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Private Equity in Space Exploration [5 Case Studies][2024]

Space’s final frontier remains a realm of scientific curiosity and an arena of immense commercial potential. “Private Equity in Space Exploration” delves into how pioneering private equity firms shape the future of space travel, infrastructure, and resource utilization through strategic investments. This article presents five compelling case studies that illustrate the dynamic role of private equity in transforming theoretical space ventures into tangible, profitable enterprises. Each case study highlights a unique aspect of space exploration, from lunar mining to orbital infrastructure, detailing the challenges, solutions, and outcomes driven by visionary capital and relentless innovation.

Private Equity in Space Exploration [5 Case Studies]

1. space exploration investment by ae industrial partners, company profile.

AE Industrial Partners (AEI), a private equity firm specializing in aerospace, defense, and government services, has actively engaged in the burgeoning space exploration sector. Established in 1998 and based in Boca Raton, Florida, AEI manages over $4 billion in assets. The firm’s commitment to advancing technology and innovation extends into space exploration, where it has invested in several key startups and established companies aiming to commercialize space travel and exploration. This strategic focus drives technological advancements and aligns with AEI’s mission to support high-growth companies in critical sectors.

The space exploration sector presents unique challenges, notably the high cost and risk of developing reliable and scalable technologies capable of long-duration space missions. AEI faced the challenge of identifying viable investment opportunities that align with their strategic goals while managing the inherent risks of space technology ventures. The task was compounded by the need to ensure a sustainable return on investment amidst a landscape filled with competitive, high-tech startups and the complexities of international space travel and commerce regulations. The firm needed to navigate these challenges to establish a foothold in the competitive space exploration market, balancing innovation with financial prudence.

AEI approached these challenges by leveraging its deep industry expertise and extensive network to partner with promising space exploration enterprises. One significant move was the investment in Redwire Space, which specializes in developing advanced space infrastructure and robotics. AEI provided capital and strategic guidance to scale Redwire’s operations and enhance its technological capabilities. The firm facilitated key industry partnerships and supported Redwire in acquiring complementary businesses to broaden its service offerings. Furthermore, AEI emphasized sustainable growth by focusing on commercial applications of space technologies that could offer quicker returns, such as satellite deployment and space manufacturing technologies.

AEI’s strategic investments in the space exploration sector, particularly its backing of Redwire Space, have shown promising results. Redwire has successfully expanded its capabilities and market presence, becoming a notable player in the space technology industry. AEI’s approach has led to several successful launches and contracts, positioning Redwire as a critical supplier of deployable space structures and advanced digital manufacturing in space. This success has validated AEI’s investment strategy and also contributed to the firm’s reputation as a leading investor in high-growth sectors.

Related: Private Equity in Real Estate

2. Space Infrastructure Expansion by Voyager Space Holdings

Voyager Space Holdings, headquartered in Denver, Colorado, operates as a global leader in space exploration efforts, aiming to create a vertically integrated aerospace company. Since its inception in 2019, Voyager has strategically acquired several smaller companies to enhance its capabilities across various space-related domains, including space station development, satellite technology, and deep space exploration. With a focus on reducing fragmentation in the space sector, Voyager seeks to streamline operations and foster innovations that could make space more accessible and sustainable for commercial and research endeavors.

The primary challenge for Voyager Space Holdings was integrating various space technology entities to work cohesively towards pioneering next-generation space infrastructure. Each acquisition brought its own technical, cultural, and operational methodologies, often clashed, creating inefficiencies and slowing innovation. Moreover, the challenge was internal and external, as the company needed to position itself as a reliable partner in an industry where collaboration with government and international space agencies is vital. It required technological integration and aligning multiple corporate philosophies and objectives under a unified strategic vision.

Voyager Space Holdings tackled these challenges by implementing a comprehensive integration plan that included aligning technological platforms and merging corporate cultures. They established centralized operations and R&D centers to harmonize engineering efforts and foster a collaborative environment. Leadership workshops and team-building activities were initiated to blend diverse corporate cultures. Voyager also invested heavily in proprietary technology platforms that facilitated better project management and data integration across its portfolio companies. To solidify its industry standing, Voyager actively engaged with key stakeholders in the space community, including NASA and other space agencies, ensuring compliance with global space policies and securing a place in major governmental projects.

The strategic integrations and collaborative efforts led by Voyager Space Holdings have paid dividends. The company has successfully launched several collaborative projects with international space agencies, significantly boosting its market presence and operational efficiency. Their unified approach has accelerated the development of innovative space infrastructure, including new satellite deployment mechanisms and components for the planned lunar space stations. Voyager’s ability to deliver on complex space projects has reinforced its reputation as a pioneer in the space industry, driving further investment and collaboration opportunities.

Related: Private Equity in Pharma Industry

3. Enhancing Spacecraft Efficiency by Seraphim Capital

Seraphim Capital, a London-based venture firm, specializes in investments within the space industry, focusing on satellite technologies, artificial intelligence for earth observation, and other aerospace advancements. Founded in 2006, Seraphim stands out as one of the first venture funds to finance the new space ecosystem, managing over $250 million in assets. Seraphim aims to revolutionize how humanity deploys and utilizes space-born technologies by investing in high-potential startups and scaling their operations.

One of the significant challenges Seraphim Capital faced was selecting and nurturing startups that could disrupt traditional space technologies with scalable and sustainable innovations. The venture firm needed to ensure that its investments provided a substantial technological edge and aligned with long-term market demands and regulatory landscapes. The rapidly evolving nature of space technology and the high capital requirements for space ventures posed a considerable risk regarding investment returns and the practical implementation of innovative ideas.

To address these challenges, Seraphim implemented a rigorous vetting process for potential investments, focusing on companies with innovative yet practical solutions that could be quickly adapted to market needs. They provided strategic guidance and funding to Spire Global, a startup specializing in data-gathering satellites that monitor Earth’s weather and maritime traffic. Seraphim helped Spire scale its operations globally and enhance its satellite technology to offer more comprehensive data solutions. The firm facilitated partnerships between Spire and various governmental and commercial entities, ensuring a broad market for its data services. It also emphasized building robust compliance and regulatory frameworks to navigate the complex international space laws.

Seraphim Capital’s strategic investment in Spire Global yielded significant outcomes, with Spire becoming a leader in the global market for space-based data. This success has attracted additional investments and partnerships, significantly expanding Spire’s operational capacity and data offerings. The enhanced satellite capabilities have also led to new weather forecasting, maritime navigation, and environmental monitoring applications, delivering critical data that supports global sustainability efforts. Seraphim’s focused investment approach provided high returns and reinforced its reputation as a visionary leader in space venture capital, driving further innovations in the space industry.

Related: Private Equity in FMCG [Case Studies]

4. Pioneering Private Equity in Lunar Exploration: Blue Orbit Capital

Blue Orbit Capital, established in 2010 and based in Silicon Valley, has carved out a niche in the private equity landscape by focusing exclusively on the space sector, particularly lunar and asteroid mining ventures. With an asset management portfolio exceeding $3 billion, Blue Orbit is a trailblazer supporting high-risk, high-reward space ventures that promise to unlock new human resources and capabilities. The firm’s visionary approach is predicated on the belief that the moon and other celestial bodies hold the keys to solving Earth’s resource limitations.

The fundamental challenge for Blue Orbit Capital was the extremely high risk and unproven commercial viability associated with lunar exploration and mining. Investing in technologies and companies aiming to mine the moon posed significant technical and financial risks, with long lead times before achieving any potential returns. The firm had to navigate the technological uncertainties, complex international space treaties, and the logistics of supporting operations that needed to function off-planet.

To mitigate these challenges, Blue Orbit Capital took a multi-faceted approach. They strategically invested in several pioneering space firms, including MoonEx, a company developing robotic lunar landers. Blue Orbit provided capital infusion, strategic business advice, and critical industry connections to ensure technological and operational scaling. They also spearheaded a consortium with technology and mining companies to share knowledge, reduce costs, and co-develop necessary lunar mining technologies. Furthermore, Blue Orbit played an active role in shaping policy by participating in international forums on space law and advocating for regulations that support private sector participation in space resources development.

