Six-vertex model and Quantum XXZ Hamiltonian - 편집 역사

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2015-10-06T06:35:50Z

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@@ 209번째 줄: / 209번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q5985139 Q5985139]

← 이전 판		2020년 12월 26일 (토) 15:07 판
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	[[분류:math and physics]]		[[분류:math and physics]]
	[[분류:migrate]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q5985139 Q5985139]

@@ 25번째 줄: / 25번째 줄: @@
 ==integrability of the model and the Yang-Baxter equation==
-* $T(u)$ transfer matrix
+* <math>T(u)</math> transfer matrix
-* $\log T(u)=\sum_{n=0}^{\infty}Q_{n}u^n$
+* <math>\log T(u)=\sum_{n=0}^{\infty}Q_{n}u^n</math>
-* here $Q_1$ plays the role of the Hamiltonian
+* here <math>Q_1</math> plays the role of the Hamiltonian
 * necessary and sufficient codntion to have infinitely many conserved quantities
-$$[T(u), T(v)]=0$$
+:<math>[T(u), T(v)]=0</math>
-which implies $[Q_n,Q_m]=0$
+which implies <math>[Q_n,Q_m]=0</math>
-* in order to have $[T(u), T(v)]=0$, the [[Yang-Baxter equation (YBE)]] must be satisfied
+* in order to have <math>[T(u), T(v)]=0</math>, the [[Yang-Baxter equation (YBE)]] must be satisfied
@@ 37번째 줄: / 37번째 줄: @@
 ===R-matrix and Boltzmann weights===
 * [[R-matrix]]
-$$
+:<math>
 R(u,\eta)=\rho\left(
 \begin{array}{cccc}
@@ 46번째 줄: / 46번째 줄: @@
 \end{array}
 \right)
-$$
+</math>
 * multiplicative form of [[S-matrix of the quantum sine-Gordon model]]
-$$
+:<math>
 \check{R}(x)=
 \left(
@@ 59번째 줄: / 59번째 줄: @@
 \end{array}
 \right)
-$$
+</math>
 ==transfer matrix formalism and coordinate Bethe ansatz==
@@ 66번째 줄: / 66번째 줄: @@
 * one can regard the up(or down) arrows in a row as 'particles'
 * because of the ice rule, their number is conserved and one can try [[Bethe ansatz]] for the eigenvectors of the transfer matrix
-* let <math>f(x_1,\cdots,x_n)</math> be the coefficient in an eigenvector $v$ of the state with up arrows at the sites <math>x_ 1<x_ 2<\cdots<x_n</math> so that
+* let <math>f(x_1,\cdots,x_n)</math> be the coefficient in an eigenvector <math>v</math> of the state with up arrows at the sites <math>x_ 1<x_ 2<\cdots<x_n</math> so that
 :<math>v(k_1,\cdots,k_n)=
 \sum_{\substack{\mathbf{x}=(x_ 1,x_ 2,\cdots,x_n) \\ x_ 1<x_ 2<\cdots<x_n}} f(x_1,\cdots,x_n|k_1,\cdots,k_n)\sigma_{-}^{(x_1)}\cdots\sigma_{-}^{(x_n)}|0\rangle </math>
-* Bethe ansatz suggests the following form for $f$
+* Bethe ansatz suggests the following form for <math>f</math>
 :<math>f(x_ 1,\cdots,x_n)=\sum_{P\in S_n}A (P)\exp(i\sum_{j=1}^{n}x_jk _{P_j})</math>
 * Bethe ansatz equation for wave numbers : there are n conditions
 :<math>\exp(ik_jn)=\prod_{\ell \neq j}B(k_j,k_\ell)=\prod_{\ell=1}^{n}B(k_j,k_\ell),\quad \forall j=1,\cdots, n</math> where
 :<math>B(k,q)=-\frac{1+e^{ik}e^{iq}-e^{ik}}{1+e^{ik}e^{iq}-e^{iq}}</math>
-* eigenvalue $\lambda$ of $v$ is given by
+* eigenvalue <math>\lambda</math> of <math>v</math> is given by
 :<math>\lambda=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{\prod_{j=1}^{n}1-e^{ik_{j}}}=\frac{1+e^{i(k_{1}+\cdots+k_{n})}}{(1-e^{ik_{j}})\cdots(1-e^{ik_{j}})}</math>
@@ 102번째 줄: / 102번째 줄: @@
 *  the eigenvectors of the transfer matrix depended on a,b,c only via the parameter
 :<math>\Delta=\frac{a^2+b^2-c^2}{2ab}=\cos \eta</math>
-*  $\Delta$ = anisotropic parameter in [[Heisenberg spin chain model]]
+*  <math>\Delta</math> = anisotropic parameter in [[Heisenberg spin chain model]]
 ==one-point function==
 * by Baxter's corner transfer matrix method, we get
-$$
+:<math>
 G'(a)=\sum_{{\mathbb{p}\in \mathcal{P}(\Lambda_0)}\atop {W(0,\mathbb{p})=a}}q^{2\sum_{k=0}^{\infty}(k+1)(H(\mathbb{p}(k+1),\mathbb{p}(k))-H(\mathbb{p}_{\Lambda_0}(k+1),\mathbb{p}_{\Lambda_0}(k)))}
-$$
+</math>
 * one can evaluate the sum
-$$
+:<math>
 G'(a)=
 \begin{cases}
-  \frac{q^{\frac{a^2}{2}}}{\prod_{n=1}^{\infty}(1-q^{2n})}, & \text{if $a$ is even}\\
