Tunable quantum anomalous Hall effect in fullerene monolayers

TL;DR

This paper proposes a new molecular design approach using C26 fullerene monolayers to realize tunable quantum anomalous Hall effects, offering a feasible route for experimental realization.

Contribution

It introduces a 2D honeycomb lattice of C26 fullerenes as a tunable platform for QAH physics with controllable magnetic and strain parameters.

Findings

Exhibits ferromagnetic ground state breaking time-reversal symmetry

Displays a rich phase diagram with multiple Chern numbers

Offers a chemically synthesizable route for QAH materials

Abstract

Nearly four decades after its theoretical prediction, the search for material realizations of quantum anomalous Hall effect (QAHE) remains a highly active field of research. Many materials have been predicted to exhibit quantum anomalous Hall (QAH) physics under feasible conditions but the experimental verification remains widely elusive. In this work, we propose an alternative approach towards QAH materials design by engineering customized molecular building blocks. We demonstrate this ansatz for a two-dimensional (2D) honeycomb lattice of C26 fullerenes, which exhibits a ferromagnetic ground state and thus breaks time-reversal symmetry. The molecular system is found to be highly tunable with respect to its magnetic degrees of freedom and applied strain, giving rise to a rich phase diagram with Chern numbers C= +/-2, +/-1, 0. Our proposal offers a versatile platform to realize tunable QAH physics under accessible conditions and provides an experimentally feasible approach for chemical synthesis of molecular networks with QAHE.

Full text

The quantization of transverse conductivity in a two-dimensional (2D) electron gas under the effect of a perpendicular magnetic field is known as the quantum Hall effect. Soon after its experimental observation in 1980 Klitzing et al. (1980), the underlying mechanism was explained theoretically Thouless et al. (1982) by associating its remarkable robust quantization with a topological invariant called Chern number Niu et al. (1985); Kohmoto (1985). In 1988, Haldane demonstrated that the same effect could be realized on a lattice even in the absence of net magnetic flux Haldane (1988), which is referred to as the quantum anomalous Hall effect (QAHE)  Qi and Zhang (2011); Hasan and Kane (2010); Weng et al. (2015); He et al. (2018); Chang et al. (2023); Liu et al. (2016). A material realization of such a bulk-insulating system would offer immense potential from a technological perspective: Possible application range from high-precision quantum sensing devices operated near topological phase transitions Korkusinski and Hawrylak (2014); Liu et al. (2021) to ultra-low power consumption topological transistors Nadeem and Wang (2024); Gilbert (2021) or the so-called Chern networks Gilbert (2025) which can be used for information processing by exploiting the emergence dissipationless edge states.

Haldane himself, however, already questioned the feasibility to implement his model, which indeed turned out to be a challenging endeavor. The first observation of the QAHE was reported in 2013 Chang et al. (2013) at a temperature of $30~\text{mK}$ and thus clearly below ambient conditions, preventing its immediate technological applicability. It was subsequently demonstrated that the temperatures could be increased for materials with intrinsic magnetic order instead of introducing the required time-reversal symmetry breaking via magnetic dopants Deng et al. (2020). As another class of material platforms Chang et al. (2023), the QAHE has further been realized in moiré materials at $\lesssim 3~\mathrm{K}$ Serlin et al. (2020). Nevertheless, for all three successfully demonstrated implementations, observation of the QAHE at experimentally-feasible temperature remains challenging. Besides the problem of reconciling ambient, experimental conditions under which such topological phases of matter emerge, it is crucial to identify more tunable materials, which allow for sensitive switching mechanisms between different topological phases. For early realizations, this could only be achieved by applying large magnetic fields, again hindering practical applicability. Over the past years, several material platforms have been theoretically and computationally predicted to exhibit flexible quantum anomalous Hall (QAH) physics Liang et al. (2025); Zhang et al. (2025); Das et al. (2024); Zhang et al. (2023); Wu et al. (2023); Li et al. (2022); Bultinck et al. (2020); Ge et al. (2020); Zhao et al. (2019); Wang et al. (2017); Wu et al. (2014); Liu et al. (2022a); Yang et al. (2025); Zhao et al. (2023) with the highest estimated Curie temperature $\sim 550~\mathrm{K}$ Chen et al. (2021). However, precise atomic engineering necessary to synthesize such compounds remains challenging in practice Mannix et al. (2017).

