In frustrated antiferromagnets with isotropic exchange interactions, there is typically a manifold of degenerate classical ground states. This degeneracy is broken by the (free) energy of quantum or thermal fluctuations, or the uniform effects of bond disorder. We derive effective Hamiltonians to express each kind of selection effect, in both exact forms and convenient approximate forms. It is argued that biquadratic terms, representing the collinear-selecting effects of quantum fluctuations, should be included in classical simulations of large-$S$ frustrated magnets at low temperatures

The spin-hole coherent-state path integral is used to generate a systematic large-spin expansion of the t-J model on the square lattice. The single hole's classical energy is minimized by small polarons with short-ranged interactions. Intersublattice hopping of polarons is forbidden by a tunneling selection rule. We derive the low-energy Lagrangian which reduces to the model of Wiegmann, Wen, Shankar, and Lee of Neel-gauge-field-induced superconductivity.

The ‘‘Marshall–Huse–Elser’’ variational wave functions describe ordered planar (possibly noncollinear) ground states of s=1/2 Heisenberg spin-exchange Hamiltonians. We show how to generalize these wave functions to allow nonuniform states, arising from interactions which may be random and/or frustrated. In the Szi basis, the amplitude is exp[1/2H̃({Szi})] where the pseudo-Hamiltonian is given by H̃=−∑i2iθiSzi −(1/2)!∑ijKijSziSzj −(1/3!)∑ijkiLijkSzi SzjSzk. Here the classical ground-state directions {θi} (=0 or π in the Néel state) are found by minimizing an effective classical energy F=∑ij[Aij cos(θi−θj) +Bij cos2(θi−θj)], where (Aij,Jij) are functions of nearby Jij’s. Next, Kij and Lijk are taken to be functions of the values {Jij} and the angles {θi−θj}. The functions for Aij, Bij, Kij, and Lijk depend parametrically on a small set of variational parameters. Thus the dimension of parameter space does not grow with system size. We estimate the parameter values analytically, using the spin-wave approximation in a uniformly twisted square-lattice antiferromagnet. Also, the general form of the three-spin coefficient Lijk is roughly a sum of contributions ∝Jij sin(θi−θj), and j and k are both neighbors of spin i.

Type III ground states of hcp vector antiferromagnets—appropriate to the wurtzite magnetic semiconductors (MS) such as Zn1−xMnxSe —are shown to be classically unstable at long wavelengths. The Hamiltonian includes antiferromagnetic nearest-neighbor (NN) and next-nearest-neighbor (NNN) isotropic exchange (J1,J2), and NN anisotropic Dzyaloshinsky–Moriya exchange (D). The hexagonal symmetry allows Ja1 for NN in planes perpendicular to the c axis to be different from Jc1 for NN between planes. For three-component spins a five-dimensional degenerate manifold of type III ground states is found (Ja1=Jc1, D=0). Their instability is investigated through a continuum formulation of the exchange energy, treating D and ΔJ≡(Ja1−Jc1) as small compared with J1. ΔJ is found to induce a twist of a noncollinear ground state, stabilized by D to zeroth order. By fitting the shift δQ of magnetic peaks seen in neutron scattering experiments on Zn0.45Mn0.55Se the required ΔJ/J1 (∼δQ) is found to be 0.06. This value agrees with estimates of higher order superexchange processes for the two types of pairs.

The Dzyaloshinski-Moriya (DM) anisotropic superexchange constant and the resulting electron-paramagnetic-resonance (EPR) linewidth in Mn-based II-VI-compound diluted magnetic semiconductors (DMS) such as ${\mathrm{Cd}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Mn}}_{\mathrm{x}}$Te are calculated quantitatively. An Anderson Hamiltonian, developed in a previous study of isotropic superexchange, describing correlated Mn 3d states hybridized with semiconducting s- and p-derived levels, is generalized to include the anion spin-orbit coupling responsible for anisotropic superexchange. DM exchange is shown to be the dominant anisotropic interaction, with magnitude \ensuremath{\sim}5% of isotropic superexchange. The EPR line shape is calculated with use of a moment expansion of the magnetic response function to first order in inverse temperature together with a maximum-entropy ansatz. The calculated infinite-temperature linewidths are in good agreement with extrapolated experimental values. A novel fit of the theoretical temperature dependence to existing experimental linewidth data provides the first empirical value for the anisotropic exchange constant, in excellent agreement with the theoretical value. Calculated chemical trends for the exchange constants yield the experimentally expected linewidth trends.