Abstract
The MerminWagner theorem states that longrange magnetic order does not exist in one (1D) or twodimensional (2D) isotropic magnets with shortranged interactions. Here we show that in finitesize 2D van der Waals magnets typically found in lab setups (within millimetres), shortrange interactions can be large enough to allow the stabilisation of magnetic order at finite temperatures without any magnetic anisotropy. We demonstrate that magnetic ordering can be created in 2D flakes independent of the lattice symmetry due to the intrinsic nature of the spin exchange interactions and finitesize effects. Surprisingly we find that the crossover temperature, where the intrinsic magnetisation changes from superparamagnetic to a completely disordered paramagnetic regime, is weakly dependent on the system length, requiring giant sizes (e.g., of the order of the observable universe ~ 10^{26} m) to observe the vanishing of the magnetic order as expected from the MerminWagner theorem. Our findings indicate exchange interactions as the main ingredient for 2D magnetism.
Similar content being viewed by others
Introduction
The demand for computational power is increasing exponentially, following the amount of data generated across different devices, applications and cloud platforms^{1,2}. To keep up with this trend, smaller and increasingly energyefficient devices must be developed, which require the study of compounds not yet explored in datastorage technologies. The discovery of magnetically stable 2D vdW materials could allow for the development of spintronic devices with unprecedented power efficiency and computing capabilities that would, in principle, address some of these challenges^{3}. Indeed, the magnetic stability of vdW layers has been one of the central limitations for finding suitable candidates, given that strong thermal fluctuations are able to rule out any magnetism. As it was initially pointed out by Hohenberg^{4} for a superfluid or a superconductor, and extended by Mermin and Wagner^{5} for spins on a lattice, longrange order should be suppressed at finite temperatures in the 2D regime, when only shortrange isotropic interactions exist. Importantly, the theorem only excludes longrange magnetic order at finite temperature in the thermodynamic limit^{5}, i.e., for infinite system sizes. However, the common understanding is that the theorem also excludes the alignment of spins in samples studied experimentally which are a few micrometres in size^{6,7}, suggesting that such systems are indistinguishable from infinite. Previous reports^{8,9,10,11,12,13,14,15,16,17} have discussed at different levels of theoretical and experimental approaches the limitations and the potential ways to overcome the MerminWagner theorem, which provides a historical evolution of the common concepts used in the field of 2D magnetism.
The longrange order characterising infinite systems only becomes distinguishable from shortrange order describing the local alignment of the spins if the system size exceeds the correlation length at a given temperature^{18}. Previous numerical studies and the scaling analysis of 2D Heisenberg magnets^{19,20,21,22} have established that although only shortrange order is observable at finite temperature, the spin correlation length can be larger than the system size below some finite crossover temperature. An intriguing question on this longrange limit is how we can understand reallife materials, which routinely have a finite size L (Fig. 1a), in light of the MerminWagner theorem. It is known that thermal fluctuations will affect the emergence of spontaneous magnetisation at low dimensionality. Nevertheless, it is unclear which kind of spin ordering can be foreseen in thin vdW layered compounds when finitesize effects and exchange interactions play together. With recent advances in computational power and parallelisation scalability, it is possible to directly model magnetic ordering processes and dynamics of 2D materials on the micrometre lengthscale accessible experimentally.
Here, we show that shortrange order can exist in systems with no anisotropy, even down to the 1D and 2D limits. By using computerintensive atomistic spin simulations and analytical models, we demonstrate the nonapplicability of the Mermin–Wagner theorem for practical length scales and device implementations. The theorem requires that the thermodynamic limit be taken and only for distances beyond the diameter of the observable universe, as revealed by our results, it might be valid. The large distance character of shortrange interactions in 2D vdW magnets drives the formation of magnetic ordering at different lattice symmetries, flakes shapes and chemical compositions. Our results unveil that exchange interactions are the main driving force behind the stabilisation of 2D magnetism and broaden the horizons of possibilities for the exploration of compounds with low anisotropy at an atomically thin level.
Results
We start by defining the magnetisation in our systems as:
where S_{i} denotes the classical spin unit vector at lattice site i and N is the number of sites. In the absence of external magnetic fields, the expectation value of the magnetisation 〈m〉 vanishes in any finitesize system due to timereversal invariance. Yet, 3D systems of only a few nanometres in size that are far from infinite have been studied for decades and exhibit a clear crossover from a magnetically ordered to a paramagnetic phase^{23,24}. The MerminWagner theorem establishes that 〈m〉 must also be zero in infinite 2D systems with shortranged isotropic interactions. However, for practical implementations it is relevant to unveil whether the average magnetisation vanishes because the spins are completely disordered at any point in time, or if they are still aligned on short distances but the overall direction of the magnetisation m strongly suffers timedependent variation. Shortrange order may be characterised by the intrinsic magnetisation^{25}:
which is always positive by definition. The intrinsic magnetisation is 〈∣m∣〉 ≈ 1 in the shortrangeordered regime and converges to zero when the spins become completely disordered^{6,26,27}.
