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Reminiscence of a Magnetization Plateau in a Magnetization Processes of Toy-Model Triangular and Tetrahedral Clusters

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Abstract

We discuss magnetization curves of a toy-model trigonal and tetrahedral clusters. Nonlinearity of magnetization with local minimum of differential susceptibility resembling known magnetization plateaus of triangular-lattice and pyrochlore lattice antiferromagnets is observed at intermediate temperature range JT ≲ Θ (here, J is the exchange coupling constant and Θ is a Curie–Weiss temperature). This behavior is due to increased statistical weight of the states with intermediate total spin of the cluster, which is related to the “order-by-disorder” mechanism of plateau stabilization of a macroscopic frustrated magnet.

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Notes

  1. However, numerical analysis shows that the “local” Curie–Weiss temperature Θ = (gμΒ)2S(S + 1)/(3χ) – T approaches its high temperature limit within 1% only at T > 50JS(S + 1).

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ACKNOWLEDGMENTS

Author thanks Prof. L.E. Svistov (Kapitza Institute) for drawing attention to this problem and Prof. A.I. Smirnov (Kapitza Institute) for providing experimental data for comparison and useful discussions.

The work was supported by Russian Science Foundation Grant no. 17-02-01505, author work at HSE was supported by Program of fundamental studies of Higher School of Economics.

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Correspondence to V. N. Glazkov.

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Appendices

DETAILS OF WEIGHT FACTOR CALCULATIONS

We calculate the weight factor for quantum model by simple “brute force” counting. For finite quantum mechanical model one can easily count all possible spin projections, which runs from –NS to NS. Each of these projections can be constructed in several ways. Total spin of the cluster can take values from NS to 0 or 1/2. To calculate weight factor one simply has to note that if some nonnegative spin projection Sz can be constructed by \({{n}_{{{{S}_{z}}}}}\) ways and higher projection (Sz + 1) can be constructed by \({{n}_{{{{S}_{z}} + 1}}}\) ways, then weight factor for total spin S = Sz is exactly DN(S) = \({{n}_{{{{S}_{z}}}}}\)\({{n}_{{{{S}_{z}} + 1}}}\), as all total spin states that have the spin projection (Sz + 1) have to include spin projection Sz in their spin multiplet as well. Results are summed up in the Tables 1–3. We do not obtained compact combinatorial expressions for the weight factors, however we can note that for N = 3 found values correspond to D(Stot) = 1 + 2Stot for StotS and to D(Stot) = 1 + 3SStot for Stot > S. It is also interesting to note here that weight factor for the singlet (S = 0) state of tetrahedral cluster increases with increasing spin as (2S + 1), being a precursor of the macroscopic degeneracy of pyrochlore lattice antiferromagnets.

For classical model we will start from the case N = 2.We are interested in total spin of the pair, which does not change on simultaneous rotation of both spins. Thus, we can fix one spin and weight factor can be calculated from all possible orientations of the second spin with respect to the first. If Θ is a polar angle describing second spin direction selected in such a way that Θ = 0 corresponds to antiparallel orientations of spins, then total spin of the pair is

$${{S}^{2}} = 2(1 + \cos \Theta )$$

(spin vectors of the classical model are unit vectors). Then we differentiate this equation

$$SdS = - \sin \Theta d\Theta $$

and compare it with the fraction of realization of spin configurations within the same angle interval

$$\frac{{dn}}{n} = \frac{{2\pi \sin \Theta d\Theta }}{{4\pi }}.$$

This gives directly the weight factor

$${{D}_{2}}(S) = \frac{1}{n}\frac{{dn}}{{dS}} = \frac{S}{2}$$
((13))

here, 0 ≤ S ≤ 2.

The result could look counterintuitive: the probability of S = 0 configuration appears to be much less then the probability of S = 2 configuration. This is actually due to the fact that for antiparallel orientation (S = 0) small deviations from the exact antiparallel orientation results in the appearance of the total spin linear in deviation angle, while for parallel orientation (S = 2) small deviations lead to decrease of the total spin quadratic in deviation angle. Also one can note on this simple example that classical and quantum weight factors differ strongly here: in the quantum case for the pair of equal spins each value of the total spin from zero to 2S is unique.

For the case of three spins N = 3 we, again, can fix the direction of one spin S1. Then we sum up two remaining spins σ = S2 + S3, the length distribution for spin σ is found above. Total spin is equal to

$${{S}^{2}} = 1 + {{\sigma }^{2}} + 2\sigma \cos \Theta ,$$

here, polar angle Θ is selected as above.

