String hypothesis - 편집 역사

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@@ 2번째 줄: / 2번째 줄: @@
 * [[Bethe ansatz]]
 * the roots in a string are all equally spaced in the imaginary direction
-* a Bethe root consists of strings (as $L\to \infty$)
+* a Bethe root consists of strings (as <math>L\to \infty</math>)
-* $n$-string
+* <math>n</math>-string
-$$
+:<math>
 \{\lambda_{\alpha}+\frac{i}{2}(n-1)+\epsilon_{\alpha,1},\lambda_{\alpha}+\frac{i}{2}(n-3)+\epsilon_{\alpha,2},\cdots, \lambda_{\alpha}-\frac{i}{2}(n-1)+\epsilon_{\alpha,n}\}
-$$
+</math>
-where $\lambda_{\alpha}$ is a real number and $\epsilon$ small deviation
+where <math>\lambda_{\alpha}</math> is a real number and <math>\epsilon</math> small deviation
 ==Bethe Ansatz equations==
 * recall
-$$
+:<math>
 \begin{eqnarray}\label{bae}
 \left(  {\lambda_{j} - {i\over 2}
@@ 24번째 줄: / 24번째 줄: @@
 \,, \qquad j = 1 \,, \cdots \,, n \,.
 \end{eqnarray}
-$$.
+</math>.
-* assume that $\Im{\lambda}>0$
+* assume that <math>\Im{\lambda}>0</math>
-* fix $n$ and take $L\to \infty$. Then the LHS goes to $\infty$
+* fix <math>n</math> and take <math>L\to \infty</math>. Then the LHS goes to <math>\infty</math>
-* this implies that there exists $l$ such that $\lambda_{j} - \lambda_{l} + i$ must be close to 0
+* this implies that there exists <math>l</math> such that <math>\lambda_{j} - \lambda_{l} + i</math> must be close to 0
 * this suggests the existence of strings
@@ 51번째 줄: / 51번째 줄: @@
 * Kirillov, Anatol N., and Reiho Sakamoto. 2014. “Singular Solutions to the Bethe Ansatz Equations and Rigged Configurations.” arXiv:1402.0651 [math-Ph], February. http://arxiv.org/abs/1402.0651.
 * Nepomechie, Rafael I., and Chunguang Wang. 2013. “Algebraic Bethe Ansatz for Singular Solutions.” Journal of Physics A: Mathematical and Theoretical 46 (32): 325002. doi:10.1088/1751-8113/46/32/325002.
-* Volin, Dmytro. 2012. “String Hypothesis for $\mathfrak{gl}(n|m)$ Spin Chains: A Particle/Hole Democracy.” Letters in Mathematical Physics 102 (1) (October 1): 1–29. doi:[http://dx.doi.org/10.1007/s11005-012-0570-9 10.1007/s11005-012-0570-9].
+* Volin, Dmytro. 2012. “String Hypothesis for <math>\mathfrak{gl}(n|m)</math> Spin Chains: A Particle/Hole Democracy.” Letters in Mathematical Physics 102 (1) (October 1): 1–29. doi:[http://dx.doi.org/10.1007/s11005-012-0570-9 10.1007/s11005-012-0570-9].
-* Mukhin, E., V. Tarasov, and A. Varchenko. 2009. “Bethe Algebra of Homogeneous $XXX$ Heisenberg Model Has Simple Spectrum.” Communications in Mathematical Physics 288 (1): 1–42. doi:10.1007/s00220-009-0733-4.
+* Mukhin, E., V. Tarasov, and A. Varchenko. 2009. “Bethe Algebra of Homogeneous <math>XXX</math> Heisenberg Model Has Simple Spectrum.” Communications in Mathematical Physics 288 (1): 1–42. doi:10.1007/s00220-009-0733-4.
 * Hagemans, R., and J.-S. Caux. 2007. “Deformed Strings in the Heisenberg Model.” Journal of Physics A: Mathematical and Theoretical 40 (49): 14605–47. doi:10.1088/1751-8113/40/49/001.
 * Wal, A., T. Lulek, B. Lulek, and E. Kozak. “THE HEISENBERG MAGNETIC RING WITH 6 NODES: EXACT DIAGONALIZATION, BETHE ANSATZ AND STRING CONFIGURATIONS.” International Journal of Modern Physics B 13, no. 28 (November 10, 1999): 3307–21. doi:10.1142/S0217979299003039.

← 이전 판		2020년 11월 14일 (토) 09:00 판
62번째 줄:		62번째 줄:
	==questions==		==questions==
	* http://mathoverflow.net/questions/136594/roots-of-the-xxz-bethe-ansatz-equation		* http://mathoverflow.net/questions/136594/roots-of-the-xxz-bethe-ansatz-equation
		+	[[분류:migrate]]

@@ 47번째 줄: / 47번째 줄: @@
 ==articles==
 * Deguchi, Tetsuo, and Pulak Ranjan Giri. “Non Self-Conjugate Strings, Singular Strings and Rigged Configurations in the Heisenberg Model.” arXiv:1408.7030 [cond-Mat, Physics:hep-Th, Physics:math-Ph, Physics:nlin], August 29, 2014. http://arxiv.org/abs/1408.7030.
 * Kirillov, Anatol N., and Reiho Sakamoto. 2014. “Singular Solutions to the Bethe Ansatz Equations and Rigged Configurations.” arXiv:1402.0651 [math-Ph], February. http://arxiv.org/abs/1402.0651.