For simplicity we first consider a 2D honeycomb lattice (Fig. 1a) to model the magnetic ordering process for a large flake of 1000 × 1000 nm^{2}. Such a symmetry is very common in several vdW materials holding magnetic properties and interfaces^{3,28}, such as Cr_{2}Ge_{2}Te_{6} (CGT) or CrI_{3} in which 2D magnetic ordering was first discovered^{29,30}. The system consists of 8 million atoms with nearestneighbour Heisenberg exchange interactions J_{ij} and no magnetic anisotropy (K) described via highly accurate Monte Carlo simulations (see Supplementary Sections 1–2 for details). We use an isotropic Heisenberg spin Hamiltonian \({{{{{{{\mathcal{H}}}}}}}}={\sum }_{i { < }j}\,{J}_{ij}{{{{{{{{\bf{S}}}}}}}}}_{i}\cdot {{{{{{{{\bf{S}}}}}}}}}_{j}\) as stated in the Mermin–Wagner theorem^{5}. As it is shown below, our conclusions do not depend on the magnitude of the exchange interactions chosen. Nevertheless, to give a flavour of a potential material to study, we set J_{ij} to similar values to those obtained for CGT layers^{29} where a negligible magnetic anisotropy (< 1 μeV) was observed for thin layers but yet a stable magnetic signal was measured at finite temperatures ( ~ 4.7 K). We begin by assessing the existence of any magnetic order at nonzero temperatures by equilibrating the system for 39 × 10^{6} Monte Carlo steps using a uniform sampling^{31} to avoid any potential bias before a final averaging at thermal equilibrium for a further 10^{6} Monte Carlo steps.
Strikingly, a crossover between the lowtemperature shortrangeordered regime and the completely disordered state (〈∣m∣〉 ≈ 0) is observed at nonzero temperatures (Fig. 1b) and zero magnetic anisotropy (K = 0). To estimate the crossover temperature (T_{x}), the simulation data was fitted by the Curie–Bloch equation in the classical limit^{6}:
where T is the temperature and β is an exponent in the fitting. From the fitting one obtains T_{x} = 23.342 ± 0.237 K (β = 0.54 ± 0.020), which is about onethird of the meanfield (MF) critical temperature \({T}_{{{{{{{{\rm{c}}}}}}}}}^{{{{{{{{\rm{MF}}}}}}}}}=z{J}_{ij}/\left(3{k}_{{{{{{{{\rm{B}}}}}}}}}\right)=70.8\) K (where z = 3 is the number of nearest neighbours) even for this considerable system size. The simulations were then repeated, including magnetic anisotropy (K = 1 × 10^{−24} J/atom), which resulted in a slight increase in the crossover temperature (T_{x} = 26.543 ± 0.320 K, β = 0.427 ± 0.021) (Fig. 1b). We observed that this difference in T_{x} between isotropic and anisotropic cases becomes negligible as the flake size is reduced (100 × 100 nm^{2}) with minor variations of the curvature of the magnetisation versus temperature (Supplementary Section 3 and Supplementary Fig. 1). We also checked that different Monte Carlo sampling algorithms (i.e., adaptive) and starting spin configurations (i.e., ordered, disordered) do not modify the overall conclusions (Supplementary Section 4 and Supplementary Fig. 2). Taking dipolar interactions into account only has a minor effect on the intrinsic magnetisation curve (Supplementary Fig. 3). Although the magnetocrystalline anisotropy K or the dipolar interactions circumvent the MerminWagner theorem and lead to a finite critical temperature, this indicates that systems up to lateral sizes of 1 μm are not suitable for observing the critical behaviour. Instead the crossover in the shortrange order defined by the isotropic interactions dominates in this regime, regardless of whether the anisotropy is present or absent. Previous studies on finite magnetic clusters on metallic surfaces^{32,33} suggested that anisotropy is not the key factor in the stabilisation of magnetic properties at low dimensionality and finite temperatures, but rather it determines the orientation of the magnetisation.
