Altermagnetic Shastry-Sutherland fullerene networks

TL;DR

This paper introduces altermagnetic Shastry-Sutherland fullerene networks, a new class of molecular quantum materials exhibiting diverse quantum phases and unique magnetic properties, accessible through strain tuning.

Contribution

It presents the design and analysis of a novel molecular network with altermagnetic order, expanding the landscape of quantum magnetic materials with potential for scalable applications.

Findings

Discovery of an altermagnetic ground state with compensated spins.

Identification of a rich phase diagram including quantum spin liquid and dimer phases.

Demonstration of strain-tunable quantum phases in molecular networks.

Abstract

The interplay between quantum magnetism and many-body physics is of fundamental importance in condensed matter physics. %Magnetic exchange interactions in frustrated lattices give rise to rich phase diagrams. Molecular building blocks provide a versatile platform for exploring the exotic quantum phases arising from complex orderings in frustrated lattices. Here we demonstrate a showcase system based on altermagnetic Shastry-Sutherland fullerene networks, which can be constructed from a C$_{40}$ molecular synthon with two effective spin-1/2 sites due to the resonance structures. The charge-neutral, pure-carbon systems exhibit an altermagnetic ground state with fully compensated spins arranged in alternating C$_{40}$ units in a 2D rutile-like lattice, leading to $d$-wave splitting of the spin-polarised electronic band structure and strong chiral-split magnons. We report a rich phase diagram including altermagentic, quantum spin liquid, plaquette, and dimer phases, which can be accessed via moderate strains. Our findings open a new avenue for exploring quantum many-body physics based on scalable, chemically-feasible, molecular quantum materials.

Equations

$$J_{\mathrm{eff}}=\frac{\sum S_i^{(1)}{J}_{ij}^{(1,2)}{S}_j^{(2)}}{\sum S_i^{(1)}\sum S_j^{(2)}},$$

Full text

Magnetism in low-dimensional systems provides fertile ground for the discovery of exotic quantum phases Sun et al. (2025); Su et al. (2025); Fu et al. (2025); Peng and Lu (2025); Song et al. (2025). Beyond conventional ferromagnetism and antiferromagnetism, altermagnetism has emerged as a newly-identified class of collinear magnetism with compensated spin sublattices but finite momentum-dependent spin polarisation Hayami et al. (2019); Yuan et al. (2020); Šmejkal et al. (2020); Ma et al. (2021); Mazin et al. (2021); Šmejkal et al. (2022a, b); Fedchenko et al. (2024); Krempaský et al. (2024). The vanishing net magnetisation and symmetry-protected spin splitting offer tantalising opportunities for next-generation quantum devices with robust spin-polarised transport. Moreover, the interplay between altermagnetism and many-body physics leads to a rich variety of quantum phenomena such as superconductivity Bose et al. (2024); Pupim and Scheurer (2025), spin liquid Sobral et al. (2025), Hubbard model Sato et al. (2024); Das et al. (2024); Giuli et al. (2025); He et al. (2025), and Kondo lattice Zhao et al. (2025). In this context, quantum magnets in the Shastry-Sutherland model Sriram Shastry and Sutherland (1981); Miyahara and Ueda (1999); Knetter et al. (2000); Koga and Kawakami (2000); Läuchli et al. (2002); Dorier et al. (2008); Brassington et al. (2024) represent a showcase model to realise altermagnetism, as their neighbouring sublattices are connected by rotations and glide reflections in the absence of neither translational nor inversion symmetry Ferrari and Valentí (2024). Furthermore, the Shastry-Sutherland lattice provides an ideal platform to explore the exotic quantum physics and complex many-body effects in altermagnets such as superconductivity Chung and Kim (2004); Yang et al. (2008), strong correlation Liu et al. (2007), quantum phase transition Lee et al. (2019); Shi et al. (2022), and spin frustrations Pula et al. (2024); Corboz et al. (2025). Yet, realising altermagnetism in the Shastry-Sutherland lattice remains a challenge.

