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The perspective of topology provides useful insights into the lifetime of a many-body system that is not in the most thermodynamically stable form. In an aggregate of carbon atoms, for instance, whereas the most thermodynamically stable form at standard temperatures and pressures is graphite, other forms, such as diamond and fullerene-molecule crystals, also have long lifetimes, despite a thermodynamic free-energy penalty15. From a topological perspective16, the carbon networks of the crystalline allotropes are topologically distinct from each other. Therefore, diamond and fullerene-molecule crystals are unable to transform into graphite unless thermal agitation enables the breaking and reformation of the carbon networks. States with this absence of continuity in topology are often termed topologically protected, or topologically stable, and this condition guarantees a finite lifetime for the thermodynamically less stable states of matter, thereby qualitatively accounting for their metastable nature.

Despite the many examples of topologically distinct crystals16, only recently has a topologically stable long-range magnetic order been identified experimentally in the form of a magnetic SkL, which is a hexagonal arrangement of magnetic vortex lines5,6,7,8,9 (for the schematic spin configuration, see Fig. 1a). Because of its topology, the SkL is topologically protected from other competing non-topological spin configurations, such as ferromagnetic and spin-conical orders (Fig. 1a), thus providing a unique platform to observe how long a topologically stable long-range ordering of spins can persist beyond its thermodynamic stability. To address this issue, the lifetime of a metastable SkL requires investigation. In reality, however, unless randomly positioned dopant atoms6 or pressure inhomogeneities17 are introduced, the SkL in various chiral magnets is readily annihilated when the SkL becomes thermodynamically costly by passing through the SkL-to-conical transition line5,7,10,11,12,13,14 (for a typical phase diagram, see Fig. 2a). Thus, the intrinsic lifetime of the metastable SkL remains unknown.

Figure 1: Metastable skyrmion lattice stabilized under rapid cooling.
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a, Schematic of the cooling-rate-dependent bifurcation of an equilibrium skyrmion lattice (SkL). Whereas the equilibrium SkL changes into the thermodynamically stable conical phase under slow cooling, rapid cooling causes kinetic avoidance of the SkL-to-conical transition, resulting in a quenched metastable SkL. Under an intermediate cooling rate, a mixture of the two phases is expected. b, Magnetic field dependence of the Hall resistivity, ρyx, measured before and after quenching at 0.22 T. By applying a current pulse to the equilibrium state under a magnetic field of 0.22 T (the cross), the ρyx value changes to an enhanced value (the open circle). c, Cooling-rate dependence of ρyx measured at 10 K and 0.22 T after quenching is complete. d, Contour plot of ρyx measured at 10 K post-quenching using various current densities under selected magnetic fields. The deduced sample temperature during the electric current application is also indicated on the upper horizontal axis (see also Supplementary Fig. 1). Contour plots measured at 5 and 23 K are shown in Supplementary Fig. 2.

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Figure 2: Magnetic state diagrams of MnSi under equilibrium and quenched conditions.
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a, Phase diagram under thermodynamic equilibrium. b, Magnetic state diagram including the metastable SkL created by quenching under a magnetic field. Red symbols in b were determined from the data shown in Fig. 1b and Supplementary Fig. 2. Whereas the boundaries of the equilibrium SkL were determined from the ρxx data, the ferromagnetic transition line was determined from the ρyx data (see Supplementary Figs 3 and 4).

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In pursuing a metastable SkL that should potentially form even in a disorder-free system, we noted that, in the creation process of natural diamonds, thermodynamically stable diamonds grown in the mantle are quenched when they move to the surface of the Earth and, hence, the diamond-to-graphite transition is kinetically avoided. On the basis of this realization, we envisaged that the SkL-to-conical transition, which normally occurs during cooling, can also be kinetically avoided if the cooling rate is sufficiently high (Fig. 1a). To test this working hypothesis, we chose nominally dopant-free MnSi as an archetypal system of a skyrmion-hosting material and exploited rapid cooling following electric-pulse heating applied to the sample. For instance, when a rectangular current pulse of 3.1 × 106 A m−2 with a duration of 100 ms is applied to the sample at 10 K, the sample temperature is observed to quickly increase and reach a quasi-steady state with a temperature of approximately 33–34 K. After the pulse ceases, the sample is rapidly cooled to the environmental temperature at a cooling rate of approximately 700 K s−1 (for the estimation of the time-varying sample temperature and the cooling rate, see Supplementary Information and Supplementary Fig. 1). Using the quenching technique described, the impact of quenching on the magnetic structure can be explored beyond the normally used cooling rate, 10−2–100 K s−1. To this end, we focused on the Hall resistivity, ρyx, as a sensitive probe for SkL formation, particularly in MnSi (refs 18, 19), and found that −ρyx at 10 K under a magnetic field of 0.22 T indeed changes from −2 to 32 nΩ cm after applying the current pulse described. Whereas the enhanced ρyx value remains constant as a function of time (verified up to one week), it exhibits transition-like behaviour in response to a slow magnetic field sweep (5.0 × 10−4 T s−1) and then decreases to the values corresponding to the equilibrium state (Fig. 1b). These observations are reasonably understood by considering that a long-lived metastable magnetic structure is formed as a result of heating and subsequent rapid cooling.

