In TASI Lectures on Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems there is a statement that 2d supersymmetry can can emerge from the dilute Ising model:

$$ \beta H = -J \sum_{<i,j>} \sigma_i \sigma_j - \mu \sum_i \sigma_i^2 $$ where $\sigma = ±1$ represents a site with spin up or down, and $\sigma = 0$ represents a vacant site.

The tricritical point is described by the $\phi^6$-theory:

$$ S = \int d^2x\, [(\partial_\mu \phi)^2 + \lambda_6 \phi^6] $$ where $\phi ∼ \langle\sigma\rangle $ describes the magnetic order parameter.

Although there is no fermion in this action, one can construct a fermion field $\psi$ from a string of spins through the Jordan-Wigner transformation. At the tricritical point, the scaling dimensions of $\phi^2$ and $\psi$ differ exactly by 1/2.

This is not an accident and these two fields form a multiplet under an emergent supersymmetry. More generally, the operators which are even (odd) under the $Z_2$ spin symmetry form the Neveu-Schwarz (Ramond) algebra. The dilute Ising model provides deformations of the underlying superconformal theory within $(−1)^F = 1$ sector, where $F$ is the fermion number.

I don't understand two statements:

1) Why $\phi^6$ theory? How to understand this? Why not $\phi^4$? or $\phi^8$? or $\phi^4+\phi^6$?

2) One can construct fermions from strings of spins even in ordinary Ising model. Why does SUSY emerge in the dilute Ising model model but not in the Ising model?