Phy5670/Bethe Ansatz for many particle systems

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Introduction

Bethe ansatz is a very powerful method for finding the exact solutions of one-dimensional quantum many-body systems. Experts conjecture that each universality class in one dimension contains at least one model solvable by the Bethe ansatz. It was invented by Hans Bethe in 1931 to find the exact eigenvalues and eigenvectors of the one-dimensional antiferromagnetic Heisenberg model Hamiltonian. Since then the method has been extended to other models in one dimension: Bose gas, Hubbard model, etc. The exact solutions of the s-d model (by P.B. Wiegmann [2] in 1980 and independently by N. Andrei,[3] also in 1980) and the Anderson model (by P.B. Wiegmann [4] in 1981, and by N. Kawakami an A. Okiji [5] in 1981) are also both based on the Bethe ansatz. The method is also applicable to some of the 2-dimensional classical systems as the transfer matrices of 2D classical system sometimes have common eigenfunctions as some 1-dimensional quantum systems. Some examples of the application of Bethe ansatz in 2D system are stated in Ref[6,7,8].

Models solvable by the Bethe ansatz can be compared to free fermion models. One can say that the dynamics of a free model is one-body reducible: the many-body wave function for fermions (bosons) is the anti-symmetrized (symmetrized) product of one-body wave functions. Models solvable by the Bethe ansatz are not free: the two-body sector has a non-trivial scattering matrix, which in general depends on the momenta. On the other hand the dynamics of the models solvable by the Bethe ansatz is two-body reducible: the many-body scattering matrix is a product of two-body scattering matrices. Many-body collision happen as a sequence of two-body collisions and the many-body wave function can be represented in a form which contains only elements from two-body wave functions. The many-body scattering matrix is equal to the product of pairwise scattering matrices.

In using this method for N-body problem, the N-body wavefunction is represented as a linear combination of N! plane waves with N qusai-momenta. The energy eigenvalue of the lowest energy state in the thermodynamics limit is reduced to a distribution function of the quasi-momenta. The distribution function must satisfy a linear integral equation. The energy per unit length is obtained by solving this integral equation. Elementary excitations from the ground state are expressed by the deviation of the distribution of quasi-momenta from the equilibrium. For general eigenstates, the quasi-momenta are complex numbers, but in many cases they are real number. The total energy is represented by the distribution functions of quasi-momenta.



Reference

1. http://en.wikipedia.org/wiki/Bethe_ansatz.

2. P.B. Wiegmann, Soviet Phys. JETP Lett., 31, 392 (1980).

3. N. Andrei, Phys. Rev. Lett., 45, 379 (1980).

4. P.B. Wiegmann, Phys. Lett. A 80, 163 (1981).

5. N. Kawakami, and A. Okiji, Phys. Lett. A 86, 483 (1981).

6. Mehran Kardar, Nuclear Physics B, Volume 290, 1987, Pages 582-602.

7. S. Park and K. Moon, Solid State Communications, Volume 132, Issue 12, December 2004, Pages 851-856.

8. M.J. Martins, Phys. Rev. E 59, 7220–7223 (1999).

9. http://de.wikipedia.org/wiki/Bethe-Ansatz.

10. Minoru Takahashi, Thermodynamics of one-dimensional solvable models, 1999.

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