+  \frac{q^{\frac{a^2}{2}}}{\prod_{n=1}^{\infty}(1-q^{2n})}, & \text{if </math>a<math> is even}\\
-, & \text{if $a$ is odd} \\
+, & \text{if </math>a<math> is odd} \\
 \end{cases}
-$$
+</math>
-$$
+:<math>
 ==thermodynamic properties==
@@ 184번째 줄: / 184번째 줄: @@
 * Kozlowski, K. K. “On Condensation Properties of Bethe Roots Associated with the XXZ Chain.” arXiv:1508.05741 [math-Ph, Physics:nlin], August 24, 2015. http://arxiv.org/abs/1508.05741.
 * Martins, M. J. ‘The Symmetric Six-Vertex Model and the Segre Cubic Threefold’. arXiv:1505.07418 [math-Ph], 27 May 2015. http://arxiv.org/abs/1505.07418.
-* Morin-Duchesne, Alexi, Jorgen Rasmussen, Philippe Ruelle, and Yvan Saint-Aubin. ‘On the Reality of Spectra of $\boldsymbol{U_q(sl_2)}$-Invariant XXZ Hamiltonians’. arXiv:1502.01859 [cond-Mat, Physics:hep-Th, Physics:math-Ph], 6 February 2015. http://arxiv.org/abs/1502.01859.
+* Morin-Duchesne, Alexi, Jorgen Rasmussen, Philippe Ruelle, and Yvan Saint-Aubin. ‘On the Reality of Spectra of </math>\boldsymbol{U_q(sl_2)}<math>-Invariant XXZ Hamiltonians’. arXiv:1502.01859 [cond-Mat, Physics:hep-Th, Physics:math-Ph], 6 February 2015. http://arxiv.org/abs/1502.01859.
 * Vieira, R. S., and A. Lima-Santos. “Where Are the Roots of the Bethe Ansatz Equations?” arXiv:1502.05316 [cond-Mat, Physics:math-Ph, Physics:nlin], February 18, 2015. http://arxiv.org/abs/1502.05316.
 * Hamel, Angèle M., and Ronald C. King. “Tokuyama’s Identity for Factorial Schur Functions.” arXiv:1501.03561 [math], January 14, 2015. http://arxiv.org/abs/1501.03561.
 * Tavares, T. S., G. A. P. Ribeiro, and V. E. Korepin. “The Entropy of the Six-Vertex Model with Variety of Different Boundary Conditions.” arXiv:1501.02818 [cond-Mat, Physics:math-Ph, Physics:nlin], January 12, 2015. http://arxiv.org/abs/1501.02818.
-* Garbali, Alexander. ‘The Scalar Product of XXZ Spin Chain Revisited. Application to the Ground State at $\Delta=-1/2$’. arXiv:1411.2938 [math-Ph], 11 November 2014. http://arxiv.org/abs/1411.2938.
+* Garbali, Alexander. ‘The Scalar Product of XXZ Spin Chain Revisited. Application to the Ground State at </math>\Delta=-1/2<math>’. arXiv:1411.2938 [math-Ph], 11 November 2014. http://arxiv.org/abs/1411.2938.
 * Ribeiro, G. A. P., and V. E. Korepin. “Thermodynamic Limit of the Six-Vertex Model with Reflecting End.” arXiv:1409.1212 [cond-Mat, Physics:hep-Th, Physics:math-Ph, Physics:nlin], September 3, 2014. http://arxiv.org/abs/1409.1212.
 * Mangazeev, Vladimir V. “Q-Operators in the Six-Vertex Model.” arXiv:1406.0662 [hep-Th, Physics:math-Ph], June 3, 2014. http://arxiv.org/abs/1406.0662.
 * António, N. Cirilo, N. Manojlović, and Z. Nagy. 2013. “Trigonometric Sl(2) Gaudin Model with Boundary Terms.” arXiv:1303.2481 (March 11). http://arxiv.org/abs/1303.2481.
 * Szabo, Richard J., and Miguel Tierz. 2011. “Two-Dimensional Yang-Mills Theory, Painleve Equations and the Six-Vertex Model”. ArXiv e-print 1102.3640. http://arxiv.org/abs/1102.3640.
-* Deguchi, Tetsuo. 2006. “The Six-vertex Model at Roots of Unity and Some Highest Weight Representations of the $\rm Sl_2$ Loop Algebra.” Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics 7 (7-8): 1531–1540. doi:[http://dx.doi.org/10.1007/s00023-006-0290- 10.1007/s00023-006-0290-8]
+* Deguchi, Tetsuo. 2006. “The Six-vertex Model at Roots of Unity and Some Highest Weight Representations of the </math>\rm Sl_2<math> Loop Algebra.” Annales Henri Poincaré. A Journal of Theoretical and Mathematical Physics 7 (7-8): 1531–1540. doi:[http://dx.doi.org/10.1007/s00023-006-0290- 10.1007/s00023-006-0290-8]
 * De Vega, H.J., and F. Woynarovich. 1985. “Method for Calculating Finite Size Corrections in Bethe Ansatz Systems: Heisenberg Chain and Six-vertex Model.” Nuclear Physics B 251: 439–456. doi:[http://dx.doi.org/10.1016/0550-3213(85)90271-8 10.1016/0550-3213(85)90271-8].
 * Kazuhiko Minami, [http://dx.doi.org/10.1063/1.2890671 The free energies of six-vertex models and the n-equivalence relation]

