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ising [2021/04/14 03:28]
Scott Lawrence [Ising model] 		ising [2021/09/23 17:34] (current)
Scott Lawrence [Fractional dimensions]
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 All above data are from [[https://arxiv.org/pdf/1603.04436.pdf]]. All above data are from [[https://arxiv.org/pdf/1603.04436.pdf]].
  
-The lowest-lying irrelevant operator is $\epsilon'$.+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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 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: 2021/04/14 03:28
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