Blue Orbit Capital’s strategic initiatives have begun to yield promising results. MoonEx has successfully launched several missions demonstrating the capabilities of its lunar landers and has positioned itself as a leader in lunar exploration technology. These successes have significantly enhanced investor confidence and attracted further funding. The consortium’s collaborative efforts have led to breakthrough technologies essential for lunar surface operations, such as regolith processing and resource extraction. Blue Orbit’s advocacy has also influenced policy frameworks to better accommodate private investments in space resources, paving the way for a new era of off-planet resource utilization and securing Blue Orbit’s position at the forefront of lunar exploration ventures.

Related: Private Equity Interview Questions and Answers

5. Revolutionizing Orbital Infrastructure: Starlight Ventures

Starlight Ventures, a private equity firm based in Houston, Texas, specializes in investments that expand and enhance orbital infrastructure, such as satellite communications and space station development. Founded in 2015, Starlight Ventures has rapidly grown to manage over $2 billion in assets, driven by a mission to support sustainable and scalable space operations. Their investment portfolio is diverse, including companies developing next-generation satellite technologies and those working on constructing and operating private space stations.

Starlight Ventures faced the challenge of effectively scaling operations and technologies in an industry characterized by excessive costs and technical complexity. The firm needed to ensure that its investments advanced technological boundaries and remained financially viable and operationally sustainable. Additionally, the rapid pace of technology evolution in space operations meant that investment decisions had to be forward-thinking and resilient to industry dynamics and regulatory environment shifts.

Starlight Ventures adopted an integrative approach to these challenges. They focused on companies with innovative yet practical orbital technologies that promised quicker market readiness and revenue generation, such as Orbital Assembly Corporation, which aims to construct space stations. Starlight provided the necessary funding and strategic industry positioning by facilitating partnerships with aerospace giants and international space agencies. They also emphasized the importance of developing modular and scalable technologies that could adapt to various space missions, enhancing the utility and longevity of the orbital infrastructure developed by their portfolio companies.

The initiatives led by Starlight Ventures have significantly advanced the development of orbital infrastructure. Orbital Assembly Corporation has made notable progress in its space station projects, with plans for the first commercial space station modules slated for deployment within the decade. These developments have helped solidify Starlight Ventures’ reputation as a key player in expanding humanity’s presence in orbit. The firm’s strategic investments have attracted additional partnerships and funding, further accelerating innovation in orbital technologies and making space operations more accessible and commercially viable.

Related: Private Equity in eCommerce [Case Studies]

The exploration of space, once dominated by governmental agencies, is increasingly being propelled by the private sector, with private equity playing a crucial role in its expansion. The case studies showcased in this article highlight the financial courage and the strategic insight of firms investing in the vast potential of outer space. These firms are not just funding missions but are actively shaping the future of space travel and exploitation. The results from these investments demonstrate a promising horizon where private equity continues to enable technological advancements and infrastructural expansions, ensuring that the space industry remains at the forefront of innovation and commercial viability.

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Conduct a world-first trailer drop in space for YouTube Original’s flagship sci-fi horror show 'Origin'

Ahead of the release of their new show 'Origin', YouTube came to us to put together a cosmic look at the show's trailer, presented by starring actor Tom Felton.

We modified a Google Pixelbook to operate in the harsh environmental conditions of space, before launching it 100,000 feet above the Earth. We launched the laptop alongside a 360° camera, creating an immersive VR video experience to highlight YouTube’s recently introduced support for 360° content. Heads-up display-style CG infographics and Easter eggs were added in post-production, matching the designs used in the world of the show for maximum relevance. The entire launch process was documented as extending legacy content to build anticipation for the show. The promotional video was posted on Origin’s YouTube channel, amassing over 60,000 views, and the first episode of the show has garnered over a million views.

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Reaching more creators and artists through YouTube Spaces

By Robert Kyncl , Chief Business Officer, YouTube

Feb 18, 2021 – minute read

A new flexible strategy will combine virtual and in-person Pop-up programming to reach even more communities and better meet the needs of our global creators and artists.

“ In just over four years, we hosted over 45 Pop-up events reaching over 15,000 creators and artists...in over 20 cities that normally wouldn’t have access to a physical Space.”

One of these efforts was the creation of YouTube Spaces. Nine years ago, we launched Spaces in major cities around the world. These physical locations across key markets provided our creators and artists with important resources, including state-of-the-art studios, events, and classes. Most importantly, they offered a place to build valuable creative connections. In 2016, we began seeing that community stretched beyond the four walls of our physical facilities, and we launched YouTube Pop-up Spaces: smaller, more nimble physical locations to bring the best programming of our Spaces temporarily to over 20 additional countries.

In just over four years, we hosted over 45 Pop-up events reaching over 15,000 creators and artists, as well as NGOs and educational institutions. These were located in over 20 cities that normally wouldn’t have access to a physical Space, from Buenos Aires and Mexico City, to Madrid, Milan, Cairo, Jakarta, Taiwan, and Mumbai. And they all brought with them a unique local flavor, like our music-focused Pop-ups in Stockholm and Nashville. 

When the COVID-19 pandemic hit, we refocused our partner programming and plans even further to a completely virtual model. Over the course of 2020, we ran over a thousand online events, from introductory Shorts workshops for creators in India, to a U.S. virtual series helping creators produce great content from home, to programs like Boss & Bloom in the UK, which focused on supporting diverse female creators. All in all, in spite of closing our facilities last year, our virtual events reached over 70,000 people across 145 countries worldwide. 

“ This flexible new strategy will allow us to reach more regions, positively impact more new and existing creators and artists by giving them the guidance and resources they need to take their craft to the next level.”

Today, our YouTube partner community continues to expand at an exponential pace, and we need to better meet the needs of our creators and artists, no matter where they might be. Moving forward, we’re doubling down on the more scalable and nimble strategy to reach more creators and artists with the tools and workshops we offer. As a result, we won't be reopening our physical YouTube Spaces locations in Berlin, London, Los Angeles, New York, Paris, Rio, and Tokyo, but will instead focus on a hybrid model that combines virtual and in-person Pop-up programming.

We truly believe this flexible new strategy will allow us to reach more regions, and positively impact more new and existing creators and artists by giving them the guidance and resources they need to take their craft to the next level. 

In 2021, we are committing to the following:

Multi-week virtual development programs to invest in the growth and success of our first-ever  #YouTubeBlack Voices Fund  for creators and artists—recipients span the U.S., UK, Kenya, South Africa, Nigeria Brazil and Australia. 

Ongoing investment in the NextUp program to support fast-rising creators in markets like Russia, Japan, the Philippines and Germany.

Multiple live and recorded online workshops as part of the Creator Academy’s  Learning Toolkits . These offerings help creators produce great content at home, explore new ways to earn money with YouTube tools, and learn best practices for livestreaming.

YouTube artist and label workshops to preview the latest product developments, content strategies, and case studies.

Once in-person events are allowed, we will once again bring Pop-up events and experiences to even more new communities. Programming will include trainings on new products like Shorts, and events like Music Nights.

Our vision at YouTube is to give everyone a voice and show them the world. This new approach will help us deliver more than ever on this promise.

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YouTube Space, London

Project summary.

YouTube Space London is a specially created facility where YouTubers and vloggers can learn, connect and create. Its purpose is to allow content contributors to improve the quality of their posts by providing them with high quality studio and editing spaces and technical assistance from studio staff. Set inside parent company Google’s office at 6 Pancras Square, YouTube Space London has two fully-isolated double-height studios and a third single height isolated studio, an isolated sound control room, production control room and training and edit facilities. The primary studios were designed with a focus on broadcasting with the capability for music performance. The largest Studio can also be used as a cinema/screening room with a Dolby Atmos surround sound system and 4k screening facility. Replacing their previous facilities at Central St Giles, YouTube Space London spans 20,000 sq. feet, making it the largest YouTube Space in Europe.

Services Provided

Sandy Brown was appointed to provide the acoustic design for Google’s office at 6 Pancras Square. As part of the project, we were also commissioned to advise on the YouTube Space at the basement, basement mezzanine and lower ground floors of the building, from conception to completion.

