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[1] 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 of the Bethe ansatz for a many-body wavefunction is

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

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.[2] The recently developed algebraic Bethe ansatz[3] led to essential progress, stating 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[4] in 1980 and independently by N. Andrei,[5] also in 1980) and the Anderson model (by P.B. Wiegmann[6] in 1981, and by N. Kawakami and A. Okiji[7] 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[8] and by C.J. Bolech and N. Andrei[9]). 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.

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${\displaystyle (N)}$).

Chronology

• 1928: Werner Heisenberg publishes his model.[10]
• 1930: Felix Bloch proposes an oversimplified Ansatz which miscounts the number of solutions to the Schrödinger equation for the Heisenberg chain.[11]
• 1931: Hans Bethe proposes the correct Ansatz and carefully shows that it yields the correct number of eigenfunctions.[1]
• 1938: Lamek Hulthén (de) obtains the exact ground-state energy of the Heisenberg model.[12]
• 1958: Raymond Lee Orbach uses the Bethe Ansatz to solve the Heisenberg model with anisotropic interactions.[13]
• 1962: J. des Cloizeaux and J. J. Pearson obtain the correct spectrum of the Heisenberg antiferromagnet (spinon dispersion relation),[14] showing that it differs from Anderson’s spin-wave theory predictions[15] (the constant prefactor is different).
• 1963: Elliott H. Lieb and Werner Liniger provide the exact solution of the 1d δ-function interacting Bose gas[16] (now known as the Lieb-Liniger model). Lieb studies the specturm and defines two basic types of excitations.[17]
• 1964: Robert B. Griffiths obtains the magnetization curve of the Heisenberg model at zero temperature.[18]
• 1966: C.N. Yang and C.P. Yang rigorously prove that the ground-state of the Heisenberg chain is given by the Bethe Ansatz.[19] They study properties and applications in[20] and.[21]
• 1967: C.N. Yang generalizes Lieb and Liniger's solution of the δ-function interacting Bose gas to arbitrary permutation symmetry of the wavefunction, giving birth to the nested Bethe Ansatz.[22]
• 1968 Elliott H. Lieb and F. Y. Wu solve the 1d Hubbard model.[23]
• 1969: C.N. Yang and C.P. Yang obtain the thermodynamics of the Lieb-Liniger model,[24] providing the basis of the Thermodynamic Bethe Ansatz (TBA).

