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

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<h5>introduction</h5>
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==introduction==
  
−
*  energy perturbation '''[Kau49]''', '''[MTW77]'''<br>
+
*  energy perturbation '''[Kau49]''', '''[MTW77]'''
 
** related to A1
 
** related to A1
 
** Ising field theory
 
** Ising field theory
−
*  magnetic perturbation'''[Zam89]'''<br>
+
*  magnetic perturbation'''[Zam89]'''
 
** related to E8
 
** related to E8
  
−
 
 
  
−
 
+
==Ising field theory==
  
−
<h5>Ising field theory</h5>
+
*  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)
−
 
 
−
*  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)<br>
 
 
** "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
−
* [Zam89]
+
* 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>
  
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+
 +
==constant TBA equation==
 +
===Y-system===
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* [[Thermodynamic Bethe ansatz (TBA)]]
 +
* Let <math>X=E_8</math>
  
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<h5>history</h5>
 
  
−
* Soon after Zamolodchikov’s first paper appeared, Fateev and Zamolodchikov conjectured in [FZ90] that if you take a minimal model CFT constructed from a compact Lie algebra g via the coset construction and perturb it in a particular way, then you obtain the affine Toda field theory (ATFT) associated with g, which is an integrable field theory. This was confirmed in [EY] and [HoM]. If you do this with g = E8, you arrive at the conjectured integrable field theory<br> investigated by Zamolodchikov and described in the previous paragraph. That is, if we take the E8 ATFT as a starting point, then the assumptions (Z1)–(Z4) become deductions. This is the essential role of E8 in the numerical predictions relevant to the cobalt niobate experiment. (In the next section, we will explain how the masses that Zamolodchikov found arise naturally in terms of the algebra structure. But that is just a bonus.)
+
===constant Y-system solution===
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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* constant Y-system
 +
:<math>
 +
y_{i}^2=\prod _{j\in I} (1+y_{j})^{\mathcal{I}(X)_{ij}}
 +
</math>
 +
* 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\}
 +
</math>
  
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−
 
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===Klassen-Melzer solution===
 +
* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
 +
* Let <math>N=(N_{ij})</math> be the matrix given by
 +
:<math>
 +
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 \\
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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)
 +
</math>
 +
* note that this is equivalent to
 +
:<math>
 +
N=2\mathcal{C}(E_8)^{-1}-I_8
 +
</math>
 +
* The TBA equation is
 +
:<math>
 +
\epsilon_i=\sum_{j}N_{ij}\log (1+e^{-\epsilon_j})
 +
</math>
 +
or
  
−
<h5>related items</h5>
+
:<math>
 +
e^{\epsilon_i}=\prod_{j}(1+e^{-\epsilon_j})^{N_{ij}}
 +
</math>
 +
* we have the relationship <math>y_i=e^{\epsilon_i}</math>
  
−
* [[massive integrable perturbations of CFT and quasi-particles|massive integrable perturbations and quasi-particles]]
+
==history==
−
* [[exact S-matrices in ATFT]]
 
  
−
 
+
* 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 <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]'''.
 +
* 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 <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=
  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">encyclopedia</h5>
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==related items==
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* [[(3,4) Ising minimal model CFT]]
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* http://en.wikipedia.org/wiki/
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* [[Massive integrable perturbations of CFT and quasi-particles]]
−
* http://www.scholarpedia.org/
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* [[Y-system]]
−
* http://www.proofwiki.org/wiki/
+
* [[exact S-matrices in ATFT]]
−
* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
+
* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
−
 
+
* [[Dilute A model]]
−
 
 
−
 
 
−
 
 
−
 
 
−
<h5>books</h5>
 
−
 
 
−
 
 
  
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* [[2010년 books and articles]]<br>
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==computational resource==
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* http://gigapedia.info/1/
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* https://docs.google.com/file/d/0B8XXo8Tve1cxSWIwb1l2YkoyNDg/edit
−
* http://gigapedia.info/1/
 
−
* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
  
−
 
 
  
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<h5>expositions</h5>
 
  
−
* David Borthwick and Skip Garibaldi, “Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?,” 1012.5407 (December 24, 2010), http://arxiv.org/abs/1012.5407. 
+
==expositions==
−
*  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].<br>
+
* [[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].
 +
* 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.
 +
* Dorey, Patrick. 1992. “Hidden Geometrical Structures in Integrable Models.” arXiv:hep-th/9212143 (December 23). http://arxiv.org/abs/hep-th/9212143.
 +
  
−
 
+
==articles==
−
 
+
* Koca, Mehmet, and Nazife Ozdes Koca. “Radii of the E8 Gosset Circles as the Mass Excitations in the Ising Model.” arXiv:1204.4567 [hep-Th, Physics:math-Ph], April 20, 2012. http://arxiv.org/abs/1204.4567.
−
 
+
* Kostant, Bertram. “Experimental Evidence for the Occurrence of E8 in Nature and the Radii of the Gosset Circles.” arXiv:1003.0046 [math-Ph], February 28, 2010. http://arxiv.org/abs/1003.0046.
−
 
+
* Coldea, R., D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer. 2010. Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry. Science 327, no. 5962 (January 8): 177 -180. doi:[http://dx.doi.org/10.1126/science.1180085 10.1126/science.1180085].  
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
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* Alessandro Nigro [http://dx.doi.org/10.1088/1742-5468/2008/01/P01017 On the integrable structure of the Ising model] J. Stat. Mech. (2008) P01017
−
 