Even though shortrange interactions can stabilise shortrange magnetic order in 2D vdW magnetic materials, this does not necessarily imply that the direction or the magnitude of the magnetisation is stable over time. As thermally activated magnetisation dynamics may potentially change spin directions^{34}, it is important to clarify whether angular variations of the spins are present. Hence we compute the time evolution of the magnetisation along different directions (x, y, z) and its angular dependence (Fig. 1c, d) through the numerical solution of the LandauLifshitzGilbert equation (see Methods for details). Over the whole simulation (40 ns), all components of the magnetisation assume approximately constant values which deviate by ± 5° from the mean direction θ_{av}. Similar analyses undertaken for different flake sizes (L × L, L = 50, 100, 500 nm) show that the spin direction is very stable at each temperature considered (2.5 K, 10 K, 20 K, 30 K, 40 K) and follows a Boltzmann distribution (Supplementary Section 5 and Supplementary Fig. 4). These results show that the magnetisation in a 2D isotropic magnet is not only stable in magnitude but its direction only negligibly varies over time.
An outstanding question raised by the modelling of the 2D finite flakes is whether other kind of common lattice symmetries (i.e., hexagonal, square), lower dimensions (i.e., 1D) and different sizes may follow similar behaviour to that found in the honeycomb lattice. Figure 2 shows that the effect is universal regardless of the details of the lattice or the dimension considered. We find persistent magnetic order for T > 0 K at zero magnetic anisotropy for the cases considered. There is a consistent reduction in the crossover temperature as a function of the system size L → ∞ in agreement with the general trend of the temperature dependence of the correlation length discussed above (Fig. 2a–c). The 1D model (atomic chain) displays a similar trend (Fig. 2d) although the variation of 〈∣m∣〉 with T is different due to the lower dimensionality. We have also checked that several additional factors do not affect these conclusions, such as i) the type of boundary conditions, e.g., open; ii) flake shape (e.g., circular), and iii) strength of the exchange interactions. Supplementary Figs. 5 and 6 provide a summary of this analysis. Indeed, the stabilisation of magnetism in 2D is independent of the magnitude of the exchange interactions considered, as a linear rescaling of the temperatures is obtained for different J_{ij} values. This indicates the generality of the results which are valid regardless of the chemical details of the 2D material and its corresponding J_{ij} interactions. Moreover, if the exchange coupling between atoms could be engineered via chemical synthesis^{35,36,37}, then magnets with either low or high crossover temperatures might be fabricated depending on the target application. Such a procedure would not require heavy elements with sizeable spin orbitcoupling for the generation of magnetic anisotropy since it is not necessary for 2D magnetism.
To give an analytical description of these effects, we use the anisotropic spherical model (ASM) for the calculation of the finitesize effects on the intrinsic magnetisation^{25,38,39} (see Supplementary Section 6 for details). The ASM takes into account Goldstone modes in the system and selfconsistently generates a gap in the correlation functions which avoids infrared divergences responsible for the absence of longrange order for isotropic systems in dimensions d ≤ 2 as L → ∞ as per the MerminWagner theorem. We applied the formalism to 1D and 2D systems for the isotropic Heisenberg Hamiltonian in the absence of an external magnetic field^{25}. The results of our analytical calculations are shown as shaded regions in Fig. 2 (see Supplementary Section 6 for the definition of the regions). At low temperatures both limits agree well with our Monte Carlo calculations within the statistical noise and clearly show the existence of a finite intrinsic magnetisation at nonzero temperature for finite size. At higher temperatures there is a systematic difference between the degree of magnetic ordering between the simulations and the analytical calculations due to the ASM only becoming exact in the limit of infinitely many spin components. The large number of Monte Carlo steps and strict convergence criteria to the same thermodynamic equilibrium for ordered and disordered starting states (Supplementary Section 4) rule out critical slowing down^{40} as a source of difference between the analytical calculations and the simulations.
One may also argue in terms of the correlation length ξ which is comparable to the system size at the crossover temperature. It has been demonstrated^{20} that \(\xi \propto \exp (cJ/T)\), where c is a constant, meaning that the inverse crossover temperature \({T}_{x}^{1}\) only logarithmically increases with the system size. Although our simulations are at the limit of the capabilities of current supercomputers, this effect is expected to persist for larger sizes of 2–10 μm. These values represent typical sizes of continuous 2D microflakes in experiments, and much larger than the ideal nanoscale devices likely to be used in future 2D spintronic applications. Fitting a scaling function to the crossover temperatures for different lattice symmetries (Fig. 2), we can plot the scaling of the crossover temperature with size (Fig. 3a), which can then be extrapolated to larger scales. The crossover temperature is still approximately 30 K for 2–10 μm flakes (Fig. 3b). The graph can be extrapolated to show that only at the 10^{15} − 10^{25} m range does the crossover temperature become lower than ~ 1 K. To put these numbers into perspective for physical systems, these length scales lie between the distance of the Earth to the Sun and the diameter of the observable universe. Therefore, the often asserted notion^{3} that experimental 2D magnetic samples can be classified as infinite and therefore display no net magnetic order at nonzero temperatures, as expected from the Mermin–Wagner theorem, is not applicable. Surprisingly, simple estimations by Leggett^{41} for the stability of graphene crystals following the Mermin–Wagner theorem would require sample sizes of the order of the distance from the Earth to the Moon, which are in sound agreement with our simulation results.
