Abstract
Historically, the Heisenberg chain has been the first exactly solved (interacting) model. It is a fairly realistic model for a one-dimensional quantum magnet and it has been instrumental in moving beyond the classical Ising-type models. We will solve this chain following the same steps we introduced in the previous chapter for the Lieb-Liniger model: in Sect. 3.2 we introduce the chain’s fundamental excitation, the magnon, in Sect. 3.3 we study the two-body problem and in Sect. 3.4 we write the coordinate Bethe Ansatz solution and the Bethe equations. We will see that the latter admit a mixture of complex and real solutions and discuss the challenges this implies. For infinitely long chains, it is believe that the string hypothesis allows to organize the different bound solutions: this is the topic of Sect. 3.4.1. Having organized the Hilbert space of the model, we discuss the ground state and the low energy excitations for the ferromagnetic and antiferromagnetic order in Sects. 3.5 and 3.6, respectively. For the latter, we introduce and discuss the role of spinons as emergent quasi-particles. Finally, in Sect. 3.7 we study the effect of switching on an external magnetic field.
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Notes
- 1.
We shall see however that states with differing quantum numbers can have equal quasi-momenta, if these lie at the edge of the Brillouin zone.
- 2.
Note that, as proven in [64], complex solutions to the Bethe equations always appear in conjugated pairs: \(\{ \lambda _j \} = \{ \lambda _j^*\}\).
- 3.
The reader must have noticed that we use notations reminiscent of the representations of SU(2).
- 4.
To be clear in the notation used, \(\lambda _{M, j}\) is the real part of the j-th complex of length M.
- 5.
We assume that N is even. For the odd case there are two degenerate ground states in the \(S^z = \pm 1/2\) sectors, but we will not discuss this case further.
- 6.
Note that negative magnetization states cannot be reached in this formalism: one starts from the completely negatively polarized states and excites magnons out of it. On general ground, the model is invariant under the reversal of every spin across the \(x-y\) plane (a particle/hole duality) and thus any positive magnetization state is related by this symmetry to one with a negative one.
- 7.
These excitations are holes with respect to the description we have been using, but they should be considered as particle excitations on top of the vacuum state.
- 8.
Such configuration is also a beautiful physical proof of the mathematical identity \(\sum _{n=1}^\infty (-1)^n = - {1 \over 2}\), which is otherwise obtained through analytical continuation.
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Franchini, F. (2017). The Heisenberg Chain. In: An Introduction to Integrable Techniques for One-Dimensional Quantum Systems. Lecture Notes in Physics, vol 940. Springer, Cham. https://doi.org/10.1007/978-3-319-48487-7_3
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DOI: https://doi.org/10.1007/978-3-319-48487-7_3
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