"Integrable perturbations of Ising model"의 두 판 사이의 차이

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13번째 줄: 13번째 줄:
 
** "kink" states (boundaries between regions of differing spin) = basic objects of the theory
 
** "kink" states (boundaries between regions of differing spin) = basic objects of the theory
 
** called quasiparticle
 
** called quasiparticle
 +
* an entry of S-matrix
 +
:<math>
 +
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)}
 +
</math>
 +
* it has poles with positive residue when <math>\theta=i y,\, 0<y<\pi</math> at <math>y=\pi/15,2\pi/5,2\pi/3</math>
  
 
   
 
   
18번째 줄: 23번째 줄:
 
===Y-system===
 
===Y-system===
 
* [[Thermodynamic Bethe ansatz (TBA)]]
 
* [[Thermodynamic Bethe ansatz (TBA)]]
−
* Let $X=E_8$
+
* Let <math>X=E_8</math>
  
  
24번째 줄: 29번째 줄:
 
===constant Y-system solution===
 
===constant Y-system solution===
 
* constant Y-system
 
* constant Y-system
−
$$
+
:<math>
 
y_{i}^2=\prod _{j\in I} (1+y_{j})^{\mathcal{I}(X)_{ij}}
 
y_{i}^2=\prod _{j\in I} (1+y_{j})^{\mathcal{I}(X)_{ij}}
−
$$
+
</math>
 
* solution
 
* solution
−
$$
+
:<math>
 
\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\}
 
\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\}
−
$$
+
</math>
  
  
 
===Klassen-Melzer solution===
 
===Klassen-Melzer solution===
 
* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
 
* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
−
* Let $N_{ij}$ be the matrix
+
* Let <math>N=(N_{ij})</math> be the matrix given by
−
$$
+
:<math>
−
\mathcal{I}(E_8)\cdot(\mathcal{C}(E_8))^{-1}=
+
N=\mathcal{I}(E_8)\cdot(\mathcal{C}(E_8))^{-1}=
 
\left(
 
\left(
 
\begin{array}{cccccccc}
 
\begin{array}{cccccccc}
50번째 줄: 55번째 줄:
 
\end{array}
 
\end{array}
 
\right)
 
\right)
−
$$
+
</math>
 +
* note that this is equivalent to
 +
:<math>
 +
N=2\mathcal{C}(E_8)^{-1}-I_8
 +
</math>
 
* The TBA equation is
 
* The TBA equation is
−
$$
+
:<math>
 
\epsilon_i=\sum_{j}N_{ij}\log (1+e^{-\epsilon_j})
 
\epsilon_i=\sum_{j}N_{ij}\log (1+e^{-\epsilon_j})
−
$$
+
</math>
 
or
 
or
  
−
$$
+
:<math>
 
e^{\epsilon_i}=\prod_{j}(1+e^{-\epsilon_j})^{N_{ij}}
 
e^{\epsilon_i}=\prod_{j}(1+e^{-\epsilon_j})^{N_{ij}}
−
$$
+
</math>
−
* we have the relationship $y_i=e^{\epsilon_i}$
+
* we have the relationship <math>y_i=e^{\epsilon_i}</math>
  
 
==history==
 
==history==
66번째 줄: 75번째 줄:
 
* Soon after Zamolodchikov’s first paper '''[Zam]''' appeared,
 
* Soon after Zamolodchikov’s first paper '''[Zam]''' appeared,
 
*  Fateev and Zamolodchikov conjectured in '''[FZ90]''' that
 
*  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.
+
** if you take a minimal model CFT constructed from a compact Lie algebra <math>\mathfrak{g}</math> via the coset construction and perturb it in a particular way, then you obtain the affine Toda field theory (ATFT) associated with <math>\mathfrak{g}</math>, which is an integrable field theory.
 
** This was confirmed in '''[EY]''' and '''[HoM]'''.
 
** 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.
+
* If you do this with <math>\mathfrak{g}=E_8</math>, 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.
+
* That is, if we take the <math>E_8</math> ATFT as a starting point, then the assumptions (Z1)–(Z4) become deductions.
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
  
79번째 줄: 88번째 줄:
 
* [[exact S-matrices in ATFT]]
 
* [[exact S-matrices in ATFT]]
 
* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
 
* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
−
 
+
* [[Dilute A model]]
  
 
==computational resource==
 
==computational resource==
89번째 줄: 98번째 줄:
 
* [[Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?]]
 
* [[Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?]]
 
*  Affleck, Ian. 2010. “Solid-state physics: Golden ratio seen in a magnet”. <em>Nature</em> 464 (7287) (3월 18): 362-363. doi:[http://dx.doi.org/10.1038/464362a 10.1038/464362a].
 
