Over the past two decades, 2D materials have rapidly evolved into a diverse and expanding family of material platforms. Many members of this materials class have demonstrated their potential to deliver transformative impact on fundamental research and technological applications across different fields. In this roadmap, we provide an overview of the key aspects of 2D material research and development, spanning synthesis, properties and commercial applications. We specifically present roadmaps for high impact 2D materials, including graphene and its derivatives, transition metal dichalcogenides, MXenes as well as their heterostructures and moir\'e systems. The discussions are organized into thematic sections covering emerging research areas (e.g., twisted electronics, moir\'e nano-optoelectronics, polaritronics, quantum photonics, and neuromorphic computing), breakthrough applications in key technologies (e.g., 2D transistors, energy storage, electrocatalysis, filtration and separation, thermal management, flexible electronics, sensing, electromagnetic interference shielding, and composites) and other important topics (computational discovery of novel materials, commercialization and standardization). This roadmap focuses on the current research landscape, future challenges and scientific and technological advances required to address, with the intent to provide useful references for promoting the development of 2D materials.

Though extensively studied, hardness, defined as the resistance of a material to deformation, still remains a challenging issue for a formal theoretical description due to its inherent mechanical complexity. The widely applied Teter's empirical correlation between hardness and shear modulus has been considered to be not always valid for a large variety of materials. Here, inspired by the classical work on Pugh's modulus ratio, we develop a theoretical model which establishes a robust correlation between hardness and elasticity for a wide class of materials, including bulk metallic glasses, with results in very good agreement with experiment. The simplified form of our model also provides an unambiguous theoretical evidence for Teter's empirical correlation.

Atomically thin layers of transition-metal dicalcogenides (TMDCs) are often known to be metastable in the ambient atmosphere. Understanding the mechanism of degradation is essential for their future applications in nanoelectronics, and thus has attracted intensive interest recently. Here, we demonstrate a systematic study of atomically thin WTe$_{2}$ in its low temperature quantum electronic transport properties. Strikingly, while the temperature dependence of few layered WTe$_{2}$ showed clear metallic tendency in the fresh state, degraded devices first exhibited a re-entrant insulating behavior, and finally entered a fully insulating state. Correspondingly, a crossover from parabolic to linear magnetoresistance, and finally to weak anti-localization was seen. Real-time Raman scattering measurement, together with transmission electron microscopy studies done before and after air degradation of atomically thin WTe$_{2}$ further confirmed that the material gradually form amorphous islands. It thus leads to localized electronic states and explains the low temperature Coulomb gap observed in transport measurements. Our study reveals for the first time the correlation between the unusual magnetotransport and disorder in few-layered WTe$_{2}$, which is indispensable in providing guidance on its future devices application.

We show by micromagnetic simulations that spontaneous skyrmion ground state can exist in Co/Ru/Co nanodisks without the Dzyaloshinsky-Moriya interaction (DMI), which can remain stable in the applied magnetic field along +z direction even up to 0.44 T. The guiding center ($R_x$,$R_y$) of skyrmion defined by the moments of the topological density presents a novel gyration with a star-like trajectory in a pulsed magnetic field and a hexagonal trajectory after the field is switched off, which is different from that of vortex or bubble. One of the coupled skyrmions could move without an external magnetic field, but only induced by the motion of the other one due to strong inter-layer magnetostatic interactions. This work sheds light on how novel skyrmions can be discovered in various (not limited to magnetic) systems with competing energies and contributes to the understanding of the dynamical properties of skyrmion.