Here we pursue an alternative approach to realize QAH materials using stable molecular building blocks. We refer to this method as the superatom ansatz: The essential idea is to design material properties based on features exhibited by stable molecular units which behave like atoms. This ansatz allows for increased chemical tunability while facilitating less challenging implementation compared to atomic building blocks Peng (2022, 2023); Peng and Pizzochero (2025a, b); Wu and Peng (2025); Shaikh et al. (2025). In light of recent progress in synthesizing 2D fullerenic networks Hou et al. (2022); Meirzadeh et al. (2023), we demonstrate this approach in a honeycomb structure of $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ molecules. The origin of magnetization in fullerenic materials has recently been rationalized Wu et al. (2025a); Peng and Pizzochero (2025c); Wu et al. (2025b). Here, individual $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ molecules exhibit non-vanishing magnetization due to a spin-polarized ground state. Ferromagnetic exchange interactions facilitate time-reversal symmetry breaking, which is mandatory in order to observe QAH physics. The high degree of symmetry facilitated by the underlying hexagonal lattice commonly stabilizes massless quasiparticles of Weyl type, which can open topologically non-trivial band gaps upon considering spin-orbit coupling (SOC)  Zhang et al. (2025). We confirm this behavior for the direct band gap from first-principles. The weak SOC in carbon systems implies a high degree of magnetic anisotropy, enabling topological phase transitions by controlling the orientation of magnetization. Uniaxial strain provides further degrees of freedom, which can be exploited to induce transitions between distinct topological phases. Based on theoretical considerations and Wannier tight-binding (TB) models, we establish the rich phase diagram of honeycomb $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ polymeric fullerene monolayers. Upon inducing a global band gap for topological regimes, phase boundaries resemble 2D Weyl half-semimetals (WHSMs), featuring Weyl-like dispersion in a single spin channel around the Fermi energy You et al. (2019). The plethora of topological phases in combination with easily-accessible control mechanisms in molecular systems provide an unprecedented material realization for exploring QAH physics in practice – in particular regarding ease of synthesis and external control. Our results demonstrate the strengths of the superatom ansatz, which is anticipated to present a versatile approach towards designing functional quantum materials.

*Methods– * Our numerical results are based on a quasi-2D $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ monolayer structure which is relaxed using the conjugate gradients method Payne et al. (1992) implemented in the Vienna Ab initio Simulation Package (VASP) Kresse and Hafner (1993); Kresse and Furthmüller (1996, 1996); Kresse and Joubert (1999) based on density functional theory (DFT) Hohenberg and Kohn (1964); Kohn and Sham (1965) within the generalized gradient approximation (GGA) using the PBEsol functional Perdew et al. (2008). A plane-wave cutoff of $800~e\text{V}$ is used on a $5\times 5\times 1$ Monkhorst-Pack $\mathbf{k}$-point mesh for the 2D Brillouin zone (BZ). Electrons in 1s orbitals are frozen using the projector augmented wave (PAW) method. A vacuum layer of $\sim 35~\text{\AA }$ is introduced between two monolayers to study the quasi-2D system. Phonon band structures are obtained using the implementation of density functional perturbation theory (DFPT) Baroni et al. (2001); Gonze and Lee (1997) in VASP for a $2\times 2$ supercell and visualized using PHONOPY Togo et al. (2023); Togo (2023). To find the magnetic ground state, a TB model is constructed using Wannier90 Pizzi et al. (2020) by projecting onto $\sigma$ bond centres between all carbon atoms and one extra $p_{z}$ orbital of each $sp^{2}$ carbon atom, as widely used in fullerene systems Mostofi et al. (2008). The exchange interactions between fractional spins are calculated using TB2J He et al. (2021) and the magnon dispersion is obtained using Magnopy Holstein and Primakoff (1940); Colpa (1978); Ivanov et al. (2021) with $1/9~\mu_{\mathrm{B}}$ on each magnetic atom.

Structural & magnetic properties–

We demonstrate the superatom ansatz by employing $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ molecules as stable building blocks. The constituting carbon atoms bond to form three hexagons separated by a total of twelve pentagons, comprising molecular symmetry group $D_{3h}$. Figure 1a shows a 2D structure built from $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ molecules arranged on a honeycomb lattice. Its dynamical stability is demonstrated in terms of the absence of imaginary-frequency phonon modes in Fig. 1b.

