A dichotomy theory for the height functions of the BKT transition
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911633602772992 |
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| author | Lammers, Piet |
| author_facet | Lammers, Piet |
| contents | This text considers the discrete height functions associated with the Berezinskii--Kosterlitz--Thouless transition (BKT) at slope zero. Our main results are as follows.
* Sharpness: If the model is localised, then the two-point function (covariance) decays exponentially fast in the distance between the points.
* Effective temperature gap: If the model is delocalised, then the variance grows at least as $c\log n$, where $n$ is the distance to the boundary and $c>0$ a universal constant not depending on the temperature. Thus, the effective temperature must jump from $0$ to at least $c$ at the transition point; values in the interval $(0,c)$ are forbidden.
* Delocalisation at the transition point: The delocalised phase includes the transition point, in the sense that it is a closed set in the phase diagram in the appropriate topology.
These results contribute to the understanding of the regime at and around the transition point which remained largely unexplored. In a follow-up paper, the sharpness derived here is used to establish that the localisation-delocalisation transition is equivalent to the BKT transition in the dual XY and Villain models. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_14365 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A dichotomy theory for the height functions of the BKT transition Lammers, Piet Probability Mathematical Physics 82B20, 82B41 (Primary) 82B30 (Secondary) This text considers the discrete height functions associated with the Berezinskii--Kosterlitz--Thouless transition (BKT) at slope zero. Our main results are as follows. * Sharpness: If the model is localised, then the two-point function (covariance) decays exponentially fast in the distance between the points. * Effective temperature gap: If the model is delocalised, then the variance grows at least as $c\log n$, where $n$ is the distance to the boundary and $c>0$ a universal constant not depending on the temperature. Thus, the effective temperature must jump from $0$ to at least $c$ at the transition point; values in the interval $(0,c)$ are forbidden. * Delocalisation at the transition point: The delocalised phase includes the transition point, in the sense that it is a closed set in the phase diagram in the appropriate topology. These results contribute to the understanding of the regime at and around the transition point which remained largely unexplored. In a follow-up paper, the sharpness derived here is used to establish that the localisation-delocalisation transition is equivalent to the BKT transition in the dual XY and Villain models. |
| title | A dichotomy theory for the height functions of the BKT transition |
| topic | Probability Mathematical Physics 82B20, 82B41 (Primary) 82B30 (Secondary) |
| url | https://arxiv.org/abs/2211.14365 |