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

• Fermionic formula and X=M=N conjecture
• Rigged configurations

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