The two upper critical dimensions of the Ising and Potts models
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914952190623744 |
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| author | Wiese, Kay Joerg Jacobsen, Jesper Lykke |
| author_facet | Wiese, Kay Joerg Jacobsen, Jesper Lykke |
| contents | We derive the exact actions of the $Q$-state Potts model valid on any graph, first for the spin degrees of freedom, and second for the Fortuin-Kasteleyn clusters. In both cases the field is a traceless $Q$-component scalar field $Φ^α$. For the Ising model ($Q=2$), the field theory for the spins has upper critical dimension $d_{\rm c}^{\rm spin}=4$, whereas for the clusters it has $d_{\rm c}^{\rm cluster}=6$. As a consequence, the probability for three points to be in the same cluster is not given by mean-field theory for $d$ within $4<d<6$. We estimate the associated universal structure constant as $C=\sqrt{6-d}+ {\cal O}(6-d)^{3/2}$. This shows that some observables in the Ising model have an upper critical dimension of 4, while others have an upper critical dimension of $6$. Combining perturbative results from the $ε=6-d$ expansion with a non-perturbative treatment close to dimension $d=4$ allows us to locate the shape of the critical domain of the Potts model in the whole $(Q,d)$ plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_01529 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The two upper critical dimensions of the Ising and Potts models Wiese, Kay Joerg Jacobsen, Jesper Lykke High Energy Physics - Theory Statistical Mechanics We derive the exact actions of the $Q$-state Potts model valid on any graph, first for the spin degrees of freedom, and second for the Fortuin-Kasteleyn clusters. In both cases the field is a traceless $Q$-component scalar field $Φ^α$. For the Ising model ($Q=2$), the field theory for the spins has upper critical dimension $d_{\rm c}^{\rm spin}=4$, whereas for the clusters it has $d_{\rm c}^{\rm cluster}=6$. As a consequence, the probability for three points to be in the same cluster is not given by mean-field theory for $d$ within $4<d<6$. We estimate the associated universal structure constant as $C=\sqrt{6-d}+ {\cal O}(6-d)^{3/2}$. This shows that some observables in the Ising model have an upper critical dimension of 4, while others have an upper critical dimension of $6$. Combining perturbative results from the $ε=6-d$ expansion with a non-perturbative treatment close to dimension $d=4$ allows us to locate the shape of the critical domain of the Potts model in the whole $(Q,d)$ plane. |
| title | The two upper critical dimensions of the Ising and Potts models |
| topic | High Energy Physics - Theory Statistical Mechanics |
| url | https://arxiv.org/abs/2311.01529 |