Griffiths-type theorems for short-range spin glass models
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866913242946732032 |
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| author | Itoi, Chigak Mukaida, Hisamitsu Tasaki, Hal |
| author_facet | Itoi, Chigak Mukaida, Hisamitsu Tasaki, Hal |
| contents | We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap $R^{1,2}$ implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter $λ$. In $\mathbb Z_2$ invariant models such as the standard Edwards-Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards-Anderson order parameter. This generalization of Griffiths' theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04775 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Griffiths-type theorems for short-range spin glass models Itoi, Chigak Mukaida, Hisamitsu Tasaki, Hal Mathematical Physics Disordered Systems and Neural Networks Statistical Mechanics Probability We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap $R^{1,2}$ implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter $λ$. In $\mathbb Z_2$ invariant models such as the standard Edwards-Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards-Anderson order parameter. This generalization of Griffiths' theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model. |
| title | Griffiths-type theorems for short-range spin glass models |
| topic | Mathematical Physics Disordered Systems and Neural Networks Statistical Mechanics Probability |
| url | https://arxiv.org/abs/2310.04775 |