The microscopic electric interaction is spin independent to leading order, but the Pauli exclusion principle ties the spatial symmetry of an electronic state to its spin symmetry. Different spin combinations therefore have different Coulomb and kinetic energies. Projecting those energies onto localized low-energy moments often gives an approximately rotationally invariant bilinear exchange Hamiltonian operator . For spin one half, the spin singlet state and triplet state energies are distinguished by and , so any isotropic two-level spin singlet state–triplet state splitting can be written in this form plus a constant. More general spins or orbital degeneracy can require extra interactions.
An antiferromagnetic oxide commonly has superexchange through a ligand such as oxygen. Virtual hopping between half-filled metal orbitals lowers the energy of antiparallel spins more strongly because the corresponding intermediate doubly occupied configurations are allowed by the Pauli exclusion principle. The simple one-orbital Hubbard model limit gives , equivalent to negative in the question's minus-sign convention. Bond angles and orbital occupancy can change the sign in more complicated oxides, so superexchange is not universally antiferromagnetic irrespective of geometry.
For the classical XY ferromagnet choose every spin parallel within the plane. Each nearest-neighbor and next-nearest-neighbor in-plane dot product is . The printed sum counts one bond of each range per spin, soThere is no additional factor of two from counting both directions: the Hamiltonian operator already lists only and . Alignment along would give zero XY exchange energy and is not the in-plane ordered state used for these expressions.
For collinear antiferromagnetic order in the plane, take . Nearest neighbors are antiparallel, giving , whereas second neighbors lie on the same sublattice and remain parallel, giving . ThusThe result follows from substituting into the specified XY Hamiltonian operator, regardless of whether that trial state is the global minimum for the chosen exchanges. Comparing trial-state energies does not eliminate the possibility of a lower helical state.
In both collinear trial states, a spin and its second neighbor are parallel. The next-nearest-neighbor energy is therefore per spin in either case. Making negative makes that contribution positive and raises both energies.
A negative prefers opposite second-neighbor spins. Neither all-parallel order nor the simple alternating nearest-neighbor pattern satisfies that preference: in the latter, alternating twice returns to the original direction. This competition is a source of frustration when the nearest-neighbor interaction favors one of those collinear patterns, and allows a noncollinear twist to lower the energy. Merely calling antiferromagnetic does not mean that the nearest-neighbor alternating state satisfies it.
A relative angle between neighbors gives in-plane dot products and at the two ranges. Set . ThenThe classical helical minimum of the J1-J2 XY chain follows because for this is a strictly convex quadratic in . Its unconstrained minimum isIt is a genuine noncollinear helix when . Both chiralities and have the same energy, and the pitch is defined modulo . At equality the helix merges into a collinear endpoint, so the strict inequality is required for an energy strictly lower than both collinear choices.
For a direct comparison write . The minimum and the energy differences areBoth differences are strictly positive in the stated interior region. The stronger classical global argument Fourier-transforms the translationally invariant bilinear exchange: its energy is minimized by the wave vector maximizing , and the planar single- helix realizes the fixed spin length at every site. This establishes the minimum within the classical/mean-field model, not an assertion of true long-range magnetic order at positive temperature in a one-dimensional short-range quantum chain.
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