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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.02033 |
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| _version_ | 1866914934973005824 |
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| author | Dumer, R. A. Godoy, M. |
| author_facet | Dumer, R. A. Godoy, M. |
| contents | In this work, we have employed Monte Carlo calculations to study the Ising model on a 2D additive small-world network with long-range interactions depending on the geometric distance between interacting sites. The network is initially defined by a regular square lattice and with probability $p$ each site is tested for the possibility of creating a long-range interaction with any other site that has not yet received one. Here, we used the specific case where $p=1$, meaning that every site in the network has one long-range interaction in addition to the short-range interactions of the regular lattice. These long-range interactions depend on a power-law form, $J_{ij}=r_{ij}^{-α}$, with the geometric distance $r_{ij}$ between connected sites $i$ and $j$. In current two-dimensional model, we found that mean-field critical behavior is observed only at $α=0$. As $α$ increases, the network size influences the phase transition point of the system, i.e., indicating a crossover behavior. However, given the two-dimensional system, we found the critical behavior of the short-range interaction at $α\approx2$. Thus, the limitation in the number of long-range interactions compared to the globally coupled model, as well as the form of the decay of these interactions, prevented us from finding a regime with finite phase transition points and continuously varying critical exponents in $0<α<2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | 2D additive small-world network with distance-dependent interactions Dumer, R. A. Godoy, M. Statistical Mechanics 82M31 I.6.6 In this work, we have employed Monte Carlo calculations to study the Ising model on a 2D additive small-world network with long-range interactions depending on the geometric distance between interacting sites. The network is initially defined by a regular square lattice and with probability $p$ each site is tested for the possibility of creating a long-range interaction with any other site that has not yet received one. Here, we used the specific case where $p=1$, meaning that every site in the network has one long-range interaction in addition to the short-range interactions of the regular lattice. These long-range interactions depend on a power-law form, $J_{ij}=r_{ij}^{-α}$, with the geometric distance $r_{ij}$ between connected sites $i$ and $j$. In current two-dimensional model, we found that mean-field critical behavior is observed only at $α=0$. As $α$ increases, the network size influences the phase transition point of the system, i.e., indicating a crossover behavior. However, given the two-dimensional system, we found the critical behavior of the short-range interaction at $α\approx2$. Thus, the limitation in the number of long-range interactions compared to the globally coupled model, as well as the form of the decay of these interactions, prevented us from finding a regime with finite phase transition points and continuously varying critical exponents in $0<α<2$. |
| title | 2D additive small-world network with distance-dependent interactions |
| topic | Statistical Mechanics 82M31 I.6.6 |
| url | https://arxiv.org/abs/2409.02033 |