One of the long sought-after goals in manipulation of light through light-matter interactions is the realization of magnetic-field-tuneable colouration, so-called magneto-chromatic effect, which holds great promise for optical, biochemical and medical applications due to its contactless and non-invasive nature. This goal can be achieved by magnetic-field controlled birefringence, where colours are produced by the interference between phase-retarded components of transmitted polarised light. Thus far birefringence-tuneable coloration has been demonstrated using electric field, material chirality and mechanical strain but magnetic field control remained elusive due to either weak magneto-optical response of transparent media or low transmittance to visible light of magnetically responsive media, such as ferrofluids. Here we demonstrate magnetically tuneable colouration of aqueous suspensions of two-dimensional cobalt-doped titanium oxide which exhibit an anomalously large magneto-birefringence effect. The colour of the suspensions can be tuned over more than two wavelength cycles in the visible range by moderate magnetic fields below 0.8 T. We show that such giant magneto-chromatic response is due to particularly large phase retardation (>3 pi) of the polarised light, which in its turn is a combined result of a large Cotton-Mouton coefficient (three orders of magnitude larger than for known liquid crystals), relatively high saturation birefringence (delta n = 2 x 10^-4) and high transparency of our suspensions to visible light. The work opens a new avenue to achieve tuneable colouration through engineered magnetic birefringence and can readily be extended to other magnetic 2D nanocrystals. The demonstrated effect can be used in a variety of magneto-optical applications, including magnetic field sensors, wavelength-tuneable optical filters and see-through printing.

Topological spin textures as an emerging class of topological matter offer a medium for information storage and processing. The recently discovered topological Hall effect (THE) is considered as a fingerprint for electrically probing non-trivial spin-textures. But the origin of THE in oxides has remained elusive. Here we report an observation of the THE in ultrathin (unit cells. u.c.) 4d ferromagnetic SrRuO3 films grown on SrTiO3(001) substrates, which can be attributed to the chiral ordering of spin structure (i.e., skyrmion-like) in the single SrRuO3 layer without contacting 5d oxide SrIrO3 layer. It is revealed that the RuO6 octahedral tilting induced by local orthorhombic-to-tetragonal structural phase transition exists across the SrRuO3/SrTiO3 interface, which naturally breaks the inversion symmetry. Our theoretical calculations demonstrate that the Dzyaloshinskii-Moriya (DM) interaction arises owing to the broken inversion symmetry and strong spin-orbit interaction of 4d SrRuO3. This DM interaction can stabilize the N\'eel-type magnetic skyrmions, which in turn accounts for the observed THE in transport. The RuO6 octahedral tilting-induced DM interaction provides a pathway toward the electrical control of the topological spin textures and resultant THE, which is confirmed both experimentally and theoretically. Besides the fundamental significance, the understanding of THE in oxides and its electrical manipulation presented in this work could advance the low power cost topological electronic and spintronic applications.

The common feature for a nontrivial hard problem is the existence of nontrivial topological structures, non-planarity graphs, nonlocalities, or long-range spin entanglements in a model system with randomness. For instance, the Boolean satisfiability (K-SAT) problems are nontrivial, due to the existence of non-planarity graphs, nonlocalities, and the randomness. In this work, the relation between a spin-glass three-dimensional (3D) Ising model with the lattice size N = mnl and the K-SAT problems is investigated in detail. With the Clifford algebra representation, it is easy to reveal the existence of the long-range entanglements between Ising spins in the spin-glass 3D Ising lattice. The internal factors in the transfer matrices of the spin-glass 3D Ising model lead to the nontrivial topological structures and the nonlocalities. At first, we prove that the absolute minimum core (AMC) model exists in the spin-glass 3D Ising model, which is defined as a spin-glass 2D Ising model interacting with its nearest neighboring plane. Any algorithms, which use any approximations and/or break the long-range spin entanglements of the AMC model, cannot result in the exact solution of the spin-glass 3D Ising model. Second, we prove that the dual transformation between the spin-glass 3D Ising model and the spin-glass 3D Z2 lattice gauge model shows that it can be mapped to a K-SAT problem for K > = 4 also in the consideration of random interactions and frustrations. Third, we prove that the AMC model is equivalent to the K-SAT problem for K = 3.

