#
Grassmannization of the 3D Ising Model^{ †}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

^{†}

^{‡}

## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Grassmannization of the Ising Model

- Draw all the n topologically different connected diagrams connecting the origin with site $\mathbf{r}$. Any occupied bond contributes 1 to the expansion order n.
- The n bare propagators live on the links (they are local), and are represented by a pair of Grassmann variables. A pair contributes a factor v in magnitude to the weight of the diagram. Propagation lines have no arrows.
- The $(n+1)$ interaction vertices live on the sites of the lattice, and can be of different type: the origin and end vertex belong to the ${V}_{1}$, ${V}_{3}$ or ${V}_{5}$ class, the $(n-1)$ others belong to ${V}_{2}$, ${V}_{4}$ or ${V}_{6}$ class, whose weights are in accordance with Equation (1). All j legs of the vertices ${V}_{j}$ must be connected by propagator lines.
- If a link is multiply occupied, a minus sign occurs when swapping 2 Grassmann variables. The minus signs can equivalently be inferred by identifying all fermionic loops.
- The total weight will be ${(-1)}^{P}{\left(1\right)}^{{q}_{1}+{q}_{2}}{(-2)}^{{q}_{3}+{q}_{4}}{\left(16\right)}^{{q}_{5}+{q}_{6}}{v}^{n}$, being P the signature of the exchange permutation, and ${q}_{j}$ is the sum of al vertices ${V}_{j}$ that are of type j, $j=1,\dots ,6$. It follows that the weight of a diagram is an integer number times ${v}^{n}$.

#### 2.2. Results of the simulations

## References

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**Figure 1.**Plot of Equation (3). Black dots are the ratios of the expansion coefficients, while the red solid line is a linear fit. The intercept provides information about ${T}_{c}$, while the slope is related to $\gamma $. Results are in good accordance with the results found in the literature. The red dashed line is a linear fit through the points omitting the one at $n=1$ but it leads to results for ${T}_{c}$ and $\gamma $ that deviate from the correct answer.

**Table 1.**The table shows the number of diagrams connecting the origin with the corresponding site. The second column lists the symmetry factor associated with each site on a cubic lattice.

Site | ${\mathit{S}}_{\mathit{F}}$ | ${\mathit{\nu}}^{0}$ | ${\mathit{\nu}}^{1}$ | ${\mathit{\nu}}^{2}$ | ${\mathit{\nu}}^{3}$ | ${\mathit{\nu}}^{4}$ | ${\mathit{\nu}}^{5}$ | ${\mathit{\nu}}^{6}$ | ${\mathit{\nu}}^{7}$ | ${\mathit{\nu}}^{8}$ | ${\mathit{\nu}}^{9}$ | ${\mathit{\nu}}^{10}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

(1,0,0) | 6 | 0 | 1 | 0 | 4 | 0 | 40 | 0 | 456 | 0 | 6100 | 0 |

(1,1,0) | 12 | 0 | 0 | 2 | 0 | 16 | 0 | 170 | 0 | 2144 | 0 | 30334 |

(1,1,1) | 8 | 0 | 0 | 0 | 6 | 0 | 54 | 0 | 648 | 0 | 8840 | 0 |

(4,0,0) | 6 | 0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 | 1156 | 0 | 24136 |

(4,1,0) | 24 | 0 | 0 | 0 | 0 | 0 | 5 | 0 | 202 | 0 | 5006 | 0 |

(4,1,1) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 30 | 0 | 936 | 0 | 21474 |

(4,2,0) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 15 | 0 | 748 | 0 | 18647 |

(4,2,1) | 48 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 105 | 0 | 3507 | 0 |

(4,2,2) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 420 | 0 | 13440 |

(4,3,0) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 | 0 | 2219 | 0 |

(4,3,1) | 48 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 280 | 0 | 11060 |

(4,3,2) | 48 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1260 | 0 |

(4,3,3) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4200 |

(7,0,0) | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 112 | 0 |

(7,1,0) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 8 | 0 | 802 |

(7,1,1) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |

(7,2,0) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 36 | 0 |

(7,2,1) | 48 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 360 |

(7,3,0) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 120 |

(9,0,0) | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |

(9,1,0) | 24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 |

(10,0,0) | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |

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Martello, E.; Angilella, G.G.N.; Pollet, L.
Grassmannization of the 3D Ising Model. *Proceedings* **2019**, *12*, 20.
https://doi.org/10.3390/proceedings2019012020

**AMA Style**

Martello E, Angilella GGN, Pollet L.
Grassmannization of the 3D Ising Model. *Proceedings*. 2019; 12(1):20.
https://doi.org/10.3390/proceedings2019012020

**Chicago/Turabian Style**

Martello, E., G. G. N. Angilella, and L. Pollet.
2019. "Grassmannization of the 3D Ising Model" *Proceedings* 12, no. 1: 20.
https://doi.org/10.3390/proceedings2019012020