Integrable perturbations of Ising model

수학노트
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introduction

  • energy perturbation [Kau49], [MTW77]
    • related to A1
    • Ising field theory
  • magnetic perturbation[Zam89]
    • related to E8


Ising field theory

  • the continuum limit of the Ising model is made to look like a field theory only through the application of a certain transformation (Jordan-Winger)
    • "kink" states (boundaries between regions of differing spin) = basic objects of the theory
    • called quasiparticle
  • an entry of S-matrix

\[ S_{1,1}(\theta)=\frac{\tanh \left(\frac{1}{2} \left(\theta +\frac{i \pi }{15}\right)\right)\tanh \left(\frac{1}{2} \left(\theta +\frac{2i \pi }{5}\right)\right)\tanh \left(\frac{1}{2} \left(\theta +\frac{2i \pi }{3}\right)\right)}{\tanh \left(\frac{1}{2} \left(\theta -\frac{i \pi }{15}\right)\right)\tanh \left(\frac{1}{2} \left(\theta -\frac{2i \pi }{5}\right)\right)\tanh \left(\frac{1}{2} \left(\theta -\frac{2i \pi }{3}\right)\right)} \]

  • it has poles with positive residue when \(\theta=i y,\, 0<y<\pi\) at \(y=\pi/15,2\pi/5,2\pi/3\)


constant TBA equation

Y-system


constant Y-system solution

  • constant Y-system

\[ y_{i}^2=\prod _{j\in I} (1+y_{j})^{\mathcal{I}(X)_{ij}} \]

  • solution

\[ \left\{2+2 \sqrt{2},5+4 \sqrt{2},11+8 \sqrt{2},16+12 \sqrt{2},42+30 \sqrt{2},56+40 \sqrt{2},152+108 \sqrt{2},543+384 \sqrt{2}\right\} \]


Klassen-Melzer solution

\[ N=\mathcal{I}(E_8)\cdot(\mathcal{C}(E_8))^{-1}= \left( \begin{array}{cccccccc} 3 & 4 & 6 & 6 & 8 & 8 & 10 & 12 \\ 4 & 7 & 8 & 10 & 12 & 14 & 16 & 20 \\ 6 & 8 & 11 & 12 & 16 & 16 & 20 & 24 \\ 6 & 10 & 12 & 15 & 18 & 20 & 24 & 30 \\ 8 & 12 & 16 & 18 & 23 & 24 & 30 & 36 \\ 8 & 14 & 16 & 20 & 24 & 27 & 32 & 40 \\ 10 & 16 & 20 & 24 & 30 & 32 & 39 & 48 \\ 12 & 20 & 24 & 30 & 36 & 40 & 48 & 59 \\ \end{array} \right) \]

  • note that this is equivalent to

\[ N=2\mathcal{C}(E_8)^{-1}-I_8 \]

  • The TBA equation is

\[ \epsilon_i=\sum_{j}N_{ij}\log (1+e^{-\epsilon_j}) \] or

\[ e^{\epsilon_i}=\prod_{j}(1+e^{-\epsilon_j})^{N_{ij}} \]

  • we have the relationship \(y_i=e^{\epsilon_i}\)

history

  • Soon after Zamolodchikov’s first paper [Zam] appeared,
  • Fateev and Zamolodchikov conjectured in [FZ90] that
    • if you take a minimal model CFT constructed from a compact Lie algebra \(\mathfrak{g}\) via the coset construction and perturb it in a particular way, then you obtain the affine Toda field theory (ATFT) associated with \(\mathfrak{g}\), which is an integrable field theory.
    • This was confirmed in [EY] and [HoM].
  • If you do this with \(\mathfrak{g}=E_8\), you arrive at the conjectured integrable field theory investigated by Zamolodchikov and described in the previous paragraph.
  • That is, if we take the \(E_8\) ATFT as a starting point, then the assumptions (Z1)–(Z4) become deductions.
  • http://www.google.com/search?hl=en&tbs=tl:1&q=


related items

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question and answers(Math Overflow)