Carbon-based materials provide unpaired $\pi$ electrons in delocalised $p_{z}$ orbitals, which are typically found in defects such as edges, vacancies, or heteroatom sites that disrupt homogeneous electron density Yazyev (2010); Slota et al. (2018); Ma et al. (2025). However, it is challenging to synthesise these systems due to atomically-precise, periodic assembly of the defected carbon fragments into the $\pi$-conjugated frameworks. Comparing to carbon fragments, fullerene building blocks are promising alternatives due to their stable units and highly-tuneable structures. We have recently discovered approaches to systematically introduce magnetism into pure-carbon, charge neutral, fullerene monolayers by enforcing crystalline symmetry based on molecular-orbital theory Wu et al. (2025) or by creating $\pi$ magnetism from resonating valence bond Peng and Pizzochero (2025a), opening new avenues for realising exotic quantum phases such as quantum anomalous Hall effects in otherwise non-magnetic monolayers Pingen et al. (2025). Most importantly, the inherent ability of assembling these molecules into periodic frameworks has been confirmed since the synthesis of polymeric C60 monolayers Hou et al. (2022), as demonstrated by their rich structural phases as predicted theoretically Peng (2022, 2023); Jones and Peng (2023); Wu and Peng (2025); Shearsby et al. (2025); Kayley and Peng (2025); Shaikh and Peng (2025); Peng and Pizzochero (2025b, c) and synthesised experimentally Meirzadeh et al. (2023); Wang et al. (2023); Zhang et al. (2025a). Therefore, it would be insightful to introduce altermagnetism into fullerene-based networks as a promising platform to realise exotic models such as the Shastry-Sutherland lattice.

Here, we use fullerene building blocks to design quantum altermagnets in 2D Shastry-Sutherland lattice. The key to the realisation of altermagnetism lies in the elegant design of a molecular network that incorporates both a C40 unit, with unpaired electrons for $\pi$ magnetism, and a rutile-like 2D lattice, for constructing two sublattices with opposite spins. We show that the resonance structure of the C40 cage provides six carbon sites for two unpaired electrons on the opposite sides of the molecule and stabilises two effective spin-1/2 sites, which, upon assembly into a rutile-like monolayer, realise an altermagnetic ground state. Our first-principles calculations reveal $d$-wave splitting in the electronic bands and strong chiral-split magnon dispersions. Most remarkably, the altermagnetic Shastry-Sutherland networks can be continuously tuned into the frustrated quantum spin liquid phase through moderate strain. Our work opens a new frontier that unites the chemistry of carbon-based molecules with the physics of strongly-frustrated quantum magnetism.

Altermagnetic fullerene networks. Figure 1a shows a C40 fullerene molecular synthon, which serves as a stable building block that has been synthesised decades agoTomilin et al. (2001); Enyashin and Ivanovskii (2008); Kharlamov et al. (2013). Among the thirty carbon atoms on the top surface of C40, there are twelve fully-saturated carbon atoms with $sp^{3}$ hybridisation for intermolecular bonds (marked in yellow). For the eight carbon atoms from the two five-membered rings in the centre of C40, each atom is surrounded by three single bonds due to the $sp^{2}$ hybridisation, leaving eight $p_{z}$ electrons to form four double bonds as marked by red in Fig. 1a. Consequently, all twenty carbon atoms above are non-magnetic.

We then focus on the ten carbon atoms in two W-shaped chains, as highlighted in purple boxes in Fig. 1a. For each W-shaped cluster, there are three pairing schemes for two double bonds between the five carbon atoms, leaving one unpaired electron at the leftmost, middle and rightmost sites of the chain, respectively. The resonance structure leads to an averaged distribution of 1/3 $\mu_{\rm B}$ magnetic moment per purple carbon atom and a quantised total magnetic moment of 1 $\mu_{\rm B}$ for the group of five atoms in the W-shaped chain. Therefore, a C40 unit has six magnetic carbon atoms with a total magnetic moment of 2 $\mu_{\rm B}$, as confirmed by first-principles calculations.

Afterwards, we construct closely-packed fullerene networks by rotating neighbouring C40 building units by $\pi/2$, and such $C_{4}$ rotation creates a 2D rutile-like lattice similar to RuO2 Fedchenko et al. (2024). The space-efficient arrangement of fullerene molecules is expected to stabilise the structure as found experimentally Hou et al. (2022); Meirzadeh et al. (2023). Most interestingly, the closely-packed lattice leads to an altermagnetic ground state that is energetically favourable than the ferromagnetic, non-magnetic and other antiferromagnetic order, respectively. This suggests stronger intramolecular exchange interactions than the intermolecular coupling (as discussed later). Figure 1b shows the corresponding spin densities where magenta (cyan) iso-surface represents spin-up (spin-down) states. Within each C40 unit, the evenly distributed spin densities among the six magnetic carbon atoms confirm the resonance structure of 1/3 $\mu_{\rm B}$ per magnetic carbon atom. The neighbouring C40 units are connected by $C_{4}$ rotations and $G_{x}/G_{y}$ glide reflections in the absence of neither translational nor inversion symmetry, leading to compensated antiparallel magnetic order. Such an alternating order of the magnetic moments results in a zero net magnetisation. Additionally, for fullerene units with light carbon elements, the spin-orbit coupling approaches the non-relativistic limit. Therefore, the vanishing net magnetisation has a strong non-relativistic origin.