All possible configurations are confined in the plane (σ, cosΘ) with 0 ≤ σ ≤ 2 and –1 ≤ cosΘ ≤ 1. Fraction of realization of spin configurations within element dσdcosΘ is

$$\frac{{dn}}{n} = {{D}_{2}}(\sigma )d\sigma d(\cos \Theta ){\text{/}}2 = \frac{\sigma }{4}d\sigma d(\cos \Theta ).$$

From isoline equation cosΘ = (S2 – 1 – σ2)/(2σ) one can derive d(cosΘ) = SdS/σ. This results in differential weight factor

$$\frac{1}{n}\frac{{dn}}{{dS}} = \frac{S}{4}d\sigma ,$$

which have to be integrated over possible σ.

Possible σ range depends on S value: (1 – S) ≤ σ ≤ (1 + S) for S < 1, and (S – 1) ≤ σ ≤ 2 for S > 1. This yields

$${{D}_{3}}(S) = \left\{ \begin{gathered} \frac{{{{S}^{2}}}}{2},\quad 0 \leqslant S < 1, \hfill \\ \frac{{S(3 - S)}}{4},\quad 1 \leqslant S \leqslant 3. \hfill \\ \end{gathered} \right.$$
((14))

Finally, for the case of tetrahedral cluster N = 4 we follow the same route. We fix spin vector S1 and sum up all other vectors to the spin vector σ = S2 + S3 + S4 with known length distribution and derive differential weight factor

$$\frac{1}{n}\frac{{dn}}{{dS}} = \frac{S}{2}\frac{{{{D}_{3}}(\sigma )}}{\sigma }d\sigma ,$$

which have to be integrated over possible σ. Possible σ ranges are: (1 – S) ≤ σ ≤ (1 + S) for S < 1, and (S – 1) ≤ σ ≤ (1 + S) for 1 < S < 2, and (S – 1) ≤ σ ≤ 3 for S > 2. This yields

$${{D}_{4}}(S) = \left\{ \begin{gathered} \frac{{{{S}^{2}}}}{2}\left( {1 - \frac{3}{8}S} \right),\quad 0 \leqslant S < 2, \hfill \\ S{{\left( {1 - \frac{S}{4}} \right)}^{2}},\quad 2 \leqslant S \leqslant 4. \hfill \\ \end{gathered} \right.$$
((15))

Note, that D3(S) and D4(S) are equal to zero at extreme spin values (0 and N) and reach maximal values at S = 3/2 and S = 16/9 for triangular and tetrahedral clusters, correspondingly. The weight factors are continuous functions of total spin, however their derivative are discontinuous: slope of DN(S) changes at S = 1 and S = 2 for trigonal and tetrahedral clusters, correspondingly. Obtained weight factors coincide with the values obtained using a different approach [21].

SUSCEPTIBILITY CURVES FOR THE EXTREME QUANTUM CASE OF S = 1/2

Cases of S = 1/2 spin clusters differ from the higher spin cases analyzed above. The reason is that for small S energy scales of exchange coupling J and Curie–Weiss temperature JS(S + 1) do not differ strongly and, hence, the intermediate temperatures regime is simply absent.

Modeling of magnetization and susceptibility curves (see Fig. 8) shows expected smearing of the magnetization steps on heating at T ≈ 0.3J, differential susceptibility demonstrates local minimum up to the temperatures about (0.6–0.7)J. On cooling this local minimum smoothly evolves toward position right between the magnetization steps: to B0 = Bsat/2 for the trigonal cluster and to B0 = 3Bsat/4 for the tetrahedral cluster. Consequently, plateau quality factor reaches unity at T = 0. Similar locking of the susceptibility minimum between magnetization steps also occurs for higher spins, however for higher spin temperature evolution of plateau-like feature parameters has demonstrated some sort of change (crossover-like) at locking: quality factor was starting to increase more rapidly, slope of the B0(T) curve was changing. Such a change made it possible (for spins S ≥ 1) to single outquite clearly the lower boundary of the low-temperature regime in which the local minimum position is determined by "locking" between the nearest magnetization steps and to exclude this trivial regime from analysis.

Fig. 8.
figure 8

(Color online) (a, b) Examples of differential susceptibility curves for S = 1/2 triangular (a, c) and tetrahedral (b, d) clusters. Curves are shifted for better presentation. (c) and (d) Temperature dependences of susceptibility minimum position (symbols) and its Q-factor (solid curve) for S = 1/2 triangular (a, c) and tetrahedral (b, d) clusters.

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Glazkov, V.N. Reminiscence of a Magnetization Plateau in a Magnetization Processes of Toy-Model Triangular and Tetrahedral Clusters. J. Exp. Theor. Phys. 128, 464–476 (2019). https://doi.org/10.1134/S106377611903004X

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