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

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
• exact S-matrices in ATFT
• Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer

computational resource

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

expositions

• 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.
• 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

• 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
• V. V. Bazhanov, B. Nienhuis, S. O. Warnaar Lattice Ising model in a field: E8 scattering theory, 1994
• 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]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
• [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