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Magnetoelectric Effects in the Spiral Magnets CuCl$_{2}$ and CuBr$_{2}$

The nature and symmetry of the transition mechanisms in the spin-spiral copper halides CuCl$_2$ and CuBr$_2$ are analyzed theoretically. The magnetoelectric effects observed in the two multiferroic compounds are described and their phase diagram at zero and applied magnetic fields are worked out. The emergence of the electric polarization at zero field below the paramagnetic phase is shown to result from the coupling of two distinct spin-density waves and to be only partly related to the Dzialoshinskii-Moriya interactions. Applying a magnetic field along the two-fold monoclinic axis of CuCl$_2$ yields a decoupling of the spin-density waves modifying the symmetry of the phase and the spin-spiral orientation. The remarkable periodic dependences of the magnetic susceptibility and polarization, on rotating the field in the monoclinic plane, are described theoretically.