The Bethe Ansatz

Chronology of exactly solvable models i.c

  • 1928: Heisenberg publishes his model 1928.Heisenberg.ZP.49.
  • 1930: Felix Bloch proposes an Ansatz for the wavefunctions 1930.Bloch.ZP.61: combinations of free waves. He simplifies the relative amplitudes too much. Finds too many solutions. Had it been done correctly, we'd today talk about the Bloch Ansatz.
  • 1931: Hans Bethe obtains the correct form of the wavefunctions 1931.Bethe.ZP.71. Spends lots of effort showing that the counting of solutions gives the correct number of eigenstates. End: proposes to extend method to higher dimensions (in title of this paper: a very revealing \(1\)!)
  • 1938: Hulthén obtains the ground-state energy of the Heisenberg model 1938.Hulthen.AMAF.26A.
  • 1958: Orbach generalizes the solution to the Heisenberg model with anisotropic interactions (\(XXZ\) case) 1958.Orbach.PR.112.
  • 1962: des Cloizeaux and Pearson obtain the correct spectrum of the Heisenberg antiferromagnet (spinon dispersion relation) 1962.desCloizeaux.PR.128, showing that it differs from Anderson's spin-wave theory predictions 1952.Anderson.PR.86 (the constant prefactor is different).
  • 1963: Lieb and Liniger provide the exact solution of the 1d \(\delta\)-function interacting Bose gas 1963.Lieb.PR.130.1. Lieb studies the specturm and defines two basic types of excitations 1963.Lieb.PR.130.2.
  • 1964: Griffiths obtains the magnetization curve of the Heisenberg model at \(T = 0\) 1964.Griffiths.PR.133.
  • 1966: Yang and Yang prove Bethe's hypothesis for the ground-state of the Heisenberg chain 1966.Yang.PR.150.1. They study properties and applications in 1966.Yang.PR.150.2 and 1966.Yang.PR.151.
  • 1967: Yang generalizes Lieb and Liniger's solution of the \(\delta\)-function interacting Bose gas to arbitrary permutation symmetry of the wavefunction 1967.Yang.PRL.19.
  • 1968: Lieb and Wu solve the 1d Hubbard model 1968.Lieb.PRL.20.
  • 1969: Yang and Yang obtain the thermodynamics of the Lieb-Liniger model 1969.Yang.JMP.10, providing the basis of the Thermodynamics Bethe Ansatz (TBA).
  • 1971 (9 April): Gaudin obtains coupled nonlinear equations for the thermodynamics of the (anisotropic) Heisenberg chain (\(\Delta \geq 1\)) 1971.Gaudin.PRL.26.
  • 1971 (9 April): Lai obtains the thermodynamics of interacting fermions in 1d 1971.Lai.PRL.26.
  • 1971: Takahashi proposes equations for the thermodynamics of the Heisenberg model 1971.Takahashi.PTP.46.
  • 1971 (13 September): Takahashi proposes equations for the thermodynamics of the gapless Heisenberg antiferromagnet (\(|\Delta| < 1\)) 1971.Takahashi.PLA.36.
  • 1972 (May): Baxter solves the \(XYZ\) (8-vertex) model 1972.Baxter.AP.70.1, 1972.Baxter.AP.70.2.
  • 1972 (December 1971): Johnson, McCoy and Lai criticise Takahashi's solution for \(|\Delta| < 1\) on the basis of the high-temperature expansion 1972.Johnson.PLA.38.
  • 1972 (March/October): Johnson and McCoy use Gaudin's equations to obtain the low-\(T\) expansion of the free energy for \(|\Delta| \geq 1\) 1972.Johnson.PRA.6.
  • 1972 (July/August): Takahashi and Suzuki obtain a corrected form of the coupled nonlinear integral equations for the thermodynamics of the Heisenberg chain for \(|\Delta| < 1\) 1972.Takahashi.PLA.41, 1972.Takahashi.PTP.48.
  • 1972: Gaudin's internal Saclay report contains a conjecture for the norm of Bethe states.
  • 1978-…: Faddeev, Sklyanin, Kulish, Takhtajan, etc. start developing the Algebraic Bethe Ansatz (see b-KBI and references therein).
  • 1981: Faddeev and Takhtajan properly understand spinons as spin-\(1/2\) excitations 1981.Faddeev.PLA.85.
  • 1981: Gaudin, McCoy and Wu obtain a conjecture for the norm of Bethe eigenstates of the Heisenberg chain 1981.Gaudin.PRD.23.
  • 1982: Korepin proves Gaudin's conjecture for the norm of Bethe states using the ABA 1982.Korepin.CMP.86.
  • 1988: Sklyanin extends the ABA to boundary cases 1988.Sklyanin.JPA.21.
  • 1988-1989: Slavnov obtains expressions for scalar products (giving the density operator matrix element for the 1DBG) 1989.Slavnov.TMP.79 1990.Slavnov.TMP.82
  • 1999: Solution of the quantum inverse problem. Calculation of matrix elements of the \(XXZ\) chain 1999.Kitanine.NPB.554.

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Author: Jean-Sébastien Caux

Created: 2023-06-07 Wed 16:02