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The spherical magnetization law for the three-dimensional quantum Heisenberg ferromagnet

OpenAI

Published Oct 6, 2026• 0 citations
Publisher

Abstract

We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure. The law is expressed through self-adjoint linear combinations of the spin components and requires no joint measurement of noncommuting observables.

DOI
10.83295/gv7v-kev2

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We prove the spherical magnetization law for the three-dimensional nearest-neighbor isotropic quantum Heisenberg ferromagnet at every fixed positive quantum spin and every sufficiently low fixed positive temperature. As the even periodic cubes grow, the symmetric zero-field magnetization converges in moments to a uniform direction with a deterministic positive magnitude. This magnitude equals the right derivative at zero field of the infinite-volume pressure. The law is expressed through self-adjoint linear combinations of the spin components and requires no joint measurement of noncommuting observables.
We prove Bloch's T3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ3, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion. The thermodynamic limit is taken before the right field derivative at zero, and the low-temperature limit is taken last.
We investigate analytically the magnetization of Heisenberg ferromagnetic lattices in one and two dimensions, and we derive approximate expressions that are valid at high and low temperatures. In the case of the spin-12\frac{1}{2}21​ Heisenberg XX chain in a transverse field, we show that, when the applied magnetic field hhh exceeds its critical value hc=Jh_c=Jhc​=J, where JJJ is the exchange coupling constant, the magnetization per site deviates at low temperatures from its saturation value, 12\frac{1}{2}21​, following a power series with terms involving the power laws T1/2T^{1/2}T1/2, T3/2T^{3/2}T3/2, T5/2T^{5/2}T5/2 ..etc. When h<hch<h_ch<hc​, the zero temperature magnetization per spin turns out to be equal to 1πarcsin⁡(hhc)\frac{1}{\pi}\arcsin(\frac{h}{h_c})π1​arcsin(hc​h​). In this case, the temperature dependence of the magnetization is given by a series expansion with power laws of the form TTT, T3T^3T3, T5T^{5}T5,...etc. In both cases, the coefficients of the expansions are temperature-dependent and are explicitly derived. Using he properties of the Eulerian polynomials, we show that, because of the fast convergence of the derived series for the Fermi-Dirac and the Bose-Einstein distributions, it is possible (in particular in strong magnetic fields) to express the magnetization of the Heisenberg model in a simple analytical form. Furthermore, the analytical results are compared with the exact numerical ones.
For the two-dimensional quantum Heisenberg ferromagnet with asymptotically radial power-law exchange, we prove the leading low-temperature free energy at every fixed spin, uniformly on the weak-field scale. We obtain fractional and logarithmically corrected laws associated with the dispersions predicted in [16, 17]. Quantum Monte Carlo simulations support the associated anomalous magnon dispersions for spin one-half [20]. For kernels with finite second moment, we also extend the nearest-neighbour theorem [14], generalizing Takahashi's quadratic law [21] and its weak-field dependence [11].
We study the ground-state and finite-temperature properties of the one-dimensional isotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions which compete with each other. Extrapolating the exact results for finite-size systems of up to 20 spins, we estimate the ground-state energy and the ground-state two-spin correlation function in the infinite-size limit. Using the cluster transfer matrix method, we also calculate the temperature dependence of the internal energy and the specific heat and that of the inverse correlation length and the wave number, by both of which the asymptotic behavior of the two-spin correlation function in the long-distance limit is characterized. We compare the results of these calculations with those obtained previously for the case where the nearest-neighbor interaction is antiferromagnetic.