Quantum theory of spin waves for helical ground states in a hollandite lattice

We perform spin-wave analysis of classical ground states of a model Hamiltonian proposed earlier (Mandal et al 2014 Phys. Rev. B 90 104420) for [Formula: see text] compounds. It is known that the phase diagram of the hollandite lattice (lattice of [Formula: see text] compounds) consists of four different helical phases (FH, A2H, C2H, CH phase) in the space of model parameters [Formula: see text]. The spin wave dispersion shows presence of gapless mode which interpolates between quadratic to linear depending on phases and values of J _i . In most cases, the second lowest mode shows the existence of a roton-like minima mainly from [Formula: see text] to [Formula: see text] and [Formula: see text] to [Formula: see text] path and it appears at the value of [Formula: see text] for constant [Formula: see text]. Few higher modes also show similar minima. Each helical phase has its characteristic traits which can be used to determine the phases itself. The analytical expressions of eigenmodes at high symmetry points are obtained which can be utilized to extract the values of J _i . Density of states, specific heat and susceptibilities at low temperature have been studied within spin-wave approximation. The specific heat shows departure from T ^1.5(3) dependence found in three-dimensional unfrustrated ferromagnetic(anti-ferromagnetic) system which seems to be the signature of incommensurate helical phase. The parallel susceptibility is maximum for FH phase and minimum for CH phase at low temperature. The perpendicular susceptibility is found to be independent of temperature at very low temperature. Our study can be used to compare experiments on magnon spectrum, elastic neutron scattering, and finite temperature properties mentioned above for clean [Formula: see text] system as well as determining the values of J _i .