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

2013년 4월 20일 (토) 14:48 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

Y-system

Thermodynamic Bethe ansatz (TBA)
Let $X=E_8$
Y-system is

$$ Y_{i}(u-1)Y_{i}(u+1)=\prod _{j\in I} (1+Y_{j}(u))^{\mathcal{I}(X)_{ij}} $$

in $\theta$-plane

$$ Y_{i}(\theta+i\frac{\pi}{h})Y_{i}(\theta-i\frac{\pi}{h})=\prod _{j\in I} (1+Y_{j}(\theta))^{\mathcal{I}(X)_{ij}} $$

in $t=\exp(\frac{2h}{h+2}\theta)$-plane

$$ Y_{i}(\Omega t)Y_{i}(\Omega^{-1}t)=\prod _{j\in I} (1+Y_{j}(t))^{\mathcal{I}(X)_{ij}} $$ where $\Omega=\exp(\frac{2i\pi}{h+2})$

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

Let $N_{ij}$ be the matrix

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

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

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)

@@ 49번째 줄: / 49번째 줄: @@
 * Let $N_{ij}$ be the matrix
 $$
-I_r-\mathcal{I}(E_8)=
+\mathcal{I}(E_8)(\mathcal{C}(E_8))^{-1}=
 \left(
 \begin{array}{cccccccc}
@@ 73번째 줄: / 73번째 줄: @@
 $$
 * we have the relationship $y_i=e^{\epsilon_i}$
 ==history==