The key acoustic design requirements were driven towards providing very low-noise, acoustically sensitive spaces, included:

• achieving very high levels of sound insulation between the primary studios such that amplified music performance could take place in one without giving rise to disturbance in the directly adjacent, highly sensitive, other studio.
• the design of a very quiet, low air speed ventilation system for all three studios such that air conditioning noise would not interfere with recordings
• the development of a bespoke, durable acoustic wall treatment system that provides reverberation control across the frequency range whilst maintaining the architectural aspirations and functionality

Special Acoustic Features

Isolated box-in-box structures were incorporated in order to provide very high levels of sound insulation around the studios. These comprised jack-up floating concrete floors, a combination of light and heavyweight partition elements and fully independent lids. The complex mechanical services design incorporated highly attenuated air paths, low-noise ductwork, acoustic lagging and isolated connections in order to deliver conditioned air into the high-load studios whilst maintaining very low services noise levels. Early adoption of appropriate acoustic principles was key to achieving the required zonal allowances for the strategy to be successful. The bespoke acoustic wall treatment was developed in order to provide broadband reverberation control in the studios using a combination of porous and panel absorbers, all contained within a durable architectural finish to maintain the aesthetic intent. The modular absorbers can be re-distributed within the studios in order to vary reverberation and provide a flexible acoustic environment to the end user. Acoustic commissioning was undertaken upon completion with a 13 kW PA system in order to generate sufficient low frequency sound to accurately measure the sound insulation performance of the studio walls.

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• Open access
• Published: 12 September 2024

Topological superconductivity from unconventional band degeneracy with conventional pairing

• Zhongyi Zhang   ORCID: orcid.org/0000-0003-0887-9358 1 , 2 , 3 ,
• Zhenfei Wu 4 ,
• Chen Fang   ORCID: orcid.org/0000-0002-9150-8023 2 , 5 ,
• Fu-chun Zhang 3 , 5 , 6 ,
• Jiangping Hu   ORCID: orcid.org/0000-0002-4837-7742 2 , 5 ,
• Yuxuan Wang 4 &
• Shengshan Qin   ORCID: orcid.org/0009-0003-3484-8747 7  

Nature Communications volume  15 , Article number:  7971 ( 2024 ) Cite this article

Metrics details

• Superconducting properties and materials
• Topological matter

We present a new scheme for Majorana modes in systems with nonsymmorphic-symmetry-protected band degeneracy. We reveal that when the gapless fermionic excitations are encoded with conventional superconductivity and magnetism, which can be intrinsic or induced by proximity effect, topological superconductivity and Majorana modes can be obtained. We illustrate this outcome in a system which respects the space group P 4/ n m m and features a fourfold-degenerate fermionic mode at ( π ,   π ) in the Brillouin zone. We show that in the presence of conventional superconductivity, different types of topological superconductivity, i.e., first-order and second-order topological superconductivity, with coexisting fragile Wannier obstruction in the latter case, can be generated in accordance with the different types of magnetic orders; Majorana modes are shown to exist on the boundary, at the corner and in the vortices. To further demonstrate the effectiveness of our approach, another example related to the space group P 4/ n c c based on this scheme is also provided. Our study offers insights into constructing topological superconductors based on bulk energy bands and conventional superconductivity and helps to find new material candidates and design new platforms for realizing Majorana modes.

Introduction

Topological superconductors 1 , 2 , 3 , 4 , 5 (TSCs) are renowned for hosting a special kind of quasiparticles, the Majorana modes, whose antiparticles are themselves. Owing to their potential application in fault-tolerant quantum computation 4 , 6 , 7 , a substantial effort has been made to search for the Majorana modes, and great advances have been achieved both in theory 5 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 and in experiment 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 over the past few decades. The p -wave superconductors have been suggested as promising candidates for the TSCs, and experimental signatures of p -wave superconductivity have been detected 29 , 30 , 31 . Various artificial devices have been proposed to support topological superconductivity, such as the heterostructure between a conventional superconductor and a topological insulator 8 or the Rashba electron gas 10 , 11 , and experimental evidence for the Majorana modes have been observed 21 , 27 . Despite the progress, an efficient way towards platforms realizing the numerous exotic topological superconducting phases 15 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , especially the high-order topological superconducting states, is still elusive.

In recent years, remarkable strides have been made in understanding the topological states of matter. It is realized that the topological property of a system can be indicated by the symmetry information of its occupied bands at high-symmetry points, and the system must be topologically nontrivial if its symmetry information at these points differs from that of an atomic insulator 49 , 50 , 51 . A parallel formalism has also been developed for the TSCs 52 , 53 , 54 . Motivated by these achievements, we suggest a new scheme to realize TSCs, built on the heterostructure sketched in Fig.  1 , based on symmetry-protected band degeneracies near the fermi energy and conventional superconductivity. Such fermionic modes, i.e., the band degeneracies, always carry different quantum numbers, such as rotation eigenvalues, mirror eigenvalues, etc. To make the core of our proposal clearer, let us start with the time-reversal symmetric BdG Hamiltonian with conventional superconductivity, i.e., the uniform s -wave pairing. In such a system, the chiral symmetry, which is the combined operation of the time-reversal symmetry and the particle-hole symmetry, maps a negative-energy state to a positive-energy state. Moreover, the unitary chiral symmetry commutes with the crystalline symmetries 3 , 33 , 42 , leading to that the two states related by the chiral symmetry carry the same quantum numbers. This property implies that in the system, the information of the symmetry eigenvalues corresponding to all the negative-energy states at the high-symmetry point in the Brillouin zone is always the same as the condition where the normal-state electronic states are fully occupied or fully unoccupied, which must be topologically trivial 52 . Notice that the above conclusion is always true, regardless of the location of the Fermi energy. Therefore, in the sense of the symmetry indicator, any time-reversal symmetric superconductor with uniform s -wave pairing is topologically trivial 52 , 53 , 54 . However, if the time-reversal symmetry is broken, such as by the magnetic orders, the above symmetry constraint fails. Moreover, as long as the eigenvalues of the crystalline symmetries carried by the negative-energy states are different from that in the time-reversal symmetric case, some nontrivial topology is indicated in the superconductor, and such a condition is most likely to occur when there is band degeneracy near the Fermi energy. More specifically, when the band degeneracy is encoded with magnetism, it will split; If the chemical potential resides within the split band gap, in the superconducting state, the symmetry eigenvalues carried by the positive-energy states will no longer match those of the negative-energy states, indicating the presence of nontrivial topology (more details in Supplementary Note  1 ). We illustrate this scheme in a system respecting the space group P 4/ n m m and show various topological superconducting states can be achieved in accordance with the different magnetic orders. To further show the effectiveness of our approach, we provide another example related to the space group P 4/ n c c in Supplementary Note  10 . Compared with earlier proposals 4 , 10 , 11 , 55 , the key advantage here is that by leveraging the nonsymmorphic crystalline symmetries, the resulting phases of topological superconductivity are much richer. In recent years, the distinct irreducible representations (IRs) of the little group of the crystalline symmetries can assist in identifying different types of free fermionic excitations, such as the unconventional quasiparticles beyond Dirac and Weyl fermions 56 . Based on those abundant fermionic excitations, our method can be applied to a wide range of systems, and opens up a new direction of searching for novel topological superconducting phases in these materials.

It is based on systems with gapless fermionic excitations (the intermediate layer) protected by nonsymmorphic crystal symmetries. In the system, the magnetism (top) and the conventional superconductivity (bottom) can be induced through either the proximity effect or the intrinsic properties of the intermediate layer. The colored balls, black arrows and colored cones represent the different lattice sites, the magnetic moments and energy dispersion, respectively.

In the following, we focus on the space group P 4/ n m m , which has a four-dimensional irreducible projective representation at the Brillouin zone corner. We show that the antiferromagnetic (AFM) order and ferromagnetic (FM) order can both split the fourfold degeneracy into two twofold ones. In the presence of conventional superconductivity, the AFM order drives the system into a second-order TSC state coexisting with fragile Wannier obstruction, while the FM order results in a first-order TSC, as long as the chemical potential lies in the magnetic gap. These results may be relevant to iron-based superconductors and heterostructures thereof, which host intrinsic AFM order and high- T c superconductivity.

