{[}\;\bigg{(}\eta(\mathbf{r})^{2}-\frac{\displaystyle\;1\;}{\displaystyle\;4\;}\bigg{)}^{2}-g\Big{(}{\epsilon}_{2}(\mathbf{r})^{2}-{\epsilon}^{2}_{MIT}\Big{)}\,\eta(\mathbf{r})\;\bigg{]}\,,$
(S68)

with $a>0$, $g>0$, and ${\epsilon}^{2}_{MIT}$ either positive or negative. A
minimum at $\eta<0$ corresponds to a metal, while one at $\eta>0$ to an
insulator, and both may coexist. Note that
${\epsilon}_{2}^{2}>{\epsilon}^{2}_{MIT}$ implies that the global minimum is
insulating, whereas ${\epsilon}_{2}^{2}<{\epsilon}^{2}_{MIT}$ that is
metallic.
Therefore, the total energy functional is the sum of the shear strain energy
$E\big{[}{\boldsymbol{{\epsilon}}}_{2}\big{]}$ in Eq. (S67) and of the dimer
stretching contribution $\delta E\big{[}{\epsilon}_{2},\eta\big{]}$ in Eq.
(S68), and is

$\displaystyle E\big{[}{\boldsymbol{{\epsilon}}}_{2},\eta\big{]}$
$\displaystyle=E\big{[}{\boldsymbol{{\epsilon}}}_{2}\big{]}+\delta
E\big{[}{\epsilon}_{2},\eta\big{]}\,,$ (S69)

which can actually describe rhombohedral and monoclinic phases, either
metallic or insulating, as those observed experimentally in and out-of
equilibrium.

In order to study the free energy corresponding to the classical energy
functional (S69), we decided to use the mean-field theory developed in Refs.
[67, 62].
For that, we consider a square film, and introduce a dense mesh of $N$ sites
to discretise the real space coordinate $\mathbf{r}$. Below the monoclinic
insulator spinodal point, the landscape of the energy functional (S69)
displays more than a single potential well, corresponding, within the
coexistence region, to the rhombohedral metal and the three equivalent
monoclinic twins, and only to the latter three below the rhombohedral metal
spinodal point. We expect that at thermal equilibrium the field variables,
${\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$ and $\eta(\mathbf{r})$, will
locally fluctuate inside one of those potential wells. We henceforth neglect
such thermal fluctuations, and constraint
${\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$ and $\eta(\mathbf{r})$ to locally
take only the values at the minima of the potential wells. This is equivalent
to assuming that the entropy is just contributed by the configurational term
due to the presence of different domains.
Below the monoclinic insulator spinodal point we thus assume that

  * •

the monoclinic insulating domains are characterised by

$\displaystyle{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$
$\displaystyle=\overline{{\epsilon}}_{2}(\tau)\;{\boldsymbol{S}}_{n}\,,$
$\displaystyle n$ $\displaystyle=1,2,3\,,$ (S70)

where $\overline{{\epsilon}}_{2}(\tau)$ is the value of ${\epsilon}_{2}$ at
the monoclinic homogeneous minimum of Eq. (S69), even if metastable, with
energy $E_{m}(\tau)$, and

$\displaystyle{\boldsymbol{S}}_{n}$
$\displaystyle=\begin{pmatrix}\cos\phi_{2,n}\\\
\sin\phi_{2,n}\end{pmatrix}\,,$ (S71)

are unit-length two-component vectors, with $\phi_{2,n}$, $n=1,2,3$, the
angles defined in Eq. (S43);

  * •

the rhombohedral metallic domains are instead identified by

$\displaystyle{\boldsymbol{{\epsilon}}}_{2}(\mathbf{r})$
$\displaystyle={\boldsymbol{0}}=\overline{{\epsilon}}_{2}(\tau)\,\begin{pmatrix}0\\\
0\end{pmatrix}\equiv\overline{{\epsilon}}_{2}(\tau)\;{\boldsymbol{S}}_{0}\,,$
(S72)

with the same amplitude $\overline{{\epsilon}}_{2}(\tau)$ as in Eq. (S70),
thus absorbing the vanishing strain into a zero-length spin. Its energy,
$E_{r}$, is actually independent of the reduced temperature $\tau$. We allow
for rhombohedral domains also below the rhombohedral spinodal point because
${\epsilon}_{2}=0$ is always a turning point of Eq. (S69) [67, 62]. Therefore,
it could be stabilised at the interface between two monoclinic domains,
despite being a maximum of the energy.