← 이전 판		2020년 11월 14일 (토) 04:41 판
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	[[분류:integrable systems]]		[[분류:integrable systems]]
	[[분류:math and physics]]		[[분류:math and physics]]
		+	[[분류:migrate]]

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 ==expositions==
 * Lamers, J. “A Pedagogical Introduction to Quantum Integrability, with a View towards Theoretical High-Energy Physics.” arXiv:1501.06805 [hep-Th, Physics:math-Ph, Physics:nlin], January 27, 2015. http://arxiv.org/abs/1501.06805.
 * Reshetikhin, N. 2010. “Lectures on the Integrability of the Six-vertex Model.” In Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, 197–266. Oxford: Oxford Univ. Press. http://www.ams.org/mathscinet-getitem?mr=2668647.

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 * Lamers, J. “A Pedagogical Introduction to Quantum Integrability, with a View towards Theoretical High-Energy Physics.” arXiv:1501.06805 [hep-Th, Physics:math-Ph, Physics:nlin], January 27, 2015. http://arxiv.org/abs/1501.06805.
 * Reshetikhin, N. 2010. “Lectures on the Integrability of the Six-vertex Model.” In Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, 197–266. Oxford: Oxford Univ. Press. http://www.ams.org/mathscinet-getitem?mr=2668647.
 * T Miwa [http://www.springerlink.com/content/f9961j132852j27q/ Integrability of the Quantum XXZ Hamiltonian], 2009
 * Tetsuo Deguchi [http://arxiv.org/abs/cond-mat/0304309 Introduction to solvable lattice models in statistical and mathematical physics], 2003