On projection (in)dependence of monopole condensate in lattice $SU(2)$ gauge theory

We study the temperature dependence of the monopole condensate in different Abelian projections of the $SU(2)$ lattice gauge theory in thermodynamic limit. Using the Fröhlich-Marchetti monopole creation operator we show numerically that the monopole condensate depends strongly on the choice of the Abelian projection. Contrary to the claims in the literature we observe that in the Abelian Polyakov gauge and in the field strength gauge the monopole condensate does not vanish at the critical temperature in large-volume limit. Therefore the monopole condensate in these gauges is not an order parameter of the confinement-deconfinement phase transition.