Integrable perturbations of Ising model - 편집 역사

2020년 11월 16일 (월) 18:06에 Pythagoras0님의 편집

2020-11-16T18:06:48Z

2020년 11월 16일 (월) 17:57에 Pythagoras0님의 편집

2020-11-16T17:57:00Z

2020년 11월 14일 (토) 04:40에 imported>Pythagoras0님의 편집

2020-11-14T04:40:31Z

2020년 11월 14일 (토) 04:40에 imported>Pythagoras0님의 편집

2020-11-14T04:40:31Z

imported>Pythagoras0: /* articles */

2015-04-15T08:43:38Z

articles

imported>Pythagoras0: /* related items */

2015-04-15T08:33:06Z

2015년 4월 15일 (수) 05:54에 imported>Pythagoras0님의 편집

2015-04-15T05:54:19Z

imported>Pythagoras0: /* Klassen-Melzer solution */

2015-01-02T07:55:26Z

Klassen-Melzer solution

imported>Pythagoras0: /* Klassen-Melzer solution */

2015-01-02T07:49:58Z

Klassen-Melzer solution

imported>Pythagoras0: /* articles */

2015-01-02T07:47:54Z

articles

@@ 14번째 줄: / 14번째 줄: @@
 ** called quasiparticle
 * an entry of S-matrix
-$$
+:<math>
 S_{1,1}(\theta)=\frac{\tanh \left(\frac{1}{2} \left(\theta +\frac{i \pi }{15}\right)\right)\tanh \left(\frac{1}{2} \left(\theta +\frac{2i \pi }{5}\right)\right)\tanh \left(\frac{1}{2} \left(\theta +\frac{2i \pi }{3}\right)\right)}{\tanh \left(\frac{1}{2} \left(\theta -\frac{i \pi }{15}\right)\right)\tanh \left(\frac{1}{2} \left(\theta -\frac{2i \pi }{5}\right)\right)\tanh \left(\frac{1}{2} \left(\theta -\frac{2i \pi }{3}\right)\right)}
-$$
+</math>
-* it has poles with positive residue when $\theta=i y,\, 0<y<\pi$ at $y=\pi/15,2\pi/5,2\pi/3$
+* it has poles with positive residue when <math>\theta=i y,\, 0<y<\pi</math> at <math>y=\pi/15,2\pi/5,2\pi/3</math>
@@ 23번째 줄: / 23번째 줄: @@
 ===Y-system===
 * [[Thermodynamic Bethe ansatz (TBA)]]
-* Let $X=E_8$
+* Let <math>X=E_8</math>
@@ 29번째 줄: / 29번째 줄: @@
 ===constant Y-system solution===
 * constant Y-system
-$$
+:<math>
 y_{i}^2=\prod _{j\in I} (1+y_{j})^{\mathcal{I}(X)_{ij}}
-$$
+</math>
 * solution
-$$
+:<math>
 \left\{2+2 \sqrt{2},5+4 \sqrt{2},11+8 \sqrt{2},16+12 \sqrt{2},42+30 \sqrt{2},56+40 \sqrt{2},152+108 \sqrt{2},543+384 \sqrt{2}\right\}
-$$
+</math>
 ===Klassen-Melzer solution===
 * [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
-* Let $N=(N_{ij})$ be the matrix given by
+* Let <math>N=(N_{ij})</math> be the matrix given by
-$$
+:<math>
 N=\mathcal{I}(E_8)\cdot(\mathcal{C}(E_8))^{-1}=
 \left(
@@ 55번째 줄: / 55번째 줄: @@
 \end{array}
 \right)
-$$
+</math>
 * note that this is equivalent to
-$$
+:<math>
 N=2\mathcal{C}(E_8)^{-1}-I_8
-$$
+</math>
 * The TBA equation is
-$$
+:<math>
 \epsilon_i=\sum_{j}N_{ij}\log (1+e^{-\epsilon_j})
-$$
+</math>
 or
-$$
+:<math>
 e^{\epsilon_i}=\prod_{j}(1+e^{-\epsilon_j})^{N_{ij}}
-$$
+</math>
-* we have the relationship $y_i=e^{\epsilon_i}$
+* we have the relationship <math>y_i=e^{\epsilon_i}</math>
 ==history==
@@ 75번째 줄: / 75번째 줄: @@
 * 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.
+** if you take a minimal model CFT constructed from a compact Lie algebra <math>\mathfrak{g}</math> via the coset construction and perturb it in a particular way, then you obtain the affine Toda field theory (ATFT) associated with <math>\mathfrak{g}</math>, 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.
+* If you do this with <math>\mathfrak{g}=E_8</math>, 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.
+* That is, if we take the <math>E_8</math> ATFT as a starting point, then the assumptions (Z1)–(Z4) become deductions.
 * http://www.google.com/search?hl=en&tbs=tl:1&q=
@@ 98번째 줄: / 98번째 줄: @@
 * [[Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?]]
 *  Affleck, Ian. 2010. “Solid-state physics: Golden ratio seen in a magnet”. <em>Nature</em> 464 (7287) (3월 18): 362-363. doi:[http://dx.doi.org/10.1038/464362a 10.1038/464362a].
-* Jihye Seo, [http://isites.harvard.edu/fs/docs/icb.topic572189.files/Jihye_Seo_Ising_model_in_field.pdf Solving 2D Magnetic Ising Model at $T=T_c$ Using Scattering Theory] 2009
+* Jihye Seo, [http://isites.harvard.edu/fs/docs/icb.topic572189.files/Jihye_Seo_Ising_model_in_field.pdf Solving 2D Magnetic Ising Model at <math>T=T_c</math> 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.
@@ 110번째 줄: / 110번째 줄: @@
 * G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/S0550-3213%2898%2900063-7 Non-integrable aspects of the multi-frequency sine-Gordon model], 1998
 * G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/0550-3213%2895%2900464-4 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:[http://dx.doi.org/10.1016/0370-2693%2894%2991107-X 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.
+* 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:[http://dx.doi.org/10.1016/0370-2693%2894%2991107-X 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 <math>S</math>-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