Fourfold degenerate fermion with SG 129

We begin with an introduction of the space group \({{\mathcal{G}}}=P4/nmm\) ( # . 129), i.e., the symmetry group governing the iron-based superconductors. We focus on the quasi-two-dimensional (2D) case and consider the lattice in Fig.  2 a, which is similar to the monolayer FeSe. The space group P 4/ n m m is nonsymmorphic, and it has a special group structure as follows 57

where T is the translation group, D 2 d is the point group at the lattice sites, and Z 2 is a two-element group, including the inversion symmetry, which switches the two sublattices in the lattice in Fig.  2 a. As D 2 d and Z 2 are defined on different points, Eq. ( 1 ) holds in a sense that symmetry operations are equivalent if they differ by a lattice translation, hence the quotient group on the left-hand side. According to Eq. ( 1 ), \({{\mathcal{G}}}/T\) can be generated by the generators of D 2 d and Z 2 , including the inversion symmetry { I ∣ τ 0 }, the mirror symmetry { M y ∣ 0 } and the rotoinversion symmetry { S 4 z ∣ 0 }. Here, we express the symmetry operations in the form of the Seitz operators. In the generators, the point group parts act on the Cartesian coordinates as I : ( x ,  y ,  z ) ↦ (− x , − y , − z ),  M y  : ( x ,  y ,  z ) ↦ ( x , − y ,  z ), and S 4 z  : ( x ,  y ,  z ) ↦ ( y , − x , − z ), and τ 0  =  a 1 /2 +  a 2 /2 with a 1 ( a 2 ) the primitive lattice translation along the x ( y ) direction in Fig.  2 a.

a shows a quasi-2D lattice respecting group P 4/ n m m . The green and red balls label the two sublattices. The orange and brown dashed lines indicate the different edges considered in the text. The black dashed arrows represent a bending process from the (10) edge to the [11] and \([1\overline{1}]\) edges. b presents the bands obtained from Eq. ( 4 ), with G, X, M representing (0, 0), ( π , 0), ( π ,  π ) in the Brillouin zone, respectively, with the other parameters set to be \(\{t,\, {t}^{{\prime} },\, \lambda \}=\{-1.0,0.8,0.5\}\) . The blue dashed line in ( b ) represents the chemical potential considered in the text.

For electronic systems in the presence of spin-orbit coupling, group P 4/ n m m has only one single 4D IR at ( π ,  π ), i.e., the M point in the Brillouin zone, where all the symmetry operations in \({{\mathcal{G}}}/T\) are respected. It describes the fourfold degeneracy composed of two Kramers’ doublets, J z  = ±1/2 and J z  = ±3/2, with opposite parities, where J z is the angular momentum defined according to { S 4 z ∣ 0 }. The degeneracy can be understood from the group structure in Eq. ( 1 ). The point group \({D}_{2d}^{D}\) (double group version of the point group D 2 d ) supports two different 2D IRs corresponding to Kramers’ doublet J z  = ±1/2 and J z  = ±3/2 separately. At the M point, { S 4 z ∣ 0 } in \({D}_{2d}^{D}\) and { I ∣ τ 0 } in Z 2 satisfy the following anticommutation relation

which enforces the degeneracy between the two 2D IRs labeled by J z  = ±1/2 and J z  = ±3/2 at M (more detailed analysis in Supplementary Note  2 ). In the paramagnetic state, besides crystalline symmetries, the time-reversal symmetry \({{\mathcal{T}}}\) also exists. Correspondingly, the system actually respects the type-II magnetic space group \(P4/nmm{1}^{{\prime} }\) ( # . 129.412), which reads

Notice that the time-reversal symmetry does not affect the 4D fermionic IR at M.

Assuming trivial band structure at other high-symmetry points, we describe the fourfold degenerate fermion at M by the following tight-binding model 58 , 59

where a single s orbital is assumed at each site in the lattice in Fig.  2 a. In Eq. ( 4 ), the Pauli matrices s i and σ i ( i  = 1, 2, 3) stand for the spin and sublattice degrees, respectively. t ( \({t}^{{\prime} }\) ) is the nearest-neighbor intrasublattice (intersublattice) hopping. λ is the inversion-symmetric Rashba spin-orbit coupling, which arises due to the mismatch between the lattice sites and the inversion center 60 , 61 . The band structure based on \({{{\mathcal{H}}}}_{0}({{\bf{k}}})\) is plotted in Fig.  2 b. We set the Fermi energy near the fourfold band degeneracy, as indicated in Fig.  2 b, and consider conventional superconductivity in the system. The corresponding BdG Hamiltonian takes the form

in the basis ψ † ( k ) = ( c † ( k ),  i s 2 σ 0 c (− k )). In Eq. ( 5 ), the Pauli matrix κ i describes the Nambu spinor, μ is the chemical potential, and Δ sc is the superconducting order parameter. In the superconducting state, the matrix form for the symmetry generators are \({{\mathcal{I}}}={s}_{0}{\sigma }_{1}{\kappa }_{0},\,{{{\mathcal{M}}}}_{y}=i{s}_{2}{\sigma }_{3}{\kappa }_{0}\) and \({{{\mathcal{S}}}}_{4z}={e}^{i{s}_{3}\pi /4}{\sigma }_{3}{\kappa }_{0}\) 58 , where \({{\mathcal{I}}},\,{{{\mathcal{M}}}}_{y}\) and \({{{\mathcal{S}}}}_{4z}\) correspond to { I ∣ τ 0 },  { M y ∣ 0 },  { S 4 z ∣ 0 } respectively. The time-reversal symmetry takes the form \({{\mathcal{T}}}=i{s}_{2}{\sigma }_{0}{\kappa }_{0}K\) and the particle-hole symmetry \({{\mathcal{P}}}={s}_{2}{\sigma }_{0}{\kappa }_{2}K\) , with K the complex conjugation operation. It is easy to check that the system described by \({{{\mathcal{H}}}}_{{{\rm{BdG}}}}\) in Eq. ( 5 ) is topologically trivial.

AFM order induced second-order TSCs

We study possible topological superconductivity in the structure sketched in Fig.  1 , based on the above fourfold degenerate fermion. First, we consider the checkboard AFM order preserving the translational symmetries in the system as illustrated in Fig.  3 a, and we assume the magnetic polarization along the z direction. Correspondingly, the system is described by the following Hamiltonian

with Δ AFM the strength of the AFM order. It is easy to check that, the system respects the type-III magnetic space group \(P{4}^{{\prime} }/{n}^{{\prime} }{m}^{{\prime} }m\) ( # . 129.416)

We consider the effect of the AFM order on the fourfold degeneracy at M. Obviously, all the symmetry operations in \({{{\mathcal{G}}}}_{{{\rm{AFM}}}}/T\) preserve at the M point. A direct analysis shows that the fourfold degeneracy is broken into two twofold degenerate ones. It is the J z  = 1/2 ( J z  = −1/2) state that is degenerate with the J z  = 3/2 ( J z  = −3/2) state. A detailed group analysis is presented in the Supplementary Note  3 . Such twofold band degeneracies arise from the relation \(\{{S}_{4z}| {{\bf{0}}}\}\{{M}_{xy}| {{{\tau }}}_{0}\}=\{{M}_{xy}| {{{\tau }}}_{0}\}\{{\overline{S}}_{4z}^{3}| {{{\bf{a}}}}_{{{\bf{1}}}}\}\) , which at M leads to

Recalling that \({\overline{S}}_{4z}={S}_{4z}^{5}\) , one immediately comes to the above conclusion. We simulate the bands in the presence of the AFM order numerically, and show the results at Δ AFM  = 0.5 in Fig.  3 a. Here, it is worth mentioning that the bands in Fig.  3 a are always twofold degenerate due to the symmetry \(\{I| {{{\tau }}}_{{{\bf{0}}}}\}{{\mathcal{T}}}\) which exists at every k point in the Brillouin zone and satisfies \({(\{I| {{{\tau }}}_{{{\bf{0}}}}\}{{\mathcal{T}}})}^{2}=-1\) .

a shows the normal bands for the system in Eq. ( 6 ) at Δ AFM  = 0.5, with the AFM order illustrated in the inset. The blue dashed line represents the chemical potential at μ  = 4.0. b shows the superconducting edge modes corresponding to the bands in ( a ) on the (10) edge. The edge modes on the right and left edges are degenerate. c shows an atomic insulator constructed by placing two Wannier orbits (WOs) with J z  = ±1/2 at 2 c Wyckoff positions (the center of the square formed by the red and green balls), one WO with J z  = +1/2 at one of 2 a Wyckoff positions (red balls) and one WO with J z  = −1/2 at the other 2 a Wyckoff position (green balls). d shows an atomic insulator constructed by placing one WO with J z  = +3/2 at one of 2 a Wyckoff positions and one WO with J z  = −3/2 at the other 2 a Wyckoff position. e shows the low-energy superconducting spectrum (inset) and the real-space wavefunction profiles of the zero-energy modes, corresponding to the bands in ( a ). Open boundary conditions are set in both the [11] and \([1\overline{1}]\) directions. f shows the low-energy superconducting spectrum in the presence of a single vortex in ( e ). In the shadow region, among the three zero-energy modes, there are two vortex-bound Majorana modes (V.M.) and one corner-bound Majorana mode (C.M.). g  and  h show the real-space wavefunction profiles of the two V.M. in the shadow region in ( f ), and the C.M. in ( f ) has a similar wavefunction to that in ( e ). The color bars in ( e ), ( g ), and ( h ) are in the unit of 10 −3 . In the calculations, the superconducting order is set to be Δ sc  = 0.2.

