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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?

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    $\begingroup$ I'm not sure I understand the first question. The statement is just that tuning this lattice model to its (tricritical) fixed point, the CFT is the minimal model with $c=7/10$. In this minimal model the Virasoro algebra is enhanced to super-Virasoro (see the discussion in Di Francesco). Other potentials for $\phi$ would be different continuum descriptions. $\endgroup$
    – 4xion
    Jul 7, 2020 at 20:17
  • $\begingroup$ @4xion, first question is how relate lattice model to field theory? Second question is how to understand emergemce of SUSY on field level? $\endgroup$
    – Nikita
    Jul 7, 2020 at 20:19
  • $\begingroup$ Have you read the discussion of the minimal models (and the superconformal models) in the Di Francesco CFT book (sec 7.4)? It addressed most of these questions I think $\endgroup$
    – 4xion
    Jul 7, 2020 at 20:21
  • $\begingroup$ @4xion, as I understand, they consider such question in algebraic level. I am intrested in field theory description. $\endgroup$
    – Nikita
    Jul 7, 2020 at 20:23
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    $\begingroup$ The continuum description of the lattice model is often called the Landau-Ginzburg theory, which is more general, but for the minimal models just amounts to different $\phi$ potentials. Look in 7.4.7 of Di Francesco or search around for LG theory $\endgroup$
    – 4xion
    Jul 7, 2020 at 20:28

1 Answer 1

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The best reference I know of for recognizing the `dilute Ising model', or the Blume-Capel model as it is referred to, is Chapter 14.3 in Mussardo's Statistical Field Theory Text book. The basic steps are

  1. A Hubbard Stratonovich transformation of the Ising model gives a $\phi^4$ Landau-Ginsberg model. When you include vacancies, you need to go to $\phi^6$. This is remarkable in that the LG theory is exact in some sense, rather than phenomenological.

  2. Construct the Kac table of the $\mathcal{M}_4$ minimal model with central charge $c = \frac{7}{10}$. You can identify both the even and odd sectors now with your knowledge of the $\phi^6$ LG model with both a spin and a duality $\mathbb{Z}_2$ symmetry. (See chapter 11.6 in the Reference).

  3. As a check, using the conformal weights of the primary fields you construct with $\mathcal{M}_4$ you can match the correlation scalings with the critical exponents for the tricritical Ising model universality class.

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