In this approximation, the energy functional (S69) below the monoclinic
insulator spinodal point can be written as a pseudo-spin Hamiltonian

$\displaystyle H(\tau)$ $\displaystyle=\int
d\mathbf{r}\,\bigg{\\{}E(\tau)\,{\boldsymbol{S}}(\mathbf{r})\cdot{\boldsymbol{S}}(\mathbf{r})-\overline{{\epsilon}}_{2}(\tau)^{2}\,\frac{\displaystyle\;K\;}{\displaystyle\;2\;}\,{\boldsymbol{S}}(\mathbf{r})\cdot\nabla^{2}{\boldsymbol{S}}(\mathbf{r})\bigg{\\}}$
(S73) $\displaystyle+\kappa\,\overline{{\epsilon}}_{2}(\tau)^{2}\iint
d\mathbf{r}\,d\mathbf{r}^{\prime}\,{\boldsymbol{S}}(\mathbf{r})\cdot\,\hat{U}(\mathbf{r}-\mathbf{r}^{\prime}){\boldsymbol{S}}(\mathbf{r})\equiv\frac{\displaystyle\;1\;}{\displaystyle\;N\;}\,\sum_{\mathbf{q}}\,{\boldsymbol{S}}(\mathbf{q})\cdot\,\hat{H}(-\mathbf{q},\tau)\,{\boldsymbol{S}}(-\mathbf{q})\,,$

where $E(\tau)=E_{m}(\tau)-E_{r}$, and, we remind,
${\boldsymbol{S}}(\mathbf{r})$ can take only the values
${\boldsymbol{S}}_{n}$, $n=0,\dots,3$, see Eqs. (S71) and (S72). We also note
that the Hamiltonian (S73) depends explicitly on temperature through the
parameter $\tau$. Following Refs. [67, 62] we treat such pseudo-spin
Hamiltonian in mean field. Since the key role in the domain formation is
played by the long range potential, which is diagonal in momentum space, we
assume a factorised density matrix
$\hat{\rho}=\prod_{\mathbf{q}}\,\hat{\rho}(\mathbf{q})$ that minimises the
free-energy functional

$F\big{[}\hat{\rho}\big{]}=\mathrm{Tr}\Big{(}\hat{\rho}\,H(\tau)\Big{)}-T\,\sum_{\mathbf{q}}\,\mathrm{Tr}\Big{(}\hat{\rho}(\mathbf{q})\,\ln\hat{\rho}(\mathbf{q})\Big{)}\,,$
(S74)

thus providing an upper estimate of the actual free energy. One readily finds
that the minimum is obtained for

$\displaystyle\hat{\rho}$ $\displaystyle=$
$\displaystyle\exp\bigg{[}\,-\frac{\displaystyle\;\beta\;}{\displaystyle\;N\;}\,\sum_{\mathbf{q}}\,{\boldsymbol{S}}(\mathbf{q})\cdot{\boldsymbol{B}}(-\mathbf{q},\tau)\,\bigg{]}$
$\displaystyle=$ $\displaystyle\exp\bigg{[}\,-\beta\,\int
d\mathbf{r}\,{\boldsymbol{S}}(\mathbf{r})\cdot{\boldsymbol{B}}(\mathbf{r},\tau)\,\bigg{]}\,,$

where

$\displaystyle{\boldsymbol{B}}(\mathbf{r},\tau)$
$\displaystyle=\frac{\displaystyle\;1\;}{\displaystyle\;N\;}\,\sum_{\mathbf{q}}\,\text{e}^{i\mathbf{q}\cdot\mathbf{r}}\;\hat{H}(\mathbf{q},\tau)\,\mathrm{Tr}\Big{(}\hat{\rho}(\mathbf{q})\,{\boldsymbol{S}}(\mathbf{q})\Big{)}\,.$
(S76)