As the magnetism breaks the time-reversal symmetry but preserves the particle-hole symmetry, the system belongs to class D which in the 2D case is characterized by a \({\mathbb{Z}}\) topological index, i.e., the Chern number, according to the Altland-Zirnbauer classification 5 . The Chern number can be calculated efficiently based on the symmetry eigenvalues carried by the occupied bands at the high-symmetry points. In systems respecting the fourfold rotational symmetry C 4 , in the weak-pairing condition, the Chern number C h satisfies 42

where m is the angular momentum carried by the Cooper pair, ξ ( Γ ) and ξ (M) are the products of the C 4 eigenvalues of the occupied bands at Γ and M, respectively, and N occ (Γ),  N occ (M) and N occ (X) are the number of the occupied bands at Γ , M and X, respectively. Since C 4 is equivalent to S 4 in 2D systems, the formula in Eq. ( 9 ) applies to our consideration (In fact, in the nonsymmorphic group P 4/ nmm besides the S 4 z symmetry, there is also the C 4 z symmetry which is defined at the center of the square formed by the four nearest neighboring lattice sites in Fig. 2 a, and we have specified this point in the Supplementary material. As group P 4/ nmm merely has one 4D IR at M, all the analyses related to S 4 z also work for C 4 z .). The conventional superconductivity carries zero angular momentum, i.e., m  = 0. Therefore, the Chern number is completely determined by the S 4 z eigenvalues of the occupied bands at Γ and M, and for the condition in Fig.  3 a, we find that C h  = 0, which is also confirmed by the gapped modes on the (11) and \([1\overline{1}]\) edges (see Supplementary Note  5 ). Nonetheless, the system is topologically nontrivial, as evidenced by the helical edge mode on the (10) edge in Fig.  3 b. In fact, the system is a TSC protected by the antiunitary symmetry \({{{\mathcal{M}}}}_{y}{{\mathcal{T}}}\) . We focus on high-symmetry line k y  =  π , where \({{{\mathcal{M}}}}_{y}{{\mathcal{T}}}\) and the particle-hole symmetry \({{\mathcal{P}}}\) preserve. Moreover, \({{{\mathcal{M}}}}_{y}{{\mathcal{T}}}\) serves as a pseudo time-reversal symmetry on line k y  =  π satisfying \({({{{\mathcal{M}}}}_{y}{{\mathcal{T}}})}^{2}=1\) . Therefore, the k y  =  π line can be viewed as a 1D subsystem of the whole system, which belongs to symmetry class BDI 5 . The topological property of such a system is featured by the winding number,

with \(\widetilde{{{\mathcal{C}}}}={{{\mathcal{M}}}}_{y}{{\mathcal{T}}}{{\mathcal{P}}}\) being the pseudo-chiral symmetry on k y  =  π . We calculate the winding number straightforwardly, and it turns out w  = 2 (details in Supplementary Note  4 ), which is consistent with the two zero-energy modes at k y  =  π on the (10) edge presented in Fig.  3 e.

More interestingly, the above even winding number state is actually a second-order TSC state 35 , 36 , 37 , 38 protected by \({{{\mathcal{M}}}}_{y}{{\mathcal{T}}}\) . We demonstrate it numerically. As presented in Fig.  3 e, a single Majorana mode exists at the corner between the neighboring (11) and \((1\overline{1})\) edges. To understand the phenomenon, we start with the helical mode in Fig.  3 b. On the (10) edge, the symmetry \(\{{M}_{y}| {{\bf{0}}}\}{{\mathcal{T}}}\) and the particle-hole symmetry preserve. Considering the two symmetries, we can get the effective theory on the (10) edge as \({{{\mathcal{H}}}}_{(10)}=v{k}_{y}{\eta }_{1}\) , with v the Fermi velocity and η i the Pauli matrices in the space spanned by the helical edge mode. Then, we bend edge (10) into a right angle, with the two sides along the [11] and \([1\overline{1}]\) directions, as illustrated in Fig.  2 a. The helical mode on each edge gains a mass, since \(\{{M}_{y}| {{\bf{0}}}\}{{\mathcal{T}}}\) breaks on the (11)/ \((1\overline{1})\) edge. The gapped edge modes are depicted by the following effective theory

where \({m}_{(11)/(1\overline{1})}\) is the mass term on the (11)/ \((1\overline{1})\) edge. Moreover, \(\{{M}_{y}| {{\bf{0}}}\}{{\mathcal{T}}}\) requires \({m}_{(11)}=-{m}_{(1\overline{1})}\) . Therefore, Eq. ( 11 ) describes a massive Dirac theory, with the mass changing sign at the corner between the (11) and \((1\overline{1})\) edges. The mass domain results in a single Majorana mode at the corner 62 , 63 , 64 . Due to the pseudo-chiral symmetry \(\widetilde{{{\mathcal{C}}}}\) , the corner Majorana modes carry chirality, and the modes with the same chirality cannot hybridize with each other. Thus, the classification for the second-order TSC here is \({\mathbb{Z}}\) . Moreover, it is worth pointing out the above second-order TSC state exists in the condition \({(4t+\mu )}^{2}+{\Delta }_{{{\rm{sc}}}}^{2} \, < \, {\Delta }_{{{\rm{AFM}}}}^{2}\) , i.e., the chemical potential in the AFM gap in the weak-pairing condition, and it belongs to a \({\mathbb{Z}}\) classification corresponding to the winding number along k y  =  π protected by \(\{{M}_{y}| {{\bf{0}}}\}{{\mathcal{T}}}\) . We present more detailed analyses of the above effective edge theory and the topological phase transitions in Supplementary Note  5 .

Interestingly, the negative energy states of the BdG Hamiltonian in Eq. ( 6 ) display both fragile Wannier obstruction and second-order topology. To this end, we treat the BdG band structure as an insulator, i.e., ignoring the particle-hole symmetry. Noting that particle-hole partners in the BdG bands carry opposite angular momenta, the angular momenta of the four “occupied" (negative energy) BdG bands are J z  = ±1/2, ±1/2 at G, J z  = −1/2, −1/2, −3/2, −3/2 at M, and J z  = ±1/2, ±1/2 at X. By exhaustion, one can show that no Wannier representation exist. However, if one includes two additional trivial bands (e.g., from core electrons) that are equivalent to two Wannier orbitals with J z  = ±3/2 each at one of the 2a Wyckoff positions shown in Fig.  3 d, the combined six bands, nevertheless, become Wannier representable. The six Wannier orbitals are centered at Wyckoff position 2 c with angular momenta J z  = ±1/2, ±1/2 and Wyckoff position 2 a with J z  = −1/2, 1/2, as shown in Fig.  3 c. Therefore, the occupied bands, despite not being Wannier representable, can be viewed as the difference between two Wannier representable systems, with six and two occupied bands, respectively, as shown in Fig.  3 c, d. By definition, the four occupied bands display the fragile Wannier obstruction 65 . Formally, using the modern language of magnetic elementary band representation 66 , we express the fragile Wannier obstruction protected by the magnetic space group symmetries in Supplementary Note  6 .