An overview of the mathematical structure of the three-dimensional (3D) Ising model is given, from the viewpoints of topologic, algebraic and geometric aspects. By analyzing the relations among transfer matrices of the 3D Ising model, Reidemeister moves in the knot theory, Yang-Baxter and tetrahedron equations, the following facts are illustrated for the 3D Ising model: 1) The complexified quaternion basis constructed for the 3D Ising model represents naturally the rotation in a (3 + 1) - dimensional space-time, as a relativistic quantum statistical mechanics model, which is consistent with the 4-fold integrand of the partition function by taking the time average. 2) A unitary transformation with a matrix being a spin representation in 2^(nlo)-space corresponds to a rotation in 2nlo-space, which serves to smooth all the crossings in the transfer matrices and contributes as the non-trivial topologic part of the partition function of the 3D Ising model. 3) A tetrahedron relation would ensure the commutativity of the transfer matrices and the integrability of the 3D Ising model, and its existence is guaranteed also by the Jordan algebra and the Jordan-von Neumann-Wigner procedures. 4) The unitary transformation for smoothing the crossings in the transfer matrices changes the wave functions by complex phases {\Phi}x, {\Phi}y, and {\Phi}z. The relation with quantum field and gauge theories, physical significance of weight factors are discussed in details. The conjectured exact solution is compared with numerical results, and singularities at/near infinite temperature are inspected. The analyticity in {\beta} = 1/(kB T) of both the hard-core and Ising models has been proved for {\beta} > 0, not for {\beta} = 0. Thus the high-temperature series cannot serve as a standard for judging a putative exact solution of the 3D Ising model.

The force field describing the calculated interaction between atoms or molecules is the key to the accuracy of many molecular dynamics (MD) simulation results. Compared with traditional or semi-empirical force fields, machine learning force fields have the advantages of faster speed and higher precision. We have employed the method of atomic cluster expansion (ACE) combined with first-principles density functional theory (DFT) calculations for machine learning, and successfully obtained the force field of the binary Fe-Co alloy. Molecular dynamics simulations of Fe-Co alloy carried out using this ACE force field predicted the correct phase transition range of Fe-Co alloy.

Interactions among charge carriers in graphene can lead to the spontaneous breaking of multiple degeneracies. When increasing the number of graphene layers following rhombohedral stacking, the dominant role of Coulomb interactions becomes pronounced due to the significant reduction in kinetic energy. In this study, we employ phonon-polariton assisted near-field infrared imaging to determine the stacking orders of tetralayer graphene devices. Through quantum transport measurements, we observe a range of spontaneous broken-symmetry states and their transitions, which can be finely tuned by carrier density n and electric displacement field D. Specifically, we observe a layer antiferromagnetic insulator at n = D = 0 with a gap of approximately 15 meV. Increasing D allows for a continuous phase transition from a layer antiferromagnetic insulator to a layer polarized insulator. By simultaneously tuning n and D, we observe isospin polarized metals, including spin-valley-polarized and spin-polarized metals. These transitions are associated with changes in Fermi surface topology and are consistent with the Stoner criteria. Our findings highlight the efficient fabrication of specially stacked multilayer graphene devices and demonstrate that crystalline multilayer graphene is an ideal platform for investigating a wide range of broken symmetries driven by Coulomb interactions.

Nitriding introduces nitrides into the surface of steels, significantly enhancing the surface me-chanical properties. By combining the variable composition evolutionary algorithm and first-principles calculations based on density functional theory, 50 thermodynamically stable or metastable Fe-N compounds with various stoichiometric ratios were identified, exhibiting also dynamic and mechanical stability. The mechanical properties of these structures were systemati-cally studied, including the bulk modulus, shear modulus, Young's modulus, Poisson's ratio, Pugh's ratio, Cauchy pressure, Klemen parameters, universal elastic anisotropy, Debye tempera-ture, and Vickers hardness. All identified stable and metastable Fe-N compounds were found in the ductile region, with most exhibiting homogeneous elastic properties and isotropic metallic bonding. As the nitrogen concentration increases, their bulk moduli generally increase as well. The Vickers hardness values of Fe-N compounds range from 3.5 to 10.5 GPa, which are signifi-cantly higher than that of pure Fe (2.0 GPa), due to the stronger Fe-N bonds strength. This study provides insights into optimizing and designing Fe-N alloys with tailored mechanical properties.

