Abstract: New 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.Abstract: New 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.

Jan 5, 20110 citations

SOLUTION FOR THE LMG MODEL HAMILTONIAN BASED ON THE RECURRENCE FORMULA AND JACOBI MATRIX

Z. ALIPOUR, M. A. Jafarizadeh, N. FOOLADI +1 more

Based on the recursion relation formulas and the Jacobi matrix from the theory of orthogonal polynomials, a numeric solution for the Lipkin–Meshkov–Glick (LMG) model Hamiltonian is given, without considering any restriction on the parameters specifying the strengths of the interactions included in the LMG model. Moreover, via this method, we have studied the shape phase transitions for one example of nuclei. At the end we compare the results thus obtained with those of Bethe ansatz and Hartree–Fock methods.Based on the recursion relation formulas and the Jacobi matrix from the theory of orthogonal polynomials, a numeric solution for the Lipkin–Meshkov–Glick (LMG) model Hamiltonian is given, without considering any restriction on the parameters specifying the strengths of the interactions included in the LMG model. Moreover, via this method, we have studied the shape phase transitions for one example of nuclei. At the end we compare the results thus obtained with those of Bethe ansatz and Hartree–Fock methods.

International Journal of Modern Physics EApr 5, 20112 citations

Solutions to the Bethe Ansatz equations in the XXX Heisenberg model

K. Isler, Manu B. Paranjape

Jan 1, 19930 citations

Benchmarking near-term quantum devices with the variational quantum eigensolver and the Lipkin-Meshkov-Glick model

K. Robbins, Peter J. Love

The variational quantum eigensolver is a promising algorithm for noisy intermediate scale quantum (NISQ) computation. Verification and validation of NISQ algorithms' performance on NISQ devices is an important task. We consider the exactly diagonalizable Lipkin-Meshkov-Glick (LMG) model as a candidate for benchmarking NISQ computers. We use the Bethe Ansatz to construct eigenstates of the trigonometric LMG model using quantum circuits inspired by the LMG's underlying algebraic structure. We construct circuits with depth $O(N)$ and $O({log}_{2}N)$ that can prepare any trigonometric LMG eigenstate of $N$ particles. The number of gates required for both circuits is $O(N)$ . The energies of the eigenstates can then be measured and compared to the exactly known answers.The variational quantum eigensolver is a promising algorithm for noisy intermediate scale quantum (NISQ) computation. Verification and validation of NISQ algorithms' performance on NISQ devices is an important task. We consider the exactly diagonalizable Lipkin-Meshkov-Glick (LMG) model as a candidate for benchmarking NISQ computers. We use the Bethe Ansatz to construct eigenstates of the trigonometric LMG model using quantum circuits inspired by the LMG's underlying algebraic structure. We construct circuits with depth $O(N)$ and $O({log}_{2}N)$ that can prepare any trigonometric LMG eigenstate of $N$ particles. The number of gates required for both circuits is $O(N)$. The energies of the eigenstates can then be measured and compared to the exactly known answers.

arXiv:2105.06761Physical Review AAug 16, 202111 citations

Analysis of the Bethe ansatz equations of the XXZ model

O. Babelon, Hector J. de Vega, C.-M. Viallet

Nuclear Physics BMay 1, 1983137 citations

O(N)

and

O({log}_{2}N)

that can prepare any trigonometric LMG eigenstate of

N

particles. The number of gates required for both circuits is

O(N)

. The energies of the eigenstates can then be measured and compared to the exactly known answers.