The significance of the crossover temperature T_{x} in relation to the Curie temperature T_{C} is particularly important when discussing the nature of the magnetic ordering in 2D magnets at zero anisotropy for T > 0 K. We investigate this behaviour through colour maps of the spin ordering after 40 million Monte Carlo steps comparing different system sizes and temperatures (Fig. 4). At very low temperatures T = 2.5 K, where there is a high degree of order, the spin directions are highly correlated, as indicated by a mostly uniform colouring. Although the temperatures are near zero, the system is superparamagnetic indicating that over time the magnetisation direction fluctuates, and the effect is most apparent for the smallest sizes where the average direction has moved significantly from the initial direction S∣∣z. At higher temperatures, the deviation of the spin directions within the sample increases as indicated by the more varied colouring. To quantitatively assess the spin deviations we plot the statistical distribution of angle between the spin direction and the mean direction for different temperatures for each size (Supplementary Fig. 4). For an isotropic distribution on the unit sphere there is a \(\sin (\theta )\) weighting, which is seen at the highest temperature for all system sizes. For lower temperatures where the spin directions are more correlated, the distribution is biased towards lower angles. Qualitatively there is little difference in the spin distributions for the different samples. At T = 20 K, there is, however, a systematic trend in the peak angle increasing from θ = 40° for the 50 × 50 nm^{2} flake (Supplementary Fig. 4a) to around θ = 60° at 1000 × 1000 nm^{2} (Supplementary Fig. 4d) indicating an increased level of disorder averaged over the whole sample. This effect is straightforwardly explained by the size dependence of spinspin correlations (Supplementary Fig. 7). At small sizes the spins are strongly exchange coupled, preventing large local deviations of the spin directions. At longer length scales available for the larger systems, the variations in the magnetisation direction are also larger. Surprisingly, our calculations reveal that this effect is weak: even for very large flakes of a micrometre in size, only a small increase can be observed in the position of the peak in the angle distribution at a fixed temperature. Above the crossover temperature, the spinspin correlation length becomes very small compared to the system size with rapid local changes in the magnetisation direction, indicative of a completely disordered paramagnetic state. Our analysis reveals that the spins in finitesized 2D isotropic magnets are strongly aligned due to shortrange order at nonzero temperatures and up to the crossover temperature.
Discussion
Mathematically a phase transition is defined as a nonanalytic change in the state variable for the system, such as the particle density or the magnetisation in the case of spin systems. For any finite system the state variable is continuous by definition due to a finite number of particles, forming a continuous path of intermediate states between two distinct physical phases^{42}. The same is true for a magnetic system, forming a continuous path between an ordered and a paramagnetic state. A priori then, it is impossible to have a true phase transition for any finite magnetic samples which are routinely implemented in device platforms. Yet, nanoscale magnets that are far from infinite have been studied for decades and exhibit a clear crossover from magnetically ordered to paramagnetic phases, occurring for systems only a few nanometres in size^{23,24}. The crossover temperature in a finitesize system hence can be described as an inflection point in 〈∣m∣〉. The precise definition of a phase transition is significant when considering the main conclusions of Mermin and Wagner^{5}, which explicitly only apply in the case of an infinite system. As our results clearly show, sample sizes measured experimentally are not classifiable as infinite and, therefore, not subject to the MerminWagner theorem. It is noteworthy that 3D compounds have weak dependence of their critical temperature on magnetic anisotropy^{43}. Similar analysis performed for a finite 3D bulk system (Supplementary Fig. 8a, b) show that the inclusion of anisotropy barely changes the results for T_{c}. This suggests that magnetism is an exchangedriven effect in both two and three dimensions.