Based on high-throughput density functional theory calculations, we have found 49 ferromag-netic cases in FexN1-x (0

In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan-von Neumann-Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan-von Neumann-Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry.

We present an open-source program, QR$^2$-code, that computes double-resonance Raman (DRR) spectra using first-principles calculations. QR$^2$-code can calculate not only two-phonon DRR spectra but also single-resonance Raman spectra and defect-induced DRR spectra. For defect-induced DDR spectra, we simply assume that the electron-defect matrix element of elastic scattering is a constant. Hands-on tutorials for graphene are given to show how to run QR$^2$-code for single-resonance, double-resonance, and defect-induced Raman spectra. We also compare the single-resonance Raman spectra by QR$^2$-code with that by QERaman code. In QR$^2$-code, the energy dispersions of electron and phonon are taken from Quantum ESPRESSO (QE) code, and the electron-phonon matrix element is obtained from the electron-phonon Wannier (EPW) code. All codes, examples, and scripts are available on the GitHub repository.

In this work, the computational complexity of a spin-glass three-dimensional (3D) Ising model (for the lattice size N = lmn, where l, m, n are the numbers of lattice points along three crystallographic directions) is studied. We prove that an absolute minimum core (AMC) model consisting of a spin-glass 2D Ising model interacting with its nearest neighboring plane, has its computational complexity O(2^mn). Any algorithms to make the model smaller (or simpler) than the AMC model will cut the basic element of the spin-glass 3D Ising model and lost many important information of the original model. Therefore, the computational complexity of the spin-glass 3D Ising model cannot be reduced to be less than O(2^mn) by any algorithms, which is in subexponential time, superpolynomial.

The quantum statistics mechanism is very powerful for investigating the equilibrium states and the phase transitions in complex spin disorder systems. The spin disorder systems act as an interdisciplinary platform for solving the optimum processes in computer science. In this work, I determined the lower bound of the computational complexity of knapsack problems. I investigated the origin of nontrivial topological structures in these hard problems. It was uncovered that the nontrivial topological structures arise from the contradictory between the three-dimensional character of the lattice and the two-dimensional character of the transfer matrices used in the quantum statistics mechanism. I illustrated a phase diagram for the non-deterministic polynomial (NP) vs polynomial (P) problems, in which a NP-intermediate (NPI) area exists between the NP-complete problems and the P-problems, while the absolute minimum core model is at the border between the NPI and the NP-complete problems. The absolute minimum core model of the knapsack problem cannot collapse directly into the P-problem. Under the guide of the results, one may develop the best algorithms for solving various optimum problems in the shortest time, being in subexponential and superpolynomial. This work illuminates the road on various fields of science ranging from physics to biology to finances, and to information technologies.

Altermagnets combine antiferromagnetic order with ferromagnet-like spin splitting, a duality that unlocks ultrafast spin-dependent responses. This unique property creates unprecedented opportunities for spin-current generation, overcoming the intrinsic limitations of conventional spin-transfer and spin-orbit torque approaches in magnetic memory technologies. Here, we establish a fundamental relationship between Fermi surface geometry and time-reversal-odd ($\mathcal{T}$-odd) spin currents in altermagnets through combined model analysis and first-principles calculations. We demonstrate that a $d$-wave altermagnet with a flat Fermi surface can achieve a theoretical upper limit of charge-to-spin conversion efficiency (CSE) of 100%. This mechanism is realized in the newly discovered room-temperature altermagnetic metal KV$_2$O$_2$Se, which exhibits a CSE of $\sim$78% at the charge neutrality point, nearly double that of RuO$_2$, setting a new record for $\mathcal{T}$-odd CSE. Under electron doping, this efficiency further increases to $\sim$98%, approaching the theoretical limit. Our work advances the fundamental understanding of $\mathcal{T}$-odd spin currents via Fermi surface geometry engineering and provides key insights for developing next-generation altermagnet-based memory devices.

