Classification of parameter-dependent quantum integrable models, their parameterization, exact solution and other properties
H. Owusu, E. Yuzbashyan
Jun 9, 2011
Citations
19
Citations
Journal
Journal of Physics A: Mathematical and Theoretical
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Key Takeaway Key Takeaway: This paper presents a parameterization for type M quantum integrable models, enabling exact solutions for their eigenvalues and eigenvectors, and explores the taxonomy of types in the 1D Hubbard model.
Abstract
We study general quantum integrable Hamiltonians linear in a coupling constant and represented by finite N × N real symmetric matrices. The restriction on the coupling dependence leads to a natural notion of nontrivial integrals of motion and classification of integrable families into types according to the number of such integrals. A type M family in our definition is formed by N–M nontrivial mutually commuting operators linear in the coupling. Working from this definition alone, we parameterize type M operators, i.e. resolve the commutation relations, and obtain an exact solution for their eigenvalues and eigenvectors. We show that our parameterization covers all type 1, 2 and 3 integrable models and discuss the extent to which it is complete for other types. We also present robust numerical observation on the number of energy-level crossings in type M integrable systems and analyze the taxonomy of types in the 1D Hubbard model.