Phases of the Goldstone multitrace matrix model in the large-$N$ limit

We consider the Goldstone Hermitian matrix model with a multitrace term. When defining the solution on two intervals, we introduce a special parameter $\xi$ describing the phase. We discuss the phase existence conditions at $\xi=0$ (or $1$) and at $\xi=1/2$. We calculate the propagator and the vacuum energy in the symmetric case $\xi=1/2$. In the general case, we discuss the solution structure and calculate the magnetization and other parameters expressed in terms of the sum of all the intervals.