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 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 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.
• [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
• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• massive integrable perturbations and quasi-particles
• exact S-matrices in ATFT

 

 

encyclopedia

• http://en.wikipedia.org/wiki/
• http://www.scholarpedia.org/
• http://www.proofwiki.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

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.
  

 

 

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. 
• On the integrable structure of the Ising model
  
  • Alessandro Nigro J. Stat. Mech. (2008) P01017
• Non-integrable aspects of the multi-frequency sine-Gordon model
  
  • G. Delfinoa and G. Mussardo, 1998
• The spin-spin correlation function in the two-dimensional Ising model in a magnetic field at T = Tc
  
  • G. Delfinoa and G. Mussardo, 1995
• Lattice Ising model in a field: E8 scattering theory
  
  • V. V. Bazhanov, B. Nienhuis, S. O. Warnaar, 1994
• [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]Integrable field theory from conformal field theory
  
  • A.B. Zamolodchikov, Adv. Stud. Pure Math. 19, 641-674 (1989)
• Ising Field Theory: Quadratic Difference Equations for the n-Point Green's Functions on the Lattice
  
  • Barry M. McCoy,  Craig A. Tracy, Tai Tsun Wu, Phys. Rev. Lett. 46, 757–760 (1981)

• [MTW77]Two-Dimensional Ising Model as an Exactly Solvable Relativistic Quantum Field Theory: Explicit Formulas for n-Point Functions
  
  • Barry M. McCoy,  Craig A. Tracy, Tai Tsun Wu, Phys. Rev. Lett. 38, 793–796 (1977)

• [Kau49]Statistics. II. Partition Function Evaluated by Spinor Analysis
  
  • Bruria Kaufman, Phys. Rev. 76, 1232–1243 (1949) Crystal

• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://arxiv.org/
• http://www.pdf-search.org/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://dx.doi.org/10.1038/464362a

 

 

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
• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
    
  • http://blogsearch.google.com/blogsearch?q=
• http://ncatlab.org/nlab/show/HomePage

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/