References

1. Bethe, H. (March 1931). "Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette". Zeitschrift für Physik. 71 (3–4): 205–226. doi:10.1007/BF01341708. S2CID 124225487.
2. Korepin, Vladimir E. (1982). "Calculation of norms of Bethe wave functions". Communications in Mathematical Physics. 86 (3): 391–418. Bibcode:1982CMaPh..86..391K. doi:10.1007/BF01212176. ISSN 0010-3616. S2CID 122250890.
3. Korepin, V. E.; Bogoliubov, N. M.; Izergin, A. G. (1997-03-06). Quantum Inverse Scattering Method and Correlation Functions. Cambridge University Press. ISBN 9780521586467.
4. Wiegmann, P.B. (1980). "Exact solution of s-d exchange model at T = 0" (PDF). JETP Letters. 31 (7): 364.
5. Andrei, N. (1980). "Diagonalization of the Kondo Hamiltonian". Physical Review Letters. 45 (5): 379–382. Bibcode:1980PhRvL..45..379A. doi:10.1103/PhysRevLett.45.379. ISSN 0031-9007.
6. Wiegmann, P.B. (1980). "Towards an exact solution of the Anderson model". Physics Letters A. 80 (2–3): 163–167. Bibcode:1980PhLA...80..163W. doi:10.1016/0375-9601(80)90212-1. ISSN 0375-9601.
7. Kawakami, Norio; Okiji, Ayao (1981). "Exact expression of the ground-state energy for the symmetric anderson model". Physics Letters A. 86 (9): 483–486. Bibcode:1981PhLA...86..483K. doi:10.1016/0375-9601(81)90663-0. ISSN 0375-9601.
8. Andrei, N.; Destri, C. (1984). "Solution of the Multichannel Kondo Problem". Physical Review Letters. 52 (5): 364–367. Bibcode:1984PhRvL..52..364A. doi:10.1103/PhysRevLett.52.364. ISSN 0031-9007.
9. 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): 237206. arXiv:cond-mat/0204392. Bibcode:2002PhRvL..88w7206B. doi:10.1103/PhysRevLett.88.237206. ISSN 0031-9007. PMID 12059396. S2CID 15180985.
10. Heisenberg, W. (September 1928). "Zur Theorie des Ferromagnetismus". Zeitschrift für Physik. 49 (9–10): 619–636. Bibcode:1928ZPhy...49..619H. doi:10.1007/BF01328601. S2CID 122524239.
11. Bloch, F. (March 1930). "Zur Theorie des Ferromagnetismus". Zeitschrift für Physik. 61 (3–4): 206–219. Bibcode:1930ZPhy...61..206B. doi:10.1007/BF01339661. S2CID 120459635.
12. Hulthén, Lamek (1938). "Über das Austauschproblem eines Kristalles". Arkiv Mat. Astron. Fysik. 26A: 1.
13. Orbach, R. (15 October 1958). "Linear Antiferromagnetic Chain with Anisotropic Coupling". Physical Review. 112 (2): 309–316. Bibcode:1958PhRv..112..309O. doi:10.1103/PhysRev.112.309.
14. des Cloizeaux, Jacques; Pearson, J. J. (1 December 1962). "Spin-Wave Spectrum of the Antiferromagnetic Linear Chain". Physical Review. 128 (5): 2131–2135. Bibcode:1962PhRv..128.2131D. doi:10.1103/PhysRev.128.2131.
15. Anderson, P. W. (1 June 1952). "An Approximate Quantum Theory of the Antiferromagnetic Ground State". Physical Review. 86 (5): 694–701. Bibcode:1952PhRv...86..694A. doi:10.1103/PhysRev.86.694.
16. Lieb, Elliott H.; Liniger, Werner (15 May 1963). "Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State". Physical Review. 130 (4): 1605–1616. Bibcode:1963PhRv..130.1605L. doi:10.1103/PhysRev.130.1605.
17. Lieb, Elliott H. (15 May 1963). "Exact Analysis of an Interacting Bose Gas. II. The Excitation Spectrum". Physical Review. 130 (4): 1616–1624. Bibcode:1963PhRv..130.1616L. doi:10.1103/PhysRev.130.1616.
18. Griffiths, Robert B. (3 February 1964). "Magnetization Curve at Zero Temperature for the Antiferromagnetic Heisenberg Linear Chain". Physical Review. 133 (3A): A768–A775. Bibcode:1964PhRv..133..768G. doi:10.1103/PhysRev.133.A768.
19. Yang, C. N.; Yang, C. P. (7 October 1966). "One-Dimensional Chain of Anisotropic Spin-Spin Interactions. I. Proof of Bethe's Hypothesis for Ground State in a Finite System". Physical Review. 150 (1): 321–327. Bibcode:1966PhRv..150..321Y. doi:10.1103/PhysRev.150.321.
20. Yang, C. N.; Yang, C. P. (7 October 1966). "One-Dimensional Chain of Anisotropic Spin-Spin Interactions. II. Properties of the Ground-State Energy Per Lattice Site for an Infinite System". Physical Review. 150 (1): 327–339. Bibcode:1966PhRv..150..327Y. doi:10.1103/PhysRev.150.327.
21. Yang, C. N.; Yang, C. P. (4 November 1966). "One-Dimensional Chain of Anisotropic Spin-Spin Interactions. III. Applications". Physical Review. 151 (1): 258–264. Bibcode:1966PhRv..151..258Y. doi:10.1103/PhysRev.151.258.
22. Yang, C. N. (4 December 1967). "Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction". Physical Review Letters. 19 (23): 1312–1315. Bibcode:1967PhRvL..19.1312Y. doi:10.1103/PhysRevLett.19.1312.
23. Lieb, Elliott H.; Wu, F. Y. (17 June 1968). "Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension". Physical Review Letters. 20 (25): 1445–1448. Bibcode:1968PhRvL..20.1445L. doi:10.1103/PhysRevLett.20.1445.
24. Yang, C. N.; Yang, C. P. (July 1969). "Thermodynamics of a One‐Dimensional System of Bosons with Repulsive Delta‐Function Interaction". Journal of Mathematical Physics. 10 (7): 1115–1122. Bibcode:1969JMP....10.1115Y. doi:10.1063/1.1664947.

External links

• Introduction to the Bethe Ansatz