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

2020년 11월 16일 (월) 11:06 기준 최신판

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

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”. 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

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: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
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: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 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)

@@ 13번째 줄: / 13번째 줄: @@
 ** "kink" states (boundaries between regions of differing spin) = basic objects of the theory
 ** 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 <math>\theta=i y,\, 0<y<\pi</math> at <math>y=\pi/15,2\pi/5,2\pi/3</math>
+==constant TBA equation==
+===Y-system===
+* [[Thermodynamic Bethe ansatz (TBA)]]
+* Let <math>X=E_8</math>
+===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 <math>N=(N_{ij})</math> be the matrix given by
+:<math>
+N=\mathcal{I}(E_8)\cdot(\mathcal{C}(E_8))^{-1}=
+\left(
+\begin{array}{cccccccc}
+& 4 & 6 & 6 & 8 & 8 & 10 & 12 \\
+& 7 & 8 & 10 & 12 & 14 & 16 & 20 \\
+& 8 & 11 & 12 & 16 & 16 & 20 & 24 \\
+& 10 & 12 & 15 & 18 & 20 & 24 & 30 \\
+& 12 & 16 & 18 & 23 & 24 & 30 & 36 \\
+& 14 & 16 & 20 & 24 & 27 & 32 & 40 \\
+& 16 & 20 & 24 & 30 & 32 & 39 & 48 \\
+& 20 & 24 & 30 & 36 & 40 & 48 & 59 \\
+\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 <math>y_i=e^{\epsilon_i}</math>
 ==history==
@@ 20번째 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=
 ==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]]
-==encyclopedia==
+==computational resource==
+* https://docs.google.com/file/d/0B8XXo8Tve1cxSWIwb1l2YkoyNDg/edit
-* http://en.wikipedia.org/wiki/
-* http://www.scholarpedia.org/
-* http://www.proofwiki.org/wiki/
-==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.
+* [[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.
 ==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
-* 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 <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
-* '''[Zam]'''[http://dx.doi.org/10.1142/S0217751X8900176X INTEGRALS OF MOTION AND S-MATRIX OF THE (SCALED) T = Tc ISING MODEL WITH MAGNETIC FIELD]
+* '''[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)
 * '''[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
-* http://dx.doi.org/10.1038/464362a
 ==question and answers(Math Overflow)==
@@ 88번째 줄: / 129번째 줄: @@
 [[분류:integrable systems]]
 [[분류:math and physics]]
+[[분류:migrate]]