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Skyrmion Phase Transition: Monte Carlo ResultsWe use the Metropolis algorithm [48, 199] to calculate physical quantities of the system at finite temperatures T. As said above, we use mostly the size N x N x L with N = 40 and thickness L = L_{m} + Lf = 8 [4 magnetic layers, 4 ferroelectric layers). Simulation times are 10^{5} Monte Carlo steps (MCS) per spin for equilibrating the system and 10^{5} MCS/spin for averaging. We calculate the internal energy and the layer order parameters of the magnetic (M_{m}) and ferroelectric (M/) films. The order parameter МДл) of layer n is defined as
where (...) denotes the time average. The definition of an order parameter for a skyrmion crystal is not obvious. Taking advantage of the fact that we know the GS, we define the order parameter as the projection of an actual spin configuration at a given T on its GS and we take the time average. This order parameter of layer n is thus defined as
where S,(7 t) is the /th spin at the time t, at temperature T, and S,(T = 0) is its state in the GS. The order parameter M_{m}(n) is close to 1 at very low T where each spin is only weakly deviated from its state in the GS. M,„(n) is zero when every spin strongly fluctuates in the paramagnetic state. The above definition of M_{m}[n) is similar to the EdwardAnderson (EA) order parameter used to measure the degree of freezing in spin glasses [234]. The EA order parameter, by definition, is calculated as follows. We follow each spin during the time. If it is frozen, then its time average is not zero. If it strongly fluctuates with time evolution, then its time average is zero. To calculate the overall degree of freezing, it suffices to add the square of each spin’s time average. In doing so, we see that the EA order parameter does not express the nature of ordering, but only the degree of freezing. In general, when the GS has several degenerate configurations such as the allup and the alldown spin configurations in a ferromagnetic system of Ising spins, the system chooses one of the two when T tends to 0. The coexistence of several phases is not tolerated in such a case because the resulting energy is higher than that of a pure one (due to walls). However, in frustrated systems where one can construct a ground state by random stacking of frustrated units, one does not have a longrange ordering. Here, we wish to follow the evolution of the system ordering from T = 0, so we have to compare the configuration at temperature T at the time t with the GS we have selected to do the slow heating. That was what we did: We compare the actual configuration obtained by slowly heating the selected GS by projecting it on the selected GS, see Eq. (15.15). There are several possibilities: (i) If the spin structure is not stable when T Ф 0, M_{m}[n) goes to zero with time, this is the case of the KosterlitzThouless (XY spins in 2D with NN interaction); (ii) if the spin structure is frozen or ordered (SG, ferromagnets,...), M_{m}(n) is not zero at low T. In our case of skyrmion structure, we have observed the second case, namely the GS is stable up to a finite T. If the system makes a global rotation during the simulation, then 0 • S? (T = 0) = 0 for a longtime average. But the length of this runtime depends on the nature of ordering and the size of the system used in simulations. For large disordered systems such as SG and complicated noncollinear extended skyrmion structures, the global rotation may be forbidden or the time to realize it is out of reach in MC simulations. To see if a global rotation is realized or not, we have to make a finitetime scaling to deduce properties at the infinite time. This is very similar in spirit with the finitesize scaling used to deduce properties at the infinite ciystal size. We have previously performed a finitetime scaling for the 2D skyrmion crystal [103]. In that work, we have used the same order parameter as Eq. (15.15). We have seen that skyrmions need much more than 10^{6} MC steps per spin to relax to equilibrium. The order parameter follows a stretched exponential law as in SG and stabilized at nonzero values for T Figure 15.9 Energy of the magnetic film versus temperature T for (a) )^{m}f = —0.1 ,J^{m}f = — 0.125, J^{m}f = —0.15, J ^{m}f = —0.2 (all the lines are the same, see text for comments); (b) ] "'/ = —0.45 (purple line), J ^{m}f = —0.75 (green line), 7'"/ = —0.85 (blue line) and J ^{m}/ =—1.2 (gold line), without an external magnetic field. we have made a very slow heating of a selected GS and we did not observe a global rotation. Note that the counting of topological charges around each skyrmion is numerically possible. In that case, the charge number evolves with T and goes to zero at the phase transition. The procedure is equivalent to projecting the skyrmion spin texture on its GS. We have chosen the projection one. The total order parameters M_{m} and Mf are the sum of the layer order parameters, namely M_{m} = M_{m}(n) and Mf = M/(n). In Fig. 15.9, we show the dependence of energy of the magnetic film versus temperature, without an external magnetic field, for various values of the interface magnetoelectric interaction: In Fig. 15.9a for weak values J ^{m}f = 0.1, J^{m}f = 0.125, J ^{m}f = —0.15, J ^{m}f = —0.2, and in Fig. 15.9b for stronger values ]^{m}f = 0.45,/"'/ = 0.75,7'"/ = 0.85,7 ^{m}/ = 1.2. As said in the GS determination, when J'"/ is weak, the GS is composed with large ferromagnetic domains at the interface (see Fig. 15.3). Interior layers are still ferromagnetic. The energy, therefore, does not vaiy with weak values of J'"/ as seen in Fig. 15.9a. The phase transition occurs at the curvature change, namely maximum of the derivative or maximum of the specific heat, ~ 1.25. Note that the energy at Г = 0 is equal to —2.75 by extrapolating the curves in Fig. 15.9a to T = 0. This value is just the sum of energies of the spins across the layers: 2 interior spins with 6 NN, 2 interface spins with 5 NN. The energy per spin is thus (in ferromagnetic state): E = —(2 x 6 + 2 x 5)/(4 x 2) = 2.75 (the factor 2 in the denominator is to remove the bond double counting in a crystal). For stronger values of J^{m}f, the curves shown in Fig. 15.9b indicate a deviation of the ferromagnetic state due to the non collinear interface structure. Nevertheless, we observe the magnetic transition at almost the same temperature, namely Г"’ ~ 1.25. It means that spins in interior layers dominate the ordering. We show in Fig. 15.10 the total order parameters of the magnetic film M_{m} and the ferroelectric film Mf versus T for various values of the parameter of the magnetoelectric interaction J^{m}f = —0.1, 0.125, 0.15, 0.2 and for ] ^{m}f = 0.45, 0.75, 0.85, 1.2, without an external magnetic field. Several remarks are in order:
Figure 15.10 (a) Order parameter of the magnetic film M,„ versus T; (b) Order parameter of the ferroelectric film Mf versus T, for J^{m}f = —0.1 (purple dots), J^{m}f = —0.125 (green dots), J^{m}f = 0.15 (blue dots), ) ^{m}f = —0.2 (gold dots), without an external magnetic field. observed in many systems, for example, the surface layer of a thin film can become disordered at a low temperature while the bulk is still ordered [248]. One can also mention the partial phase transition in helimagnets in a field [105]. We show in Fig. 15.11 the order parameters of the magnetic and ferroelectric films at strong values of J ^{m}f as functions of T, in zero field. We observe that the stronger J ^{m}f  is, the lower T™ becomes. This is understood by the discussion given below Eq. 15.13 for a monolayer: The stronger J ^{m}f  makes the larger angle в. In the case of many magnetic layers shown in Fig. 15.11, the larger angle causes a stronger competition with the collinear ferromagnetic interaction of the interior layers. This enhanced competition gives rise to the destruction of the ordering at a lower temperature. We examine the field effects now. Figure 15.12 shows the order parameter and the energy of the magnetic film versus T, for various values of the external magnetic field. The interface magnetoelectric interaction is ]^{m}f = 1.2. Depending on the magnetic field, the non collinear spin configuration survives up to a temperature between 0.5 and 1 (for H = 0). After the transition, spins align themselves in the field direction, giving a large value of the order parameter (Fig. 15.12a). The energy shows a sharp curvature change only for Figure 15.11 (a) Order parameter of the magnetic film