# Bethe ansatz

In physics, the Bethe ansatz is an ansatz method for finding the exact wavefunctions of certain one-dimensional quantum many-body models. 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: the (anisotropic) Heisenberg chain (XXZ model), the Lieb-Liniger interacting Bose gas, the Hubbard model, the Kondo model, the Anderson impurity model, the Richardson model etc.

## Discussion

In the framework of many-body quantum mechanics, models solvable by the Bethe ansatz can be contrasted with 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 collisions 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 generic form the Bethe ansatz for a many-body wavefunction is

${\displaystyle \Psi _{M}(j_{1},\cdots ,j_{M})=\prod _{M\geq a>b\geq 1}sgn(j_{a}-j_{b})\sum _{P\in P_{M}}(-1)^{[P]}e^{i\sum _{a=1}^{M}k_{P_{a}}j_{a}+{\frac {i}{2}}\sum _{M\geq a>b\geq 1}sgn(j_{a}-j_{b})\phi (k_{P_{a}},k_{P_{b}})}}$

in which ${\displaystyle M}$ is the number of particles, ${\displaystyle j_{a},a=1,\cdots M}$ their position, ${\displaystyle P_{M}}$ is the set of all permutations of the integers ${\displaystyle 1,\cdots ,M}$, ${\displaystyle k_{a}}$ is the (quasi-)momentum of the ${\displaystyle a}$-th particle, ${\displaystyle \phi }$ is the scattering phase shift function and ${\displaystyle sgn}$ is the sign function. This form is universal (at least for non-nested systems), with the momentum and scattering functions being model-dependent.

The Yang–Baxter equation guarantees consistency of the construction. 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 recently[when?] developed algebraic Bethe ansatz[2] led to essential progress, stating[who?] that

The quantum inverse scattering method ... a well-developed method ... has allowed a wide class of nonlinear evolution equations to be solved. It explains the algebraic nature of the Bethe ansatz.

The exact solutions of the so-called s-d model (by P.B. Wiegmann[3] in 1980 and independently by N. Andrei,[4] also in 1980) and the Anderson model (by P.B. Wiegmann[5] in 1981, and by N. Kawakami and A. Okiji[6] in 1981) are also both based on the Bethe ansatz. There exist multi-channel generalizations of these two models also amenable to exact solutions (by N. Andrei and C. Destri[7] and by C.J. Bolech and N. Andrei[8]). Recently several models solvable by Bethe ansatz were realized experimentally in solid states and optical lattices. An important role in the theoretical description of these experiments was played by Jean-Sébastien Caux and Alexei Tsvelik.[citation needed]

## Example: the Heisenberg antiferromagnetic chain

The Heisenberg antiferromagnetic chain is defined by the Hamiltonian (assuming periodic boundary conditions)

${\displaystyle H=J\sum _{j=1}^{N}{\boldsymbol {S}}_{j}\cdot {\boldsymbol {S}}_{j+1},\qquad {\boldsymbol {S}}_{j+N}\equiv {\boldsymbol {S}}_{j}.}$

This model is solvable using Bethe ansatz. The scattering phase shift function is ${\displaystyle \phi (k_{a}(\lambda _{a}),k_{b}(\lambda _{b}))=\theta _{2}(\lambda _{a}-\lambda _{b})}$, with ${\displaystyle \theta _{n}(\lambda )\equiv 2\arctan {\frac {2\lambda }{n}}}$ in which the momentum has been conveniently reparametrized as ${\displaystyle k(\lambda )=\pi -2\arctan 2\lambda }$ in terms of the rapidity ${\displaystyle \lambda }$. The (here, periodic) boundary conditions impose the Bethe equations

${\displaystyle \left[{\frac {\lambda _{a}+i/2}{\lambda _{a}-i/2}}\right]^{N}=\prod _{b\neq a}^{M}{\frac {\lambda _{a}-\lambda _{b}+i}{\lambda _{a}-\lambda _{b}-i}},\qquad a=1,...,M}$

or more conveniently in logarithmic form

${\displaystyle \theta _{1}(\lambda _{a})-{\frac {1}{N}}\sum _{b=1}^{M}\theta _{2}(\lambda _{a}-\lambda _{b})=2\pi {\frac {I_{a}}{N}}}$

where the quantum numbers ${\displaystyle I_{j}}$ are distinct half-odd integers for ${\displaystyle N-M}$ even, integers for ${\displaystyle N-M}$ odd (with ${\displaystyle I_{j}}$ defined mod${\displaystyle (N)}$).

## Notes

1. ^ Korepin, Vladimir E. (1982). "Calculation of norms of Bethe wave functions". Communications in Mathematical Physics. 86 (3): 391–418. doi:10.1007/BF01212176. ISSN 0010-3616.
2. ^ Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1997-03-06). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press. ISBN 9780521586467.
3. ^ Wiegmann, P.B. (1980). "Exact solution of s-d exchange model at T = 0" (PDF). JETP Letters. 31 (7): 364.
4. ^ Andrei, N. (1980). "Diagonalization of the Kondo Hamiltonian". Physical Review Letters. 45 (5): 379–382. doi:10.1103/PhysRevLett.45.379. ISSN 0031-9007.
5. ^ Wiegmann, P.B. (1980). "Towards an exact solution of the Anderson model". Physics Letters A. 80 (2–3): 163–167. doi:10.1016/0375-9601(80)90212-1. ISSN 0375-9601.
6. ^ Kawakami, Norio; Okiji, Ayao (1981). "Exact expression of the ground-state energy for the symmetric anderson model". Physics Letters A. 86 (9): 483–486. doi:10.1016/0375-9601(81)90663-0. ISSN 0375-9601.
7. ^ Andrei, N.; Destri, C. (1984). "Solution of the Multichannel Kondo Problem". Physical Review Letters. 52 (5): 364–367. doi:10.1103/PhysRevLett.52.364. ISSN 0031-9007.
8. ^ Bolech, C. J.; Andrei, N. (2002). "Solution of the Two-Channel Anderson Impurity Model: Implications for the Heavy Fermion UBe13". Physical Review Letters. 88 (23). arXiv:cond-mat/0204392. doi:10.1103/PhysRevLett.88.237206. ISSN 0031-9007.

## References

• H. Bethe (1931). "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette". (On the theory of metals. I. Eigenvalues and eigenfunctions of the linear atom chain), Zeitschrift für Physik, 71:205–226 (1931). SpringerLink.