To elucidate the origin of the metastable state, we first examined whether it is actually a quenched state. We tailored the cooling rate with ramp pulses (for details, see Supplementary Information and Supplementary Fig. 1) and observed how the ρyx value changes. Figure 1c summarizes the ρyx values measured after applying various cooling rates; remarkably, sharp crossover behaviour is observed from the slowly cooled to the rapidly cooled regimes, thus verifying that the enhanced ρyx value of approximately 32 nΩ cm is indeed a consequence of the quenching. Because neither the ordinary Hall nor the anomalous Hall resistivities can explain such a greatly enhanced value of ρyx (Fig. 1b), it is important to consider the contributions of the topological Hall resistivity, which is known to arise in the presence of the SkL because of its real-space Berry phase associated with the non-zero topological winding number18,19,20,21. In fact, the magnitude of the enhancement, Δρyx ≈ 31–34 nΩ cm, is similar to the reported value, 35–37 nΩ cm, observed when the metastable SkL persists at 10 K in MnSi under the influence of pressure inhomogeneity17. Given that the low-temperature value of the topological Hall resistivity depends only weakly on pressure and sample quality as long as the metastable SkL is realized17, the good agreement between the present and reported values indicates that the SkL is the most likely candidate for the present quenched state. This scenario is further supported by the observation of an enhanced ρyx value under various magnetic field/current density conditions. The contour plot of ρyx measured after the application of a pulse with a given current density is shown in Fig. 1d; here, we find that the enhancement of ρyx is limited to a magnetic field range of 0.17–0.26 T, which is in good agreement with that of the equilibrium SkL phase in our sample, 0.14–0.27 T (Fig. 2a). Furthermore, the enhancement of ρyx becomes appreciable only if the deduced sample temperature during the pulse application (see the upper horizontal axis of Fig. 1d) reaches or exceeds the temperatures of the equilibrium SkL (27.0–28.8 K). These findings lead us to conclude that the quenched magnetic state is a metastable SkL caused by the kinetic avoidance of the SkL-to-conical transition. Unexpectedly to us, the metastable SkL is formed even when the sample is quenched from the paramagnetic regime (>28.8 K), indicating that the paramagnetic-to-SkL transition has faster kinetics than the SkL-to-conical transition and, therefore, cannot kinetically be avoided, at least at the present cooling rate, 700 K s−1.

Having established a method to create a metastable SkL in a nominally clean sample, we can now address the issue of the stability beyond the equilibrium phase diagram. We found that the long-lived metastable SkL can be created at any temperature between 5 and 23 K using the same quenching method (see Supplementary Figs 2 and 3); accordingly, we could further examine the stability of the metastable SkL against a magnetic field sweep at each temperature and construct a magnetic state diagram that includes the quenched SkL, as shown in Fig. 2b. Here, two important aspects can be highlighted. First, the temperature/magnetic-field region in which the quenched SkL can persist is significantly extended compared to that of the equilibrium SkL phase, which exemplifies the remarkable stability of the metastable SkL. Second, there is a temperature gap (∼24–27 K) in which neither the equilibrium SkL nor the metastable SkL can persist, implying that the metastable SkL is short-lived in this temperature range.

Such a short-lived metastable SkL can be substantiated by the time decay of the enhanced ρyx value, as shown in Fig. 3a. The relaxation of ρyx to the equilibrium value can clearly be seen, in accord with the expectation that the metastable SkL is short-lived in this temperature range. The behaviour is characterized well by the standard relaxation equation (denoted by the broken lines in Fig. 3a):

where ρyx, 0 and ρyx, ∞ denote the initial and fully relaxed (or equilibrium) values of ρyx, respectively, and τ represents the relaxation time (or, equivalently, the lifetime of the metastable SkL). Figure 3b summarizes the τ values derived for various temperatures. Note that as the temperature approaches the SkL transition temperature of 27 K, the relaxation time decreases markedly and will probably decrease to less than 10 s. This behaviour reasonably accounts for why the SkL is readily annihilated during cooling in nominally clean systems, even though it is topologically stable5,7,10,11,12,13,14. Nevertheless, after quickly passing through the high-temperature region in which the metastable SkL is short-lived, the lifetime becomes unmeasurably long towards low temperatures. Consequently, the quenched SkL is stabilized practically. The significant increase in τ invokes the thermal activation process and suggests that the SkL-to-conical topology change proceeds via an intermediate spin configuration with a creation energy, Eg. In estimating the order of magnitude of Eg, we tentatively assume that Eg is dependent on temperature and proportional to the square of the local magnetic moment, m. Then, if we adopt a simplified m–temperature profile, m ∼ (Tc–T)0.5, with Tc ≈ 29 K (the SkL-paramagnetic transition temperature), Eg at zero temperature is estimated to be ∼2 × 103 K (for details, see Supplementary Fig. 5). Although the intermediate spin configurations cannot be clearly described from the ρyx data, the ‘emergent magnetic monopoles’ discussed in the literature22 may be relevant.