We next examine the band structures of altermagnetic C40 networks in Fig. 1c. The spin-up and spin-down bands are doubly degenerate along the $\Gamma$–X–M high-symmetry paths. Along M–$\Gamma$–M’, however, the symmetry-protected collinear compensated magnetic order in real space generates an alternating spin polarisation in the band structure in the reciprocal momentum space, and the spin-up and spin-down bands become split, breaking the time-reversal symmetry without magnetisation. We further compute the iso-energy surface at 0.025 eV below the valence band maximum (VBM), which corresponds to moderate hole doping. As shown in Fig. 1d, the spin orientations at time-reversed, opposite momenta on the iso-energy surface are the same (non-time-reversed), while the spin-up and spin-down bands are symmetric, exhibiting typical features for $d$-wave order similar to RuO2 Fedchenko et al. (2024).

To further confirm the altermagnetism, we compute the magnon dispersion. As shown in Fig. 1e, the low-frequency magnons near $\Gamma$ show linear dispersion, which is a typical antiferromagnetic/altermagnetic featureKittel (1976). The degenerate magnon bands along $\Gamma$–X–M exhibit giant chiral splitting along M–$\Gamma$–M’ in the absence of spin-orbit coupling. The chiral magnon modes have either left-handed or right-handed spin precession along a given Néel vector with opposite precessional angular momentum. Similar to the spin splitting in the electronic band structure, the chiral splitting of magnon bands in altermagnets even without relativistic effects differs altermagnets from ferromagnets and antiferromagnets Šmejkal et al. (2023); Zhang et al. (2025b), giving rise to novel applications in altermagnetic spintronics and magnonics. All magnon frequencies are real and non-negative with a global minimum at $\Gamma$, which, again, indicates a stable altermagnetic ground state Tellez-Mora et al. (2024).

Exchange interactions. We then examine the exchange interactions in altermagnetic C40 monolayers. As shown in Fig. 2a, the three magnetic carbon atoms in the W-shaped chain share one $\pi$ electron, forming an effective spin-1/2 group. The exchange interactions between atoms $i$ and $j$ within one effective spin-1/2 group are denoted as $J_{ij}$, the intramolecular interactions between the two effective spin-1/2 groups are denoted as $J_{ij^{\prime}}$, while the intermolecular interactions from the nearest neighbouring effective spin-1/2 groups are denoted as $J_{ij^{\prime\prime}}$.

Figure 2b summarises the calculated $J$ parameters as a function of distance between these magnetic atoms. Within one effective spin-1/2 group, the interactions are ferromagnetic, with much stronger coupling strength ($J_{ij}>7$ meV) than $J_{ij^{\prime}}$ and $J_{ij^{\prime\prime}}$.

Intramolecular $J_{ij^{\prime}}$ coupling is much smaller (between $-0.9$ and 0.5 meV) than $J_{ij}$. The antiferromagnetic couplings ($J_{13^{\prime}}=-0.38$ meV, $J_{11^{\prime}}=J_{33^{\prime}}=-0.90$ meV) are stronger than the ferromagnetic ones ($J_{22^{\prime}}=0.48$ meV, $J_{12^{\prime}}=J_{23^{\prime}}=0.04$ meV). Interestingly, the long-range exchange interactions $J_{11^{\prime}}$ and $J_{33^{\prime}}$ from the largest intramolecular distance of 6.49 Å have the strongest coupling strength, whilst the smallest $J_{12^{\prime}}$ and $J_{23^{\prime}}$ have a relatively shorter distance of 5.98 Å.

