arXiv2003
The apparent universality of Bloch's T^{3/2}-law for the temperature dependence of the spontaneous magnetization, and of generalizations thereof, is considered. It is argued that in the derivation one should not only consider the exchange interaction between the spins, but also the other interactions between them, leading to elliptical spin precession and deviations from the parabolic dispersion of magnons. Also interaction effects are important to explain the apparent universality of generalized Bloch law exponents e_B, defined by M_s(T)= M_s(0)-const. x T^{e_B}, valid in a wide temperature range T_1 < T < T_2, and for dimensionalities d = 1, 2, and 3. The above-mentioned temperature range, the 'Bloch range', lies above the quantum range, where magnetic long-range order (e.g. in d=2 dimensions) is nontrivially enforced by the additional interactions, but below the thermal critical region, where universal 'anomalous scaling dimensions' apply. In contrast, for the Bloch temperature region, the universality is only apparent, i.e. a crossover-phenomenon, and simple scaling considerations with 'normal dimensions' apply. However, due to interactions, the Bloch exponent e_B depends not only on the dimensionality d of the system, but also on the spin quantum number s (mod (1/2)) of the system, i.e. for given d the Bloch exponent e_B is different for half-integer s and for integer s.