Figure 3: Lifetime of the metastable skyrmion lattice.
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a, Time evolution of ρyx at selected temperatures after quenching at 0.22 T. The broken red lines are fits to the standard relaxation equation (1). b, Temperature dependence of the lifetime of the metastable SkL at 0.22 T. The lifetimes are obtained from the fitting results in a.

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Finally, on the basis of the present findings, we demonstrate reversible and deterministic switching between the topologically distinct spin configurations. Figure 4a shows phenomenological free-energy landscapes of the relevant spin configurations, illustrating the working principles of the switching. In the ‘SET process’, as has been discussed in Fig. 1b–d, the application of a current pulse to the conical state results in the metastable SkL through the kinetic avoidance of the SkL-to-conical transition. In the ‘RESET’ process, a relatively long current pulse (10 s) with a moderate intensity (1.7 × 106 A m−2) was applied to the quenched SkL to heat the sample to 25–27 K, at which temperature the metastable SkL is short-lived (<10 s; see Fig. 3b) and can therefore relax into the conical state within the pulse duration. The reversible electric switching of the topological/non-topological magnetic structure is thus feasible, as manifested in the switching of ρyx (Fig. 4b, c). The switching exhibits good repeatability (Fig. 4d), highlighting the deterministic nature of the working principle; that is, whereas thermal agitation at high temperatures allows the system to relax into the most thermodynamically stable state within a relatively short time, such relaxation is almost unexpected at low temperatures because the topological stability surpasses the thermal agitation. Whereas such a deterministic switching using heat pulses and quenching has often been demonstrated in glass-forming systems23,24,25, the switching between distinct long-range magnetic orders has, to the best of our knowledge, never been reported.

Figure 4: Creation and annihilation of the metastable skyrmion lattice through thermal control.
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a, Scheme for repeatable switching between the SkL and the conical phase in terms of a schematic free-energy landscape in a multidimensional spin-configuration space. Note that, whereas the SkL is thermodynamically stable in the red-hatched temperature region (27–29 K), the conical state is thermodynamically stable in the blue-hatched temperature region (<27 K). b,c, Single-cycle operation of the switching under the application of rectangular current pulses. The time profiles of ρyx and the current density are shown in b and c, respectively. d, Repetitive switching between the metastable SkL and equilibrium conical phases.

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Our observations clearly establish that a metastable SkL hidden behind non-topological magnetic ordering can be accessed by applying sufficiently high cooling rates. Because the local moments generally develop as the temperature decreases, SkL-derived physical properties become more accessible in the quenched SkL, facilitating experimental studies (for instance, the topological Hall resistivity is markedly enhanced at low temperatures; see Supplementary Fig. 6 and ref. 17). In the context of the possible application of skyrmions, in which the manipulation of a single skyrmion is ultimately considered26,27,28,29,30, we conjecture that a similar creation/annihilation method is also applicable to the case of a small number of skyrmions because the working principle demonstrated above is predicated on manipulating the delicate balance among the topological stability, the thermodynamic stability, and the thermal agitation, properties that remain well defined in a finite-size system. We have also shown that the topological stability is undermined at high temperatures, suggesting that the operating temperature of skyrmion devices must be chosen in the context of a competition between the topological stability and the thermal agitation.

Methods

Sample preparation.

A single crystal of MnSi was grown by the Czochralski method. The sample was oriented by Laue X-ray diffraction, cut with a wire saw, and polished to a size of 2.5 × 0.95 × 0.1 mm3, with the largest surface normal to the 〈100〉 axis. The residual resistivity ratio of the sample used in this study was approximately 55. Cu current leads of 0.43 mm diameter were attached to the sample and fixed with silver paste. Au wires for the voltage probe were soldered to the sample with indium. At 10 K, the sum of the contact resistances of the two current electrodes is approximately 615 mΩ, whereas the resistance of the whole sample is only approximately 1.6 mΩ. Hence, the Joule heating during the current-pulse application occurs mainly at the contacts. For details on the estimation of the Joule heating and the subsequent heating and cooling rates, see Supplementary Information.

Transport measurements.

The Hall resistivity ρyx was measured at a current excitation of 33 Hz with the five-probe method under a magnetic field parallel to the 〈100〉 axis using a lock-in amplifier (Signal Recovery, 7270 DSP) equipped with a transformer preamplifier (Stanford Research Systems, SR554). The current density of the a.c. excitation was set to less than 1.4 × 105 A m−2. We confirmed that the a.c. current caused no appreciable increase in the sample temperature. The pulse currents used for the sample heating were generated by a function generator (NF Corporation, WF1947) connected to a bipolar amplifier (NF Corporation, HSA4014). The voltage drop between the voltage probes attached to the sample was amplified with a low-noise preamplifier (NF Corporation, LI-75A) and was monitored as a function of time using a data logger (Data Translation, DT8824), so that we could derive the time-varying sample temperature (see Supplementary Information). During the time-varying Hall-resistivity measurements, we monitored the digital-to-analog converter output of the lock-in amplifier using the data logger.