The magnetization can be understood through the $\pi$ electrons of the fullerenes. The $\pi$ system of each molecule features two degenerate orbitals at the Fermi level that transform under the 2D irreducible representation (irrep) $E^{\prime}$ at $\Gamma$. These two orbitals are filled with three electrons, with one being unpaired and hence net $1\,\mu_{\mathrm{B}}$ magnetization per molecule (see Supplemental Material sup ). In the unit cell with two $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ molecules, these orbitals hybridise into eight bands (four spin-up and four spin-down). Filling with six electrons results in four filled spin-up and two filled spin-down bands, leaving two empty spin-down bands. The main features of these four bands near the Fermi level are well predicted by a minimum TB model (see Supplemental Material sup ).

Embedded into the monolayer structure, exchange interactions between magnetic fullerene molecules facilitate ferromagnetic order, giving rise to spontaneous time-reversal symmetry breaking. The magnetic ground state is demonstrated in terms of the absence of imaginary magnon modes with a global minimum at $\Gamma$ in Fig. 1c. Using the Multibinit package Wojdeł et al. (2013); Evans et al. (2014); Eriksson et al. (2017), we determine the magnetic order to persist up to a Curie temperature of $T_{\text{C}}\sim 25~\text{K}$.

The symmetry of the resulting quasi-2D crystal structure is described by space group (SG) $P\overline{6}2m$ (No. 189). Our forthcoming discussion focuses on the properties of the two bands in proximity of the Fermi energy in Fig. 2a. In the absence of SOC, we determine these states to transform under the 2D irreps of the little co-groups at K and $\text{K}^{\prime}$ using the IrVsp software package Gao et al. (2021). Further two-fold degeneracies are observed along all BZ boundaries connecting $\text{M}_{j\in\{1,2,3\}}$ with K and $\text{K}^{\prime}$. They are protected by both two-fold rotational symmetry around the axis perpendicular to the respective BZ boundary and reflection symmetry about the vertical plane projecting onto this axis. Consequently, degeneracies between the two relevant bands occur at a total of eight inequivalent lattice momenta. Locally, we confirm them to disperse linearly, giving rise to fermionic 2D Weyl quasiparticles with symmetry-protection.

Upon introducing SOC, the system’s ground state energy generally depends on the orientation of magnetization, which is commonly quantified in terms of the magnetic anisotropy energy (MAE). Due to the absence of heavy atoms in our system, however, this effect is negligible, such that magnetic isotropy is maintained up to practically irrelevant corrections (see Supplemental Material sup ). Nevertheless, relevant symmetries are now captured by the magnetic double SG under which the full Hamiltonian is invariant. Crucially, in the presence of SOC, the system’s reflection and rotational symmetries are preserved if and only if the spin magnetization $\mathbf{M}\propto(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)^{\text{T}}$ in spherical coordinates is directed perpendicular to the reflection plane and parallel to the rotation axes, respectively. The dependence is illustrated in Fig. 2b. Some consequences are in line:

(i) The symmetry-protection of the two Weyl points along the BZ boundary on which $\text{M}_{j\in\{1,2,3\}}$ is located becomes broken unless magnetization is in-plane, i.e. $\theta=\pi/2$, and $\phi~\text{mod}~\pi\in\mathcal{M}_{j}$. Here, we define the sets $\mathcal{M}_{j}\equiv\{(2j-1)\pi/6,(j+1)\pi/3\}$ such that for $(\theta,\phi~\mathrm{mod}~\pi)\in\{\pi/2\}\times\mathcal{M}_{j}$ the direction of magnetization is parallel or perpendicular to the BZ boundary on which $\text{M}_{j\in\{1,2,3\}}$ is located. The fact that two non-diametral values $\phi\in\mathcal{M}_{j}$ allow to restore the corresponding Weyl point relates to the latter’s protection by both reflection and rotation symmetry.

