Integrable perturbations of Ising model

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

• Thermodynamic Bethe ansatz (TBA)
• Let $X=E_8$

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

• Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer
• Let $N=(N_{ij})$ be the matrix given by

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

• 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

• (3,4) Ising minimal model CFT
• Massive integrable perturbations of CFT and quasi-particles
• Y-system
• exact S-matrices in ATFT
• Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer
• Dilute A model

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxSWIwb1l2YkoyNDg/edit

expositions

• Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?
• Affleck, Ian. 2010. “Solid-state physics: Golden ratio seen in a magnet”. Nature 464 (7287) (3월 18): 362-363. doi:10.1038/464362a.
• Jihye Seo, Solving 2D Magnetic Ising Model at $T=T_c$ 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:10.1126/science.1180085.
• Alessandro Nigro On the integrable structure of the Ising model J. Stat. Mech. (2008) P01017
• G. Delfinoa and G. Mussardo Non-integrable aspects of the multi-frequency sine-Gordon model, 1998
• G. Delfinoa and G. Mussardo The spin-spin correlation function in the two-dimensional Ising model in a magnetic field at T = Tc, 1995
• 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: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 $S$-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
• [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
• [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.
• [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]A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19, 641-674 (1989)
• Barry M. McCoy, Craig A. Tracy, Tai Tsun Wu Ising Field Theory: Quadratic Difference Equations for the n-Point Green's Functions on the Lattice, Phys. Rev. Lett. 46, 757–760 (1981)
• [MTW77]Barry M. McCoy, Craig A. Tracy, Tai Tsun Wu 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)
• [Kau49] Bruria Kaufman Statistics. II. Partition Function Evaluated by Spinor Analysis, Phys. Rev. 76, 1232–1243 (1949) Crystal

question and answers(Math Overflow)

• 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