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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2309.14562 |
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| _version_ | 1866911014055837696 |
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| author | Sedik, Muhammad Bhat, Junaid Majeed Dhar, Abhishek Shastry, B Sriram |
| author_facet | Sedik, Muhammad Bhat, Junaid Majeed Dhar, Abhishek Shastry, B Sriram |
| contents | We revisit the somewhat less studied problem of Yang-Lee zeros of the Ising antiferromagnet. For this purpose, we study two models, the nearest-neighbor model on a square lattice, and the more tractable mean-field model corresponding to infinite-ranged coupling between all sites. In the high-temperature limit, we show that the logarithm of the Yang-Lee zeros can be written as a series in half odd integer powers of the inverse temperature, $k$, with the leading term $\sim k^{1/2}$. This result is true in any dimension and for arbitrary lattices. We also show that the coefficients of the expansion satisfy simple identities (akin to sum rules) for the nearest-neighbor case. These new identities are verified numerically by computing the exact partition function for a 2D square lattice of size $16\times16$. For the mean-field model, we write down the partition function (termed the mean-field polynomials) for the ferromagnetic (FM) and antiferromagnetic (AFM) cases, and derive from them the mean-field equations. We analytically show that at high temperatures the zeros of the AFM mean-field polynomial scale as $\sim k^{1/2}$ as well. Using a simple numerical method, we find the roots lie on certain curves (the root curves), in the thermodynamic limit for the mean-field polynomials for the AFM case as well as for the FM one. Our results show a new root curve, that was not found earlier. Our results also clearly illustrate the phase transition expected for the FM and AFM cases, in the language of Yang-Lee zeros. Moreover, for the AFM case, we observe that the root curves separate two distinct phases of zero and non-zero complex staggered magnetization, and thus depict a complex phase boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_14562 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Yang-Lee Zeros of Certain Antiferromagnetic Models Sedik, Muhammad Bhat, Junaid Majeed Dhar, Abhishek Shastry, B Sriram Statistical Mechanics Mathematical Physics Computational Physics We revisit the somewhat less studied problem of Yang-Lee zeros of the Ising antiferromagnet. For this purpose, we study two models, the nearest-neighbor model on a square lattice, and the more tractable mean-field model corresponding to infinite-ranged coupling between all sites. In the high-temperature limit, we show that the logarithm of the Yang-Lee zeros can be written as a series in half odd integer powers of the inverse temperature, $k$, with the leading term $\sim k^{1/2}$. This result is true in any dimension and for arbitrary lattices. We also show that the coefficients of the expansion satisfy simple identities (akin to sum rules) for the nearest-neighbor case. These new identities are verified numerically by computing the exact partition function for a 2D square lattice of size $16\times16$. For the mean-field model, we write down the partition function (termed the mean-field polynomials) for the ferromagnetic (FM) and antiferromagnetic (AFM) cases, and derive from them the mean-field equations. We analytically show that at high temperatures the zeros of the AFM mean-field polynomial scale as $\sim k^{1/2}$ as well. Using a simple numerical method, we find the roots lie on certain curves (the root curves), in the thermodynamic limit for the mean-field polynomials for the AFM case as well as for the FM one. Our results show a new root curve, that was not found earlier. Our results also clearly illustrate the phase transition expected for the FM and AFM cases, in the language of Yang-Lee zeros. Moreover, for the AFM case, we observe that the root curves separate two distinct phases of zero and non-zero complex staggered magnetization, and thus depict a complex phase boundary. |
| title | Yang-Lee Zeros of Certain Antiferromagnetic Models |
| topic | Statistical Mechanics Mathematical Physics Computational Physics |
| url | https://arxiv.org/abs/2309.14562 |