(ii) Three-fold rotational symmetry is preserved only for $\theta\in\{0,\pi\}$, where the corresponding little co-groups at K and $\text{K}^{\prime}$ are reduced to the double group $C_{3h}$ and thus do not feature 2D irreps. Consequently, no Weyl points can occur at the BZ corner points for any values of $\theta$ and $\phi$ when SOC is non-vanishing. In the context of a $\mathbf{k}\cdot\mathbf{p}$ model, however, we show that for $\theta=\pi/2$ and those $\phi$ giving rise to symmetry-protected Weyl points on the BZ boundary on which $\mathrm{M}_{j}$ is located, SOC shifts the degeneracies from $\text{K}^{(\prime)}$ onto the same boundary (see Supplemental Material sup ). Any orientation of magnetization $(\theta,\phi)\in\{\pi/2\}\times\mathcal{M}$ with $\mathcal{M}\equiv\cup_{j}\mathcal{M}_{j}$ thus exhibits a total of four Weyl points. Their protection due to either reflection or two-fold rotational symmetry implies that the bands are fully spin-polarized even in the presence of SOC. Upon suppressing the electronic density of states at the Fermi energy, $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ monolayers thus realize a WHSM  You et al. (2019).

As it turns out, these arguments exhaust the symmetry-protected degeneracies between the two relevant bands. We remark, however, that $\theta$ and $\phi$ provide additional degrees of freedom that can be tuned to close the band gap at some point in the BZ even in the presence of only trivial little co-groups. These so-called accidental degeneracies become relevant in the forthcoming discussion.

*Topological properties– * Time-reversal symmetry-breaking exhibited by honeycomb $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ fullerene monolayers originates in the spin-channel. In the absence of SOC, the Berry curvature $\Omega_{xy}(\mathbf{k})$ thus remains odd as a function of lattice momentum $\mathbf{k}$. Since the Chern number $C\in\mathbb{Z}$ is proportional to the integral of Berry curvature over the BZ, topologically non-trivial phases, i.e. $C\neq 0$, can emerge only upon coupling spin and spatial degrees of freedom. Hence, we investigate the system’s topological properties with respect to the direct gap between the two electronic bands near the Fermi energy incorporating SOC. Away from the previously discussed orientations of magnetization $(\theta,\phi)\in\{\pi/2\}\times\mathcal{M}$ and $\theta\in\{0,\pi\}$, no symmetry is present that could enforce degeneracies anywhere in the BZ. Weyl points thus generically obtain a finite mass $m$ and contribute $\Delta C=\text{sign}({m})/2$ to the Chern number. Any smooth deformation of the system’s parameters such that $\Delta C\to-\Delta C$ is consequently accompanied by an intermediate state at which $m=0$ corresponding to a vanishing direct band gap.

We now examine the implications of two-fold rotational symmetries in $D_{3h}$. In the presence of SOC, these have to be paired with the same rotation of the spin degree of freedom Damhus (1984); Liu et al. (2022b); Jiang et al. (2024) and thus rotate both the lattice momentum and magnetization while changing the sign of Berry curvature. From the orientation of rotation axes with respect to Weyl point positions, we conclude that smoothly changing the orientation of magnetization from $\theta=0$ to $\theta=\pi$ necessitates each Weyl point to change the sign of its mass term via a gap closure. Vertical reflection and three-fold rotational symmetries further constrain how these gapless phases can emerge in the parameter space spanned by $\theta$ and $\phi$. For the detailed arguments we refer to the Supplemental Material sup . The resulting phase diagram depending on $\theta$ and $\phi$ is illustrated in Fig. 3a. In particular, for in-plane magnetization, i.e. $\theta=\pi/2$, we determine a topologically trivial phase which transits into a high Chern number phase with $\absolutevalue{C}=2$ for out-of-plane magnetization. Honeycomb $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ monolayers thus exhibit the physics described by Haldane’s model Haldane (1988).

Our theoretical insights are further verified numerically by using Wannier90 Pizzi et al. (2020) to construct a TB model representing the isolated group of twelve bands around the Fermi energy shown in Fig. 2a. Topological properties are subsequently deduced using WannierTools Wu et al. (2018), and the resulting phase diagram is exemplified in Fig. 3b. In particular, for out-of-plane magnetization, topological phases with $\absolutevalue{C}=2$ are confirmed. Due to negligible MAE, changing the orientation of magnetization using weak external magnetic fields facilitates a simple topological phase-switching mechanism.