+
* 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
−
* Coldea, R., D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer. 2010. Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry. Science 327, no. 5962 (January 8): 177 -180. doi:[http://dx.doi.org/10.1126/science.1180085 10.1126/science.1180085]. 
+
* 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
−
* [http://dx.doi.org/10.1088/1742-5468/2008/01/P01017 On the integrable structure of the Ising model]<br>
+
* 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.
−
** Alessandro Nigro J. Stat. Mech. (2008) P01017
+
* '''[EY]'''T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989), 373-8 B
−
* [http://dx.doi.org/10.1016/S0550-3213%2898%2900063-7 Non-integrable aspects of the multi-frequency sine-Gordon model]<br>
+
* '''[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
−
** G. Delfinoa and G. Mussardo, 1998
+
* '''[Zam]'''Zamolodchikov, A. B. Integrals of Motion and S-Matrix of the (scaled) T = Tc Ising Model with Magnetic Field. International Journal of Modern Physics A 04, no. 16 (10 October 1989): 4235–48. doi:10.1142/S0217751X8900176X.
−
* [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]<br>
 
−
** G. Delfinoa and G. Mussardo, 1995
 
−
* [http://dx.doi.org/10.1016/0370-2693%2894%2991107-X Lattice Ising model in a field: E8 scattering theory]<br>
 
−
** V. V. Bazhanov, B. Nienhuis, S. O. Warnaar, 1994
 
−
* [http://dx.doi.org/10.1142/S0217751X8900176X INTEGRALS OF MOTION AND S-MATRIX OF THE (SCALED) T = Tc ISING MODEL WITH MAGNETIC FIELD]
 
 
* '''[FZ90]'''V. A. Fateev and A. B. Zamolodchikov. Conformal field theory and purely elastic S-matrices. Int. J. Mod. Phys., A5 (6): 1025-1048
 
* '''[FZ90]'''V. A. Fateev and A. B. Zamolodchikov. Conformal field theory and purely elastic S-matrices. Int. J. Mod. Phys., A5 (6): 1025-1048
−
* '''[Zam89]'''Integrable field theory from conformal field theory<br>
+
* '''[Zam89]'''A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19, 641-674 (1989)
−
** A.B. Zamolodchikov, Adv. Stud. Pure Math. 19, 641-674 (1989)
+
* Barry M. McCoy,  Craig A. Tracy, Tai Tsun Wu [http://dx.doi.org/10.1103/PhysRevLett.46.757 Ising Field Theory: Quadratic Difference Equations for the n-Point Green's Functions on the Lattice], Phys. Rev. Lett. 46, 757–760 (1981)
−
* [http://dx.doi.org/10.1103/PhysRevLett.46.757 Ising Field Theory: Quadratic Difference Equations for the n-Point Green's Functions on the Lattice]<br>
+
* '''[MTW77]'''Barry M. McCoy, Craig A. Tracy, Tai Tsun Wu [http://dx.doi.org/10.1103/PhysRevLett.38.793 Two-Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n-Point Functions], Phys. Rev. Lett. 38, 793–796 (1977)
−
** Barry M. McCoy,  Craig A. Tracy, Tai Tsun Wu, Phys. Rev. Lett. 46, 757–760 (1981)
+
* '''[Kau49]''' Bruria Kaufman [http://dx.doi.org/10.1103/PhysRev.76.1232 Statistics. II. Partition Function Evaluated by Spinor Analysis], Phys. Rev. 76, 1232–1243 (1949) Crystal
  
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* '''[MTW77]'''[http://dx.doi.org/10.1103/PhysRevLett.38.793 Two-Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n-Point Functions]<br>
+
==question and answers(Math Overflow)==
−
** Barry M. McCoy,  Craig A. Tracy, Tai Tsun Wu, Phys. Rev. Lett. 38, 793–796 (1977)
 
−
 
 
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* '''[Kau49]'''[http://dx.doi.org/10.1103/PhysRev.76.1232 Statistics. II. Partition Function Evaluated by Spinor Analysis]<br>
 
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** Bruria Kaufman, Phys. Rev. 76, 1232–1243 (1949) Crystal
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://arxiv.org/
 
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* http://www.pdf-search.org/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://dx.doi.org/10.1038/464362a
 
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<h5>question and answers(Math Overflow)</h5>
 
  
 
* http://mathoverflow.net/questions/32315/has-the-lie-group-e8-really-been-detected-experimentally
 
* http://mathoverflow.net/questions/32315/has-the-lie-group-e8-really-been-detected-experimentally
 
* http://mathoverflow.net/questions/32432/does-the-quantum-subgroup-of-quantum-su-2-called-e-8-have-anything-at-all-to-do-w
 
* http://mathoverflow.net/questions/32432/does-the-quantum-subgroup-of-quantum-su-2-called-e-8-have-anything-at-all-to-do-w
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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<h5>blogs</h5>
 
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*  구글 블로그 검색<br>
 
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**  http://blogsearch.google.com/blogsearch?q=<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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* http://ncatlab.org/nlab/show/HomePage
 
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<h5>experts on the field</h5>
 
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* http://arxiv.org/
 
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<h5>links</h5>
 
  
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
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[[분류:integrable systems]]
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* [http://pythagoras0.springnote.com/pages/1947378 수식표현 안내]
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[[분류:math and physics]]
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
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[[분류:migrate]]
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* http://functions.wolfram.com/
 

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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