2017/07/09 by R. S. Vieira, Vieira, R. S., A. Lima-Santos +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum and electron transport phenomena
paper · pdf · doi:10.48550/arxiv.1707.02584
openalex publication_date 2017/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We present in this paper a comprehensive introduction to the algebraic Bethe Ansatz, taking as examples the six-vertex model with periodic and non-periodic boundary conditions. We propose a diagrammatic representation of the commutation relations used in the algebraic Bethe Ansatz, so that the action of the transfer matrix in the nth excited state gives place to labeled combinatorial trees. The analysis of these combinatorial trees provides in a straightforward way the eigenvalues and eigenstates of the transfer matrix, as well as the respective Bethe Ansatz equations. Several identities between the R-matrix elements can also be derived from the symmetry of these diagrams regarding the permutation of their labels. This combinatorial approach gives some insights about how the algebraic Bethe Ansatz works, which can be valuable for non-experts readers.