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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2510.04238 |
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| _version_ | 1866908821633368064 |
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| author | Eisenriegler, E. |
| author_facet | Eisenriegler, E. |
| contents | In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles $\langle σ({\bf r}_1) \rangle$ and $\langle ε({\bf r}_2) \rangle$, respectively. Does increasing the order at ${\bf r}_1$ always decrease the disorder at ${\bf r}_2$? Does increasing the disorder at ${\bf r}_2$ always decrease the order at ${\bf r}_1$? The answer to these questions is contained in the cumulant response function $\langleσ({\bf r}_1) \, ε({\bf r}_2) \rangle^{(\rm cum)}$. This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of $σ({\bf r}_1) \, ε({\bf r}_2)$ and exact results for the Ising model, we analyze $\langleσ({\bf r}_1) \, ε({\bf r}_2) \rangle^{(\rm cum)}$ in two-dimensional critical systems defined on the $x-y$ plane with mixed $+$ and $-$ boundary conditions. Particularly interesting behavior is found when either of the operators $σ$ or $ε$ is located on a ``zero line" in the $x-y$ plane, along which $\langleσ({\bf r})\rangle$ vanishes. Results for half-plane, triangular, and rectangular geometries are presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions Eisenriegler, E. Statistical Mechanics In spin systems such as the Ising model, the local order and disorder can be characterized by the order-parameter and energy density profiles $\langle σ({\bf r}_1) \rangle$ and $\langle ε({\bf r}_2) \rangle$, respectively. Does increasing the order at ${\bf r}_1$ always decrease the disorder at ${\bf r}_2$? Does increasing the disorder at ${\bf r}_2$ always decrease the order at ${\bf r}_1$? The answer to these questions is contained in the cumulant response function $\langleσ({\bf r}_1) \, ε({\bf r}_2) \rangle^{(\rm cum)}$. This correlation function vanishes in the unbounded bulk but not in systems with fixed-spin boundary conditions. Using the universal operator-product expansion of $σ({\bf r}_1) \, ε({\bf r}_2)$ and exact results for the Ising model, we analyze $\langleσ({\bf r}_1) \, ε({\bf r}_2) \rangle^{(\rm cum)}$ in two-dimensional critical systems defined on the $x-y$ plane with mixed $+$ and $-$ boundary conditions. Particularly interesting behavior is found when either of the operators $σ$ or $ε$ is located on a ``zero line" in the $x-y$ plane, along which $\langleσ({\bf r})\rangle$ vanishes. Results for half-plane, triangular, and rectangular geometries are presented. |
| title | Interplay of order and disorder in two-dimensional critical systems with mixed boundary conditions |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2510.04238 |