Phy5670/Bethe Ansatz for many particle systems: Difference between revisions
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==Introduction== | ==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] | |||
In the framework of many-body quantum mechanics, 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. | |||
The Yang-Baxter equation guarantees the consistency. Experts conjecture that each universality class in one dimension contains at least one model solvable by the Bethe ansatz. The Pauli exclusion principle is valid for models solvable by the Bethe ansatz, even for models of interacting bosons. | |||
The ground state is a Fermi sphere. Periodic boundary conditions lead to the Bethe ansatz equations. In logarithmic form the Bethe ansatz equations can be generated by the Yang action. The square of the norm of Bethe wave function is equal to the determinant of the matrix of second derivatives of the Yang action [1]. | |||
The exact solutions of the so-called 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. | |||
==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). | |||
Revision as of 16:30, 8 December 2010
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]
In the framework of many-body quantum mechanics, 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.
The Yang-Baxter equation guarantees the consistency. Experts conjecture that each universality class in one dimension contains at least one model solvable by the Bethe ansatz. The Pauli exclusion principle is valid for models solvable by the Bethe ansatz, even for models of interacting bosons.
The ground state is a Fermi sphere. Periodic boundary conditions lead to the Bethe ansatz equations. In logarithmic form the Bethe ansatz equations can be generated by the Yang action. The square of the norm of Bethe wave function is equal to the determinant of the matrix of second derivatives of the Yang action [1].
The exact solutions of the so-called 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.
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).