(-3,0) –(3.3,0); (0,-3)–(0,3); (0,0) circle \[radius=2\]; (3.6,0 ) node [$m$]{}; (0,3.3) node [$h$]{}; (1.8,1.8) node [$|+\rangle\!\rangle $]{}; (-1.7,1.8) node [$|+\rangle\!\rangle $]{}; (-1.7,-1.8) node [$|-\rangle\!\rangle$]{}; (1.8,-1.8) node [$|-\rangle\!\rangle$]{}; (3.8,1) – (2.1,0.1); (-2,0) node [$\bullet$]{} ; (2,0) node [$\bullet$]{} ; (3.8,1.2) node [$|+\rangle\!\rangle \oplus |-\rangle\!\rangle$]{}; (-2.4,0.3) node [$|F\rangle\!\rangle$]{};

The fact that we obtain a superposition of boundary states for $h=0, m>0$ reflects the spontaneous symmetry breaking of the low temperature phase.

We have also investigated the imaginary magnetic field. It is convenient to label the RG trajectories by the dimensionless ratio \[y\_def\] y= . As shown in [@FZ] for imaginary magnetic field when $y>y_{\rm cr}=-2.42929$ (the value is taken from [@Zam2]) the vacuum energy is complex with a two-dimensional vacuum subspace spanned by the eigenstates with complex conjugated energy eigenvalues. We find that in all such cases the RG boundary is $|+\rangle\!\rangle \oplus |-\rangle\!\rangle$. For $y<y_{\rm cr}$ the vacuum energy is real and the flows are massive. We argue that in this case the RG boundary does not approach a single conformal boundary condition as we go to the far infrared. For large negative values of $y$ the RG flow of the vacuum vector can be well approximated by the boundary magnetic 