← 이전 판	2020년 11월 14일 (토) 04:40 판
1번째 줄:	1번째 줄:
	+	==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”. <em>Nature</em> 464 (7287) (3월 18): 362-363. doi:[http://dx.doi.org/10.1038/464362a 10.1038/464362a].
	+	* Jihye Seo, [http://isites.harvard.edu/fs/docs/icb.topic572189.files/Jihye_Seo_Ising_model_in_field.pdf 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:[http://dx.doi.org/10.1126/science.1180085 10.1126/science.1180085].
	+	* Alessandro Nigro [http://dx.doi.org/10.1088/1742-5468/2008/01/P01017 On the integrable structure of the Ising model] J. Stat. Mech. (2008) P01017
	+	* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/S0550-3213%2898%2900063-7 Non-integrable aspects of the multi-frequency sine-Gordon model], 1998
	+	* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/0550-3213%2895%2900464-4 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:[http://dx.doi.org/10.1016/0370-2693%2894%2991107-X 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 [http://dx.doi.org/10.1103/PhysRevLett.46.757 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 [http://dx.doi.org/10.1103/PhysRevLett.38.793 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 [http://dx.doi.org/10.1103/PhysRev.76.1232 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
	+
	+
	+
	+	[[분류:integrable systems]]
	+	[[분류:math and physics]]
	+	[[분류:migrate]]

← 이전 판		2020년 11월 14일 (토) 04:40 판
1번째 줄:		1번째 줄:
−	~~==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”. <em>Nature</em> 464 (7287) (3월 18): 362-363. doi:[http://dx.doi.org/10.1038/464362a 10.1038/464362a].
−	* Jihye Seo, [http://isites.harvard.edu/fs/docs/icb.topic572189.files/Jihye_Seo_Ising_model_in_field.pdf 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:[http://dx.doi.org/10.1126/science.1180085 10.1126/science.1180085].
−	* Alessandro Nigro [http://dx.doi.org/10.1088/1742-5468/2008/01/P01017 On the integrable structure of the Ising model] J. Stat. Mech. (2008) P01017
−	* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/S0550-3213%2898%2900063-7 Non-integrable aspects of the multi-frequency sine-Gordon model], 1998
−	* G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/0550-3213%2895%2900464-4 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:[http://dx.doi.org/10.1016/0370-2693%2894%2991107-X 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 [http://dx.doi.org/10.1103/PhysRevLett.46.757 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 [http://dx.doi.org/10.1103/PhysRevLett.38.793 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 [http://dx.doi.org/10.1103/PhysRev.76.1232 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
−
−
−
−	~~[[분류:integrable systems]]~~
−	~~[[분류:math and physics]]~~

@@ 110번째 줄: / 110번째 줄: @@
 * G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/S0550-3213%2898%2900063-7 Non-integrable aspects of the multi-frequency sine-Gordon model], 1998
 * G. Delfinoa and G. Mussardo [http://dx.doi.org/10.1016/0550-3213%2895%2900464-4 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 [http://dx.doi.org/10.1016/0370-2693%2894%2991107-X Lattice Ising model in a field: E8 scattering theory], 1994
+* 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:[http://dx.doi.org/10.1016/0370-2693%2894%2991107-X 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

@@ 88번째 줄: / 88번째 줄: @@
 * [[exact S-matrices in ATFT]]
 * [[Purely Elastic Scattering Theories and Their Ultraviolet Limits by Klassen and Melzer]]
+* [[Dilute A model]]
 ==computational resource==

← 이전 판	2020년 11월 16일 (월) 17:57 판
(차이 없음)

← 이전 판		2015년 4월 15일 (수) 05:54 판
13번째 줄:		13번째 줄:
	** "kink" states (boundaries between regions of differing spin) = basic objects of the theory		** "kink" states (boundaries between regions of differing spin) = basic objects of the theory
	** called quasiparticle		** 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 $y=\pi/15,2\pi/5,2\pi/3$

@@ 37번째 줄: / 37번째 줄: @@
 * Let $N_{ij}$ be the matrix
 $$
-\mathcal{I}(E_8)(\mathcal{C}(E_8))^{-1}=
+\mathcal{I}(E_8)\cdot(\mathcal{C}(E_8))^{-1}=
 \left(
 \begin{array}{cccccccc}

@@ 107번째 줄: / 107번째 줄: @@
 * '''[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)
+* '''[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 [http://dx.doi.org/10.1103/PhysRevLett.46.757 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 [http://dx.doi.org/10.1103/PhysRevLett.38.793 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)