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| 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 | + | * a Bethe root consists of strings (as <math>L\to \infty</math>) |
| − | * | + | * <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}\} | \{\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 | + | where <math>\lambda_{\alpha}</math> is a real number and <math>\epsilon</math> small deviation |
==Bethe Ansatz equations== | ==Bethe Ansatz equations== | ||
* recall | * recall | ||
| − | + | :<math> | |
\begin{eqnarray}\label{bae} | \begin{eqnarray}\label{bae} | ||
\left( {\lambda_{j} - {i\over 2} | \left( {\lambda_{j} - {i\over 2} | ||
| 24번째 줄: | 24번째 줄: | ||
\,, \qquad j = 1 \,, \cdots \,, n \,. | \,, \qquad j = 1 \,, \cdots \,, n \,. | ||
\end{eqnarray} | \end{eqnarray} | ||
| − | + | </math>. | |
| − | * assume that | + | * assume that <math>\Im{\lambda}>0</math> |
| − | * fix | + | * fix <math>n</math> and take <math>L\to \infty</math>. Then the LHS goes to <math>\infty</math> |
| − | * this implies that there exists | + | * 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 | * this suggests the existence of strings | ||
| 47번째 줄: | 47번째 줄: | ||
==articles== | ==articles== | ||
| + | * Gainutdinov, A. M., Wenrui Hao, Rafael I. Nepomechie, and Andrew J. Sommese. ‘Counting Solutions of the Bethe Equations of the Quantum Group Invariant Open XXZ Chain at Roots of Unity’. arXiv:1505.02104 [cond-Mat, Physics:hep-Th, Physics:math-Ph], 8 May 2015. http://arxiv.org/abs/1505.02104. | ||
| + | * 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. | * 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. | * 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 | + | * 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 | + | * 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. | * 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. | ||
* 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]. | * 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]. | * 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]. | ||
| 59번째 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]] | ||
2020년 11월 16일 (월) 04:36 기준 최신판
introduction
- Bethe ansatz
- the roots in a string are all equally spaced in the imaginary direction
- a Bethe root consists of strings (as \(L\to \infty\))
- \(n\)-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
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
computational resource
expositions
- Nick Plantz, Bethe's Ansatz: coordinate Bethe Ansatz, Bethe-Ansatz equations
- Sato-Deguchi, Numerical analysis of string solutions of the integrable XXZ spin chains
- R.P. Vlijm Numerical solutions of the Bethe equations for the isotropic spin-1 chain
articles
- Gainutdinov, A. M., Wenrui Hao, Rafael I. Nepomechie, and Andrew J. Sommese. ‘Counting Solutions of the Bethe Equations of the Quantum Group Invariant Open XXZ Chain at Roots of Unity’. arXiv:1505.02104 [cond-Mat, Physics:hep-Th, Physics:math-Ph], 8 May 2015. http://arxiv.org/abs/1505.02104.
- 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.
- 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: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.
- 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.
- 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: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:10.1007/BF01087245.
- Takahashi, Minoru. 1971. “One-Dimensional Heisenberg Model at Finite Temperature.” Progress of Theoretical Physics 46 (2) (August 1): 401–415. doi:10.1143/PTP.46.401.
- Bethe, H. 1931. “Zur Theorie der Metalle.” Zeitschrift für Physik 71 (3-4) (March 1): 205–226. doi:10.1007/BF01341708