Heisenberg magnet with modulated exchange

Abstract

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 wave vector ${\mathbf{Q}}_{c}$ 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 $\ensuremath{\sim}{S}_{\mathbf{q}}^{\ensuremath{\alpha}}{S}_{\mathbf{q}\ifmmode\pm\else\textpm\fi{}{\mathbf{Q}}_{c}}^{\ensuremath{\alpha}}.$ As a result, for a general spin-spiral ground state of the nonperturbed exchange Hamiltonian, an incommensurate shift of the propagation vector Q and additional new magnetic Bragg peaks, at $\mathbf{Q}\ifmmode\pm\else\textpm\fi{}n{\mathbf{Q}}_{c},$ $n=1,2,\dots{},$ 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\'eel 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 ${\mathbf{Q}}_{c},$ if there is sufficient easy-axis anisotropy, or a magnetic field in an easy plane.