String hypothesis

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

related items

computational resource

https://docs.google.com/file/d/0B8XXo8Tve1cxWnJNbWE2bVF3UXM/edit

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

questions

http://mathoverflow.net/questions/136594/roots-of-the-xxz-bethe-ansatz-equation