It follows that

$\displaystyle\mathrm{Tr}\Big{(}\hat{\rho}\,{\boldsymbol{S}}(\mathbf{r})\Big{)}$
$\displaystyle=\frac{\displaystyle\;\quad\sum_{n=1}^{3}\,{\boldsymbol{S}}_{n}\;\text{e}^{-\beta\,{\boldsymbol{S}}_{n}\cdot{\boldsymbol{B}}(\mathbf{r},\tau)}\quad\;}{\displaystyle\;1+\sum_{n=1}^{3}\;\text{e}^{-\beta\,{\boldsymbol{S}}_{n}\cdot{\boldsymbol{B}}(\mathbf{r},\tau)}\;}\;,$
(S77)

is actually a self-consistency equation that we have to solve. In practice, we
start from a randomly chosen configuration
$\langle\,{\boldsymbol{S}}(\mathbf{r})\,\rangle$, which determines
$B(\mathbf{r},\tau)$ of Eq. (S76) [77], calculate by means of (S77) the new
configuration $\langle\,{\boldsymbol{S}}(\mathbf{r})\,\rangle$, and proceed
till convergence. Such mean-field scheme well inside the monoclinic insulator
phase does not even require enforcing the curl-free condition
${\boldsymbol{\nabla}}\wedge{\boldsymbol{{\epsilon}}}(\mathbf{r})=0$. Indeed,
because of the $Z_{3}$ symmetry among the three allowed monoclinic
${\boldsymbol{S}}_{n}$, $n=1,2,3$, the interface between two domains
identified by two of those pseudo-spins turns out to be automatically directed
along the third one.
Evidently, both the pseudo-spin representation and the chosen mean-field
scheme are valid only inside the monoclinic phase well below the rhombohedral
metal spinodal point. Here, in fact, the monoclinic minima of the energy
functional are deep enough to justify neglecting amplitude fluctuations.

##  S6 Supplementary Figures

Figure S4:  Temporal and spatial characterization of laser and X-rays beams
for tr-PEEM experiment. (a) Laser spot profile as reconstructed by means of
non-linear photoemission map from a surface defect. (b) Intensity of the non-
linear photoemission signal $I_{\scriptsize{\mbox{NLP}}}$(t) from a surface
defect as a function of the delay between the laser and X-ray fields. The
minimum of the total intensity is shown in correspondence with the temporal
coincidence between the near-IR pump and X-ray probe (i.e. time zero). Bottom
panels: images of the integrated area used to obtain the total intensity
$I_{\scriptsize{\mbox{NLP}}}$(t). (c) Example of the tr-PEEM image obtained
from the acquisition protocol. The red and blue areas represent, respectively,
the regions of maximum ($I^{light}$) and minimum ($I^{dark}$) photoemission
intensity contrast used to extract the relative contrast variation $\delta
I(t)$ between neighbouring monoclinic domains.

Figure S5:  Typical equilibrium resistivity hysteresis measured as function of
the temperature of a 50 $\mathrm{n}\mathrm{m}$ V2O3 on (0001)-Al2O3 substrate.
The red (blue) curves indicate the heating (cooling) branch. The curves have
been measured using the 4-points Van Der Pauw configuration with a temperature
sweep rate of 0.5 K/min.

Figure S6:  Typical transient variation of reflectivity $\delta R/R$ measured
at $T$=100 K as function of the time delay between pump and probe pulses. The
pump (probe) wavelength is 800 $\mathrm{n}\mathrm{m}$ (515
$\mathrm{n}\mathrm{m}$). The repetition rate is 10
$\mathrm{k}\mathrm{H}\mathrm{z}$ to avoid Joule heating effects. The incident
pump fluence is 8 $\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$,
enough to completely photo-induce the metallic state as shown in Fig. 2d of
the main text.

Figure S7:  Domains morphology for different V2O3 samples. a) PEEM images
taken at 120 K for the sample with (right) and without (left) the Cr2O3 buffer
layer. In the bottom we report schematics of the sample structure. The
dashed/solid coloured arrows (blue, pink) highlight the main orientations of
the monoclinic domains. b) Cuts of the two-dimensional Fast Fourier Transform
along the directions defined by the arrows in panel a). The yellow box
highlights the main peaks (in k-space) corresponding to the periodicity of the
stripe domains in the real space.

Figure S8: Hexagonal axes, ${\boldsymbol{a}}_{H}$ and ${\boldsymbol{b}}_{H}$.
Also shown are the coordinates of vertices and centre of the hexagon expressed
in terms of ${\boldsymbol{a}}_{H}$ and ${\boldsymbol{b}}_{H}$.