The elucidation of the fragile Wannier obstruction enables an alternative understanding of the second-order topology invoking only S 4 z . Ignoring the particle-hole symmetry, the stable second-order topology degenerates into the fragile Wannier obstruction. More specifically, from Fig.  3 c, the six-orbital Wannier representation displays a filling anomaly. Indeed, viewed as an insulator, if we neglect the difference between two 2a sites in Fig.  3 c, and combine both the ionic charge and electronic charge at 2a, the configuration is exactly the same as the Benalcazar-Bernevig-Hughes model 35 for higher-order topology protected by fourfold rotation symmetry (equivalent with our S 4 z ), only rotated by 45 degrees. It can be verified from ref. 67 that our model hosts a corner charge e /2 because of the mismatch of charge neutrality and rotation symmetry, which ensures a degeneracy of four corner states. In our system, corner states are pinned at zero energy by the particle-hole symmetry and they are Majorana zero modes. Since the filling anomaly requires only S 4 z , the corner zero modes are stable even when the corner is asymmetric under { M y ∣ 0 }. In fact, to reveal the corner charge in an S 4 z symmetric sample, one only needs to avoid the edge terminations (10) and (01) where gapless edge modes are present due to additional mirror symmetries { M y ∣ 0 }. Considering the various topology in the system, for clarity we summarize the relation between the symmetry and the topology in Table  1 .

Vortex-bound Majorana modes

In the the second-order TSC state in the above, each vortex can bind two Majorana modes which are stable due to the S 4 z symmetry. The phenomenon is closely related to the fact that for group P 4/ n m m , the effective theory near M in the normal state can be viewed as a direct sum of two Rashba electron gas systems with angular momenta J z  = ±1/2 and J z  = ±3/2 separately. To make it clearer, we consider the low-energy theory near M in the second-order TSC state, for instance,

where q is defined with respect to the M point and the identity matrices are omitted for simplicity. Ignoring the high-order \({t}^{{\prime} }{q}_{x}{q}_{y}\) term, it is obvious to notice that \({{{\mathcal{H}}}}_{{{\rm{eff}}}}\) can be decoupled in the σ space, i.e., the sublattice space. In the σ  = ±1 subspace, it describes a superconducting Rashba electron gas in the presence of a Zeeman field  ±Δ AFM ; and in each subspace, the vortex can bind a single Majorana mode 10 carrying \({{{\mathcal{S}}}}_{4z}\) eigenvalue 1. Notice that in the presence of a vortex, the S 4 z symmetry takes eigenvalues  ±1 and  ± i . However, the σ  = +1 subspace is spanned by the Kramers’ doublet J z  = ±1/2, while σ  = −1 subspace is spanned by J z = ∓ 3/2, which can be inferred from the basis of \({{{\mathcal{H}}}}_{{{\rm{eff}}}}\) . When we consider the \({{{\mathcal{S}}}}_{4z}\) eigenvalue of the Majorana mode, in the σ  = −1 subspace, the basis contributes an additional phase factor e i ∓ π  = −1. Therefore, the vortex-bound Majorana mode in the σ  = ±1 subspace has \({{{\mathcal{S}}}}_{4z}\) eigenvalue  ±1. The two Majorana modes are immune to perturbations preserving the \({{{\mathcal{S}}}}_{4z}\) symmetry, such as the \({t}^{{\prime} }{q}_{x}{q}_{y}\) term in Eq. ( 12 ). Namely, the second-order TSC state in the above supports two Majorana modes in each vortex protected by the S 4 z symmetry, i.e., one with S 4 z eigenvalue  +1 and the other  −1. We carry out numerical simulations for the vortex-bound states and present the results in Fig.  3 f–h. It is interesting to notice that in the second-order TSC state, the corner MZMs in Fig.  3 e coexist with the two vortex-bound MZMs. This arises from the fact that, the vortex core is far away from the corners, making the corner MZMs can hardly feel the effect of the vortex.

FM order induced first-order TSCs

We also consider FM order in the system in Eq. ( 5 ), and we assume the magnetic polarization along the z direction. Correspondingly, the whole system can be depicted by the following Hamiltonian

where Δ FM is the strength of the FM order. We first study how the FM order affects the fourfold degenerate fermion at M in the normal state. According to the real-space configuration in Fig.  4 a, the symmetry of the system is lowered to the type-III magnetic space group \(P4/n{m}^{{\prime} }{m}^{{\prime} }\) ( # . 129.417)

with \({S}_{4}^{D}\) the double group generated by { S 4 z ∣ 0 }. All the symmetry operations in \({{{\mathcal{G}}}}_{{{\rm{FM}}}}/T\) maintain at the M point. A direct group theory analysis shows that the 4D IR in the paramagnetic state at M splits into two 2D IRs, similar to the AFM case. However, differently in the FM case, one corresponds to twofold band degeneracy between the J z  = 1/2 and J z  = −3/2 states and the other between the J z  = −1/2 and J z  = 3/2 states (more details in Supplementary Note  3 ). Such degeneracies can be understood from the anticommutation relation between { S 4 z ∣ 0 } and { I ∣ τ 0 } at M, proved in Eq. ( 2 ). We confirm the above analysis numerically in Fig.  4 a.

a The bands in the presence of the FM order with Δ FM  = 0.3. The inset in ( a ) illustrates the real-space configuration of the FM order. The blue dashed line in ( a ) represents the chemical potential at μ  = 4.0. b shows the superconducting edge modes on the (10) edge corresponding to the bands in ( a ), with the edge modes on the right (left) edge marked by the red (green) color. In ( a ) and ( b ), the other parameters are the same as those in Fig.  2 .

To study the topological property in systems depicted by \({{{\mathcal{H}}}}_{{{\rm{BdG,FM}}}}\) corresponding to the normal bands in Fig.  4 a, we first calculate the Chern number. Based on the formula in Eq. ( 9 ) and the above analysis, the Chern number can be calculated to be ∣ C h ∣ = 2, whose sign depends on the sign of Δ FM . To verify this, we simulate the superconducting edge modes numerically. As shown in Fig.  4 b, two chiral modes appear on each edge corresponding to the normal state in Fig.  4 a, which is consistent with the above analysis. In fact, the above chiral TSC state arises through a gap-close-reopen process at M as the FM order becomes stronger, and the phase transition occurs at \({(4t+\mu )}^{2}+{\Delta }_{{{\rm{sc}}}}^{2}={\Delta }_{{{\rm{FM}}}}^{2}\) . Accordingly, in the weak-pairing condition, the system is a TSC with ∣ C h ∣ = 2, as long as the chemical potential is in the FM gap (details in Supplementary Note 4 ). Moreover, the vortex in the chiral TSC state can also bind two Majorana modes, and the analysis is similar to that of the AFM case. We present a more detailed analysis and simulate the vortex-bound states numerically in Supplementary Note  8 .

We discuss the effects of the symmetry-breaking perturbations (For the TSC states, more essential are the symmetries in the magnetic states rather than the specific magnetic orders considered in Figs. 3 a and 4 a. Therefore, here we refer to the perturbations breaking the magnetic group symmetries.), which may arise from tilting the magnetization off the z direction in Figs.  3 a and   4 a, on the above TSC states. Obviously, the vortex-bound Majorana modes are sensitive to the { S 4 z ∣ 0 } breaking perturbations and will gap out immediately. However, the Majorana edge and corner modes can persist against the perturbations. The chiral TSC state is robust as long as the bulk energy gap is not closed. For the second-order TSC state, perturbations breaking \(\{{M}_{y}| {{\bf{0}}}\}{{\mathcal{T}}}\) gap out the helical Majorana mode on the (10) edge and break the \({\mathbb{Z}}\) classification of the corner Majorana modes to a \({{\mathbb{Z}}}_{2}\) one. Nevertheless, the corner Majorana modes can be more robust due to a S 4 z protected filling anomaly, or due to the boundary obstruction 47 , 68 .

In the proposal, the FM order can be replaced by an external magnetic field. More difficult is to construct the antiferromagnetic heterostructures, which require well-matched lattices between the magnetic layer and the layer offering the band degeneracy. A possible candidate is the heterostructure between the antiferromagnetism A Co 2 As 2 ( A  = Ca, Ba, Sr) and the iron-based superconductors, whose lattice constants are similar 69 , 70 . A more feasible scheme lies in the magnetic materials. For example, in Eu 1− x La x FeAs 2 71 and Sr 2 VO 3− δ FeAs 72 , magnetic layers exist next to the superconducting FeAs layer; and in Ba 1− x Na x Fe 2 As 2 73 and Ba 1− x K x Fe 2 As 2 74 , a tetragonal AFM phase may coexist with the superconductivity. By methods of doping or gating, one may tune the chemical potential in the iron-based superconductors near the fourfold band degeneracy, and topological superconductivity can possibly be realized. We use the genuine bands of the iron-based superconductors to simulate the topological superconductivity in Supplementary Note  9 .

