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Spin-gapped magnets with weak anisotropies I: Constraints on the phase of the condensate wave function

We study the thermodynamic properties of dimerized spin-gapped quantum magnets with and without exchange anisotropy (EA) and Dzyaloshinsky and Moriya (DM) anisotropies within the mean-field approximation (MFA). For this purpose we obtain the thermodynamic potential $Ω$ of a triplon gas taking into account the strength of DM interaction up to second order. The minimization of $Ω$ with respect to self-energies $Σ_n$ and $Σ_{an}$ yields the equation for $X_{1,2}=Σ_n\pmΣ_{an}-μ$, which define the dispersion of quasiparticles $E_k=\sqrt{ε_k+X_1}\sqrt{ε_k+X_2}$ where $ε_k$ is the bare dispersion of triplons. The minimization of $Ω$ with respect to the magnitude $ρ_0$ and the phase $Θ$ of triplon condensate leads to coupled equations for $ρ_0$ and $Θ$. We discuss the restrictions on $ρ_0$ and $Θ$ imposed by these equations for systems with and without anisotropy. The requirement of dynamical stability conditions $(X_1>0, X_2>0)$ in equilibrium, as well as the Hugenholtz-Pines theorem, particularly for isotropic Bose condensate, impose certain conditions to the physical solutions of these equations. It is shown that the phase angle of a purely homogenous Bose-Einstein condensate (BEC) without any anisotropy may only take values $Θ=πn $ (n=0,$\pm 1,\pm 2$...) while that of BEC with even a tiny DM interaction results in $Θ=π/2+2πn$. In contrast to the widely used Hartree-Fock-Popov approximation, which allows arbitrary phase angle, our approach predicts that the phase angle may have only discrete values, while the phase of the wave function of the whole system remains arbitrary as expected. The consequences of this phase locking for interference of two Bose condensates and to their possible Josephson junction is studied.