versus T; (b) Order parameter of the ferroelectric film versus T for J ^{m}f = 0.45 (purple dots), )^{m}f = —0.75 (green dots),/= 0.85 (blue dots) and J^{m}f = 1.2 (gold dots), without an external magnetic field. H = 0, meaning that the specific heat is sharp only for H = 0 and broadened more and more with increasing H. We consider now the case of very strong interface couplings. Figure 15.13a shows the magnetic order parameter versus T. The purple and green lines correspond to M for J ^{m}f =  2.5 with H^{z} = 1.0 and H^{z} = 1.5, respectively; the blue and gold lines correspond to M for J^{m}f = m  6 with H^{z} = 0 and H^{z} = 1. These curves indicate firstorder phase transitions at T” = 1.05 for C!^{mf} = 2.5, H^{z} = 1) (purple), at T_{c}^{m} = 1.12 for (J^{m}f = 2.5, H^{z}= 1.5) (green) and at T_{c}^{m} = 1.25 for (7 ^{m}f = 6, H^{z} = 1) (blue). In the case of zero field, namely (/ ^{m}f = —6, H^{z} = 0) (gold), one has a firstorder phase transitions occurring at T_{c} = 2.30. Let us discuss the nature of the transition in shown in Fig. 15.13a. When H Ф 0, the first transition at low temperature (Г ~ 1.05  1.25) is due to the destruction of the skyrmion structure. After this transition, the z spin components being not zero under an applied field come close to zero only at high T (~ 2.3). This is not a phase transition because the z components will never be zero in a field if J^{m}f is not so strong. When J ^{m}f is veiy strong (J'^{n}f = 6, blue data points), the DM interaction is so strong that the spins will lie in the xy plane in spite of H : We see that the z spin components are zero after the loss of the ferroelectric ordering at Г ~ 2.3. Note Figure 15.12 (a) Temperature dependence of (a) the magnetic order parameter; (b) the magnetic energy for H =0 (purple dots), H = 0.25 (green line), H = 0.5 (blue line), H =0.75 (gold line), H = 1 (yellow line). The interface magnetoelectric interaction is J = —1.2. that when H = 0 (gold data points), there is no skyrmion, the spin configuration is chiral (helical) as shown in Section 15.3. The single transition to the paramagnetic phase occurs at Г ~ 2.3 where the chiral ordering and the ferroelectric ordering are lost at the same time (see Fig. 15.13b). Figure 15.13b shows the magnetic (purple) and ferroelectric (green) energies versus T for (J^{m}f = — 6, H^{z} = 0). One sees the discontinuities of these curves at T_{c}~ 2.3, indicating the first order transitions for both magnetic and ferroelectric at the same temperature. In fact, with such a strong ] ^{m}f the transitions in both Figure 15.13 (a) Order parameter of magnetic film versus T. The purple and green dots correspond to M for (J ^{m}f = —2.5, H^{z} = 1) and (J ^{m}f = —2.5, H^{z} = 1.5), blue and gold dots correspond to M for (J ^{m}f = —6, H^{z} = 1) and {J^{m}f = —6, H^{z} = 0). (b) Energies of magnetic (purple dots) and ferroelectric (green dots) subsystems versus T for (/ = —6, H = 0). magnetic and ferroelectric films are driven by the interface, this explains the same T_{c} for both. The firstorder transition observed here can be understood because the present system is a frustrated system due to the competing interactions. So far all frustrated non collinear spin systems have been found possessing a firstorder transition (see Ref. [85] and references in Ref. [251]). Let us show the effect of an applied electric field. For the ferroelectric film, polarizations are along the z axis so that an applied electric field E along this direction will remove the phase transition: The order parameter never vanishes when E Ф 0, similar to the case of a ferromagnet in an applied magnetic field. This is seen in Figure 15.14 (a) Order parameter and (b) energy of ferroelectric film, versus temperature for £ = 0 (purple dots), £ = 0.25 (green line), E = 0.5 (blue line), £ = 0.75 (gold line), £ = 1 (yellow line). The interface magnetoelectric interaction is J ^{m}f = —1.2 Fig. 15.14. Note that the energy has a sharp change of curvature for E = 0 indicating a transition, other energy curves with E Ф 0 do not show a transition. One notices some anomalies at Г ~ 1 —1.1 which are due to the effect of the magnetic transition in this temperature range. 
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