Zero-Temperature Configurations of Short Odd-Numbered Classical Spin Chains with Bilinear and Biquadratic Exchange Interactions

The lowest energy configurations of short odd open chains with classical spins are determined for antiferromagnetic bilinear and biquadratic nearest-neighbor exchange interactions. The zero field residual magnetization generates differences with the magnetic behavior of even chains, as the odd chain is like a small magnet for weak magnetic fields. The lowest energy configuration is calculated as a function of the total magnetization M, even for M less than the zero field residual magnetization. Analytic expressions and their proofs are provided for the threshold magnetic field needed to drive the system away from the antiferromagnetic configuration and the spin polar angles in its vicinity, when the biquadratic interaction is relatively weak. They are also given for the saturation magnetic field and the spin polar angles close to it. Finally, an analytic expression along with its proof is given for the maximum magnetization in zero magnetic field for stronger biquadratic interaction, where the lowest energy configuration is highly degenerate.Comment: 17 pages, 9 figure

Thermalization away from Integrability and the Role of Operator Off-Diagonal Elements

We investigate the rate of thermalization of local operators in the one-dimensional anisotropic antiferromagnetic Heisenberg model with next-nearest neighbor interactions that break integrability. This is done by calculating the scaling of the difference of the diagonal and canonical thermal ensemble values as function of system size, and by directly calculating the time evolution of the expectation values of the operators with the Chebyshev polynomial expansion. Spatial and spin symmetry is exploited and the Hamiltonian is divided in subsectors according to their symmetry. The rate of thermalization depends on the proximity to the integrable limit. When integrability is weakly broken thermalization is slow, and becomes faster the stronger the next-nearest neighbor interaction is. Three different regimes for the rate of thermalization with respect to the strength of the integrability breaking parameter are identified. These are shown to be directly connected with the relative strength of the low and higher energy difference off-diagonal operator matrix elements in the symmetry eigenbasis of the Hamiltonian. Close to the integrable limit the off-diagonal matrix elements peak at higher energies and high frequency fluctuations are important and slow down thermalization. Away from the integrable limit a strong low energy peak gradually develops that takes over the higher frequency fluctuations and leads to quicker thermalization.Comment: 11 pages, 9 figure

Classical magnetization of a four-dimensional Platonic solid

The 600-cell is a regular 4-polytope that is a four-dimensional analog of a Platonic solid. Three-dimensional Platonic solids with icosahedral $I_h$-symmetry have been shown to have a discontinuous ground-state magnetization response in an external field at the classical and quantum level, when spins mounted on their vertices interact according to the antiferromagnetic Heisenberg model. The discontinuities are not due to anisotropy in spin space, but rather to the special connectivity of the molecules. Here the nearest-neighbor antiferromagnetic XX and Heisenberg models in a magnetic field are considered for classical spins mounted on the 120 vertices of the 600-cell. The ground-state magnetization response is rich, characterized by six magnetization discontinuities in the XX case, and six magnetization and three susceptibility discontinuities in the Heisenberg case. This demonstrates that going from three to four spatial dimensions enriches the ground-state magnetization response.Comment: 4 pages, 2 figures, 3 table