In the above analysis, we have mainly focused on the TSC states in the space group P 4/ n m m and the possible material realization. However, as pointed out, our method can be applied to a wide range of systems with band degeneracy near the Fermi energy. To further demonstrate the effectiveness of our method, we analyze another case where the lattice respects the space group P 4/ n c c . The group protects an eightfold band degeneracy at ( π ,  π ,  π ) in the normal state. When conventional superconductivity is introduced, both the FM order and the C-type AFM order drive the system into the nodal TSC states, but the topological properties are different. More detailed analyses are presented in the Supplementary Note  10 . Another interesting point worth mentioning is that the symmetry of the system in the presence of the magnetic order is determined by both the type of the magnetic order and the direction of the spin polarization, and it is possible that the higher-order TSC states can be realized by the simpler FM order, which deserves further study in the future.

In summary, we propose a general method that is based on the bulk energy bands and the conventional superconductivity to realize topological superconductivity. We show that by manipulating systems with crystal symmetry-protected fermionic excitations with magnetism, TSCs, including the high-order ones, can be generally obtained when conventional superconductivity is introduced, and the property of the TSCs is thoroughly determined by the property of the magnetism. Thus, our study provides a new method to realize the various types of topological superconductivity and can help to find new platforms to realize the Majorana modes.

Near the end of the paper, we became aware of a work 75 in which the vortex-bound states in high-order TSCs are studied, and the conclusion of the work is consistent with our results in the second-order TSC state in the AFM case.

Symmetries in time-reversal invariant superconductors

Generally, a superconductor can be described by the following BdG Hamiltonian

in the basis \({\psi }^{{\dagger} }({{\bf{k}}})=({c}_{{{\bf{k}}},\uparrow }^{{\dagger} },\, {c}_{{{\bf{k}}},\downarrow }^{{\dagger} },\, {c}_{-{{\bf{k}}},\uparrow },\, {c}_{-{{\bf{k}}},\downarrow })\) . Notice that we neglect other indices except for the spin index here. For a time-reversal symmetric superconductor, it respects the following three symmetries: the time-reversal symmetry \({{\mathcal{T}}}\) , the particle-hole symmetry \({{\mathcal{P}}}\) and the combined chiral symmetry \({{\mathcal{C}}}={{\mathcal{P}}}{{\mathcal{T}}}\) . These symmetries act on the Hamiltonian as follows

Moreover, in the basis for \({{{\mathcal{H}}}}_{{{\rm{BdG}}}}({{\bf{k}}})\) in Eq. ( 15 ), the above symmetries take the form \({{\mathcal{T}}}=i{s}_{2}{\kappa }_{0}K,\,{{\mathcal{P}}}={s}_{0}{\kappa }_{1}K\) and \({{\mathcal{C}}}=i{s}_{2}{\kappa }_{1}\) . Besides the above local symmetries, the system also respects the crystalline symmetries. The crystalline symmetry \(\widetilde{g}\) transforms the BdG Hamiltonian as \(\widetilde{g}{{{\mathcal{H}}}}_{{{\rm{BdG}}}}({{\bf{k}}}){\widetilde{g}}^{-1}={{{\mathcal{H}}}}_{{{\rm{BdG}}}}({\widetilde{g}}^{-1}{{\bf{k}}})\) , and has the form

In the above equation, η is determined by the pairing symmetry, i.e., g Δ( k ) g T  =  η Δ( k ). In the present study, we focus on the conventional superconductivity, which belongs to the trivial irreducible representation of the crystalline symmetry group. Namely, η always equals 1 for \(\widetilde{g}\) in Eq. ( 17 ) in our consideration.

Then, we consider the commutation relation between the unitary chiral symmetry \({{\mathcal{C}}}\) and the crystalline symmetries. It can be directly shown

where we have taken use of the fact η  = 1 in \(\widetilde{g}\) . Recall that in a time-reversal symmetric system, the time-reversal symmetry commutes with all the crystalline symmetries, and in the normal state, it demands \(Tg{T}^{-1}=g=(i{s}_{2}K)g{(i{s}_{2}K)}^{-1}={s}_{2}{g}^{*}{s}_{2}\) where T stands for the time-reversal symmetry in the normal state. Therefore, we have \({{\mathcal{C}}}\widetilde{g}{{{\mathcal{C}}}}^{-1}=\widetilde{g}\) in Eq. ( 18 ), namely \([{{\mathcal{C}}},\, \widetilde{g}]=0\) . The above commutation relation leads to that for any eigenstate \(\left\vert \phi ({{\bf{k}}})\right\rangle\) of \({{{\mathcal{H}}}}_{{{\rm{BdG}}}}({{\bf{k}}})\) carrying energy E ( k ), its chiral partner \({{\mathcal{C}}}\left\vert \phi ({{\bf{k}}})\right\rangle\) possesses energy  − E ( k ) but the same symmetry eigenvalue with \(\left\vert \phi ({{\bf{k}}})\right\rangle\) for any crystalline symmetry. This means that in the level of the symmetry indicator, the system must be equal to the topological trivial superconductor. More detailed analyses are presented in Supplementary Note  1 .

The calculation of winding number

To analytically calculate the winding number at k y  =  π in the AFM case, we rewrite \({{{\mathcal{H}}}}_{{{\rm{BdG,AFM}}}}\) in Eq. ( 6 ) in the basis diagonalizing the pseudo-chiral symmetry \(\widetilde{{{\mathcal{C}}}}\) . After the basis transformation, \({{{\mathcal{H}}}}_{{{\rm{BdG,AFM}}}}\) takes an off-diagonal form in the Nambu space. In the specific AFM case, the off-diagonal block matrix Q ( k x ) in the upper right corner is

with \({q}_{\pm }({k}_{x})=-[2t(\cos {k}_{x}-1)-\mu ]{s}_{0}\pm 2\lambda \sin {k}_{x}{s}_{2}\pm {\Delta }_{{{\rm{AFM}}}}{s}_{3}\pm i{\Delta }_{{{\rm{sc}}}}{s}_{2}\) . Accordingly, the winding number along k y  =  π can be calculated as

Here, ν ( q ± ) characterizes the winding number of \(\det {q}_{\pm }({k}_{x})\) around the origin point in the complex plane. See more details in Supplementary Note  4 .

A short review of MEBR

When we place the bases \(\{{\phi }_{i}^{\alpha }\}\) of the irreducible co-representations u i of these on-site magnetic point groups \({{{\mathcal{G}}}}_{{{\bf{x}}}}\) at their corresponding Wyckoff positions x , the induced co-representation \({({u}_{i})}_{{{\bf{x}}}}\uparrow {{\mathcal{G}}}\) of the space group \({{\mathcal{G}}}\) from the irreducible co-representations of the subgroup \({{{\mathcal{G}}}}_{{{\bf{x}}}}\) is referred to as magnetic elementary band representations (MEBR). In the AFM case, the four negative-energy bands host the co-representations

At G point: \({\bar{\Gamma }}_{7}\oplus {\bar{\Gamma }}_{7}\)

At M point: \({\bar{M}}_{7}\oplus {\bar{M}}_{7}\)

At X point: \({\bar{X}}_{3}{\bar{X}}_{5}\oplus {\bar{X}}_{2}{\bar{X}}_{4}\)

Therefore, our target band can only be expressed as a combination of MEBRs with the negative integer

which implies the fragile topology. See more details in Supplementary Note  7 .

Model Hamiltonian used for SG 130

To illustrate the effectiveness and generality of our method, we introduce a more complex example for space group P 4/ n c c ( # . 130). We start with the paramagnetic normal state, where the system actually respects the type-II magnetic space group \(P4/ncc{1}^{{\prime} }\) . The group can be generated by the following symmetry operations

The magnetic space group \(P4/ncc{1}^{{\prime} }\) has one and only one eightfold irreducible representation at the A point, i.e., the ( π ,  π ,  π ) point, in the spinful condition. Namely, all the bands are eightfold degenerate and respect the same low-energy effective model in the spinful case. In the lattice model condition, the eightfold band degeneracy can be captured by the following tight-binding model 76

Based on this model, we study the possible TSC states in the system when conventional superconductivity and different magnetic orders are introduced. More details are presented in Supplementary Note  10 .