On the practical side, heterostructures with conventional metallic magnetic materials could establish preferential directions of the magnetisation through anisotropic exchange and dipolar couplings. However, it is important to point out that the shortrange order is enforced by the isotropic exchange couplings, and even a low anisotropy may suffice for stabilizing the direction of the magnetisation in the vdW layers, i.e., from underlying magnetic substrates. We can imagine micrometresized samples where all spins are still correlated at finite temperatures so it could represent a single bit. However, for miniaturization purposes multiple nanometresized bits are required on the same sample in order to be implemented in recording media. This is typically achieved by magnetic domains, but there are no domains in an isotropic model since the domain wall width is infinite. However, if vdW layers can be grown with grain boundaries, like in 2D mosaics^{44}, which are large enough that each grain area would have a uniform magnetisation, then a magnetic monolayer would have as many bits as available on the material surface. The underlying substrate hence would set the magnetisation direction for further implementations. This spininterface engineering would be a considerable step towards ondemand magnetic properties at the atomic level given the flexibility on the orientation of the magnetic moments without a predefined direction at the layer. While the anisotropy circumvents the MerminWagner theorem and causes the critical temperature T_{c} to be nonzero in infinitely large systems, in finite samples the shortrange order persists up to much higher temperatures (T_{x} > T_{c}) since T_{x} is proportional to the isotropic exchange rather than the anisotropy^{45,46}. Indeed, the long tail features observed in the intrinsic magnetisation (Fig. 2) extending above the crossover temperature suggest that shortrange order is present. In addition, the existence of shortrange order in bulk magnetic systems near and above the Curie temperature has been experimentally and theoretically discovered in elemental transition metals^{47,48,49}. These studies indicate the persistence of magnetic ordering within the supposedly disordered phase above the Curie temperature, where any ordered phase is primarily controlled by exchange interactions as in the case for 2D magnets. For instance, in bccFe a shortrange order within 5.4 Å was found^{47} which is much smaller than the magnitudes obtained in our simulations for vdW materials.
In conclusion, we presented largescale spin dynamics simulations and analytical calculations on the temperature dependence of the intrinsic magnetisation in 2D magnetic materials described by an isotropic Heisenberg model. We found that shortrange magnetic order at nonzero temperature is a robust feature of isotropic 2D magnets even at experimentally accessible length and time scales. Our data show that the often asserted MerminWagner limit^{5} does not apply to 2D materials on real laboratory sample sizes . Since the spins are aligned due to the exchange interactions already in the isotropic model, the direction of the magnetisation may be stabilized by geometrical factors or finitesize effects. These findings open up possibilities for a wider range of 2D magnetic materials in device applications than previously envisioned. Furthermore, the limited applicability of the analytical Mermin–Wagner theorem opens similar possibilities in other fields such as superconductivity^{9} and liquid crystal systems^{50}, where the relevant length scale of correlations is known to be much greater than that required for experimental measurements and applications. Our results suggest that if the magnetic anisotropy can be controlled to a certain degree^{51} until it completely vanishes, new effects of strongly correlated spins or more unusual disordered states may be observed.
Methods
We used atomistic simulations methods^{6,27,52,53,54,55,56} implemented in the VAMPIRE software^{57} to compute the magnetic properties of 2D magnetic materials. The energy of our system is calculated using the spin Hamiltonian:
where S_{i,j} are unit vectors describing the local spin directions on magnetic sites i, j and J_{ij} is the exchange constant between spins. An easyaxis magnetocrystalline anisotropy constant K can be included as well, with negligible modifications of the results as described in the text. Simulations were run for system sizes of 50 nm, 100 nm, 500 nm and 1000 nm laterally along the x and y directions with periodic boundary conditions (PBCs), and 1 atomic layer thick along the z direction. Similar PBCs were used in the analytical model. However, simulation results using open boundary conditions (OBCs) ended up in similar conclusions (Supplementary Fig. 5). For the honeycomb lattice, the simulations were initialised in either a perfectly ordered state aligned along the z direction or a random state corresponding to infinite temperature. For these simulations the final 〈∣m∣〉(T) curves were identical to each other. However, at low temperatures it took ten times as many steps to reach the final equilibrium state from the random state, so for the remaining structures only simulations starting from the ordered states were run. The systems were integrated using a Monte Carlo integrator using a uniform sampling algorithm^{57} to remove any bias introduced from more advanced algorithms^{31}. To investigate the temperature dependence, the simulation temperature was varied from 0 to 90 K in 2.5 K steps. 40 × 10^{6} Monte Carlo steps were run for each temperature step. This was split into 39 × 10^{6} equilibration steps and then 10^{6} time steps from which the statistics were calculated. The Monte Carlo simulations use a pseudorandom number sequence generated by the Mersenne Twister algorithm^{58} due to its high quality, avoiding correlations in the generated random numbers and with an exceptionally long period of 2^{19937} − 1 ~ 10^{6000}. The parallel implementation generates different random seeds on each processor to ensure no correlation between the generated random numbers.
