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

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(피타고라스님이 이 페이지의 이름을 integrable perturbations of Ising model로 바꾸었습니다.)
 
(사용자 3명의 중간 판 57개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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==introduction==
  
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*  energy perturbation '''[Kau49]''', '''[MTW77]'''
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** related to A1
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** Ising field theory
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*  magnetic perturbation'''[Zam89]'''
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** related to E8
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==Ising field theory==
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*  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
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* an entry of S-matrix
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:<math>
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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)}
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</math>
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* 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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 +
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==constant TBA equation==
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===Y-system===
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* [[Thermodynamic Bethe ansatz (TBA)]]
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* Let <math>X=E_8</math>
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===constant Y-system solution===
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* constant Y-system
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:<math>
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y_{i}^2=\prod _{j\in I} (1+y_{j})^{\mathcal{I}(X)_{ij}}
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</math>
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* solution
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:<math>
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\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\}
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</math>
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===Klassen-Melzer solution===
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* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
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* Let <math>N=(N_{ij})</math> be the matrix given by
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:<math>
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N=\mathcal{I}(E_8)\cdot(\mathcal{C}(E_8))^{-1}=
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\left(
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\begin{array}{cccccccc}
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3 & 4 & 6 & 6 & 8 & 8 & 10 & 12 \\
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4 & 7 & 8 & 10 & 12 & 14 & 16 & 20 \\
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6 & 8 & 11 & 12 & 16 & 16 & 20 & 24 \\
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6 & 10 & 12 & 15 & 18 & 20 & 24 & 30 \\
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8 & 12 & 16 & 18 & 23 & 24 & 30 & 36 \\
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8 & 14 & 16 & 20 & 24 & 27 & 32 & 40 \\
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10 & 16 & 20 & 24 & 30 & 32 & 39 & 48 \\
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12 & 20 & 24 & 30 & 36 & 40 & 48 & 59 \\
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\end{array}
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\right)
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</math>
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* 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
 +
 +
:<math>
 +
e^{\epsilon_i}=\prod_{j}(1+e^{-\epsilon_j})^{N_{ij}}
 +
</math>
 +
* we have the relationship <math>y_i=e^{\epsilon_i}</math>
 +
 +
==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 <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.
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** 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.
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
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 +
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==related items==
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* [[(3,4) Ising minimal model CFT]]
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* [[Massive integrable perturbations of CFT and quasi-particles]]
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* [[Y-system]]
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* [[exact S-matrices in ATFT]]
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* [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
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* [[Dilute A model]]
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxSWIwb1l2YkoyNDg/edit
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==expositions==
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* [[Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?]]
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*  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].
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* 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
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* 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.
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* Dorey, Patrick. 1992. “Hidden Geometrical Structures in Integrable Models.” arXiv:hep-th/9212143 (December 23). http://arxiv.org/abs/hep-th/9212143.
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==articles==
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* 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.
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* 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.
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* 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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* 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
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* 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
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* 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
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* 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.
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* '''[EY]'''T. Eguchi and S.-K. Yang, Deformations of conformal field theories and soliton equations, Phys. Lett. B 224 (1989), 373-8 B
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* '''[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
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* '''[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.
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* '''[FZ90]'''V. A. Fateev and A. B. Zamolodchikov. Conformal field theory and purely elastic S-matrices. Int. J. Mod. Phys., A5 (6): 1025-1048
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* '''[Zam89]'''A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19, 641-674 (1989)
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* 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)
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* '''[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)
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* '''[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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==question and answers(Math Overflow)==
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* http://mathoverflow.net/questions/32315/has-the-lie-group-e8-really-been-detected-experimentally
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* 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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[[분류:integrable systems]]
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[[분류:math and physics]]
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[[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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