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We study the stability against thermal phonon fluctuations of the magneticdual chiral density wave (MDCDW) phase, an inhomogeneous phase arising in colddense QCD in a magnetic field. Following a recent study that demonstrated theabsence of the Landau-Peierls (LP) instability from this phase, we calculatethe (threshold) temperature at which the phonon fluctuations wash out thelong-range order over a range of magnetic fields and densities relevant toastrophysical applications. Using a high-order Ginzburg-Landau expansion, wefind that the threshold temperature is very near the critical temperature forfields of order $10^{18}$ G, and still a sizable fraction of the criticaltemperature for fields of order $10^{17}$ G. Therefore, at sufficiently largemagnetic fields, the long-range order of the MDCDW phase is preserved over mostof the parameter space where it is energetically favored; at smaller magneticfields, the long-range order is still preserved over a considerable region ofparameter space relevant to compact stars.

Thermal phonon fluctuations and stability of the magnetic dual chiral density wave phase in dense QCD