The timedependent simulations in Fig. 1c, d were performed by solving the stochastic Landau–Lifshitz–Gilbert equation:
which models the interaction of an atomic spin moment S_{i} with an effective magnetic field \({{{{{{{{\bf{B}}}}}}}}}_{{{\mbox{eff}}}}=  1/{{{\mathrm{\mu}}}}_{{{\mathrm{s}}}} \, \partial {{{{{{{\mathcal{H}}}}}}}}/\partial {{{{{{{{\bf{S}}}}}}}}}_{i}\). The effective field causes the atomic moments to precess around the field, where the frequency of precession is determined by the gyromagnetic ratio of an electron (γ_{e} = 1.76 × 10^{11} rad s^{−1}T^{−1}) and λ = 1 is the damping constant. The large value of λ was used to accelerate the relaxation dynamics in order to be computationally achievable ( ~ 72 hours). For a different damping, one has to wait longer or shorter for this to happen. Based on the system sizes used in our computations, this can vary between ~ 5 days up to several weeks, which is not practical. However, once the system is at equilibrium, the value of the damping is not important. Moreover, a large damping would correspond to large fluctuations on the magnitude of the magnetisation and its direction. Lower damping would lead to naturally slower dynamics of the magnetisation. Nevertheless, we barely noticed any at the timescale included in our work (Fig. 1c–d). It is worth mentioning that no damping parameter are used in the Monte Carlo calculations which support our conclusions. The effect of temperature is taken into account using Langevin dynamics^{59} (as in Eq. (5)), where the thermal fluctuations are represented by a Gaussian white noise term. At each time step the instantaneous thermal field acting on each spin is given by
where k_{B} is the Boltzmann constant, T is the system temperature and Γ(t) is a vector of standard (mean 0, variance 1) normal variables which are independent in components and in time. The thermal field is added to the effective field in order to simulate a heat bath. The system was integrated using a Heun numerical scheme^{57}.
Data availability
The data that support the findings of this study are available within the paper and its Supplementary Information.
References
Dieny, B. et al. Opportunities and challenges for spintronics in the microelectronics industry. Nat. Electron. 3, 446–459 (2020).
Sander, D. et al. The 2017 magnetism roadmap. J. Phys. D: Appl. Phys. 50, 363001 (2017).
Wang, Q. H. et al. The magnetic genome of twodimensional van der Waals materials. ACS Nano. 16, 6960–7079 (2022).
Hohenberg, P. C. Existence of longrange order in 1 and 2 dimensions. Phys. Rev. 158, 383–386 (1967).
Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one or twodimensional isotropic Heisenberg models. Phys. Rev. Lett. 17, 1133–1136 (1966).
Wahab, D. A. et al. Quantum rescaling, domain metastability, and hybrid domainwalls in 2d CrI_{3} magnets. Adv. Mater. 33, 2004138 (2021).
Kim, M. et al. Micromagnetometry of twodimensional ferromagnets. Nat. Electron. 2, 457–463 (2019).
Kapikranian, O., Berche, B. & Holovatch, Y. Quasilongrange ordering in a finitesize 2d classical Heisenberg model. J. Phys. A: Math. Theor. 40, 3741–3748 (2007).
Palle, G. & Sunko, D. K. Physical limitations of the HohenbergMerminWagner theorem. J. Phys. A: Math. Theor. 54, 315001 (2021).
Jongh, L. J. D. & Miedema, A. R. Experiments on simple magnetic model systems. Adv. Phys. 50, 947–1170 (2001).
Pomerantz, M. Experiments on literally twodimensional magnets. Surf. Sci. 142, 556–570 (1984).
Wehr, J., Niederberger, A., SanchezPalencia, L. & Lewenstein, M. Disorder versus the MerminWagnerHohenberg effect: From classical spin systems to ultracold atomic gases. Phys. Rev. B. 74, 224448 (2006).
Niederberger, A. et al. Disorderinduced order in twocomponent BoseEinstein condensates. Phys. Rev. Lett. 100, 030403 (2008).
Niederberger, A. et al. Disorderinduced order in quantum XY chains. Phys. Rev. A. 82, 013630 (2010).
Crawford, N. On random fieldinduced ordering in the classical xy model. J. Stat. Phys. 142, 11–42 (2011).
Crawford, N. Random field induced order in low dimension. EPL (Europhys. Lett.) 102, 36003 (2013).
Imry, Y. & Ma, S.k Randomfield instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35, 1399–1401 (1975).
Halperin, B. I. On the Hohenberg–Mermin–Wagner theorem and its limitations. J. Stat. Phys. 175, 521–529 (2019).
Stanley, H. E. & Kaplan, T. A. Possibility of a Phase Transition for the TwoDimensional Heisenberg Model. Phys. Rev. Lett. 17, 913–915 (1966).
Shenker, S. H. & Tobochnik, J. Monte Carlo renormalizationgroup analysis of the classical Heisenberg model in two dimensions. Phys. Rev. B. 22, 4462–4472 (1980).
Blöte, H. W. J., Guo, W. & Hilhorst, H. J. Phase transition in a twodimensional Heisenberg model. Phys. Rev. Lett. 88, 047203 (2002).
Tomita, Y. Finitesize scaling analysis of pseudocritical region in twodimensional continuousspin systems. Phys. Rev. E. 90, 032109 (2014).
