Heisenberg magnet with modulated exchange

A modification of the ground state of the classical-spin Heisenberg Hamiltonian in the presence of a weak superstructural distortion of an otherwise Bravais lattice is examined. It is shown that a slight modulation of the crystal lattice with wavevector $\bQ_c$ results in a corresponding modulation of the exchange interaction which, in the leading order, is parametrized by no more than two constants per bond, and perturbs the spin Hamiltonian by adding the ``Umklapp'' terms $\sim S^\alpha_{\bq} S^\alpha_{\bq \pm \bQ_c}$. As a result, for a general spin-spiral ground state of the non-perturbed exchange Hamiltonian, an incommensurate shift of the propagation vector, $\bQ$, and additional new magnetic Bragg peaks, at $\bQ \pm n\bQ_c$, $n = 1,2,...$, appear, and its energy is lowered as it adapts to the exchange modulation. Consequently, the lattice distortion may open a region of stability of the incommensurate spiral phase which otherwise does not win the competition with the collinear N\'{e}el state. Such is the case for the frustrated square-lattice antiferromagnet. In addition, the ``Umklapp'' terms provide a commensuration mechanism, which may lock the spin structure to the lattice modulation vector $\bQ_c$, if there is sufficient easy-axis anisotropy, or a magnetic field in an easy plane.