Ising model in half-space: A series of phase transitions in low magnetic fields

For the Ising model in half-space at low temperatures and for the “unstable boundary condition,” we prove that for each value of the external magnetic field $\mu$, there exists a spin layer of thickness $q(\mu)$ adjacent to the substrate such that the mean spin is close to $-1$ inside this layer and close to $+1$ outside it. As $\mu$ decreases, the thickness of the $(-1)$-spin layer changes jumpwise by unity at the points $\mu_q$, and $q(\mu)\to\infty$ as $\mu\to+0$. At the discontinuity points $\mu_q$ of $q(\mu)$, two surface phases coexist. The surface free energy is piecewise analytic in the domain $\operatorname{Re}\mu>0$ and at low temperatures. We consider the Ising model in half-space with an arbitrary external field in the zeroth layer and investigate the corresponding phase diagram. We prove Antonov's rule and construct the equation of state in lower orders with the precision of $x^7$, $x=e^{-2\varepsilon}$. In particular, with this precision, we find the points of coexistence of the phases $0,1,2$ and the phases $0,2,3$, where the phase numbers correspond to the height of the layer of unstable spins over the substrate.