Roduner, E. Size matters: why nanomaterials are different. Chem. Soc. Rev. 35, 583–592 (2006).
Singh, R. Unexpected magnetism in nanomaterials. J. Magn. Magn. Mater. 346, 58–73 (2013).
Kachkachi, H. & Garanin, D. A. Boundary and finitesize effects in small magnetic systems. Phys. A: Stat. Mech. its Appl. 300, 487–504 (2001).
Sun, Q.C. et al. Magnetic domains and domain wall pinning in atomically thin CrBr_{3} revealed by nanoscale imaging. Nat. Commun. 12, 1989 (2021).
AbdulWahab, D. et al. Domain wall dynamics in twodimensional van der Waals ferromagnets. Appl. Phys. Rev. 8, 041411 (2021).
Miao, N. & Sun, Z. Computational design of twodimensional magnetic materials. WIREs Computational Mol. Sci. 12, e1545 (2022).
Gong, C. et al. Discovery of intrinsic ferromagnetism in twodimensional van der Waals crystals. Nature 546, 265–269 (2017).
Huang, B. et al. Layerdependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature 546, 270–273 (2017).
AlzateCardona, J. D., SabogalSuárez, D., Evans, R. F. L. & RestrepoParra, E. Optimal phase space sampling for Monte Carlo simulations of Heisenberg spin systems. J. Phys.: Condens. Matter 31, 095802 (2019).
Minár, J., Bornemann, S., Šipr, O., Polesya, S. & Ebert, H. Magnetic properties of CO clusters deposited on Pt(111). Appl. Phys. A. 82, 139–144 (2006).
Šipr, O. et al. Magnetic moments, exchange coupling, and crossover temperatures of CO clusters on Pt(111) and Au(111). J. Phys.: Condens. Matter 19, 096203 (2007).
Brown, W. F. Thermal fluctuations of a singledomain particle. Phys. Rev. 130, 1677–1686 (1963).
Bussian, D. A. et al. Tunable magnetic exchange interactions in manganesedoped inverted core–shell ZnSe–CdSe nanocrystals. Nat. Mater. 8, 35–40 (2009).
Rinehart, J. D., Fang, M., Evans, W. J. & Long, J. R. A n23–radicalbridged terbium complex exhibiting magnetic hysteresis at 14 K. J. Am. Chem. Soc. 133, 14236–14239 (2011).
Baumgarten, M. Tuning the magnetic exchange interactions in organic biradical networks. Phys. status solidi (b) 256, 1800642 (2019).
Garanin, D. A. Spherical model for anisotropic ferromagnetic films. J. Phys. A: Math. Gen. 29, L257–L262 (1996).
Garanin, D. A. Ordering in magnetic films with surface anisotropy. J. Phys. A: Math. Gen. 32, 4323–4342 (1999).
Nightingale, M. P. & Blöte, H. W. J. Dynamic exponent of the twodimensional Ising model and Monte Carlo computation of the subdominant eigenvalue of the stochastic matrix. Phys. Rev. Lett. 76, 4548–4551 (1996).
Leggett, A. J.Lecture 9 from Lecture Notes ’Physics in Two Dimensions’ delivered at the University of Illinois at UrbanaChampaign36003 https://courses.physics.illinois.edu/phys598PTD/fa2013/L9.pdf (2013).
Stanley, H.Introduction to Phase Transitions and Critical Phenomena. International series of monographs on physics (Oxford University Press, 1971). https://books.google.co.uk/books?id=4K_vAAAAMAAJ.
Coey, J. M. D. J. Magn. Magn. Mater. (Cambridge University Press, Cambridge, 2010).
Yao, W., Wu, B. & Liu, Y. Growth and grain boundaries in 2d materials. ACS Nano 14, 9320–9346 (2020).
Irkhin, V. Y., Katanin, A. A. & Katsnelson, M. I. Selfconsistent spinwave theory of layered Heisenberg magnets. Phys. Rev. B. 60, 1082–1099 (1999).
Grechnev, A., Irkhin, V. Y., Katsnelson, M. I. & Eriksson, O. Thermodynamics of a twodimensional Heisenberg ferromagnet with dipolar interaction. Phys. Rev. B. 71, 024427 (2005).
Haines, E. M., Clauberg, R. & Feder, R. Shortrange magnetic order near the Curie temperature iron from spinresolved photoemission. Phys. Rev. Lett. 54, 932–934 (1985).
Maetz, C. J., Gerhardt, U., Dietz, E., Ziegler, A. & Jelitto, R. J. Evidence for shortrange magnetic order in Ni above T_{c}. Phys. Rev. Lett. 48, 1686–1689 (1982).
Antropov, V. Magnetic shortrange order above the curie temperature of Fe and Ni. Phys. Rev. B. 72, 140406 (2005).
