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I was studying the algebraic Bethe ansatz for the spin-1/2 XXZ model. In the end one ends up with $2^L$ integrals of motion $Q_k$ that commute with the Hamiltonian, (https://doi.org/10.1103/PhysRevLett.125.090602) and (https://doi.org/10.1088/1751-8121/ac0961).

However the way they are defined is very dense, and I just wanted to know if the integrals of motions are written down explicitly anywhere, at least for a new sites, $n=2,3,4$, etc, in terms of the Pauli/spin matrices. There should be (4, 8, 16) of them, so in principle you should be able to write them down.

I guess I wanted to see if for the few site integrals of motion you can start to see that they are somehow local, and independent from the projectors $\lvert n \rangle \langle n \rvert$.

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  • $\begingroup$ Does the question, my answer and the comments at physics.stackexchange.com/q/669182 help? $\endgroup$
    – Jules Lamers
    Commented Nov 14, 2023 at 23:54
  • $\begingroup$ Hi Jules, thanks for the reply. I already knew that Q_1 is the magnetisation, and Q_2 is the Hamiltonian, but I was hoping to find at least Q_3, the nontrivial conserved quantity. I went through your notes but couldn't find an explicit construction of Q_3. Is it possible to write it down for the XXZ model? $\endgroup$
    – purestate
    Commented Nov 16, 2023 at 5:43
  • $\begingroup$ The next charge can be e.g. found in Exercise 2.7 of the book C. Gómez, M. Ruiz-Altaba, and G. Sierra, "Quantum Groups in Two-Dimensional Physics" (Cambridge University Press, 1996) $\endgroup$
    – Jules Lamers
    Commented Nov 17, 2023 at 11:01
  • $\begingroup$ (I'm just talking about the log derivative of the transfer matrix, which gives $L$ charges, if one also includes $S^z$. I have not read the first reference that you cite.) $\endgroup$
    – Jules Lamers
    Commented Nov 17, 2023 at 11:40
  • $\begingroup$ See also physics.stackexchange.com/a/796554 $\endgroup$
    – Jules Lamers
    Commented Jan 9 at 1:45

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