Abstract
We consider quantum integrable systems associated with reductive Lie algebra gl(n) and Cartan-invariant non-skew-symmetric classical r-matrices. We show that under certain restrictions on the form of classical r-matrices “nested” or “hierarchical” Bethe ansatz usually based on a chain of subalgebras gl(n)⊃gl(n−1)⊃...⊃gl(1) is generalized onto the other chains or “hierarchies” of subalgebras. We show that among the r-matrices satisfying such the restrictions there are “twisted” or Zp-graded non-skew-symmetric classical r-matrices. We consider in detail example of the generalized Gaudin models with and without external magnetic field associated with Zp-graded non-skew-symmetric classical r-matrices and find the spectrum of the corresponding Gaudin-type hamiltonians using nested Bethe ansatz scheme and a chain of subalgebras gl(n)⊃gl(n−n1)⊃gl(n−n1−n2)⊃gl(n−(n1+...+np−1)), where n1+n2+...+np=n.
Highlights
Quantum integrable models with long-range interaction play important role in the nonperturbative physics
In the papers [12,13] we have proposed a generalization of classical and quantum Gaudin models with [13] and without [12] external magnetic field associated with arbitrary non-skew-symmetric g ⊗ g-valued non-dynamical classical r-matrices with spectral parameters that satisfy the so-called generalized or “permuted” classical Yang–Baxter equation [29]
The proof is achieved by the direct calculation, using the formulas (36), taking into account that in the case of the generalized Gaudin models in an external magnetic field kk(u) = N λ(km)rkk(νm, u) + ckk(u) and taking the residue in the point u = νm in the formula (36). 2 m=1
Summary
Quantum integrable models with long-range interaction play important role in the nonperturbative physics. The main example of integrable models with long-range interaction are the famous Gaudin spin chains [11] associated with simple (reductive) Lie algebras g and skew-symmetric g ⊗ g-valued classical r-matrices with spectral parameters. In our previous paper [23] we have considered quantum integrable systems associated with the Lie algebra gl(n) and Cartan-invariant non-skew-symmetric classical 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) ⊃ ... In the case when n = 4, n1 = n2 = 2 such Gaudin-type model produce integrable “p + ip” proton–neutron model [18], whose exact solution is not provided by standard nested Bethe ansatz scheme based on the chain gl(4) ⊃ gl(3) ⊃ gl(2) ⊃ gl(1). In the fourth section we consider example of Zp-graded classical r-matrix and the corresponding Gaudin models with and without external magnetic field
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