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We study the stability against thermal phonon fluctuations of the magnetic dual chiral density wave (MDCDW) phase, an inhomogeneous phase arising in cold dense QCD in a magnetic field. Following a recent study that demonstrated the absence of the Landau-Peierls (LP) instability from this phase, we calculate the (threshold) temperature at which the phonon fluctuations wash out the long-range order over a range of magnetic fields and densities relevant to astrophysical applications. Using a high-order Ginzburg-Landau expansion, we find that the threshold temperature is very near the critical temperature for fields of order 10^18 G, and still a sizable fraction of the critical temperature for fields of order 10^17 G. Therefore, at sufficiently large magnetic fields, the long-range order of the MDCDW phase is preserved over most of the parameter space where it is energetically favored; at smaller magnetic fields, the long-range order is still preserved over a considerable region of parameter space relevant to compact stars.



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