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Effective field theories and spin-wave excitations in helical magnets

We consider two classes of helical magnets. The first one has magnetic ordering close to antiferromagnet and the second one has magnetic ordering close to ferromagnet. The first case is relevant to cuprate superconductors and the second case is realized in FeSrO$_3$ and FeCaO$_3$. We derive the effective field theories for these cases and calculate corresponding excitation spectra. We demonstrate that the "hourglass" spin-wave dispersion observed experimentally in cuprates is a fingerprint of the "antiferromagnetic spin spiral state". We also show that quantum fluctuations are important for the "ferromagnetic spin spiral", they influence qualitative features of the spin-wave dispersion.