field model that is exactly solvable and provides further insight into the non-convergent behaviour. At $y=y_{\rm cr}$ the RG trajectory approaches the Yang-Lee edge singularity [@YL1], [@YL2] that is described by a non-unitary conformal minimal model ${\cal M}(2,5)$ [@Cardy2]. In this case our findings point as well to the picture in which the RG interface does not approach a single conformal interface but demonstrates an oscillatory non-convergent behaviour. The case $y=-\infty$ is the exceptional case of no magnetic field and the RG boundary is given by the free boundary condition as before. We summarise these answers on the diagram below.

(-3,0) –(3.3,0); (0,-3)–(0,3); (-1.41,-1.41) arc (225:135:2); (-1.41,-1.41) arc (-135:135:2); (3.6,0 ) node [$m$]{}; (0,3.3) node [$ih$]{}; (1.8,2.1) node [$|+\rangle\!\rangle \oplus |-\rangle\!\rangle$]{}; (1.8,-2.1) node [$|+\rangle\!\rangle\oplus |-\rangle\!\rangle$]{}; (-3.6,-1) – (-2.1,-0.1); (-2,0) node [$\bullet$]{} ; (-1.3,2.2) node ; (-3.95,-1) node [$|F\rangle\!\rangle$]{}; (0,0) parabola (-1.7,2.04); (0,0) parabola (-1.7,-2.04);

The main body of the paper is organised as follows. In section 2 we work out in detail the analytically solvable case of vanishing magnetic field. In section 3 we explain the basic TCSA setup for the perturbed Ising model using free fermions. In section 4 we present numerical results obtained using TCSA for real valued 