A quantum integrable spin chain model associated with the G 2 exceptional Lie algebra is studied. By using the fusion technique, the closed recursive relations among the fused transfer matrices are obtained. These identities allow us to derive the exact energy spectrum and Bethe ansatz equations of the system based on polynomial analysis. The present method provides a unified treatment to investigate the Bethe ansatz solutions for both the periodic and the non-diagonal open boundary conditions associated with exceptional Lie algebras.A quantum integrable spin chain model associated with the G 2 exceptional Lie algebra is studied. By using the fusion technique, the closed recursive relations among the fused transfer matrices are obtained. These identities allow us to derive the exact energy spectrum and Bethe ansatz equations of the system based on polynomial analysis. The present method provides a unified treatment to investigate the Bethe ansatz solutions for both the periodic and the non-diagonal open boundary conditions associated with exceptional Lie algebras.

arXiv:2408.04201Nuclear Physics BDec 13, 20243 citations

The Bethe Ansatz Equation

Chapman & Hall/CRC research notes in mathematics seriesJan 28, 20040 citations

Explicit Solutions of the Bethe Ansatz Equations for Bloch Electrons in a Magnetic Field

Yasuhiro Hatsugai, Mahito Kohmoto, Yong-Shi Wu

For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is $\frac{1}{Q}$ with $Q$ an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit $Q\ensuremath{\rightarrow}\ensuremath{\infty}$, the wave function is ${|\ensuremath{\psi}(x)|}^{2}=(\frac{2}{sin \ensuremath{\pi}x})$, which is critical and unnormalizable. For the golden-mean flux, the distribution of roots has exact self-similarity and the distribution function is nowhere differentiable. The corresponding wave function also shows a clear self-similar structure.For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is $\frac{1}{Q}$ with $Q$ an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit $Q\ensuremath{\rightarrow}\ensuremath{\infty}$, the wave function is ${|\ensuremath{\psi}(x)|}^{2}=(\frac{2}{sin \ensuremath{\pi}x})$, which is critical and unnormalizable. For the golden-mean flux, the distribution of roots has exact self-similarity and the distribution function is nowhere differentiable. The corresponding wave function also shows a clear self-similar structure.

arXiv:cond-mat/9405028Physical Review LettersAug 22, 199444 citations

COMPLETENESS OF BETHE ANSATZ SOLUTIONS

E. KOZAK, R. NIEMIEC, D. GOŁOJUCH +1 more

International School on Symmetry and Structural Properties of Condensed Matter (SSPCM)May 1, 20010 citations

Universal Bethe ansatz solution for the Temperley–Lieb spin chain

Rafael I. Nepomechie, Rodrigo A. Pimenta

We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin- s representation of quantum-deformed s l ( 2 ) . We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.We consider the Temperley–Lieb (TL) open quantum spin chain with “free” boundary conditions associated with the spin- s representation of quantum-deformed s l ( 2 ) . We construct the transfer matrix, and determine its eigenvalues and the corresponding Bethe equations using analytical Bethe ansatz. We show that the transfer matrix has quantum group symmetry, and we propose explicit formulas for the number of solutions of the Bethe equations and the degeneracies of the transfer-matrix eigenvalues. We propose an algebraic Bethe ansatz construction of the off-shell Bethe states, and we conjecture that the on-shell Bethe states are highest-weight states of the quantum group. We also propose a determinant formula for the scalar product between an off-shell Bethe state and its on-shell dual, as well as for the square of the norm. We find that all of these results, except for the degeneracies and a constant factor in the scalar product, are universal in the sense that they do not depend on the value of the spin. In an appendix, we briefly consider the closed TL spin chain with periodic boundary conditions, and show how a previously-proposed solution can be improved so as to obtain the complete (albeit non-universal) spectrum.

arXiv:1601.04378Nuclear Physics BMay 4, 201610 citations

For Bloch electrons in a magnetic field, explicit solutions are obtained at the center of the spectrum for the Bethe ansatz equations of Wiegmann and Zabrodin. When the magnetic flux per plaquette is

\frac{1}{Q}

with

Q

an odd integer, distribution of the roots of the Bethe ansatz equation is uniform except at two points on the unit circle in the complex plane. For the semiclassical limit $Q\ensuremath{\rightarrow}\ensuremath{\infty}$, the wave function is ${|\ensuremath{\psi}(x)|}^{2}=(\frac{2}{sin \ensuremath{\pi}x})$, which is critical and unnormalizable. For the golden-mean flux, the distribution of roots has exact self-similarity and the distribution function is nowhere differentiable. The corresponding wave function also shows a clear self-similar structure.