CFT Zoo

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ising [2021/04/28 14:21]
Scott Lawrence [Two dimensions] 		ising [2026/03/20 15:48] (current)
Ludo Fraser-Taliente [Higher dimensions]
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 S = \sum_r \left[ J_x s_r s_{r+\hat x} + J_y s_r s_{r+\hat y} + J_z s_r s_{r+\hat z} \right] S = \sum_r \left[ J_x s_r s_{r+\hat x} + J_y s_r s_{r+\hat y} + J_z s_r s_{r+\hat z} \right]
 \] \]
-The case where $J_y = J_z \gg J_x$ is termed the Hamiltonian limit, as it is connected by the Suzuki-Trotter expansion (and the [[transfer matrix|transfer matrix]]) to a quantum mechanical system.+The case where $J_y = J_z \gg J_x$ is termed the Hamiltonian limit, as it is connected by the Suzuki-Trotter expansion (and the [[transfer matrix|transfer matrix]]) to a quantum mechanical system, usually termed the "transverse-field Ising model".
 \[ \[
 H = -\mu \sum_i \sigma_x(i) -J \sum_{\langle i j \rangle} \sigma_z(i) \sigma_z(j) H = -\mu \sum_i \sigma_x(i) -J \sum_{\langle i j \rangle} \sigma_z(i) \sigma_z(j)
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 In $d > 4$ dimensions this was [[https://projecteuclid.org/journalArticle/Download?urlid=cmp%2F1103921614|proven]] by Michael Aizenman. In $d > 4$ dimensions this was [[https://projecteuclid.org/journalArticle/Download?urlid=cmp%2F1103921614|proven]] by Michael Aizenman.
  
 +However, there do exist non-unitary CFTs that can be considered as the analytic continuation of the $d$-dimensional Ising model to $d>4$: see [[on_model#general_dimension | the discussion on the O$(N)$ model page for more]].
 ==== Fractional dimensions ==== ==== Fractional dimensions ====
 +
 +See the discussion [[list_of_cfts#fractional_continuous_dimension | here]] for more, or:
  
 [[https://arxiv.org/abs/cond-mat/9802018]] [[https://arxiv.org/abs/cond-mat/9802018]]
 +
 +[[https://hal.archives-ouvertes.fr/jpa-00210418/document]]
 +==== Supersymmetrization ==
 +
 +[[https://arxiv.org/pdf/1502.04124 | The 3d supersymmetric Ising model]].
 +
 ===== External links ===== ===== External links =====
   * [[https://arxiv.org/abs/0902.0045|An improved lattice measurement of the critical coupling in $\phi^4_2$ theory]] (Schaich)   * [[https://arxiv.org/abs/0902.0045|An improved lattice measurement of the critical coupling in $\phi^4_2$ theory]] (Schaich)
   * [[https://projecteuclid.org/journalArticle/Download?urlid=cmp%2F1103921614|Geometric analysis of $\phi^4$ fields and Ising models]] (Aizenman)   * [[https://projecteuclid.org/journalArticle/Download?urlid=cmp%2F1103921614|Geometric analysis of $\phi^4$ fields and Ising models]] (Aizenman)
Last modified: 2021/04/28 14:21
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