Data availability

All data needed to evaluate the conclusions in the study are present in the paper and/or the  Supplementary Information . The data that support the findings of this study are available from the corresponding authors upon request.

Code availability

The computer code used for numerical calculation and theoretical understanding is available upon request from the corresponding authors.

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Acknowledgements

This work is supported by the Ministry of Science and Technology (Grant No. 2022YFA1403901 and No. 2022YFA1403902), National Natural Science Foundation of China (Grant No. NSFC-12304163, NSFC-12325404, and NSFC-11920101005), National Key R & D Program of China (Grant No. 2022YFA1403800 and No. 2023YFA1406700), Innovation program for Quantum Science and Technology (Grant No. 2021ZD0302500), Chinese Academy of Sciences (Grant No. XDB33020000 and No. JZHKYPT-2021-8), the New Cornerstone Investigator Program, and the Beijing Institute of Technology Research Fund Program for Young Scholars. Z.F.W. and Y.W. are supported by NSF under award number DMR-2045781.

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Zhongyi Zhang

Beijing National Research Center for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, 100190, Beijing, China

Zhongyi Zhang, Chen Fang & Jiangping Hu

University of Chinese Academy of Sciences, 100049, Beijing, China

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Department of Physics, University of Florida, Gainesville, Florida, 32601, USA

Zhenfei Wu & Yuxuan Wang

Kavli Institute for Theoretical Sciences and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, 100190, Beijing, China

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Zhang, Z., Wu, Z., Fang, C. et al. Topological superconductivity from unconventional band degeneracy with conventional pairing. Nat Commun 15 , 7971 (2024). https://doi.org/10.1038/s41467-024-52156-1

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Quality and health impact of groundwater in a coastal region: a case study from west coast of southern India

• Research Article
• Published: 12 September 2024

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• Ayushi Agarwal 1 , 2 &
• Ratnakar Dhakate 1 , 2  

Seawater intrusion seriously threatens the quality of coastal groundwater, affecting nearly 40% of the world’s population in coastal areas. A study was conducted in the Kamini watershed situated in the Udupi district of Karnataka to assess the groundwater quality and extent of seawater intrusion. During the pre-monsoon period, 57 groundwater and 3 surface water samples were analyzed to understand the impact of seawater on the groundwater and surface water. The analysis revealed that the groundwater in the study area is slightly alkaline. The weighted overlay analysis map indicated that 11% of the study area is unsuitable for drinking water due to the influence of seawater. The Piper plot analysis revealed that the groundwater is predominantly CaMgCl facies. The hydrogeochemical facies evolution diagram (HFED) showed that 62% of the groundwater is affected by seawater. The HFED and Piper plots also indicate that the surface water is also affected by seawater. These results are also supported by various molar ratios such as Cl − vs. Cl⁻/HCO 3 ⁻, Cl⁻ vs. Na⁺/Cl⁻, Cl − vs. SO 4 2− /Cl − , and Cl⁻/HCO 3 − vs. Mg 2+ /Ca 2+ , suggesting that the majority of the water sample has been affected by seawater. The saturation indices indicated that mineral dissolution has significantly contributed to groundwater salinization. The correlation between sulfate concentration and calcite and dolomite dissolution suggested the influence of seawater intrusion in the coastal aquifer. The process of reverse ion exchange mainly influences the groundwater chemistry according to chloroalkali indices. The total hazard index (THI) values of nitrate and fluoride exceeded limits, posing health risks to adults and children. Studies suggest that with time and space, seawater intrusion is increasing in some pockets of the study area, especially along the west coast.

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Acknowledgements

The authors express their sincere gratitude to the Director, CSIR-NGRI, Hyderabad, for his continuous support of the research activity. The authors express their sincere thanks to the Editor-in-Chief for his encouragement and support. The author also thanks the anonymous reviewers for their constructive and scientific suggestions for improving the manuscript standard. The manuscript Reference No. is NGRI/Lib/2024/Pub-009.

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Ayushi Agarwal: formal analysis, investigation, methodology, software, figure preparation, validation, and manuscript writing.

Ratnakar Dhakate: conceptualization, supervision, data curation, methodology, validation, review and editing.

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Agarwal, A., Dhakate, R. Quality and health impact of groundwater in a coastal region: a case study from west coast of southern India. Environ Sci Pollut Res (2024). https://doi.org/10.1007/s11356-024-34930-2

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DOI : https://doi.org/10.1007/s11356-024-34930-2

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Malaysian police rescue 400 children from care homes after sex abuse claims

Police say victims were the children of employees working for a business group allegedly in charge of the homes.

Warning: The story below contains details about abuse in care homes.

Malaysian authorities have rescued more than 400 children suspected of being sexually and physically abused at charity homes run by a prominent business group, police said.

Following coordinated raids on 20 premises across two states on Wednesday, police rescued 402 children and arrested 171 adults – including religious teachers and caretakers, according to Inspector-General of Police Razarudin Husain.

The homes were run by Global Ikhwan Services and Business (GISB), Razarudin said.

Religious authorities in Selangor state on Thursday widened their investigation into GISB, which said it did not run the homes.

Those rescued included 201 boys and 201 girls, aged between one and 17, after reports were filed this month that alleged neglect, abuse, sexual harassment and molestation, Razarudin told a news conference.

He did not say who the reports were from.

The Selangor Islamic Religious Department (JAIS) said it had asked police to hand over any teaching material seized during the raids in Selangor and Negeri Sembilanto states, so it could decide if any offences were committed under Islamic laws, which are implemented alongside secular laws in the country’s dual-track legal system.

GISB said in a statement: “It is not our policy to do things that go against Islam and the laws.”

Preliminary police investigations determined that the affected children were the sons and daughters of Malaysian employees of GISB, a self-described “Islamic” company involved in businesses ranging from supermarkets to laundromats.

Razarudin said the children were sent to the homes shortly after they were born, and that they had been subjected to multiple forms of abuse, allegedly sexually abused by adult guardians and later taught to sexually abuse other children.

“Those who were sick were not allowed to seek medical attention until their condition became critical,” he said. Some young children were also burned with a hot spoon when they made mistakes, and caretakers had touched the children’s bodies as if to conduct medical checks, he said.

He said the children would be temporarily housed at a police training centre in the capital Kuala Lumpur and would undergo health checks.

Police believed that GISB, which operates in a number of countries, including Indonesia, Singapore, Egypt, Saudi Arabia and France, exploited the children and used religious sentiments to collect donations, said Razarudin.

‘Shocked and appalled’

The case is being investigated under laws covering sexual offences against children and human trafficking.

Two of the premises raided were registered with the state government as Islamic schools, JAIS said in a statement on Thursday.

The department had monitored the schools as recently as July but found no offences had been committed. It said it would investigate further and take appropriate action should any violations be found.

A GISB spokesperson said on Thursday that the group will cooperate with authorities.

GISB has been linked to the now-defunct Malaysia-based Al-Arqam religious sect, which was banned by the government in 1994. On its website, the company says that its aim is “to develop the Islamic way of life”.

Robert Gass, a representative in Malaysia for the United Nations Children’s Fund (UNICEF), said on Thursday that the organisation was “deeply shocked and appalled” by the alleged abuse and called for long-term professional medical and psychosocial support for the children.

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Sustainable spatial development of multifunctional villages: a case study of eastern poland.

1. Introduction

Spatial characteristics of a sustainable village, 2. materials and methods, 3.1. changes in the compactness of development, 3.2. preservation of traditional rural spatial arrangements, 3.3. availability of services, 4. discussion, 5. conclusions, author contributions, data availability statement, conflicts of interest.

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Soszyński, D.; Kociuba, P.; Tucki, A. Sustainable Spatial Development of Multifunctional Villages: A Case Study of Eastern Poland. Sustainability 2024 , 16 , 7965. https://doi.org/10.3390/su16187965

Soszyński D, Kociuba P, Tucki A. Sustainable Spatial Development of Multifunctional Villages: A Case Study of Eastern Poland. Sustainability . 2024; 16(18):7965. https://doi.org/10.3390/su16187965

Soszyński, Dawid, Piotr Kociuba, and Andrzej Tucki. 2024. "Sustainable Spatial Development of Multifunctional Villages: A Case Study of Eastern Poland" Sustainability 16, no. 18: 7965. https://doi.org/10.3390/su16187965

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