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

• 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.
• 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