@@ 47번째 줄: / 47번째 줄: @@
 ==articles==
 * Kirillov, Anatol N., and Reiho Sakamoto. 2014. “Singular Solutions to the Bethe Ansatz Equations and Rigged Configurations.” arXiv:1402.0651 [math-Ph], February. http://arxiv.org/abs/1402.0651.
 * Nepomechie, Rafael I., and Chunguang Wang. 2013. “Algebraic Bethe Ansatz for Singular Solutions.” Journal of Physics A: Mathematical and Theoretical 46 (32): 325002. doi:10.1088/1751-8113/46/32/325002.

@@ 52번째 줄: / 52번째 줄: @@
 * Mukhin, E., V. Tarasov, and A. Varchenko. 2009. “Bethe Algebra of Homogeneous $XXX$ Heisenberg Model Has Simple Spectrum.” Communications in Mathematical Physics 288 (1): 1–42. doi:10.1007/s00220-009-0733-4.
 * Hagemans, R., and J.-S. Caux. 2007. “Deformed Strings in the Heisenberg Model.” Journal of Physics A: Mathematical and Theoretical 40 (49): 14605–47. doi:10.1088/1751-8113/40/49/001.
 * Kirillov, A. N. 1985. “Combinatorial Identities, and Completeness of Eigenstates of the Heisenberg Magnet.” Journal of Soviet Mathematics 30 (4) (August 1): 2298–2310. doi:[http://dx.doi.org/10.1007/BF02105347 10.1007/BF02105347].
 * Faddeev, L. D., and L. A. Takhtadzhyan. 1984. “Spectrum and Scattering of Excitations in the One-dimensional Isotropic Heisenberg Model.” Journal of Soviet Mathematics 24 (2) (January 1): 241–267. doi:[http://dx.doi.org/10.1007/BF01087245 10.1007/BF01087245].

@@ 47번째 줄: / 47번째 줄: @@
 ==articles==
 * Volin, Dmytro. 2012. “String Hypothesis for $\mathfrak{gl}(n|m)$ Spin Chains: A Particle/Hole Democracy.” Letters in Mathematical Physics 102 (1) (October 1): 1–29. doi:[http://dx.doi.org/10.1007/s11005-012-0570-9 10.1007/s11005-012-0570-9].
 * Mukhin, E., V. Tarasov, and A. Varchenko. 2009. “Bethe Algebra of Homogeneous $XXX$ Heisenberg Model Has Simple Spectrum.” Communications in Mathematical Physics 288 (1): 1–42. doi:10.1007/s00220-009-0733-4.
@@ 54번째 줄: / 56번째 줄: @@
 * Takahashi, Minoru. 1971. “One-Dimensional Heisenberg Model at Finite Temperature.” Progress of Theoretical Physics 46 (2) (August 1): 401–415. doi:[http://dx.doi.org/10.1143/PTP.46.401 10.1143/PTP.46.401].
 * Bethe, H. 1931. “Zur Theorie der Metalle.” Zeitschrift für Physik 71 (3-4) (March 1): 205–226. doi:[http://dx.doi.org/10.1007/BF01341708 10.1007/BF01341708]
 ==questions==
 * http://mathoverflow.net/questions/136594/roots-of-the-xxz-bethe-ansatz-equation

@@ 41번째 줄: / 41번째 줄: @@
 ==expositions==
 * Sato-Deguchi, [http://cfim11.sciencesconf.org/conference/cfim11/Dijon_JS.pdf Numerical analysis of string solutions of the integrable XXZ spin chains]
 * R.P. Vlijm [http://www.science.uva.nl/onderwijs/thesis/centraal/files/f1093904327.pdf Numerical solutions of the Bethe equations for the isotropic spin-1 chain]

← 이전 판		2014년 4월 8일 (화) 01:20 판
8번째 줄:		8번째 줄:
	$$		$$
	where $\lambda_{\alpha}$ is a real number and $\epsilon$ small deviation		where $\lambda_{\alpha}$ is a real number and $\epsilon$ small deviation
		+
		+
		+	==Bethe Ansatz equations==
		+	* recall
		+	$$
		+	\begin{eqnarray}\label{bae}
		+	\left( {\lambda_{j} - {i\over 2}
		+	\over \lambda_{j} + {i\over 2}} \right)^{L}
		+	= \prod_{\substack{
		+	l=1\\
		+	l\neq j}}^n
		+	{\lambda_{j} - \lambda_{l} - i
		+	\over
		+	\lambda_{j} - \lambda_{l} + i }
		+	\,, \qquad j = 1 \,, \cdots \,, n \,.
		+	\end{eqnarray}
		+	$$.
		+	* assume that $\Im{\lambda}>0$
		+	* fix $n$ and take $L\to \infty$. Then the LHS goes to $\infty$
		+	* this implies that there exists $l$ such that $\lambda_{j} - \lambda_{l} + i$ must be close to 0
		+	* this suggests the existence of strings

← 이전 판		2014년 4월 8일 (화) 00:55 판
2번째 줄:		2번째 줄:
	* [[Bethe ansatz]]		* [[Bethe ansatz]]
	* the roots in a string are all equally spaced in the imaginary direction		* the roots in a string are all equally spaced in the imaginary direction
		+	* a Bethe root consists of strings (as $L\to \infty$)
		+	* $n$-string
		+	$$
		+	\{\lambda_{\alpha}+\frac{i}{2}(n-1)+\epsilon_{\alpha,1},\lambda_{\alpha}+\frac{i}{2}(n-3)+\epsilon_{\alpha,2},\cdots, \lambda_{\alpha}-\frac{i}{2}(n-1)+\epsilon_{\alpha,n}\}
		+	$$
		+	where $\lambda_{\alpha}$ is a real number and $\epsilon$ small deviation