Self-similarity on 4d cubic lattice
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916775988297728 |
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| author | Korepanov, Igor G. |
| author_facet | Korepanov, Igor G. |
| contents | A phenomenon of "algebraic self-similarity" on 3d cubic lattice, providing what can be called an algebraic analogue of Kadanoff--Wilson theory, is shown to possess a 4d version as well. Namely, if there is a $4\times 4$ matrix $A$ whose entries are indeterminates over the field $\mathbb F_2$, then the $2\times 2\times 2\times 2$ block made of sixteen copies of $A$ reveals the existence of four direct "block spin" summands corresponding to the same matrix $A$. Moreover, these summands can be written out in quite an elegant way. Somewhat strikingly, if the entries of $A$ are just zeros and ones -- elements of $\mathbb F_2$ -- then there are examples where two more "block spins" split out, and this time with different $A$'s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20140 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-similarity on 4d cubic lattice Korepanov, Igor G. Quantum Algebra Mathematical Physics 15A24 (Primary), 82B20, 82B28 (Secondary) A phenomenon of "algebraic self-similarity" on 3d cubic lattice, providing what can be called an algebraic analogue of Kadanoff--Wilson theory, is shown to possess a 4d version as well. Namely, if there is a $4\times 4$ matrix $A$ whose entries are indeterminates over the field $\mathbb F_2$, then the $2\times 2\times 2\times 2$ block made of sixteen copies of $A$ reveals the existence of four direct "block spin" summands corresponding to the same matrix $A$. Moreover, these summands can be written out in quite an elegant way. Somewhat strikingly, if the entries of $A$ are just zeros and ones -- elements of $\mathbb F_2$ -- then there are examples where two more "block spins" split out, and this time with different $A$'s. |
| title | Self-similarity on 4d cubic lattice |
| topic | Quantum Algebra Mathematical Physics 15A24 (Primary), 82B20, 82B28 (Secondary) |
| url | https://arxiv.org/abs/2412.20140 |