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?

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?

May be another formulation of my question is: How to identify dilute Ising model with $c=7/10$ minimal model?

In TASI Lectures on Emergence of Supersymmetry, Gauge Theory and String in Condensed Matter Systems there is 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 $σ = ±1$ represents a site with spin up or down, and $σ = 0$ 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 $\phi ∼< σ >$ describes the magnetic order parameter.

Although there is no fermion in this action, one can construct a fermion field $ψ$ from a string of spins through the JordanWigner transformation. At the tricritical point, the scaling dimensions of $\phi^2$ and $ψ$ 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) Whi $\phi^6$ theory? How to understant this? Why not $\phi^4$? or $\phi^8$? or $\phi^4+\phi^6$?

2) One can construct fermion from string of spins even in ordinary Ising model. Why SUSY emerge in the delute Ising model model, but don't emerge in Ising model??

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?