CFT Zoo

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ising [2021/04/11 18:13]
Scott Lawrence [Ising model] 		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.+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".
  
 ===== 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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 ===== Computational methods ===== ===== Computational methods =====
  
 +==== Epsilon expansion ====
 +
 +[[https://arxiv.org/pdf/cond-mat/9803240.pdf]]
 ==== Lattice simulations ==== ==== Lattice simulations ====
  
 +The standard [[https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm|Metropolis-Hastings algorithm]] works for the Ising model. HMC is not available due to the discrete degrees of freedom of the standard lattice model.
 +
 +Near the quantum phase transition one encounters "critical slowing down". This is not exponentially bad, but still inconvenient, so it's better to use the [[https://doi.org/10.1103/PhysRevLett.58.86|Swendsen-Wang algorithm]] ([[https://en.wikipedia.org/wiki/Swendsen%E2%80%93Wang_algorithm|Wikipedia]]) for serious work. Note that that algorithm can be generalized to speed the mixing of scalar field theory as well.
 ==== Renormalization group ==== ==== Renormalization group ====
  
 +[[https://arxiv.org/abs/cond-mat/9803240]]
 ==== Conformal bootstrap ==== ==== Conformal bootstrap ====
  
 +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 =====
  
 As mentioned [[#liquid-gas transition|above]], some properties of the liquid-gas transition are believed to be described by the Ising model. As mentioned [[#liquid-gas transition|above]], some properties of the liquid-gas transition are believed to be described by the Ising model.
  
 +[[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.35.4823]]
  
 +It is sometimes conjectured that the thermodynamic behavior of the critical point of QCD is described by the Ising model.
 ===== Other dimensions ===== ===== Other dimensions =====
 Lattice Ising models can be defined in any number of dimensions. The critical point of such a theory is referred to as an Ising CFT in that number of dimensions. Only in three dimensions does an interacting theory result. The other possibilities are described below. Lattice Ising models can be defined in any number of dimensions. The critical point of such a theory is referred to as an Ising CFT in that number of dimensions. Only in three dimensions does an interacting theory result. The other possibilities are described below.
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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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 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.
  
 +==== Fractional dimensions ====
 +
 +[[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: 2021/04/11 18:13
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