integrable systems - R-matrix for spin chains

In algebraic Bethe ansatz procedure, one of the central objects is the R-matrix satisfying the Yang-Baxter equation, but all the papers/books give directly its expression without deriving it, so my question is How can i derive the R-matrix for XYZ/XXZ Heisenberg model?

Answer

This is essentially an answer to your questions R-matrix for spin chains, Elliptic R-matrix and Yang Baxter solution for XYZ model, $R$ matrix for XYZ spin chain, Algebraic Bethe Ansatz and $R$-matrices, which all basically ask the same question anyway.

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In short: to the best of my knowlegde, coming up with an R-matrix is an art, not a derivation. (Cf. the notion of a Lax pair in classical integrability.)

The quantum inverse-scattering method (QISM) was developed as a synthesis of the classical ISM, spin chains and lattice models. The best way to understand it is from this multi-topic point of view, rather than focussing just on spin chains. Faddeev's How Algebraic Bethe Ansatz works for integrable model [arXiv:hep-th/9605187] focusses on spin chains mostly, which makes several constructions --- such as the introduction of an auxiliary space --- appear somewhat ad hoc; at least it certainly felt so to me when I first read them. The vertex-model point of view makes these constructions much more natural; this is also why I organized my lecture notes A pedagogical introduction to quantum integrability, with a view towards theoretical high-energy physics, [arXiv:1501.06805] in the way I did.

Some more comments:

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  Once you know the Lax matrix (containing the vertex weights) of the six- or eight-vertex model you can solve for the R-matrix (solving the "RTT-relation" with $T=L$ for the case of one site), see e.g. Sections 9.6 and 10.4 in Baxter, Exactly solved models in statistical mechanics (or Appendix C in my lecture notes mentioned above).

  
  



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  Alternatively, you can look for solutions of the Yang--Baxter equation, and then interpret each R-matrix you get as a vertex model or see which spin chain it yields by computing the logarithmic derivative of the associated transfer matrix.

  
  


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  It might be instructive to read up on another example: Shastry's R-matrix for the Hubbard model. See e.g. Section 12.2 in Essler, Frahm, Göhmann, Klümper, Korepin, The one-dimensional Hubbard model [e-print].