Student Seminar: Classical and Quantum Integrable Systems

• Coordinate Bethe ansatz. This technique was originally introduced by H. Bethe
to solve the XXX Heisenberg model.
• Algebraic Bethe ansatz. It was realized afterwards that the Bethe ansatz can
be formulated in such a way that it can be understood as the quantum analogue
of the classical inverse scattering method. Thus, “Algebraic Bethe ansatz” is
another name for “Quantum inverse scattering method”.
• Functional Bethe ansatz. The algebraic Bethe ansatz is not the only approach
to solve the spectral problems of models connected with the Yang-Baxter algebra.
It is only applicable if there exists a pseudo-vacuum. For models like the
Toda chain, which has the same R-matrix as XXX Heisenberg magnet (spin-
1
chain), but has no pseudo-vacuum, the algebraic Bethe ansatz fails. For
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these types of models another powerful technique, the method of “sepration of
variables” was devised by E. Sklyanin. It is also known as “Functional Bethe
Ansatz”
• Nested Bethe ansatz. The generalization of the Bethe ansatz to models with
internal degrees of freedom proved to be very hard, because scattering involves
changes of the internal states of scatters. This problem was eventually solved
by C.N. Yang and M. Gaudin by means of what is nowadays called “nested
Bethe ansatz”.
• Asymptotic Bethe anzatz Many integrable systems in the finite volume cannot
be solved by the Bethe ansatz methods. However, the Bethe ansatz provides the
leading finite-size correction to the wave function, energy levels, etc. for systems
in infinite volumes. Introduced and extensively studied by B. Sutherland.
• Thermodynamic Bethe ansatz. This method allows to investigate the thermodynamic
properties of integrable systems.
5.1 Coordinate Bethe Ansatz (CBA)
Here we will demonstrate how CBA works at an example of the so-called onedimensional
spin- 1 XXX Heisenberg model of ferromagnetism.
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Consider a discrete circle which is a collection of ordered points labelled by the
index n with the identification n ≡ n + L reflecting periodic boundary conditions.
Here L is a positive integer which plays the role of the length (volume) of the space.
The numbers n = 1, . . . , L form a fundamental domain. To each integer n along the
chain we associate a two-dimensional vector space V = C 2 . In each vector space we
pick up the basis
( )
( )
1 0
| ↑〉 = , | ↓〉 =
0 1
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