Boundary conditions and universal finite-size scaling for the hierarchical $|φ|^4$ model in dimensions 4 and higher
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arXiv
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| Format: | Preprint |
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2023
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| author | Michta, Emmanuel Park, Jiwoon Slade, Gordon |
| author_facet | Michta, Emmanuel Park, Jiwoon Slade, Gordon |
| contents | We analyse and clarify the finite-size scaling of the weakly-coupled hierarchical $n$-component $|φ|^4$ model for all integers $n \ge 1$ in all dimensions $d\ge 4$, for both free and periodic boundary conditions. For $d>4$, we prove that for a volume of size $R^{d}$ with periodic boundary conditions the infinite-volume critical point is an effective finite-volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order $R^{-2}$. For both boundary conditions, the average field has the same non-Gaussian limit within a critical window of width $R^{-d/2}$ around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount $R^{-2}$. In particular, at the infinite-volume critical point the susceptibility scales as $R^{d/2}$ for periodic boundary conditions and as $R^{2}$ for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non-hierarchical) models on $\mathbb{Z}^d$ in dimensions $d \ge 4$. For $d=4$ we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00896 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Boundary conditions and universal finite-size scaling for the hierarchical $|φ|^4$ model in dimensions 4 and higher Michta, Emmanuel Park, Jiwoon Slade, Gordon Mathematical Physics Probability 82B27, 82B28, 60K35 We analyse and clarify the finite-size scaling of the weakly-coupled hierarchical $n$-component $|φ|^4$ model for all integers $n \ge 1$ in all dimensions $d\ge 4$, for both free and periodic boundary conditions. For $d>4$, we prove that for a volume of size $R^{d}$ with periodic boundary conditions the infinite-volume critical point is an effective finite-volume critical point, whereas for free boundary conditions the effective critical point is shifted smaller by an amount of order $R^{-2}$. For both boundary conditions, the average field has the same non-Gaussian limit within a critical window of width $R^{-d/2}$ around the effective critical point, and in that window we compute the universal scaling profile for the susceptibility. In contrast, and again for both boundary conditions, the average field has a massive Gaussian limit when above the effective critical point by an amount $R^{-2}$. In particular, at the infinite-volume critical point the susceptibility scales as $R^{d/2}$ for periodic boundary conditions and as $R^{2}$ for free boundary conditions. We identify a mass generation mechanism for free boundary conditions that is responsible for this distinction and which we believe has wider validity, in particular to Euclidean (non-hierarchical) models on $\mathbb{Z}^d$ in dimensions $d \ge 4$. For $d=4$ we prove a similar picture with logarithmic corrections. Our analysis is based on the rigorous renormalisation group method of Bauerschmidt, Brydges and Slade, which we improve and extend. |
| title | Boundary conditions and universal finite-size scaling for the hierarchical $|φ|^4$ model in dimensions 4 and higher |
| topic | Mathematical Physics Probability 82B27, 82B28, 60K35 |
| url | https://arxiv.org/abs/2306.00896 |