"Integrable perturbations of Ising model"의 두 판 사이의 차이
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| + | ==constant TBA equation== | ||
| + | * Let $X=E_8$ | ||
| + | $$ | ||
| + | Y_{i}(u-1)Y_{i}(u+1)=\prod _{j\in I} (1+Y_{j}(u))^{\mathcal{I}(X)_{ij}} | ||
| + | $$ | ||
| + | * solution | ||
| + | $$ | ||
| + | \left\{2+2 \sqrt{2},5+4 \sqrt{2},11+8 \sqrt{2},16+12 \sqrt{2},42+30 \sqrt{2},56+40 \sqrt{2},152+108 \sqrt{2},543+384 \sqrt{2}\right\} | ||
| + | $$ | ||
| + | |||
==history== | ==history== | ||
2013년 4월 20일 (토) 09:53 판
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
constant TBA equation
- Let $X=E_8$
$$ Y_{i}(u-1)Y_{i}(u+1)=\prod _{j\in I} (1+Y_{j}(u))^{\mathcal{I}(X)_{ij}} $$
- solution
$$ \left\{2+2 \sqrt{2},5+4 \sqrt{2},11+8 \sqrt{2},16+12 \sqrt{2},42+30 \sqrt{2},56+40 \sqrt{2},152+108 \sqrt{2},543+384 \sqrt{2}\right\} $$
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=
- (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
computational resource
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