We further examine the effect of in-plane uniaxial strain. When applied along any of the high-symmetry directions with azimuthal angle $\varphi\in\mathcal{M}$, one vertical reflection and one two-fold rotational symmetry are preserved, reducing the SG to $Amm2$ (No. 38). Away from these strain-directions, the symmetry protection of all Weyl points is broken and no further topologically non-trivial phases are expected. For $\varphi\in\mathcal{M}$ and $(\theta,\phi~\text{mod}~(\pi/2))=(\pi/2,\varphi)$, on the other hand, Weyl points along the BZ boundary perpendicular or parallel to the axes of strain remain protected. Hence, under small symmetry-preserving perturbations, they can only be shifted along the lines of mirror and reflection invariant lattice momenta. We will now focus on the three equivalent directions $\varphi=2j\pi/3$ with $j\in\{0,1,2\}$ for concreteness and discuss our numerical results. Depending on the type of structural deformation, we observe two distinct mechanisms: Under tensile strain, Weyl points are shifted away from the $\mathrm{M}_{j}$ lattice momenta until they meet pairwise in reciprocal space. Upon further increasing exerted strain, the band degeneracy is lifted. The pairing is such that driving $(\theta,\phi)$ away from the symmetry-preserving values, always two annihilating Weyl points develop mass terms with opposite signs. This process is thus not accompanied by a topological phase. Under compressive strain, on the other hand, Weyl points are shifted towards $\mathrm{M}_{j}$ and annihilate in pairs of equal mass signs such that two band inversions are observed: For out-of-plane magnetization, the system transits from $C=0$ to $\absolutevalue{C}=1$ for $\sim 1.2\%$ compression and back to $C=0$ for $\sim 3.4\%$. Topological phase boundaries realize Weyl points in the dispersion of the two relevant bands, again giving rise to fine-tuned WHSM phases. These results are visualized in terms of a 2D slice of the associated phase diagram in Fig. 3c.

*Global Band Gap– * At this point, our discussion is still to be taken with a grain of salt: While we have analyzed the direct band gap topology in $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ monolayers and found the structure to realize a Chern metal Cook and Paramekanti (2014); Liang et al. (2025), realization of the QAHE and fine-tuned emergence of the WHSM phase rely on introducing a global band gap and transiting the system into a bulk-insulating state. For the $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ monolayer, we find that uniformly exerting tensile strain within the plane levels the energetic gap between valence and conduction band in reciprocal space. The increased ionic distance reduces the overlap of bonding and anti-bonding atomic orbitals, increasing and decreasing their energy, respectively, which generally yields a reduced band gap Huang et al. (2015). As moderate uniform strains preserve the symmetry, however, the SOC -induced band-repulsion mechanism maintains the direct band gap away from in-plane orientations of magnetization with $\phi\in\mathcal{M}$, causing the Weyl points to localize in energy.

Since the band gap is opened solely due to SOC, the energetic separation of valence and conduction bands in uniformly strained $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ monolayers is below room temperature energy scale. Practical application under ambient conditions thus urges heavier ions to contribute to electronic states close to the Fermi energy, which can be accomplished via endohedral doping Zhao et al. (2020) or the substrate effect Giovannetti et al. (2007), for example. We demonstrate the general mechanism by simulating this proximity effect via rescaled SOC in the Supplemental Material sup . The resulting direct band gap $\sim\mathcal{O}(10~\mathrm{m}e\mathrm{V})$ gives rise to well-known manifestations of the QAHE such as dissipationless conducting edge states under experimentally readily accessible conditions.

*Summary & Discussion– * We demonstrate the superatom ansatz in the context of the QAHE. Taking 2D honeycomb $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ fullerene networks as an example, we show how time-reversal symmetry breaking can be achieved by the choice of molecular constituents. The band topology of the resulting structure is thoroughly investigated, establishing its phase diagram from both theoretical and computational insights. Easily accessible phase-control mechanisms are identified as the direct band gap increases (i) under moderate strain and (ii) as a consequence of SOC.

While proposals for realizations of the QAHE are numerous, most materials are based on complex structures engineered on atomic level, thus evading chemical synthesis capabilities. Our proposal is solely based on carbon atoms, and the underlying superatom paradigm is anticipated to further facilitate implementation. While in the absence of heavy atoms only a small band gap emerges, the $\text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{26}}}$ monolayer provides a versatile platform to realize various topological phases with several options to tune observation temperature as required.

*Acknowledgments– * We are grateful to Prof. Frederick Duncan Michael Haldane at Princeton, Dr Gunnar F. Lange at Oslo, and Mr Wojciech J. Jankowski at Cambridge for helpful discussions and comments. L.W.P. acknowledges support from the Klaus Höchstetter-Stiftung at Munich and from the Trinity College Cambridge Studienstiftung des deutschen Volkes exchange studentship. 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.