As it is described in this question the BCS wave function has definite phase but indefinite electron number. Therefore the superconducting order parameter $\langle \phi| c_{k \uparrow}c_{-k \downarrow} |\phi\rangle$ can be nonzero. However one can construct a wave function with definite particle number. In that case the order parameter $\langle N|c_{k \uparrow}c_{-k \downarrow} |N\rangle$ will be zero!

Why does the order parameter depend on our choice of the wave function?

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    $\begingroup$ The value of order parameters always depend on the choice of the wavefunction. Why would you think this is an issue? $\endgroup$ – Norbert Schuch Mar 30 '19 at 7:36
  • $\begingroup$ Compared to e.g ferromagnet I assume the order parameter should have a definite value! $\endgroup$ – richard Mar 30 '19 at 7:51
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    $\begingroup$ For the Ising ferromagnet you can also write states with zero magnetization. $\endgroup$ – Norbert Schuch Mar 30 '19 at 13:43
  • $\begingroup$ @NorbertSchuch Yes but for $T<T_c$ magnetization (order parameter) must be nonzero, right? $\endgroup$ – richard Mar 30 '19 at 15:12
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    $\begingroup$ No, there can are always (well, almost) states with zero order parameter, take e.g. |all up>+|all down>. The point is that (i) random states have a non-zero order parameter, and more importantly (ii) there is long-range order (long-range correlations), which are independent of the ground state chosen, and which are closely linked to the maximum value of the order parameter. $\endgroup$ – Norbert Schuch Mar 30 '19 at 23:12

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