The intermolecular interactions are all antiferromagnetic, with much weaker interaction strength than that of the intramolecular coupling. The largest intermolecular coupling comes from atom 1 and atom 3” ($J_{13^{\prime\prime}}=-0.42$ meV) at a distance of 7.27 Å, whereas the smallest intermolecular coupling corresponds to $J_{22^{\prime\prime}}=-0.10$ meV at a smaller distance of 5.65 Å. The intermolecular interactions beyond the nearest neighbouring group quickly decay to zero.

Phase diagram of the Shastry-Sutherland lattice. Having understood the exchange interactions of monolayer C40 networks, we can build a lattice spin model by treating the three magnetic atoms in the effective spin-1/2 group as one spin-1/2 site, as these three atoms have dominant ferromagnetic interactions.

The effective spin-1/2 sites form the well-known Shastry-Sutherland lattice, as shown in Fig. 3a. The effective exchange interactions between these effective spin-1/2 sites can be divided into two groups – intramolecular exchange $J_{0}$ (marked in dark red) and intermolecular exchange $J_{1}$ (marked in blue). The effective exchange interactions can be defined as

[TABLE]

where $S^{(1)}_{i}$ and $S^{(2)}_{j}$ indicate electron spin (taken as 1/6) from atomic site $i$ from effective spin-1/2 group 1 and atomic site $j$ from effective group 2 respectively, and $J_{ij}^{(1,2)}$ represents their exchange interactions. The intramolecular $J_{0}$ can be computed by choosing two effective spin-1/2 groups within the same molecule (i.e., $J_{ij^{\prime}}$ in Fig. 2a), while the $J_{1}$ can be calculated from the intermolecular, nearest neighbouring effective spin-1/2 groups (i.e., $J_{ij^{\prime\prime}}$ in Fig. 2a).

For monolayer C40 networks, we obtain effective exchange interactions $J_{0}=-0.21$ meV and $J_{1}=-0.25$ meV, leading to a ratio $J_{1}/J_{0}$ of 1.18. According to the phase diagram for the Shastry-Sutherland lattice in Fig. 3 b, this corresponds to the Néel phase where $J_{1}$ dominates over $J_{0}$, and the Shastry-Sutherland lattice effectively reduces to the altermagnetic ground state as we find in ab initio calculations. In the Néel state, the two spins within one C40 unit are the same, whereas neighbouring fullerene molecules have opposite spins, which is exactly the ground-state altermagnetic phase.

The phase diagram can be tuned by the ratio between intermolecular and intramolecular coupling $J_{1}/J_{0}$. As shown in Fig. 3 b, decreasing the $J_{1}/J_{0}$ ratio to 0.82 leads to a quantum spin liquid phase due to the frustrations between the intramolecular and intermolecular exchange couplings. This exotic phase has promising applications such as error-resistant topological qubits Klocke et al. (2024). When the $J_{1}/J_{0}$ ratio lies between 0.77 and 0.675, a phase of independent “plaquette” units is formed, where each unit consists of four spins. This is known as the plaquette phase. In 2D C40 networks, this phase corresponds to four spins located on four molecules around an interstice. Further decreasing the $J_{1}/J_{0}$ ratio below 0.675 results in the dimer valence bond solid phase. In the dimer phase, the two spins on each fullerene pair up into a singlet state as the intramolecular $J_{0}$ dominates, and the neighbouring singlet pairs stop to interact with each other.

To tune the ratios between $J_{1}$ and $J_{0}$, we introduce bi-axial strains. Figure 3 c shows the $J_{1}/J_{0}$ ratios computed for monolayer C40 networks under varied strain. For bi-axial strains between $-3$ and 2%, the systems remain in the Néel order, indicating an altermagnetic ground state. When further increasing the strains to 3$\%$, the C40 system passes through both the quantum spin liquid and plaquette phases, until reaching the dimer region. We use a strain step of 0.1% between $2-3$% strains. In the quantum spin liquid region, $J_{1}$ and $J_{0}$ become ill-defined, and our first-principles calculations show strong frustrations in the computed exchange interactions. In the plaquette phase, the four calculated $J_{1}$ terms around one spin-1/2 site are no longer symmetry equivalent but exhibit strong anisotropy, which agrees with the phase behaviours. Upon reaching the dimer phase, the four $J_{1}$ terms around one spin-1/2 site are equal again, leading to a well defined $J_{1}/J_{0}$ ratio.