Illing, B. et al. MerminWagner fluctuations in 2d amorphous solids. Proc. Natl Acad. Sci. USA 114, 1856–1861 (2017).
Verzhbitskiy, I. A. et al. Controlling the magnetic anisotropy in Cr_{2}Ge_{2}Te_{6} by electrostatic gating. Nat. Electron. 3, 460–465 (2020).
Kartsev, A., Augustin, M., Evans, R. F. L., Novoselov, K. S. & Santos, E. J. G. Biquadratic exchange interactions in twodimensional magnets. npj Comput. Mater. 6, 150 (2020).
Alliati, I. M., Evans, R. F. L., Novoselov, K. S. & Santos, E. J. G. Relativistic domainwall dynamics in van der Waals antiferromagnet MnPS_{3}. npj Comput. Mater. 8, 3 (2022).
Strungaru, M., Augustin, M. & Santos, E. J. G. Ultrafast laserdriven topological spin textures on a 2d magnet. npj Comput. Mater. 8, 169 (2022).
Augustin, M., Jenkins, S., Evans, R. F. L., Novoselov, K. S. & Santos, E. J. G. Properties and dynamics of meron topological spin textures in the twodimensional magnet CrCl_{3}. Nat. Commun. 12, 185 (2021).
Dabrowski, M. et al. Alloptical control of spin in a 2d van der waals magnet. Nat. Commun. 13, 5976 (2022).
Evans, R. F. L. et al. Atomistic spin model simulations of magnetic nanomaterials. J. Phys.: Condens. Matter 26, 103202 (2014).
Matsumoto, M. & Nishimura, T. Mersenne Twister: a 623dimensionally equidistributed uniform pseudorandom number generator. ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).
Brown, W. Thermal fluctuation of fine ferromagnetic particles. IEEE Trans. Magn. 15, 1196–1208 (1979).
Acknowledgements
We thank David Mermin, Mikhail Katsnelson, and Bertrand Halperin for valuable discussions. L.R. gratefully acknowledges funding by the National Research, Development and Innovation Office of Hungary via Project No. K131938 and by the Young Scholar Fund at the University of Konstanz. U.A. gratefully acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)ProjectID 328545488TRR 227, Project No. A08; and grants PID2021122980OBC55 and RYC2020030605I funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and “ESF Investing in your future”. E.J.G.S. acknowledges computational resources through CIRRUS Tier2 HPC Service (ec131 Cirrus Project) at EPCC funded by the University of Edinburgh and EPSRC (EP/P020267/1); ARCHER UK National Supercomputing Service (http://www.archer.ac.uk) via Project d429. E.J.G.S. acknowledges the Spanish Ministry of Science’s grant program “EuropaExcelencia” under grant number EUR2020112238, the EPSRC Early Career Fellowship (EP/T021578/1), and the University of Edinburgh for funding support. K.S.N. is supported by the Ministry of Education, Singapore, under its Research Centre of Excellence award to the Institute for Functional Intelligent Materials (IFIM, project No. EDUNC3318279V12) and by the Royal Society (UK, grant number RSRP\R\190000). For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.
Author information
Authors and Affiliations
Contributions
E.J.G.S. conceived the idea and supervised the project. S.J. performed the atomistic simulations with inputs from E.J.G.S. and R.F.L.E. L.R. and U.A. developed the semianalytical model and undertook the numerical simulations. E.J.G.S. wrote the paper with a draft initially prepared by S.J. and R.F.L.E. and also with inputs from K.S.N., U.A. and L.R. All authors contributed to this work, read the manuscript, discussed the results, and agreed on the included contents.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Communications thanks Antonio T. Costa, Zhimei Sun and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Additional information
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary information
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Jenkins, S., Rózsa, L., Atxitia, U. et al. Breaking through the MerminWagner limit in 2D van der Waals magnets. Nat Commun 13, 6917 (2022). https://doi.org/10.1038/s41467022343890
Received:
Accepted:
Published:
DOI: https://doi.org/10.1038/s41467022343890
This article is cited by

Multistep magnetization switching in orthogonally twisted ferromagnetic monolayers
Nature Materials (2024)

Unraveling effects of electron correlation in twodimensional FenGeTe2 (n = 3, 4, 5) by dynamical mean field theory
npj Computational Materials (2023)

Magnetic properties of intercalated quasi2D Fe3xGeTe2 van der Waals magnet
npj 2D Materials and Applications (2023)

Finitetemperature critical behaviors in 2D longrange quantum Heisenberg model
npj Quantum Materials (2023)

Thermalactivated escape of the bistable magnetic states in 2D Fe3GeTe2 near the critical point
Communications Physics (2023)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.