Van der Waals In$_2$Se$_3$ has garnered significant attention due to its unique properties and wide applications associated with its rich polymorphs and polymorphic phase transitions. Despite extensive studies, the vast complex polymorphic phase space remains largely unexplored, and the underlying microscopic mechanism for their phase transformations remains elusive. Here, we develop a highly accurate, efficient, and reliable machine-learning potential (MLP), which not only facilitates accurate exploration of the intricate potential energy surface (PES), but also enables us to conduct large-scale molecular dynamics (MD) simulations with first-principles accuracy. We identify the accurate structure of the $\beta''$ polymorph and uncover several previously unreported $\beta'$ polymorph variants exhibiting dynamic stability and competing energies, which are elucidated by characteristic flat imaginary phonon bands and the distinctive Mexican-hat-like PES in the $\beta$ polymorph. Through the MLP-accelerated MD simulations, we directly observe the polymorphic phase transformations among the $\alpha$, $\beta$, $\beta'$, and $\beta''$ polymorphs under varying temperature and pressure conditions, and build for the first time an ab initio temperature-pressure phase diagram, showing good agreement with experiments. Furthermore, our MD simulations reveal a novel strain-induced reversible phase transition between the $\beta'$ and $\beta''$ polymorphs. This work not only unveils diverse polymorphs in van der Waals In$_2$Se$_3$, but also provides crucial atomic insights into their phase transitions, opening new avenues for the design of novel functional electronic devices.

Remarkable exploitation of valence and lattice mismatch in epitaxial ferroelectric heterostructures generates physical effects not classically expected for perovskite oxides, such as 2D electron gas and polar skyrmions. However the widespread application of these interfacial properties and functionalities is impeded by the ultrathin layered structure and essential presence of underlying lattice-matched substrates for the deposition of epitaxial thin films. Here, we report a bottom-up pathway to synthesize bulk ferroelectric heterostructures (BFH) with periodic composition fluctuation (8 nm in wavelength) using elemental partitioning by cation diffusion, providing opportunities to exploit novel characteristics of hetero-epitaxial oxide thin films in bulk materials. Exemplar monolithic BiFeO3-BaTiO3 BFH ceramics described herein share common features with their thin film heterostructure counterparts, which facilitates control and stabilisation of ferroelectric polarisation along with a significant enhancement in Curie temperature, Tc, and functionality. BFH ceramics exhibit a record Tc (up to 824 °C) and a piezoelectric coefficient (d33 = 115 pC N-1 ), in comparison with other perovskite or non-perovskite solid solutions, providing sustainable solutions for emergent high temperature piezoelectric sensing, actuation and energy conversion applications. By creating BFH ceramics using different electromechanical boundary conditions, distinct morphologies of aliovalent A-site cation segregated regions along with different types of ferroelectric order are achieved. This formation mechanism provides unprecedented control over local ferroelectric ordering and domain stabilisation in BFH ceramics; it also paves the way to explore new types of functionality, beyond those achievable in both bulk ferroelectrics and thin film heterostructures.

In this work, we prove the equivalence between the pair correlation functions of primes, and of spins in a two-dimensional (2D) Ising model with a mixture of ferromagnetic and randomly distributed competing interactions. At first, we prove that the correlation function between a pair of spins in a distance l within the 2D Ising model is larger than zero at whole temperature region. Second, we prove that the pair correlation function of spins in the model is equivalent to the pair correlation function of its energy levels. Third, we prove that the energy-energy correlation function of the model is equivalent to the pair correlation function of nontrivial zeros of the Dirichlet function (including the Riemann zeta function). Fourth, we prove that the pair correlation function between the nontrivial zeros of the Dirichlet function is equivalent to the correlation function between a pair of primes p and p+q for every even q. In a conclusion, we have proven that the pair correlation function of primes p and p+q for every even q is larger than zero.