Further unscreened hybrid functional calculations suggest that the dimer phase is stable at all strains between $\pm 3\%$ in the zero-screening limit, which provides the lower bound for $J_{1}/J_{0}$ in free-standing monolayers without any substrates. This suggests that the phase transitions can be induced through control of screening effects such as the application of substrates with varied dielectric constants. Additionally, the dimer phase itself also exhibits rich physics such as triplon-bound state and spin-nematic ordering Zayed et al. (2017); McClarty et al. (2017); Wang and Batista (2018); Wulferding et al. (2021), which can be further tuned by chemical decoration and exohedral doping. This will unlock future opportunities in realising room-temperature, organic quantum altermagnets to explore the complex many-body effects.

Methods. Density functional theory (DFT) calculations Hohenberg and Kohn (1964); Kohn and Sham (1965) are performed using the Vienna ab initio Simulation Package (VASP) Kresse and Furthmüller (1996a, b) under the generalised gradient approximation (GGA) formalism with projector augmented wave (PAW) basis sets Blöchl (1994); Kresse and Joubert (1999). The Perdew-Burke-Ernzerhof functional corrected for solids (PBEsol) Perdew et al. (2008) is used along with C $2s^{2}2p^{2}$ as valence electrons. A plane-wave cutoff of $800\,\mathrm{eV}$ is employed and the self-consistent field energy convergence criterion is set to be $10^{-6}\,\mathrm{eV}$. The Brillouin zone is sampled with a converged $\mathbf{k}$-mesh of $3\times 3$. Spin polarisation is included throughout all the calculations. The crystal structures are fully relaxed using the conjugate gradient method Payne et al. (1992) until the Hellmann-Feynman forces are less than $10^{-2}\,\mathrm{eV/\AA }$. The vacuum spacing between monolayers is set to be more than 20 Å, and dipole corrections are employed perpendicular to the monolayers to eliminate interactions between periodic images Makov and Payne (1995). A Wannier tight-binding model Marzari and Vanderbilt (1997); Souza et al. (2001); Marzari et al. (2012) is constructed using Wannier90 Mostofi et al. (2008, 2014); Pizzi et al. (2020) by projecting Wannier functions onto each $\sigma$ bond centre, with $sp^{2}$ carbon atoms having an extra Wannier function to mimic the $p_{z}$ orbitals. This bond-centre approach has been widely used in fullerene molecules Mostofi et al. (2008) and leads to 188 Wannier functions in a unit cell of two C40 molecules. The exchange interactions are computed from the Green’s function method Liechtenstein et al. (1987); Korotin et al. (2015) using the TB2J package He et al. (2021) where the positions of Wannier centres and the magnetic moments are carefully checked. The magnon dispersion is computed under the Holstein-Primakoff transformation Holstein and Primakoff (1940) based on the bosonic Hamiltonian Colpa (1978), as implemented in the Magnopy package that has been widely used to study low-dimensional antiferromagnets Rybakov et al. (2024); Boix-Constant et al. (2025). The chirality of magnons with band index $j$ is defined as the right-handed (RH) or left-handed (LH) spin precession along a given Néel vector with opposite precessional angular momentum Šmejkal et al. (2023); Yuan et al. (2025). To analyse the effect of strain, the lattice parameters are modified (with respect to the fully relaxed structure) and fixed, while the atomic coordinates are allowed to fully relax. Unscreened hybrid functional PBEsol0 calculations Adamo and Barone (1999) are also performed to study the exchange interactions in the zero-screening limit.

Acknowledgment. We thank Samzi Tishler at the University of Cambridge for proof reading. J.W. acknowledges support from the Cambridge Undergraduate Research Opportunities Programme and from Peterhouse for the James Porter Scholarship. B.P. acknowledges support from Magdalene College Cambridge for a Nevile Research Fellowship. The calculations were performed using resources provided by the Cambridge Service for Data Driven Discovery (CSD3) operated by the University of Cambridge Research Computing Service (www.csd3.cam.ac.uk), provided by Dell EMC and Intel using Tier-2 funding from the Engineering and Physical Sciences Research Council (capital grant EP/T022159/1), and DiRAC funding from the Science and Technology Facilities Council (http://www.dirac.ac.uk), as well as with computational support from the UK Materials and Molecular Modelling Hub, which is partially funded by EPSRC (EP/T022213/1, EP/W032260/1 and EP/P020194/1), for which access was obtained via the UKCP consortium and funded by EPSRC grant ref EP/P022561/1.