*  Affleck, Ian. 2010. “Solid-state physics: Golden ratio seen in a magnet”. <em>Nature</em> 464 (7287) (3월 18): 362-363. doi:[http://dx.doi.org/10.1038/464362a 10.1038/464362a].
−
* Jihye Seo, [http://isites.harvard.edu/fs/docs/icb.topic572189.files/Jihye_Seo_Ising_model_in_field.pdf Solving 2D Magnetic Ising Model at $T=T_c$ Using Scattering Theory] 2009
+
* Jihye Seo, [http://isites.harvard.edu/fs/docs/icb.topic572189.files/Jihye_Seo_Ising_model_in_field.pdf Solving 2D Magnetic Ising Model at <math>T=T_c</math> Using Scattering Theory] 2009
 
* Delfino, Gesualdo. 2003. “Integrable Field Theory and Critical Phenomena. The Ising Model in a Magnetic Field.” arXiv:hep-th/0312119 (December 11). doi:10.1088/0305-4470/37/14/R01. http://arxiv.org/abs/hep-th/0312119.
 
* Delfino, Gesualdo. 2003. “Integrable Field Theory and Critical Phenomena. The Ising Model in a Magnetic Field.” arXiv:hep-th/0312119 (December 11). doi:10.1088/0305-4470/37/14/R01. http://arxiv.org/abs/hep-th/0312119.
 
* Dorey, Patrick. 1992. “Hidden Geometrical Structures in Integrable Models.” arXiv:hep-th/9212143 (December 23). http://arxiv.org/abs/hep-th/9212143.
 
* Dorey, Patrick. 1992. “Hidden Geometrical Structures in Integrable Models.” arXiv:hep-th/9212143 (December 23). http://arxiv.org/abs/hep-th/9212143.
101번째 줄: 110번째 줄:
 
* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/S0550-3213%2898%2900063-7 Non-integrable aspects of the multi-frequency sine-Gordon model], 1998
 
* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/S0550-3213%2898%2900063-7 Non-integrable aspects of the multi-frequency sine-Gordon model], 1998
 
* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/0550-3213%2895%2900464-4 The spin-spin correlation function in the two-dimensional Ising model in a magnetic field at T = Tc], 1995
 
* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/0550-3213%2895%2900464-4 The spin-spin correlation function in the two-dimensional Ising model in a magnetic field at T = Tc], 1995
−
* V. V. Bazhanov, B. Nienhuis, S. O. Warnaar [http://dx.doi.org/10.1016/0370-2693%2894%2991107-X Lattice Ising model in a field: E8 scattering theory], 1994
+
* Bazhanov, V. V., B. Nienhuis, and S. O. Warnaar. ‘Lattice Ising Model in a Field: E8 Scattering Theory’. Physics Letters B 322, no. 3 (17 February 1994): 198–206. doi:[http://dx.doi.org/10.1016/0370-2693%2894%2991107-X 10.1016/0370-2693(94)91107-X].* Braden, H. W., E. Corrigan, P. E. Dorey, and R. Sasaki. 1990. “Aspects of Perturbed Conformal Field Theory, Affine Toda Field Theory and Exact <math>S</math>-matrices.” In Differential Geometric Methods in Theoretical Physics (Davis, CA, 1988), 245:169–182. NATO Adv. Sci. Inst. Ser. B Phys. New York: Plenum. http://www.ams.org/mathscinet-getitem?mr=1169481.
−
* Braden, H. W., E. Corrigan, P. E. Dorey, and R. Sasaki. 1990. “Aspects of Perturbed Conformal Field Theory, Affine Toda Field Theory and Exact $S$-matrices.” In Differential Geometric Methods in Theoretical Physics (Davis, CA, 1988), 245:169–182. NATO Adv. Sci. Inst. Ser. B Phys. New York: Plenum. http://www.ams.org/mathscinet-getitem?mr=1169481.
 
 
* '''[EY]'''T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989), 373-8 B
 
* '''[EY]'''T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989), 373-8 B
 
* '''[HoM]'''T.J. Hollowood and P.Mansfield, Rational conformal theories at, and away from criticality as Toda field theories, Phys. Lett. B226 (1989) 73-79
 
* '''[HoM]'''T.J. Hollowood and P.Mansfield, Rational conformal theories at, and away from criticality as Toda field theories, Phys. Lett. B226 (1989) 73-79
121번째 줄: 129번째 줄:
 
[[분류:integrable systems]]
 
[[분류:integrable systems]]
 
[[분류:math and physics]]
 
[[분류:math and physics]]
 +
[[분류:migrate]]

2020년 11월 16일 (월) 11:06 기준 최신판

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

computational resource


expositions


articles

question and answers(Math Overflow)

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