Cellular automata in $d$ dimensions and ground states of spin models in $(d+1)$ dimensions
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866913965348487168 |
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| author | Sfairopoulos, Konstantinos Causer, Luke Mair, Jamie F. Garrahan, Juan P. |
| author_facet | Sfairopoulos, Konstantinos Causer, Luke Mair, Jamie F. Garrahan, Juan P. |
| contents | We show how the trajectories of $d$-dimensional cellular automata (CA) can be used to determine the ground states of $(d+1)$-dimensional classical spin models, and we characterise their quantum phase transition, when in the presence of a transverse magnetic field. For each of the 256 one-dimensional elementary CA we explicitly construct the simplest local two-dimensional classical spin model associated to the given CA, and we also describe this method for $d>1$ through selected examples. We illustrate our general observations with detailed studies of: (i) the $d=1$ CA Rule 150 and its $d=2$ four-body plaquette spin model, (ii) the $d=2$ CA whose associated model is the $d=3$ square-pyramid plaquette model, and (iii) two counter-propagating $d=1$ Rule 60 CA that correspond to the two-dimensional Baxter-Wu spin model. For the quantum spin models, we show that the connection to CAs implies a sensitivity on the approach to the thermodynamic limit via finite size scaling for their quantum phase transitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08059 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Cellular automata in $d$ dimensions and ground states of spin models in $(d+1)$ dimensions Sfairopoulos, Konstantinos Causer, Luke Mair, Jamie F. Garrahan, Juan P. Statistical Mechanics Quantum Physics We show how the trajectories of $d$-dimensional cellular automata (CA) can be used to determine the ground states of $(d+1)$-dimensional classical spin models, and we characterise their quantum phase transition, when in the presence of a transverse magnetic field. For each of the 256 one-dimensional elementary CA we explicitly construct the simplest local two-dimensional classical spin model associated to the given CA, and we also describe this method for $d>1$ through selected examples. We illustrate our general observations with detailed studies of: (i) the $d=1$ CA Rule 150 and its $d=2$ four-body plaquette spin model, (ii) the $d=2$ CA whose associated model is the $d=3$ square-pyramid plaquette model, and (iii) two counter-propagating $d=1$ Rule 60 CA that correspond to the two-dimensional Baxter-Wu spin model. For the quantum spin models, we show that the connection to CAs implies a sensitivity on the approach to the thermodynamic limit via finite size scaling for their quantum phase transitions. |
| title | Cellular automata in $d$ dimensions and ground states of spin models in $(d+1)$ dimensions |
| topic | Statistical Mechanics Quantum Physics |
| url | https://arxiv.org/abs/2309.08059 |