Abstract
We consider quantum integrable systems associated with the Lie algebra gl(n) and Cartan-invariant non-dynamical non-skew-symmetric classical r-matrices. We describe the sub-class of Cartan-invariant non-skew-symmetric r-matrices for which exists the standard procedure of the nested Bethe ansatz associated with the chain of embeddings gl(n)⊃gl(n−1)⊃gl(n−2)⊃⋯⊃gl(1). We diagonalize the corresponding quantum integrable systems by its means. We illustrate the obtained results by the examples of the generalized Gaudin systems with and without external magnetic field associated with three classes of non-dynamical non-skew-symmetric classical r-matrices.
Highlights
Quantum integrable spin models with long-range interaction play important role in the nonperturbative physics
In our previous papers [13,14] we have proposed a generalization of classical and quantum Gaudin models with [14] and without [13] external magnetic field associated with arbitrary nonskew-symmetric g ⊗ g-valued non-dynamical classical r-matrices with spectral parameters that satisfy the so-called generalized or “permuted” classical Yang–Baxter equation
In the present paper we have considered quantum integrable systems associated with the Lie algebra gl(n) and Cartan-invariant non-skew-symmetric classical r-matrices
Summary
Quantum integrable spin models with long-range interaction play important role in the nonperturbative physics. We have shown that the corresponding models are applied in order to construct new integrable fermion models of reduced BCS-type [15] and integrable proton–neutron Richardson’s-type models of nuclear physics [16] This makes a study of the generalized Gaudin models associated with non-skew-symmetric classical r-matrices important from the mathematical and from the physical point of view. The structure of the present paper is the following: in the second section we remind general facts about quantum integrable systems associated with non-skew-symmetric classical r-matrices, in the third section we consider nested Bethe ansatz (for such the r-matrices) and prove the main theorem. At last in the fourth section we present three classes of examples of the classical r-matrices for which our construction is applicable
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