The extended Heine–Stieltjes polynomials associated with a special LMG model

Abstract

The extended Heine-Stieltjes polynomials associated with a special Lipkin-Meshkov-Glick (LMG) model corresponding to the standard two-site Bose-Hubbard model are derived based on the Stieltjes correspondence. It is shown that there is a one-to-one correspondence between zeros of this new polynomial and solutions of the Bethe ansatz equations for the LMG model. A one-dimensional classical electrostatic analogue corresponding to the special LMG model is established according to Stieltjes early work. It shows that any possible configuration of equilibrium positions of the charges in the electrostatic problem corresponds uniquely to one set of roots of the Bethe ansatz equations for the LMG model, and the number of possible configurations of equilibrium positions of the charges equals exactly to the number of energy levels in the LMG model. Some relations of sums of powers and inverse powers of zeros of the new polynomials related to the eigenenergies of the LMG model are derived.

Exact solution of the two-axis countertwisting Hamiltonian

“…It can be inferred from (22) that the spectrum of the model after the Jordan-Schwinger two-boson realization is generated from the non-linear boson-quartet excitations based on the single-boson and the boson-pairing excitations, where the single-boson excitation affects both the scaling of the energy and the boson-quartet excitations, while the boson-pairing excitation energies contribute to the total energy linearly. Moreover, as shown previously [32][33][34], though the eigenstates provided in (14) are not normalized, they are always orthogonal with…”

“…k } with ζ = 1, 2, · · · , k + 1, corresponding to global minimums of the total electrostatic energy of the system [32]. It follows from this that the total number of these configurations is exactly the number of ways to put the k zeros into the two open intervals, which is k + 1.…”

“…Similar to what was shown in [35], a Wolfram Mathematica package according to (32)- (35) is compiled, which is very efficient even when J is a large number due to the fact that to generate and diagonalize a tridiagonal matrix are easier and more CPU time saving than other more complicated sparse matrices. When J ≤ 1, the solutions are trivial with k = 0, of which the eigen-energies are simply given by…”

Bethe ansatz and ordinary differential equation correspondence for degenerate Gaudin models

“…In order to solve the sets of quadratic equations defined by (8) we use a combination of Taylor expansion to generate an approximative solution at g + δg and Newton's method to refine this approximation. For both methods we need to solve a linear system of equations defined by the block-N × N −matrix…”

Exact solution of the mean-field plus separable pairing model reexamined

“…It has been shown that either spherical or deformed mean-field plus the standard (orbit-independent) pairing interaction can be solved exactly by using the Gaudin-Richardson method [3][4][5]. The Gaudin-Richardson equations in this case can be solved more easily by using the extended Heine-Stieltjes polynomial approach [6][7][8][9]. The deformed and spherical mean-field plus the extended pairing models have also been proposed, which can be solved more easily than the standard pairing model, especially when both the number of valence nucleon pairs and the number of single-particle orbits are large [10,11].…”