We show that correlations between the phases of the galaxy density field in redshift space provide additional information about the growth rate of large-scale structure that is complementary to the power spectrum multipoles. In particular, we consider the multipoles of the line correlation function (LCF), which correlates phases between three collinear points, and use the Fisher forecasting method to show that the LCF multipoles can break the degeneracy between the measurement of the growth rate of structure $f$ and the amplitude of perturbations $sigma_8$ that is present in the power spectrum multipoles at large scales. This leads to an improvement in the measurement of $f$ and $sigma_8$ by up to 220 per cent for $k_{rm max} = 0.15 , hmathrm{Mpc}^{-1}$ and up to 50 per cent for $k_{rm max} = 0.30 , hmathrm{Mpc}^{-1}$ at redshift $z=0.25$, with respect to power spectrum measurements alone for the upcoming generation of galaxy surveys like DESI and Euclid. The average improvements in the constraints on $f$ and $sigma_8$ for $k_{rm max} = 0.15 , hmathrm{Mpc}^{-1}$ are $sim 90$ per cent for the DESI BGS sample with mean redshift $overline{z}=0.25$, $sim 40$ per cent for the DESI ELG sample with $overline{z}=1.25$, and $sim 40$ per cent for the Euclid H$alpha$ galaxies with $overline{z}=1.3$. For $k_{rm max} = 0.30 , hmathrm{Mpc}^{-1}$, the average improvements are $sim 40$ per cent for the DESI BGS sample and $sim 20$ per cent for both the DESI ELG and Euclid H$alpha$ galaxies.
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