CFT Zoo

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ising [2021/04/13 21:22]
Scott Lawrence [Lattice simulations] 		ising [2021/09/23 17:34] (current)
Scott Lawrence [Fractional dimensions]
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 {{tag>3d}} {{tag>3d}}
  
-The Ising model is an interacting CFT in three dimensions. For related models in other dimensions, see [[#other dimensions]].+The Ising model is an interacting CFT in three dimensions. For related models in other dimensions, see [[#other dimensions]]. The Ising model is the $N=1$ case of the [[on_model]].
  
 The Ising CFT also goes by the name "Wilson-Fisher fixed point" (in any $d < 4$). Another terminological difficulty: "Ising model" can mean either the lattice Ising model (discussed below) or the CFT itself. Even when the lattice model is tuned to criticality, these are different objects. To disambiguate, one can refer to the "Ising CFT" or the "critical Ising model". The Ising CFT also goes by the name "Wilson-Fisher fixed point" (in any $d < 4$). Another terminological difficulty: "Ising model" can mean either the lattice Ising model (discussed below) or the CFT itself. Even when the lattice model is tuned to criticality, these are different objects. To disambiguate, one can refer to the "Ising CFT" or the "critical Ising model".
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 ===== Operator content ===== ===== Operator content =====
  
 +The Ising CFT has two relevant operators, conventionally denoted $\sigma$ and $\epsilon$. Their scaling dimensions are given by
 +$$
 +\Delta_\sigma = 0.5181489(10)\text{ and }
 +\Delta_\epsilon = 1.412625(10)\text.
 +$$
 +The nonvanishing [[operator product expansion|OPE]] coefficients are
 +$$
 +\lambda_{\sigma\sigma\epsilon} = 1.0518537(41)\text{ and }
 +\lambda_{\epsilon\epsilon\epsilon} = 1.532435(19)
 +$$
  
 +All above data are from [[https://arxiv.org/pdf/1603.04436.pdf]].
 +
 +Both relevant operators have spin $l=0$. The stress tensor of course has $l=2$ and $\Delta = 3$. The lowest-lying irrelevant operator is $\epsilon'$ (also $l=0$), with $\Delta_{\epsilon'} \approx 3.8$.
 ===== Lattice model ===== ===== Lattice model =====
  
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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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 See [[https://arxiv.org/pdf/2007.14315.pdf|this review]] by Slava Rychkov. See [[https://arxiv.org/pdf/2007.14315.pdf|this review]] by Slava Rychkov.
 +
 +See also [[https://arxiv.org/pdf/1203.6064.pdf]] and [[https://arxiv.org/pdf/1403.4545.pdf]] and [[https://arxiv.org/pdf/1603.04436.pdf]].
 ===== Physical realizations ===== ===== Physical realizations =====
  
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 The two-dimensional lattice Ising model was solved exactly by Onsager. The two-dimensional lattice Ising model was solved exactly by Onsager.
  
 +The Hamiltonian limit of this model is a spin chain (referred to as the "transverse Ising model"):
 +\[
 +H = -\mu \sum_r \sigma_x(r) - J \sum_r \sigma_z(r) \sigma_z(r+1)\text.
 +\]
 +This model can be transformed to a quadratic theory of fermion fields, and thence solved, via a Jordan-Wigner transformation.
 ==== Higher dimensions ==== ==== Higher dimensions ====
  
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 [[https://arxiv.org/abs/cond-mat/9802018]] [[https://arxiv.org/abs/cond-mat/9802018]]
 +
 +[[https://hal.archives-ouvertes.fr/jpa-00210418/document]]
 ===== 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: 2021/04/13 21:22
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